A16:
NUMERICAL, COMPUTATIONALAND THEORETICAL ACOUSTICS

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022 in Gyeongju, Korea

ABS-0829

Computational study of ultrasonic flow meter by lattice Boltzmann
methodHyungkeun CHUNG1; Young J. MOON2*; Shang-yoon HWANG3; Yong-min KIM4; Joo-hyun KIM5
1,2

School of Mechanical Engineering, Korea University, Korea, Rep. of
3,4,5

HSCMT Co., LTD, Korea, Rep. of

ABSTRACTIn this study, computational analysis of ultrasonic flowmeter was performed with regularized lattice
Boltzmann method (RLBM). Because RLBM has the advantage of having explicit properties, parallel
computing using general-purpose computing on the graphics processing unit (GPGPU) can significantly
reduce computational time. A grid-convergence test was first conducted to find the minimum grid resolution
required for ultrasonic wave calculation using RLBM. Based on the time-transit method, computational
analyses of ultrasound flow meter were performed for various conditions of uniform velocity flow, wallbounded shear flow, and grid-generated turbulent flow. The present computational method yields an error of
about 3-5% for uniform velocity and wall-bounded shear flow conditions. However, in the grid-generated
turbulent condition, the error is larger than in the previous cases. The computational results seem to indicatethat the time difference of ultrasound arrival was highly influenced by the interaction of ultrasound waves
with complex flow such as turbulence.
Keywords: Ultrasonic flow meter, Ultrasound, Lattice Boltzmann method (LBM), GPGPU

1. INTRODUCTION
Ultrasonic flow meter is a device that can measure the flow rate inside the pipe with ultrasonicwaves. It is frequently used because of its versatile applications to various fluids with a known speedof sound. In a time-transit ultrasonic flow meter, the velocity of flow is obtained with the timedifference between two ultrasonic waves traveling in opposite directions. There have been attempts
to analyze ultrasound flow meter computationally. Computational analysis enables us to quantify andvisualize the interaction between geometry of the flow meter, flow inside of the tube, and ultrasonicwave. However, ultrasound computation requires high resolution of the mesh, which leads to excessive
computational power and calculation time (1). For instance, it is commonly known that around 20
meshes per single wavelength of ultrasound are required to resolve acoustic waves in a field domain.
Therefore, resolving 0.5MHz ultrasound in a full-scale ultrasonic flow meter with a diameter of 50mm
would require 333 mesh points lengthwise, and thus approximately 3.6×107 mesh points in a cubic
domain.
In this study, we propose the lattice Boltzmann method (LBM) for calculating ultrasonic waves ina full-scale ultrasonic flow meter. LBM has advantages over traditional computational methods in thatit uses a distribution function based on the discretized Boltzmann equation to solve for physicalquantities, while adopting an explicit solution procedure. The explicit properties of LBM enable
methods to be applied in parallel structures using general-purpose computing in graphics processing
units (GPGPUs). The parallelization of the GPU allows simulating ultrasound propagation in the flow
meter on a mesh grid size of approximately 1×108 in about 10 minutes.
The objectives of this study are to conduct computations for ultrasonic flow meter using LBM andryu02129@korea.ac.kr
yjmoon@korea.ac.kr; *corresponding author
3
envata@naver.com
4
ieeekim@naver.com
5jhkim@hscmt.co.kr
12

to validate the accuracy of the present computational methods in a full-scale flow meter with uniform
velocity flow and wall-bounded shear flow. In addition, it examines the accuracy of the present timetransit method with grid-generated turbulent flow.

2. NUMERICAL METHODS
2.1 Lattice Boltzmann Method (LBM)
Lattice Boltzmann Method (LBM) solves physical quantities such as density and velocity explicitlywith a distribution function f (2).
fi (x + ci ∆t, t + ∆t) = fi (x, t) + Ωi (x, t)(1)
The most basic LBM method is the Bhatnagar-Gross-Krook (BGK) model with its collision operator
𝛺 shown as follows.
2(f - f eq )u⋅ ci (u⋅ ci ) u⋅ u(2,3)Ωi (x, t) = - i i ∆t ,f ieq = wi ρ (1 + 2 +
)
τ
cs
2c4s
2c2s
The distribution ideally converges to the equilibrium distribution function f ieq at a rate determinedby the relaxation time τ . The equilibrium function f ieq is determined by the weight wi regarding
direction, density ρ, velocity u, lattice velocity ci , and lattice speed of sound cs = √1/ 3. The calculationconsists of a collision and streaming process of distributions through discrete time steps. The time step dt isdefined by the speed of sound in lattice unit dv and the unit lattice length dx (dt = dx/dv, dv = 1500 ×√ 3
(m/s⋅lattice speed)). Density and velocity are updated after streaming as follows.
𝜌 = ∑ fi ,

ρu = ∑ ci fi

i

i

(4,5)

For the grid structure model, the current work selected D3Q19 structure, which uses 19 directions(direction counts denoted as i) to depict the 3D grid domain.
2.2 Regularized Lattice Boltzmann Method (RLBM)
The original LBM using the Bhatnagar-Gross-Krook (BGK) model is unstable in high Reynolds
numbers. Therefore, various measures have been proposed to improve stability of the method. One ofthese measures is the Regularized LBM (RLBM) (3). The RLBM introduces a regularization process
to reduce the discrepancy between the non-equilibrium part of distribution function f ineq = fi - f ieq andthe first order term of fi , fi (1). The regularization process uses the non-equilibrium momentum flux tensor
Πneq to obtain fi (1) and the resulting distribution function after collision fi out . RLBM reduces theinstability to the extent that it can be used to compute ultrasound using LBM.
wi1
neqfi (1) = 4 Qiαβ Πiαβ ,fi out =f ieq + (1 ) f (1)
(6,7)
τ i
2cs
2.3 Immersed Boundary Lattice Boltzmann Method (IB-LBM)
To simulate boundary of the geometry, the current work selects IB-LBM, which is easy to
implement with only a CAD geometry export (STL file format in current work) required. IB-LBM
utilizes Lagrangian geometry to impose its velocity on the Eulerian field with a velocity correction.
The correction procedure is conducted after the update of density and velocity in the normal LBM
procedure. As a way for the velocity correction, multi direct-forcing method proposed by Kang (4)and modified version of direct-forcing method by Li (5) was used in this study. Density and velocity
at the position of Lagrangian nodes are interpolated by the surrounding Eulerian nodes, using thediscretized Dirac delta function (denoted D (xE - xL )).
ρL = ∑ ρE D(xE - xL )(∆x)d ,i(ρu)L = ∑(𝜌𝑢)𝐸 D(xE - xL )(∆x)d

(8,9)

i

These interpolated properties are used to obtain the external force needed from Lagrangian points
to Eulerian points to impose wanted velocity from the boundary, where 𝑢𝑠 denotes the supposed fluid
velocity at a given Lagrangian point.

ρL us = (ρu)L +

∆t
F
2 s

(10)

The external force is then spread out again using the Dirac delta function to be used in velocity
correction using the momentum equation for the LBM, where ∆s is the length between the Lagrangian
points in 2D and area in 3D.
∆t
𝐹E = ∑ Fs D(xE - xL )∆s ,ρu= ∑ ci fi + 𝐹E
(11,12)2ii

The modified velocity is used again to re-modify the velocity for a predefined number of iterations.
This iterative procedure is intended to improve accuracy compared to the single-iteration direct
forcing. Multi direct-forcing method uses explicit measures to impose a velocity close to that of agiven geometry in the field, which is relatively inexpensive compared to implicit methods that requirecomputing inverse matrices of enormous size at every time step. The calculated external force isfurther used as an external force term using Guo’s forcing scheme (6).

3. A SIMPLE HARMONIC ULTRASOUND SOURCE
3.1 Computational Setup
To simulate ultrasonic acoustic wave in water with LBM, appropriate level of grid resolution isnecessary to provide reliable results. A grid-convergence test with ultrasonic wave of 0.5MHz is
conducted with maximum amplitude of 1Pa to investigate optimal and yet minimal resolution forsimulation. A cubic domain of 3mm is used, and three levels of resolution are tested: 10, 15, 20 grid
lengths per single wavelength. The acoustic pressure profile in the center plane is then expressed inlogarithmic scale to check for its attenuation rate. The attenuation rate should be -1 log (Pa)/log (m)according to the inverse square law.
3.2 Grid-convergence test results
Figure 1 shows the computed three-dimensional acoustic pressure at the center plane of a test casewith 20 grids per wavelength. In each test case, four crests are compared with a dotted line with aninclination of -1 log (Pa)/log (m) and these results are presented in Figure 2. The first case with 10
grids per wavelength clearly shows that it does not match the correct attenuation rate. The second caseof 15 grids per wavelength shows a huge improvement, but the magnitude of the last crest is slightly
off the line indicating that the minimal resolution is close to this second test case. The last case of 20
grids per wavelength matches perfectly with the line of the correct attenuation rate. Therefore, it canbe concluded that the minimal resolution for simulating 0.5MHz is between 15 to 20 grid lengths per
single wavelength.

Figure 1 – Acoustic pressure distribution of center plane depicted in 3D

Figure 2 – Acoustic pressure (Pa) versus radius (m) in logarithmic scale. (Left) Resolution of 10 grids per
wavelength. (Middle) Resolution of 15 grids per wavelength. (Right) Resolution of 20 grids per
wavelength.

4. COMPUTATION OF ULTRASONIC FLOW METER
4.1 Computational Setup
In ultrasonic flowmeters, ultrasonic pulses are generated from one actuator to the othe r receiver. A
pulse is then generated in reverse to measure the time difference of arrival between the two cases . The
time difference obtained with two simulations is then used to calculate the flow velocity.
Computational simulations are performed in a parallel multi-GPU platform to measure the differencein pulse arrival times for various flow conditions in an ultrasonic flow meter.
In the present computation, the diameters of the flowmeter and actuator are 53.5mm and 16mm,
respectively. And physical properties of water such as kinematic viscosity (1.2 × 10-6 m2 /s) and speedof sound (1500 m/s) are used. A computational domain of 54.08 × 75.84 × 96 (mm) is used with a gridsize of 339 × 475 × 601 to simulate a full-scale ultrasonic flow meter. In computations, ultrasoundwave of 0.5MHz is simulated with 18.75 grids per wavelength.
The actuator is simulated using the IB-LBM, where the actuator moves according to the frequencyof the ultrasound. The travelling distance of the pulse is 81mm, and the angle of the actuator andreceiver with respect to the axial direction is 40°. Three consecutive pulses are generated at theactuator to obtain time differences more consistently, and a sponge layer is used to avoid any
reflections of waves from the inlet and outlet boundaries. In the present computational analyses, four
flow conditions are tested: quiescent, uniform velocity, wall-bounded shear, and grid-generated
turbulence.

Figure 3 – Computational domain of ultrasonic flow meter simulation. Actuator is colored in red

4.2 Quiescent Flow
Ultrasound simulation in a quiescent condition is conducted to serve as a reference. Figure 4 shows
ultrasound waves at different instances at 6.15×10-6 s , 2.46×10-5 s , and 5.54×10-6 s . It shows
ultrasound wave propagations, reflections from the side walls, and scattering of waves in the flow
meter. Any reflection of waves at the inlet and outlet boundaries are avoided by the sponge layerboundary condition applied to the inlet and outlet boundaries.

Figure 4 –Ultrasound waves at different instances (at 6.15×10-6 s, 2.46×10-5 s, 5.54×10-6 s)

Figure 5 – Pressure probed at the center of the receiver, versus time
Figure 5 shows the pressure probed at the center of the receiver over time. The first three pulses in
the figure are the initially generated pulses and the following pulses are produced by reflections fromthe side wall. Note that the wavefronts of the three first generated pulses are not affected by any
geometric disturbances before reaching the receiver.
4.3 Uniform Velocity Flow
A uniform velocity flow of 0.5m/s is tested to verify the accuracy of the ultrasound simulation bymeasuring the time difference of pulse arrival between opposite directions. Due to the structure of thecavities, there is some pressure and velocity disturbance in the flow, as shown in Figure 6 (left). Since
the velocity is not uniform along the path of the ultrasound waves, it is necessary to find the reference
average velocity in the direction of wave travel. The reference average velocity along the pathway of
the wave is obtained as 0.445m/s by integrating the velocity profile shown in Figure 6 (right).
Figure 7 shows the pressure probed at the center of the receiver, which clearly shows a phase shift
created by two ultrasound waves travelling in the opposite directions. The calculated time differenceis 2.0×10-8 (s), from which the velocity is obtained as 0.432m/s. The computational result is consistent
with the physical relationship between the time difference and the flow velocity calculated from thetransit time, and an error between the computed and averaged velocities is 2.92%. It clearly showsthat the cavity effect is negligible except for the decrease in mean velocity. This also proves that thecurrent grid resolution setting is applicable to simulate the ultrasound wave of 0.5MHz.

Figure 6 – (Left) Velocity profile on the center plane of the flow meter. The dotted line indicates thecenterline of the path. (Right) Streamwise direction velocity (m/s) on the centerline of the path, versus
vertical distance (m) across the flow meter. The location of cavity is indicated by dotted lines.

Figure 7 – Pressure probed at the center of the receiver, versus time. The time difference between the twocases is 2.0×10-8 (s).
4.4 Wall-bounded Shear Flow
A wall-bounded shear flow is computed with the same procedure. In this case, the imposed flow
velocity is maximum at the centerline, which is 0.245m/s. This case is to investigate the effect ofcontinuously varying flow velocity on the transit time of ultrasound (see Figure 8 (left)). The averagevelocity along the path is obtained as 0.130m/s by integrating the velocity profile shown in Figure 8(right). Figure 9 shows the pressure probed at the center of the receiver and clearly shows a phaseshift created by two waves traveling in opposite directions. The calculated time difference is
0.63×10-8 (s), from which the velocity is obtained as 0.136m/s. The error between the computed andthe reference velocities is found to be 4.62%. The slightly increased error compared to the previous
case indicates that the non-uniform velocity along the path interferes with propagation of theultrasonic waves. Note, however, that the interaction between the non-uniform flow and the ultrasoundwave is not so significant, and this is due to the fact that the flow is symmetric about the centerline
so that the downstream and the upstream propagating waves have the same time-shifting delay effect.

Figure 8 – (Left) Velocity profile on the center plane of the flow meter. (Right) Streamwise direction
velocity (m/s) on the centerline of the path, versus vertical distance (m) across the flow meter. The locationof cavity is indicated by dotted lines.

Figure 9 – Pressure probed at the center of the receiver, versus time. The time difference between the twocases is 0.63×10-8 (s).

4.5 Grid-generated Turbulent Flow
A grid-generated turbulent flow is computed to investigate the effect of turbulence generated bystructural disturbances on ultrasound waves. The diameter of a cylindrical grid structure is 5mm andthe distance between the grid structures is 5.6mm. The undisturbed velocity is 0.14m/s and the
Reynolds number of the flow is 583 based on the grid diameter. Figure 10 (top left) shows the gridturbulence generated by the grid structure inside the flow meter.
Figure 10 (top right, bottom) clearly shows that the velocity field is highly three -dimensional andnon-uniform in both the streamwise and cross-sectional directions, which is strongly affected by thegrid turbulence. Table 1 shows the average velocities of seven different cross-sectional slices of planes(indicated by dotted lines on Figure 10 (top right)). Note that the average velocity of a cross-sectionalslice is obtained with velocities along the three different paths. The reference average velocity is foundto be 0.132m/s, that is, the average of the velocities in Table 1. Note also that this average velocity isdifferent from the reference velocity of 0.14m/s.
As shown in Figure 11, the measured time difference from the pulses is 0.73×10-8 (s), from which
the velocity is obtained as 0.158m/s. The difference between the computed and the average velocities
is 19.7% and that between the computed and reference velocities is 12.9%. Both are larger than in theprevious cases for several reasons. One of the reasons is that ultrasound propagation is influenced by
the complex flow of turbulence, in which the wavefront of the ultrasonic wave is distorted by thethree-dimensional turbulence. Furthermore, the downstream and upstream propagations of ultrasoundwave are noticeably different due to the random nature of the turbulent eddies.

Figure 10 – (Top left) Velocity iso-magnitude surfaces of the grid turbulences. (Top Right) Streamwise
velocity profile on a perpendicularly sliced plane at the actuator. (Bottom) Streamwise direction velocities(m/s) along the centerline of the path, versus vertical distance (m) at three cross-sectional slices of planes;-0.0048(m), 0(m) and 0.0048(m). The location of cavity is indicated by a dotted line.
Table 1 – Average velocities from seven different cross-sectional slices of planes
Slice Location, m-0.0056 -0.0048 -0.0032
00.00320.0048Average Velocity, m/s

0.155

0.165

0.133

0.0954

0.121

0.124

0.00560.128

Figure 11 – Pressure probed at the center of the receiver, versus time. The time difference between the twocases is 0.73×10-8 (s)5. CONCLUSIONSIn this computational study, the RLBM computation of ultrasound wave in water was validated bygrid-convergence tests. The grid-convergence test shows that at least 15-20 grid points per single
wavelength of ultrasound are required to properly simulate the ultrasound waves. The computational
results of the ultrasound flow meter show 3~5% error for uniform velocity and wall-bounded shear
conditions. On the other hand, the grid-generated turbulent flow condition shows an error of 19.7%
between the computed velocity and the velocity averaged in both the wave path direction and crosssectional direction. The error is 12.9% if the reference velocity of undisturbed flow is used. The
relatively large error is due to the fact that grid-generated turbulent flow is fully three-dimensionaland velocity field is superimposed with fluctuations of turbulent eddies. The results seem to indicatethat the time difference of ultrasound arrival was highly influenced by the interaction of ultrasoundwaves with complex flow such as turbulence.

ACKNOWLEDGEMENTS
This work was supported by the HSCMT Co., LTD, Korea, Rep. of through the grant (Q2116131).

REFERENCES
1. COMSOL Multiphysics. Ultrasound Flowmeter with Generic Time-of-Flight Configuration.
2. Kruger T, Kusumaatmaja H, Kuzmain A, Shardt O, Silva G, Viggen EM. The Lattice Boltzmann Method.,
Springer; 2017.
3. Latt J, Chopard B. Lattice Boltzmann Method with regularized non-equilibrium distribution functions.
Physical Review Letters. 2005;72.
4. Kang SK, Hassan YA. A comparative study of direct-forcing immersed boundary-lattice Boltzmann
methods for stationary complex boundaries. Int. J. Numer. Meth. Fluid; 2011;66.
5. Li Z, Favier J, D ’Ortona U, Poncet S. An immersed boundary-lattice Boltzmann method for single- andmulti-component fluid flows. Journal of Computational Physics, Elsevier, 2016;304.
6. Guo Z, Zheng C, Shi B. Discrete lattice effects on the forcing term in the lattice Boltzmann method.
Physical Review Letters. 2002;65.

ABS-0325

Vibro-acoustic shape optimization for radiation problems: Application tocompact loudspeakers
Peter Risby Andersen(1) , Vicente Cutanda Henríquez(2) , Niels Aage(3) , Seongyeol Goo (4) , Junghwan Kook(5)(1) Audio Research, GN Audio A/S & Jabra, Denmark, prandersen@jabra.com(2) Centre for Acoustic-Mechanical Micro Systems, Department of Electrical and Photonics Engineering, Technical Universityof Denmark, Denmark, vcuhe@dtu.dk(3) Centre for Acoustic-Mechanical Micro Systems, Department of Civil and Mechanical Engineering, Technical University ofDenmark, Denmark, naage@mek.dtu.dk(4) Innovative SMR System Development Division, Korea Atomic Energy Research Institute, Republic of Korea,

ksy1124@kaeri.re.kr(5) Audio Research, GN Audio A/S & Jabra, Denmark, jkook@jabra.com

ABSTRACTShape optimization is a promising tool for improving and guiding the design of computationally large-sizedvibro-acoustic devices such as compact loudspeaker systems and hearing aids. Typically, such devices are
subject to unwanted structural vibrations that can degrade the acoustic performance, i.e., contributing withdestructive and constructive interference or creating feedback problems. Numerical-based shape optimization
can potentially help engineers to limit or control these vibrational effects. This work presents i) a method to
perform gradient-based vibro-acoustic shape optimization and ii) an investigation of the impact of different
cost function formulations. The shape optimization framework is based on a coupled vibro-acoustic Finite Element and Boundary Element utilizing a Free Form Deformation parametrization approach. The framework
will be used to control the acoustic radiation characteristics of vibrating structures, having the design of compact loudspeakers in mind. When performing shape optimization, a difficult task is to formulate appropriate
cost functions that can create well-performing designs in a broadband frequency range. Therefore, the work
will discuss different cost function definitions, mainly focusing on manipulating the directivity performanceof the acoustic radiation from vibrating structures.

Keywords: loudspeakers, shape optimization, boundary element method, free form deformation

1

INTRODUCTION

This work presents the application of shape optimization for the design of vibro-acoustic devices, with thefocus on the design of compact loudspeaker systems such as speakerphones. Speakerphones are communication devices used for online conference meetings typically with multiple participants in a single meetingroom. They consist of a loudspeaker system used to radiate sound to the meeting participants and several microphones to pickup speech signals. It is often a requirement that speakerphones should be portable. Hence,
it is desirable to make them light and compact. Therefore, their cabinets are typically constructed from e.g.
thin plastic materials. A potential shortcoming of this is that not only will the loudspeaker unit vibrate, thecabinet can also vibrate and create unwanted interference in the acoustic signal. Also, microphones mounted
on the cabinet might vibrate and therefore deteriorate the voice pickup performance.
Shape optimization based on a full vibro-acoustic discription of the loudspeaker system can be used to alterand improve the acoustic characteristics. Several examples of the utilization of shape optimization within thearea of acoustics exist. Especially, the optimization of horn loudspeakers has been studied [1, 2, 3]. Also,
shape, material and topology optimization have been applied to conventional loudspeaker design [4, 5, 6].
Additionally, vibro-acoustic parameter and level-set based shape optimization have also been applied improve
the performance of hearing instruments [7, 8].
In the following, the implementation of a full vibro-acoustic model of a quarter symmetric loudspeaker system is presented. To simplify the modelling of the loudspeaker unit, the model includes a partially lumped
element modelling (LPM) approach of the loudspeaker magnet, basket and spider. Typically, the loudspeaker
unit in a speakerphone is not free of exterior objects in front of the loudspeaker, i.e., often a speaker grill,
fabric, or other obstructions are affecting the loudspeaker unit. Here, the presented model will also include
an acoustic lens that is placed in front of the loudspeaker unit. The lens is used to control the sound field,
but also protect the loudspeaker unit. The underlying numerical modelling approach is mainly based on a
structural and acoustic Finite Element Method (FEM) description with an acoustic-acoustic coupling between
the FEM and the Boundary Elemement Method (BEM) to fulfill the exterior radiation conditions. Hereafter,
the geometry is parameterized based on a cylindrical Free Form Deformation (FFD) approach. Finally, theFFD approach is used to optimize the acoustic lens and cabinet shape.

2

LOUDSPEAKER GEOMETRY, MATERIAL AND BOUNDARY CONDITIONS

To reduce the computational burden, the model of the loudspeaker is based on a quarter symmetric description. The structural part of the loudspeaker system is shown in Figure 1a. In the figure, the gray domain
represents the cabinet and the lens geometry, the black domain is the loudspeaker surround, the blue domain is the loudspeaker diaphragm, and the green domain is the dustcap. Moreover, the dark red surfaces
in Figure 1a are the surfaces where symmetry boundary conditions are applied. To simplify the modeling of
the loudspeaker unit the remaining parts (voice-coil, basket, magnet and spider) are simply modeled througha lumped model. The lumped model creates two forces, a surface force acting on the cabinet through thereaction force from the basket and an edge force from the voice-coil applied to the edge connecting thediaphragm and the dustcap. Figure 1b shows the location of the surface force (purple surface) and the edge
force (red line). The development of the lumped model is further described in Section 5. The structural
material parameters used to model the cabinet, surround, diaphragm and dustcap are given in Table 1. Also,
the surface of the loudspeaker at z = 0 has a fixed displacement boundary condition.
Table 1. The material properties used to model the loudspeaker system
Young’s modulus (GPa)

Density (kg/m3 )Loss factor

Poisson’s ratio

Cabinet and lens

2.34

1190

0.019

0.35

Surround

0.023

67

0.04

0.45

Diaphragm

10

10

0.04

0.42

Dustcap

2.62

1050

0.04

0.37

Parts

3

VIBRO-ACOUSTIC FEM AND BEM MODEL OF THE LOUDSPEAKERThe underlying model of the loudspeaker is based on a FEM description of both the mechanical parts of
the loudspeaker, but also the interior and exterior acoustic domains are modelled with the FEM. However,to fullfill the Sommerfeld radiation condition the exterior acoustic FEM domain is coupled to the BEM.
Applying classical Galerkin FEM assuming the solution to be time-harmonic, the matrix equations for themechanical domains become(1 + iη)Ks − ω 2 Ms u = fs ,
(1)

a)

b)
z

Lens

Hl = 8 mm
Hcl = 1.3 cm

R5
R4
R3

eaker
LoudspHc = 3.7 cm

t1
t2
y

Surface force from LPM

t3

R2

R1
x

cm
5.5
=
Rc

CabinetEdge force from LPM

Figure 1. The structural part of the initial loudspeaker geometry. In the figure, a) is the entire structuralpart of the geometry and b) is a top view of the loudspeaker excluding the acoustic lens. Figure a .Figureb) also includes the location of the lumped element force conditions derived in Section 5. The size ofthe loudspeaker unit is given by the radii R1 = 9.5 mm, R2 = 18 mm, R3 = 22.9 mm, R4 = 23.9 mm andR4 = 26.4 mm. Moreover, the thicknesses of the cabinet are given by t1 = 2 mm, t2 = 5 mm and t3 = 6 mm.
where Ks and Ms are the mechanical stiffness and mass matrices, respectively. Moreover, the angular frequency is given by ω, u is the displacement vector, η is the stiffness proportional isotropic damping factor
and fs is the load vector. Similarly, the acoustic finite element domains can be described in matrix form as∂p
K f − ω 2M f p = S ,∂n(2)where K f is the acoustic stiffness matrix, M f is the acoustic mass matrix, p is a vector with the acousticpressure, S is the boundary mass matrix and ∂∂ pn is a vector with the surface normal derivative of the acousticpressure. The mechanical and acoustic coupling is performed using conformal mapping with the coupling
matrix beingZ
NTs nN f dΓs f ,Cs f =(3)Γs f

where N is the shape function matrix with the subscripts s and f representing either the structural or the
fluid domains, respectively. Integration is carried out on the surface interface between the fluid and thestructure,i.e. , Γs f . The model is extended to include exterior radiation conditions by an acoustic FEM-BEM
coupling. Applying classical collocation BEM of the Helmholtz equation, the boundary element matrices canbe written as
∂p
=0
(4)
∂n
where H and G are the matrices arising from the double and single layer potentials, respectively. It should
be noted that the BEM kernels are created from the assumption of quarter symmetry and an infinite baffle atz = 0. The acoustic FEM and BEM coupling is achieved by assuming continuity between p and ∂∂ pn . Hereby,
it is possible to assemble a fully coupled system of equations for the vibro-acoustic behaviour as
Hp − G

esh

tic m

a)
y

cous
rior a

Inte

b)

y

sh
c me
ousti
c
a
r
io
Exter

c)

z
y

Structural mes
hzz

Figure 2. The computational mesh (a) and the FFD parametrization of the cabinet (b) and the lens (c).
It should be noted that (b) and (c) exclude the exterior acoustic FEM mesh. However, this is done forvisualization purposes, and the mesh is included and mapped to the FFD parametrization.



As



Qsys x = −ω 2 ρ f CTs f

0

−Cs f
Af
H

 
f
    s   −Sb   p  =  0 
   
∂p
0
−G
∂n
0



u



(5)

where the stiffness and mass matrices from the structural and acoustic domains are condensed into the matrices As and A f , respectively. Also, ρ f is the density of air originating from utilizing Euler’s relation and
Sb is the boundary mass matrix evaluated at the FEM-BEM coupling surface. The finite element matrices
are implemented within MATLAB and the assembly of the BEM matrices rely on a compiled C++/MEX
implementation discussed in the Refs. [10, 11]. For the particular problem, the matrices are stored as sparse
matrices and solved using the MUltifrontal Massively Parallel sparse direct Solver (MUMPS). It should be
noted that this approach is only reasonable for problems where the BEM matrices are relatively small. The
meshed geometry is shown in Figure 2a, where the dark blue, cyan and white domains are the structural,interior acoustic, and exterior acoustic finite element domains, respectively. The two light green surfaces
correspond to the location of the acoustic-acoustic coupling between FEM and BEM. The complete meshconsists of 83135 quadratic tetrahedral elements.

4

PARAMETRIZATION USING FREE FORM DEFORMATION

The geometry is parameterized using FFD in cylindrical coordinates similar to the approach in Ref. [11]where Bernstein polynomials are used as basis functions. Here, two FFD regions are created; a region controlling the the shape of the lens (Figure 2c) and a region controlling the shape of the top of the loudspeakercabinet (Figure 2b). For simplicity the control points in the FFD regions are only allowed to move in the zdirection. Also, to avoid unacceptable mesh deformations only control points inside the FFD region are free
to move, i.e., the control points on the outer sides of the FFD region remain fixed. Moreover, the control
points are only allowed to move ±8 mm and ±6 mm in the FFD regions respectively covering the lens andthe cabinet. In total, the optimization problem consists of 72 design variables.

5

PARTIALLY LUMPED LOUDSPEAKER

To simplify the overall numerical model, some parts of the loudspeaker driver are modelled using LPM. In
this case, the basket and the motor system of the driver are lumped to describe the forces acting on the
loudspeaker diaphragm and the cabinet. From an impedance based derivation, the following two equations
can be formulated
Zcone v1,z = Fv1,z = F0 − ZT (v1,z − v2,z ) − ZMc v1,z ,(6)

Zbox v2,z = Fv2,z = ZT (v1,z − v2,z ) − F0 − ZMMm,ch v2,z .

(7)Here, Fv1,z is the total force applied to the edge connecting the diaphragm and the dust cap, and Fv2,z isthe total force acting on a surface on top of the cabinet. Also, F0 is the driving force, ZMm,ch , ZMc and ZT
are mechanical impedances, and Zb is an electrical impedance. In Eqs. (6) and (7), the different forces and
impedances are given by the lumped elements as
1
Bl 2
ZMc , = iωMc ,
+
iωCc
Zb
ω 2 L22 R2 + iωL2 R22
ZMm,ch = iω (Mm + Mch ) ,Zb =+ Re + iωLe .
ω 2 L2 R22√
In the above equations, Bl = 2.558 Wb/m is the force factor, V0 = 8 V is the driving voltage, Re = 2.61 Ω
is the electrical voice coil resistance at DC, Le = 8.5 · 10−5 H is the frequency independent part of the voice
coil, L2 = 8.1 · 10−5 H is the para-inductance of the voice coil, R2 = 0.67 Ω is the electrical resistance due toeddy current losses, Mm = 0.0434 kg is the mass of the magnet, Mch = 3.47 g is the mass of the loudspeakerchassis, and Cc = 0.0025 m/N is the compliance of the spider. The loudspeaker geometry and the location of
the two forces are depicted in Figure 1b. In Eqs. (6) and (7), the velocities v1,z and v2,z represent the averagevelocity on the edge or surface, respectively. The average velocities are given from the shape functions as
F0 =

BlV0
,
Zb

ZT =

Ns,z,1
Ns,z,2u and v2,z = iω
,u
(8)
L
A
where L and A are the length and the area of the interface where the total force is applied, respectively.
Also, the subscript z denotes that the shape functions only describe the z-component of the displacement.
Substitution of the forces into the right-hand-side of the Eq. (5), yield
Ns,z,1
Ns,z,2
Ns,z,1
F0
T
TT
fs,1 = Ns,z,1 − iωZT Ns,z,1 2 − Ns,z,2 2u − iωZMc NTs,z,1 2 u,
(9)
L
L
A
L


Ns,z,1
Ns,z,2
Ns,z,2F0
TT
fs,2 = iωZT Ns,z,1 2 − Ns,z,2 2u − iωZMm,ch NTs,z,2 2 u − NTs,z,2
(10)
L
A
A
A
where
fs = fs,1 + fs,2 .
(11)
v1,z = iωRearranging, the system of equations for the structural part in Eq. (5) becomes
F0
F0− NTs,z,2 ,(12)LA
where D1 and D2 contains the terms in Eqs. (9) and (10) which are multiplied with u. A verification of
the implementation is shown in Figure 3 where the SPL frequency response at the field point (0,0,1) m iscomputed and compared with a similar implementation in the simulation software COMSOL Multiphysics.
The result shows a good agreement between the in-house implementation and the commercial simulation
tool. It should be noted that the approach has been tested against a full vibro-acoustic model of the same
loudspeaker driver showing reasonable agreement between the partially lumping approach and a full model,
but with less computational effort. Nevertheless, the lumping approach might become less accurate at higher
frequencies.
(As + iω (D1 + D2 )) u = NTs,z,1

Figure 3. Comparison between the custom MATLAB C++/MEX FEM-BEM implementation and a similar
implementation in the simulation tool COMSOL multiphysics. The SPL is evaluated at the field point (0,0,1)m.

6

OPTIMIZATION PROBLEMS

Two different cost functions are investigated. The first cost function is trying to direct the exterior acoustic
energy to a specific direction, more specifically the pressure magnitude is maximized at the field point (0.82,
0, 0.57) m. The second cost function minimizes the difference between a target pressure and the pressure
magnitude in the field point in an attempt to flatten the frequency response. A potential way to formulate
this is by using the two cost functions given by
φk,1 = − |pi |2and

φk,2 = (Tp − |pi |)2(13)

where k indicates that the cost function is evaluated at the k-th frequency, pi is the complex pressure at
the field point, and Tp = 0.3 (corresponds to a SPL of 80.5 dB) is the target pressure magnitude. The
design sensitivities are obtained following the semi-analytical adjoint approach in Ref. [11]. For multifrequency optimization it is still an open research question how to formulate good performing cost functions
and optimization approaches. However, unpublished research indicate that using a minimax formulation is agood candidate. Therefore, the cost functions are minimized using the minimax formulation in the MATLAB
function fminimax. For both test cases the optimization is performed at 5 discrete frequencies logarithmicallyspaced between 1 kHz and 3 kHz.

7

OPTIMIZATION RESULTS

The results of the two shape optimization cases are shown in Figure 4. Figure 4a and 4b show the SPL
frequency responses for the designs obtained using φk,1 and φk,2 , respectively. The figures also includes the
response of the initial design. As it is observed, it has been difficult for the optimization to drasticallychange the overall behaviour of the responses. This is to some extend attributed to the limited freedom
given to control points in the FFD regions. Also, most of the optimization effort is put towards changing
the response at 3 kHz. Inspecting the overlayed displacement magnitude at 3kHz seems to indicate that the
optimizer uses a combined cabinet and loudspeaker driver resonance to increase the pressure output. At 3
kHz the top of the cabinet is vibrating but also a break-up resonance of the loudspeaker diaphragm can beobserved. Even though optimizing with two different cost function definitions gives very similar results, it isobserved that the location of the first cabinet resonance (around 800 Hz) can be manipulated as is evident
when optimizing with φk,1 . Hence, it might be interesting to focus the optimization to this region.

a)x

749 Hz95

SPL (dB)

90

max

x

3 kHz

max

z ZeroOptimized
Initial

z

Zero

85
80
75

x

200 Hz

max

70
65
60
100

z Zero 500
1000
Frequency (Hz)

2000

3000

b)
x

795 Hzmaxx95

SPL (dB)

90

3 kHz

max

z ZeroOptimized
Initialz Zero

85
80
75

x

200 Hz

max

70
65
60
100

z Zero500
1000
Frequency (Hz)

2000

3000

Figure 4. The SPL frequency response of optimized designs and the initial design. In the figure, a) andb) are the frequency responses of the designs optimized using φk,1 and φk,2 , respectively. The overlaying
plots are the magnitude of the z-component of the displacement and the vertical dashed black lines are theoptimization frequencies.

8

DISCUSSION AND CONCLUSIONSThis work has presented implementation aspects of a three dimensional full vibro-acoustic shape optimization
framework and its application to loudspeaker simulation and optimization. This includes the development of
a partially lumping approach of the loudspeaker basket, magnet and spider that reduces the overall computational complexity of the loudspeaker model. Event though the optimization framework is working as
intended, the results also showcase some of the many difficulties of broadband vibroacoustic optimization.
One of the difficulties is the computational burden of 3D vibroacoustic optimization. For the given optimization problems, each of the optimization cases takes approximately 2 days to complete. Moreover, structural
resonance phenomena are very local meaning that a much finer frequency resolution is required as comparedto what is presented here. Even with a fine frequency resolution it might be difficult to change the frequency response. Therefore, it will most likely be required to also play with different materials, damping
and stiffening mechanisms, and starting guesses to limit the effect from structural resonances. Also, it has
been decided to use a rather conservative range at which the control points in FFD region can move. Albeit
increasing the range of the control points can result in more design freedom. However, it also brings therisk of distorting the elements too much. Hence, incorporating a more robust parametrization method suchas level-set parametrzaiton perhaps in combination with a cut-element approach might be beneficial. Alternatively, nonlinear constraints can be incorporated that insures a sufficient element quality at all optimization
iterations. Another approach is to use re-meshing that ensures sufficient element quality, which would be atechnique well suited with FFD as the parameterization and the physics modeling is completely decoupled.
Additionally, only two cost functions have been tested, a much larger range of cost functions should be tested
in a comparative review, e.g. minimizing the displacement magnitude, directivity index etc.

REFERENCES[1] Udawalpola R., Wadbro E., Berggren M. Optimization of a variable mouth acoustic horn. (2011) Int J
Numer Methods Eng. 85(5) 591-606[2] Udawalpola R., Berggren M. Optimization of an acoustic horn with respect to efficiency and directivity.
(2008) Int J Numer Methods Eng. 73(11) 1571-1606[3] Schmidt S., Wadbro E., Berggren M. Large-scale three-dimensional acoustic horn optimization. (2016)
Siam Journal on Scientific Computing. 38(6) B917-B940.
[4] Bezzola A. Numerical optimization strategies for acoustic elements in loudspeaker design. (2018) AudioEngineering Society, 145th Audio Engineering Society International Convention.
[5] Christensen R. Shape and Topology Optimization of Loudspeaker Drivers. (2020) Proceedings fromCOMSOL Conference, Grenoble/Online.
[6] Nielsen D. G., Agerkvist F. T., Jensen J. S. Achieving a flat, wideband frequency response of a loudspeaker unit by numerical optimization with requirements on its directivity. (2021) Journal of the Acoustical Society of America. 150(2) 663-672[7] Creixell Mediante E., Jensen J. S., Naets F., Brunskog J., Larsen M. Adaptive parametric model order
reduction technique for optimization of vibro-acoustic models: Application to hearing aid design. (2018)
Journal of Sound and Vibration. 424 208-223[8] Dilgen S. B., Aage N., Jensen, J. S. Three dimensional vibroacoustic topology optimization of hearing
instruments using cut elements. (2022) Journal of Sound and Vibration. 532 116984[9] Sederberg T. Free-form Deformation of Solid Geometric Models. (1986) Computer Graphics. 20(4) 151160[10] Andersen P, Kook J, Cutanda Henríquez V, Aage N. Towards large-scale acoustic shape optimization forindustrial applications using the boundary element method. Proceedings of 2020 International Congresson Noise Control Engineering, Inter-noise, Soul, South Korea, Online.
[11] Andersen P, Kook J, Cutanda Henríquez V, Aage N. 3D shape optimization of loudspeakers. Proceedingsof 2021 International Congress on Noise Control Engineering, Inter-noise, Washington, United States ofAmerica, Online.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022 in Gyeongju, Korea

ABS-0761

Directivity optimization of MEMS ultrasonic transducers byimplementing acoustic horns
Tenia MEISEL1; Anton MELNIKOV1; Adrian ALEXANDER1, Tim BRÄNDEL1, Jorge M
MONSALVE1, Bert KAISER1, Harald SCHENK1,2
1
2

Fraunhofer IPMS, Dresden, Germany

BTU Cottbus-Senftenberg, Cottbus, Germany

ABSTRACTThe applications of microscopic ultrasonic transducers are often limited due to their non-optimal
directivity pattern. The aim of this work is to design passive structures to adapt the directivity pattern
to an intended application, e.g. range finder. The horns were designed and optimized using a FiniteElement-Method (FEM) model and manufactured by a conventional 3d-printing technique. The
experimental validation was done using a novel MEMS-based ultrasonic transducer based on lateral
actuation developed by Fraunhofer IPMS. This transducer type generates sound waves by displacing
air inside a 3x3 mm² silicon chip using microscopic sized beams instead of using a diaphragm. This
article presents several horn structures that exhibit a pronounced main lobe, inter alia an exponential
horn and a folded horn with reduced overall dimensions, both optimized for an operatin g frequencyof 40 kHz. We have shown numerically and experimentally that directivity properties of the transducer
were significantly improved considering peak pressure, reduction of the side lobes and main lobe
width by simple horn geometries. The implementation of the designed horns will enable additional
applications where a specific directivity pattern is required. Furthermore, the given results imply th at
the presented design strategies can be used to create various directivity patterns.
Keywords: acoustics, acoustic horn, directivity optimization, ultrasonic transducer, ultrasound

1. INTRODUCTION
Ultrasonic transducers have a wide range of applications in research and industry. They are used,among other things, for gesture recognition, collision warning, non-destructive material testing, andlevel measurements. Micromachined Ultrasonic Transducers (MUTs), which are fabricated withtechniques developed for the microelectronics industry, benefit from a small size and have a be tter
acoustic coupling with the surrounding air than the traditional bulk piezoelectric transducers.
Recently, a new sort of MUT has been developed at Fraunhofer IPMS, which is not based on thevertical (“out-of-plane”) deflection of a membrane, but instead it induces a lateral (“in-plane”)
deflection an array of microbeams (1). A thourough description of this transducer can be found in (2).
The core element of this transducer is a thin, rectangular beam placed in front of a fixed electrode that
attracts the beam with an electrostatic force once a voltage difference is applied between them. This
beam-stator arrangement was laid out in such a way that several beams can be placed next to eachother in a single chip while ensuring their electrical connection. Figure 1 shows a layout of the inplane electrostatic MUT that was analysed in this work.
In a previous study, we analysed the radiation pattern of this ultrasonic device, finding a highagreement of the experimental radiation pattern and that obtained by a COMSOL simulation with theboundary-element method (3). This simulation, which assumed the “subchips” in figure 1 as flat,square pistons led, to the conclusion that the radiation pattern of this ultrasonic device was heavily
dependent on the immediate surroundings (i.e. the housing) of the chip, and not on the shape of its
elementary beams. This is consistent with the fact that the length of the beams is less than λ/6, theycount thus as point sources at the operation frequency. We observed, nonetheless, that at some of the1

jorge.mario.monsalve.guaracao@ipms.fraunhofer.defrequencies of interest there were unwanted minima in the desired propagation direction, as well as
undesired side lobes. Therefore, a further step in the design of this device is the enhancement of thedirectivity by means of passive structures (i.e. acoustic horns).
The aim of this work is to design an acoustic horn as a passive structure that maximizes acousticpressure of the main lobe and at the same time reduces side lobes, which results in improved soundfocusing. Additionally, the horn should not only meet the acoustic requirements, but also the physicalones. These include, above all, minimization of the installation space and simple production. The
acoustic horns were designed using the COMSOL Multiphysics software and, in order to determine
the accuracy the simulation, a verification experiment was conducted. Finally, this work will give an
insight into the possibilities that arise when sound propagation can be manipulated in a desired manner.

Figure 1 – Layout of the electrostatic in-plane MUT with five independent subchips with an area of
3x3 mm2, each containing 258 microbeams connected in parallel.

2. METHODS
2.1 The FEM model
To simulate the propagation of the sound waves generated by the transducer and the influence of
acoustic horns on its directivity pattern, a Finite-Element-Method (FEM) model was set up using the
COMSOL Multiphysics software. Figure 2 shows the model, which was designed in a Cartesian
coordinate system, with the point (0,0,0) representing the center of the transducer surface. It resembles
the air domain above the transducer in positive z-direction. The implemented acoustic horns are
adapted to the shape of the transducer. Within the defined domain of the model the Helmholtz equationin the frequency domain was solved (4, p. 202). Then the model used the Helmholtz-Kirchhoff integral
to obtain the acoustic pressure in the exterior field outside of the defined domain (4, p. 214). The
sound waves are assumed to be radiating with a frequency of 40 kHz from an infinite baffle toward
an open half space.

Figure 2 – FEM model of the transducer representing the air domain in positive z-direction.
The sound waves are generated by applying a normal velocity of 1 m/s at the outer transducer
surface (Huygens principle, see 5, p. 27), which coincides with the x-y-plane at z = 0. Two different
operating cases of the transducer units are used in the simulation: simultaneous operation of all units
as a basis for optimization process, and individual operation of unit 2 for comparison purposes
between simulation and experiment. The domain was discretized by λ/10 hexahedral elements with
Lagrangian shape function of second order. To reduce the time of calculation a symmetry plane isdefined on the y-z-plane at x = 0 and on the x-z-plane at y = 0. A Cartesian perfectly matched layer(PML) is defined on the domains that confine the air space. It absorbs the outgoing sound waves insuch a way that this artificial termination has no influence on the near field. The
2.2 Analytical design method – The Webster horn equation
Initial approach for a focusing horn design is the one-dimensional Webster horn equation (6, p.
135 and 7, p. 191)
∂2 𝑝 d(ln 𝐴) ∂𝑝
1 ∂2 𝑝(1)+=∂𝑧 2
d𝑧 ∂𝑧 𝑐 2 ∂𝑡 2
with acoustic pressure p, speed of sound c and cross section A. It describes the acoustic pressure as afunction of the distance from the source for radially symmetrical horn shapes of infinite length andconsiders the cross section of the acoustic horn. With the approach for an exponential horn we obtain
the partial solution for a diverging sound wave emerging from the horn
𝑎𝑧 −𝑖𝑘𝑧√1− 𝑎2
4𝑘 2

𝑝(𝑧) = 𝑝0 𝑒 − 2 𝑒(2)where the parameter a > 0 describes the exponential increase of the horn. With constant distance fromthe radiator z and constant wave number k, equation 2 is now only dependent on the parameter a. Now,
the question is for which a the acoustic pressure p(a) becomes a maximum. After differentiation andnumerical solving for z = 100 mm and k with an operating frequency of f = 40 kHz, we obtain
1(3)mm
Equation 3 then results in an exponential horn whose cross section is matched to the shape of thetransducer and therefore is not radially symmetric.
𝑎 = 1.036

2.3 Numerical designs method – The parametric sweep
Another design strategy is the numerical study about the influence of different design properties
that define the shape of the horn. The number of parameters that are varied determines the degree ofthe numerical study. The parameters that currently describe the shape of the acoustic horn are theexponent a and the length of the horn l.
First, based on the initial design from section 2.2, a numerical study was conducted to investigate
the influence of an additional tube attachment with length b on the directivity of the acoustic horn.
For this purpose, a parameter variation was performed for a and l, respectively for the horn without
and the horn with tube attachment. This parameter variation results in a 𝑀 ∈ |𝑎| × |𝑙| matrix. From
the results of the simulation, each entry of the matrix, i.e. each parameter combination describing theacoustic horn, can be assigned a value for different directivity metrics under investigation. From theresulting numerical plots of the metrics over the parameter matrices (5, p. 43), we can see the effectof the additional tube attachment on the directivity of the transducer. In addition, we can select theparameter combination that achieves the desired results: a maximization of acoustic pressure of themain lobe and a minimization of acoustic pressure of the first order side lobes.
Next, a parameter variation of higher degree, including the additional tube attachment b as aparameter, was applied. To obtain the parameter combination, which results in a horn that results inthe desired directivity of the transducer, a parametric sweep as a build in COMSOL function, is applied.
For each parameter, an individual range is specified (i.e. see section 3.4 Parametric sweep horn) in
which each parameter is varied in equidistant intervals. To find the most ideal parameter combinationin the solution space an evaluation function is applied. It calculates and sorts the directivity metrics
for each parameter combination. The resulting vectors assign the value of their index to eachparameter combination, so that the combination with the smallest sum represents the optimal solution(see 5, p. 53).
2.4 The folded horn method
Since the Webster horn equation from section 2.2 assumes an infinite long horn, its solution implies
that in reality the acoustic pressure increases for longer horns. In order to achieve a further increaseof acoustic pressure without enlarging the overall dimensions of the horn, a second opposite
exponential horn is introduced into the existing horn (6, p. 424). This inner horn reflects the soundwaves inside the horn back to the transducer surface and this way increase s the acoustic pressure.

Thus, the sound waves do not have to propagate through a longer exponential horn but the acousticpressure is increased already in the structure itself. The intended horn design is presented in figure 3.

Figure 3 – Basic principle of the folded horn design. The inner horn is positions inside the outer horn withtotal length l+b, l length of the horn and b length of the tube attachment.
To allow the resonances to occur that lead to an increase in acoustic pressure, a multiple of the half
wavelength above the transducer surface is chosen for the location of the inner exponential horn. The
aperture of the inner horn is chosen bigger than half wavelength, in order to allow an interaction insuch a way that reflection and not only diffraction effects occurs between the sound waves generated
by the transducer and the inner acoustic horn.
2.5 Experimental Setup
The resulting horns are manufactured using a conventional 3d-printing technique and then attachedto the board with the transducer. The measurement signal is generated by an audio analyzer, and isthen amplified by a Krohn-Hite 7602M amplifier and provided with a constant offset. The sound waves
are generated by unit 2 of the transducer only, which is operated at a bias voltage of 30 V and an
amplitude voltage of 5 V. The acoustic field is measured with a Microtech Gefell MM310 free-fieldmicrophone, which is amplified by Microtech Gefell MV310 microphone amplifier. The microphone
is positioned oppositely to the transducer with the horn. Then, at a constant distance z, the transducer
is rotated by an angle of Φ = 360° azimuth while the microphone is rotated by an angle θ = 59° altitude,with the Linear Stage M-403.8/6DG and the Precision Rotation Stage M-060.DG respectively.
2.6 Directivity metrics
For comparison purposes, different metrics are obtained from the directivity patterns of simulationand experiment. The absolute acoustic pressure p ml, the beam width of the main lobe θ b and the degreeof the first order side lobes p ml/p sl. The absolute acoustic pressure quantifies the acoustic pressure of
the main lobe determined at Φ = θ = 0°, and z = 100 mm. The beam width of the main lobe isdetermined for 6 dB level down from Φ = θ = 0°. The degree of the first order side lobes is definedby the quotient of the absolute acoustic pressure of the main lobe p ml and the absolute acoustic pressureof the first order side lobe psl.

3. RESULTS
3.1 No horn
The directivity patterns of normalized acoustic pressure on azimuth Φ at θ = 0° and z = 100 mm
shown in figure 4 show a general agreement of simulation and experiment.

Figure 4 Directivity patterns of the transducer without acoustic horn for FEM simulation and experiment
standardized to mean value.

The main lobes align well while the directivity patterns differ in their side lobes. The plotemphasizes the potential for improvement: For both plots, the main lobe shows a lower acousticpressure than the side lobes that radiate at ±55° from the main lobe. The aim is therefore to minimize
or even eliminate the side lobes and to maximize the acoustic pressure of the main lobe.
3.2 Webster horn
To investigate the correlation between the acoustic pressure at Φ = θ = 0° and z = 100 mm andexponent a, the exponent is varied at constant horn length of 4 mm. The graph for the absolute acousticpressure p ml dependent on exponent a presented in figure 5a shows a local maximum at the exponent
obtained from Webster’s horn equation (eq. 3). Figure 5b shows the frequency at which the directivity
pattern shows the highest acoustic pressure at Φ = θ = 0°, and z = 100 mm. This frequency is locatedaround 40 kHz and therefore confirms that the Webster horn designed according to section 2.2 isoptimized for the operating frequency of the transducer.

Figure 5 – a) Acoustic pressure pml over exponent a showing a local maximum around a = 1.036.
b) Frequency chart (polar diagrams for all measured frequencies) showing the optimal operating frequency
at 40 kHz.
For individual operation unit 2, the directivity patterns of simulation and experiment are presented infigure 6a. The beam widths show a deviation of only 5 % with 36.1° and 34.4°, for simulation andexperiment respectively. The main lobes match well, while the side lobes with ratio psl/p ml = 0.62 are
more pronounced in reality than they are assumed in the simulation. Compared to the transducer
without an acoustic horn, this still means a reduction to almost one third of the previous value of
psl/p ml = 1.68 (see figure 6b). Additionally, the acoustic pressure of the main lobe was more than tripled.

Figure 6 – a) FEM and experimental directivity patterns standardized to mean value in comparison.
b) Experimental directivity patterns of transducer and Webster horn standardized to maximum value of
Webster horn in comparison.
While this is an improvement compared to the directivity pattern of the transducer without an
acoustic horn, it still means that undesired side lobes continue to occur.
3.3 Folded horn
To further reduce side lobes, an additional structure in shape of a tube attachment with the lengthof b = 1 mm is implemented in the horn. A numerical study according to section 2.3 is conducted andthe resulting numerical plots in figure 7 show the influence of the tube attachment on the side lobes.
It increases the acoustic pressure of the main lobe (see 5, p. 43) and, at the same time, re duces theside lobes. It is now possible to determine the parameter combination for which the directivity pattern
fulfills the intended conditions. In this case, a horn with the parameters a = 0.6 1/mm and l = 2.5 mm

Figure 7 – Numerical plots for the degree of the first order side lobes psl/pml for the variation of a and lwithout and with additional tube attachment.
is selected. The directivity patterns of simulation and experiment presented in figure 8a show a very
good match. The comparison of the Webster horn and the horn with the additional tube attachmentshows an increase by 31 % of the acoustic pressure of the main lobe and a decrease by 48 % of theratio p sl/p ml , resulting in psl/p ml = 0.33 (figure 8b).

Figure 8 – a) FEM and experimental directivity patterns standardized to mean value in comparison.
b) Experimental directivity patterns of Webster horn and horn with tube attachment standardized tomaximum value of Horn with tube attachment in comparison.
Another numerical study is conducted for design parameter of the inner exponential horn. The
directivity patterns of the resulting folded horn with a = 0.5 1/mm and l = 2 mm show a good matchof the main lobes (figure 9a). However, the side lobes do not match, with the asymmetry of the side
lobes in the experiment being particularly prominent. Figure 9b shows that this design method does
not result in an increase of sound pressure for individual operation of unit 2. Nevertheless, simulationand experiment both show that the sound pressure is decreased by 50 % compared to the acousticpressure of the horn with tube attachment.

Figure 9 – a) FEM and experimental directivity patterns standardized to mean value in comparison.
b) Experimental directivity patterns of horn with tube attachment and folded horn standardized tomaximum value of horn with tube attachment in comparison.

3.4 Parametric sweep horn
The total length of the previously presented horns ranged from three to four millimeters. We now
allow bigger overall dimensions but limit the total length l+b to 12 mm. For the parametric sweep ofhigher degree, including b as a parameter, the parameter ranges shown in Table 1 are specified.
Table 1 – Specified ranges for the parametric sweep
Parameter

Lower Boundary

Scale

Upper Boundary

Description

P1 = a0.14 1/mm

0.0025 1/mm

0.2 1/mm

Exponent

P2 = l

6 mm

0.25 mm

10 mm

Length of the horn

P3 = b00.25 mm

2 mm

Length of tube attachment

Again, as shown in figure 10a, the directivity patterns of simulation and experiment show very good
match. The beam width and the ratio psl/p ml both only show a deviation of 3.2% from the measured
value. The implementation of the parametric sweep horn nearly quintuples the acoustic pressure of
the main lobe, while the side lobes that now occur at an angle of ±28° are reduced by 80 %. The beamwidth remains unchanged compared to the transducer without an acoustic horn (figure 10b).

Figure 10 – a) FEM and experimental directivity patterns standardized to mean value in comparison.
b) Experimental directivity patterns of transducer without acoustic horn and parametric sweep hornstandardized to maximum value of parametric sweep horn in comparison.

4. DISCUSSION
The previously determined directivity patterns show a good agreement between simulation andexperiment for all presented acoustic horns. Especially the accuracy of the simulation results of themain lobe was proven. In the cases where the side lobes do not show good match, they are larger inthe experiment than they are assumed in the simulation. This is mainly due to the assumption of an
infinite half space and an infinite baffle in contrast to the real geometry of the board. Apart from theWebster horn, the ratios of the measured sound pressures between the individual horns relative to thetransducer without an acoustic horn show a deviation of 6% – 15%. Table 2 shows the resulting
experimentally measured values of the directivity metrics.
Table 2 – Experimentally measured values of directivity metrics for all presented acoustic horns
horn design

pml [Pa]θb [°]

psl/pmll + b [mm]no horn (transducer)

47.4

30.2

1.68

-

Webster horn

153

34.4

0.62

4

tube attachment horn

199

27.8

0.33

4

double horn

99.6

24.2

0.42

4

parametric sweep horn

229

30.4

0.33

11

In sections 3.1 to 3.4, the directivity of the transducer was improved successively starting with theinitial directivity of the transducer without an acoustic horn until the parametric sweep horn in section

3.4. The directivity pattern with individual operation of unit 2 of the transducer showed the
optimization potential especially in the maximization of the main lobe and the minimization of theside lobes. The Webster horn as the initial approach increased acoustic pressure of the main lobe, but
the limitations of this analytical approach lead to a reoccurrence of side lobes. The parameter
optimization of the exponential horn according to the Webster horn equation refers to the
maximization of the acoustic pressure on the z-axis of radially symmetrical horn shapes. Therefore,
no conclusions can be drawn about the occurrence or the degree of the side lobes.
The tube attachment was then able to reduce the reoccurrence of side lobes while increasing theacoustic pressure of the main lobe (8). The presented folded exponential horn did not show the desired
increase of the acoustic pressure. The main reason for this is that the fine structures of the folded horn
could not be manufactured in a sufficient quality, which also leads to the asymmetry of the side lobes.
With the condition of higher resolution manufacturing and simultaneous operation of all units, theFEM simulation shows the highest increase of acoustic pressure for the folded horn, in that casemaking it a promising prospect for a focusing horn design with reduced overall dimension.
Compared to the transducer without an acoustic horn the parameteric sweep horn shows the biggest
increase in acoustic pressure of all the acoustic horns presented, in both simulation and experiment.
With a total length of 11 mm, it takes up the most installation space among all the horns presented.
Now, the question is why the final design of the acoustic horn differs from the analytical method. As
mentioned earlier, Webster’s horn equation is restricted to radially symmetrical horn shapes. Whereas,
the parametric sweep horn uses impedance matching and constructive interference to maximize theacoustic pressure of the main lobe. If you consider constructive interference of a sound wave
𝜆propagating through a single slit with d ~ λ, tan(𝛼) = 𝑑 = 27.7° gives the angle at which the side
lobes of the first order occur for the parametric sweep horn. Ultimately, the decision, which design
feature and directivity metrics to prioritize, depends on the type of operation of the transducer and theintended application of the horn.

5. CONCLUSIONS
By designing an acoustic horn using a FEM model, the directivity pattern of the MEMS-based
ultrasonic transducer could be significantly improved. The influence of different horn parameters(including an additional tube attachment) on the directivity pattern was shown numerically andexperimentally. Ultimately, various approaches to modeling acoustic horns were investigated, theapplication of which result in improved sound focusing. The designed horns in this article thus referto a minimization of the beam width and the acoustic pressure of the side lobes, and to a maximizationof the acoustic pressure of the main lobe.
Application offered by the implementation of focusing acoustic horns include improved distanceand level measurements, where a wide directivity pattern leads to disturbing re flections fromsurrounding objects.
Since the generation of the intended directivity pattern by the presented acoustic horns has
succeeded, this indicates that with further approaches or with the expansion of the presented
approaches, different directivity patterns can be addressed. Since the sound is always received as it isemitted, the generation of sound propagation with a spherical characteristic could be of interest. It isalso possible to optimize the shape of an acoustic horn that influences the directivity pattern in a wide
range of acoustic medium parameters (e.g. the speed of sound). Furthermore, the results indicate thatthe ideal horn shape depends on the combination of simultaneously operated transducer units. The
potential of this work lies not only in the fact that the acoustic horns show the intended directivity
pattern, but also that they were manufactured using a conventional 3d-printing technique, whichmakes them inexpensive and easy to manufacture.

REFERENCES
1. Monsalve J.M., Melnikov A., Stolz M., Mrosk A., Jongmanns M., Wall F., Langa S., Marica-Bercu I.,
Brändel T., Kircher M., Schenk H.A.G., Kaiser B., Schenk H. Proof of Concept of an Air-Coupled
Electrostatic Ultrasonic Transducer Based on Lateral Motion. Sensors and Actuators A: Physical; 2022.
2. Monsalve J.M., Kircher M., Wall F., Krenkel M., Kaiser B., Langa S., Schenk H. First Time of nanoscopic
electrostatic drives pushing for ultrasonic transmission for gesture recognition. IEEE International
Ultrasonics Symposium (IUS); 2020. DOI: https://doi.org/10.1109/IUS46767.2020.9251316
3. Melnikov A., Marica-Bercu I., Monsalve J. M., Jongmanns M., Kaiser B. Strahlcharakteristik eineselektrostatischen,membranlosen
MEMS-Ultraschallwandlers
fürUmgebungserfassung,MikroSystemTechnik Kongress 2021; 2021.
4. COMSOL. The Acoustics Module User’s Guide.
5. Meisel T. Optimierung der Richtcharakteristik von MEMS-Ultraschallwandlern durch passive Strukturen.
Bachelor Thesis. BTU Cottbus-Senftenberg, Fraunhofer IPMS. Cottbus, Dresden; 2022.
6. Lerch R., Sessler G., Wolf D. Technische Akustik. Springer Berlin Heidelberg; 2009.
7. Ginsberg J. H. Acoustics-A Textbook for Engineers and Physicists. Cham: Springer International
Publishing; 2018.
8. De Boer B. Acoustic tubes with maximal and minimal resonance frequencies. The Journal of the AcousticalSociety of America; 2008. DOI: 10.1121/1.2935231

ABS-0768

Efficient placement of damping layers on an aircraft fuselage for cabin noise
reduction
Sebastian ROTHE(1) , Christopher BLECH(2) , Steffen HOFFMANN(3) , Sabine C. LANGER(4)(1) TU Braunschweig, Institute for Acoustics, Germany, sebastian.rothe@tu-braunschweig.de(2) TU Braunschweig, Institute for Acoustics, Germany, c.blech@tu-braunschweig.de(3) TU Braunschweig, Institute for Acoustics, Germany, steffen.hoffmann@tu-braunschweig.de(4) TU Braunschweig, Institute for Acoustics, Germany, s.langer@tu-braunschweig.de

ABSTRACT

A consideration of sound reduction measures in early design phases is expected to have a decisive advantage
compared to design changes in later phases. The combination of numerically solved vibroacoustic models androbust identification criteria for the placement of measures enables one-shot design proposals with respect toacoustic objective functions. The sound pressure level within a passenger aircraft cabin is exemplarily considered within this contribution while a vibroacoustic finite element model of a fuselage section delivers theobjective function. The sound pressure levels in the cabin are reduced by the application of damping layers on
the outer skin of an aluminium airframe, whereby the distribution of such passive damping measures should
be well-considered for cost, weight and efficiency reasons. In order to efficiently identify a distribution, acomparison of three approaches for the placement is shown: Sophisticated one-shot design criteria such as thedeformation work and the mean squared velocity in skin fields and an intuitive engineering approach. Based on
the results, recommendations for the efficient placement of damping layers in early design phases are delivered
and potential extensions of the approaches are discussed.

Keywords: damping layer, placement, aircraft, acoustic-oriented design

1

INTRODUCTION

Vibrations in plate-like structures are often the cause of noise generated by technical products. Especially in
automobiles and aeroplanes, passengers are surrounded by thin-walled structures. Exciting dynamic forces of thestructures are unavoidable. It is therefore important to design and integrate acoustic measures already during
the design and development phase that reduce or prevent the generation of noise.
One frequently used acoustic measure are damping layers, which can be designed as unconstrained or constrained layer damping. In order to use them efficiently, the placement and distribution should always be adapted
to the vibroacoustic problem. An approach to meet this challenge is investigated in this paper.
Especially for noise prediction of large systems like aircraft cabins, which require a complex model withmany degrees of freedom to predict the acoustic response with good accuracy, innovative and efficient approaches are needed. Complex optimisation loops to identify suitable areas and the design of the measure are
to expensive and time-consuming. Therefore, the use of identification criteria, which can provide one-shot designs after one calculation, can support the early design process. On an realistic example the use of an acoustic
identification criterion is demonstrated in this work. Criteria based on physical system quantities are used to
identify a well suited distribution of damping layers on an aircraft outer skin. The aim is to reduce the soundpressure in the cabin for a defined critical frequency range in which, for example, a strong excitation of theengines is expected. A finite element model of a realistic aircraft segment serves as the basis.

2

PROBLEM STATEMENT

As motivated above, a complex vibroacoustic system is studied in order to derive one-shot design proposals forthe placement of damping measures by use of acoustic identification criteria. The problem of sound transmitted
within an aircraft fuselage segment is chosen as exemplary demonstrator. The model, shown in Figure 1, issolved in frequency domain by the finite element method (A BAQUS 2018) and directly delivers the sound pressure inside the passenger cabin. As excitation, a single point source is placed near the outer skin (2 m distance).
The aircraft fuselage model comprises four fully coupled structural and acoustic domains:• Airframe | Shell formulation (8-node quadrilaterals) for 3 mm aluminium modelled by a linear elastic
isotropic material with structural damping. For simplicity, linear elastic properties of aluminium are considered within this paper. Future aircraft are mainly expected to be made of carbon-fibre reinforced plastic(CFRP), which allows the direct integration of damping layers. In Figure 1, 78 skin fields within the airframe are highlighted as design space for the proposed damping measure.

airframepoint source

insulationinterior trim
passenger cabin

Figure 1. Finite element model of an aircraft fuselage segment with highlighted (orange) design space (78 skinfields).
• Insulation | Frequency-dependent complex-valued 3D Helmholtz domain for aircraft-grade glass wool
• Interior trim panels | Sandwich made of glass-fibre reinforced plastic (GFRP) top layers modelled byshells (8-node quadrilaterals, linear elastic orthotropic material with structural damping) and of a honeycomb core modelled by a 3D continuum (20-node hexahedrons, linear elastic orthotropic material withstructural damping)• Passenger cabin | 3D Helmholtz domain (20-node hexahedrons) with frequency-dependent complex-valued
bulk modulus in order to consider homogenised damping
Details on the airframe material parameters are given in Table 1, while information on the three remaining
mechanical domains can be found in [2].
Concerning the mesh size, non-conforming meshes are applied in the introduced domains fulfilling at least
10 nodes per wave length at the highest frequency of interest (580 Hz), respectively. As critical frequency range,
250 to 400 Hz is chosen exemplary as jet engine excitations and low transmission loss characteristics can be
expected within this range (compare [2]).

Table 1. Considered material parameters (aluminium) for airframe
parameter

symbolvalue

Young’s modulusE

7 · 1010 N/m2

densityρ

2700 kg/m3

Poisson’s ratio

ν

0.34

loss factor

η

0.02

The objective function is defined as the mean squared sound pressure in the cabin. For this purpose, anodeset of 90 nodes is selected at head height in the area of one seat row. The nodeset is highlighted in red
for different views on the fuselage segment in Figure 2.

Figure 2. Defined nodeset for the calculation of the mean squared sound pressure from different views of thefuselage segment; from left to right: front view, iso view, side view.
In order to reduce the objective function — the mean squared sound pressure (MSP) in the critical frequency
range — nine of the 78 skin fields (see Figure 1) are specifically damped. This corresponds to about 12% of thedefined design area. Two acoustic identification criteria are used to identify suitable fields. First, the deformation
work in the skin fields is used. The deformation work Wr is derived from the volume integral of the stresses σ
and strains ε (Equation (1)).
ZWr =σ ε dV(1)

V

According to ROTHE [1], this quantity is suitable to specifically place damping layers on structures. To do so,
the deformation work is calculated for each finite element in the centroid and averaged for the respective skinfields. Since these are frequency-dependent quantities, the next step is to sum over the critical frequency range.
This results in an assessment quantity according to which the area with the highest deformation work is themost suitable for placing damping. More details about this procedure can be found in [1] and [4]. It should be
mentioned that the approach proposed in the publications are shown to efficiently reduce structure-borne sound.
The use of this approach for the reduction of an objective function in the fluid (MSP) is a new step. The same
applies to the excitation of the structure via a point source.
Since the surface velocities are significantly responsible for the sound radiation, the previously described
procedure is also carried out with the mean squared velocity (MSV) in the skin fields. It serves as anotheridentification quantity, which results in the second identification criterion investigated in this study. Also in thiscase, the areas with the highest value (max.) are defined as the most suitable.
An intuitive engineering approach is used as the third alternative to be compared. For this purpose, the skinfields are selected intuitively according to engineering common sense. In addition, the variant in which all skin

fields are damped is also investigated in order to illustrate the maximum possible reduction.
As a passive acoustic measure, a highly damping polymer as an intermediate layer in the skin fields isassumed. This could be realised, for example, by combining an aluminium base and top layer with K RAIBON DAMP as an intermediate layer. Since the frequency-dependent behaviour of the storage and loss modulus is
unknown, this is determined by vibroacoustic measurement in order to take a realistic loss factor into account.
The measurement of the sandwich structures made of 3.0 mm aluminium and K RAIBON DAMP (ηComp ≈ 0.2)
results in a thousandfold increase of the loss factor of the basic structure (ηAl ≈ 0.0002) for several samples in
the interesting frequency range around 0.5 kHz. The procedure for determining the material parameters of thecomposite and the inverse determination of the material properties of the intermediate layer is described in [3].
To model the damped skin fields, only the loss factor (structural damping) is increased in these areas andassumed to be constant over frequency (ηdamp = 0.2). This has the advantage that only the position of thedamping is rated and no additional influences due to the frequency-dependent damping modelling. The rest of
the structure is modelled with a loss factor of η = 0.02. This is deliberately chosen to be higher than for purealuminium, in order to take into account additional damping of joints. The difference in loss factors between
damped areas (ηdamp ) and the basic structure (η) is therefore a factor of 10.

3

RESULTSThis section summarise the results of the investigations. The reference configuration and the fully damped
variant (full damped) are shown in Figure 1. The resulting three variants using the identification criteria as wellas the intuitive approach are shown in Figure 3. For the intuitive variant (intuitive damped), the 9 skin fields thatare as close as possible to the sound source are damped (see Figure 3, left). The skin fields of the approacheswith identification criteria according to maximum MSV (MSV (max.) damped) and maximum deformation work
(DW (max.) damped) in the critical frequency range from 250 to 400 Hz are shown comparatively. The damped
skin fields are highlighted in orange. In total, the same area is damped for all three variants.
intuitive damped

MSV (max.) damped

DW (max.) damped

Figure 3. Design variants with highlighted damped skin fields; from left to right: Intuitive engineering solution
near the sound source, maximum mean squared velocity, maximum deformation work.
In Figure 4, the resulting mean squared pressure is plotted in third octave band for the three proposed design
variants. In addition, a full damping treatment of the entire design space is studied, which yields a total level
reduction of 9.6 dB within the selected frequency range. By applying both the intuitive engineering approachand the two identification criteria (mean squared velocity, deformation work), a lower reduction is yielded as
only 12% of the area is treated. For the intuitive approach, a reduction of 4.8 dB results, while the criteria yielda better improvement of 5.5 dB in the cabin. The difference is small but expected to increase for more complex

models. For instance, an aircraft fuselage with varying skin field thickness, distributed stochastic loads and fulllength hardly allow a simple intuitive engineering solution for a damping measure design.

MSP (dB re 2 · 10−5 Pa)

100
90
80
referencefull damped (∆ MSP = -9.6 dB)
intuitive damped (∆ MSP = -4.8 dB)
MSV (max.) damped (∆ MSP = -5.5 dB)
DW (max.) damped (∆ MSP = -5.5 dB)

70
60
50
40

100

200

300
frequency (Hz)

400

500

Figure 4. Mean squared pressure over frequency (third octave band) at selected seat row (ear height volume)for different design variants; critical frequency range highlighted in grey.

MSP (dB re 2 · 10−5 Pa)

To illustrate the detailed frequency response of the sound pressure, it is shown in Figure 5 as a comparisonof the reference configuration and the variant with 9 damped skin fields according to the deformation work. Thisshows that some peaks can be highly reduced. How far this reduction also brings a psychoacoustic advantage
needs to be investigated with psychoacoustic metrics and cannot be assessed from this.
80
60
40
20
reference
DW (max.) damped (∆ MSP = -5.5 dB)

0
100

200

300
frequency (Hz)

400

500

Figure 5. Mean squared pressure over frequency at selected seat row (ear height volume) for three design
variants; critical frequency range highlighted in grey.

4

SUMMARY AND CONCLUSION

The investigations in this paper focus on the efficient placement of damping layers on an airframe to reduce
cabin noise. A realistic finite element model of an aircraft fuselage segment is used for this purpose. The aim
is to reduce the average sound pressure at a row of seats (ear height) by targeted damping of certain skin fieldsof the outer skin. For the efficient identification of suitable skin fields, two identification criteria and an intuitiveengineering approach are used in comparison.

The results show that a targeted distribution of damping in relation to a critical frequency range is advantageous. In comparison to the intuitively dampened variant, an additional reduction in the mean squared soundpressure level of approx. 1 dB can be achieved with the same damped area. Both identification criteria provide
a design proposal that results in a similar reduction — they also deliver a similar distribution pattern.
It should be noted that the identification criteria used in this paper are originally designed to reduce structureborne sound within a structure (see [1]). In order to be able to apply them validly to objective functions that
affect airborne sound, many more investigations are required. The investigations in this paper are only intended
to show the basic methodology. Nevertheless, the results already show that suitable areas for damping placement
can be identified very efficiently with the help of identification criteria.
In future investigations, the restriction to individual skin fields could be resolved in order to achieve a freedesign or distribution of the damping. This would possibly increase the efficiency of the applied damping measure. Additional expansion options also exist in the modeling of the outer skin. CFRP is of particular interest
as a material for future aircraft designs. Here, damping material such as K RABON DAMP can be integrated directly into the CFRP layers, as in [5]. In addition, a closer look at the frequency-dependent material behaviour(stiffness, damping) is recommended and should be considered for further investigations.

REFERENCES[1] Rothe S.: Design and placement of passive acoustic measures in early design phases. Schriften des Instituts
für Akustik, Braunschweig, Shaker, 2022, 2.
[2] Blech C.: Wave-resolving aircraft cabin noise prediction. Schriften des Instituts für Akustik, Braunschweig,
Shaker, 2022, 1.
[3] Hoffmann S., Rothe S., Langer S. C.: Vorgehen zur Charakterisierung und Modellierung von Mehrschichtbelägen. Conference proceedings of DAGA, Stuttgart, 2022.
[4] Rothe S., Hoffmann S., Langer S. C.: Efficient placement of damping patches based on an identification
criterion. Proceedings of ISMA, 2022.
[5] Fasana, A., Ferraris A., Airale A. G., Polato D. B., Carello M.: Experimental characterisation of damped
CFRP materials with an application to a lightweight car door. Shock and Vibration, 2017.

ABS-0810

Energy-based non-negative surface contributing approximated for the volumeof an inward radiating sphere
Caglar GURBUZ(1) , Steffen MARBURG(2)(1) Chair of Vibroacoustics of Vehicles and Machines, Germany, caglar.guerbuez@tum.de(2) Chair of Vibroacoustics of Vehicles and Machines, Germany, steffen.marburg@tum.de

ABSTRACT
Investigating vibrating surfaces is a continuing concern within the analysis of acoustic cavities. One valuable
tool to determine vibrating surfaces is the contribution analysis. However, a contribution analysis for an entire
cavity is usually accompanied with the cumbersome evaluation of the related volume integral. In order to bypass
that, this study aims to embed the approximative evaluation of the cavity in a contribution analysis. Therefore,
the boundary element method is deployed for solving the three-dimensional Helmholtz equation. Moreover, thesound energy is used as the objective function, since the sound energy is sensitive to the sound pressure as well as
the particle velocity. To validate the proposed method, the example of an inward radiating sphere is investigated.
Initial results already show that cavity volumes can be analyzed within an energy-based contribution analysis.
In this way, the proposed method designates an efficient technique to evaluate contributing surfaces inside an
acoustic cavity.

Keywords: Sound energy, Contribution analysis, Boundary element method

1

INTRODUCTION

Panel or surface contributions are a central concern in the design of vehicle cabins, particularly, in terms ofcustomer comfort. For this reason, various finite element software, e.g. COMSOL, LS-DYNA, or MSC Nastran,
adopted this method in their repertoire. In these packages, the sound pressure is considered as the objectivefunction. For interior acoustics, however, the sound energy density is suggested, since it is sensitive to thesound pressure and to the particle velocity [1, 2, 3]. In this way, contributing surfaces can be also recovered inregions with low sound pressure. In analogy to the non-negative intensity [4], the energy surface contributions
are also non-negative. since they are formulated in a quadratic expression. By this means, acoustic shortcircuits due to alternating signs are circumvented. Another similarity is that the energy-based contributions
are also implemented in a boundary element formulation. While the concept of energy densities has been
already implemented in a contribution analysis scheme, it has been yet limited to the evaluation at a single field
point [5]. This contribution aims to expand the energy-based contributions to the evaluation of an entire volume.
Therefore, the cumbersome volume integral is bypassed by evaluating the energy at a grid with multiple field
points. In this study, the proposed method is validated for an inward radiating sphere. Initial results emphasizethat contributing surfaces regarding an entire volume can be retrieved by evaluating multiple field points, which
are regularly distributed within the acoustic domain.

2

METHODSBy assuming the harmonic time dependence e−iωt , the three-dimensional acoustic problem in the frequency
domain is governed by the Helmholtz equation∆p(~x ) + k2 p(~x ) = 0

x ∈ Ω ⊂ R3 ,(1)with p(~x ) denoting the sound pressure in Ω and k = ω/c the wavenumber, which is composed by the angularfrequency ω and the speed of sound c. The Helmholtz equation fulfills the Neumann boundary condition (BC)v f (~x ) =1 ∂ p(~x )iωρ ∂ n(~x )x ∈ Γ ⊂ R2

(2)

and the Robin BCx ∈ Γ ⊂ R2 .

v f (~x ) − vs (~x ) = Y (~x )p(~x )(3)with v(~x ) denoting the particle velocity. The subscripts s and f denote the structural and the fluid particle
velocity.
The sound pressure at internal field points can be obtained byZZ ∂ G(~x ,~y i )~y i ∈ Ω,− iωρ G(~x ,~y i )Y (~x ) p(~x )dΓ(~x )(4)pi (~y i ) = iωρ G(~x ,~y i )vs (~x )dΓ(~x ) −∂ n(~x )ΓΓwith ~y i denoting the position of the field point. The field point has the subscript i = 1, 2, ...N, where N is thenumber of field points. Similarly, the particle velocity at internal field points can be computed by the normal
derivative of Eq.(4)ZZ
1 ∂ G(~x ,~y i ) ~~~~y i ∈ Ω.
~v i (~y i ) = ∇G(~x ,~y i )vs (~x )dΓ(~x ) −− ∇G(~x ,~y i )Y (~x ) p(~x )dΓ(~x )(5)∇ΓΓ iωρ ∂ n(~x )The sound pressure and the particle velocity at internal field points can be alternatively expressed in matrixformpi (~y i ) = b Ti (~y i )vvs

~y i ∈ Ω(6)T~v i (~y i ) = b̃bi (~y i )vvs

~y i ∈ Ω,(7)

where b i (~y i ) and b̃bi (~y i ) denote the influence coefficient vectors with respect to the sound pressure and theparticle velocity at the internal field points. The total sound energy in the acoustic field can be obtained by
Et = E p + Ek =

1
2ρc2

Z
Ω

p(~x )2 dΩ +

ρ4Z~v f (~x ) ·~v f (~x )∗ dΩ.

(8)ΩWithin the scopeof this work, the volume integral is approximated by evaluating the energy density at numerousRfield points, Ω e dΩ ≈ V · ∑Ni=1 ei . The surface contributions with respect to the sound energy can be thenobtained by
Et = E p + Ek =

ρ N T
1 N T ∗ TT(bbi v s ) (bbi v s ) + ∑ (b̃bi v s )∗ (b̃bi v s )∑22ρc i=1
4 i=1

H
H
= vH
s A p v s + v s A k v s = v s At v s(9)with A p , A k , and At denoting the coupling matrix regarding the potential energy, the kinetic energy, and thetotal energy. Finally, the energy-based surface contributions can be expressed by
Nn
Nn
√ H√
H
E = vHA
A v s = ∑ α ∗j α j = ∑ µ j ,s Avs = vs
j=1

(10)

j=1

with µ j denoting the contribution from the node with the identity j. For further details on the derivation of thesurface contributions, the interested reader is referred to [5].

3

NUMERICAL EXAMPLE

The sphere is of radius 1 m and is discretized by 384 boundary elements with constant, discontinuous interpolation
functions. We assume full-reflecting boundaries, Y = 0, and excite all elements with the real-valued structural
particle velocity, vs = 1.0 m/s. The relevant frequency is f = 100 Hz and we assume air as the acoustic medium(ρ = 1.21 kg/m3 , c = 343 m/s). Regarding the field point grids, we consider a single field point and a grid with
32 field points, which are regularly distributed. The mesh of the sphere and the related field point grids are
depicted in Fig. 1.

(b) Multiple field points(a) Single field point

Figure 1. Sphere mesh and the field point grids: The first grid consists of only a single field point (a), thesecond grid contains 32 regularly distributed field points (b). The contribution analysis is performed 100 Hz.
For the sake of overview, not all 32 evaluation points are depicted.
The single field point analysis indicates that surfaces contributing to the potential energy density can be
identified at one side of the sphere, see Fig. 2a. Regarding the kinetic energy contributions, all surfaces appear
inactive (Fig. 2b). As a result, the total energy surface contributions are very similar to those regarding thepotential energy (Fig. 2c). The multi-point analysis, on the other hand, reveals high contributions from each(a) Potential(b) Kinetic(c) Total

Figure 2. Surface contributions regarding the potential (a), kinetic (b), and total (c) energy at 100 Hz. The
contributions are evaluated for a single field point.
surface to the potential energy, see Fig. 3a. Almost no contributions can be observed regarding the kinetic
energy contributions (Fig. 3b). The total energy contributions, in turn, also indicates that all surfaces contributeto the total energy (Fig. 3c). By comparing the single field point analysis with the multi-point analysis, one
can observe that the ratio between the potential energy and the kinetic energy are nearly the same. Moreover,it becomes apparent that contributing surfaces spread from one side of the sphere to the entire area. This result

(a) Potential(b) Kinetic(c) Total

Figure 3. Surface contributions regarding the potential (a), kinetic (b), and total (c) energy at 100 Hz. The
contributions are evaluated for a grid with 32 regularly distributed field points.
already indicates that the surface contributions regarding the entire sphere volume can be obtained by evaluating
the energy contributions at a grid with multiple field points. In this way, the contribution analysis with regard
to multiple field points provides an efficient method to evaluate entire volumes.

4

CONCLUSIONSIn this work, a formulation was introduced for the evaluation of the surface contributions regarding an entire
volume. For this purpose, the cumbersome computation of the volume integral was bypassed by the evaluationof the energy density at multiple field points, which were regularly distributed within the acoustic field. By
using the energy density as an objective function, surface contributions can be also detected in regions withsound pressure. Furthermore, the energy-based contributions are non-negative by definition. In this way, acoustic
short circuits can be effectively circumvented. The proposed method was implemented in a boundary element
formulation and demonstrated for an academic example, the inward radiating sphere. The findings alreadyindicate that the evaluation of a volume can be efficiently approximated by evaluating a grid with multiple field
points. Future work will concentrate on the transfer to industrial application. In this context, a very interesting
field of application, which also motivated this research, is the vehicle interior noise problem.

ACKNOWLEDGEMENTS
This research benefited from a grant from the German Research Foundation (DFG) under award 418936727.

REFERENCES[1] Sommerfeldt S, Nashif PJ. An adaptive filtered-x algorithm for energy-based active control. J Acoust Soc
Am. 1994;96:300-306.
[2] Cazzolato B, Hansen S, Colin H. Active control of sound transmission using structural error sensing. J
Acoust Soc Am. 1998;104:2878-2889.
[3] Cazzolato B, Hansen S, Colin H. Errors arising from three-dimensional energy density sensing in onedimensional sound fields. J Sound Vib. 2000;236:375-400.
[4] Marburg S, Lösche E, Peters H, Kessissoglou N. Surface contributions to radiated sound power. J Acoust
Soc Am. 2013;133:3700-3705.
[5] Gurbuz C, Schmid JD, Luegmair M, Marburg S. Energy density-based non-negative surface contributions ininterior acoustics. J Sound Vib. 2022;527:116824.

ABS-0839
A reduced order model including viscothermal losses
Mikkel PALTORP SCHMITT(1) ; Vicente CUTANDA HENRÍQUEZ(2) ; Niels AAGE(3) ; Peter RISBY ANDERSEN(4)(1) Department of Electrical & Photonics Engineering, Technical University of Denmark, Denmark, mpasc@dtu.dk(2) Centre for Acoustic-Mechanical Microsystems, Department of Electrical & Photonics Engineering, Technical University ofDenmark, Denmark, vcuhe@elektro.dtu.dk(3) Centre for Acoustic-Mechanical Microsystems, Department of Civil & Mechanical Engineering, Technical University ofDenmark, Denmark, naage@mek.dtu.dk(4) Audio Research, GN Audio A/S & Jabra, Denmark, prandersen@jabra.com

ABSTRACTThe Boundary Element Method (BEM) is a well-known numerical method for solving time harmonic acoustical problems. While the BEM has attractive features e.g. automatically satisfying the free field conditions,the frequency dependence of the Greens function makes is inconvenient for broadband simulations due to excessive computational costs. This problem becomes worse yet when viscous and thermal effects are includedin the BE computations. Recently it has been suggested that a series expansion of the Greens function as wellas its directional derivatives together with model order reduction techniques can relieve some of the computational demands. This paper applies similar ideas in the setting of the so-called boundary layer impedanceboundary conditions used to approximate viscous and thermal effects. The final computational model can beused to efficiently perform broadband simulations including viscothermal losses of devices on the centimeterscale, thereby paving the path towards e.g. broadband shape optimization of small acoustical devices such astransducers, metamaterials and hearing aids.

Keywords: Boundary Element Method, Viscothermal Effects, Reduced Order Model1

INTRODUCTION

The Boundary Element Method (BEM) is a well-known simulation technique where the underlying (partial)differential equation is solved through the discretization of a boundary integral. It is a popular choice forproblems concerning semi-infinite and infinite domains, as the solution computed from the BEM automatically fulfills the so-called free-field conditions. As a result the method is widespread within the fields ofelectromagnetics and acoustics as these commonly deal with such domains. As opposed to the Finite Element
Method (FEM) the discretization of the BEM result in frequency-depended and dense linear systems. The
latter problem can be alleviated by utilizing compression techniques such as the Fast Multipole Method[1](FMM) and H -matrices[2]. While some details of the two approaches differ they are both based on thesame ideas of so-called near and far field splits of the underlying integral operators. While these approaches
solves the memory issue of the BEM they still require a reevaluation of the underlying boundary integrals
for every frequency of interest. As a result the BEM is cumbersome to apply when one is interested in awide range of frequencies. Several solutions to this problem has been proposed during the years, all withdifferent pros and cons. These techniques are almost exclusively based on interpolations of either the discrete or continuous form of the boundary integral [3, 4, 5]. In this text the interpolation scheme chosen isthat of a Taylor series expansion of the (derivatives of the) Green’s function. This approach transforms thetime-consuming boundary integral discretization into a sum of matrices. For small orders of Taylor expansions this reduces the assembly significantly, however the total memory used to store the derivatives matrices

makes this approach close to unusable in many cases. This memory increase can be diminished by following
the approach taken in Panagiotopoulos et al. [6]. Here a projection matrix is created from the collection of
Krylov vectors from specified frequencies. The full system is then projected resulting in a reduced numberof DOFs. This approach works under the assumption the solution is spanned by said Krylov vectors, which
is true for problems for which a Krylov based iterative solver converges quickly.
The main result presented in this paper is an extension of the model proposed in Panagiotopoulos et al.
[6]. This extension makes it possible for the model to also handle the so-called Boundary Layer Impedance(BLI) boundary conditions [7]. The text is structured as follows: In section 2 we describe how the discrete
BEM system representing the BLI boundary condition can be formulated as a Taylor series of frequencydecoupled matrices. Next, in section 3 the model order reduction procedure is explained. Finally, in section 4
the proposed model is numerically evaluated using a simple test case.

2

BOUNDARY ELEMENT METHOD WITH BLI BOUNDARY CONDITIONIn this section a rudimental description of the BEM, the Boundary Layer Impedance (BLI) boundary condition and the Series Expansion Boundary Element Method (SEBEM) is given.
2.1 The Boundary Element method
The BEM for time-harmonic acoustical problems is based on the discretization of the Kirchoff-Helmholtz
representation formula for the scalar Helmholtz equation. The representation formula has the following form
Zc(x)p(x) =∂ p(y)∂ G(x, y)dSy −p(y) dSy ,
∂ ny
∂ ny
Γ
Z

G(x, y)Γx ∈ Ω,(1)

where Ω is the domain with boundary Γ, c(x) is the free coefficient depending only on the geometry at point
x and (∂n G) G is the (normal derivative of the) Green’s function for the scalar Helmholtz equation, with theform
G(x, y) =exp (−ikkx − yk2 ),4πkx − yk2∂ G(x, y) exp (−ikkx − yk2 ) (1 + ikkx − yk2 ) (x − y)> ny=,∂ ny
4πkx − yk32

(2)

where k = ωc is called the wavenumber. Due to the nonlinear coupling between the frequency-depended
wavenumber and the variable of integration inside the exponential, the frequency can not be decoupled fromthe integration in (1). As a result the integral needs to be recomputed for every frequency of interest, making
the use of BEM for large frequency sweeps inconvenient. In subsection 2.3 we describe how these can be
decoupled through a Taylor series expansion.
2.2 The Boundary Layer Impedance
The Boundary Layer Impedance (BLI) boundary condition, described in Berggren et al. [7], is an impedance
like boundary condition in the way that it linearly relates the pressure and its normal derivative. However, in
addition to this relation there is also the addition of the tangential Laplacian∂p
ik2 i∆T= (γ − 1)
−
p,
(3)
∂n
kh
kv
where kh and kv are respectively the thermal and viscous wavenumbers and γ is the ratio of specific heats.
The two additional wavenumbers are computed using the frequency and the physical properties of the fluid
such as the thermal conductivity, specific heat capacity under constant pressure and the shear and bulk viscosity coefficients [8].
In the following the boundary of the domain is split into two parts. The first will be applied a Neumannboundary condition while the second will be applied the BLI boundary condition (Figure 1).

n

∂Γ

Ω

Γv

ΓBLI

∂Γ
Figure 1. Domain of an interior problem. The boundary is split into two parts: One with a Neumannboundary condition (Γv ) and one with the BLI boundary condition (ΓBLI ).
Using the boundary splitting and inserting (3) into the representation formula we attain the following


Z
Z
Z
∂ p(y)(γ − 1)ik2 i∆T∂ G(x, y)c(x)p(x) =G(x, y)dSy +G(x, y)−p(y) dSy −p(y) dSy . (4)
∂ ny
kh
kv
∂ ny
Γv
ΓBLI
Γ
Due to the polynomial representation of the pressure on each element the tangential Laplacian restricts theelements to be at least second order. To relax this restriction we apply integration by parts on the term
including the tangential Laplacian. Setting the additional term equal to 0, similar to what is described in
Berggren et al. [7], while rearranging a few terms we end up with the following equation of interestZΓv∂ p(y)∂ G(x, y)dSy = c(x)p(x) +p(y) dSy
∂ ny
∂ ny
Γ
Z

G(x, y)−(γ − 1)ik2
kh

Z
ΓBLI

G(x, y)p(y) dSy −

i
kv

Z

(5)
∇T G(x, y) · ∇T p(y) dSy .

ΓBLI

In the above only the tangential gradient of the pressure is needed, meaning that elements need only be of atleast linear order. Finally, applying the element discretization on (5) the following linear system of equations
is computed(γ − 1)ik2
idiag(c) + F(k) −H(k) − T(k) p = b(k).
(6)
kh
kv
2.3 Series Expansion BEM
As mentioned previously a drawback of the BEM is the frequency-depended integrals. In order to resolve
this issue all relevant kernels in (5) is expanded in terms of their respective Taylor series. This approach issometimes referred to as the Series Expansion BEM (SEBEM), although other series representations of thekernels could have been utilized. The simplest of the Taylor series expansions is that of the Green’s function
G(x, y) =exp (−ikkx − yk2 )1 ∞ (k − k0 )m (−ikx − yk2 )m=exp (−ik0 kx − yk2 ) ,∑ m!
4πkx − yk2
4π i=0kx − yk2(7)

where k0 is the expansion wavenumber. Applying the analogous on the remaining kernels in (5) we havethat the discrete form of (6) can be approximated as

!
M−1
M−1
(k − k0 )m(γ − 1)ik2
i(k − k0 )mdiag(c) + ∑Fm −Hm − Tm p = ∑
bm .
(8)
m!
kh
kv
m!
m=0
m=0
Note that we already here have truncated the series the Taylor series to be order M (degree M − 1). As aresult the above should strictly speaking be an approximation. While the Series Expansion Boundary Element
Method (SEBEM) reduces the assembly time by reducing the integral discretization to matrix additions, theresulting model have a large memory footprint, stemming from the need to store intermediate matrices. A
first fix could be to Taylor expand the full expression in (5) instead of the kernels separately. This would
result in only needing to store one derivative matrix (for the total derivative) instead of the 3 required in (8)(one for each integral). However, the implementation of this approach would be significantly more complicated while most likely being less accurate.

3

BOUNDARY ELEMENT METHOD WITH MODEL ORDER REDUCTION

In order to reduce the large memory footprint of the SEBEM a Model Order Reduction (MOR) approach
can be utilized. There exist several possible MOR techniques that can be applied with the one used here
following that of Panagiotopoulos et al. [6]. In short this approach works under the assumption that the true
solution can be well approximated by a lower-dimensional subspace, meaning that the total size of linear
system can be reduced by projection onto this subspace.
3.1 Krylov projection
In general a projection based MOR utilizes the assumption that the solution as well as the right-hand sideof the linear system of equations can be well-approximated by a lower-dimensional subspace. As such thefollowing are assumed to be good approximationsp ≈ U` p` ,bm ≈ U` b`m ,U` ∈ CN×` ,(9)where the columns of U` forms a basis for the lower-dimensional subspace. The Krylov projection technique
is based on setting these columns equal to the first ` Krylov vectors of the system matrix (subsection 3.2).
In order to reduce the system further, one can introduce a projection of the rows. If the projection of therows is equal to the Hermitian transpose of the column projection, then the approach is called Galerkin. Thisis the approach taken here. As a result (8) can be approximated as

!
M−1
M−1
m
2
(k
−
k
)
(
γ
−
1)ik
i
(k − k0 )m0UH
diag(c) + ∑Fm −Hm − Tm U` p` = UHU` b`m .
(10)
`
` ∑
m!
kh
kv
m!
m=0
m=0
Moving U` and UH` inside the parentheses and using that U` is unitary, the above can be written as

!
M−1
M−1
(k − k0 )m(γ − 1)ik2
i(k − k0 )mC` + ∑F`m −H`m − T`m p` = ∑
b`m ,
m!
kh
kv
m!
m=0
m=0

(11)

where all matrices are in C`×` and vectors in C` . Given that `  N the above is a very efficient way ofstoring the information. In many practical cases one find that storing the full reduced SEBEM requires lessmemory than storing just a single of the original N × N matrices.
3.2 Creating a projection matrix
Computing a projection matrix U` using just the information available at a single frequency does not result ina good approximation on the full frequency range (Figure 3). To resolve this issue information from multiple
frequencies is utilized. The collection of frequencies for which this information is extracted will be referred
to as primary frequencies (denoted by fi ). The total number of primary frequencies is denoted by L. At eachprimary frequency the full BEM system (A fi , b fi ) is computed and the first q Krylov vectors computed using
the Arnoldi Algorithm (Algorithm 1).

Algorithm 1 Arnoldi Algorithm
Assemble A fi , b fi
k1 ← bhfi /kb
i fi k2

K fi ← k1
for j = 2 : q do
j−1
k j ← A fi k j−1 − ∑l=1
kH
l A fi k j−1 kl
k j ← khj /kk j k2 i
K fi ← K fi
end forkj

The idea is that by collecting all ` = qL Krylov vectors will result in a projection matrix that covers
the full frequency range of interest. However, doing this without any additional computation will result in a
projection matrix that is not necessarily unitary, which was a requirement in order to arrive at (11). In order
to arrive at a unitary projection matrix the Singular Value Decomposition (SVD) is applied to the collectionof the Krylov subspaces
hiU` Σ` VH
(12)
K f1 K f2 . . . K fL ,` = svd
where U` is the final projection matrix.

4

NUMERICAL EVALUATIONIn this section the accuracy of the Reduced Order Series Expansion Boundary Element Method (ROSEBEM)
for the BLI boundary conditions is numerically evaluated. All simulation were performed using the Julia
programming language [9, 10] on computational resources provided by the DTU Computing Center [11].
The core of the code was inspired by the OpenBEM software written in MATLAB [12, 13].
4.1 Simulation setups
In this section we numerically study a small tube of length 10 cm and radius 1 cm (Figure 2). The usedmesh contained 1916 quadratic elements resulting in a total of 3834 DOFs. The frequency of interest is[20 Hz, 10 kHz], meaning that the mesh with hmax ≈ 0.033 cm had at minimum 10 elements per wavelength,
satisfying the standard rule-of-thumb of [14]. The object is excited by applying a Neumann boundary condition at one of the tube ends (Figure 2) while the remaining part of the boundary were applied the BLI
boundary condition. In order to evaluate the response the acoustic pressure were evaluated at a point on the
opposite side of the Neumann condition.

vn = ρ1c
Figure 2. Cylinder with length 10 cm and radius 1 cm. The mesh has 1916 quadratic elements resulting in
3834 DOFs. The pressure is evaluated at the point shown in red, located at x = (0 cm, −1 cm, 0 cm).
Two different ROSEBEM with expansion frequency f0 = 5 kHz where evaluated. The first used only the
information available at the expansion frequency when creating the projection matrix whereas the second
used two additional primary frequencies, meaning that fi ∈ {1.5 kHz, 5 kHz, 8.5 kHz}. In both cases the total
size of the projection space was 450, resulting in a total reduction of DOFs of approximately 88%.

4.2 Frequency response
The acoustic response is evaluated at increments of 1 Hz in the interval [20 Hz, 10 kHz] resulting in a totalof 9981 evaluations. Around the expansion frequency the two ROSEBEM setups are indistinguishable fromthe full BEM computations (Figure 3). However, at the two ends of the frequency range of interest the setupthat only uses the information from the expansion frequency drifts away from the true solution (Figure 3).
The pointwise relative error is significantly lower over the full domain when using three primary frequencies
as compared to only one (Figure 3), albeit both setups experience large relative errors close to resonances.
BEM

L=1

L=3

|p|

102
101
100
.
log |p − pt |/|pt |

101

10−7

10−15
0

1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000
Frequency [Hz].

Figure 3. Top: Absolute value of the pressure at point x as a function of frequency. Bottom: Relative errorof the ROSEBEM setups compared to the full BEM solution.
Comparing the computational efforts of the model can easily lead to skewed results as the efficiency
greatly depend on the problem at hand. One of the more prominent of these dependencies is the numberof frequencies evaluated. The more frequencies evaluated the better the ROSEBEM will perform comparedto the full model. It is therefore better to think in terms of the computational and memory scaling of thevarious components that goes into the ROSEBEM. A good summary of these can be found in Table 1 in
Panagiotopoulos et al. [6]. A general rule-of-thumb is that the ROSEBEM should only be utilized if thenumber of frequencies evaluated is larger than the order of the Taylor expansion.
For our specific setup there were almost 10,000 frequencies evaluated resulting in the ROSEBEM beingaround 106 times faster than the full solution (Table 1). In practical terms this meant that the total computational time went from close to 5 days to a little over an hour. Comparing the total memory, it was seenthat the ROSEBEM used 370.789 MB which is 1.65 times more than the full BEM. Either decreasing or
increasing the size of the projection matrix could change these results. However, had the ROM not been
applied the SEBEM it would have used close to 27 GB of memory which highlights why the ROM is a
crucial part of making the model applicable on regular machines.

Table 1. Computation efforts: Full BEM vs. ROSEBEM.

5

Full BEMROSEBEM

Computational Time6.914 min

65 minMemory Consumption

224.297 MB

370.789 MB

CONCLUSIONSIn this short work we extended a Model Order Reduction (MOR) technique for the Series Expansion Boundary Element Method (SEBEM) to handle the Boundary Layer Impedance (BLI) boundary condition. Similar
to the previous work the series expansion was based on the Taylor expansion of all relevant kernels and the
reduced order model based on the Galerkin projection. Lastly the model was numerically verified and analyzed using a cylinder as a test case. It was found that information from a single frequency was not enoughto create a suitable projection basis.

REFERENCES[1] Liu Yijun. Fast multipole boundary element method: Theory and applications in engineering, volume 9780521116596. Cambridge University Press, 2009. ISBN 9780511605345, 0521116597,
9780521116596. doi: 10.1017/CBO9780511605345.
[2] Wolfgang Hackbusch. Hierarchical Matrices: Algorithms and Analysis.
3662473232, 3662473240, 9783662473238, 9783662473245.

Springer, 2015.

ISBN[3] Suhaib Koji Baydoun, Matthias Voigt, and Steffen Marburg. Low-rank iteration schemes for the multifrequency solution of acoustic boundary element equations. Journal of Theoretical and Computational
Acoustics, 29(3), 2021. doi: 10.1142/S2591728521500043.
[4] SM Kirkup and DJ Henwood. Methods for speeding up the boundary element solution of acoustic
radiation problems. Journal of Vibration and Acoustics-transactions of the Asme, 114(3):374–380, 1992.
doi: 10.1115/1.2930272.
[5] Dionysios Panagiotopoulos, Elke Deckers, and Wim Desmet. A two-step reduction method for acoustic
BEM systems. In Forum Acusticum, pages 2291–2298, Lyon, France, December 2020. doi: 10.48465/
fa.2020.0712. URL https://hal.archives-ouvertes.fr/hal-03235474.
[6] Dionysios Panagiotopoulos, Elke Deckers, and Wim Desmet. Krylov subspaces recycling based modelorder reduction for acoustic bem systems and an error estimator. Computer Methods in Applied Mechanics
and Engineering, 359, 2020. doi: 10.1016/j.cma.2019.112755.
[7] Martin Berggren, Anders Bernland, and Daniel Noreland. Acoustic boundary layers as boundary conditions. Journal of Computational Physics, 371:633–650, 2018. doi: 10.1016/j.jcp.2018.06.005.
[8] Vicente Cutanda Henriquez. Numerical Transducer Modeling. PhD thesis, Technical University of Denmark, 2002.
[9] Jeffrey Werner Bezanson, Alan Edelman., Massachusetts Institute of Technology. Dept. of Electrical Engineering, and Massachusetts Institute of Technology. Dept. of Electrical Engineering. Julia : an efficient
dynamic language for technical computing - efficient dynamic language for technical computing. 2012.
[10] Jeff Bezanson, Alan Edelman, Stefan Karpinski, and Viral B Shah. Julia: A fresh approach to numericalcomputing. SIAM review, 59(1):65–98, 2017. URL 10.1137/141000671.
[11] DTU Computing Center. DTU Computing Center resources, 2021. URL https://doi.org/10.48714/
DTU.HPC.0001.

[12] Vicente Cutanda Henríquez and Peter Møller Juhl. Openbem - an open source boundary element method
software in acoustics. Proceedings of Internoise 2010, 7:5796–5805, 2010.
[13] MATLAB. version 9.10.0 (R2021a). The MathWorks Inc., Natick, Massachusetts, 2021.
[14] Steffen Marburg. Six boundary elements per wavelength: Is that enough?
Acoustics, 10(1):25–51, 2002. doi: 10.1142/S0218396X02001401.

Journal of Computational

ABS-0862

Acoustic design through similitude and sensitivity analysis
Christian ADAMS1(1) Technical University of Darmstadt, Mechanical Engineering Department, System Reliability, Adaptive Structures, andMachine Acoustics SAM, Germany, christian.adams@sam.tu-darmstadt.de

ABSTRACT
Every physical equation must be dimensionally homogeneous. This statement seems self-evident. In fact, it is
the starting point of similitude theory. In fluid mechanics and thermodynamics similitude methods were alreadydeveloped in the 19th and 20th centuries to carry out model tests on scaled models and to predict the behaviourof the original. In vibroacoustics model tests based on similitude theory have not yet been able to establish
themselves. On the other hand, model tests are well suited to validate vibroacoustic properties of mechanicalengineering systems in early design phases. This paper describes a method that combines similitude theory withsensitivity analysis. This simplifies the derivation of scaling laws in early phases of product design processes
since the method requires only input parameters and output parameters such as geometrical dimensions or material properties and, respectively, modal quantities or acoustic transfer functions. Two case studies, a vibrating
car undercarriage structure and gear box housings, demonstrate how model tests can contribute to the acoustic
design of these structures. Besides numerical simulations experimental validations will be discussed for the gearbox housings as well.

Keywords: Acoustic Design, Similitude, Sensitivity analysis

1

INTRODUCTION

Similitude theory is founded on dimensional homogeneity [1]. Every physical relation on a model scale f(m)can be related to that of the original scale (prototype) f(p) by a scaling factor
φ=

f(m)
.
f(p)

(1)

The scaling factors can be determined from dimensional analysis [1] or from equations of motion [2]. In vibroacoustics, a generalised modal approach can also be utilised [3, 4]. These approaches exactly replicate theprototype behaviour from that of a scaled model if model and prototype are completely similar. In practice, thesimilitude is often distorted, e.g. geometrical dimensions are scaled with different factors. The similitude methods can still replicate the prototype behaviour, but the accuracy reduces more and more the more the similitude
is distorted [5].
This paper demonstrates in two case studies how similitude analysis can be used in early phases of productdesign to deepen the understanding of the vibration behaviour of structures. It is the combination with sensitivity
analysis that enables similitude analyses in early phases of acoustic design. The first case study will show how
sensitivity analysis is used to derive the scaling laws of a generic car undercarriage (generic means that it issimplified in its geometry) from numerical simulations. A parameter study will reveal the limits of applicabilityof scaling laws. The second case study is a gear box housing in different sizes. Natural frequencies are
replicated for gear box housings in various sizes. Numerical analyses contribute to the derivation of scaling
laws. The sensitivities are used to master geometrical distortion, and the proposed scaling law is validated inexperiments.

2

METHODThe method has been developed in [5, 6, 7] and is summarised in Figure 1. A power law is posed as scaling
approach, which is motivated from Buckingham’s Π-theorem of similitude theory [1]. Rewriting the power law
in terms of scaling factors of input parameters and output parameters, which is motivated from sensitivity analysis, leads to a scaling law for each output parameter of interest Yk depending on the input parameters to bescaled X j .
power lawsimilitude
theoryscaling approach
N

N

∏ϕ
j=1

αj
j

ϕ Y = ∏ (ϕ X )α
k

j=1

k, j

j

scaling laws
local or global
sensitivity analysis
to obtain αk,jinput parameters Xj
output parameters Yksensitivity
analysis

Figure 1. Flow chart of the method to derive scaling laws from similitude theory and sensitivity analysis
The exponents αk, j are calculated from sensitivity analysis, where local or global sampling techniques canbe used [5, 6]. It is a feature of the method that input parameters X j are weighed by their influence on the
output parameters Yk , e.g. a certain input parameter is directly removed from the scaling law if the output
parameter is insensitive (corresponding α equals zero) to that input parameter. Details on the method can befound in [5, 6].

3

CASE STUDY 1

The first case study derives scaling laws for a generic car undercarriage. First, the model is briefly described.
Second, scaling laws are derived for the generic car undercarriage. Third, a parameter study shows how thelimits of applicability of scaling laws can be determined if the structure is distorted in thickness.
3.1 Model
The generic car undercarriage is illustrated in Figure 2 and its geometrical dimensions are listed in Table 1. We
assume a carbon fibre reinforced composite material with Ex = 1.39 · 1011 N m−2 , Ey = 1.06 · 1010 N m−2 and
Gxy = 5.14 · 109 N m−2 being Young’s modulus in x- and y-direction and shear modulus, respectively. Poisson’s
ratio, mass density and loss factor are µ = 0.28, ρ = 1524 kg m−3 and 0.005, respectively. A finite element (FE)model is set up, see [5], with free boundary conditions at the edges. In this paper, only the free vibrations aretaken into account, i.e. the excitation force F(aF , bF ) = 0. Double eigenvalues are avoided by detuning one sideof the structure with an additional mass mA that equals 10% of the mass of the generic car undercarriage.
Table 1. Geometrical dimensions of the prototype generic car undercarriage in mm [5]symbolvalue

symbolvalue

symbolvalue

symbol

value

a

1350

a1

546

a1m

170

bF

500

b

1470

a2

610

a2m

440

bR

830

h

130

a3

740

aR

235

b1m

1080

t

4

a4

804

aF

1175

b2m

1250

y
a1m

a2m
mA

b2m
h

ṽR

b1m
bR

F(aF , bF )

z

z
aR

thickness: t
b
aF
a

bF

x

a1
a2
a3
a4

x

a

(b) profile view(a) 3-D view

Figure 2. Prototype of the generic car undercarriage [5]3.2 Scaling laws
Scaling laws are derived from a local sensitivity analysis of the structure. A central difference scheme is used toapproximate the local sensitivities. Geometrical dimensions and material properties are the input parameters of
the local sensitivity analysis and the first 20 natural frequencies are the output parameters. Complete similitude
requires that the geometrical dimensions are scaled by the same factor. However, in order to investigate theeffect of geometrical distortion, all length dimensions are scaled by the common factor φl , whereas the thickness
is scaled with the factor φt . Material properties are assumed to remain in complete similitude to retain the modalbase of the prototype in the model, i.e. Young’s moduli and shear modulus are scaled by φE , mass density byφρ and Poisson’s ratio is assumed constant (φµ ≈ 1). The scaling law for the natural frequencies fn can then be
determined from local sensitivity analysis [5]
(m)

fn

(p)
fn

= φ fn = φl−2 φt1 φE0.5 φρ−0.5 .

(2)

3.3 Parameter study
The geometrical distortion, i.e. φl 6= φt , is investigated in two different manners. First, only the thickness isscaled by the factors φt = {1.5, 2.0, . . . , 7.0}, whereas all length dimensions are retained, i.e. φl = 1. Second,
the thickness remains constant, but all length dimensions are scaled down by the factors φl = {0.9, 0.8, . . . , 0.1}.
It is assumed that the material for prototype and all models is the same, i.e. φE = φρ = 1. Figure 3 shows the
maximum error in percent of the first 20 natural frequencies replicated from Equation (2) compared to thoseobtained from FE calculations [5].
The maximum error more and more increases if the geometry is more and more distorted, which corresponds
to an increase of φt or a decrease of φl . This behaviour can be expected since complete similitude conditions are
more and more violated. However, the amount of error remains below 10% if φt is altered and if φl is altered
between 0.2 and 1.0. Such small error appears acceptable, particular in early phases of product design, so thatthe scaling laws can replicate the prototype natural frequencies from those of scaled models with acceptableaccuracy. In conclusion, complete similitude conditions can still be considered valid, although the geometry is
actually distorted. This finding is in line with [8], where scaling laws for vibrating thin shells are derived.

4

CASE STUDY 2

The second case study uses scaling laws to replicate natural frequencies of gear box housings with various
sizes (size range). First, the numerical model is briefly described. Second, scaling laws are derived, where the

25

6

20
ε f , max in %ε f , max in %

8

4
2
0

15
10
5

1

2

3

4

5

6

7

0

0

0.2

0.4

0.6

0.8

1.0

φl

φt

Figure 3. Maximum error of the first 20 replicated natural frequencies, left: distortion of the thickness, right:
scale down of all length dimensions [5]sensitivity is used to master geometrical distortions. Third, experimental results demonstrate that the method
applies in practice sufficiently well.
4.1 Model
Three FE models are built as depicted in Figure 4. The material considered is cast iron (E = 1011 N m−2 ,ρ = 7200 kg m−3 , µ = 0.26) and the boundary conditions are free suspensions. The models comprise the main
geometrical features of the gear box housings used in the experiments only, but the geometrical dimensions
are in distorted similitude. Due to the distorted geometrical similitude the models are referred to as avatar
according to [3]. Details on the geometrical setup and the FE models can be found in [5]. Figure 5 illustrates
the first 20 natural frequencies, which are calculated from the FE models.
model (avatar a1)

prototypemodel (avatar a2)

Figure 4. FE models of the three gear box housings [5]fn in kHz

avatar a1
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0

1

2

3

4

5

6

7

8

prototypeavatar a2

9 10 11 12 13 14 15 16 17 18 19 20
mode number

Figure 5. First 20 natural frequencies of the gear box housings [5]4.2 Scaling laws
The scaling laws are obtained from a local sensitivity analysis of the gear box housings, where each geometricaldesign parameter is altered separately according to [5]. This leads to the scaling law
(m)

fn

(p)
fn

−1

= φ fn = φ l .

(3)

The natural frequencies of the models are replicated from those of the prototype by empirically defining asuitable scaling factor φ l in Equation (3). A first scaling factor can be determined from the mean of all distorted
geometrical design parameter, which yields
(a1)

φl

= 0.90(a2)

and φ l= 1.48(4)

for avatar a1 and avatar a2, respectively [5]. A second scaling factor can be determined from the most sensitive
distorted geometrical design parameters, which yields
(a1)

φl

= 0.80(a2)

and φ l= 1.59(5)

for avatar a1 and avatar a2, respectively [5]. Figure 6 shows the replicated natural frequencies versus thosefrom the FE calculation (cf. Figure 5). The accuracy of the replicated natural frequencies increases if only themost sensitive geometrical design parameters are considered. This can be seen from the fact that the points
in Figure 6 are closer to the main diagonal if the scaling factors from Equation (5) are used [5]. Only thosegeometrical dimensions influence the scaling behaviour of a given natural frequency that exhibit significant
vibrations of the corresponding mode shape in the vicinity of these geometrical dimensions. In conclusion,acoustic design can benefit from the sensitivity twofold: (1) Sensitivity analysis can be used to determine thescaling law, i.e. Equation (3). (2) Sensitivity analysis supports engineers to determine a suitable scaling factor
in case of distorted geometrical similitude.
4.3 Experimental validation
Finally, the scaling law from Equation (3) is validated through experimental modal analyses. The scaling factors
are empirically derived as
(a1)
(a2)
φ l = 0.72 and φ l = 1.56.
(6)
They are different to those from the numerical calculation since the geometry has been simplified in the FEmodels [5]. Figure 7 illustrates the three gear boxes on the test stand, where free boundary conditions are
realised through air springs. Figure 8 shows the scaled natural frequencies versus those from experimental
modal analysis. The numbers indicate the mode numbers since not all prototype modes could be identified in
the experimental modal analyses of the models. Although some replicated natural frequencies deviate from thereplicated from Eqs. (3) and (4)replicated from Eqs. (3) and (5)avatar a2

2.5

4

2.0

fn (replicated) in kHz

fn (replicated) in kHz

avatar a1
5

3
2
1
0

0

1

2

3

4

fn (calculated) in kHz

5

1.5
1.0
0.5
0.0
0.0

0.5

1.0

1.5

2.0

2.5

fn (calculated) in kHz

Figure 6. Scaled natural frequencies of the avatars a1 and a2 replicated from those of the prototype [5]measured ones, the scaling law accurately replicates the natural frequencies of the models from those of the
prototype. Thus, the scaling law can be experimentally validated. It should be noted that the modal base of
the prototype can only be partially retained in the models due to distorted similitude. This can be seen as areason for deviations in the scaled natural frequencies [5]. From the viewpoint of acoustic design, this is an
important step since the validated method allows to design complex structures with virtual design tools (such as
FE models) and to transfer the vibration behaviour to scaled models of a prototype.

Figure 7. Test stand for experimental modal analysis of gear box housings from left to right: avatar a1, prototype, avatar a2 [5]5CONCLUSIONSThis paper demonstrates how similitude analysis can contribute to acoustic design in early design phases bycombining similitude theory with sensitivity analysis. Two case studies show the applicability of the method
developed in [5, 6, 7]. The combined method allows for deriving scaling laws directly from a sensitivity analysis. This makes the method suitable for early design phases, where numerical models are used and sensitivity
analyses can easily be carried out. Furthermore, the sensitivity information can be used to master distorted
geometrical similitude. It allows to prioritise geometrical parameters that are relevant for scaling so that relevantgeometrical parameters can be selected for the calculation of a scaling factor. Numerical and experimental
analyses show that the method applies to structures in complete and distorted similitude. It enables to design
structures in a early phase of a product design process and to transfer the vibration behaviour on scaled models
of a prototype. This can help to deepen physical understanding of the vibration behaviour of complex structuresavatar a2

avatar a1
3.0

8
5

4
2

3

9

fn (replicated) in kHz

fn (replicated) in kHz

5

3
4

1

2
1
0

0

1

2

3

4

fn (measured) in kHz

5

2.5

8

2.0

6
4
2

1.5

9
7

1
1.0
0.5
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
fn (measured) in kHz

Figure 8. Scaled natural frequencies of avatar a1 and avatar a2 (numbers refer to mode numbers) [5]so that an acoustic design can be implemented. Nevertheless, the accuracy of the similitude method reduces themore the similitude is distorted [5]. Future research will further improve the similitude method by incorporating
the new concept of space scaling. A first study in [9] shows that the method is promising to overcome current
limitations of similitude methods based on Buckingham’s Π-theorem. But more research is required to develop
the method towards distorted similitude in vibroacoustics. Particularly, the distortion of geometry and distortiondue to damping will be in the focus.

REFERENCES[1] Langhaar, H.L.: Dimensional Analysis and Theory of Models. John Wiley & Sons, New York, 1951.
[2] Kline, S.J.: Similitude and Approximation Theory. Springer, Berlin, 1986.
[3] De Rosa, S., Franco, F., Meruane, V.: Similitudes for the structural response of flexural plates. Proceedingsof the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 230(2):174–
188, 2016[4] Casaburo, A.; Petrone, G.; Franco, F.; De Rosa, S. A review of similitude methods for structural engineering. Applied Mechanics Reviews, 71(3), 2019, 030802-1–030802-32.
[5] Adams, C.: Similitudes and sensitivities as contributions to scaling laws in machine acoustics. Dissertation,
Technische Universität Darmstadt, Darmstadt, 2019[6] Adams, C., Bös, J., Slomski, E.M., Melz, T.: Scaling laws obtained from a sensitivity analysis and applied
to thin vibrating structures. Mechanical Systems and Signal Processing 110(2018):590–610, 2018
[7] Adams, C., Bös, J., Melz, T.: Performance of similitude methods for structural vibration analyses of rectangular plates. Mechanical Systems and Signal Processing 162:108074, 2022
[8] Soedel, W.: Similitude approximations for vibrating thin shells. The Journal of the Acoustical Society ofAmerica, 49(5):1535–1541, 1971.
[9] Davey, K., Sadeghi, H., Adams, C., Darvizeh, R.: Anisotropic scaling for thin-walled vibrating structures.
Journal of Sound and Vibration, 2022, https://doi.org/10.1016/j.jsv.2022.117182, accepted for
publication

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0264

Observation of underwater noise during the operation of windturbine in the Korea Southwest offshore wind farmYoung Geul YOON1; Dong-Gyun HAN2; Dae Hyeok LEE3; Jee Woong CHOI4
1,3,4

Department of Marine Science and Convergence Engineering, Hanyang University ERICA, Korea
2
4

Division of Ocean Sciences, Korea Polar Research Institute, Korea

Department of Military Information Engineering, Hanyang University ERICA, Korea

ABSTRACTAs the number of offshore wind power plants continues to increase worldwide, there is concern that
underwater noise generated by offshore wind farms may have a negative impact on the marine ecosystem.
Although there have been many efforts to measure and predict the impact pile driving noise that occurs during
the construction of offshore wind power plants, the measurement and characteristic analysis of underwater
noise during operation has hardly been carried out. In this study, the underwater noise generated during theoperation of a wind turbine in the Korea southwest offshore wind farm was measured for 10 days using amoored acoustic recorder. Underwater noise caused by turbine operation was characterized by several tonalcomponents of which the frequencies changed with the rotor speed of the turbine at frequencies below 200
Hz. In addition, a strong tonal component was observed at a frequency of 198 Hz when the rotor speed
reached the rated speed.
Keywords: Offshore wind turbine, Underwater noise, Operational noise

1. INTRODUCTION
An increase in offshore wind farm can lead to an increase in underwater noise levels and has thepotential to have short- and long-term negative impacts on the marine environment (1,2,3). Until
recently, the underwater noise caused by the operation of wind turbines was treated as insignificant
because the noise level was relatively low compared to pile driving noise. However, the continued
increase in the number and size of offshore wind turbines can lead to significant increases inoperational noise levels (4). In this study, in order to understand the potential impact on the marine
environment, the operational noise of wind turbine according to changes in wind speed and operating
conditions was measured and analyzed.

2. MEASUREMENTS
Underwater operational noise measurements of wind turbine were made in February and March
2021 at the Southwest oﬀshore wind farm, which was located in shallow water off the southwest coastof Korea. A self-recording hydrophone (SM3M/standard, Wildlife Acoustics, Inc., USA) wa s mooredapproximately 75m away from the wind turbine to measure the underwater noise from operating
offshore wind turbine. Turbine operation information including wind speed and rotor speed during
underwater noise measurement was provided by Korea Offshore Wind Power Co., Ltd.

3. RESULT
Operational noise was mainly generated at the low frequencies below several hundred Hz, andstrong tonal components were observed at frequencies below 200 Hz. As wind speed increased, theygyoon@hanyang.ac.kr
edh0921@naver.com
3dghan@kopri.re.kr
4choijw@hanyang.ac.kr
1
2

tonal frequency below 100 Hz tended to increase with rotor speed from the lowest rotor speed of 6.4rpm to the rated speed of 10.7 rpm. The rated speed was reached when the wind speed was ~8.1 m/s.
However, after the rotor speed reached the rated speed, the tonal frequency did not increase even if
the wind speed was further increased, and a very strong tonal component with a frequency of ~200
Hz was observed. As shown in Fig. 1(a), the periodicity in the spectrum observed at frequencies below
50 Hz seems to be flow noise caused by tidal changes.

Figure 1 – Examples of (a) the time series data for wind speed and rotor speed measured SCADA systemand (b) spectrogram of operational noise measured about 75 m from wind turbine.

4. CONCLUSIONSIn the Southwest offshore wind farm, which was located in shallow water off the southwest coastof Korea, the underwater noise radiated into the water during operation of a jacket -type wind turbine
was measured. It was confirmed that the spectral properties of the operational noise depend on therotational speed of the rotor due to the change in wind speed. The tonal frequency of the operational
noise is estimated to be due to the gear mesh frequencies of the gearbox and their harmonics (4,5).

ACKNOWLEDGEMENTS
This work was supported by Korea Institute of Energy Technology Evaluation and
Planning(KETEP) grant funded by the Korea Government (MOTIE) (20203030020080,
Environmental Impact Analysis on the Oﬀshore Wind Farm and Database System Development).

REFERENCES
1. Yang CM, Liu ZW, Lu LG, Yang GB, Huang LF, Jiang Y. Observation and comparison of tower vibration
and underwater noise from offshore operational wind turbines in the East China Sea Bridge of Shanghai.
J Acoust Soc Am. 2018;144(6):EL522.
2. Popper AN, Hice-Dunton L, Jenkins E, Higgs DM, Krebs J, Mooney A, Rice A, Roberts L, Thomsen F,
Vigness-Raposa K, Zeddies D, Williams KA. Offshore wind energy development: Research priorities forsound and vibration effects on fishes and aquatic invertebrates. J Acoust Soc Am. 2022;151(1):205-215.
3. Han DG. Choi JW. Measurements and Spatial Distribution Simulation of Impact Pile Driving Underwater
Noise Generated During the Construction of Offshore Wind Power Plant Off the Southwest Coast of
Korea. Frontiers in Marine Science, 2022;8:654991.
4. Tougaard J, Hermannsen L, Madsen PT. How loud is the underwater noise from operating offshore windturbines?. J Acoust Soc Am. 2020;148(5):2885-2893.
5. Pangerc T, Theobald PD, Wang LS, Robinson SP, Lepper PA. Measurement and characterisation ofradiated underwater sound from a 3.6 MW monopile wind turbine. J Acoust Soc Am. 2016;140(4):29132922.

ABS-0285

Influence of the pile geometry on the underwater noise during offshore piledriving.
Elin KLAGES, Stephan LIPPERT, Otto VON ESTORFF
Hamburg University of Technology (TUHH), Institute of Modelling and Computation, Germany, mub@tuhh.de

ABSTRACT
Offshore wind energy is evolving worldwide. Very often, impact-driven monopiles are used as foundations foroffshore wind turbines. However, driving piles into the seabed causes very high underwater sound pressurelevels, which might endanger animals living in the sea. Therefore, several prediction methods for the occurring
underwater sound pressure levels from pile driving have been developed in the past years. Furthermore, different
noise mitigation systems exist to decrease the noise emission into the water. Additional knowledge of the soundsource, i.e., the driven pile, would simplify predictions and help the development of impact hammers or piles
emitting less noise. The present paper applies findings from the theory of axisymmetric vibrations of cylindricalshells as well as their sound radiation to the field of offshore pile driving.

Keywords: Offshore pile driving, underwater noise, finite element method, shell theory, sound radiation

1

INTRODUCTION

Offshore wind energy is on the rise in many countries. In 2022, the German government decided to almost
quadruple the nowadays installed capacity of offshore wind energy [1, 2]. Similar goals have been set by othercountries. Although offshore wind energy is a renewable energy source, placing an offshore wind park in the sea
also affects negatively its environment. One aspect is the sound emission during construction using monopiles,
which are a very common foundation for offshore wind turbines. To drive a pile into the sea bed, an impact
hammer is used. Due to the hammer impact on the pile head, waves are excited and travel along the pile. The
vibrations of the pile excite sound waves in the surrounding air, water and sea bed. The resulting underwatersound pressure might be very high, possible dangerous for mammals and fishes [3]. Therefore many countries
have imposed limits for resulting underwater sound pressure levels. Throughout this paper, the sound exposurelevel (SEL) is used as an exemplary limit [4]!
Rt 2
1 t12 p (t) dt
SEL = 10 log10
.
(1)
t0
p20
Often, noise mitigation systems to hinder sound propagation underwater, e.g. bubble curtains, are used toensure these limits. Additionally, state of the art hammers cause a longer, smoother impact than hammers froma decade ago. However, the capacity, and therefore also the size as well as the mass of the installed offshore
wind turbines increase and the pile dimensions increase as well. Unfortunately, larger piles cause higher soundpressure levels [5].
The present paper aims for explaining possible reasons why larger piles cause higher sound pressure levels,
even if the same ram energy is used. Here, the effect of the pile diameter on the propagation of waves in thepile and the radiation into the water is taken into account.
To simplify the analysis, neither the conical top of the pile nor the influence of the wall thickness is considered. The pile is assumed to be a hollow cylinder of constant wall thickness of 100 mm and a diameter range
from 1 m to 12 m.

A shell model for the vibration of the pile and and a radiation model for infinitely long cylinders areapplied to realistic diameters and a frequency range up to 100 Hz. The results are compared with results froma numerical model and consequences for the reduction of pile driving noise are discussed.

2

MODELSIn the following, the three models applied here are presented. The first two are analytical models, namely ashell model to infer properties of the vibrations of the pile and a radiation model to estimate how much of theenergy of the pile converts to sound energy in the water. The third one is a numerical model, a finite-element(FE) model, which is capable of taking a more realistic scenario into account, especially the finite length of thepile and the sea bed.
2.1 Shell model
A shell is a curved structure where one dimension is small compared to the other two. Different models exist
for cylindrical shells. For the present paper, a by Means modified version of the Donnell model of a thin shellis used [6], [7]. The Donnell model is based on the bending theory. Means modified the model with respect tothree aspects: the model was extended for the dynamic case; the model was further simplified for a very thinshell, i.e. all higher order terms were omitted, and the model was extended to include fluid loading, i.e. wateroutside of the shell. The model is represented by the equations


1+ν
1−ν
uy0 y0 +vx0 y0 + ν wx0 = 0,(2)uττ − ux0 x0 +
2
2


1−ν
1+ν
vττ − vy0 y0 +vx0 x0 +ux0 y0 + wy0 = 0,(3)22wττ + [ν ux + vy + w] + εO4 w = iΩ Z 0 w.

(4)Here, x is the axial and y the circumferential direction, i.e. y = R φ where R is the radius of the middle surface,i.e. R = ri + (re − ri )/2 if re is the outer radius and ri the inner radius. The angle is denoted φ , x0 , y0 stand for
x/R, y/R respectively. The displacements u, v and w of the middle surface refer to the directions x, y and theradial direction r. The dimensionless frequency is Ω defined as ω/ωr with the ring frequency
s
E
ωr =
,
(5)
((1 − ν 2 ) ρ R2 )the Young’s modulus E and the density ρ. The symbol τ refers to the dimensionless time ωr t. The poisson
ratio is denoted ν and ε stands for h2 /(12 R2 ), where h refers to the wall thickness. The fluid loading outsideof the pile is included via the resulting force in radial direction, i.e. iΩ Z 0 w, where i is the imaginary unit andZ 0 equal to Zrad /(ρ h ωr ), where Zrad is the radiation impedance. The radiation impedance may be approximatedby that of a flat plate, in this case
ρf R

Z0 ≈
ρh

r

ωr2 R2
1
− 1−ν
2
c2

,

(6)

where ρ f is the density and c the sound velocity of the fluid.
To solve these equations, planar waves, inside the cylinder cross section, of constant frequencies are assumed,

 

u
A

 

0
0

 

(7)
 v  =  B  ei (α x +β y +Ω τ) ,
 


w
C

where, the dimensionless wave numbers are introduced, α = kx R and α = ky R. Based on the two equations
above, a system of algebraic equations and the corresponding dispersion equation (i.e. characteristic polynomial)
can be derived. The here relevant case is for β = 0, i.e. waves propagating in direction of x. Finding the roots
for this equation yields to possible waves travelling along the pile. For these waves, the ratio of C to A, as
defined in Eq. (7), can be obtained, which is the reason why this model is used here. For quasi-longitudinal
waves, an exemplary ratio of C to A with respect to the dimensionless frequency is shown in Figure 1s.

C/A

2
1
0

0

0.2

0.4

0.8

0.6

1

1.2

1.4

Ω
Figure 1. Amplitude of radial to axial displacement of quasi-longitudinal waves propagating in x-direction of
the cylinder shell with respect to the dimensionless frequency based on [7]. The wall thickness to radius ratio
is 0.01. The material parameters are: E=210 N/m2 , ρ=7850 kg/m3 ,ν=0.3, ρ f =1000, c=1500 m/s.

radiation efficiency [dB]

2.2 Radiation
The sound radiation of a cylinder can be estimated using Hankel functions to solve the linear wave equation incylindrical coordinates as shown for example in Möser, 2010. The here relevant case in for axisymmetric waves
propagating in x-direction. Here it is irrelevant if the cylinder is hollow or full.
Based on the sound pressure field, the radiation efficiency 10 log10 (σ ) can be determined [8]. The ratio σ
is defined as
P
,
(8)
σ=
ρ f c S mean {v2rms }
where P is the radiated power, S the surface of the radiating body and mean v2rms the spatial mean of thesquare of the root mean square particle velocity [8]. For a cylinder of infinite length, the radiation efficiency
depends on the radial wave number in the surrounding medium kr and the outer radius of the pile re [8], asshown in Figure 2.

0
−5
−10

0

2

4
kr re

6

8

Figure 2. Radiation efficiency for an infinitely long cylinder for axisymmetric waves propagating in the directionof x [8].

2.3 Numerical model
A finite element model in the time domain is used to simulate the wave propagation due to the ram impulse in
the pile, water and sea bed. The velocity at the pile head is defined as boundary condition to model the ram

impulse. The output of interest is the acoustic pressure, here in 10 m distance to the pile, from which the mean
SEL can be calculated and the deformation velocity at the pile in z and r-direction, here exemplary at 30 m
below the pile head. The pile is axisymmetric. The same can be assumed for the sea bed without affecting theanalysis in the present paper, therefore a discretization of half the cross section is sufficient. The model was
implemented within the program Abaqus.
The pile and sea bed are modelled as linear elastic. The latter is modelled with horizontal, parallel layers,
where each layer is described using the following material parameters: the Young’s modulus, the density, thepoisson number, and rayleigh damping parameters. Here, the values obtained for wind park Bard Offshore I [9]were used. Rayleigh damping is also defined in the embedded part in the pile to model effects of damping of
the sea bed which cannot be modelled by the linear elastic model of the sea bed. The linear wave equationis the governing equation for the water. The interface of water and pile, and water and sea bed are tied. The
radius of the water and sea bed is 20 m. To avoid reflections at the border of the numerical model, which do
not represent borders in reality, a matching impedance condition in water and infinite elements in the soil areused. The boundary condition at the water surface is p = 0.
Bilinear elements are used for the pile and soil. The size of the elements in the soil is approximately 20 cm
x 20 cm. The height of the elements in the pile is as well 20 cm, but four elements per wall thickness are used.
For the water, linear acoustic elements, approximately 20 cm x 20 cm, are employed. The total simulation time
is t = 0.15 s.
The model is largely based on the models developed during the project BORA [10], which were extensively
validated [9].

3ANALYSIS OF THE PILE AS SOUND SOURCE

In the following three subsections, the general theory introduced above, is applied to the exemplary pile dimensions (d = 1, 2, . . . , 12 m, h = 100 mm) and frequency range (up to 100 Hz) used in the present paper.
3.1 Application of the shell model
The condition h/R < 0.1 for a thin shell is for a pile not necessarily true. For the chosen exemplary dimensions,this conditions are not fulfilled for the outer diameter of d = 1 m and 2 m. However, the condition h/λ  1 isfulfilled for all considered frequencies. The most important difference might be that fluid loading is taken into
account only outside the cylinder, not inside, since fluid loading has a significant effect on the vibrations of thepile. Also important to note is, that due to the assumption of the infinitely large shell, the effects of reflections
at the ends are not taken into account.
Keeping these limitations in mind, the curve shown in Figure 1 can be applied for a specific pile diameter,
using
s
1
E
1
Ω ωr =
Ω
.
(9)
f=
2π
2π
(1 − ν 2 ) ρ R2
The results for three exemplary pile diameters are shown in Figure 3, adapted to the here exemplary considered frequency range. These curves show that the ratio of the radial and axial component of the deformation
increases significantly with frequency and diameter.
3.2 Radiationq
Using kx = k02 − kr2 [8] and the corresponding approximate sound velocities for water and steel, cw = 1500m/s
and cs = 6000 m/s, one can determine the radiation efficiency for axisymmetric waves propagating in x-directionof the cylinder. The results are shown in Figure 4 for the pile diameters d = 1, 6, 12 m. The curves support the
hypothesis that more energy is radiated as acoustic energy into the surrounding water the higher the frequenciesare. Also, because of the decreasing ratio of wave length to pile diameter, the radiation efficiency increases
significantly with larger pile diameters.

C/A

0.6

d = 12 m
d = 6md = 1m

0.4
0.2
0
0

20

40

60
f [Hz]

80

100

radiation efficiency [dB]

Figure 3. Amplitude of radial to axial displacement of quasi-longitudinal waves propagating along the axis of
the cylinder for three exemplary diameters.

0

d = 12 m
d = 6md = 1m

−5
−10
−15

0

20

40

60
f [Hz]

80

100

Figure 4. Radiation efficiency for an infinitely long cylinder,[8], made of steel and adapted for radiation into
water for three exemplary diameters.
3.3 Numerical results
An artificial impulse was used for the ram impulse, i.e. the velocity at the pile head. The artificial impulse andits frequency spectrum are shown in Figure 5.
The numerical model allows to determine the axial and radial displacement as well as the sound pressurelevel. The mean ratio of the radial to axial velocity for each pile diameter is shown in Figure 6. For each pile
diameter, exactly the same pile head velocity, as given in Figure 5, was used. Therefore, the results show that
although the frequencies did not change, the ratio depends strongly on the pile diameter.

4

DISCUSSION

ram impulse [m/s]

The SEL increases with an increasing pile diameter, even though the frequency spectrum does not change, as
shown in Figure 6. This means, that a larger portion of the energy of the ram impulse is radiated as acoustical

0

1

−1
0.5

−2
−3

0
0

10

20
30
time [ms]

40

50

0

20

40

60
f [Hz]

80

100

Figure 5. Artificial impulse which is set as the boundary condition at the pile head and the corresponding
frequency spectrum.

∆SEL [dB]mean(vr (t)/vx (t))∆SEL - 10 log10 (A p (d)/A p (1 m))
∆SEL
0.1
0.05
2

4
8
10
6
Outer pile diameter [m]

12

30
20
10
0

2

4
8
10
6
Outer pile diameter [m]12

Figure 6. Numerical results: Mean ratio of radial to axial velocity at 30 m below the pile head. (left) ∆SEL at
15 m distance to the pile center for different radii but the same ram impulse. (right)
energy into the water. One reason might be the increasing relative radial amplitude. Based on the shell model,
the ratio increases with increasing pile diameter, as shown in Figure 3. The results of the FE-simulations, for theratio of the axial and radial velocity vr /vx , show a similar trend. The mean ratio increases with increasing pile
diameter, as shown in Figure 6. Another possible reason is the increase in radiation efficiency with increasing
pile diameters. Based on the theory for an infinitely long cylinder surrounded by water, the radiation efficiency
increases, as shown in Figure 4.
The finite length of the pile, i.e. reflections at the ends, the influence of the embedded part of the pile andthe water inside the pile are only considered in the FE-model. Despite of this, the results shown in Figure 6
match rather well with the expected qualitative behaviour based on the shell and radiation model.
To decrease noise during pile driving, based on the shown results, a smaller pile diameter and a ram impulsein the lower frequency spectrum appear to be beneficial. The former seems to be impractical, pile diameters
for offshore wind turbines have been rising. The latter, in the form of a prolongation of the ram impulse to
decrease noise, has been applied in state of the art hammers. The slower energy transmission due to a longer
impulse itself is unlikely to be the reason for the decrease in the SEL, as long as the sound radiation still falls
into the time period defined in Eq. (1). More likely are the reasons described here, namely a decreasing radial
deformation and a decreasing radiation efficiency for lower frequencies.
Therefore, the approach might be to move the frequency spectrum of the impulse towards lower frequencies
to reverse the effect of the larger pile diameter. However, two aspects cause difficulties here. First, a longer
impulse, using the same ram energy, causes lower forces which leads to a conflict with the drivability. Second,although less energy in the lower frequency domain might be converted to acoustic energy, damping in thewater, sea bed and also due to noise mitigation systems, e.g. a bubble curtain, might be less effective for lower
frequencies.

5

CONCLUSIONS

Already known was that larger pile diameters tend to increase and that longer, smoother ram impulses tend to
decrease sound pressure levels. The present paper provides a compelling explanation for both of these observations. The results from the shell model indicate that the radial component of the vibration of the pile increases
with higher frequencies. The radiation model indicates that the same is true for the radiation of sound waves
into the water. Depending on the pile diameter, the same frequency might be rather high or low, e.g. f = 40 Hz
is a rather low frequency for a diameter equal to 1 m but a rather high frequency for a diameter equal to 12 m.
Based on these results it seems beneficial to prolong the impulse, if the pile diameter increases, to maintainthe ram energy in the frequency range where less energy is converted to acoustic energy in the water. However,additional analysis is required. Most importantly the SEL at 750 m distance in combination with secondary
noise mitigation should be taken into account, because for lower frequencies, damping in the water and sea bed
might be lower as well, which could reverse, at least partly, the noise reduction due to the longer impulse.

REFERENCES[1] Koalitionsvertrag zwischen SPD, Bündnis 90/Die Grünen und FDP, 2021
[2] Status des Offshore-Windenergieausbaus in Deutschland - Jahr 2021, Deutsche WindGuard GmbH, 2022
[3] Thomsen, F.; Lüdemann, K,; Kafemann, R.; Piper, W., Effects of offshore wind farm noise on marine
mammals and fish, biola, Hamburg, Germany on behalf of COWRIE Ltd, 2006
[4] Juretzek, C; Schmidt, B; Boethling, M., Turning scientific knowledge into regulation: effective measuresfor noise mitigation of pile driving. Journal of Marine Science and Engineering, 2021
[5] Koschinski, S.; Lüdemann, K., Noise mitigation for the construction of increasingly large offshore windturbines, Bundesamt für Naturschutz (BfN), 2020
[6] Donnell LH, Stability of thin-walled tubes under torsion. NACA Rep. No. 479, 1935
[7] Means S.L., Wave phenomena of fluid loaded cylindrical shells. The Pennsylvania State University, 1994
[8] Möser M., Körperschall, Springer, 2010[9] von Pein, J.; Lippert, S.; von Estorff, O., Validation of a finite element modelling approach for mitigatedand unmitigated pile driving noise prognosis. The Journal of the Acoustical Society of America, 2021
[10] Heitmann, K.; Mallapur, S.; Lippert, T.; Ruhnau, M., Lippert, S.; von Estorff, O., Numerical determinationof equivalent damping parameters for a finite element model to predict the underwater noise due to offshore
pile driving. Proceedings of Euronoise 2015, Maastricht, Netherlands, 2015

ABS-0591

Numerical simulation of pile-driving induced-noise propagation into theocean: effects of sedimentary basins and interaction of two distant piledriving operations
Paul Cristini(1) , Louise Pacaut(1) and Nathalie Favretto-Cristini(1)(1) Aix-Marseille Univ, CNRS, Centrale Marseille, LMA, Marseille, France

ABSTRACT

Pile driving is a major source of the noise generated by human activities in the ocean. The complex wavefield
which is generated by this activity depends not only on the water column characteristics but also on the
structure of the sediment layers beneath. In general, this structure is considered horizontally stratified witha gradient of mechanical properties. In this presentation, we propose to investigate a sediment configuration
with a heterogeneous structure including two sedimentary basins. For this purpose, we performed several
numerical simulations using a 2D time-domain spectral element method which allows for accurate handlingof the characteristics of the complex marine environment (complex bathymetry and elastic sediments). We
emphasize the presence of interface waves of the Stoneley-Scholte type. Additionally, we propose a method togenerate 3D interaction maps of two distant pile driving operations using only one 2D simulation. Our results
show the importance of taking into account the phase of the signals to correctly evaluate the interferences
between the two operations.

Keywords: Pile driving, Wave propagation, Interface waves, Numerical modeling

1

INTRODUCTION

Pile driving is a major source of anthropogenic noise in the ocean. Many studies have been conductedon this topic, see for example the review article by A. Tsouvalas [1] and the references cited in this paper.
Accurately predicting the noise levels generated in the water column by this type of source is fundamental fora reliable assessment of its potential hazards. In order to improve the understanding of the distribution of theocean noise level generated by a pile-driving operation, this work aims to study the influence of a complex
geological environment on the observed signals. Usually, a nearly horizontally stratified environment with avelocity gradient is considered, but this configuration may not reflect some situations where the sedimentarylayer structure comes from a more complex geology. For example, sedimentary basins are known to strongly
modify the acoustic wave field in the water column [3].
In this work, we chose to consider an ocean floor with two sedimentary basins and study how theirpresence can affect the propagation of acoustic waves in the water layer.
In a first step, a 2D model is built and used to perform numerical simulations in the time domain of
acoustic and seismic wave propagation. Then, in a second step, the results of these numerical simulations
are analyzed and some conclusions are drawn.

2

CONFIGURATION

We first design a 2D model of the structure of the ocean bottom with two sedimentary basins (se Figure1) lying on a rock basement. The length of the model is 13 km. The first sedimentary basin is 10m deep
with en extension of 3 km while the second sedimentary basin is 3m deep with a 2 km extension. In thevicinity of the pile, the sediment layer thickness is 2m. The water depth increases steadily with distancestarting at 15m and ending at 50m. At the end of the model, far from the source, the layer structure is range
independent.
In order to perform the numerical simulations, the entire domain is meshed with quadrilateral elements
using the Gmsh Software[4].

Figure 1. Bathymetry.
The numerical simulations are performed using the open-source software SPECFEM2D1 . SPECFEM2D
implements the spectral-element method. It is based upon a high-order piecewise polynomial approximationof the weak formulation of the wave equation. It combines the accuracy of the pseudospectral method withthe flexibility of the finite-element method. In this method, the wavefield is represented in terms of highdegree Lagrange interpolants, and integrals are computed based upon Gauss–Lobatto–Legendre quadrature.
This combination leads to a perfectly diagonal mass matrix, which in turn leads to a fully explicit timescheme that lends itself very well to numerical simulations on parallel computers. It is particularly well
suited to handling complex geometries and interface conditions. As a consequence, the accurate simulationof surface wave propagation is straightforward without any additional cost. This is a very important featurein our configuration since the influence of interface waves (Stoneley-Scholte, Leaky Rayleigh, etc. ) is known
to be important. The numerical simulations are performed using the cylindrical coordinate system.
The source model of pile driving is taken from reference [2]. The source signal is a Ricker wavelet witha central frequency of 85 Hz. This means that the maximum frequency which is considered in the numericalsimulations is 212.5 Hz.
The bottom of the ocean is considered as an elastic medium with the following physical characteristics:• Water: ρ = 1000 kg/m3 , c p = 1450 m/s• Sediment: ρ = 2200 kg/m3 , c p = 1750 m/s, cs = 600 m/s• Basement: ρ = 2700 kg/m3 , c p = 3550 m/s, cs = 2200 m/s
where ρ, c p and cs are respectively the density, the velocity of longitunal waves and the velocity of shear
waves.

3

NUMERICAL RESULTS

In this section, we present the numerical results obtained with the configuration presented in section 2. First,
we show the results of the evaluation of the Sound Exposure Level (SEL) at a depth of 14m. This parameteris given by the following relation:
Z T

2
2
SEL = 10 log
p (t)dt/p00Then, we present the spectra of several signals recorded along the path from the source and finally theevaluation the interaction of two pile driving operations is presented. Finally, the interaction between two
distant pile driving operations is studied.
1 https://github.com/geodynamics/specfem2d

3.1 Sound exposure levels
The amplitude of the source is adjusted to give an SEL value of 190 dB at a distance of 100 m from thesource.

Figure 2. Variation of the SEL values with the distance to the source for receivers situated at a depth of
14m
A strong decrease followed by a strong increase in SEL values can be observed in the first sedimentarybasin. The increase reaches a value of 10 dB at the end of the first sedimentary basin. The effect of thesecond sedimentary basin is much smaller with a slight increase in SEL values. Nevertheless, it is clear thata site effect occurs, radically changing the behavior of the sound pressure decay. A complex geometry of thesedimentary structure can strongly affect the distribution of acoustic energy in the water column.
3.2 Spectrum evolution
In addition, it is also interesting to analyze how the spectral content of the signals evolves with the distanceto the source. Indeed, it is known that site effects are characterized by a shift towards low fequencies of thespectra of the signals and also by a lengthening of the signal duration [5]. In this extended abstract, we will
only tackle the frequency aspect of the site effect.

(a) Receiver positions(b) Spectra

Figure 3. Spectra evolution of the horizontal component of the displacement. The color of the points indicated in subfigure a) corresponds to the color of the plots in subfigure b)
Figure 3 depicts the evolution of the spectral content of signals recorded at several receivers situated on
the interface between water and sediments (see Figure 3a for the receiver positions). The shift towards low
frequencies starts to be seen in the middle of the first sedimentary basin. At the end of the first sedimentarybasin, it is very clear and this shift remains afterward. The frequency interval where the acoustic energy is
the strongest shifts from the interval [30Hz-70Hz] to [10-15Hz]. It confirms the presence of a site effect due
to the presence of a sedimentary basin.
This site effect may be due to the conversion of interface waves that may exist in this configuration.
Depending on the frequency, the sedimentary basin may or may not affect wave propagation depending on
its relative size to the wavelength of the interface wave and thus an energy transfer between the existing
interface waves may be generated. In this configuration, we can have a Stoneley-Scholte wave at the watersediment interface, a Stoneley-Scholte wave at the water-basement interface, and, a leaky Rayleigh wave at
the water-basement interface. With the increase of the depth of the first sedimentary basin, we make thehypothesis that the high -requency part of the leaky Rayleigh wave at the water-basement interface may be
converted into a Stoneley-Scholte wave at the water-sediment interface while the low-frequency part remains
unchanged. This, of course, needs to be confirmed by a more in-depth analysis.

3.3 Interaction of two distant pile driving sources
As a preliminary step to the study of the interaction of two distant pile driving operations in 3D, we propose
to generate pseudo 3D SEL maps from a single 2D simulation in cylindrical coordinates. For this purpose,
we consider a common area corresponding to the planar part of the 2D simulation where we evaluate theinteraction between the signals emitted by the two distant sources. We first store the signals received by
80 receivers located, in what we defined as the common area, between 8 km and 10 km at a given depth.
Then, we construct the 3D map by adding these signals as shown in Figure 4. The resulting non-uniform
3D spatial sampling is made uniform using "Neareast Neighbor" interpolation.

Figure 4. Schematic of the construction of a 3D map from one 2D simulation
The SEL maps are built in two ways to show the importance of taking into account the phase of thesignals when there is an interaction. If p1 and p2 are respectively the sound pressure generated by Source
1 and Source 2, then the SELs can be evaluated using the following relation if we take into account theinteraction between the two signals:!
RT2
0 (p1 (t) + p2 (t)) dt
SEL = 10 log
p20
Alternatively, if we do not take into account the interaction, the SELs are evaluated using:
!
RT 2
RT 2
p
(t)dt
+
p
(t)dt
0 2
SEL = 10 log 0 1
p20
The generated pseudo-3D maps are shown in Figure 5. Figure 5a represents the SELs for two sources
which are at the same distance to the center of the common area. Stronger SELs are observed along andnear the diagonal of the image corresponding to the places where constructive interferences between the two
emitted signals occur. Figure 5b represents the SELs for two sources whose distance to the center of thecommon area is different. One of the sources is moved back by a distance of 250m. This results in a shiftof the line where constructive interferences occur.
If the SELs are evaluated without taking into account the interaction between the signals the SEL map isvery different (see Figure 5c). The SELS are uniforms.

(a) dx=0m(b) dx=250m(c) without interaction (dx=0m)

Figure 5. 3D SEL maps at a depth of 14m in the common area

4

CONCLUSION

In this expanded abstract, we investigated the structure of the acoustic wavefield generated by pile drivingin the presence of a complex sediment structure. It was shown that the presence of sedimentary basins canstrongly affect the distribution of acoustic energy on the water column both in terms of amplitude and. frequency content. In addition, we generated pseudo 3D SEL maps of the interaction of two distant pile driving
operations using a single 2D simulation in cylindrical coordinates. These maps highlighted the importance oftaking into account the interference of the two emitted signals.

ACKNOWLEDGEMENTS
This work was granted access to the HPC/AI resources of CINES under the allocation A0110410305 madeby GENCI and of the Aix-Marseille Supercomputing Mesocenter under allocations h025.

REFERENCES[1] A. Tsouvalas, Underwater Noise Emission Due to Offshore Pile Installation: A Review, Energies 2020,
13, 3037.
[2] P. G Reinhall and P. H Dahl. Underwater mach wave radiation from impact pile driving : Theory andobservation. The Journal of the Acoustical Society of America, 130(3) :1209– 1216, 2011.
[3] N. Favretto-Cristini et al. Assessment of risks induced by counter-mining unexploded large-charge his-

torical ordnance in a shallow water environment: Part 2. Modeling of seismo-acoustic wave propagation.
IEEE J Ocean Eng. 2022; 47(2): 374-398.
[4] C. Geuzaine and J.-F. Remacle. Gmsh: a three-dimensional finite element mesh generator with built-in
pre- and post-processing facilities. International Journal for Numerical Methods in Engineering 79(11),pp. 1309-1331, 2009.
[5] P.-Y. Bard and M. Bouchon. The two-dimensional resonance of sediment-filled valleys. Bulletin of the
Seismological Society of America, 75(2) :519–541, 1985.

ABS-0777

Scaling offshore pile driving noise: Estimating the relevant sound pressurelevelsJonas VON PEIN(1) , Tristan LIPPERT(2) , Stephan LIPPERT(1) , Otto VON ESTORFF(1)(1) Hamburg University of Technology, Institute of Modelling and Computation, Denickestrasse 17, 21073 Hamburg, Germany,jonas.pein@tuhh.de(2) AQUSTIX GbR, Katharinenstrasse 4, 20457 Hamburg, Germany

ABSTRACT
Underwater noise due to construction activities can reach sound pressure levels that are potentially harmful tothe marine fauna. Percussive pile driving is the state-of-the-art installation technique for the foundations ofoffshore wind farms. The installation process leads to very high noise emissions. To protect the environment,
limits have to be fulfilled for offshore pile driving noise in many countries. The assessment of the occurringsound pressure levels can be either done by numerical models or by simplified scaling methods. The scaling
methods are especially useful at early stages of the planning phase, since a lot of input data that is necessary forthe numerical models may not be available at that point. Within this contribution, the scalability of the relevant
sound pressure levels such as the sound exposure level (SEL) and the peak sound pressure level (SPLpeak ) isshown with and without the application of a big bubble curtain (BBC).

Keywords: pile driving noise, underwater noise, scaling laws

1

INTRODUCTION

The most common foundation types for offshore wind farms are based on piles. The standard installation
procedure is percussive pile driving. Every single strike on the pile head leads to the emission of underwater
noise with comparably high sound pressure levels, when not properly mitigated. These noise emissions arepotentially harmful to the marine environment [1]. Therefore, many countries have defined limits for the emitted
sound pressure levels [2, 3].
Two important metrics for emitted pile driving noise are the sound exposure level (SEL) and the peak soundpressure level (SPLpeak ). The SEL is defined with the sound exposure E p and the reference value E p0 = 1 µPa2 s
by
SEL = 10 log10

E 
p

E p0

[dB re 1µPa2 s] .

(1)

The sound exposure can be computed with the pressure time series p(t) by E p = tt12 p2 (t) dt. The times t1 andt2 mark the beginning and the end of the sound event [4]. The SPLpeak is defined by the maximal amplitude ofp(t) and p0 = 1 µPa withR

SPLpeak = 20 log10

 max(|p(t)|) p0[dB re 1µPa] .

(2)

The prediction of these sound pressure levels can be done by empirical or numerical models, c.f. Lippert
et al. [5, 6] or Tsouvalas [7].
Both approaches, however, have the disadvantage of being only available to a few number of experts andinstitutions. Therefore, simple to apply scaling laws for the SEL have been developed by von Pein et al. forunmitigated [8] as well as mitigated [9] pile driving noise. The contribution at hand is going to show theextension and application of such scaling laws for the estimation of the SPLpeak .
The paper is organized as follows: Within Section 2 the used modelling approach, which is based on the
combination of a close range finite element method (FEM) model and a Parabolic Equation (PE) model, isbriefly described. The scaling laws for the SEL as derived by von Pein et al. [8, 9] are provided in Section 3. Despite these already published dependencies for the SEL, scaling laws derived for the estimation of
the SPLpeak are given. The application of the scaling laws for the sound pressure levels for unmitigated andmitigated pile driving scenarios is shown in Section 4. Therein, the scaled and measured results of six different
sites are compared. A conclusion and an outlook on further research is given within Section 5.

2

MODELLING APPROACH

The modelling approach is described in more detail in von Pein et al. [9]. For reasons of completeness it isalso summarized in the following:
The derivation of the scaling laws for unmitigated scenarios is done by applying a pile driving noise model
based on the FEM [10].
The derivation of scaling laws for mitigated scenarios is done by a hybrid modelling approach. The closerange model is based on the FEM and the far-range model on the PE technique [11].
Within the FEM model the pile, soil, and water are discretized with a high level of detail. The pile headexcitation is usually derived within a pre-calculation considering the original geometries of the hammer components. Within the acoustical model the derived pile head excitation is applied as a boundary condition at thepile head. At the lateral and lower end of the domain non-reflecting boundary conditions are applied. For adetailed description and validation of this modelling approach, see [10].
The far-range model is based on the split-step Padé approximation of the outgoing wave equation. A description of the hybrid modelling approach as well as a detailed validation is provided in von Pein et al. [12].
Noise mitigation systems are included into the PE model by multiplying the pressure field in the frequency
domain by a frequency dependent transmission coefficient at its dedicated position. The depth and frequency
dependent transmission coefficient of the BBC is derived with an approach suggested by Huisman et al. [13]. It
is based on the differences of the impedance of the air water mixture and water as well as a first approximateof the bubble resonances. The bubble rise speed, the width of the BBC at the sea surface as well as the air
supply are three parameters that need to be defined for the BBC model. Furthermore, the distance of the BBC
to the pile is necessary for the computation.

3

SCALING LAWS FOR PILE DRIVING NOISE

The considered scaling laws for unmitigated pile driving noise are based on the dependencies derived and validated in von Pein et al. [8, 9]. However, only scaling laws for the SEL are presented therein. The corresponding
dependencies of the SPLpeak are derived with the same data sets and compared to the dependencies on the SEL
in the following.
Within von Pein et al. [8] it is shown that the found dependencies can be added up to derive an estimate of
the SEL in the following way:
SELi = SEL0 + ∆SELE + ∆SELd + ∆SELmr + ∆SELh ,(3)

where SEL0 is the measured or computed value of the reference site, ∆SELE is the difference caused by different strike energies, ∆SELE is the difference related to the pile diameters, ∆SELmr is the difference induced due
to different ram weights, and ∆SELh corresponds to the difference influenced by different water depths. The
same approach is also valid for the scaling of the SPLpeak .
In the following, the dependency of the SEL and the SPLpeak for unmitigated as well as mitigated scenarios
is scaled by∆E = 10 log10 Ei /E0 .
(4)

3.1 Scaling of unmitigated scenarios
The influence of the pile diameter d for unmitigated pile driving on the sound pressure levels is scaled by(5)∆d = kd log10 di /d0 .
The factors kd,SEL = 16.7 and kd,SPLpeak = 12.7 have been derived with a best-fit approximation. Therefore, theSPLpeak increases slightly less than the SEL with increasing pile diameter, leading to a small difference between
SPLpeak and SEL for bigger piles. The influence of the ram weight mr was found to be scalable by(6)∆mr = −kmr log10 mr,i /mr,0with kmr ,SEL = 10 and kmr ,SPLpeak = 11.5. This result indicates that a greater ram weight leads to a higher
reduction of the SPLpeak compared to the SEL.
The influence of the water depth on the SEL and SPLpeak can be approximated with the damping term of
the damped cylindrical spreading model [14] and is defined with the reflection coefficient between water andsoil R, in dependence on the water depth h, the propagation angle ϕ, and the distance ri by 10 log (|R |2 ) 10 log (|R |2 ) 
i
0
10
10
−
ri .
(7)
2 cot(ϕ)hi
2 cot(ϕ)h0
The propagation angle ϕ is approximately 17° as shown by Reinhall and Dahl [15] and the reflection coefficient can be computed with the acoustical parameters of soil and water.
∆h =3.2 Scaling of mitigated scenarios
As described in von Pein et al. [9] the scaling laws with the consideration of a BBC have been derived witha best-fit approximation of the mean of 50 runs of the detailed numerical model with different soil and BBC
properties. The multiple runs have been conducted in order to take the influence of uncertain soil and BBC
properties into account.
The scaling of the influence of the diameter with the application of a BBC slightly differs compared to thetrends derived for unmitigated pile driving. The influence of the diameter can be scaled by∆d, BBC = kd, BBC log10 di /d0(8)with the corresponding parameters of kd,SEL,BBC = 18.3 and kd,SPLpeak ,BBC = 15.3. This trend is slightly higher
than the relation considered for unmitigated pile driving. The presence of a BBC leads to a shift of the propagating frequency spectra to lower frequencies as the BBC has a greater insertion loss at higher frequencies.
The change in pile diameter directly leads to a higher radiation efficiency at low frequencies. The interactionof these effects lead to the differences in the dependencies.
The best-fit of the mean of the ram weight influence is resulting in a scaling of∆mr, BBC = −kmr, BBC log10 mr,i /mr,0(9)with kmr, SEL,BBC = 9.2 and a similar result of kmr, SPL ,BBC = 9.6. Therefore, the SPLpeak decreases a little morepeakthan the SEL with increasing ram weight. This difference is slightly smaller compared to the unmitigatedscenarios.
The greatest difference between the unmitigated and mitigated scaling laws can be found in the water depth
dependency. For the unmitigated scenarios the water depth effect is mainly induced by the differences in propagation loss which can be computed with the damped cylindrical spreading model. This effect is still present in
the cases of the application of a BBC. However, it is combined with the water depth dependent insertion loss
behavior of the BBC. An important influence on the insertion loss of the BBC is the water depth dependent bubble size. It effects the volume fraction of air within the air-water mixture and the resonance effects. The best-fit
approximation of the results evaluated at 750 m leads to parameters of khw, SEL,BBC = 16.2 and khw, SPL ,BBC = 16.5
peak
to be used in∆hw, BBC = khw, BBC log10 hw,i /15 m .
(10)

3.3 Summary
All in all, the main difference between the scaling of the SEL and the SPLpeak can be identified in the dependence on the ram weight as well as on the pile diameter. This is the case for the dependencies found for
mitigated as well as unmitigated scenarios. The SPLpeak is directly linked to the peak of the first stress wave
within the pile. Both of these parameters have a direct influence on the stress within the pile and therefore
show a different dependency for the SPLpeak compared to the dependencies of the SEL.
The greatest difference between the mitigated and unmitigated scaling laws are identified with respect to thewater depth scaling.

4

APPLICATION OF THE SCALING LAWS

The applications of the scaling laws are conducted with the following measurement data sets: The data of thewind farms Sandbank (SB), Veja Mate (VM), Trianel (TR), and Deutsche Bucht (DB) are all taken from thedata base MarineEARS [16]. Furthermore, data of Borkum West 2 (BW2) [17] and Global Tech I (GTI) [18]are used. The according data sets are listed in Table 1. They all have been derived with the application of aBBC. Furthermore, for all of these sites also measured data of an unmitigated scenario is available.
Table 1. Measurement data derived with the application of a BBC.
name

piles

SEL05 [dB]

SPLpeak [dB]range [m]strike energy [kJ]

pile diameter [m]water depth [m]ram weight [t]hammer typeSB3

162 - 163

182 - 185

663 - 757

1013 - 1522

6.8

26.8 - 29.6

175

MHU 3500S

VM

1

171

193

803

1893

7.8

39

200

Hydrohammer S-4000

TR

1

163

181

749

1457

8

28.3

150

Hydrohammer S-3000

DB

1

168

187

749

2150

8

39.3

200

Hydrohammer S-4000

BW2

2

160 - 163187 - 188

750

1200

2.44

30

60

Hydrohammer S-1200

GTI

2

163 - 164186 - 187

600 - 700

583

2.48

40

66

MHU 1200S

A common way of estimating the SEL and SPLpeak for mitigated scenarios is to derive an estimate for theunmitigated scenario and subtracting a noise mitigation specific insertion loss. The presented scaling laws allow
to directly estimate the SEL and SPLpeak for mitigated as well as for unmitigated scenarios. To demonstrate
this, the six sites of Table 1 are taken and five of the six sites are scaled to the considered site. The average
is displayed together with the minimum and maximum value in Figure 1. These results are plotted right next
to the actual measured sound pressure levels. The sound pressure levels of the unmitigated scenario are derived
with 20 of 21 sites as shown by von Pein et al. [8] with the scaling laws provided within Section 3. This data
set is now extended with the SPLpeak .
The comparison of the estimated sound pressure levels of the mitigated and unmitigated scenarios in Figure 1 allows new insights into the measured data:
The size of the errorbars indicating the lowest and highest estimate is in a similar range for the unmitigatedand mitigated SEL. However, for the SPLpeak it is much greater for the mitigated scenarios. Since usually only
the highest occurring SPLpeak is reported, the SPLpeak is much more likely to be influenced by dynamic effects
related to the BBC performance as well as statistical outliers.
The estimations of the BBC results of SB show a very good agreement with the measurements and also
indicate that the measurements were at the lower end of the considered data sets. At the same time the comparison of estimated and measured SEL and SPLpeak for the unmitigated scenario shows that these are thehighest measured values in the data base. This new perspective allows a better classification of the actual BBC
performance.
For VM the unmitigated SEL and SPLpeak estimation matches the measured result very well. The comparison of the measured and estimated SEL and SPLpeak with a BBC shows that the BBC performance is at thehigher end of the range compared to the other sites.
The comparison of the TR results shows that for the unmitigated and the mitigated scenario the measured
SEL and SPLpeak are at the lower end of the scaled range. However, the cause of this comparably low noise

200
190

DB

BW
2

GT
I

2
BW

GT
I

TR
TR

DB

VM

180
170
160

SB

SEL [dB re 1µPa2 s]

VM

180

SB

SPLpeak [dB re 1µPa]

site remains unclear. Furthermore, the measured mitigated SPLpeak is the lowest in the data base and the reason
for the comparably great range of the errorbar.
The estimated results of DB, BW2, and GTI are also in good agreement with the derived estimation range.
The comparisons demonstrate that the approach of directly scaling the SEL and SPLpeak for scenarios witha BBC can provide new insights into the actual BBC performance and are another way of estimating the soundpressure levels with the application of a BBC.

scaled unmitigatedscaled BBC

meas. unmitigatedmeas. BBC

Figure 1. Comparison of the scaled and measured SEL and SPLpeak of six different sites with and without theapplication of a BBC.

5

CONCLUSIONS AND OUTLOOKThis contribution presents the possibility of scaling the SEL and the SPLpeak from one site to another. This isdone by taking the influence of the strike energy, pile diameter, ram weight, and water depth into account. The
scaling laws and their application are shown for unmitigated as well as for mitigated scenarios using a BBC.
With a decent data base of measurements the derived scaling laws are a useful tool for the estimation of
the SEL and SPLpeak . Furthermore, they can be employed to get a good insight and benchmark of a BBC
performance.
The next steps in the development of scaling laws for offshore pile driving noise are the adaptation toscenarios with a noise mitigation system in the direct vicinity of the pile, a double big bubble curtain, and their
combination. Furthermore, the investigation of the according influences of the four parameters on the emitted
frequency spectra will be conducted.

REFERENCES[1] Tougaard, J.; Beedholm, K.; Madsen, P. T. Thresholds for noise induced hearing loss in harbor porpoises
and phocid seals. J. Acoust. Soc. Am., 151(6), 2022, 4252-4263.

[2] National Marine Fisheries Service. 2018 revision to: Technical guidance for assessing the effects of anthropogenic sound on marine mammal hearing (version 2.0). National Oceanic and Atmospheric Administration,
Washington DC, 167, 2018.
[3] Juretzek, C.; Schmidt, B.; Boethling, M. Turning Scientific Knowledge into Regulation: Effective Measuresfor Noise Mitigation of Pile Driving. Journal of Marine Science and Engineering, 9(8), 2021.
[4] International Organization for Standardization. DIN ISO18406:2018-08 - Underwater acoustics - Measurement of radiated underwater sound from percussive pile driving. 2018.
[5] Lippert, S.; Nijhof, M.; Lippert, T.; Wilkes, D.; Gavrilov, A.; Heitmann, K.; Ruhnau, M.; von Estorff,
O.; Schäfke, A.; Schäfer, I.; Ehrlich, J.; MacGillivray, A.; Park, J.; Seong, W.; Ainslie, M.A.; de Jong,
C.; Wood, M.; Wang, L.; Theobald, P. COMPILE — A generic benchmark case for predictions of marine
pile-driving noise. IEEE J. Ocean. Eng., 41(4), 2016, 1061–1071.
[6] Lippert, S.; von Estorff, O.; Nijhof, M.; Lippert, T. COMPILE II — A benchmark of pile driving noisemodels against offshore measurements. In proceedings of Inter-Noise 2018 Chicago, 2018.
[7] Tsouvalas, A. Underwater noise emission due to offshore pile installation: A review. Energies, 13(12), 2020.
[8] von Pein, J.; Lippert, T.; Lippert, S.; von Estorff, O. Scaling Laws for Unmitigated Pile Driving: Dependence of Underwater Noise on Strike Energy, Pile Diameter, Ram Weight, and Water Depth. Journal ofApplied Acoustics, 2022, (in print).
[9] von Pein, J.; Lippert, T.; Lippert, S.; von Estorff, O. Scaling offshore pile driving noise: A case study for
scenarios with and without noise mitigation. Proceedings of Meetings on Acoustics, International Conference
on Underwater Acoustics, Southampton, 2022, (in print).
[10] von Pein, J.; Lippert, S.; von Estorff, O. Validation of a finite element modelling approach for mitigatedand unmitigated pile driving noise prognosis. J. Acoust. Soc. Am., 149(3), 2021, 1737-1748.
[11] von Pein, J.; Klages, E.; Lippert, S.; von Estorff, O. A hybrid model for the 3D computation of piledriving noise. In proceedings of OCEANS 2019, Marseille, France, 2019.
[12] von Pein, J.; Lippert, S.; von Estorff, O. Validation of a 3D pile driving noise model with noise mitigation
measures. In proceedings of 6th Underwater Acoustics Conference and Exhibition, Crete, Greece, 44, 2021.
[13] Huisman, M.; Lederwasch, L.; Smidt Lützen, R. Bubble curtain modelling: analytical prediction of underwater piling noise mitigation. In proceedings of Inter-Noise 2021.
[14] Lippert, T.; Ainslie, M.A.; von Estorff, O. Pile driving acoustics made simple: Damped cylindrical spreading model. J. Acoust. Soc. Am., 143(1), 2018, 310-317.
[15] Reinhall, P.; Dahl, P. Underwater Mach wave radiation from impact pile driving: Theory and observation.
J. Acoust. Soc. Am., 130(3), 2011, 1209-1216.
[16] MarinEARS – Marine Explorer and Registry of Sound; specialist information system for underwater noise
and national noise-register for the notification of impulsive noise events in the German EEZ of the Northand Baltic Sea to the EU according to the MSFD, 2021.
[17] Diederichs, A.; Pehlke, H.; Nehls, G.; Bellmann, M.; Gerke, P.; Oldeland, J.; Grunau, C.; Witte, S.;
Rose, A. Entwicklung und Erprobung des Großen Blasenschleiers zur Minderung der Hydroschallemissionen
bei Offshore-Rammarbeiten (Development and testing of the Big Bubble Curtain to reduce hydro-acoustic
emissions during offshore pile driving). 2014.
[18] Bellmann, M.; Gündert, S.; Remmers, P. Offshore Messkampagne 2 (OMK2) für das Projekt BORA im
Offshore-Windpark Global Tech I (Offshore measurement campaign 2 (OMK2) for the BORA project at thewind farm Global Tech I). 2015.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0888

Numerical investigation of acoustic radiation efficiency of a wedgeshaped structure surrounded by air bubble layers
Sung-Ju PARK1; Cheolsoo PARK2; Kookhyun KIM3; Jaehyuk LEE4; Keunhwa LEE5; Cheolwon LEE6
1
2

Korea Research Institute of Ships & Ocean Engineering, Korea
3

4

Tongmyong University, Korea
Tongmyong University, Korea

Daewoo Shipbuilding & Marine Engineering CO., LTD., Korea
5

Sejong University, Korea
6

AL, Korea

ABSTRACT
Underwater noise pollution can affect marine life. This study explores the effect of acoustic radiation
efficiency with a fluid-loaded structure based on the numerical investigation. The effect of the bubble layer
is considered using the semi-numerical method. For validation of the proposed procedure, the acousticstructure interaction analyses using the commercial finite element program COMSOL Multiphysics were
performed for fluid-loaded plates with an air bubble layer. The semi-numerical method results show good
agreement with the numerical one for ‘plate-bubble-fluid’ model in terms of sound power level from all thefrequency domains. Subsequently, the proposed method was used to investigate the acoustic radiation
efficiency of a wedge-shaped structure surrounded by air bubble layers. The numerical result provides theprincipal mechanism of underwater noise reduction depending on the characteristic of the bubble layer on
the thin structure.
Keywords: Acoustic radiation efficiency, Air bubble layer, Equivalent bubble layer method, Acousticstructure interaction

1. INTRODUCTION
The underwater noise is produced by a combination of mechanical and hydrodynamic noise. The
air-bubble curtain is applied to reduce the sound radiated from the noise source such as the propeller,
engine, offshore exploration, underwater structure, etc. The interaction of the air bubble and radiatedsound has been studied for application to the engineering design. Würsig et al.(1) measured soundintensities near pile-driving in 6-8 m depth sea with the installation of a bubble curtain and presented
that the sound reduction by the bubble curtain was evident. Park et al.(2) suggested the analysis
solution for examining the insertion loss of fluid-loaded plate with bubble layers depending on thecharacteristic of bubble population, void fraction, and the thickness of the bubble layer. Lee et al.(3)studied elastic shell model surrounded by the bubble layer with various bubble distribution condition.
In this study, we proposed a semi-numerical method procedure and carried out the finite element
analysis to investigate the acoustic radiation efficiency of structures surrounded by air-bubble. The
behavior of the air-bubble layer was considered using the effective medium theory proposed byCommander and Properetti (4). The proposed procedure was validated through simulation of acousticstructure interaction for the fluid-loaded plates. The effect of air bubbles to the structure wasinvestigated in terms of acoustic radiation efficiency.
1parksj0314@tu.ac.kr
parkcs@kriso.re.kr
3
kimk@tu.ac.kr
4leehel7@naver.com
5
nasalkh2@sejong.ac.kr
6
lcw7376@gmail.com
22. THEORYTICAL BACKGROUND
2.1 Finite element method for acoustic-structure interaction
The behavior of an acoustic region in the frequency domain uses the inhomogeneous Helmholtz
equation as:
1
∇𝑝
𝜌

∇∙

𝑘 𝑝
𝜌

𝐪

𝑄

(1)

The 𝑝 is the total pressure which is the sum of the background pressure field 𝑝 and thescattered field 𝑝 . The wave number 𝑘 denotes the ratio of the angular frequency 𝜔 to the soundspeed 𝑐. The sound pressure wave is propagating with density 𝜌 at the sound speed. The terms 𝑄
and 𝐪 are the monopole and dipole domain sources, respectively. In equation 1, the sound pressure𝑝 𝐱, 𝜔
𝑝 𝐱 𝑒 .
can be expressed in pressure varies with time as 𝑝
The acoustic-structure coupling includes the fluid load on the structure and the structuralacceleration caused by the fluid. The exterior boundaries expressed as:
𝐧∙

1
∇p
𝜌

𝐧 ∙ 𝐮𝐭𝐭

𝐪𝐝

(2)

(3)
𝐅𝐀 𝑝 𝐧where 𝐮𝐭𝐭 is the structural acceleration, 𝐧 is the normal vector to the surface. 𝐅𝐀 is the force perunit area experienced by the structure. The internal boundary condition follows as:1∇p𝐪𝐝
𝐧 ∙ 𝐮𝐭𝐭
𝐧∙
(4)
𝜌
1
∇p
𝜌

𝐧∙

𝐪𝐝

𝐧 ∙ 𝐮𝐭𝐭𝑝,𝑝, 𝐧𝐅𝐀The up and down subscripts mean the two sides of the interior boundary.

(5)
(6)

2.2 Acoustic insertion loss by a bubble layer
As shown in Commander and Prosperetti (4), the equivalent wave number 𝑘 of bubbly fluid
give the following relation for pressure wave with a time dependence proportional to:
𝑘

𝜔
𝑐

4𝜋

𝜔

𝑎𝑓 𝑎
𝑑𝑎
𝜔
2𝑖𝛿𝜔

(7)

The 𝑓 𝑎 is the function of the bubble distribution, the 𝛿 is a damping coefficient of bubble, 𝜔 isthe resonant frequency of bubble. The damping coefficient of the mixed medium of bubble and wateris expressed by an imaginary part of equivalent wave number:(8)nepers/m
𝛼 𝐼𝑚𝑎𝑔 𝑘
When the propagated to ℎ m , the attenuation 𝛼 is correspond 8.686 𝛼ℎ dB , approximately. The
insertion loss was determined by the comparing the input sound pressure 𝑝 and output soundpressure 𝑝 , which is the transmission by the bubble layer:𝑝𝐼𝐿
20 log(9)𝑝
To permit a simplification of the bubble layer in finite element analysis, this approximation can beassumed that the insertion loss is equivalent to attenuation as shown in Park et al. (2020).

3. VALIDATION OF NUMERICAL METHODSThis paper carried out the acoustic structure interaction analysis for fluid-loaded finite and infinite
plate models without a bubble layer. The sound pressure level was obtained from simulation, theresults considered a bubble layer was calculated under the assumption that insertion loss is equal toattenuation, i.e., the output sound pressure, which is after passed through the bubble layer, wascalculated from the difference of the insertion loss and input sound pressure. This method was defined
as semi-numerical simulation. To validate the proposed method, the simi-numerical simulation wascompared with a fluid-loaded plate model with a bubble layer. The infinite plate model refers to Parket al. (2). The parameters for numerical simulation are presented in Table 1 and the model is illustrated
in figure 1. Where the H_1=0.0 m only consists of “plate-water-PML” (without bubble layers).
Table 1 – parameters for fluid-loaded plate with bubble layer simulation
MediumParameter

Value

Thickness 𝑡

0.015 mYoung’s modulus 𝐸
Plate

Water

bubble19.5

10 N/mPoisson’s ratio 𝜈

0.3

Density 𝜌

7,800 kg/m

Compressional sound of speed 𝑐5801.2 m/s

Structural loss factor 𝜁

0.005

Depth 𝐻

3.0 m

Density 𝜌

1,000 kg/m

Sound of speed 𝑐

1,500 m/s

Bubble radius 𝑎

0.5 mm ~ 2.5 mm

Specific heat ratio 𝛾

1.4

Layer thickness (𝐻 )0.0 m, 0.1 m

Wave Speed 𝑐𝑐 /𝑓Density (𝜌 )999 kg/m(a) Infinite plate model(b) Finite plate model
Figure 1 – The illustration of fluid-loaded plate with bubble layer.

The 2D axisymmetric model was used for numerical simulation. The perfectly matched layer (PML)
was added to the absorbing boundary for mimicking the open infinite domain. The pressure acoustic
model of the bubble layer was defined as the speed of sound and attenuation coefficient as shown infigure 2.

(a) Sound speed(b) Attenuation
Figure 2 – The pressure acoustic model parameters for bubble layer
The point load was applied at the center of the top surface of the plate. The convergence study was
performed to determine the element size for finite element analysis, the one bending wave of the plate
was discretized with eight elements. As shown in figure 3, the sound power level was presented in
0.1 m (plate-bubble-water), H
0.0 m (plate-water), and semi-numericalcomparison to H
method results from 200 Hz - 20kHz. The presented procedure shows good agreement with the mode
solution, approximately.

(c) Infinite plate model(d) Finite plate model
Figure 3 – The comparison of the sound power level between numerical and semi-numerical simulation

4. APPLICATION TO THE WEDGE-SHAPED STRUCTURE SURROUNDED BY AIR
BUBBLE LAYERS
4.1 Model
The geometry and finite element model of the wedge-shaped structure were illustrated in figure 4.
For structure, the material density, Young’s modulus, and Poisson’s ratio were 7,085 kg/m , 200e9
Pa, and 0.3, respectively. The density and speed of sound in water were 1,025 kg/m and 1,500 m/s.
The unity load was applied to the inner stiffener in the structure.

(a) Geometry(b) Finite elelemt model
Figure 4 – The geometry and finite element model of wedge structure
4.2 Acoustic radiation efficiency
The radiation efficiency is defined as:𝑊𝜎𝜌𝑐𝑆〈𝑣̅ 〉where 𝑊 is the sound power emitted by the vibrating structure, 𝑣̅ is the mean square velocity inthe entire area of the surface 𝑆. Figure 5 shows the sound power level and acoustic radiation of thewater-loaded structure (𝐻
0.0 m) and structure considered the effect of air-bubble (𝐻
0.1 m).
the characteristic of bubble layer refers to Table 1.

(a) Sound power level(b) Radiation efficiency
Figure 5 – The comparison of sound power level and radiation efficiency between water-loaded structureand structure surrounded air-bubble layer

5. RESULTS
The semi-numerical method of acoustic radiation efficiency for the structure surrounded by an air
bubble layer is proposed based on finite element analysis and analytical method (2). The method iscompared with the finite element analysis results through a fluid-loaded infinite and finite plate cases
and then applied to the wedge structure surrounded by the air bubble layer. The results show the
behavior of the noise reduction caused by the air-bubble layer in fluid-loaded structure in the water.

ACKNOWLEDGEMENTS
This work was supported by the Ministry of Trade, Industry and Energy (No. 20008585) andKorea Research Institute of Ships and Ocean engineering (2021B027), to which deep gratitude isexpressed.

REFERENCES
1.

Würsig B, Greene CR, Jefferson TA. Development of an air bubble curtain to reduce underwater noiseof percussive piling. Mar Environ Res. 2000 Feb;49(1):79–93.

2.

Park C, Jeong SW, Kim G Do, Moon I, Yim G. Acoustic insertion loss by a bubble layer for theapplication to air bubble curtain and air masker. J Acoust Soc Korea. 2020;39(4):227–36.

3.

Lee K, Lee C, Park C. Insertion loss by bubble layer surrounding a spherical elastic shell submerged inwater. J Acoust Soc Korea. 2022;41(2):147–83.

4.

Commander KW, Prosperetti A. Linear pressure waves in bubbly liquids: Comparison between theory
and experiments. J Acoust Soc Am. 1989 Feb;85(2):732–46.

vv

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022 in Gyeongju, Korea

ABS-0120

Comparison of linear and logarithmic sampling for a sensitivity
analysis of a NVH vehicle body simulation
Sophie CRAM1,2; Marinus LUEGMAIR1; Johannes D. SCHMID2; Steffen MARBURG2
1
2BMW Group, Germany

Technical University of Munich, Germany

ABSTRACT
In vibroacoustics, uncertainties are important for vehicle body simulations in an early development stage. A
deterministic finite element model is typically evaluated several times with stochastic input parameters togain stochastic results of the mobility of the vehicle body. It is our aim to reduce the number of stochastic
parameters in this noise vibration harshness simulation for a vehicle body, i.e. on the one hand side to savesimulation costs of the expensive car body model with many million degrees of freedom and on the other
hand side to reduce the effort for parameter uncertainty quantifications. For this reduction, we conduct asensitivity analysis with the generalized polynomial chaos expansion to get the Sobol’ indices for which weneed to sample the stochastic input parameters. Furthermore, it is our aim to focus on the most sensitive andtherefore most relevant parameters of the noise vibration harshness simulation. The kind of sampling – linear
or logarithmic – of widespread parameters influences the mean and standard deviation of the mobility curve.
The resulting differences are shown and discussed.
Keywords: Vibroacoustics, Stochastic, Finite element car model, Parameter reduction

1. INTRODUCTION
Uncertainties significantly affect the vibroacoustics of a vehicle body as shown with differently
conducted acoustic measurements:
Measurements are compared with an uncertainty-filled structural-acoustic car model by Durand etal. (1).
In 1993, Kompella et al. (2) evaluated the two transfer pathways of 57 pickup trucks and 99 cars,
which were structure-borne and airborne, respectively. Each set of vehicles has the exact same
configuration. The deviations in the frequency response functions range from 5 to 10 dB.
Hills et al. (3) conduct a larger investigation of measured vehicle acoustic data: They examine datafrom fleets of 316, 403, and 411 automobiles and display the resulting noise distributions.
In addition to these measurements, there are contributions which deal with uncertainties insimulations to cover the measured variations:
The NVH simulation of a vehicle is the main topic of Luegmair ’s and Schmid’s (4) description ofthis process in terms of uncertainty. Uncertainty in the simulation ’s input parameters stems fromdifferences in the manufacture, use, life cycle, and features that the client has chosen for his car.
The generalized polynomial chaos (gPC) expansion is added to this NVH finite element (FE)simulation procedure by Schmid et al. (5) . According to (6), we develop a two-step sensitivity analysis
which reduces the uncertain parameters as well as the computation time based on the simulation
method with gPC expansion.
The gPC expansion, the simulation process and the two-step sensitivity analysis will be covered
later on in this contribution.
The considerable variance of the results, as shown in (7), necessitates the inclusion of unknown
parameters in these simulations: To identify the types of distributions, we quantify the dynamic
stiffness and damping of a car door seal and use the Pearson chart to assess the results.
According to (8), statistical factors are taken into consideration with a margin of safety in
traditional design based on the manufacturing, ambient circumstances, and design procedures. This
1

Sophie.Cram@bmw.demargin causes the system to be designed in a way that exceeds its intended functio n. Because of this,a stochastic analysis is performed to investigate the impact on the vehicle shaking response (8).
Furthermore, we find comparisons of logarithmic and linear transformations (9), averaging (10)and sampling (11) in the literature. The latter compare phase approximations in the frequency domain
which are logarithmic and linear.
The following queries are addressed in this contribution: How does a logarithmic or linear
sampling of widespread uncertain parameters influence the admittance curves ? How does the linear
sampling affect the reduction of the uncertain parameters regarding the two-step sensitivity analysis
as presented in (6)?

2. METHODSIn this section, a brief description of the sensitivity analysis is presented along with a short
introduction to the gPC expansion. Then, we show the computation of the sensitivities using the Sobol’
indices relying on the gPC expansion. To further aid in the comprehension of the two -step sensitivity
analysis, the superelement method is also shortly introduced.
2.1 Sensitivity Analysis
We work with a sensitivity analysis in this article to show the dependencies between the input andoutput of a model (12). Especially, we focus on a so-called variance-based sensitivity analysis – aglobal analysis – that is dependent on the output variance. This variance is made up of ratios which
are linked to sets of inputs or individuals (13). The first order sensitivity index, often known as the
Sobol’ index 𝑆𝑖 , defines the averaged partial decrease for a model of k factors 𝑓(𝑥) = (𝑥1 , 𝑥2 , … , 𝑥𝑘 ) ∈ℝ𝑘 in the unconditional variance 𝑉𝑎𝑟(𝑦) of the output 𝑦 for a set 𝑥𝑖 and is stated as (14)𝑆𝑖 =𝑉𝑎𝑟𝑥𝑖 [𝐸𝑥~𝑖 (𝑦|𝑥𝑖 )].
𝑉𝑎𝑟(𝑦)(1)

𝐸𝑥~𝑖 represents the inner mean across all factors except 𝑥𝑖 (14).
2.2 Polynomial Chaos Expansion
The polynomial chaos expansion which can be used to calculate the Sobol’ index analytically (15)is suggested for cases requiring intense computational power (13). Based on Wiener (16), thepolynomial chaos expansion – a spectral based method – is enhanced by Cameron and Martin (17),
Xiu and Karniadakis (18) and Ghanem and Spanos (19) to the generalized polynomial chaos (gPC)expansion. This method depicts the uncertainty in a system which is stochastic with infinite series
expansion. A random variable 𝑋(𝝃) is presented in the spectral representation as∞

𝑋(𝝃) = ∑ 𝑥𝑖 𝛹𝑖 (𝝃), 𝑥𝑖 ∈ ℝ

(2)

𝑖=0

given a stochastic subspace where the basis polynomials 𝛹𝒊 (𝝃) are orthogonal with respect to eachother. 𝑥𝑖 are the deterministic coefficients. We recommend (5, 6) to find detailed information on thegPC expansion and its implementation with a deterministic FE model.
2.3 Computation of the Sobol’ Indices with the Generalized Polynomial Chaos Expansion
Following the use of the gPC expansion, it just requires a minimal amount of additional work to
calculate the so-called polynomial chaos based Sobol’ indices, which Sudret (15, 20) published. The
first order sensitivity index is defined as

𝑆𝑖 =∑𝛼∈𝕴 𝑦𝛼2 ⟨𝛹𝛼2 (𝜉𝑖 )⟩
𝐢

𝑁

∑
𝑗=1

𝑦𝑗2 ⟨𝛹𝑗2(𝜉)⟩(3)and describes the impact of the 𝑖 𝑡ℎ input variable 𝜉𝑖 (21). According to Eq. (1), the numerator isthe variance of the terms which have the single parameter 𝜉𝑖 included and it is divided by the output’s
overall variance. The terms in the expansion that only use the single variabl e 𝜉𝑖 are extracted using
the set of indices 𝕴𝐢 (21).
2.4 Superelement
The linkage of substructures for dynamic analyses was first introduced by Craig and Bampton in1968 (22). We use the method of superelements in Nastran as the study’s FE solver which reduces thecomputation time by modal reduction. For interested readers, more information is provided in the
Superelements User’s Guide (23).

3. State-Of-The-Art NVH FE Simulation
Figure 1 depicts the FE NVH model that we study which has around 2.6 million elements,
18.4 million degrees of freedom and 3.1 million nodes.

Figure 1 – FE NVH model in its entirety, including the physical part (body in white) and the additional
vehicle components indicated by pink dots.
We divide this model in two parts: the body in white which we call the physical part and the
additional vehicle components which are represented by cuboids with physical properties to reduce
the simulation time.
This FE model is excited by four forces at the rear axle attachment points. Since the goal of the
NVH simulation is to forecast NVH behavior prior to the vehicle ’s construction, it is noteworthy that
uncertain parameters are inserted in the process. Therefore, a deterministic simulation is enabled dueto the sampling of the distributions of the unknown parameters. To obtain a statistical outcome of theadmittance, we iteratively run the deterministic simulation. More detailed information regarding thestate-of-the-art NVH FE model and simulation process can be found in (6).

4. Parameter reduction regarding the sampling approach for a two-stepsensitivity analysis
In this section, we describe the uncertain parameters and their sampling and the two-step sensitivity
analysis which both were introduced in (6). Then, we compare the results of the parameter reduction.
4.1 Uncertain parameters and their samplingFirst of all, we introduce the uncertain parameters with their ranges and numbers which we explainin (6) in detail: For the body in white, we chose the Young’s moduli and the thicknesses of the shells
and for the additional vehicle components the translational and rotational stiffnesses, the positions of
the center of gravity of the cuboids, their moments of inertia, their products of inertia and their masses .
In sum, we have got 1,300 uncertain parameters which we vary for the sensitivity analysis. In theprevious work (6), we sample the translational and rotational stiffnesses logarithmically because thevalue range varies in a wide range. Logarithmic sampling means that we sample linear on a logarithmic
scale and therefore receive equal numbers of samples between 100 and 1,000, 1,000 and 10,000, and10,000 to 100,000. With a linear sampling approach, we sample on a linear scale and therefore getequal numbers of samples as in the logarithmic example between 100 and 33,400, 33,400 and 66,700,
and 66,700 and 100,000. The 1,300 uncertain parameters, their ranges, numbers and the sampling arepresented in Table 1.

Table 1 – Uncertain parameters with their ranges, numbers and sampling.
Parameter

Range

NumberSampling

Young’s modulus

210 GPa ±6 % (197.4 to 222.6 GPa)

25

linearThickness

±5 %

606

linearTranslational stiffness

100 to 100,000 N/mm

246

logarithmic or linear

Rotational stiffness

100 to 100,000 Nmm/rad

246

logarithmic or linear±50 mm63

linearMoments of inertia

±5 %

63

linearProducts of inertia±5 % or neglected if equal 0

63 / 30

linear

Mass

±10 %

21

linearPosition of the centerof gravity

In this contribution, we provide the results for linear sampling which we compare to the logarithmic
results.
4.2 Comparison of the parameter reduction with the two-step sensitivity analysis
To answer the question if the two-step sensitivity analysis also achieves a high reduction for the
linear sampling of the stiffnesses, we proceed this analysis again with the linear sampling which is
described in detail in (6).
In the first step of the analysis, we start with all parameters of the additional vehicle components(669 parameters). This set of parameters has got the advantage that we are allowed to use a super element of the physical part and receive an around 25 times faster computation times. We set a limitof 1 % and filter the Sobol’ indices of each parameter if it is greater than this limit. Next, we compare
the admittances of all 669 parameters with the reduced set of 135 uncertain parameters in Figure 2.

Figure 2 – First step of the sensitivity analysis with linear sampling: Comparison of the admittances of 669
uncertain parameters and the 135 sensitive parameters over the frequency.
The error of the mean is 1.9 % and 12.9 % of the standard deviation. Compared to the logarithmic
sampling with 177 sensitive parameters, cf. (6), the mean value is 5.9 % smaller (logarithmic case:
7.8 %) and the standard deviation is 1.6 % higher (logarithmic case: 11.3 %). Because of these verysmall errors and the good fitting, we continue with the second step: We add all parameters of the bodyin white – 631 ones – to the 135 sensitive parameters of the step before. We cannot use thesuperelement any more due to the parameter set of the physical part. In Figure 3, these results werecompared with the 106 sensitive parameters which we get if we set the limit of the Sobol’ indicesagain to 1 %.

Figure 3 – Second step of the sensitivity analysis with linear sampling: Comparison of the admittances of
766 uncertain parameters and the 106 sensitive parameters over the frequency.
Again, the error of the mean is very small – only 1 % – as well as of the standard deviation –10.4 %. In the end, we receive a parameter reduction of 91.9 %. In the logarithmic case, cf. (6), we
receive an error of the mean of 3.5 % which is 2.5 % higher than in the linear case and an error of thestandard deviation of 9.3 % which is 1.1 % smaller than the linear case by a parameter reduction of
92.2 %. We validate this result with a comparison of the results from the 106 sensitive parameters of
the two-step sensitive analysis with the results from all 1,300 uncertain parameters in Figure 4.

Figure 4 – Validation step of the sensitivity analysis with linear sampling: Comparison of the admittancesof 1,300 uncertain parameters and the 106 sensitive parameters over the frequency.
The error of the mean is 2.2 % and of the standard deviation 15 %. The mean error is 6.8 % smallerthan in the logarithmic case (9 %) and the error of the standard deviation is only 0.5 % higher than in
the logarithmic case (14.5 %), cf. (6).
4.3 Results
In Table 2, we sum up the linear and logarithmic sampling results: The total reduction of theuncertain parameters and the errors of the mean and the standard deviation for the first and second
step of the two-step sensitivity analysis as well as for the validation.
Table 2 – Overview of the parameter reduction and the errors of the mean and of the standard deviation for
the first step, second step and the validation.
Total reduction of parameters

Linear sampling

Logarithmic sampling

91.9 %

92.2 %First step
Error of the mean

1.9 %

7.8 %

Error of the standard deviation

12.9 %

11.3 %

Second step
Error of the mean

1%

3.5 %

Error of the standard deviation

10.4 %

9.3 %Validation
Error of the mean

2.2 %

9%

Error of the standard deviation

15 %14.5 %We see that the mean error of the linear sampling is for each comparison between 2.5 % and 6.8 %smaller and the error of the standard deviation of the linear sampling is for each comparison between
0.5 % and 1.6 % higher.
Furthermore, we compare the influence of the logarithmic and linear sampling of the translationaland rotational stiffnesses on the admittance curve over the frequency for the mean value and thestandard deviation in Figure 5 for the first step of the sensitivity analysis with the parameters of theadditional vehicle components.

Figure 5 – Comparison of the admittance for logarithmic (red) and linear (blue) sampling of the stiffnesses
over the frequency.

As we compare the curves of the admittance in Figure 5, we see extreme differences: The standard
deviation of the linear sampling is smaller and the peaks of the linear case have got higher values
compared to the logarithmic sampling. Due to the linear sampling, we got less values of small
stiffnesses so that the samples are stiffer than in the logarithmic case.
Overall, we state that the sampling of the stiffnesses has got a huge influence on the admittancecurve which must be taken into consideration if the result curves are further investigated . In addition,
the linear sampling produces smaller errors but both sampling approaches show a very good reductionof the uncertain parameters (91.9 % and 92.2 %).

5. CONCLUSIONSIn this article, we gave a short introduction about the used methods as the sensitivity analysis, thegPC expansion and the computation of the first order Sobol’ indices with the gPC expansion as wellas the definition of a superelement. Next, we introduced the NVH FE process with its model that split
up in two parts. These two parts were the basis for the two-step sensitivity analysis with in sum 1,300
uncertain parameters. These uncertain parameters were the Young’s moduli, the thicknesses of theshells, the translational and rotational stiffnesses, the positions of the center of gravity of the cuboids,
their moments of inertia, their products of inertia and their masses. Furthermore, we conducted thetwo-step sensitivity analysis with a linear sampling. In the results, we compared the admittance curvesof the linear and logarithmic sampling of the translational and rotational stiffnesses and noted extreme
differences. Over all comparisons, the mean error was smaller for the linear sampling and the error of
the standard deviation was smaller for the logarithmic sampling. The parameter reduction was for bothcases satisfying.
Henceforward, we use for these as unimportant classified parameters deterministic values and savea lot of computation time. Future research will focus on determining distributions of the unknown
parameters and investigating the sensitive FE NVH model parameters. It would therefore be
reasonable to examine the sensitivities using the new distributions and contrast them with thiscontribution, which made use of uniform distributions.

ACKNOWLEDGEMENTS
The contribution of Marinus Luegmair and Johannes D. Schmid is part of the joint project
newAIDE under the consortium leadership of BMW AG with the partners Altair Engineering GmbH,divis intelligent solutions GmbH, MSC Software GmbH, Technical University of Munich, TWT
GmbH. The project is supported by the Federal Ministry for Economic Affairs and Climate Action(BMWK) on the basis of a decision by the German Bundestag.
The authors thank Rafaella Dantas for adding a graphical user interface to the code.

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ABS-0235

Model-based machine learning in acoustics applications from a Bayesian
perspective
Ning Xiang(1) and Ziqi Chen(1) Jiahua Zhang(1) Mélanie Nolan(1)(1) Graduate Program in Architectural Acoustics, Rensselaer Polytechnic Institute, Troy, New York USA, xiangn@rpi.edu

ABSTRACT
Recent acoustics investigations increasingly employ machine learning methods to analyze experimental data.
Using well-understood models or hypotheses, acoustics analyses often embody two levels of inference: model
selection and parameter estimation based on the chosen model. Model-based Bayesian analysis provides solutions of both inference levels centered around Bayes’ theorem within a unified framework. The data-driven
machine learning method represents a learning process from data in order to update our knowledge. This isexactly consistent with the core of Bayesian probability theory, which represents how our prior knowledge isupdated in presence of the data. This paper discusses the model-based machine learning methods in a wayof thinking like a Bayesian. In particular, it emphasizes that the well-understood models are considered as apart of the prior knowledge which are readily incorporated in the learning process. This paper highlights datadriven machine learning methods from a Bayesian perspective based on recent acoustics investigations, suchas acoustic array signal processing, normal incident impedance tube measurements, chamber-based absorption
measurements and wavenumber spectral analysis in enclosed spaces. Numerical challenges and promising solutions for two-levels of Bayesian learning are also discussed.

Keywords: machine learning, Bayesian inference, data-driven algorithms, model-based analysis

1

INTRODUCTION

Machine Learning (ML) refers to the process by which computers or algorithms are able to make predictions
based on data and learn from data, possibly adjust some pertinent properties without being specifically programmed to do so. In recent machine learning literature, sometimes it is also referred to as data-driven machinelearning. This paper sheds some light on the data-driven machine learning from Bayesian perspective, primarily
based on Bayesian probability theory. One of core issues within Bayesian paradigms is that the probability isinterpreted as degree/strength of implication or confirmation. In details, degree of implication can quantitatively
encode state of knowledge or state of information in probabilities [1]. While Bayesian probability theory, centered around Bayes’ theorem represents how our prior knowledge is updated in the presence of data, computersor algorithms equipped with the machine learning capability are able to learn from data and make predictions
based on data. This exactly represents the Bayesian viewpoint, that we empower the computers and the specificalgorithms to update the prior knowledge when involving (experimental) data based on Bayesian probability
theory, that also includes all valid rules of statistics for relating and manipulating probabilities. Therefore weview this fundamental scientific exploration, to learn from real-world by conducting experiments in acquiring
data, as Bayesian learning.
This paper highlights data-driven machine learning methods from a Bayesian perspective based on recent
acoustics applications, discusses the two-levels of Bayesian inference to carry out the data-driven machine learning, exploiting well established and/or well understood models, specifically model-based Bayesian learning fromexperimental data. In the following we formulate two levels of Bayesian inference within a unified learning
framework. Then we discuss a number of recent investigations using either the first level of inference, Bayesianparameter estimation or the second level, Bayesian model selection.

2

A UNIFIED INFERENTIAL FRAMEWORK OF BAYESIAN ANALYSISThis paper focuses on a specific class of Bayesian learning, namely model-based inference. Many acoustical applications have in common the existence of well established, or at least well understood models for predicting
the processes under investigation. In order to learn the process, acousticians conduct experimental measurements
to obtain a set of experimental data. The prediction models can be physical or phenomenological, they can also
be analytical or numerical. This Section discusses the two levels of model-based Bayesian learning, namely
model selection and parameter estimation. The model selection represents higher (second) level of inference,
while the parameter estimation is the lower (first) level one. Many machine learning approaches, such as optimizations are within the first level of learning. However, if one engages a wrong model or a redundant model
for any sort of optimizations, the estimations/optimizations become also wrong or redundant, no matter how
well the model-based optimization will be. The following Section applies a top-down approach in formulating
the two levels of probabilistic inference.
2.1 Bayesian model selection
In some applications the investigators are challenged to consider a handful number of models M, they arecompeting to each other to predict the data D. Among them the investigators want to learn from the data
which model among the considered set of models is preferred. This situation can be formulated as backgroundinformation I as‘the models specified by index n = 1, 2, 3, . . . may all describe the data reasonably well.’Taking this background information, the model selection applies the Bayesian theorem to evaluate the probabilityof Mn among a set of models M = {M1 , . . . , Mn , . . . , MN } for n = 1, 2, . . . , N given the data D, [1]p(M|D, I) =p(D|M, I) p(M|I).
p(D|I)(1)

The Bayesian theorem in Eq. (1) as applied to this set of models M represents how the investigator’s priorknowledge p(M|I) about the models, given the background information I as stated above, is updated in presenceof the data as represented by p(D|M, I). It is termed marginal likelihood within the context of Bayesian model
selection, while p(M|I) is the prior probability of the models, often referred to prior in short. Quantity p(D|I)
in the denominator is a normalization constant. Upon the presence of the data, this learning process updates
the prior via the data based on the model incorporated in p(D|M, I), which leads to p(M|D, I). It becomesthe posterior (probability) of the models given the data and the above stated background information I. The
background information in this specific model learning includes a number (N) of the competing models withmodel index n .
At the model selection level, which is a higher level of inference, detailed model parameters Θ are deferred
into the background. Among the posteriors of the limited number of models MN , the Bayesian model selection
evaluates Bayes’ factor, a posterior ratio [1],Bi j =

p(Mi |D, I)
p(D|Mi , I) p(Mi |I)=.
p(M j |D, I)
p(D|M j , I) p(M j |I)(2)

The Bayes’ factor Bi j indicates how much the model Mi is preferred over model M j given the data D, while
the second ratio on the right-hand side of Eq. (2) represents the investigator’s prior preference of model Mi overmodel M j before involving the data, termed prior ratio. If the investigator has no prior preference to any models
among the model set, the prior ratios being 1 for all model pairs exactly encode this state of preference. This
represents that the model selection only relies on the first ratio on the right-hand side. It is termed Bayesianevidence ratio (evidence ratio in short).
When implementing the model selection in the machine learning, a logarithm of Bayes’ factors is often
calculated
ζi j = 10 log10 (Bi j ) = (ζi − ζ j ) + [10 log10 (Πi ) − 10 log10 (Π j )], [db],(3)with Zi = p(Mi |D, I), ζi = 10 log10 (Zi ), and Πi = p(Mi |I) to simplify the notations . When encoding ‘no preference’ to any model, the second term of Eq. (3) would become zero, so as to rely solely on the evidence ratio.
The logarithmic Bayes’ ratio is expressed in unit ‘decibans’, [1] or ‘db’ in short. In some large-scale applications, instead of directly estimating the evidence, the inverse Bayesian information criteria (IBIC) or inverse
Deviance information criteria (IDIC) will allow for more computationally feasible approximation of the evidence [1], rather computationally prohibitive. The Bayesian model selection quantitatively implements Occam’s
razor ’when two models predict the data equally well, the simpler one is preferred’ [1].
2.2 Bayesian parameter estimation
After a specific model is selected from the pre-selected model set through Bayes’ factor discussed above, theparameter estimation pursues further learning of the model parameters Θ given model MN from the data D. The
first (lower) level of inference, the parameter estimation, engages once again the Bayesian theorem to evaluate
the probability of the parameters Θp(D|Θ, MN , I) p(Θ|MN , I).
p(D|MN , I)p(Θ|D, MN , I) =(4)

Within this parameter estimation level, the background information I becomes more specific, including that‘A specific model noted as MN has been selected given the data D. This model specified now
by value N encapsulates model parameters Θ. The model then will predict the data well if theparameters are well estimated.’
Beyond the background information along with the well established model, no further information is available
at this point.
2.3 Unified learning framework
The denominator p(D|MN , I) in Eq. (4) is a normalization constant to ensure that the posterior probability ofthe parameters is integrated to unity over the entire parameter space. Integration of both sides of Eq. (4) resultsinZ

ZN = p(D|MN , I) =p(D|Θ, MN , I) p(Θ|MN , I) dΘ.

(5)Θ

Although the denominator is of less importance at the level of the Bayesian parameter estimation, it exactly
equals the marginal likelihood in Eqs. (1,2). Applying short-hand notations Π(Θ) = p(Θ|MN , I) and L (Θ) =p(D|Θ, MN , I), Equation (4) is simplified toposteriorevidence

likelihood

prior

z
}|
{
z }| { z }| {z}|{p(Θ|D, M, I) × Z = L (Θ) × Π(Θ),(6)

where the subscript of model is dropped from now on for simplicity and Bayesian evidence Z = p(D|MN , I) is
determined using Eq. (5). Yet readers should still bear in mind that the model has been selected from now on.
Equation (6) indicates that both levels of inference (model selection and parameter estimation) can be pursued within one unified framework. Furthermore, Eq. (6) also explicitly expresses the relationship among thefour quantities of Bayes’ theorem; the likelihood and the prior are two probabilistic inputs to the Bayesiananalysis while the posterior and the evidence are two outputs. More specifically, the likelihood also represents
one of priors which is the quantity to incorporate the data and the prediction model into the learning process.
On one hand, the model is part of prior knowledge. Some models are often established based on generations oflearning and education in many applications. On the other hand, the likelihood offers a mechanism for the process to be driven by the data. In order to proceed, two priors (the prior and the likelihood) need to be assigned.
The prior probabilities encode what the investigator knows a priori. At the start of the learning process, notmuch is known (otherwise there is no need to learn), therefore, the assignment based on Bayesian probabilistic
theory follows the principle of the maximum entropy [1].

Figure 1. Two-microphone method in an impedance tube for normal incidence reflection/absorption coefficientmeasurement [2]. D is diameter of the tube, λ is the acoustic wavelength used for the measurements
Furthermore, the evidence represents the key output quantity for the Bayesian model selection [?], it quantitatively implements Occam’s razor. In many applications, the Bayes’ factor in Eqs. (2,3) needs to be evaluatedamong the pre-selected model set, this evaluation predominantly relies on the Bayesian evidence of each individual model [1]. The posterior represents the key output for estimating the corresponding model parameters
given the specified model.

3

APPLICATION EXAMPLES

Acoustics has increasingly seen model-based Bayesian inference applications, some apply the first level of inference (for example in Sec. 3.1 and Sec. 3.2), while some others the both levels of inference (for example inSec. 3.3 and Sec. 3.4). This paper uses the most recent examples to demonstrate a wide variety of data-driven
machine learning applications.
3.1 Dissipation and sound speed estimation in impedance tubes
The recent work [2] can also be classified as a typical data-driven machine learning application, where theexperimental data are obtained by experimentally measuring the transfer functions between two microphone
positions inside the tube. Figure 1 schematically illustrates the tube measurement set up. The measurement data
at two microphone positions (labeled 1, and 2) are processed to yield the reflection coefficient of a ’material’
under test [3]αζ c + jω
H 12 − e−γs 2γL
e , with γ =,(7)RD = γse − H 12cwhere H 12 = P2 /P1 is the transfer function between the pressure spectrum at microphone position ’2’ and thatof position ’1’. γ represents the complex-valued propagation coefficient, αζ is the damping coefficient, ω is theangular frequency and c is the sound speed.
These data obtained using Eq. (7) drive the machine learning algorithm to accurately estimate the tube
wall dissipation, the sound speed, and optionally the microphone positions. In order to facilitate the modelbased learning, Chen et al.[3] established a prediction model for a hypothesized air layer in front of a rigid
termination of the impedance tube
(8)
R M = e−2γd .
In this data-driven machine learning application, the prediction model encapsulates model parameters Θ = [c, ζ ]in form of the surface reflection coefficient RM of the air layer in Eq. (8) with a Bayesian parameter estimation
approach, an application for the first level of inference in this study. In the context of Bayesian parameter
estimation as expressed in Eq. (4), the reflection coefficient RM represents the given model MN , while thereflection coefficient of Eq. (7) is the data D in Eq. (4). Detailed formulations of Bayesian learning and its
implementation are beyond the scope of the current paper, see a recent publication [3] for more details.

Figure 2. Flow diagram of data-driven machine learning for model-based Bayesian absorption coefficient estimation. [4].
3.2 Chamber-based absorption coefficient measurement
Chamber-based measurements of random incidence absorption coefficients have recently emerged on a debatable stage. Some challenging issues have been debated for possible solutions. The standardized measurement
technique overlooks the non-diffused sound field in the room acoustics. That may result in decreased accuracy
and even conflicting results. Zhang et al. [4] apply a diffusion equation model to room-acoustic simulationsof standardized reverberation chambers (see Fig. 2). They numerically solve the room-acoustic finite difference
simulations in time domain (Fig. 2 right). This machine learning example engages a numerical model of asimulated reverberant chamber, while the data-driven machine learning relies on experimentally measured soundenergy decays in the standardized reverberant chamber with/without the absorber of standardized size (Fig. 2left).
In case of the empty chamber measurement, the experimentally measured energy decay function drives thelearning algorithm. The learning process iteratively adjusts the absorption coefficient throughout the interior
surfaces of the simulated chamber of the same geometry/dimensions at the same source/receiver locations until
the maximum a posteriori absorption coefficient is reached. Then the machine learning algorithm switches its
focus on the absorber measurement. The energy decay data experimentally measured in the chamber drive
again the algorithm during the second adjustment of absorption coefficient only on the absorber under test in
the simulated chamber, while keeping all the other interior surfaces fixed at the previously obtained estimate.
This machine learning task represents an estimation of one-parameter value, yet it runs twice in a sequential
manner based on the diffusion equation model which is solved numerically using the finite difference approach.
The Bayesian learning (Fig. 2 middle) is accomplished with a high computational efficiency since the diffusion
equation-based modeling relies on a highly sparse meshing condition of the domain which is dictated by themean-free path length rather wavelengths. This method essentially conducts a machine learning driven by theexperimentally measured data in the reverberation chamber. Interpreted in a Bayesian view, the process learns
from the experimental data, iteratively updates an initial vague knowledge of sound absorption coefficient toarrive at sharply accurate estimates. More details can be obtained from a paper in this proceedings [4].

Figure 3. Comparison between experimentally measured data maps with those predicted by the beamforming
model. In one source scenario, the sound source finds itself at 5 cm distance from the origin of the sphere,
while for two source scenario, the second source finds itself at 5.5 cm distance [5].
3.3 Source enumeration and localization using a spherical microphone array
This application intends to localize potentially multiple co-existing sound sources in near-field around a sphericalmicrophone array. Spherical microphone data are processed to analyse incoming sound fields. Sound pressure
signals Pmic (k, a, θ i ) of M microphones flush-mounted on the rigid spherical surface of radius a are Fouriertransformed by [5]N

D(θ ) = ∑n

4π M

∑ Ynm (θ )Pnm2 , with Pnm = M ∑ Pmic (k, a, θ i )[Ynm (θ i )]∗ ,n=0 m=−n(9)i=1where θ , θ i represent (beamforming) looking directions, the microphone positions on the spherical array, respectively. Ynm (·) represents spherical harmonics and Pnm the transformed microphone signals in spherical harmonics
domain. k = ω/c is the propagation coefficient with ω and c being the angular frequency and sound speed,
respectively. In this data-driven processing D(θ ) is in two-dimensional matrix form, denoted as D when using
Eq. (6) for the two-levels of Bayesian learning. Figure 3 illustrates so achieved beamforming data for one or
two co-existing sound sources.
For the model-based Bayesian inference, the prediction models need to incorporate an unknown numberof sound sources along with localization parameters including source distances rs and directions of arrivals
θ s = [ϑ s , φ s ] over a potential looking angles θ = [ϑ , φ ] of interest,
S

N

MS (Θ) = ∑ As g2 (Θ), with g(θ s , rs ) = ∑s=1n∑ Bns (k, rs ) [Ynm (θ s )]∗Ynm (θ ),(10)

n=0 m=−nwhere Θ = [ϑS , φS , rS ] collectively includes all the localization parameters for potentially S number of soundsources, the near-field modal strength,j′ (k a)Bsn (k, r s ) = j k jn (k a) − n′hn (k a) hn (k rs ),(11)
hn (k a)Figure 4. Wavenumber spectral analysis based on a set of sound pressures experimentally measured in an
undamped room. Bayesian learning leads to 634 plane waves for this set of data. (a) Regularized learning [7].
(b) Bayesian learning [6],where jn (), hn () and jn′ (), h′n () are spherical Bessel, Hankel functions and their derivatives, respectively. Figure 3 also illustrates the predicted data using the model M2 (Θ) of Eq. (10).
3.4 Wavenumber spectral analysis in reverberant spaces
This application intends to quantify the degree of isotropy of sound fields in reverberation chambers [6]. For
this purpose, a wavenumber spectrum is analysed based on decomosition of plane waves. Experimental data D
are acquired via a robot-arm driving a microphone for measuring sound pressures at Q positions in the soundfield. Based on the plane wave expansion, the sound pressures at these measuring positions can be predicted asa large number (L) of plane waves [6]hiM(rq ) = e−jkl ·rq P̃(kl ) , with, 1 ≤ l ≤ L, 1 ≤ q ≤ Q,
(12)
where the plane wave propagation e−jkl ·rm on the right-hand side is expressed as a matrix of dimension Q × L
and the plane wave pressure amplitude P̃ is expressed as column vector of L × 1 elements. The left-hand side
contains a column vector of Q × 1 elements. M in Eq. (12) represents the prediction models M = [M1 , M2 , . . .],when two-levels of inference are involved using Eq. (6). The task is to infer complex-valued amplitudes ofthe plane waves from experimental measurements D with an array of microphones or a robot-arm holding a
microphone. Nolan et al. [6] apply the data-driven Bayesian learning using the two levels of inference to estimate the complex-valued amplitudes in the plane-wave expansion. The unified framework learns the plane wave
parameters from the data. Figure 4 illustrates comparison of machine learning results between the regularizedlearning in Fig. 4 (a) and the Bayesian inferential learning in Fig. 4 (b).
The applications discussed in Sec. 3.3 and 3.4 represent the data-driven two-levels of Bayesian learning
examples, since the higher level of inference, the model selection needs to be involved. In Sec. 3.3, the model
selection has to be first applied to a number of models MS (x) for S = 1, 2, . . . for the data to drive the estimationof how many sound sources are present before the localization parameters of the right number of sound sources
are inferred, through the lower level of inference. In Sec. 3.4, the number (L) of plane waves has to be
estimated. Without the higher level of inference, any algorithms at the lower level, including some efficient
optimizations, may lead to wrongly, at least redundantly ’optimized’ parameters. In accomplishing the twolevels of inference Eq. (6) plays a central role, explicitly indicates that the two levels of machine learning canbe carried out within the unified Bayesian framework.

4

SUMMARY

This paper reviews recent applications of Bayesian learning in acoustics investigations. A wide variety of machine learning algorithms can be put in context of Bayesian learning, particularly when well established, wellunderstood models for the prediction of the data are incorporated in the analysis. They are model-based machinelearning methods, intended to learn from data, then update the initial hypotheses (prior knowledge) to improve
the learning outcomes, whether it be the prediction models to be selected or the parameters to be estimated.
Such approaches represent the data-driven learning processes which are consistent with the core of Bayesian
probabilistic theory, learning from the data in order to update our prior knowledge.

ACKNOWLEDGEMENTS
The authors are grateful to Mr. S. Weikel and Mr. T. Metzger for their contributions to the sound source
localization using a spherical microphone array.

References[1] Xiang N, Model-based Bayesian analysis in acoustics – A tutorial, J. Acoust. Soc. Am. 2020;148:1101-20.
[2] Chen Z, Xiang N, Bayesian inference for sound speed and dissipation estimations in impedance tubes, 24th
Intl. Con. Acoust. (this proceedings), Gyeongju, Korea, Aug. 2022
[3] Chen Z, Xiang N, Fackler CJ, Bayesian estimations of dissipation, sound speed, and microphone positionsin impedance tubes, J. Acoust. Soc. Am. Express Lett.;2(8):083601-6 (DOI: 10.1121/10.0012993), 2022.
[4] Zhang J, Nolan M, Xiang N, Navarro JM, Diffusion equation-based modelling of reverberation chambers forsound absorption measurements, 24th Intl. Con. Acoust. (this proceedings), Gyeongju, Korea, Aug. 2022.
[5] Xiang N, Metzger T, Near-field localization of sound sources using a spherical microphone array, 24th Intl.
Con. Acoust. (this proceedings), Gyeongju, Korea, Aug. 2022.
[6] Nolan M, Xiang N, Bayesian wavenumber spectrum inference for sound field analysis in reverberant spaces,
24th Intl. Con. Acoust. (this proceedings), Gyeongju, Korea, Aug. 2022.
[7] Nolan M, Berzborn M, Fernandez-Grande E, Isotropy in decaying reverberant sound fields, J. Acoust. Soc.
Am.2020;148(2):1077-88,[8] Xiang N, Fackler CJ, Hou Y, Schmitt A, Bayesian design of broadband multilayered microperforated panel
absorbers, J. Acoust. Soc. Am.2022;151:3091-3103.

ABS-0731

Multi-fidelity modeling using Gaussian Processes for the Helmholtz Equation

Caglar GURBUZ(1) , Martin ESER(2) , Johannes SCHAFFNER, Steffen MARBURG(3)(1) Chair of Vibroacoustics of Vehicles and Machines, Germany, caglar.guerbuez@tum.de(2) Chair of Vibroacoustics of Vehicles and Machines, Germany, martin.eser@tum.de(3) Chair of Vibroacoustics of Vehicles and Machines, Germany, steffen.marburg@tum.de

ABSTRACTIn engineering practice, accurate physical experiments as well as large-scale numerical simulations are often
accompanied with high resource and time expenses. Therefore, the data generation process can only be performed
for a limited number of sample size which leads to insufficient model results. The idea of multi-fidelity modelingis to holistically evaluate data from different sources with varying complexity and costs. For instance, highfidelity models expose highly accurate results at high costs, whereas low-fidelity models are rather inaccurate but
fast to evaluate. In general, physical experiments and large-scale numerical simulations belong to the group of
high-fidelity models, while analytical solutions as well as numerical simulations at small scale can be considered
as low-fidelity models. For an ideally positive correlation between the low-fidelity and the high-fidelity models,
multi-fidelity models allow to obtain additional data at lower costs. The aim of this contribution is to present a
multi-fidelity model applied on a sound propagation problem in interior acoustics. Regarding the high-fidelity
model, a boundary element model with a finer mesh is considered, whereas a model with a coarser mesh is
adopted as the low-fidelity model. The multi-fidelity model is realized as a vector-valued Gaussian process.

Keywords: Multi-fidelity modeling, Gaussian Processes, Helmholtz equation, Boundary element method

1

INTRODUCTION

To assess the behavior of an acoustic system, engineers and specialists are usually faced with the challenge
to choose an appropriate model for the problem at hand. Ideally, the model should efficient to evaluate andhighly accurate. However, in practice, this is rarely the case. Models, which are usually fast to evaluate, lackof accuracy, whereas models with a high prediction capability are expensive in terms of computational costs
or time expenses. Multi-fidelity models seek to combine the advantages of both worlds meaning the efficiency
of a so-called low-fidelity model and the accuracy of a so-called high-fidelity model. In this context, lowfidelity models can be analytical solutions or numerical models at small scale. On the other hand, large-scale
numerical models as well as cumbersome physical experiments are considered as high-fidelity models. For thispurpose, Gaussian Processes (GPs) are deployed to melt the characteristics of the low- and the high-fidelity
model. GPs designate a regression method combining the observed data with a user-defined kernel function.
By this means, they provide an intuitive and at the same time data-driven approach. In addition to this, GPsalso account for the quantification of uncertainties, which can arise due to manufacturing tolerances or model
assumptions. To evaluate the frequency response of an acoustic system with the boundary element method(BEM), numerous evaluations at various frequencies are usually required. For highly accurate results, however,time and resource expenses are enormously high. This contribution aims at the development of a multi-fidelity
model for accurate predictions with a reduced the number evaluations. Therefore, GPs are adopted allowing
for an additional quantification of uncertainties. The multi-fidelity model is applied on a vehicle interior noiseproblem, for which a model with two different numbers of degrees of freedom is investigated. The modelwith a large number of DOFs is considered as the high-fidelity model, whereas the model with a low number

of DOFs is considered as the low-fidelity model. Initial results already demonstrate the effectiveness of the
proposed method, as the multi-fidelity model allows for accurate predictions with a drastically reduced numberof forward simulations.

2

METHODSIn this section, the fundamentals GPs and multi-fidelity GPs are briefly introduced. The interested reader isreferred to for further details on GPs [2]. A thorough presentation of the theory of multi-fidelity models is
detailed in [1].
2.1 Gaussian Processes
A Gaussian Process (GP) is defined to be a collection of random variables which are jointly Gaussian distributed
and indexed by an infinite index set
f (x) ∼ G P m(x), k(x, x0 ) ,(1)
where f is the GP with the mean function m and the covariance function k. These functions expect the input
variables x and x0 , which are both from the input space X. In practice, Gaussian Processes can be interpreted as
infinite dimensional joint normal distributions, which are reduced to a finite dimensional joint normal distribution
in a numerical implementation:
k11 (x, x0 ) k12 (x, x0 ) · · · k1N (x, x0 )
m1 (x1 )






 m2 (x2 )   k21 (x, x0 ) k22 (x, x0 ) · · · k2N (x, x0 ) 
, 
 ,

(2)
f∼N 
..
..
..

 ..  ..
 .  
.
.
.
.



kN1 (x, x0 ) kN2 (x, x0 ) · · · kNN (x, x0 )mN (xN )where x is a vector that collects N inputs. The vector f collects the random variables fi with i = 1, ..., N withthe mean values mi and the elements of the covariance matrix ki j .
The entries of the covariance matrix are defined by the covariance function or kernel k(x, x0 ), which describes
how the random variables fi are correlated. In this contribution, the focus is on the squared exponential kernel:


1
0 T
0
0
2
(3)
k(x, x ) = σ f exp − 2 (x − x ) (x − x ) .
2lThis kernel implements the expectation that the closer together two inputs are in the input space, the more
the maps should correlate [2]. The width of the kernel and the distance in the input space is defined by thecharacteristic length l. The amplitude of the kernel is given as σ 2f . These variables are called the hyperparametersof the kernel. Note that the kernel does not depend on the output of any measured data. In this way, priorbeliefs on the unknown function can be expressed without introducing any measured data. The prior distributionof the measured outputs (training points) f can be then written asf ∼ N 0, k(x, x0 ) .
(4)
In this initial study, the mean is set to be zero. This prior can be extended to express the joint distribution off and the predicted outputs f∗ , where the training inputs are collected in an array x and the test inputs in x∗ .
These data arrays can be then used to express the joint distribution of f and f∗ :   
f0k(x, x) k(x, x∗ )  ∼ N  , 
 .
(5)
f∗
0
k(x∗ , x) k(x∗ , x∗ )By conditioning the GP on the observed data, the posterior probability distribution can be defined asp(f∗ |x∗ , x, f) = N (k(x∗ , x)T k(x, x)−1 f, k(x∗ , x∗ ) − k(x∗ , x)T k(x, x)−1 k(x, x∗ )).

(6)

Typically, the training data is not noiseless. In this case, the observations are expressed by y = f (x) + ε, whereε ∼ N (0, σn2 I). The joint distribution of y and f∗ is then   
y0k(x, x) + σn2 I k(x, x∗ )  ∼ N  , 
 ,
(7)
f∗
0
k(x∗ , x)k(x∗ , x∗ )and the posterior becomes:p( f∗ |x∗ , x, y) = N (k(x∗ , x)T (k(x, x) + σn2 I)−1 y, k(x∗ , x∗ ) − k(x∗ , x)T (k(x, x) + σn2 I)−1 k(x, x∗ )).

(8)

To find the optimal hyperparameters that define the kernel, the negative log marginal likelihood function:1
1n
log p(y|x) = − yT (K + σn2 I))y − log |K + σn2 I| − log 2π
2
2
2

(9)

is minimized. The negative log marginal likelihood indicates how likely the training data is for the given
hyperparameters. This optimization process determines the training process of the GP [2].
2.2 Multi-fidelity using Gaussian Processes
The multi-fidelity scheme in this contribution is closely related to the model in [3]. In this multi-fidelity model,
two different fidelity levels are in general related withfh (x) = ρ fl (x) + δ (x),(10)

such that the searched for high fidelity function consists in part of the low fidelity solution. How large thatpart is, is denoted by ρ. If ρ would be zero, the correlation between low- and high-fidelity data would be
zero. With this assumption the prior Gaussian Process for the combined low and high fidelity function can be
expressed as:   0kl (xl )ρkl (xl , xh )fl
  ∼ N  , 

(11)
fh
0
ρkl (xh , xl ) ρ 2 kl (xh ) + kδ (xh )For the sake of brevity, the detailed presentation of the formulation and its adaptation to noisy data is omitted
here.

3

NUMERICAL EXAMPLE

The problem at hand is closely related to the numerical example in [4, 5], where the acoustic behavior ofa vehicle cabin was investigated. Therefore, the three-dimensional Helmholtz equation for interior problems
is solved with the boundary element method. The cabin mesh is depicted in Fig. 1a. This mesh has 24036
DOFs and is considered as the high-fidelity model. The same model with a coarse resolution of 4420 DOFs
is considered as the low-fidelity model. The multi-fidelity GP for the energy density at the driver’s positionis shown in Fig. 1b. In this study, we choose 8 high-fidelity points and 181 low-fidelity points. It becomesapparent that the low-fidelity model exposes higher energy density levels than the high-fidelity model. Interestingly,
one can observe a good agreement between the multi-fidelity GP approximation and the ground truth result,which is the evaluation of the high-fidelity model at each frequency. This initial result already shows thatmulti-fidelity GPs can be used as an approximation for the high-fidelity model. In comparison with the reference
solution, the proposed method requires only 8 forward simulations, whereas 181 simulations were required forthe reference solution. In this way, the proposed method provides an efficient technique to approximate thebehavior of an acoustic system. In addition to this, uncertainties within the system are inherently quantified.

45

J/N

40
35

predicted LF
LF training points
selection strategy
Groundtruth
predicted HF
HF training points

3025(a) Cabin mesh

50

75

100

Hz

125

150

175

200

(b) 181 LF and 9 HF data points

Figure 1. The cabin mesh with 24036 DOFs (a) and the multi-fidelity model for 181 low-fidelity (black crosses)
and 9 high-fidelity data points (red crosses). The approximation of the multi-fidelity model (orange) is comparedto the reference solution (blue). The uncertainty area refers to the 95% confidence interval (light orange).

4

CONCLUSIONSIn this contribution, a method was introduced for the approximation of tedious data generation processes suchas industrial numerical simulations. Therefore, we implemented a multi-fidelity model combining the prediction
capabilities of a high-fidelity and a low-fidelity model. This model is based on Gaussian Processes, which
provide an intuitive and at the same time data-driven technique. Due to their probabilistic nature, the quantificationof uncertainties is inherent to Gaussian Processes. The proposed method was investigated for a boundary
element method setup, where a cabin mesh with two different degrees of refinement was examined. The initial
findings already show that the acoustic behavior of the cabin can be efficiently approximated by using multifidelity GPs. Future work will focus on further industrial applications and on the influence of noisy observations.

ACKNOWLEDGEMENTS
This research was funded by the German Research Foundation (DFG) under award 418936727.

REFERENCES[1] Kennedy, MC.; O’Hagan, A. Predicting the output from a complex computer code when fast approximations
are available. Biometrika 87.1, 2000. p. 1-13.
[2] Williams, CKI.; Rasmussen CE. Gaussian Processes for machine learning, MIT press, Cambridge (MA),
vol.2, 2006.
[3] Guo, M.; Manzoni, A.; Amendt, M.; Conti, P.; Hesthaven, JS. Multi-fidelity regression using artificial
neural networks: efficient approximation of parameter-dependent output quantities. Comput Methods Appl
Mech Eng. 389:114378, 2022.
[4] Gurbuz, C.; Eser, M.; Schmid, JD.; Marburg, S.; Luegmair, M. Gaussian Processes for Transfer Path
Analysis Applied on Vehicle Body Vibration Problems. SAE Technical Paper. 2022
[5] Gurbuz, C.; Schmid, JD.; Luegmair, M.; Marburg, S. Energy density-based non-negative surface
contributions in interior acoustics. J Sound Vib. 527:116824. 2022.

ABS-0926

Deterministic and statistical reconstruction of rough surfaces from acoustic
scattering
Jacques CUENCA(1) , Timo LÄHIVAARA(2) , Giulio DOLCETTI(3,4) , Mansour ALKMIM(1,6) , Laurent DE RYCK(1) ,Michael-David JOHNSON(5) , Anton KRYNKIN(5)(1) Siemens Industry Software NV, Leuven, Belgium, jacques.cuenca@siemens.com(2) Department of Applied Physics, University of Eastern Finland, Kuopio, Finland(3) Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento, Italy(4) Department of Civil and Structural Engineering, the University of Sheffield, Sheffield, United Kingdom(5) Department of Mechanical Engineering, the University of Sheffield, Sheffield, United Kingdom(6) Department of Mechanical Engineering, KU Leuven, Leuven, Belgium

ABSTRACTThis work proposes a model inversion framework for the reconstruction of the elevation profile of a rough surface
from acoustic scattering. The latter is measured at a microphone array while the surface is insonified with abroadband source. The forward model makes use of the Kirchhoff approximation, which provides reasonableaccuracy at high frequencies or in the far field, while enabling an efficient exploration of the design space. A
particular challenge in the inversion is the detailed reconstruction of an arbitrary surface of relatively large size
from a limited set of microphones, leading to an under-determined problem. An initial estimate of the surface
roughness profile is obtained in a deterministic manner by solving the inverse problem using a gradient-based
optimiser. The inverse problem is then formulated in a statistical sense and solved using Bayesian inference. The
sequential use of deterministic and statistical inversion yields a feasible solution and an uncertainty envelope.
The procedure is demonstrated for a surface with a one-dimensional roughness profile.

Keywords: Inverse scattering, surface reconstruction, optimisation, Bayesian inference

1

MOTIVATION

Non-contact reconstruction of the elevation profile of rough surfaces is of crucial interest in applications such as sea
floor mapping, river and sea surface monitoring [1]. Recent work demonstrates the use of a sound source and a linear
array of microphones for the reconstruction of static rigid rough surfaces using inverse scattering. The techniques
used therein include the linearisation of the propagation problem allowing for its inversion in matrix form [1] as wellas the use of a random forest algorithm [2] or numerical optimisation [3]. While the current techniques allow forrobust and efficient reconstructions, the efforts have thus far focused on obtaining deterministic point estimates.
The present work proposes a methodology for reconstructing the elevation profile of a rough surface, together withan uncertainty estimate. In order to do so, the inverse scattering problem is formulated in a statistical sense using a
Bayesian framework, allowing to sample the desired posterior distribution using a Markov chain Monte Carlo method.
In practice, the number of unknown elevation points on the surface is larger than the available number of microphones,
thus yielding an under-determined problem. An assumption of surface smoothness is introduced, which mitigates theill-posedness of the problem without restricting the scope of the method. In addition, a deterministic estimate is used
as an initial guess for the statistical inversion procedure. This brief report summarises the methodology and provides
an experimental application example.

2

MODEL
z

r0

rs

0

x

L

0
Figure 1. Scattering problem.

Rough surface; source; microphones; a point on the surface.

The purpose of the work is to obtain an estimate of the elevation profile ζ of a rough surface from scattered sound.
Fig. 1 illustrates the experimental setup used, consisting of a sound source and an array of microphones above thesurface of interest. The present study is restricted to surfaces whose roughness varies along one dimension, namelyζ = ζ (x). The acoustic field is modelled using an integral formulation and approximated by means of a stationary
phase expansion [4]. Furthermore, by using a directive loudspeaker oriented towards the centre of the surface, thedirect field can be ignored at the microphones. The acoustic pressure at a microphone position r is then obtained asriπ/4p(r) ' e2πks

Z xs =Lxs =0

A(rs )|rs − r0 ||r − rs |G(r, rs )G(rs , r0 )|rs − r0 | + |r − rs |





1
1
cos(|r − rs |, ns ) − ik −cos(|rs − r0 |, ns ) dxs , (1)ik −|r − rs ||rs − r0 |where k is the wavenumber in air, A is the directivity of the source, G is the Green function in free field, r0 is thesource position, rs = (xs , ζ ) is a point along the surface, and ns is the outgoing normal to the surface at rs . In practice,
the surface is discretised along x, yielding a finite number of unknown elevation points, gathered in the vectorhz = ζ1· · · ζs

· · · ζN

iT

.

(2)

The surface is assumed to be smooth, and the smoothness is modelled as a parameter β ∈ [0, 1], implemented as a lowpass Butterworth filter [5] and considered as an additional unknown in the problem. The introduction of a smoothness
parameter renders the surface elevation points interdependent and thus has a regularising effect. It is worth notingthat the limiting case β = 0 corresponds to uncorrelated elevation points and therefore the introduction of the extra
unknown does not restrict the generality of the problem.

3

INVERSE METHOD

The acoustic pressure is acquired at the microphones across a discrete range of frequencies. By gathering all themeasured data in a vector p, an observation model can be proposed, asp(meas) = p(z) + e,(3)

where e is the modelling and measurement error that separates the model from the ground truth. This error is here
assumed Gaussian, with a diagonal covariance matrix Γε = σε2 I, with σε the error standard deviation.
The first strategy for solving the inverse problem of estimating z from the measured pressure data p(meas) is tominimise the error e in a least-squares sense. In the present work this is done by means of a numerical optimiser using
the globally convergent method of moving asymptotes [6].

The second strategy consists in formulating the inverse problem in a statistical sense using a Bayesian framework.
Here, the standard deviation of the error σε can be estimated as well by considering it as a random variable itself.
Using Bayes’ formula [7], the posterior probability of the unknowns z, β and σε can be expressed as
P(z, σε , β |p(meas) ) ∝ P(p(meas) |z, σε , β )P(z, σε , β ),(4)where P(p(meas) |z, σε , β ) is the likelihood of the measurement given the unknowns, and P(z, σε , β ) is the prior knowledge on the unknowns. Owing to the properties of the error, the likelihood is Gaussian. Furthermore, uniform priors
are assumed for all unknowns in the present study. In order to reduce the computational burden of the statisticalsampling procedure, the solution obtained by deterministic inversion is used as a starting point. The statistical inversion procedure is carried out using an adaptive Metropolis scheme [8], from which the maximum a posteriori (MAP)
estimate as well as 95% credible intervals are extracted.

4

RESULTS

The proposed method is here applied to a medium density fibreboard surface of length L = 60 cm, milled with acontrolled roughness profile. The setup consisted of a directive Visaton G 25 FFL dome tweeter as the source and 34
linearly spaced 1/4" G.R.A.S. microphones. The experimental data for the present inversion consists of 21 frequency
lines between 16 kHz and 21 kHz. The surface is discretised into N = 150 points, yielding a total of 151 and 152
unknowns for the deterministic and statistical procedures, respectively. The sampling procedure consists of 100000
usable samples, with a further 20000 discarded as burn-in. Figure 2 shows the elevation profile of the surface as wellas the deterministic and statistical estimates. Figure 3 depicts the probability distributions of the additional unknowns,
namely the standard deviation of the error, σε , and the surface smoothness, β .

z (mm)

4
2
0
-2
-4
0

0.1

0.2

0.3

0.4

0.5

0.6

x (m)
Figure 2. Estimated surface elevation profile. — True profile; —◦ deterministic estimate; —• MAP estimate;credible intervals.
1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0
0.454

0
0.456

0.458

0.46

0.462

0.464

0

0.2

0.4

0.6

σε (-)

β (-)

(a)

(b)

0.8

1

Figure 3. Probability distribution of the additional unknowns (a) σε and (b) β .

95%

5

DISCUSSION

It can be observed that the elevation profile of the surface of interest is accurately retrieved. Overall, the 95% credibleinterval is 1.69 mm and the 75% credible interval is 1.01 mm. It is worth noting that the precision of the reconstruction
is higher in the central region of the specimen. This is due to the finite aperture of the microphone array as well as
to the directivity of the source. A further notable outcome is the existence of a single maximum in the probability
distributions of the additional unknowns. The results here presented show that the proposed methodology stands as a
viable technique for the estimation of surface roughness. In particular, the statistical inversion procedure can be usedin conjunction with robust deterministic tools as a means to gain an insight on the estimation uncertainty.

REFERENCES[1] G. Dolcetti, M. Alkmim, J. Cuenca, L. De Ryck, and A. Krynkin, “Robust reconstruction of scattering surfaces
using a linear microphone array,” Journal of Sound and Vibration, vol. 494, p. 115902, 2021.
[2] M.-D. Johnson, A. Krynkin, G. Dolcetti, M. Alkmim, J. Cuenca, and L. De Ryck, “Surface shape reconstruction
from phaseless scattered acoustic data using a random forest algorithm,” Journal of the Acoustical Society ofAmerica, vol. 152, no. 2, pp. 1045–1057, 2022.
[3] Y. Li, J. Cuenca, L. De Ryck, M. Alkmim, O. Atak, W. Desmet, G. Dolcetti, and A. Krynkin, “Robust reconstruction of scattering surfaces using a linear microphone array,” Submitted to Journal of Sound and Vibration,
2022.
[4] C. M. Bender and S. A. Orszag, Advanced mathematical methods for scientists and engineers I: Asymptotic
methods and perturbation theory. Springer Science & Business Media, 2013.
[5] A. V. Oppenheim and R. W. Schafer, Discrete-time signal processing. Pearson, 3rd ed., 2010.
[6] K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable
approximations,” SIAM Journal on Optimization, vol. 12, no. 2, p. 555, 2002.
[7] J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems. Springer-Verlag, 2005.
[8] H. Haario, E. Saksman, and J. Tamminen, “An adaptive Metropolis algorithm,” Bernoulli, vol. 7, pp. 223–242,
2001.

ABS-0932

Physics-informed deep learning in computational elastodynamics
Johannes D. SCHMID* , Philipp BAUERSCHMIDT, Caglar GURBUZ and Steffen MARBURG (1)(1) Technical University of Munich, Chair of Vibroacoustics of Vehicles and Machines, Germany

ABSTRACT
Conventional numerical methods in computational elastodynamics are primarily based on discretization likethe finite element method. However, it is challenging to incorporate observed data into the model, resulting
in a recurring loop of updating the input parameters until the model fits measured data. Purely data-driven
methods, on the other hand, may fit observations well, but predictions outside the labeled training data are often
physically inconsistent. Physics-informed neural networks (PINNs) combine the advantages of both approaches
and in this contribution, PINNs are used to solve the equations of elastodynamics in a data-driven manner. It is
the idea to incorporate the residual of the governing partial differential equation into the loss function of a neuralnetwork. The training process of the neural network minimizes the residual of the differential equation and thediscrepancy of the boundary conditions by only considering training data on the boundary. The applicabilityof PINNs is shown on an elastodynamic benchmark model and its performance is compared to an ordinaryneural network and a finite element reference solution. The additional physical prior knowledge allows for good
generalization of the displacement field inside the domain and provides a robust machine learning method evenin the small training data regime.

Keywords: Machine learning, data-driven methods, elastodynamics, structural dynamics

1

INTRODUCTION

The numerical methods in computational elastodynamics and vibroacoustics are dominated from purely physicsbased models, which mainly rely on discretization like the finite element method (FEM) [1]. Those methods arewell established, but it is the drawback that the computational costs considerably increase with finer discretization, which makes tasks with a high number of model evaluations, e.g. uncertainty quantification, optimizationand inverse problems very hard to solve [2, 3]. Furthermore, with deterministic physics-based methods it isvery challenging to incorporate measured and potentially noisy data into the models. This makes it very hard to
identify model parameters based on observed data and it results in an ever recurring loop of updating the input
parameters until the simulation model fits experimental data. In contrast, a constantly increasing amount of data
is being collected from simulation and experiment and the computational power of the hardware continuously
increases, which strengthens the need for data-driven methods.
In the past decades, machine learning (ML) methods and especially the methods based on artificial neural networks, called deep learning (DL), have shown great potential in the fields of computer vision, speech andimage recognition and natural language processing. Though, purely data-driven methods may fit observations
from the training data well, they often suffer from poor generalization, meaning that they fail in making predictions outside the labeled training data. Specifically in the field of computational physics, the main drawbackof purely data-driven methods is that the predictions may be physically inconsistent and the model parameters
are not physically interpretable. Raissi et al. [4] introduced physics-informed neural networks (PINNs), which
combine the advantages from both, physics-based methods and data-driven methods. The main idea is to incorporate physics information in the form of a differential equation into the loss function of an artificial neuralnetwork. This is done by incorporating the residual of the governing partial differential equation (PDE) into
* johannes.d.schmid@tum.dethe loss function of a neural network. By training the neural network, the loss function is minimized, whichmeans that the PDE residual will be minimized simultaneously. This way, the PINN learns a solution, which
implicitly satisfied the PDE and the boundary conditions. Karniadakis et al. [5] give a good review over the
method and practical applications in the fields of fluid flow and thermodynamics. PINNs have been applied indifferent field of computational physics including fluid dynamics [6, 7], acoustics [8], heat transfer [9] and photonics [10]. In elastodynamics, PINNs have been used to solve selected forward problems, including Kirchhoff
plate bending [11], elastic plate and bar problems [12, 13], hyperelasticity and plasticity [14] and continuum
micromechanics [15]. Furthermore, PINNs have been applied to the solution of inverse problems for parameter
identification [16] and fault detection [17].
In this contribution, PINNs are used to solve the three-dimensional elastodynamics equation in the frequency
domain applied on the so called Cook’s Membrane, a benchmark model in solid mechanics [18]. The displacement field within the structure is solved using PINNs by only considering the PDE and training data on theboundary of the domain. By incorporating the additional physics prior information, PINNs are able to generalize better compared to purely supervised neural networks, even it only a limited number of training data on theboundary is available.

2

ELASTODYNAMICS MODEL

The model chosen in this investigations is a benchmark model in solid mechanics, called Cook’s Membrane [19].
The special geometry and load configuration results in a combined bending and shear load [18]. The geometryof the Cook’s membrane is shown in Fig. 1. It consists of a three-dimensional trapezoid cantilever, where theleft side of the structure (in blue) is clamped and the right side (in red) is loaded with a harmonic shear loadof uz = 0.001 m at a frequency of f = 2180 Hz in vertical direction. All other boundaries are considered to be
traction free.

Figure 1. Cook’s Membrane benchmark model in elastodynamics with clamped boundary condition on the left (in blue) andharmonic vertical shear load on the right (in red). The green surface (CS1) represents a cross-section at x = 0.34 m.

The equations to be solved are the elastodynamics equations in three dimensions, known as the Navier-Laméequations, in the frequency domain. The boundary conditions (BCs) are Dirichlet conditions on the boundary Γ.
The boundary value problem can consequently be formulated, asω 2 ρ u + µ ∆u + (λ + µ)∇(∇ · u) = 0

in Ωu = ū

on Γ,(1)with the displacement vector field u, the angular frequency ω = 2π f , the Lamé parameters µ and λ andthe material density ρ. The structure is considered to be made out of steel with the material properties ρ =7850 kg m−3 , λ = 10.2 × 1010 N m−2 and µ = 8 × 1010 N m−2 . It should be noted that the traction free Neumann
BCs are transformed into a Dirichlet problem, by computing the displacement field via FEM and using thesolution as training data for the PINN. The FEM reference solution is computed using COMSOL Multiphysics.

3

PHYSICS-INFORMED NEURAL NETWORKS

PINNs are typically built upon a feed-forward neural network, also known as multilayer perceptron and the only
difference is the loss function to be minimized during the training process. The loss function of a PINN contains at least one term, which incorporates physics information about the system. The standard way of including
the physical knowledge is to add the residual of the PDE and the mean square error (MSE) of the boundary
conditions to the loss function. The structure of a PINN is schematically illustrated in Fig. 2, starting withthe spatial input parameters of a multilayer perceptron on the left-hand side. On the right, the elastodynamics
equation indicates the physics information, formulated as a residual, which is evaluated by computing the relevant spatial derivatives. The advantage of having a neural network model is that the gradients can be computed
easily using backpropagation and automatic differentiation. The residuals of both, the PDE and the boundary
conditions are combined into the loss function, which is minimized during training. The PDE residual R is
evaluated at a certain number of collocation points xcp , which are randomly sampled within the computational
domain Ω.

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𝑁𝑜

ℒ<𝜀
𝐷𝑜𝑛𝑒

𝑌𝑒𝑠

ℒ = ℒΩ + 𝛾Γ ℒΓ
𝐿𝑜𝑠𝑠 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛

Figure 2. Schematic illustration of a PINN with a multilayer perceptron (left) and the physics information in the form of theelastodynamics equations (right). The residuals of both, the PDE and the boundary conditions are incorporated into the loss
function, which is minimized during training.

Johannes Schmid (TUM)

The PDE loss can be written as the MSE residual over all collocation points, as
N

LΩ =

1 cp
∑ ∥ρω 2 û(xcp,i ) + µ ∆û(xcp,i )) + (λ + µ) ∇(∇ · û(xcp,i ))∥2 ,Ncp i=1(2)

where û stands for the PINN approximation of the displacement vector u. As the PDE has an infinite numberof solutions, but we are only interested in the unique solution that satisfies the boundary value problem, we
have to also include the BCs into the loss function. The Dirichlet BC loss is written as the MSE between the
networks predictions û and the prescribed Dirichlet displacements ū at the boundary and results in
LΓ =

1 Nbc
∑ ∥û(xbc,i ) − ūi ∥2 .
Nbc i=1(3)

Both loss terms are combined to the total PINN loss function, as
L = LΩ + λΓ LΓ ,(4)

where λΓ represents a scaling factor for adjusting the proportion of the BC loss term compared to the PDE
loss. Our investigations have shown that it is very important for the convergence towards the correct solutionthat both loss terms are well balanced and within the same order of magnitude. During training, the weights
and biases θ of the neural network are adjusted in the way that the loss function L is minimized. This is doneusing gradient based optimization algorithms and the minimization problem of training the network can finallybe expressed asθ̂ = arg min L (D , θ),(5)θ

where θ̂ is the converged set of neural network parameters and D is the set of training data.

4

NETWORK ARCHITECTURE

PINNs are used to predict the displacement field in Ω by only utilizing deformation data on 1000 uniformly
distributed training data points xbc on the boundary. The PDE residual is evaluated at 10,000 randomly sampled
collocation points xcp within the computational domain Ω. The neural network consists of an input layer withthree neurons, representing the spatial coordinates x, y and z. The output layer has three neurons, associatedto the displacement field in the three spatial dimensions ûx , ûy and ûz . The neural network is schematicallyillustrated on the left-hand side of Fig. 2. The neural network is built using PyTorch [20] and the optimal
network architecture was found by performing a hyperparameter optimization using Ray Tune [21]. A gridsearch method was used and the search space as well as the optimal choice of hyperparameters are summarizedin Tab. 1.
Table 1. Search space for the grid search hyperparameter optimization and optimized hyperparameters.
Hyperparameter

Grid search space

Optimized parameter

Number of layers{8, 12, 16}

12

Neurons per layer
Learning rate

{10, 15, 20}

20{0.1, 0.01, 0.004, 0.001}

0.004

Learning rate decay{0.3, 0.5}

0.5

Epochs for decay{200, 800}

800{(50, 500), (200, 2000)}(50, 500)

Batch size

The activation functions are chosen to be hyperbolic tangents and all models are trained using the Adam optimization algorithm for a total number of 9000 epochs. During training, the learning rate is reduced by thefactor learning rate decay every time, the number of epochs for decay in Tab. 1 is reached. The batch size listed
in Tab. 1 contains two values with the first one representing the batch size for the boundary conditions and thesecond one the batch size for the collocation points.

5

RESULTS

The displacement field of the Cook’s membrane is computed by solving the Navier-Lamé equations using
PINNs. Fig. 3 shows the displacement field on the cross section CS1 at x = 0.34 m, which is marked in green
in Fig. 1. The PINN prediction is compared to the FEM reference solution and a deviation plot indicates theabsolute prediction error.
FEM reference

Deviation FEM - PINNPINN prediction

Figure 3. Magnitude of the displacement field vector u at the cross section CS1 of the Cook’s membrane. The FEM reference
solution (top) is compared to the PINN prediction (bottom) and the deviation plot (right) shows the spatial absolute error.

The PINN method is able to predict the displacement magnitude with an mean absolute error of 7.78 × 10−6 m
within the computational domain. This corresponds to an average percentage error of 1.4 %, which is considered
to be a very accurate approximation. For comparison, also an ordinary neural network with the exact same
network architecture and hyperparameters was trained on the boundary displacement data. This corresponds
to a purely supervised training of the boundary displacements xbc , without any physics information. The loss
function in this case only consists of a single MSE term, as the physical loss LΩ is neglected. It turns out thatthe mean absolute error of the supervised neural network compared to the FEM solution is 27 %, which is about
an order of magnitude worse than the PINN and considered to be insufficient in this case. This proves that apurely supervised neural network fails in capturing the elastodynamic field and that the physics information
within PINNs is essential for a correct and physically consistent prediction of the displacement field in thiselastodynamics problem. The computational times for training the PINN in the offline step takes about 3 h,compared to one FEM solution of 3 s. However, once the PINN is trained once, one forward pass will only
take about 0.001 s.

6

CONCLUSIONS AND OUTLOOKThe method of physics-informed neural networks (PINNs) have been used to solve the Navier-Lamé equationsof elastodynamics in the frequency domain on the three-dimensional benchmark case of the Cook’s membrane.
A Dirichlet boundary value problem was considered with displacement training data on the boundary of thestructure. The studies show that PINNs are able to solve the PDE with a very high accuracy of less than one
percent in comparison to a FEM reference solution. Even if only training data on the boundary were used, thePINN is able to generalize the displacement solution inside the domain very well. The comparison to a neuralnetwork with purely supervised training shows that the physics information highly improves the prediction not
only in terms of accuracy but also in terms of physical interpretability.
As an outlook, PINNs show a high potential for data-driven but yet physics-informed surrogate modeling. As
the forward evaluation of a pre-trained PINN is very efficient, it could be highly advantageous for applications
with many model evaluations, e.g. optimization, uncertainty quantification or the sampling-based solution of
inverse problems. In conclusion, the authors would like to emphasize that PINNs may not replace traditional
solvers, but instead they offer a promising approach to seamlessly integrate simulated and experimental data
within one consistent framework.

ACKNOWLEDGEMENTS
This contribution is part of the joint project newAIDE under the consortium leadership of BMW AG with thepartners Altair Engineering GmbH, divis intelligent solutions GmbH, MSC Software GmbH and TWT GmbH.
The project is supported by the Federal Ministry for Economic Affairs and Climate Action (BMWK) on thebasis of a decision by the German Bundestag.

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ABS-0948

Design of Acoustic Metamaterials Using Conditional Wasserstein Generative
Adversarial Networks
Feruza A. Amirkulova(1) , Peter Lai(2) , Diego Soto(3)(1) Mechanical Engineering Department, San José State University, San José, 95192, USA, feruza.amirkulova@sjsu.edu(2) Mechanical Engineering Department, San José State University, San José, 95192, USA, peter.lai@sjsu.edu(3) Mechanical Engineering Department, San José State University, San José, 95192, USA, diego.sotolopez@sjsu.edu

ABSTRACTIn this talk, we present a method for the reduction of the total scattering cross section (TSCS) for a planar
configuration of cylinders through multi-scattering, generative modeling and deep learning. We first summarize
and illustrate existing approaches in the inverse design of metamaterials then discuss current algorithmic limitations and open challenges to preview possible future developments in metamaterial design. We then show anovel method [1, 2] to design broadband acoustic metamaterial configurations using deep learning and generative modeling targeting low TSCS responses. We combine a conditional Wasserstein Generative Adversarial
Network (cWGAN) with a standard convolutional neural networks (CNN) to generate metacluster configurations with a specified TSCS. The proposed cWGAN model is enhanced by adding to it a coordinate convolution
layer [3]. Particularly placed scattering elements can achieve minimal TSCS producing a broadband cloaking effect at discrete sets of wavenumbers. The method is demonstrated giving examples for planar uniform
configurations of rigid cylinders.

Keywords: Acoustic Metamaterials, Deep Learning, Generative Modeling, Wasserstein Generative Adversarial Networks, Convolutional Neural Networks

1

INTRODUCTION

Metamaterials [4–7] are engineered materials with unique properties not found in nature such as negative refraction, negative mass density, and negative Poisson’s ratio, producing exotic effects such as cloaking, superfocusing, beam forming, wave steering, and absorption. Even though the traditional optimization techniquessuch as gradient based optimization, shape optimization, topology optimization and stochastic algorithms, andcomputational methods in direct metamaterial design have shown great achievements, such approaches are inefficient in inverse design of complex structures. To the contrary, the potential that machine learning offers
for the discovery of new materials, devices and functionalities, is drawing increasing attention from a growing
community of researchers worldwide [8–15]. The generative networks such as generative adversarial networks(GAN) [16–18] and variational autoencoder (VAE) [19] have had a huge success since they were introduced in
2014. Generative modeling have been applied in inverse design of material microstructure [20, 21] and metamaterials [22–31], and in solving inverse scattering problems [32, 33]. The generative networks have ability
to generate better optimized complex devices and structures [34–37] enabling an automated design process ofmetamaterials and metadevices.
In this talk, we will present a novel data-driven method [1, 2] based on generative networks to inversely
design broadband acoustic metamaterials. To illustrate the approach, specifically, we will design acoustic cloaks
consisting of uniform planar configurations of cylindrical structure by means of multiple scattering theory [38]implementing cWGAN model that targets low TSCS responses. We will present the model performance ofproposed cWGAN model for two- and four-scatterer configurations.

2

METHOD

2.1 Problem formulation
We consider acoustic multiple scattering by an arbitrary planar configuration of M cylindrical scatterers in thecontext of the acoustic time harmonic wave equation [39, 40]. For simplicity of implementation, we consider
rigid cylinders of constant radii a = 1m situated in water within the circular region of radius Rout = 15m asshown in Figure 1. As a measure of scattering we consider the total scattering cross section (TSCS), denotedby σ that can be written in terms of the forward scattering amplitude via optical theorem [1, 12, 39]: σ =−2 Re f (0). where f (0) = f (θ = 0) is the forward scattering amplitude. Here, the far-field amplitude form
function, f (θ ) = f (θ , r1 , . . . , rM ), θ = arg(x), is defined by the scattered pressure psc in the far-field, [39, 40]
where r1 , . . . , rM are the position vectors and M is the total number of cylinders.
2.2 Data collection

Figure 1. Physical scatterer configuration (left) and the corresponding scaled pixel images (right) that are scaled
to 64x64 pixel resolution size with scattering elements (yellow) and exterior fluid medium (dark magenta).
We evaluate the TSCS, σ , in MATLAB using our multiple scattering solver [40, 41] for each configuration at
discrete values of normalized wavenumber ki a ∈ [0.35, 0.45], i = 1, . . . , 11 with an interval of 0.01, and reserve
them as labels for the semi-supervised learning process. To implement cWGAN and CNN models we generate
two types of datasets, one with random configurations of scatterers and one with optimal configurations obtainedin MATLAB using MultiStart from Global optimization toolbox with f mincon nonlinear programming solver.
We generate data-sets consisting of 60, 000 samples for random two-scatterer and four-scatterer uniform configurations separately and compute their corresponding TSCS values. We also generate two datasets of optimal
configurations of M = 2 and M = 4 cylinders that produce near zero values of σ (ka) at 11 values of ka. Each
optimal dataset consist of 105 samples. For two-scatterer configurations and 105 data samples, only 7 uniqueoptimal TSCS solutions are observed. It is difficult to optimize the TSCS at low frequencies in general, and,especially using only the interaction between two rigid scatterers. However, considering four-scatterer configurations, given the larger variety of four-scatterer configurations, the optimal dataset in not limited to only a few
unique solutions.
In this work, we design planar metamaterial structures. Therefore, we use the scatterer coordinates to generate a planar configuration image and convert it into a 64x64 pixel resolution image array. Figure 1 illustrates
the physical scatterer configuration (left) and the corresponding scaled pixel images (right). The incident normal
plane wave is propagating in the positive x direction indicated by blue arrows in Figure 1 in the physical configuration image. The image is scaled to 64x64 pixel resolution size with scattering elements denoted by yellow
color and dark magenta color stands for exterior fluid medium. Each scatterer in the physical space is alteredfrom its cylindrical shape to fit within the discrete pixel space. To further improve training and increase thenumber of samples in the dataset, both the original image and the flipped Y-axis image shown in Figure 1 canbe used as they give the same TSCS but this useful information has not been used in cWGAN model training
and will be considered elsewhere.

2.3 Training with generative adversarial networks
In this work, we build and train cWGAN model to propose images on planar configuration of scatterers given
expected TSCS value. We enhance the cWGAN architecture and its performance by adding to it a coordinate
convolution (CoordConv) layer [3]. As illustrated Figure 2, the proposed cWGAN model comprises of a CoordConv implemented generator and critic, and the externally trained regressor that has CNN architecture [29].
A Gaussian distributed noise vector of size 256 is concatenated with the desired TSCS label and is fed into thegenerator. The generator produces a fake image and passes it to both the critic and the regressor. The critic
computes the Wasserstein distance between the real image and the generated image and determines weather
the image is real or fake. The regressor predicts the TSCS value from the generated image and the MSE between predicted and desired TSCS is evaluated. Both Wasserstein loss and MSE loss is backpropagated throughthe network to train the generator. The full description of the proposed cWGAN model including the loss
functions and network architecture for each component of cWGAN, and the details of its training process andhyper-parameters found are provided in [1].

Figure 2. Schematic of cWGAN model [1] which comprises of a CoordConv implemented generator and critic,
as well as the externally trained regressor.

3

NUMERICAL RESULTS

3.1 The optimal results generated by cWGAN model
We train the cWGAN model using datasets described in Section 2.2 and after training the cWGAN, we investigate the performance of the model for two- and four-scatterer configurations by generating an image andverifying the values of σ , TSCS function within and outside the optimized range of wavenumbers. Once generative networks are properly trained, they can achieve the optimal performance beyond the parameter givenin the training dataset. The presented cWGAN model can produce optimal configurations that bring about nearzero TSCS values. The cWGAN model, previously trained on random configurations of 2-scatterer or 4-scatterer
configurations, is tested with the optimal configuration dataset. The f mincon minimized σ labels are concatenated to a noise vector and passed to the generator to produce new images. The generator repeatedly generates
scatterer configurations and images with a regressor predicted σ MAE of 0.07 and 0.15, for two-scatterer andfour-scatterer configurations respectively are collected. Out of 7 unique optimal TSCS solutions for two-scatterer
configurations, the cWGAN model produced design for six of them.
Figure 3 illustrates the generated two-scatterer configuration images of six samples and the variation of theircorresponding TSCS values with wavenumbers, where the red continuous curved lines correspond to TSCS
values produced by generated configurations and the blue dashed lines depict results for real configurations.
Note that cWGAN model is only trained for wavenumbers ka ∈ [0.35, 0.45]. Figures 3b illustrate the variationof TSCS with wavenumber ka in optimized range: ka ∈ [0.35, 0.45] and 3c show the results for TSCS versus ka
for extended wavenumber range ka ∈ [0, 0.50]. Most samples show a strong correlation between the evaluatedTSCS, σ , from generated configurations and real configurations. The MAE for all generated images of samples
1 through sample 6 in Figure 3 across the specified wavenumber range is 0.0266 and the average error acrossthe samples is -0.0173. The negative average error shows that the majority of generated images have values of

(a) Generated configurations of M = 2 scatterers(b) The variation of σ with wavenumber ka within the optimized wavenumber range(c) The variation of σ with wavenumber ka beyond the optimized wavenumber range

Figure 3. Sample cWGAN generated images for M = 2 optimized two-scatterer configurations (a), resulting
TSCS comparison for the optimized wavenumber range: ka ∈ [0.35, 0.45] (b) and for extended wavenumber
range ka ∈ [0, 0.50] (c). In Figures 3b and 3c, the red continuous curved lines correspond to TSCS produced bygenerated configurations and the blue dashed lines represent results for real configurations.
TSCS, σ , that is lower than the expected σ .
Figure 4 presents the generated four-scatterer images and their corresponding TSCS graphs versus wavenumbers, correspondingly for six samples. Here, the red continuous and blue dashed lines correspond to TSCS
produced by generated and real configurations respectively. Although the average MAE for all tested samples is0.1561, we examined some of the best performing generated configurations. The average MAE for the six samples shown in Figure 4 is 0.0494 with a standard deviation of 0.0173. The TSCS comparison in the extended
range show that the model is capable of archiving good performance even outside the the range at which thedevice was optimized as shown for several presented samples. In addition, as can be seen from Figures 3 and4, when viewing within the optimized range, few samples show even lower evaluated TSCS than the expected
result. Figures 3 and 4 demonstrate that for two- and four-scatterer configurations cWGAN can produce images
with responses similar, or better, than the expected σ within and outside the optimized range.

4

CONCLUSIONS

A novel data-driven method using deep learning and generative modeling instead of traditional optimization
algorithms is introduced to design broadband acoustic metamaterial configurations targeting near zero TSCS.
By implementing the conditional functionality to the standard WGAN [17, 42] architecture, the model is able
to learn the relation between scatterer configurations and the corresponding TSCS. Applying the CoordConv
layer to the generator and critic noticeably improved the quality and accuracy of generated designs. We use
the random configuration dataset to train the cWGAN to produce TSCS optimized designs. Resulting images

(a) Generated configurations of M = 4 scatterers(b) The variation of σ with wavenumber ka within the optimized wavenumber range(c) The variation of σ with wavenumber ka beyond the optimized wavenumber range

Figure 4. Sample cWGAN generated images for M = 4 optimized four-scatterer configurations (a), resulting
TSCS comparison for the optimized wavenumber range ka ∈ [0.35, 0.45] (b) and for extended wavenumber rangeka ∈ [0, 0.50] (c). In Figure 4b and 4c, the red continuous and blue dashed lines represent the TSCS values
produced by generated and real configurations correspondingly.
and evaluated TSCS show the model’s capability to produce comparable or, sometimes, even better performing
metacluster designs in regards to minimal TSCS. This work is a necessary step toward more complex acoustic
applications with metaclusters, materials, and devices. The presented cWGAN model is generalizable can be applied in design of various metamaterial structures including pentamode metamaterials [43,44], to manipulate andcontrol the propagation and scattering of various types of waves including flexural, elastodynamic, ultrasound,
optical and electromagnetic waves.
Further work will improve the capability of cWGAN model to automatically generate configurations with amuch larger number of scatterers to allow for model scaling and to be compatible with more complex acoustic
applications such as cloaking, superfocusing, and wave steering effects. To increase the degrees of freedom andthe number of material properties stored in each scatterer visually, tunable color channels may be introduced
where different hues of color will correspond to a normalized material property value.

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ABS-0961

The 2D-global optimization networks (2D-GLOnets) model for
acoustic metamaterial design
Wei-Ching Wang(1) , Feruza Amirkulova(2)(1) San Jose State University, USA, wei-ching.wang@sjsu.edu(2) San Jose State University, USA, feruza.amirkulova@sjsu.edu

ABSTRACTThis study aims to contribute to the development of innovative data-driven approaches for metamaterial design using
global optimization, deep learning, and generative modeling. The 2D-Global Optimization Networks (2D-GLOnets)
technique is implemented to design acoustic metamaterials. The 2D-GLOnets optimize the probability of providing
globally optimized meta-devices in the design space. A reparametrization technique is developed and is applied toconstrain the scatterers into a feasible region. The gradients of the loss (objective) function with respect to the weights arecalculated from backpropagation, and they update the weights of the generator. The physics-based gradients are calculated
analytically from a multiple scattering solver [1] and used in backpropagation. The 2D-GLOnets generate a batch ofmetamaterials in each iteration and calculate the loss (objective) function over the entire training. The 2D-GLOnets
gradually use the information provided to the model. The method is employed for sound localization and acoustic lensdesign. In a broadband acoustic lens design, the 2D-GLOnets maximize the absolute pressure at the focal point at discrete
wavenumber values. The method is illustrated with examples of planar configurations of cylindrical scatterers producing
sound localization and focusing effects.

Keywords: Global Optimization, Machine Learning, Generative Modeling, Acoustic Metamaterials

1

Introduction

Metamaterials [2]are artificially engineered materials capable of controlling and directing wave energy flow, allowing
sound, electromagnetic, and mechanical wave manipulations beyond typical materials. There are several ways inwhich these effects can be manifested, including super-focusing, cloaking, negative refraction, and negative massdensity. These special properties of metamaterials are useful for applications, such as biomedical imaging [3], healthmonitoring of structure [4], etc. In recent years, the research on nanophotonic devices and acoustic metamaterials
using deep learning (DL) and generative modeling has produced remarkable results [5–9]. In generative modeling,data patterns can be found without supervision. The models have the ability to generate data on their own basedon the original training dataset for the model [10]. Inverse design can benefit from generative models since they
can learn a real distribution [10]. Nowadays, automated design synthesis can revolutionize engineering design withgenerative modeling [10]. Hence, generative modeling can be applied to solve various inverse design problems
when generating expensive data [11]. Examples of generative models include Generative Adversarial Networks(GAN) [12], GLOnets [13, 14], and Variational Autoencoders (VAE) [15]. Data-driven generative models can assist
inverse materials design with high-efficiency computations and material databases. GANs have been employed ininverse design of organic molecules [11], photonic devices [16] and acoustic metamaterials [9, 17]. VAE, a deepgenerative model, reduces dimensions and extracts features [11]. A supervised VAE (SVAE) and a conditional VAE(CVAE) [18, 19] were implemented for the total scattering cross section suppression.
Fan et al. [13] presented an alternative global optimization approach introducing a generative model 1D-GLOnetsapplicable to gradient-based optimization. The 1D-GLOnets [13] take into account a population of optimal devices
simultaneously and optimize the deflection efficiency in relation to wavelength and outgoing angle. Chen etal. [16] developed reparameterization for 1D-GLOnets to impose geometric constraints on devices. Zhuo andAmirkulova [14] generalized the 1D-GLOnets of Fan et al. [13, 16] by extending the model from a one-dimensional
to a two-dimensional representation and developed a 2D-Global Optimization Networks (2D-GLOnets) introducing
the reparametrization scheme to design the acoustic cloak. In comparison to conventional optimization solvers suchas f mincon, the results obtained by these researchers were thrilling as the 2D-GLOnets model [14] outperformed
traditional optimization algorithms f mincon, and inspired us to employ the idea for acoustic lens design. We
will use DL and generative methods and employ 2D-GLOnets to design acoustic lenses. The optimization of
the parameters of the generator’s network architecture is performed with gradient information provided to the
2D-GLOnets model.

2

Methodologies

1 Problem Formulation(a) An arbitrary configuration of scatterers;(b) A near-field view of the focal point x f ;Figure 1: Figure 1a illustrates an arbitrary configuration consisting of M rigid cylinders Sm with radius am , m = 1, M, and position
vector rm ; the x vector indicates the position of arbitrary point P with respect to origin O. Figure 1b depicts a near-field view of thelens; the incident plane wave of unit amplitude is denoted by pinc ; the focal point is denoted x f vector, which has a magnitude of r f .
Acoustic multiple scattering is considered, employing multiple scattering theory in the context of the 2D acoustic
wave equation by assuming time harmonic dependence e−iωt but omitting in the following. The governing equation
for acoustic pressure p(x), x ∈ R 2 , is the Helmholtz equation:∇2 p + k2 p = 0,

(2.1)

where k = ω/c is the wavenumber, c is the acoustic speed, and ω is the frequency.
The total acoustic pressure field p(x) is defined as: p = pinc + psc , the sum of incident pinc and scattered psc pressure
fields. The absolute value of total pressure is defined as: |p| = (pp∗ )1/2 where the asterisk ∗ denotes the complex
conjugate. The incident field comprises a planar wave that interacts with a configuration of M cylindrical scatterers
as shown in Figure 1a. For simplicity, rigid cylinders are utilized.
We aim to maximize the amplitude of acoustic pressure at the focal point by rearranging scatterers’ positions as
indicated by the vectors {rm } in Figure 1a. Figure 1b depicts the schematics of lens region and the location of theFigure 2: The 2D-GLOnets architecture includes a Gaussian noise vector, generator, reparametrization, loss function,
and a multiple scattering solver evaluating the RMS of pressure amplitude |p f |RMS at the focal point and its gradient.
focal point x f which is fixed and predefined. The total acoustic pressure at the focal point p f ({rm }) = p(x f ) isdefined [20] for a given set of the position vectors {rm } and given by Equation 1.4 in Appendix A.
2 2D-GLOnets Architecture
Figure 2 illustrates a machine learning framework for the 2D-GLOnets, which could be used for inverse design.
The Gaussian noise of size 1 by 128 is used as an input. The generator, a device to generate unconstrained
configurations of the scatterers, has an architecture of the FCNN of 8 layers. The input is aggregated in thegenerator with a size of 1 by 128, i.e., U × M = 128, where U is the batch size, and M is the number of thescatterers. From the 1st to 7th layers, the LeakyReLU is used as an activation function. The tanh activation function
is used in the last layer.
3 Loss Function(rms)

Our goal for the acoustic lens design is to maximize the RMS of absolute pressure amplitude |p f | at the focal
point. Adapting conceptions from Zhuo and Amirkulova [14] and Chen et al. [16], we formulated the loss function
to train the acoustic lens model as follows:
(rms)

L=−

|p f (x )| (γ) RMS
1 Γ(1/σ )exp
n · qj ,
∑
γ γ=1
σ
γ

(2.2)

where γ is the number of unconstrained devices in a batch, σ is tunable hyperparameter, |p f ({rm })| the total
absolute pressure field at the focal point is given in vector form by Equation 1.4, the gradients qRMS
are computedj(γ)
analytically given by Equation 1.7 in the vector form, and n are gradients of the unconstrained device with respect
to the constrained device in the vector form. It is necessary to tune the hyperparameters until the loss function
approaches a range of minor fluctuations because of the nonlinearity of the optimization problem considered. During
training, the loss function may follow the trend of pressure amplitude. The loss function calculates the gradients∂ |p f |RMS∂w

and updates the weights w through backpropagation.

3

Results

Zhuo and Amirkulova [14, 20]’s concepts were adapted, and we used the capabilities of the 2D-GLOnets andgradient information to automate acoustic lens design. The 2D-GLOnets are implemented using PyTorch Python
libraries calling MATLAB engine from Python libraries. The water density ρ0 = 1000kg/m3 and the speed of soundin water c0 = 1480m/s are considered for rigid cylinders submerged underwater. The wavenumbers considered areka ∈ [0.35, 0.45] and ka ∈ [1.35, 1.45]. Numerical results for the design of acoustic lenses are presented using the
2D-GLOnets model for Nk = 11 discretized values with an increment of 0.1. A comparison between the results ofthe 2D-GLOnets and that of f mincon with the MultiStart solver will be analyzed.
1 Numerical Results for 2D-GLOnets at Lower Values of Wavenumbers ka ∈ [0.35, 0.45]

Figure 3: Comparison of pressure fields between arbitrary random configuration (left column), 2D-GLOnets (middle
column), and f mincon (right column). The variation of real part of total pressure, Re(p), (top figures) and absolutepressure amplitude, |p|, (bottom figures) at plane wave incidence on configurations of M = 7 cylinders at ka = 0.40.
The 2D plots of total pressure fields are shown in Figure 3 for configurations of M = 7 rigid cylinder at ka = 0.40.
The upper and bottom figures indicate the real part of the total acoustic pressure field, Re(p), and the total absolute
acoustic pressure field |p|, correspondingly. The left, middle, and right figures display the random configuration,
2D-GLOnets optimized configuration, and f mincon optimized configuration, respectively. The Re(p) illustrates theinteraction of the plane waves with the configuration during wave propagation in the fluid. The absolute pressure
amplitude |p| exhibits the magnitude of pressure amplitude. The color spectrum bar from blue to yellow on theright side of each figure shows the magnitude of the absolute pressure values. An asterisk denotes the focal point(12, 0) at which the waves propagate in a left-to-right direction with normal incidence get focused. Figure 3 shows
the outperformance of the 2D-GLOnets compared to f mincon at ka = 0.40. The focusing effects are distinct forthe 2D-GLOnets. The 2D-GLOnets and f mincon display symmetrical scatterer patterns oriented along the x-axis.
We plot the variance of the absolute pressure amplitude in Figure 4a and the loss in Figure 4b with number of
epochs at ka ∈ [0.35, 0.45] for configuration of M = 7 scatterers optimized by 2D-GLOnets given in Figure 3. In
Figure 4a, the absolute pressure amplitude |p f | shows a converging trend with a decreasing variance as the number

(a) absolute pressure amplitude;(b) loss function;

Figure 4: The variation of absolute pressure amplitude |p f | and loss function versus the number of epochs using the 2D-GLOnets for
M = 7 at ka ∈ [0.35, 0.45]
of epochs increases. The 2D-GLOnets have an optimized absolute pressure amplitude of |p f | ≈ 1.36, which isgreater than |p f | = 1.32 by f mincon. From Figure 4b, we can see fluctuations in the loss function. Because of
the loss function formulation, it is satisfactory as long as it converges to a range of numbers. Figure 5 shows(a) M = 8, |p|max = 1.38;(b) M = 9,|p|max = 1.46;(c) M = 11, |p|max = 1.51;(d) M = 12, |p|max = 1.52;

Figure 5: 2D-GLOnets optimized results for the absolute pressure field |p| with M = 8, 9, 11, and 12 at ka = 0.40.
the absolute pressure amplitude with a varying number of cylinders. The absolute pressure amplitudes at the focal
point |p f | for M = 8, 9, 11, and 12 scatterers were plotted. In these four examples, the 2D-GLOnets perform better
than f mincon, showing the focusing effects. The 2D-GLOnets and f mincon results show a symmetry pattern about
the x-axis. Figure 6 illustrates the absolute pressure amplitude at the focal point is optimized at ka ∈ [0.35, 0.45]
for the optimized configurations of M = 6, 8, 10, and 12 cylinders at plane wave incidence. Here, the comparisonof 2D-GLOnets results with f mincon results is conducted, and the results for non-optimized random configuration
are given in green color as a reference. A non-optimized configuration’s absolute pressure amplitude appears much
lower than that of a 2D-GLOnets configuration and that of f mincon at ka ∈ [0.35, 0.45]. The 2D-GLOnets results arecomparable with f mincon results at ka ∈ [0.35, 0.45], which is the optimized frequency range. A similar performanceis achieved by 2D-GLOnets and f mincon for M = 6 and M = 10. Compared with f mincon, the 2D-GLOnets perform
slightly worse for M = 8 and M = 12.
2 Numerical Results for 2D-GLOnets at Higher Values of Wavenumbers ka ∈ [1.35, 1.45]
Figure 7 shows a comparison between a random configuration on the left column and optimized configurations
consisting of the 2D-GLOnets results on the middle column and f mincon results on the right column at highwavenumbers ka ∈ [1.35, 1.45]. Re(p) is the real part of total pressure, and |p| is the absolute pressure amplitude. A

(a) M = 6;(b) M = 8;(c) M = 10;(d) M = 12;

Figure 6: The dependency of absolute pressure amplitude on ka ∈ [0.35, 0.45] for fixed numbers of scatterers: M = 6, 8, 10, 12.

Figure 7: Comparison of pressure fields between random configuration, 2D-GLOnets, f mincon at ka = 1.40. At
normal plane wave incidence, the real part of total pressure Re(p) is shown on the top figures, and absolutepressure amplitude |p| is shown on the bottom figures for configurations M = 9 rigid cylinders ka = 1.40. The left,
middle, and right columns show an arbitrary random configuration, a 2D-GLOnets configuration, and a f mincon
configuration, respectively.
more scattered configuration is achieved as a result of optimizing the cylinders. This particular instance shows thatthe 2D-GLOnets produces |p|max = 2.28 that outperforms the f mincon solver producing |p|max = 2.07. In contrast to
the random configuration, the 2D-GLOnets model produces distinct focusing effects at the focal point as identified
with an asterisk in Figure 7.

(a) absolute pressure amplitude;(b) loss function;

Figure 8: The variation of absolute pressure amplitude |p f | and loss function with the number of epochs using 2D-GLOnets for M = 9
at ka ∈ [1.35, 1.45].
The training curve for |p f | the loss function versus the epoch numbers are given correspondingly in Figure 8a andin Figure 8b. We used a batch size U of 15. A fluctuating amplitude of the absolute pressure is observed at the
focal point at the beginning of the training. Due to the 2D-GLOnets being able to exit local minima and reachglobal maxima, it is ascertained that there is a sudden drop and rise in absolute pressure. Successful convergenceof the loss function is achieved after the oscillations. Around the 9500th epoch of the training, the absolute totalpressure amplitude and loss function show significant reductions in noise.
Figure 9 displays the 2D-GLOnets results for absolute total pressure amplitude |p| for M = 7, 8, 10, and 11 at aselected wavenumber ka = 1.40. The normal incident wave has an absolute pressure of |pinc | = 1.0. These examples
demonstrate that 2D-GLOnets optimized configurations outperform f mincon configurations. The asterisk indicatesthe focal point where focusing effects of the optimized configurations are observed. The sound localization andfocusing effects at high frequencies are clearly visible in the absolute total pressure field.

(a) M = 7, |p|max = 2.02;(b) M = 8, |p|max = 2.04;(c) M = 10, |p|max = 2.16;(d) M = 11, |p|max = 2.18;

Figure 9: The absolute value of the total acoustic pressure field |p| for the optimized configurations of M = 7, 8, 10, and 11 optimized
by 2D-GLOnets at ka = 1.40.

Figure 10 depicts examples of the absolute value of the total acoustic pressure fields |p| that are outside theoptimized wavenumber range, i.e., ka ∈ [1.35, 1.45]. For ka = 1.45, 1.5, and 1.6, which are outside the optimal
wavenumber range, the number of scatterers M = 8 (top row) and M = 10 (bottom row) are considered. The middle
column figures show that lens devices still produce a focusing effect at wavenumber at ka = 1.5, which is outside
the range where the device was optimized. As the wavenumber ka increases, the focusing effects diminish.

Figure 10: The absolute total acoustic pressure |p| for M = 8 is shown on the top row and M = 10 is depicted on
the bottom row; the results at ka = 1.45 give on the left, ka = 1.5 shown on the middle, and at ka = 1.6 depictedon the right columns. Here, the wavenumbers ka outside the optimized range ka ∈ [1.35, 1.45] are investigated.

4

Conclusion and Future Work

The acoustic lenses have been successfully designed using the 2D-GLOnets. Focusing effects can be achieved at
two ranges of frequencies, i.e. at low (ka ∈ [0.35, 0.45]) and high (ka ∈ [1.35, 1.45]) values of wavenumbers. Thereis a successful convergence of the absolute pressure amplitude. Model performance is validated by state-of-theart gradient-based optimization algorithms of the f mincon solver. The problem becomes more challenging when
there are more scatterers, as the reparametrization consumes much memory in GPUs. There is a possibility that
2D-GLOnets may produce better results than current state-of-the-art optimization algorithms with a more powerfulcomputational resource.
Future work includes exploring a variety of non-uniform configurations considered with different material properties,
scatterers’ radii, shell structures, etc. We will consider optimizing the 2D-GLOnets by alternating the radii andpositions of the scatterers in non-uniform configurations. We can employ high-performance computing to speed
up the computing time. By doing this, the optimized configurations can accommodate more scatterers. The
reparametrization scheme will not be needed if radii variance is considered by removing the scatterer adjustment
and enabling practical designs with a larger number of scatterers in the low-frequency and high-frequency ranges.

A

APPENDIX

1 Position Dependent Acoustic Pressure at the Focal Point
The incident wave is the normal plane wave of unit amplitude, and the incident pressure at fixed focal point Pf is:pinc (x f ) = eikeψ ·x f . The scattered field in the neighborhood of cylinder Sm is given by [20]:M∞

psc (x) = ∑(m)∑ Bn Vn+ (xm ),(1.3)

m=1 n=−∞

where xm is a position vector of point P with respect to the cylinder center at Om : xm = x − rm , and the function(1)(1)
Vn± (x) is defined as: Vn± (x) = Hn (k|x|)e±in arg x , where Hn is the Hankel function of the first kind of order n.
The total pressure at the fixed focal point Pf [20] is:M

p f = eikeψ ·x f + ∑∞(m)∑ Bn Vn+ (x f − rm ) = eikeψ ·x f + vT b,(1.4)

m=1 n=−∞( j)where we introduced vectors b, v ∈ C M×(2N+1) . The elements of the dual vector v = {Vn } = {Vn (r j )} were defined(m)into the form [20]: Vn = Vn+ (r f m ), where r f m = x f − rm is the position of focal point with respect to local
coordinate at Om and rm is the position of scatterer with respect to origin at O. The components of the scattering
( j)
( j)
coefficient vector b = {Bn }, j ∈ (1, M), n ∈ (−N, N), are Bn = {Bn (r j )}, and written in column vector form [20]:

T

T
j) ( j)( j)b = b(1) b(2) . . . b(M) , b( j) = B(−N
where T means transpose, to execute this, we let N be the
B−N+1 . . . BN
truncation value of the infinite sum in Equation 1.4. We select the value as: N = 2ka.
2 Absolute Pressure Amplitude Function and its Gradients for Acoustic Lens
Broadband optimization was performed using root mean square (RMS) of absolute pressure at the focal point |p f |evaluated over some range of normalized wavenumbers ki a (i = 1, 2, ..., Nk ), i.e., |p f ({rm })|RMS [20]:|p f ({rm })|RMS =

1
1/2
|p f (k1 a, {rm })|2 + |p f (k2 a, {rm })|2 + ... + |p f (kNk a, {rm })|2
,
Nk

(1.5)

where each of absolute pressure amplitude |p f ({ki a, rm })| can be computed for each ki a value (i = 1, Nk )using Equation 1.4 in Appendix A. In our simulation, we consider Nk = 11, i.e. is 11 discrete values of normalized
wavenumbers ka where a is the cylinder radius. The vector q j defined as gradient of the absolute value of totalpressure field |p f ({rm })| with respect to positions r j and is evaluated in [20] as:qj =∂ |p f |1=p∗f p j + p f p∗j ,∂rj
2|p f |

(1.6)

∂pwhere ∗ indicates the complex conjugate and p j = ∂ r jf . The broadband gradient vectors are defined as:
=
qRMS
j


∂ |p f ({rm })|RMS
1
1
=
|p
(k
a)|q
(k
a)
+
...
+
|p
(k
a)|q
(k
a)
,
j
N
j
N
1
1
f
f
k
k
∂rj
|p f |RMS Nk

j = 1, 2, . . . M.

(1.7)

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https://doi.org/10.3397/IN–2021–2431, aug 2021.
[15] M. Kingma, D. Welling. Auto-encoding variational bayes. arXiv:1312.6114v10, 2014.
[16] Mingkun Chen, Jiaqi Jiang, and Jonathan Fan. Design space reparameterization enforces hard geometricconstraints in inverse-designed nanophotonic devices. ACS Photonics, 15, 11 2020.
[17] Caglar Gurbuz, Felix Kronowetter, Christoph Dietz, Martin Eser, Jonas Schmid, and Steffen Marburg.
Generative adversarial networks for the design of acoustic metamaterials. Journal of the Acoustical Societyof America, 149(2):1162–1174, feb 2021.
[18] Thang Tran, Feruza Amirkulova, and Ehsan Khatami. Acoustic cloak design via machine learning. Journal of
Theoretical and Computational Acoustics, (accepted), 07 2022.
[19] Thang Tran, Ehsan Khatami, and Feruza Amirkulova. Total multiple scattering cross section evaluation using
convolutional neural networks for forward and inverse designs of acoustic metamaterials. Journal of the
Acoustical Society of America, 149, 06 2021.
[20] F. A Amirkulova, S. Gerges, and A. N. Norris. Sound localization through multi-scattering and gradient-based
optimization. MDPI Mathematics, 2021.

ABS-0965

Deep learning assisted PM metamaterial design
Anam Abbas(1) , Cheng Qiu(2) , Feruza Amirkulova(3)(1) Mechanical Engineering Department, San José State University, San José, 95192, USA, anam.abbas@sjsu.edu(2) Mechanical Engineering Department, San José State University, San José, 95192, USA, cheng.qiu@sjsu.edu(3) Mechanical Engineering Department, San José State University, San José, 95192, USA, feruza.amirkulova@sjsu.edu

ABSTRACTIn this work, we present deep learning-assisted models for design of tetrahedral PM unit cells to construct a
3D lattice structure that is highly anisotropic and mimics the acoustic properties of water. We optimize unit cell
design via forward techniques by altering the design parameter and building datasets to train deep learning models. We implement conditional Wasserstein generative adversarial networks (cWGAN) and convolutional neuralnetworks (CNN) models utilizing TensorFlow Python libraries with Keras API. The purpose of establishing thedeep learning framework is to create more data on the parameters of unit cells such that we can select the cells
whose mechanical properties fit our design goal.

Keywords: Pentamode Metamaterials, Deep Learning, Generative Modeling, Wasserstein Generative Adversarial Networks, Convolutional Neural Networks

1

INTRODUCTION

1.1 The development of pentamode (PM) metamaterials
The theoretical model of pentamode (PM) metamaterial first introduced by Milton and Cherkaev in 1995, isa model of a three-dimensional lattice of a solid structure formed with tetrahedral PM unit cells [1]. This
PM structure has a high bulk modulus and low shear modulus. Kadic et al. [2] used the finite connection to
replace the strictly point-like tips and found the ratio of bulk modulus to shear modulus of the 3D structure
can realistically be made as large as about 1000. The feasibility of PM metamaterials acoustic cloaks wastheoretically conceived by Milton et al. [3] and further improved by Norris [4].
1.2 Applying deep learning and generative adversarial networks (GAN) to metamaterial design
Deep learning [5, 6] is a subset of machine learning widely applied to the design of metamaterials. Wang etal. [7] invented a framework that has a variational autoencoder and a regressor for property prediction. The
regressor plays a unique role to predict the labels for generated images in the deeper learning network. Shah et
al. [8] introduced a semi-analytical method of suppressing acoustic scattering in which a reinforcement learning agent is used to control the parameters of cylindrical scatterers in water. Wiest et al. [9] studied how to
improve the feasibility of using additive manufacturing technology for acoustic metamaterials and metasurfaces
by multiple methods, including finite element method and appropriate metamodel for Monte Carlo simulations.
Zaghloul et al. [10] built anisotropic radio-frequency metamaterial unit cells for metasurface arrays by employing deep convolutional GAN. Marburg et al. [11] invented a new method that uses conditional GAN to build aframework to generate unit cells for sound insulation purposes. Lai et al. [12] presented a method of developing the conditional Wasserstein GAN model to reduce the total scattering cross-section to near zero for a planar
configuration of cylinders.

2

METHOD

The aim of this work is to design tetrahedral PM cells to construct a 3D lattice structure. The lattice structureis expected to have a high bulk modulus, low shear modulus, and acoustic properties similar to water.
2.1 PM model and data collection
The PM model studied by Matthew Kelsten et al. [13, 14] from Rutgers University is used as a reference forour design. In Figure 1, the unit cell is a face-centered cubic diamond-shaped lattice structure made up ofcone linkages. The unit cells are modeled in COMSOL Multiphysics Commercial software using finite element
analysis and Floquet-Bloch periodicity boundary conditions. The dimension and mechanical properties of theunit cells including the height, radius and connection angle of the cone, the density, impedance, bulk modulus
and shear modulus, cutoff frequencies of upper and lower boundary of bandgap, are recorded in the excel fileand split by the ratio of 6:1 into a training and testing set, and then loaded to the training networks as a CSV
file.

Figure 1. An example of the cell parameters and appearance. The dimensions of the cell in the left table areused to build the cell shown in the right image. The mechanical properties are obtained in COMSOL simulationand SOLIDWORKS.
2.2 Training with generative adversarial networks
In the inverse design process, we adapt ideas from Lai et al.’s conditional WGAN model [12] to fit our design.
The flow for our network is shown in Figure 2. The generator is provided with random input vectors with reallabels as a guide for image generation. The generated images and real images are provided to the critic. The
critic learns to distinguish the images and generates a linear score between -1 and 1 which is fed back to thegenerator. The generator then learns to generate images such that the score decreases with each epoch, thismakes the generator produce images that are more realistic. The fake images are fed to the auxiliary regressor,
CNN model which is trained to predict the labels given the parameters of PM unit cell depicted in Figure
1. To assist the critic and generator to identify images, a CoordConv layer is added prior to the networks’convolutional layers. As the training loop executes, the GAN model loss is minimized as the generator and thecritic losses converge and approach zero.

Figure 2. Conditional Wasserstein GAN (cWGAN) framework [12, 15].

3

RESULT

3.1 The result of cWGAN and CNN model
From the generated output, we carefully choose several sets of the unit cell (Table 1) whose mechanical properties fit the set requirements: impedance and density are closed to water. We find the ratio of bulk modulus
vs. shear modulus is 35-45, and the gap between the cutoff frequency is about 40 kHz.

Table 1. Properties of the generated unit cell. The left four columns show the dimension of the cell which
determines the cell’s mechanical properties shown in the five right columns. The impedance and the density arevery close to the water.
3.2 The result of optimized cell and lattice design
We choose one of the optimized cells for PM lattice construction. The optimized unit cell is constructed andassembled into a lattice structure in SOLIDWORKS before importing to COMSOL. The full-wave simulation isconducted on the optimized PM lattice. The background pressure field of amplitude 1Pa is propagated from theleft water body to the right. The amplitude of the reflected wave in the left water body is much smaller than
the amplitude of the transmitted wave in the right water body at 22.5kHz as shown in Figure 3.

Figure 3. The left image is the lattice constructed by the optimized cells which are built by the generated
parameters from the networks. The right image shows the scattered pressure field at 22.5 kHz during the fullwave simulation.

4

CONCLUSIONS

PM cell with high bulk modulus and low shear is designed with tetrahedral cones. The lattice built with PM cellis density matched with water initially. It is later matched with the acoustic impedance of water by inverse design using cWGAN. The cWGAN and CNN provide images of the cells along with the design parameters. The
trained network is able to generate reasonably accurate designs for impedance matching. We utilize COMSOLMultiphysics, which proved to be an effective tool for conducting FEA analysis of the structures, performing

the acoustic study and full wave simulation. The result of full-wave transmission in COMSOL shows that thetitanium lattice structure can transmit waves through it with minimum backward scattering waves.

REFERENCES[1] Milton G. W. and Cherkaev A. V. Which Elasticity Tensors are Realizable?
Materials and Technology, 117(4):483–493, 10 1995.

Journal of Engineering[2] Kadic M., Bückmann T., Stenger N., Thiel M., and Wegener M. On the practicability of pentamodemechanical metamaterials. Applied Physics Letters, 100(19):191901, 2012.
[3] Milton G., Briane M., and Willis J. On cloaking for elasticity and physical equations with a transformation
invariant form. New Journal of Physics, 8(10):248–248, oct 2006.
[4] Norris A. Acoustic cloaking theory. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 464(2097):2411–2434, 2008.
[5] F. Chollet. Deep Learning with Python. Shelter Island, NY : Manning Publications Co.„ 2018.
[6] J. Brownlee. Deep Learning With Python: Develop Deep Learning Models on Theano and TensorFlow Using
Keras. Machine Learning Mastery, 2016.
[7] Wang L.W., Chan Y. C., Ahmed F., Liu Z., Zhu P., and Chen W. Deep generative modeling formechanistic-based learning and design of metamaterial systems. Computer Methods in Applied Mechanics
and Engineering, 372:113377, 2020.
[8] Shah T., Zhuo L. W., Lai P., De La Rosa-Moreno A., Amirkulova F., and Gerstoft P. Reinforcement
learning applied to metamaterial design. The Journal of the Acoustical Society of America, 150(1):321–338,
2021.
[9] Wiest T., Seepersad C., and Haberman M. Robust design of an asymmetrically absorbing willis acoustic
metasurface subject to manufacturing-induced dimensional variations. The Journal of the Acoustical Societyof America, 151(1):216–231, 2022.
[10] Hodge J. A., Mishra K. V., and Zaghloul A. I. Rf metasurface array design using deep convolutional
generative adversarial networks. presented at the 2019 IEEE International Symposium on Phased Array System& Technology, Waltham MA, USA, pages 1–6, Oct.2019.
[11] Gurbuz C., Kronowetter F., Dietz C., Eser M., Schmid J., and Marburg S. Generative adversarial networks
for the design of acoustic metamaterials. The Journal of the Acoustical Society of America, 149(2):1162–
1174, 2021.
[12] Lai P., Amirkulova F., and Gerstoft P. Conditional wasserstein generative adversarial networks applied toacoustic metamaterial design. The Journal of the Acoustical Society of America, 150(6):4362–4374, 2021.
[13] Cushing C.W., Kelsten M.J., Su X.S., Wilson P.S., Haberman M.R., and Norris A.N. Design and characterization of a three-dimensional anisotropic additively manufactured pentamode material. The Journal of
the Acoustical Society of America, 151(1):168–179, 2022.
[14] Kelsten M. Controlling wave propagation with acoustic and elastic metamaterials. PhD Dissertation, Rutgers
University, Oct. 2021.
[15] Abbas A. Pentamode metamaterial design via full wave simulation and deep learning. MSME Project
Report, San Jose State University, May 2021.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0036

Viscosity-Induced Nonreciprocity in Fluid-Solid Phononic Crystals
With Broken Space Inversion Symmetry
Arkadii KROKHIN 1; Hyeonu HEO1; Yuqi JIN1; Yurii ZUBOV1; Dmitrii SHYMKIV1; Tae-Youl
CHOI1; Ezekiel WALKER2; Arup NEOGI3
1

University of North Texas, USA
2

3

Echonovus Inc., USA

University of Electronic Science and Technology of China, China

ABSTRACT
Nonreciprocal propagation of sound becomes possible if the time reversal symmetry (T symmetry) of elastic
medium is broken. Recent methods of T symmetry breaking are based on scattering actuated by nonlinear
electric circuits, spinning of fluid, or nontrivial topological bands in acoustic spectrum. Here we reportnonreciprocity in sound propagation in a phononic crystal (PhC) with broken PT symmetry. T symmetry isnaturally broken in any fluid due to viscosity, however it is currently not recognized as a source ofnonreciprocity. We show that dissipation leads not only to irreversibility but also to nonreciprocity if space
inversion symmetry (P symmetry) is also broken. We study nonreciprocal propagation of sound through a
PhC of asymmetric metal rods imbedded in water. Experiment is supported by numerical solution of theNavier-Stokes equation. It is shown that the viscous dissipation in a PhC with broken P-symmetry isnonreciprocal. Nonreciprocal contribution is separated from asymmetry by considering the limit of inviscid
fluid.
Keywords: Boundary layer, Viscous dissipation, Acoustic diode1. INTRODUCTION

In an ideal (inviscid) fluid sound transmission is reciprocal, i.e., acoustic pressure from the source
emitting at point r A and measured at point r B is invariant with respect to switching of the positions ofemitter and receiver,
𝑝𝑝𝐴𝐴 (𝐫𝐫𝐵𝐵 ) = 𝑝𝑝𝐵𝐵 (𝐫𝐫𝐴𝐴 ).

(1)

As it was shown by Rayleigh [1], this property is due to potential flow of ideal fluid, where ∇ × 𝐯𝐯 =0, while the fluid velocity v in general case does not possess the reciprocal symmetry (1). The luck ofthis symmetry is attributed to asymmetry in scattering, caused by broken inversion symmetry (Psymmetry) in the distribution of scatterers in space. As a result, the sound intensity 𝐼𝐼(𝐫𝐫) = 𝑝𝑝(𝐫𝐫)𝑣𝑣(𝐫𝐫)is also asymmetric, 𝐼𝐼𝐴𝐴 (𝐫𝐫𝐵𝐵 ) ≠ 𝐼𝐼𝐵𝐵 (𝐫𝐫𝐴𝐴 ), as well as the transmission coefficient obtained by integration of
intensity over some area. The asymmetry of the transmission coefficient has been used for the designof sound rectification devices [2-4]. While they provide almost unidirectional propagation of sound,they are reciprocal by nature [5] since they do not break time-reversibility, if dissipative losses are
neglected. Viscosity is a natural factor that makes fluid dynamics irreversible and nonreciprocal.
However, for a long time it was considered that viscosity does not violate reciprocity in transmissionof acoustic signals. Even very recent reviews claim that since Eq. (1) remains valid for a viscous fluid,
viscous damping breaks reversibility but not reciprocity [6,7]. Here we demonstrate that broken P
symmetry, that leads to asymmetry in transmission in an ideal fluid, serves as a source ofnonreciprocal dissipation in a viscous fluid, thus adding truly nonreciprocal contribution to thetransmission.
1

arkady@unt.edu

2. NONRECIPROCAL TRANSMISSION TROUGH PHONONIC CRYSTAL WITH
ASYMMETRIC SCATTERERS

The power dissipated within some volume V due to viscous attenuation of sound wave depends on
all spatial derivatives of the velocity vector v
2𝜂𝜂 𝜕𝜕𝑣𝑣
𝜕𝜕𝑣𝑣
𝑄𝑄̇ = − ∫ � �𝜕𝜕𝑥𝑥 𝑖𝑖 + 𝜕𝜕𝑥𝑥𝑘𝑘 � + 𝜉𝜉(∇ ∙ 𝐯𝐯)2� 𝑑𝑑𝑑𝑑.
2

𝑘𝑘

𝑖𝑖

(2)

Here 𝜂𝜂 and 𝜉𝜉 are the viscosity coefficients. In a system of scatterers with broken P symmetry the
distribution of velocities in space is different for the opposite directions of propagation of sound. This
leads to different levels of dissipated power, i.e., transmission is not reciprocal. In an inviscid fluid
the transmission contains the asymmetric part which should be subtracted in order to get truly
nonreciprocal transmission caused by nonreciprocal dissipation. In presence of solid scatterers
dissipation occurs mainly within a narrow boundary layer 𝛿𝛿 = �2𝜂𝜂/𝜔𝜔𝜔𝜔 formed around each scatterer.
Here the level of turbulency reaches its maximum since the value of ∇ × 𝐯𝐯 exceeds much that value
in the bulk. This means that dissipative losses in the bulk can be neglected, and attenuation of soundoccurs due to viscous losses in the layer 𝛿𝛿 [8].
For experimental demonstration of nonreciprocal transmission [9], a phononic crystal of aluminumrods in water environment was used, see Fig. 1. The period of the phononic crystal lattice is 5.5 mm,
Fig. 1. (a) View of the sample with anodized
rods. (b) The P symmetry is broken along thevertical axis. (c) Square unit cell. The angle α
is a measure of broken P symmetry. (d)Sample with 4×7 rows of unanodized
aluminum rods [9].
Fig. 2. Lower panel: Band structure for soundpropagating along the direction of broken P
symmetry. Gaps between the even zones(black) are shaded. The odd zones (grey) are
def at normal incidence. Upper panel: Thin
(thin) red and blue lines are experimental(numerical) spectra for forward and backward
propagation. Insert: orientation of the unit cellwith respect to the incoming wave [9].

and the radius of the 120° circle sector is 2.2 mm. Two V301 1`` Panametrics 0.5 MHz immersion
transducers in a bistatic setup were arranged to measure forward and backward transmission. First,
the transmission was measured along the symmetric direction. Due to the inverse symmetry, thetransmission does not exhibit any regular feature of nonreciprocity. The sound waves experience
anisotropic scattering at each rod, but they follow the same “path” propagating forward and backward.
Unlike this, the transmission spectra for propagation along the line of broken P symmetry exhibit
regular features of nonreciprocity. In Fig. 2, experimental (thin lines) and numerical results (thicklines) for the transmission in two opposite directions are plotted together. Theoretical andexperimental results are in reasonable agreement. All the gaps and passing bands of the band structurein Fig. 2 are seen in the measured transmission spectra. The experimental curve exhibits fast
oscillations which originate from weak irregular fluctuations of the transmitted energy in Fig. 3. While
there is a general agreement between the theory and experiment, it is clearly seen that the measured
nonreciprocity exceeds the numerically simulated one. We attribute this difference to microscopicroughness of the aluminum rods. The rods were anodized to increase their resistance against oxidationin water. It is known that the surface of an anodized sample may have roughness of the size orderfrom a few to dozens of microns. At this scale, the surface of the rods is not flat. At the frequencies𝜔𝜔~106 𝑠𝑠 −1 the thickness of the viscous boundary layer 𝛿𝛿 is estimated to be ideally about a few
microns. Roughness not only changes fluid dynamics within the boundary layer but it also increases
the effective area where the energy dissipates. Thus, surface roughness increases the dissipation ofsound energy that leads to stronger nonreciprocity. In the experiment with unanodized rods possessing

more rough surfaces the level of nonreciprocity reached 30 dB [9]. Random roughness also can beconsidered as a stochastic element of the system that breaks the P symmetry at the microscopic level.
At the same time, the micron-size roughness does not contribute much to scattering, because thewavelength of sound in water is about 4-5 mm. A phononic crystal where each cylinder has half of its
surface flat and the other half is rough is an example of a system with weakly broken P symmetry andweak asymmetry in transmission [10].
Controlled roughness was introduced to the cylindrical rods using sandblasting and sandpaper ofvarying grain sizes where the average roughness 𝜎𝜎 = 140 μm [10]. Stainless steel rods of diameter D= 6.35mm were cut from quarter-inch stock and arranged in a square lattice with period a = 10.3 mm.
Each rod was masked using a mould and a razor blade to uniformly cover half of the rod with tape.
The tape acted as a resist when the rods were sandblasted to create lower levels of surface roughness.
For increased roughness, a grit of sandpaper was chosen for testing with the thickness of the sandpaper
measured as a tenth of a millimetre. The sandpaper covering precisely half of the cylindrical surface
was cut using a laser cutter. After the sandpaper strips were cut to size, they were glued to the rods
using cyanoacrylate adhesive to have the best bond between the sandpaper and the steel rod.
For frequencies within the interval 400–600 kHz, the wavelength λ changes from 3.7 to 2.5 mm.
For any frequency from this region, the inequality 𝜎𝜎, 𝛿𝛿 ≪ 𝜆𝜆 is satisfied, that guarantees weak random
scattering at the rough side of each cylinder. In Fig. 3a the transmission spectra measured along the

Fig. 3. Experimental spectra forphononic crystal with roughness
𝜎𝜎 = 140 μm . (a) Transmission
along the direction with broken P
symmetry.
Nonreciprocity
isclearly seen. (b) The transmission
is reciprocal along the P
symmetric direction. (c) The
directions of the incident wave areshown with respect to theorientationofasymmetriccylindrical scatterer.

direction of broken P symmetry exhibit nonreciprocity [10]. Since the asymmetry in transmission isminimal for the cylindrical scatterers, the difference in transmission is mostly due to nonreciprocity
in viscous absorption. For the P symmetric direction the transmission spectra are identical, see Fig.
3b.
To separate exactly the asymmetric and nonreciprocal contributions, numerical simulations were
performed for a hard cylindrical scatterer of the same diameter D = 6.35 mm as was used in theexperiment. One half of the cylinder was flat, and the other is covered by regular roughness witheither σ=10 μm or σ=100 μm, see Fig. 4a. Sound emitter and receiver were placed at the symmetricalpoints A and B. To verify the reciprocal relation (1) the following parameter

Fig. 4. Numerically simulated nonreciprocity for a single scatterer. (a) Cylindrical hard scatterer with roughness. (b)Non-reciprocity for pressure versus normalized viscosity η/ηwater calculated for the source and receiver placed at thesymmetrical points A and B situated 5mm away from the center. (c) Non-reciprocity for velocity v versus normalizedviscosity 𝜂𝜂/𝜂𝜂𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 calculated for the source and receiver placed at the symmetrical points A and B situated 5 mm
away from the center. The left (right) vertical axis is for roughness with σ = 100μm (σ = 10μm).
𝑝𝑝 (𝐫𝐫 )−𝑝𝑝 (𝐫𝐫 )Δ𝑝𝑝 = 2 𝑝𝑝𝐴𝐴 (𝐫𝐫𝐵𝐵 )+𝑝𝑝𝐵𝐵(𝐫𝐫𝐴𝐴)
𝐴𝐴

𝐵𝐵

𝐵𝐵

𝐴𝐴

(3)

was plotted as a function of normalized viscosity 𝜂𝜂/𝜂𝜂𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 in Fig. 4b. Similar parameter, Δ𝑣𝑣 , wascalculated for the modulus of velocity v of water oscillations and it was plotted in Fig. 4c. The
numerical values for Δ𝑝𝑝 presented in Fig. 4b are very small and they do not grow with viscosity. This
result suggests that pressure is a reciprocal quantity even in a viscous fluid. The graphs in Fig. 4cclearly demonstrate that the non-reciprocity in v, which exists even in ideal fluid 𝜂𝜂 = 0, grows fastwith 𝜂𝜂. The difference 𝑣𝑣(𝜂𝜂) − 𝑣𝑣(𝜂𝜂 = 0) is the contribution of the vorticity mode with ∇ × 𝐯𝐯 ≠ 0 of
velocity to non-reciprocity. Two curves for σ = 10 and 100μm demonstrate the same monotonic
behavior. Both curves exhibit square-root dependence at 𝜂𝜂 → 0, which is due to 𝛿𝛿~�𝜂𝜂. The values of
non-reciprocity obtained for σ = 10μm (right vertical axis in Fig. 4c) are approximately 10 times lessthan the values obtained for σ = 100μm (left vertical axis), i.e., Δ𝑣𝑣 ∼ σ. This linear scaling of velocity
corresponds to quadratic scaling of dissipation in Eq. (2), where 𝑄𝑄̇ is a quadratic functional of v.

3. CONCLUSIONS

Nonreciprocal acoustic transmission through a dissipative phononic crystal with asymmetric scatterers wasdemonstrated experimentally and numerically. The time-reversal symmetry is naturally broken by viscous
losses. Nonreciprocal contribution to acoustic transmission appears due to asymmetry in the field of fluid
velocities that leads to nonequal viscous dissipation for the opposite directions of sound propagation. The
asymmetry in distribution of velocities appears even in ideal (inviscid) fluid. Presence of viscosity leads toformation of the viscous boundary layers near each scatterer, where vortex component of velocity gives truly
nonreciprocal contribution to attenuation of sound. This natural mechanism of nonreciprocity opens a
possibility for engineering of acoustic nonreciprocal devises, like acoustic diode, which operate without
external sources of energy.

ACKNOWLEDGEMENTS
This work was supported by the National Science Foundation under EFRI grant no. 1741677.

REFERENCES
1. Rayleigh JWS. The theory of sound. 2nd ed. New York, USA: Macmillan;1896.
2. Liang B, Guo XS, Tu J, Zhang D, Cheng JC. An acoustic rectifier. Nat. Mater. 2010; 9: 989–992.
3. Li X-F, Ni X, Feng L, Lu M-H, He C, Chen Y-F. Tunable unidirectional sound propagation through asonic-crystal-based acoustic diode. Phys. Rev. Lett. 2011; 106: 084301.
4. Zhu J, Zhu X, Yin X, Wang Y, Zhang X. Unidirectional extraordinary sound transmission with modeselective resonant materials. Phys. Rev. Appl. 2020; 13: 041001.
5. Maznev AA, Every AG, Wright OB. Reciprocity in reflection and transmission: what is a ‘phonon diode’?
Wave Motion. 2013; 50: 776.
6. Nassar H, Yousefzadeh B, Fleury R, Ruzzene M, Alú A, Daraio Ch., Norris AN, Huang G, Haberman MR.
Nonreciprocity in acoustic and elastic materials. Nat. Rev. Mater. 2020; 5: 667–685.
7. Rasmussen C, Quan L, Alù A. Acoustic nonreciprocity. J. Appl. Phys. 2021; 129: 210903.
8. IbariasM, Zubov Yu., Arriaga J, Krokhin AA. Phononic crystal as a homogeneous viscous metamaterial.
Phys. Rev. Res. 2020; 2: 022053.
9. Walker E, Neogi A, Bozhko A, Zubov Yu., Arriaga J, Heo H, Ju J, Krokhin AA. Nonreciprocal linear
transmission of sound in a viscous environment with broken P symmetry. Phys. Rev. Lett. 2018; 120:204501.
10. Heo H, Walker E, Zubov Y, Shymkiv D, Wages D, Krokhin A, Choi T-Y, Neogi A. Non-reciprocal
acoustics in a viscous environment. Proc. R. Soc. A. 2020; 476: 20200657.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0051

Hybrid analytic model between the transfer-matrix method and themodal decomposition to design Acoustic Metamaterials:
Application to degenerate resonators
Vicente ROMERO-GARCÍA1,*, Jean-Philippe GROBY2, Noé JIMÉNEZ3,
Aurelien MERKEL4, Vincent TOURNAT2, Georgios THEOCHARIS2,
Olivier RICHOUX2 and Vincent PAGNEUX2
1

Instituto Universitario de Matemática Pura y Aplicada, Departamento de Matemática Aplicada, Universitat
Politècnica de València, Camino de Vera s/n, 46022, València, Spain
2

Laboratoire d’Acoustique de l’Université du Mans (LAUM), LAUM UMR CNRS 6613, Institutd’Acoustique—Graduate School (IA-GS), CNRS, Le Mans Université, France

3

Instituto de Instrumentación para Imagen Molecular, Consejo Superior de Investigaciones Científicas,
Universitat Politècnica de València, Camino de vera s/n, València 46022, Spain, Spain
4

Institut Jean Lamour, University of Lorraine, 2 allée André Guinier, 54011 Nancy, France, France

ABSTRACT
Plane waves are usually assumed to design acoustic metamaterials. This assumption is most of the times
satisfying, but prevents from accounting for the possible evanescent coupling between the elements of theacoustic metamaterial. In this talk, we present a hybrid analytical model between the transfer-matrix method
inside the structured material and the modal decomposition in the surrounding domains. We apply this method
to the analysis of the scattering properties of a mirror-symmetric absorber made of several slits loaded by
Helmholtz resonators. The transfer-matrix model is used to describe the propagation within each slit, while
the modal decomposition is used to described the acoustic field in the surrounding domains. Modal
decomposition assumes Bloch waves; thus, it accounts for the possible evanescent coupling between the slits.
The domains are matched through the continuity conditions between the exterior media and each slit. Thismethod is coupled to an optimization procedure to design single frequency perfect mirror symmetricabsorbers. The results are numerically and experimentally validated showing good agreement with theproposed hybrid model.
Keywords: Sound, Insulation, Transmission

1. INTRODUCTION
Metamaterials made of Helmholtz resonators (HRs) have been exploited during the last year tocontrol acoustic waves [1-4]. Different applications have been developed to attenuate [5], to control
the directivity [6, 7] or to absorb acoustic waves [8-11]. In most of the cases the resulting structures
are periodic to simplify its modeling. Usually, the transfer matrix method is used to analyticallydescribe the behavior of these systems, i.e., assuming propagation of plane waves and neglecting thehigher orders. Moreover, the periodicity implies the existence of Bloch waves that can also produce
possible evanescent coupling between the building blocks of the metamaterials. Sometimes, and insome regions of frequencies these Bloch waves can be neglected, but in general and in reality, they
produce couplings that make the modeling by the transfer matrix method wrong.

In this work we show an anytically a hybrid model combining transfer matrix and modal
decomposition methods used to model a metascreen consisting of a periodic array of slits loaded bydifferent HRs. The transfer matrix model [12] is used to describe the propagation within each slit
while the modal decomposition is used to obtain the propagation in the exterior domain considering
the Bloch waves accounting for the possible evanescent coupling between the slits. The two domains
are coupled through the continuity conditions between the exterior medium and each slit. A full-wave
numerical simulation is used to validate the analytical model. A design of a metascreen made ofmirror-symmetric resonant building blocks possessing degenerate resonances is used to show thedramatic effect of the evanescent coupling on the acoustic perfect absorption. The metascreen isexperimentally tested in a square cross-sectional impedance tube showing very good agreement withthe analytical and numerical predictions.

2. THE SETUP
The schematic view of the metascreen design that we analyze in this work, shown in Fig. 1(a), is
a combination of slits loaded by one and two resonators [11]. The HRs are made by combining two
different elements playing the role of the neck and the cavity. Due to the symmetry of the system, thescattering is reciprocal and symmetric. The metascreen is radiated by a plane wave from the top to thebottom in the medium Θ("#) . Viscothermal losses are accounted for by using the Stinson’s equivalent
expressions for the complex and frequency dependent of the density and wavenumber in the slits,
necks and cavities of the structure [13].
(a)

1

p(0+)
x2
(2)
1

x
h(1)

h(2)

Γn

1

(c)
£(0+)

0.8

Γs

0.6

0.6

... L

0.4

0.4

x1

0.2

0.2

...
h

(a)
(b)

0.8

p(0{) £(0{)
0
700

750

800
850
Frequency (Hz)

0

900

750

800

850

Figure 1. (a) Schematic view of the analyzed metascreen made of two slits loaded by Helmholtz
resonators. (b) Analytical, numerical and experimental scattering coefficients of the metascreen. (c)Scattering coefficients considering only the plane wave approximation, i.e., the transfer matrix method.

3. THEORETICAL MODELINGThe acoustic field in the exterior domains (Θ("#) and Θ("%) as defined in Fig. 1(a)), when the#⃗= (k& , k ' ) can be written as
system is excited by a plane wave k###𝑝"# = * +𝑒 %()!"(*!%+) 𝛿, + 𝑅, 𝑒 ()!" (*!%+) 0 𝑒 ()$"*$,##

𝑝("%) = * 𝑇, 𝑒 ()$"*$%()!"*!
,

𝑖𝑛

in

Θ"#

Θ"% ,

"
where the subindex 𝑞 indicates the order of the Bloch wave, with 𝑘&,= 2𝑞𝜋/2𝑑" + 𝑘&- and
'

"
" '
"
####⃗" ?@ − B𝑘&,𝑘',
= =>?𝑘C , with 𝑅𝑒B𝑘',
C ≥ 0, 𝑅, and 𝑇, are the reflection and transmission coefficientsof the 𝑞-th Bloch wave, 𝛿, is the Kronecker's delta and 𝐿 is the thickness of the acoustic metascreen.
Wave propagation in each slit is modeled by the transfer matrix method (TMM), in which the HRs are
considered as 1D point scatterers. Matrices describing the propagation in the slit, to the side branch resonatorsand to the section mismatch are assembled giving the total transfer matrix. The TMM assumes the planewave propagation in each element, which is valid in the low frequency regime. The radiation corrections areincluded in the impedances of the resonators to mimic the effect of the higher order modes. Therefore, thewave propagation through the slit n can be modeled as

(.)
𝑝(.)
𝑇
G (.) I
= J &&(.)
𝑣' * /+
𝑇'&
!

(.)

𝑝(.)
G
,
K
(.) I
(.)
𝑇'' 𝑣' * /"𝑇&'!

where the final transfer matrix is given by the product of the transfer matrices of the N elements in the n-th
slit.
Once the acoustic fields inside each domain of the problem are defined, the continuity boundary conditions
must be applied. These conditions read as𝑝("#) = 𝑝. 𝑜𝑛 Γ . 𝑎𝑡 𝑥' = 𝐿,
𝑝("-) = 𝑝. 𝑜𝑛 Γ . 𝑎𝑡 𝑥' = 0,("#)
𝑣'
= 𝑣'. 𝑜𝑛 Γ . 𝑎𝑡 𝑥' = 𝐿,("#)
𝑣'
= 0 𝑜𝑛 Γ1 𝑎𝑡 𝑥' = 𝐿,("-)
𝑣' = 𝑣'. 𝑜𝑛 Γ . 𝑎𝑡 𝑥' = 0,("-)
𝑣' = 0 𝑜𝑛 Γ1 𝑎𝑡 𝑥' = 0,
where Γ2 represents the rigid interfaces on the two faces of the metascreen, and Γ . is the interface between%&
the n-th slit and the surrounding media Θ(" ) and Θ(" ) . These conditions lead to a system of equations
after projection on the adequate orthogonal modes. The system is solved for the scattering coefficients of thesystem.
In order to validate this model, we have used a finite element method (FEM) simulation. A plane wave
impinges the system and the complete geometry is considered using the radiation conditions that simulate
the Sommerfeld conditions at the limits of the numerical domain.

4. RESULTS
The model is used to calculate the absorption coefficient of the designed metascreen. Figure 1(b) shows thescattering coefficients of the full transmission problem. The agreement between the model, the numericalsimulations and the experimental results is very good. Note here that the presence of the Bloch waves is
crucial in this kind of systems: if only plane waves had been accounted for in the system, the model would
not have reproduced the whole wave process because the evanescent coupling would have been neglected asshown in Fig. 1(c). When the Bloch waves are considered, a perfect absorption peak is observed at 800 Hz
as analytically, numerically and experimentally observed. It is worth noting here that perfect absorption isvery sensible to the geometry of the resonators, and this would explain the slight discrepancies between theanalytical or numerical predictions and the experimental results.

5. CONCLUSIONS
The resonant scattering produced by the metascreen is analytically studied by a hybrid model mixing thetransfer matrix method and the modal decomposition accounting for the Bloch waves in order to consider the
possible coupling between the slits. Note that this coupling is particularly important for the present topology.
If no higher order Bloch waves had been accounted for, i.e., if only the plane wave had been accounted foroutside the metascreen, the analytical model would not have been in agreement with neither the full-wave
numerical simulation nor the experimental results. The structure is experimentally tested in a square
impedance tube showing good agreement with the analytical and numerical predictions. The Physics behind
this work will motivate the development of broadband and perfect absorption for ventilation problems using
cascade processes similar to the case of the rainbow trapping for the unidirectional absorption.

ACKNOWLEDGEMENTS
The authors gratefully acknowledge the ANR-RGC METARoom (ANR-18-CE08-0021) project.
This publication is part of the R&D&I Project/Grant No. PID2020-112759GB-I00 funded by
MCIN/AEI/10.13039/501100011033/.

REFERENCES
1. A. Merkel, G. Theocharis, O. Richoux, V. Romero-García, and V. Pagneux, “Control of acoustic absorption in one-dimensional scattering by resonant scatterers,” Appl. Phys. Lett. 107, 244102 (2015),
https://doi.org/10.1063/1.4938121.
2. J.-P. Groby, W. Huang, A. Lardeau, and Y. Aurégan, “The use of slow waves to design simple soundabsorbing materials,” J. Appl. Phys. 117, 124903 (2015), https://doi.org/10.1063/1.4915115.
3. J.-P. Groby, R. Pommier, and Y. Aurégan, “Use of slow sound to design perfect and broadband passive
sound ab- sorbing materials,” J. Acoust. Soc. Am. 139, 1660–1671 (2016),
https://doi.org/10.1121/1.4945101.
4. P. Wei, C. Croënne, S. Tak Chu, and J. Li, “Symmetrical and anti-symmetrical coherent perfectabsorption
foracoustic
waves,”
Appl.
Phys.
Lett.
104,
121902(2014),
https://doi.org/10.1063/1.4869462.
5. G. Theocharis, O. Richoux, V. Romero-García, A. Merkel, and V. Tournat, “Limits of slow sound andtransparency in lossy locally resonant periodic structures,” New J. Phys. 16, 093017 (2014).
6. N. Jiménez, T.J. Cox, V. Romero-García, J.-P. Groby, Metadiffusers: Deep-subwavelength sounddiffusers, Scientific reports 7 (1), 1-12, (2017).
7. E. Ballestero, N. Jiménez, J.-P. Groby, H. Aygun, S. Dance, V. Romero-García, Metadiffusers for quasiperfect and broadband sound diffusion, Applied Physics Letters 119 (4), 044101, (2021)
8. N. Jiménez, W. Huang, V. Romero-García, V. Pag- neux, and J.-P. Groby, “Ultra-thin metamaterial for
perfect and quasi-omnidirectional sound absorption,” Appl. Phys. Lett. 109, 121902 (2016),
https://doi.org/10.1063/1.4962328.
9. N. Jiménez, V. Romero-García, V. Pagneux, and J.-P. Groby, “Rainbow-trapping absorbers: Broadband,
perfect and asymmetric sound absorption by subwavelength panels for transmission problems,” Sci.
Rep. 7, 13595 (2017).
10. M. Yang, C. Meng, C. Fu, Y. Li, Z. Yang, and P. Sheng, “Subwavelength total acoustic absorption withdegenerate resonators,” Appl. Phys. Lett. 107, 104104 (2015), https://doi.org/10.1063/1.4930944.
11. V. Romero-García, N. Jiménez, J.-P. Groby, A. Merkel, V. Tournat, G. Theocharis, O. Richoux, V.
Pagneux, Perfect absorption in mirror-symmetric acoustic metascreens, Physical Review Applied 14 (5),054055, (2020)
12. Jiménez N., Groby JP., Romero-García V. (2021) The Transfer Matrix Method in Acoustics. In: JiménezN., Umnova O., Groby JP. (eds) Acoustic Waves in Periodic Structures, Metamaterials, and PorousMedia. Topics in Applied Physics, vol 143, pp. 103-164. Springer, Cham. https://doi.org/10.1007/9783-030-84300-7_4
13. M. R. Stinson, “The propagation of plane sound waves in narrow and wide circular tubes, andgeneralization to uniform tubes of arbitrary cross-sectional shape,” J. Acoust. Soc. Am. 89, 550–558(1991), https://doi.org/10.1121/1.400379.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0053

Shannon entropy applied to acoustics: Characterization of avoided
crossings in acoustic superlattices
José SÁNCHEZ-DEHESA 1 and J. Ricardo ARIAS-GONZALEZ 2
1

Department of Electronic Engineering, Wave Phenomena Group, Universitat Politècnica de València, Spain

2

Department of Applied Physics, Centro de Tecnologías Físicas, Universitat Politècnica de València, Spain

ABSTRACT
Shannon’s information entropy is here employed as a characterization tool in acoustics. The displacement
field u(r) is used to define the probability distribution function P(r), which is further embedded in the wellknown formula of Shannon entropy, Su(r). We show that Su(r) describes fairly well the delocalization of theacoustic modes appearing in the Zener-like effect characterized in ultrasonic superlattices [see Phys. Rev.
Lett., 98, 134301 (2007)]. We analyze the case of a multilayer made of coupled water cavities separated bymetamaterial slabs. The results demonstrate that the acoustic’s Shannon entropy maximizes at the criticalgradient where the two eigenmodes anticross, indicating maximum spreading of both modes. Our proposal
can be considered as an alternative to the inverse participation ratio, which is usually employed tocharacterize localization effects.
Keywords: Shannon entropy, Zener-like effect, acoustic superlattices.

1. INTRODUCTION

Shannon’s informational entropy [1] has been applied in the study of novel and counterintuitive
ways of processing and transmitting information in the quantum realm. More recently, the applicationof Shannon’s formula to different physical phenomena in crystallography [2] and atomic physics [3]opened new avenues of interpretation.
This work introduces Shannon entropy in specific situations in the acoustic realm where thelocalization of acoustic eigenmodes plays a relevant role. To characterize the spatial spreading of
eigenmodes there are several indicators like the inverse participation ratio [4]. However, since theShannon entropy is a more global concept, we explore here its application as a new characterization
tool in acoustics.
The proposed formula for the Shannon entropy in acoustics is based on the displacement fields ofthe corresponding eigenmodes and it is applied to characterize the delocalization effect taking placein the Zener-like effect discovered in acoustic superlattices [5, 6].

2. SHANNON ENTROPY IN ACOUSTICS𝑙𝑙

The formula for the Shannon entropy in atomic physics, 𝑆𝑆𝜚𝜚 , employs the probability density
distribution of a given electronic state 𝜚𝜚(𝑟𝑟⃗) = |Ψ(𝑟𝑟⃗)|2 to define 𝑆𝑆𝜚𝜚 = − ∫ 𝜚𝜚(𝑟𝑟⃗) 𝑙𝑙𝑙𝑙 (𝑟𝑟⃗)𝑑𝑑𝑟𝑟⃗. To introduce
the corresponding formula in acoustics, we exploit the analogy between bound or quasi-bound statesin electronic superlattices with the localized or quasi-localized states in slabs made of cylindrical rods
embedded in a fluid-like background [7]. The displacement field,𝑢𝑢(𝑟𝑟⃗), characterize the localizationof the acoustic eigenstates, which will decay after some delay time. Therefore, the corresponding
probability distribution function is:1
2

jsdehesa@upv.es
ricardo.ariasgonzalez@upv.es

𝑃𝑃(𝑟𝑟⃗) =|𝑢𝑢(𝑟𝑟⃗)|2,
� |𝑢𝑢(𝑟𝑟 2∫ ⃗)| 𝑑𝑑𝑟𝑟⃗(1)which is obtained by normalizing the square of the displacement field of the acoustic level underconsideration. Therefore, the Shannon entropy is
𝑆𝑆𝑢𝑢 = − ∫ 𝒫𝒫(𝑟𝑟⃗)𝑙𝑙𝑙𝑙𝑙𝑙(𝑟𝑟⃗)𝑑𝑑𝑟𝑟⃗(2)This quantity should be considered as a measure of the uncertainty of the localization of sound. Noticethat this quantity increases with the spreading of the displacement field of the acoustic eigenstate and,
consequently, increases with increasing uncertainty. Since we are in the approach of linear acoustics,
the Shannon entropy in (1) can be also obtained by using alternative expressions for the probability𝒫𝒫(𝑟𝑟⃗) in (1), involving the particle velocity or the acoustic pressure.

3. CHARACTERIZATION OF AVOIDED CROSSING IN ACOUSTICSUPERLATTICES

Avoided crossing between two electronic states has been characterized, for example, in atomic
physics when the levels are submitted to strong parallel magnetic and electric fields [3]. In acoustics,
the phenomenon of avoided crossing between two acoustics eigenmodes has been described in the
acoustic analog of the Zener effect, where one component of an acoustic superlattice consists of a
gradient of thicknesses [5-6]. In what follows, it is shown how the expressions (1)-(2) defining the
Shannon entropy in acoustics describe fairly well the localization effects taking place around thecritical gradient where the avoided crossing appears.
Let us consider the case of a finite acoustic superlattice consisting of m coupled water cavities
W m separated by m+1 slabs A made of a two-dimensional periodic distribution of rigid rods embeddedin the water background [6,8. In the homogenization limit, slabs A can be tailored with the following
dynamical mass density and sound speed: 𝜌𝜌𝐴𝐴 = 5𝜌𝜌𝑊𝑊 and 𝑐𝑐𝐴𝐴 = 0.78𝑐𝑐𝑊𝑊 , where ρ A and cW are,respectively, the density and sound speed in the water; i.e., ρ W=1 g/cm3 and c W=1480 m/s. The results
for the case of a perfect superlattice (with m=8 and dW=2d A) are represented in Figure 1(a), where thetransmission spectrum (right panel) shows two clear minibands MB1 and MB2. The left panel
represents the log|𝑢𝑢(𝑧𝑧)|2 and shows that modes in MB1 are strongly localized in the water cavities
while the localization of modes in MB2 is not clearly defied. In order to observe the avoided crossing,
we follow the procedure described in [5] and introduce a gradient in the thicknesses of water cavities,∆(1/d Wℓ ), which is defined as∆ (1/d W) = [(1/d Wℓ ) – (1/dWℓ−1] / (1/dW1),(3)

where dW1 = 0.08 cm is the thickness of the first (ℓ = 1) cavity.
Figure 1(b) shows the results corresponding to the critical gradient (10.04%) where the Zenertunneling (ZT) like effect takes place. The transmission profile shows a double peak defined by ZT in
Fig. 1(b).

Figure 1: (a) Transfer matrix calculation for the square of the displacement field amplitude of thesound in the acoustic superlattice schematically deployed on top. White (grayed) stripes representthe water (metamaterial) slabs. (b) The results correspond to the case in which the water layers havea critical gradient of thicknesses and the Zener tunneling (ZT) appears.
The double peak indicating the avoided crossing is better described in Fig. 2, where the shape of
the ZT peak is represented for several values of the gradient (in %). The minimum separation between
levels corresponds to the critical gradient.

Figure 2. Transmission spectrum of an acoustic
multilayer made of eight layers of water
surrounded by nine layers of metamaterial slabs.
The transmission is plotted for several values of
the gradient (in %) of thicknesses applied to thewater cavities. All the metamaterial slabs havethe same thickness d A. The frequencies are in
reduced units, where ω0 is the central frequency
at each gradient.

The frequency and Shannon entropy on the pair of levels involved in the avoided crossing are
represented in Fig. 3(a) and 3(b), respectively. In Fig. 3(a), it is observed how the lower level u 1 (blueline) and the upper-level u2 (red line) anticross at the critical gradient of 10.04%. On the other hand,in Fig. 3(b), it is observed how the Shannon entropy takes maximum values for both levels at thecritical gradient, indicating the maximum delocalization of both levels.

Figure 3: (a) Frequency of the two interacting modes calculated as a function of the gradient of thewater cavities defining the acoustic superlattice. (b) The corresponding Shannon entropy calculatedwith the expressions (1)-(2).
In order to support the claim of maximum delocalization described by the values of the Shannon
entropy at the critical gradient, Fig. 4 depicts the amplitude of the acoustic modes for three values of
the gradient. Notice how the spreading of the modes is dependent on the gradient defining the
thicknesses of the water layers. Particularly, at the critical gradient, where a “bonding-antibonding”
combination of the respective “wavefunction” takes place, the spreading of both levels is maximum.
Therefore, the Shannon entropy can be considered a useful characterization tool to describe the anticrossing of levels appearing in acoustic structures. A maximum value of the Shannon entropy indicatesthe maximum spreading of the acoustic levels and, therefore, the exact location of the avoided-

crossing effect [9].

Figure 4: Amplitude of the acoustic modes u 1 (blue lines) and u 2 (red lines) involved in the Zenerlike acoustic phenomenon. The amplitude is depicted for three values of the gradient defining the
thicknesses, d Wℓ, of the water cavities. The vertical grayed stripes define the metamaterial slabs
enclosing the water cavities. At the critical gradient of 10.04% (central panel), both modes show
the maximum spreading across the acoustic structure.

ACKNOWLEDGEMENTS

Both authors acknowledge Helios Sanchis-Alepuz for his contribution to this work. JS-D
acknowledges the financial support by the RDI grant PID2020-112759GB-2I100 funded by
MCIN/AEI/10.13039/501100011033. The work of JRA-G has been supported by the grant with Ref.
PID2019-107391RB-I00.

REFERENCES

1. Shannon CE. A mathematical theory of communications Bell Syst. Tech. J. 1948; 27: 379-656.
2. Menendez-Velazquez A, Gracía-Granda SB, Stearns SD. Adaptive Signal Processing. Englewood Cliffs,
New Jersey, USA: Prentice-Hall; 1985.
3. Gonzalez-Férez R, Dehesa JS. Shannon-entropy as an indicator of atomic avoided crossing in strong
parallel magnetic and electric fields. Phys. Rev. Lett., 2003; 91: 113001.
4. Thouless DJ. Electrons in disordered systems and the theory of localization. Phys. Rep 1974; 13: 93-142.
5. Sánchez-Dehesa J, Sanchis-Alepuz H, Kosevich YA, Torrent D. Acoustic analog of electronic Blochoscillations and Zener tunneling, J. Acoust. Soc. Am. 2006; 120(1):3283-3283.
6. Sanchis-Alepuz H, Kosevich YA, Sánchez-Dehesa J. Acoustic Analogue of Electronic Bloch Oscillations
and Resonant Zener Tunneling in Ultrasonic Superlattices. Phys. Rev. Lett. 2007; 98: 134301.
7. Sanchis L, Håkansson, Cervera F, Sánchez-Dehesa J. Acoustic interferometers based on two-dimensional
arrays of rigid cylinders in air. Phys. Rev. B. 2003; 67: 035422.
8. He Z, Peng S, Cai F, Ke M, Liu Z. Acoustic Bloch oscillations in a two-dimensional phononic crystal,Phys. Rev. E. 2007; 76: 056605
9. Sánchez-Dehesa J, Arias-Gonzalez JR. Characterization of avoided crossings in acoustic superlattices: The
Shannon entropy in acoustics. Frontiers in Physics 2022; 10: 971171.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0212

Deep learning models for acoustic scattering problems
Waqas W. Ahmed1; Mohamed Farhat1; Pai-Yen Chen2; Xiangliang Zhang1,3; Ying Wu1
1

King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia
2

University of Illinois at Chicago, Chicago, IL60607, United States of America

3

University of Notre Dame, Notre Dame, IN 46556, United States of America

ABSTRACTWe develop deep learning (DL) models based on discriminative and generative networks to solve the forward
and inverse acoustic scattering problems and show how these models streamlines the inverse design processby eliminating the degenerate solution space. Specifically, we present DL frameworks for designing
broadband acoustic cloaks and arbitrarily-shape acoustic object recognition for underwater applications.
Keywords: Inverse scattering, Deep Learning, Acoustic Cloaks, Shape recognition

1. INTRODUCTION
Wave physics models based on analytical and numerical schemes often rely on hit and trial
modeling of physical variables or physics-informed optimization for 'on demand' scattering properties.
In general, those methods are computationally expensive, and become prohibitively slow especiallywhen devices become more complex such that the degrees of freedom significantly increase in thedesign space. As a result, intelligent models that can capture the underlying correlation between
various physical quantities and streamline the inverse design process are desirable. Recently, data
driven approaches were on the main stream of research due to their ability to solve a wide range ofphysical problems with unprecedented speed and accuracy[1,2]. Despite of this progress, there havebeen limited efforts to using deep learning techniques to learn input-output mappings with diverse
designs and inverse design insights. In this work, novel deep learning models are developed for thedesign of broadband acoustic cloak and arbitrary object recognition in water. As a surrogate physicalmodel, discriminative neural networks are designed for forward process of one -to-one mapping.
However, generative neural networks based on variational inference and adversarial learning aredesigned for the inverse process of one-to-many mapping, where the fundamental issue of
nonuniqueness is addressed by learning the latent distribution of data structures.

2. Results
2.1 Broadband Acoustic Cloaks
Acoustic cloaks, making an object invisible for sound waves, are usually build from artificiallystructured acoustic metamaterials. The realization of metamaterial-based cloak faces multiple
challenges, such as stringent requirements on the material parameters and narrow operating bandwidth.
To address these issues, we propose a design of acoustic cloak with a four-layer core-shell
configuration driven by machine learning as shown in Fig. 1(a). The performance of the system isquantified by the ratio of total scattering cross-section (TSCS) spectra for the cloaked object and thebare object, computed using transfer matrix method (TMM). The design space (D) consists of eight
design parameters, including material and geometrical parameters, and response space (R) is
discretized TSCS spectrum in the frequency range 0 ≤ 𝑘0 𝑎 ≤ 𝜋. To achieve broadband invisiblity,
we first train a forward neural network (FNN) to learn the mapping between the design and response
space using training data. Once trained, the pretrained FNN is cascaded behind the inverse neural(INN) network forming a probabilistic encoder-decoder-like structure to capture the underlying physics
1

ying.wu@kaust.edu.sa

to generate a variety of design parameters for one desired spectral response. In additions, it helps to
identify the sensitivity of design parameters through statistical learning and uncovers that material
parameters (bulk moduli) are less sensitive than geometrical paramters ( shell thickness) on the cloaking
behavior [3].
2.2 Acoustic Object Shape Recognition
Shape recognition, which relies on inverse scattering, has a wide range of applications, such as
sonar detection, remote sensing, underwater imaging, and nondestructive testing. For underwater
objects, however, accurate shape recognition is extremely challenging because of high nonlinearity,
large background index contrast, and one-to-many mapping in the solution space. Here, we consider
a 2D arbitrarily shaped steel scatterer submerged in water and propose the idea of a latent space tolearn the nonlinear mapping between acoustic objects and its far-field scattering patterns once
illuminated [see Fig. 1(b)]. The non-unique solution space is eliminated by training the probalistic
networks with multi-frequency phaseless far-field amplitudes. The scatterer is discretized as a pixelbased binary image, with zeros and ones representing water and steel, respectively. A commercial
software based on finite-element, COMSOL Multiphysics, is used to simulate the far-field patterns ofrandom geometries at five different frequencies across the full angular range. The proposed deeplearning model employs an adversarial autoencoder as a training method for Generator, which
generates geometric patterns from the latent distribution of arbitrary-shaped objects. The inverse
neural network (INN) predicts the 2D object with the matrix of 64 × 64 for given the multifrequency
far-field patterns. The method can be used to determine the shape of arbitrary objects using
multifrequency far-field data in half-angular space, which allows it to be applied to detecting
submarines, fish species, and other underwater activity [4].

Fig. 1. Deep learning frameworks for inverse acoustic scattering predictions (a) broadband cloak
design (b) arbitrary-shape objection detection.

To conclude, we present generative models for learning latent distributions of discrete data structures(for acoustic cloaks) and image data structures (for acoustic object recognition). With a wide variety ofdesign parameters, deep learning models demonstrate excellent performance over test data, making themhighly appealing for feasible acoustic devices. It opens up new opportunities for solving complex
inverse scattering problems since no complex analytical calculations are required.

ACKNOWLEDGEMENTS
The work described in here is supported by King Abdullah University of Science and Technology(KAUST) Artificial Intelligence Initiative Fund and KAUST Baseline Research Fund No. BAS/1/162601-01.

REFERENCES
1. Jiang J, Chen M, Fan JA. Deep neural networks for the evaluation and design of photonic devices.
2021; Nat Rev Mater 6, 679–700.
2. Bianco MJ, Gerstoft P, Traer J, Ozanich E, Roch MA, Gannot S, Deledalle CA. Machine learning inacoustics: Theory and applications. 2019: J Acoust Soc Am 146(5); 3590-3628.
3. Ahmed WW, Farhat M, Zhang X, and Wu Y. Deterministic and probabilistic deep learning models forinverse design of broadband acoustic cloak. 2021; Phys Rev Research, 3(1): 013142.
4. Ahmed WW, Farhat M, Chen P.-Y, Zhang X, Wu Y. A Generative deep learning approach for shaperecognition of arbitrary objects from phaseless acoustic scattering data 2022; arXiv preprint
arXiv:2207.05433

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0241

Decay Coefficient of Sound in Phononic Crystals with Viscous
Background
Jesús ARRIAGA1; Martín IBARIAS2; Arkadii KROKHIN3
1

Institute of Physics, Benemérita Universidad Autónoma de Puebla, México

2

Institute of Physics, Benemérita Universidad Autónoma de Puebla, México
3

Department of Physics, University of North Texas, USA

ABSTRACT
Using the analytical results in phonon crystals, we calculate the decay coefficient of sound waves
propagating at low-frequencies in a 2D phononic crystal with viscous background. The analytical result is
valid for structures with arbitrary Bravais lattices and arbitrary cross sections of the cylinders. It isdemonstrated that the effective viscosity of phononic crystal may exceed by 2-4 orders of magnitude theviscosity of the background fluid. Moreover, the decay coefficient exhibits dependence on the direction of
propagation, i.e., a homogenized phononic crystal behaves like anisotropic viscous fluid. Strong dependence
on the filling fraction of solid scatterers gives the possibility of tuning the effective viscosity of phononic
structure that may be used for evaluation of the efficiency of phononic devices.
Keywords: Phononic Crystals, Homogenization

1. INTRODUCTION
The pressure in a plane sound wave propagating in a viscous homogeneous fluid decays
exponentially with distance according to, p(x) ∼ e −γ 0 x , The decay coefficient γ0 grows quadraticallywith frequency ω, and depends on fluid density ρ, two viscosity coefficients η and ξ, and speed ofsound c [1]. The decay length of sound (1/γ 0) in water at frequency of 50 kHz is about15 km, meaningthat in water, dissipative losses can be ignored. However, dissipative losses is strongly increased if
sound wave meets a solid object on its way. Oscillating fluid sticks to solid boundaries, forming anarrow viscous layer of thickness δ, where velocity gradients greatly exceed the gradients in a free
fluid, leading to much higher viscous losses. In a phononic crystal, the volume occupied by the viscousboundary layers formed around solid scatterers usually constitutes a small part of the volume of thewhole sample. It, however, strongly reduces acoustic transmission. For example, if sound propagates
through a 2D periodic lattice of cylindrical rods embedded in water, the decay length is reduced by
two orders of magnitude. Here, we use the recent develop analytical approach, for exact calculationof the attenuation coefficient in a two-dimensional (2D) phononic crystal in the low-frequency limit[2], to tune the decay coefficient for different 2D Bravais lattices and different cross sections of thescatterers.

2. DISSIPATION OF ACOUSTIC ENERGY
Propagating sound waves generate vibrations of pressure p(r,t)=p(r)exp( -iωt) and velocityv(r,t)=v(r)exp(-iωt). Within the boundary layer δ, velocity in a viscous fluid decays exponentially fromits value in the bulk to zero at the interface of a motionless hard scatterer. The acoustic power (per unit
length of a cylinder) dissipated around a hard scatterer is given by the following contour integral [1]:arriaga@ifuap.buap.mx
mibarias@ifuap.buap.mx
3
Arkadii.Krokhin@unt.edu
1
2

1

2

Q = 2 2   v(r) dl

(1)

l0

The integration runs over contour l0, which is the circumference of the scatterer. This equation isused for the calculation of the dissipated power within a unit cell of 2D phononic crystal, and is valid
if the interface can be considered as flat within the lengths ~, i.e., <<L0 where L 0 is the length ofthe contour l0 that separates fluid from solid. This inequality, together with the condition ofhomogenization, ka 0<<1, defines the frequency interval where the proposed theory is valid. The decaycoefficient for a wave propagating in a homogeneous dissipative medium is given by the ratio,
 ph =

Q(2)

2c eff E c

The explicit for of the factors of eq. (3) appears in Ref. [3]. These factors take into account the
microstructure of the phononic crystal and the details of the formation of the boundary layer.
2.1 Decay coefficient of an anisotropic lattice
We apply the previous results for a phononic crystal for three different lattices of solid cylinders
in viscous water background for different scatterers but the same filling fraction, f = 30%. The decaycoefficient given by Eq. (2) is plotted in a polar diagram in Fig. 1, showing the angular dependenceof the decay coefficient. The three curves appearing are ellipses with semiaxes representing maximum
and minimum attenuation of sound along the corresponding directions. The plots in Fig. (1)demonstrate that viscous losses strongly depend on the shape of the scatterers. The main factors thatdefine viscous attenuation are the narrow channels between the neighboring scatterers, the number ofcorners per unit length and their sharpness. Combination of these factors leads to a complicatedanisotropic pattern of viscous losses which is impossible to analyze using only qualitative or
phenomenological approach.

Figure 1. Normalized decay coefficient of sound for anisotropic unit ce lls as function of angle θ.
Figures inset: square lattice with rectangular rods oriented by its longer side along y -direction,
hexagonal lattice with rectangular rods, and square lattice with equilateral triangular rods.

ACKNOWLEDGEMENTS
This work was partially supported by CONACyT (México) through Grant No. A1-S-23120.
[1]. [2] [3] M. Ibarias, Yu. Zubov, J. Arriaga, and A. A. Krokhin, Phys. Rev. Res., 2, pp. 022053(R),
2020.

REFERENCES
1. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. Elsevier, Oxford, 1984.
2. A. A. Krokhin, J. Arriaga, and L. N. Gumen, Speed of Sound in Periodic Elastic Composites, Phys. Rev.
Lett. 2003; 91:264302.
3. M. Ibarias, Yu. Zukov, J. Arriaga, and A. A. Krokhin, Phononic Crystal as a homogeneous viscous
metamaterial, Phys. Rev. Res. 2020; 2:022053.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022 in Gyeongju, Korea

ABS-0402

Design and experiment of metaliner for sound insulation in a ductwith flow
Tae Seok Oh 1; Wonju JEON1
1

Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Republic of
Korea

ABSTRACT
We propose a metaliner that insulates duct noise while allowing fluid passage with little flow resistance. A
unit cell of a metaliner consists of two different subwavelength Helmholtz resonators, and the unit cells arealternately arranged along the direction of sound propagation underneath the duct. To predict sound insulationof the metaliner in a duct with grazing flow, we establish an effective impedance model of the metaliner,
considering the effect of flow. We design metaliners with target frequencies of 500Hz or 1000Hz that achieve
high transmission loss for various flow speeds. The experimental results show that designed metaliners
exhibit transmission loss above 45 dB/m within ±5% of target frequencies for different flow speeds of 0,17, and 34 m/s.

Keywords: Metaliner, Sound insulation, Flow

1. INTRODUCTION

Noise attenuation in ducts is important in a variety of industrial applications, such as air intake ofcooling towers or vehicle ventilation systems. Acoustic treatments, such as conventional acoustic
louvers or mufflers, have been widely used to reduce duct noise (1-2), but they have poor soundattenuation in the low-frequency range and worsen the functionality of the products or facilities byblocking the air passage due to a low air-flow area. Recently, acoustic metamaterials have beendeveloped to solve the problem of low-frequency sound insulation while allowing fluid passage (3-6).
However, the previous studies focused on improving the sound insulation of the acoustic
metamaterials for a stationary medium, so they could not guarantee sound insulation in a duct with aflow.
In this study, we propose a metaliner that can insulate duct noise in the presence of grazing flow.
First, we establish a theoretical model for the effective impedance of the metaliner that accounts forthe effects of flow, based on the recent study (7). The transmission loss of the metaliner in a duct witha flow is predicted by numerical simulation. Then, we design metaliners that exhibit transmission loss
per unit length above 45 dB/m within ± 5% of the target frequencies of 500 and 1000 Hz for different
flow speeds of 0, 17, and 34 m/s. The sound insulation for different flow speeds is experimentally
validated.

2. RESULTS AND DISCUSSIONS

Figure 1 describes the geometry of the metaliner. Two different Helmholtz resonators ofsubwavelength scale are arranged alternately along the direction of sound propagation under therectangular duct with an edge length of 51 mm. The metaliner is cuboid with dimensions of
406 mm × 51 mm × 30 mm.
wonju.jeon@kaist.ac.kr

Figure 1 – Geometry of a metaliner installed under a rectangular duct
Figure 2 shows the transmission loss spectra of designed metaliners with target frequencies of 500
Hz and 1000 Hz for different average flow speeds of 0, 17, and 34 m/s. The experimental results
show good agreement with the numerical results. Experiments confirm that the metaliners maintaintransmission loss per unit length above 45 dB/m in the target frequency range (475–525 Hz or 950–
1050 Hz) for average flow speeds of 0 , 17 , and 34 m/s. In addition, the sound insulation was
realized not only at target flow speeds (0, 17, 34 m/s) but also at flow speeds between 0–34 m/s, asdetailed in reference (8).

Figure 2 – Transmission loss per unit length according to the flow speeds of 0, 17, and 34 m/s for the
metaliner with target frequencies of (a) 500 Hz and (b) 1000 Hz

REFERENCES

1. A. Craggs, A finite method for modelling dissipative mufflers with a locally reactive lining, J. Sound Vib.,
54 (1977) 285-296.
2. E. B. Viveiros, B. M. Gibbs, S. N. Y. Gerges, Measurement of sound insulation of acoustic louvers by an
impulse method, Appl. Acoust., 63 (2002) 1301-1313.
3. H. -L. Zhang, Y. -F. Zhu, B. Liang, J. Yang, J. Yang, and J. -C. Cheng, Sound insulation in a hollow pipewith subwavelength thickness, Sci. Rep., 7 (2016) 44106.
4. X. Yu, Z. Lu, L. Cheng, and F. Cui, On the sound insulation of acoustic metasurface using a sub-structuring
approach, J. Sound Vib., 401 (2017) 190-203.
5. R. Ghaffarivardavagh, J. Nikolajczyk, S. Anderson, and X. Zhang, Ultra-open acoustic metamaterial
silencer based on Fano-like interference, Phys. Rev. B, 99 (2019) 024302.
6. J. W. Jung, J. E. Kim, and J. W. Lee, Acoustic metamaterial panel for both fluid passage and broadband
soundproofing in the audible frequency range, Appl. Phys. Lett., 112 (2018) 041903.
7. H. Ryoo and W. Jeon, Dual-frequency sound-absorbing metasurface based on visco-thermal effects withfrequency dependence, J. Appl. Phys., 123 (2018) 115110.
8. T. S. Oh and W. Jeon, Acoustic metaliners for sound insulation in a duct with little flow resistance, Appl.
Phys. Lett., 120 (2022) 044103.

ABS-0448

Manipulation of scattered sound fields with spatiotemporally modulated
acoustic metasurfaces
Janghoon KANG(1) , Michael R. HABERMAN(1)(1) Walker Department of Mechanical Engineering, The University of Texas at Austin; Austin, Texas, USA;jh3010.kang@utexas.edu, haberman@utexas.edu

ABSTRACT
Spatiotemporal modulation of material properties has been used in unbounded electromagnetic and acoustic
domains to provide enhanced control of propagating waves via nonreciprocity. This idea was recently expanded
to consider the vibration of finite-length Euler beams with spatiotemporally modulated material properties byintroducing a modal expansion technique that includes modulation harmonics to show that losses are required
to observe nonreciprocal vibrational motion [Goldsberry et al., Phys. Rev., B 102, 2020]. Using a similar same
modal expansion method, we have shown that spatiotemporally modulated surface admittance at the boundary ofan acoustic domain improves sound diffuser performance by altering the direction of scattered diffraction modes
via the introduction of modulation harmonics for all diffracted orders [Kang et al., J. Acoust. Soc. Amer., 150(4),A108, (2021)]. This work employs the same generalized semi-analytical model to investigate the effects ofthe modulation function on the acoustic field scattered from spatiotemporally modulated surfaces. We consider
parameters such as the modulation frequency and amplitude as well as functional forms, e.g. superposition ofsinusoids. Results are presented in terms of the angular dependence of the scattered field and we discuss how the
scattered sound field may be manipulated for either improved diffusion or directional dependence solely basedon choice of modulation function.

Keywords: Spatiotemporal modulation, Nonreciprocity, Acoustic Metasurface

1

INTRODUCTION

The principle of reciprocity for acoustic and elastic waves restricts the transmission between to points in anarbitrary domain to be symmetric when the source and receiver positions are interchanged (1, 2). One consequence of the Onsager-Casimir principle from thermodynamics is that transmission between any two points
in an acoustic domain will be reciprocal when the system is linear and time-invariant and obeys time-reversal
symmetry at scales that are much smaller than propagating wavelengths (1, 2). Based on this principle and later
macroscopic extensions, recent studies on the creation of systems that enable nonreciprocal wave transport havefocused on violating the assumptions such as a linearity and a time-invariance. One mechanism to generate nonreciprocal wave motion is to break time-invariance through the spatiotemporal modulation of material properties(2). Inspired by early work that imposed spatiotemporal modulation in the form of a pump wave in a medium(3), recent research has studied the case of spatiotemporal modulation of material properties, such as a density
or a stiffness, in the form of a traveling wave acoustic and elastic wave propagation (4, 5, 6, 7, 8, 9, 10).
In metasurfaces, the modulation has also been shown to have a potential for a nonreciprocal transmission andreflection (10, 11) as well as Doppler-like frequency shifting (12, 13) and total power combining (14). In thiswork, we focus on the temporal and spatial harmonics caused by the modulation of an input admittance of aperiodic reflective surface and its effects on scattered sound field. We show that the scattered sound energy canbe redirected into a range of directions due to the fact that surface impedance modulation introduce frequency
harmonics into the reflected field. As such, we observe changes in the backscattered directivity that differs fromthe diffraction orders that are proportional to the spatial periodicity of the surface. The result presented here is a

(a)

(b)

Figure 1. (a) The geometry of a flat surface with the incident sound pressure, pI (x, z,t), and the scatteredsound ps,n (x, z,t) of nth diffracted order and pth modulation harmonic propagating into the direction of θnp
with respect to the surface normal. The modulation function of a surface is represented as Y (x,t) = Y0 (x) +jYm cos (ωmt − km x) and the direction of modulation depends on the sign of km . For modulation propagating in
the positive x-direction, km > 0. (b) The modulation function for one spatial period and the case of Y0 = 0 and
Ym = 1 at t = 0.
generalization of recent work by our group (15) which showed that spatiotemporal modulation of the surface admittance of a sound diffuser can drastically improve the diffusion performance of a conventional sound diffusersuch as Schroeder diffuser (16) through the introduction of harmonics in both the frequency and wavenumberof the backscattered field (15). That work showed that the far field scattering pattern depends on the directionof the direction of spatiotemporal modulation, which indicates the possibility of generating highly nonreciprocal
reflective acoustic metasurfaces (AMS) by tuning the spatiotemporal modulation of the input acoustic admittance. Here we extend the a semi-analytical approach introduced to analyzing nonreciprocal vibration in Euler
Beams (7) to analyze the sound field scattered from a spatiotemporally modulated reflective AMS with spatial periodicity. We assume that the surface admittance is modulated in the form of a travelling wave withthe modulation frequency ωm and the amplitude Ym . We then perform parametric studies of the modulation to
identify pairs of modulation parameters that maximize the degree of the nonreciprocity in the reflection fromthe surface by calculating a nonreciprocity metric based on the difference in the angle-dependent backscatteredsound pressures depending on the direction of modulation. We show that the far field scattering response calculated with the semi-analytical model is nonreciprocal and that the degree of nonreciprocity can be varied withmoderate changes in the modulation function. The predictions from the semi-analytical model are then validated using a time-domain finite element method (FEM) where we observe similar results as obtained using thesemi-analytical model. The results demonstrate that, by using the proper selection of spatiotemporal modulationof surface admittance, one may tune the sound field scattered from a flat surface to achieve nonreciprocal or
diffuse directivity in the reflected field.

2

SEMI-ANALYTICAL MODEL

Previous work by the authors has resulted in a semi-analytical model to calculate the acoustic field scatteredfrom a sound diffuser whose input impedance is modulated spatiotemporally (15). Here we use the same approach to calculate the reflected field at each point on a flat surface that has spatiotemporally modulated input
admittance, as indicated in Fig. 1(a). The far-field, angle-dependent backscattered field is then determined using the approach outlined below. Given a normalized acoustic surface admittance, Y = ρ0 c0 /Z, where Z is thesurface input impedance, and the produce of the background fluid density, ρ0 , and sound speed, c0 , yields its
characteristic acoustic impedance, the flat surface is assumed to be modulated in the form of a travelling wave
represented in Fig, 1(b) in the form Y (x,t) = Y0 (x) + jYm cos (ωmt − km x). Here Y0 (x) is the uniform surface

admittance of the unmodulated sound diffuser, km = 2π/Λ is the modulation wavelength, ωm is the modulation
frequency, and Λ is the spatial period of the modulation. To represent the temporal modulation, the terms ofboth the incident and scattered sound pressure at the interface of the flat AMS and the background fluid are
expressed as a function of space and time using the following expansions on the modulation frequency andwavenumber,∞p (x, z,t) = pI (x, z,t) + ps (x, z,t) = aI e j(ω0 t−kx x+kz z) + ∑ an (t)e− j(βn x+γn z) .

(1)

n=−∞

In Eq. 1, ω0 represents the frequency of the incident wave, kx = k sin θI and kz = k cos θI represent the projectionof the incident wavenumber on the x- and z-directions, θI is the angle of incidence, βn and γn are the wave
numbers of the nth diffracted order in the x- and z-direction, respectively. Further, the classical grating equation
requires that βn = kx + 2nπ/Λ. Considering that the modulation function can be expressed as an expansion ofharmonic terms, the modal amplitude an (t) can also be represented with a Fourier expansion of the harmonics
of the modulation frequency as∞an (t) =∑ ânp e j(ω0 +pωm )t ,

(2)

p=−∞

where ânp is the scattering amplitude of nth diffracted order and pth modulation harmonic. The wavenumber
2
p 2
associated with the scattering amplitude, ânp , in the z-direction can be expressed as γn = ωp /c0 − βn2 ,where ω p = ω0 + pωm and ω0 is the incident frequency. This shows that the number of propagating modes, i.e.
pthe modes where γn is real, is a function of the diffracted order and the modulation harmonic. Therefore, thepropagating diffraction modes can be shifted into multiple directions and then related to backscattering angles
associated with the propagating diffraction modes and the harmonics.
!


λ0
βn
−1 sin θI + Λ n
p
−1
= sin
,
(3)
θn = sinω p /c0
1 + p ωωm0
Conservation of momentum must be established at the interface between the AMS and the background fluidin order to determine the backscattered field. For this configuration, we can use the linearized momentum
equation and the definition of input impedance to relate the normal component of input impedance of a flat
surface, and the total acoustic pressure and normal component of the velocity, uz (x,t), at z = 0 as
ρ0

∂p
∂ uz
=−
,
∂t z=0∂ z z=0uz (x,t)|z=0 = −p(x,t)
Z(x,t) z=0(4)

Equations 1, 2, and 4, are then used to determine the scattering amplitude, ânp , from a large matrix equation forthe (2N + 1)(2P + 1) solutions with the truncated terms of the diffracted orders and the modulation harmonics
as n ∈ [−N, N] and p ∈ [−P, P] (15). The scattering amplitudes are then used to calculate the scattered soundpressure from the flat surface with spatiotemporally modulated input impedance with the summation!
Pps (x, z = 0,t) =

∑

p=−P

N

P

n=−N

p=−P

∑ ânp e− jβn x e jω pt = ∑ â p (x)e jω pt .

(5)

We next employ the Fraunhofer approximation to determine the backscattered directivity for each harmonicin the far-field by using the spatial Fourier transform of â p (x). Finally, by summing the far-field responses for
all modulation harmonics, the far-field sound power due to contributions from all propagating modes can be
determined for a single incident plane wave with prescribed frequency and direction. Given the symmetric geometry of a flat surface, the difference of the far-field responses associated with the direction of the modulationshows the degree of the nonreciprocity (7), which can be calculated using the metric +2
Li /10 − 10Li− /10∑mi=1 10ϒ=(6)
 + 2 − 2 ,m
Li /10Li /10
10
+
10
∑m
∑
i=1
i=1

(b)

(a)

Perfectly matched layer

Backgroundpressure

0.0921A(c)

Infinite sound soft boundary(No baffle condition)

Rigid boundary

Impedance boundaryfrom modulation function(d)Figure 2. (a) Degree of nonreciprocity with respect to the modulation speed, which is a function of the modulation frequency, and the modulation amplitude. The case marked as "A" is selected for the demonstration of anonreciprocal sound field. (b) The finite element model in Comsol including the impedance boundary condition
and the infinite sound soft boundary. The far-field response for case A is shown in (c) when using the semianalytical model, and in (d) using a time-domain finite element model.
where Li± is the sound pressure level for two opposite modulation directions, forward(+) and backward(-), and mis the index of the positions in the polar angle, which we have assumed to be equally spaced for θ ∈ [−90◦ 90◦ ].

3

PARAMETRIC STUDY

The semi-analytical model and the modulation function for the surface admittance of a flat surface in Fig. 1
can be used to perform a parametric study on the two modulation parameters: the modulation frequency ωmand the modulation magnitude, Ym . We note that the spatial wavenumber of the modulation is fixed with thepredetermined length of one period, Λ, which we fix as L/2. The plot of the result from a parametric study in
Fig. 2(a) shows that some modulation parameter pairs generate a high degree of nonreciprocity, while others do
not. We further investigated the case of cm /c0 = 0.092 and Ym = 1.97 which is labeled with "A" in Fig. 2(a).
The far-field scattered directivity response for this case determined using the semi-analytical model as describedabove is provided in Fig. 2(c), which shows that the response changes significantly depending on the directionof spatial modulation. These results demonstrate that the symmetric (flat) geometry of the surface enables theuse of spatiotemporal modulation and a specific set of modulation parameters to produce a high degree of nonreciprocal scattering. The plot of the magnitude of the scattering amplitudes provided in Fig. 3(a) shows that forthe case considered here, the sound energy is concentrated in the direction of the mode of n = 1 and p = 1 andthat when the direction of modulation reverses, sound is scattered primarily in the direction of the n = −1 andp = −1 mode, leading to strongly nonreciprocal scattering. The iso-frequency curves (i.e. slowness contours)
and the dispersion diagram in Figs. 3(c) and (b), respectively, shows a convenient representation of how the spatiotemporal modulation leads the control of backscattered sound energy and mode selection for propagating or
evanescent modes. Finally, we have compared the far-field response determined using the semi-analytical modelwith the results obtained using time-domain finite element modeling with the same geometry and the same
modulation function as shown in Fig. 2(d). The finite element model was implemented in Comsol Multiphysics
in Fig. 2(b). Modulated surface admittance was realized using an impedance boundary condition defined with

(a)

(b)

(c)
Incident

Propagating modes(n=1, p=1)

(n=-1, p=-1)(n=1,p=1)(n=1,p=1)−−

Incident(n=-1,p=-1)(n=0,p=0)(n=-1,p=-1)(evanescent)2Λ−Figure 3. (a) Magnitudes of the scattering amplitudes as a function of the diffraction mode and the modulation
harmonic. (b) The dispersion diagram represents the propagating modes and the evanescent modes dependingon the wavenumber of the scattered sound in the x-direction and the shifted frequencies. Modes inside the“sonic lines” will propagate, while those outside them will be evanescent. (c) Propagating and scattered soundmodes shown on a slowness surface (wavenumber space). The concentric half-circles represent iso-frequency−1
contours in wavenumber-space for the incident and the frequency-modulated frequencies. The angles θ−1=◦1◦−37.1 and θ1 = 32.9 are determined from the location of scattered propagating modes with the diffracted
order and modulation frequency pairs (p, n) = (−1, −1) and (p, n) = (1, 1) on the iso-frequency contours.
the modulation function in Fig. 1. The far-field polar responses were then determined using “exterior field calculation” using the acoustic field at the boundary between the simulation domain and a perfectly matched layer(PML), which is used to approximate an infinite half-space above the reflecting plane. The “exterior field calculation” calculates the far-field using exact Kirchhoff-Helmholtz integral and a “sound-soft” (i.e. pressure release)boundary condition is employed within the PML. The responses predicted using FEM are in good agreementwith those from the semi-analytical model, clearly showing a strong nonreciprocal scattering of the sound fielddepending on the modulation direction and validating the semi-analytical model. As a visual aid for the effectsof the spatiotemporal modulation, the geometry in Fig. 2(b) was modified to simulate a more realistic geometry. Specifically, a finite surface with a length of two periods. From the transient numerical simulation withthis revised geometry, we can show the non-reciprocal response by inspecting the scattered acoustic pressureby plotting the scattered acoustic pressure a single instant in time as in shown in Fig. 4. Those results clearly
indicate the difference scattered field near the AMS depending on the direction of the modulation.

Figure 4. The scattered sound field for two different modulation directions obtained using a time-domain finite
element simulation.

4

CONCLUSIONS

A semi-analytical model developed in previous work to model the performance of a spatiotemporally modulated
sound diffuser was employed in this work to show that flat surfaces with spatiotemporal modulation of the input impedance can be used to generate nonreciprocal scattering behavior. This was explored by investigating
the frequency and wavenumber shifting of the reflected field caused by the time and space modulation of thesurface response. Continuity of momentum at the AMS-fluid interface imposes the scattering of energy into thedirections of various diffraction modes at frequencies shifted from the incident wave by integer multiples of themodulation frequencies. We have shown that the diversification of propagating modes can be used to improve
diffusion performance or to generate a nonreciprocal sound scattering depending on the selection of the appropriate modulation parameters. We have selected a representative nonreciprocal scattering case by performinga parametric study of the modulation parameters and evaluating the performance with a nonreciprocity metric.
The far-field responses obtained from the semi-analytical model and finite element model clearly show that the
scattered sound field becomes asymmetric under modulation and that the direction of asymmetric is controlled
by the modulation direction.

ACKNOWLEDGEMENTS
J.K. acknowledges support from Samsung Electronics.

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Journal of the Acoustical Society of America. 1979;65:958-63.

ABS-0607

Extending the angle range of high absorption of metasurfaces by adding a
multilayered medium
Diana Maria GARZA-AGUDELO(1) * , Vicente CUTANDA HENRIQUEZ(1) , Cheol-Ho JEONG(1) , Peter Risby ANDERSEN(2) ,Martin IBARIAS(3)(1) Technical University of Denmark, Acoustic Technology, DTU Electrical and Photonics Engineering, Ørsteds Plads, Building

352, Kongens Lyngby 2800, Denmark, dmgag@dtu.dk(2) GN Audio A/S & Jabra, Audio Research, Lautrupbjerg 7, Ballerup 2750, Denmark(3) Universitat Politècnica de València, Department of Electronic Engineering, Camino de Vera s.n., Building 7F, ES-46022

Valencia, Spain

ABSTRACT
Surfaces consisting of periodic arrangements of resonators can be considered to be locally reacting, and thus
have a limited range of angles for which they can achieve high absorption. In particular, achieving significant
absorption in a wide angle range becomes more difficult in the region approaching grazing incidence. For
some applications such as reflection control in disproportionate rooms, reflection control at large angles can bedesirable. In this work, it is shown how the angle range of high absorption of periodic surfaces of resonators
can be significantly extended by adding a multilayered media above them. Two cases are demonstrated, one thatconsiders a single-frequency performance and another that considers the average performance over an octave
band. The behavior of the surfaces is estimated analytically using the transfer matrix method and numericallyusing the finite element model.

Keywords: Oblique incidence absorption, Absorbing metasurface, Multilayered media

1

INTRODUCTION

Periodic surfaces of Helmholtz or quarter wavelength resonators have been shown to be effective absorbers
at frequencies with wavelengths that are very large compared to their thickness [1, 2, 3, 4]. At the sametime, the combination of resonators tuned to different frequencies allows to extend the frequency range of highabsorption [3, 5, 6]. Thus, this type of metasurfaces represents promising solutions to control low frequency
acoustic reflections. Nevertheless, their behavior is approximately locally reacting, and therefore the angle range
for which they can achieve high absorption is limited [7, 8]. In particular, achieving significant absorption in awide angle range becomes more difficult in the region approaching grazing incidence given the high variationof the radiation impedance as a function of the angle. For different applications, it is desirable to control waves
impinging the surface from several directions. For instance, for sound level control inside rooms where there
are several sound source positions and/or reflecting objects. Reflection control for large incidence angles is of
special interest in disproportionate rooms [9].
In this work, a metasurface that consists on a parallel arrangement of 2D Helmholtz resonators topped bya multilayered medium is presented. Its key feature is that it allows to extend the angle range where highabsorption is achieved compared to that of a surface that includes only the resonators. More explicitly, themultilayered medium is realized using an array of cylinders distributed following a hexagonal lattice.
* dmgag@dtu.dk

2

ANALYTIC MODEL FOR THE MULTILAYER ABSORBER

The type of absorbers proposed in this work are a combination in series of different media. An example of aunit cell of a surface of this kind, which includes NHR = 3 different resonators and Nl = 4 layers of cylinders,is shown in Figure 1. From top to bottom, the elements of the series assembly are: layer Nl , Nl -1, ..., 1 of thecylinder arrangement, and the periodic surface of resonators. Using a transfer matrix model (TMM), the system
is described by a global transfer matrix TG as(N ) (N −1)

TG = TL l TL L

(1)...TL TP ,where the transfer matrix of each of the layers is of the form [10]!
!(l) (l)ceff ρeff
ω
ω
(l)
(l)
sin (l) cos θ h icos (l) cos θ hcos θ (l)
ceff
ceff
(l)
,

!
!
TL = 
(l)
 cos θω cos θ (l) h
(l) h
cos
θ
cos
i (l) (l) sin ω(l)(l)
ceff ρeff
ceff
ceff
(l)

(1)

(2)

(l)

ω, ceff , ρeff , and θ (l) are respectively the angular frequency, the complex effective speed of sound and density,and the propagation angle inside layer l. From Snell’s law, the propagation angle inside the first layer canbe found from the wavenumber in air k0 = ω/c0 , the incidence angle of the wave with respect to the surface(l)normal θi , and the equality k0 sin θi = k(1) sin θ (1) , where k(l) = ω/ceff . For all subsequent layers, it can be found
(l−1)
(l−1)
(l)
(l)
from the equality ksin θ= k sin θ .

Figure 1. Unit cell of a metasurface consisting of a periodic arrangement of three parallel 2D Helmholtz
resonators which are equidistantly placed, and a multilayer arrangement of cylinders following a hexagonallattice distribution.
(l)

(l)

The complex effective wave propagation properties (ceff , and ρeff ) inside each row of cylinders are obtained
from homogenized properties of the sonic crystal. More explicitly, from the speed of sound (chom ), mass density(ρhom ) and viscous attenuation coefficient of the sound wave (γhom ), which are obtained from the models by
Krokin et al. [11], Gumen et al. [12] and Ibarias et al. [13] respectively. They describe the crystal’s low
frequency behavior and are a function of the frequency, lattice parameter and filling fraction ( f f ). Then, thecomplex effective properties, which include viscous losses, for a fluid that is equivalent to a row of cylinders
are be obtained as

ω
ω
=
keff
ω/chom − iγhom ( f )chom 2ρeff ( f ) = ρhom,ceff ( f )ceff ( f ) =

(3)
(4)

In turn, the transfer matrix of the parallel arrangement of equally spaced resonators, is defined as [14]

TP =

−1




(N

ry22HR(N )  (N )ry21HR ry22HR)

NHR −1 y(m) y(m)(n)∑ ry11 − ∑ r 12 (m)21
n=1
m=1
y22

NHR



−1
!
(N ) (N )− r2 y12HR y21HR(m) (m)NHR −1y12 y21

m=1

(m)
y22

∑ r

NHR




(n) 
− ∑ ry11
n=1

(5)

(i)

where r = 1/NHR is the fraction of the unit cell length that is occupied by each different resonator, and yi j
is the i j-th element of the admittance matrix of n-th resonator. This matrix is defined as(n)(n) (n)(n) (n)
t
t
t
−
t
t
1
12 21
22 11 Y(n) = (n)  22
,
(6)
(n)
t12
1
−t11
where the elements are related to the the transfer matrix of each resonator



 

(out)
cos(kc lc )cos(kn ln ) iZn sin(kn ln ) 1 ikn ∆ln(in) Zn
t11 t12
1 ikn ∆ln Zn


=

THR = sin(k l )sin(k l )i Zc ccos(kn ln )i Zn n0
1
t21 t2201cniZc sin(kc lc ) , (7)cos(kc lc )where the subscripts n and c indicate whether the quantities are related to the neck or cavity of the resonator,
and k and Z are respectively the wavenumber and characteristic impedance of a homogeneous fluid layer that isequivalent to the neck and cavity sections of the resonator. The wavenumber and characteristic impedance inside
the neck and cavity can be estimated using Stinson’s expressions for sound propagation through narrow slits
(out)
(in)
[15]. In turn, ∆ln
and ∆ln are the inner and outer end corrections of the neck’s length. They are estimatedusing Smits and Kostens expressions for surfaces with slit-like perforations assuming constant pressure over theslit [16].
Once the global matrix of the system TG is obtained, the absorption coefficient follows as
2
zs
−1
z
= 1−
α( f , θi ) = 1 − zsr+1zr

2
TG,11 cos θi−1
TG,21 ρ0 c0,
TG,11 cos θi+1
TG,21 ρ0 c0(8)where ρ0 is the density of air, zs = TG,11 /TG,21 is the surface impedance, and zr = ρ0 c0 / cos θi is the radiation
impedance of an infinite surface.

3

DESIGN BY OPTIMIZATION

We considered the design of surfaces with broad angular absorption at: (1) a single frequency of 250 Hz, and(2) over the octave band with center frequency at 250 Hz. For the first case, only one resonator was includedin the unit cell, for the second case six resonators were included in the unit cell in order to cover the broader
frequency range. The lattice parameter for the cylinder arrangements was kept at a constant value of 0.111
cm, and 10 layers of cylinders were included. The parameters defined as variables in the design were the fourdimensions of each of the resonators, as shown in Figure 1, and the filling fraction ( f f ) of each of the layers.
Their final values were obtained by solving an optimization problem defined as
Φ=

min
x


1 Nθ 
( j)
1 − ᾱ(θi , x)∑Nθ j=1(9)subject to 0 ≤ xk ≤ 1, k = 1, ..., 4NHR + Nl ,where Nθ is the number of incidence angles included in the optimization, ᾱ is the absorption coefficient value
averaged over the frequencies of interest, and the design variables were interpolation variables related to themetasurfaces’ parameters as(k)

wn = wn,min + xk (wn,max − wn,min )for k = 1, ..., NHR ,(10)(k−N )
ln HR = ln,min + xk (ln,max − ln,min )for k = NHR + 1, ..., 2NHR ,(11)

for k = 2NHR + 1, ..., 3NHR ,(12)

for k = 3NHR + 1, ..., 4NHR ,(13)

for k = 4NHR + 1, ..., 4NHR + Nl .

(14)

(k−2NHR )wc= wc,min + xk (wc,max − wc,min )(k−3NHR )lc= lc,min + xk (lc,max − lc,min )ff(k−4NHR )= f f min + xk ( f f max − f f min )The minimum and maximum allowable values for the parameters can be found in Table 1.
Table 1. Box constraints for the parameters of the metasurfaces designed for broad angular absorption. The
dimensions of the resonator’s components are given in mm.
wn

ln

wc

lc

ff

Min.

0.8

8

20

40

0

Max.

1.2

30

90

150

0.8

For the single frequency case the distance between resonators was set to 10 cm, and for the multifrequency
case the distance between resonators was included as an additional design variable. Its allowable value wasconstrained to the range [5, 8] cm. The set of angles used to calculate the cost function included the range
0◦ ≤ θi ≤ 60◦ discretized in steps of 15◦ , and the range 61◦ ≤ θi ≤ 89◦ discretized in steps of 1◦ . An overrepresentation of the angles close to grazing incidence was used since it is the range where it is more challenging to maintain high absorption [8]. The solution to the problem was found using the sequential quadratic
programming algorithm available in the fmincon function of MATLAB® [17]. The gradient estimation was doneusing forward finite difference with a step size of 1.49 × 10−8 , and the problem was solved using 10 initial
random guesses for the single-frequency problem, and 5 different initial random guesses for the multi-frequency
problem.

4

RESULTSThe optimized values of the metasurface’s parameters for the two cases considered can be seen in Table 2. The
performance of the absorber designed for single-frequency excitation can be seen in Figure 2. Figure 2a shows
the absorption coefficient at 250 Hz as a function of the incidence angle. For reference, the angle-dependent
performance of an optimized surface consisting only of a periodic arrangement of resonators, without the added
cylinder structure, is overlayed. It can be seen that there is a significant improvement on the performance,
especially for the incidence angles close to the normal. The range for which the absorption coefficient takes
values equal or higher than 0.9 corresponds to 63◦ ≤ θi ≤ 83 for the metasurface with only resonators, andto 0◦ ≤ θi ≤ 83 for the metasurface that combines the resonators and the array of cylinders. Nevertheless,a drawback from this design, is that the cylinder structure increases the absorber’s thickness. The optimized

surface has a total thickness of 17.3 cm, about 1/8 times the wavelength at the design frequency (250 Hz),
from which 9.7 cm corresponds to the multilayered media. This aspect can be a limitation for the use of thistype of surfaces for applications were there are high space limitations.
Table 2. Optimized values for the parameters of the metasurfaces with broad angle range of high absorption.
The dimensions of the resonators’ components are given in mm.
Surface optimized for a single frequency
i

(i)
wn

(i)
ln

(i)
wc

(i)
lc

d

f f (1)f f (2)f f (3)f f (4)f f (5)f f (6)f f (7)f f (8)f f (9)f f (10)

1

0.8

30

20

46.8

100

0

0

0

0

0

0

0.8

0.8

0.8

0.8

i

(i)
wn

(i)
ln

(i)
wc

(i)
lc

1

1.2

8.0

45.0

41.0

2

1.2

8.0

45.0

41.0

3

0.8

17.0

36.8

40.0

4

0.8

17.3

45.0

46.6

5

0.8

9.4

45.0

120.6

6

0.8

15.1

29.3

40.0

Surface optimized for an octave banddf f (1)f f (2)f f (3)f f (4)f f (5)f f (6)f f (7)f f (8)f f (9)f f (10)

50

0

0

0

0

0

0

0

0.8

0.8

0.8

(a)

(b)

(c)

(d)

Figure 2. Performance of the optimized metasurface for single frequency excitation. 2a: Angle dependent absorption coefficient for the 250 Hz frequency for the proposed absorber and an optimized surface including only
resonators. 2b: Frequency-dependent absorption coefficient for selected incidence angles. Analytic and numeric
results are overlayed for verification. 2c: Real part of the normalized surface impedance. 2d: Imaginary partof the normalized surface impedance. The vertical dashed lines identify the frequency used for the optimizationand the horizontal lines mark the condition to achieve an absorption coefficient of one according to Eq. 8.

Figure 2b shows the absorption coefficient as a function of frequency for selected incidence angles. Resultsfrom a finite element model (FEM) are also included. In the FEM model, a solution was found for the acousticpressure for a unit cell of each of the optimized metasurfaces. Floquet periodicity boundary conditions whereassigned at both sides of the unit cell, a perfectly match layer was placed at 2.4 m from the absorber to simulate
an open field, and viscous and thermal losses happening at the boundaries of the cylinders and the resonators
were included using the boundary layer impedance boundary condition [18, 19].
The combination of the resonators and the cylinder structure creates a multiresonant system that has a behavior that is not locally reacting, i.e. the surface impedance is not independent from the wave incidence direction
anymore. This can be seen on Figures 2c and 2d, where the real and imaginary parts of the normalized surface
impedance can be seen for the same three incidence angles used in Figure 2b. The normalization is done withrespect to the radiation impedance. The multiple zero-crossings of the imaginary part of the surface impedanceshow the multiresonant nature of the system. Moreover, the frequency of the second zero-crossing of each of
the curves changes with the incidence angle. This explains the shift in frequency that is observed for the second
peak in the absorption coefficient curves in Figure 2b. This is because a better impedance matching between
the outer field and the absorber is obtained when Im (zs /zr ) = 0, as can be seen from Eq. (8).
The performance of the metasurface designed for multifrequency absorption is shown in Figure 3. Its performance as a function of the incidence angle is similar to the one for the single frequency case. Namely,
frequency averaged absorption coefficient values above 0.9 can be seen in the angle range between 0◦ and 83◦ .
Thus showing that it is possible to use this type of surfaces also for multifrequency applications.

(a)

(b)

Figure 3. Performance of the optimized metasurface for octave-band excitation. 3a: Frequency averaged angle
dependent absorption coefficient for the octave band with center frequency at 250 Hz. 3b: Frequency-dependent
performance for selected incidence angles. Analytic and numeric results are overlayed for verification. The
vertical dashed lines indicate the limits of the octave band considered.

5

CONCLUSIONSIn this work, a type of metasurface has been proposed which is able to achieve high absorption coefficientvalues at a broad range of incidence angles for both single and multifrequency cases. These type of surfaces are
realized as a combination of a periodic surface of resonators topped by a multilayer arrangement of cylinders.
Results show that they can achieve a significantly broader angular performance than a surface that consists only
on a parallel arrangement of resonators.

ACKNOWLEDGMENTS
The authors would like to thank José Sanchez-Dehesa for the useful discussions. This research was carriedout within the Signature project, a collaboration between the Technical University of Denmark and the Korea

Advanced Institute of Science and Technology in the research of metamaterials for sound insulation, radiation,
and absorption.

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j.jcp.2018.06.005.

ABS-0763

Interaction of acoustic meta-atoms as additional degrees of freedom formetamaterial tuning
Felix Kronowetter(1) , Steffen Marburg(2)(1) Chair of Vibro-Acoustics of Vehicles and Machines, Department of Engineering Physics and Computation, Technical

University of Munich, TUM School of Engineering and Design,Germany, felix.kronowetter@tum.de(2) Chair of Vibro-Acoustics of Vehicles and Machines, Department of Engineering Physics and Computation, Technical

University of Munich, TUM School of Engineering and Design, Germany, steffen.marburg@tum.de

ABSTRACT
Acoustic metamaterials have a widespread range of application, e.g. acoustic cloaking and acoustic lenses, andusually consist of an arrangement of meta-atoms. These meta-atoms are often based on the principle of local
resonances. Herein, we theoretically study the case of two meta-atoms in an unbounded two-dimensional fluid
domain in order to analyze their mutual interaction. Furthermore, we vary distance, orientation and detuning
while observing the eigenfrequencies of two interacting modes. Certain configurations of the varying parameters
mentioned before lead to regions of modal degeneracy, hence to maxima in the transmission loss if excited by aplane wave. Thus, we make use of this phenomenon in order to improve the wave attenuation properties of themeta-atoms..

Keywords: Acoustic metamaterial, Metamaterial tuning, Sound insulation

1

INTRODUCTION

Methods to increase sound attenuation are usually determined by increasing mass (low frequencies) or by increasing stiffness (high frequencies). A new type of materials called acoustic metamaterials are able to breakthe mass rule and hence are subject of numerous scientific studies. They commonly consist of an arrangementof mostly periodic meta-atoms that lead to non-natural material properties beneficial for noise control applications [1–4]. Non-natural material properties are, for example, a negative effective mass density [5, 6], negative
bulk modulus [7] or negative refractive index [8]. Meta-atoms work like local resonators interacting with theincoming sound waves. Furthermore, since meta-atoms are sub-wavelength structures and arranged close to eachother, pressure coupling between the individual meta-atoms occurs [9]. Pressure coupling affects each individual
meta-atom, can be used as tuning parameter and hence lead to enhanced sound transmission [10]. Coupling
effects have been used for tuning electro-magnetic metamaterials [11, 12]. Another analogy comes from thestudy of organ pipes, where coupling is a well known effect [13–16]. There are various studies about couplingof acoustic resonators in waveguides [9, 17–19]. Coupling as tuning parameter of local resonators within a unit
cell are investigated in recent years [20–22]. Furthermore, coupling is applied to tune acoustic lenses [10] andmetagratings [23].
We investigate the interaction of two C-shaped meta-atoms in a two-dimensional domain of infinite extend.
The eigenfrequencies of the system are studied for parameters like distance, orientation, detuning and radiation
losses. We analyze underlying effect and use the coupled mode theory to get a better understanding of thecoupling mechanisms. We demonstrate how the sound attenuation of meta-atoms can be improved and tuned.
The aim is to show how the interaction of meta-atoms affects the performance of the metamaterial in terms ofsound attenuation.

2

RESULTS

Figure 1(a) shows the basic geometry and dimensions of the pair of resonators, which are simplified as twodimensional C-shapes [24, 25]. The influence of the two ends for real C-shaped cylinders of finite height ishence neglected. Furthermore, the system is considered lossless. Thermo-viscous boundary layer losses are
significant for structures with narrow geometry features. However, the chosen aperture width is large comparedto the expected boundary layer thickness. The C-shapes have an inner radius r = 18.77 mm, an aperture width
w = 4 mm, and a thickness t = 2 mm. This configuration leads to the complex individual resonant frequencyω̄ = 1000 + 53i Hz, with ω̄ = ω/2π. The length l denotes the shortest distance between the two C-shapes.
Initially, both apertures face the incident plane wave. The resonator pair is surrounded by a large circular air
domain, which is in turn surrounded by a perfectly matched layer. All studies in this section are conducted as
finite element simulations in COMSOL Multiphysics [26], first and foremost in form of modal analyses, then by
the transmission response for harmonic excitation by a plane wave. The symmetric and antisymmetric modes
are depicted in Figure 1(b).

1

r
l
w

0
t

pr

0.5

α

−0.5
−1

(a) Geometry.

(b) Antisymmetric (left) and symmetric (right) mode.

Figure 1. Resonator pair geometry and dimensions, specified by inner radius r, aperture width w, wall thicknesst, and inter-resonator distance l are shown in Figure 1(a). Figure 1(b) displays the antisymmetric mode (mode 1)
and symmetric mode (mode 2), where pr denotes the real part of the total pressure normalized to its maximum
value.

2.1 Inter-resonator distance
The first parameter to be studied is the distance l between two resonators. It is varied from zero to twice theresonant wavelength. The real part of the complex eigenfrequencies ω̄ = f + iγ is the resonant frequency f , theimaginary part γ characterizes the radiation loss. Figs. 2(a) and 2(b) show the eigenfrequencies of the cavity
resonant modes (modes with high quality factors) ω̄ as a function of the inter-resonator distance l.
Both real and imaginary parts in the coupled system oscillate around the complex resonant frequency of
the individual resonators. The oscillation period matches the resonant wavelength λ = 343 mm, similar to the
Fabry-Pérot interference observed by Hein et al. [27] for duct-cavity systems. We obtain crossing points of thetwo modes at periodic distances of ∆l = λ /2 and observe that the crossing points of real and imaginary parts
are shifted against each other by about a quarter wavelength. Furthermore, the mode with the smaller imaginary
part dominates the decay process.
To explain the degeneracy of coupled modes for small detuning we use the coupled mode theory (CMT).
Assuming that a1 , a2 are the modal amplitudes with the time dependence exp (iωit) of two weakly uncoupledresonators, the dynamics of the system are given by Haus et al. [28]:
  
 
d a1  iω1 iκ  a1 
=
.
(1)
dt a2
iκ iω2a2
The coupling coefficient is denoted κ. For a better understanding of the underlying effects causing synchro-nization and coupling, we derive a surrogate model representing κ:
#
"
h


π
π
1 i hl0 ii + sin(l − l0 ) 1 +(l − l0 ) 1 +.
κ = cκ cos
cp
l
cp
l

(2)

The period constant is defined by c p = λ̄ /2 − 2(r + t)/pc , with pc being an adjustable coefficient implying
scattering effects.
Analogously, the distance of the first crossing is formulated as l0 = λ̄4 − 2 r+t
cc , whereat cc is an adjustablecrossing coefficient.
To obtain the coupling coefficient, both complex resonant frequencies are put into relation via their corresponding quality factors cκ = Q1 /Q2 . Figure 3 shows the cavity resonant frequencies as well as the resultsof the CMT dependent on the inter-resonator distance l. We here consider three cases. In case 1, the two
resonators are identical. In Figs. 2(a) and 2(b) we can see, that the results of the CMT match the simulated
data quite accurately. The modal crossings are replicated by the analytical model. Furthermore, the period issustained and so is the decay of the amplitude. Same can be observed regarding the imaginary part. In thesecond case, we introduce a detuning of ∆ = 5 % in resonator two. The two modes no longer cross in the realpart but do so in the imaginary part. Hence, the state of degenerate modes is not realizable (except for the
first crossing). The detuning leads to poorer sound attenuation properties compared to the case of identical resonators. The analytical model again replicates the modal anti-crossings in the real part as well as the increased
periodicity of the imaginary part. We merely note, that the antisymmetric fluctuations in the imaginary part of
Figure 2(d) show an enhanced deviation from the simulated data compared to Figure 2(b). In the third case, we
increase the size of the aperture to increase the coupling, hence to counteract the anti-crossings. By increasing
the radiation loss, we manage to restore the modal crossings and hence obtain degenerate modes again. Thus,
increased radiation loss stabilizes the system against detuning. As we can extract from Figure 2(e), the analytical model also represents this case. Minor inaccuracies can be identified in the imaginary part in Figure 2(f).
This is due to increasing inaccuracy of the model coming along with increased inter-resonator distance l.
A crucial parameter to understand the fundamental coupling effects is the complex coupling coefficient κ.
In Figure 3 the behavior of κ is displayed for the second and third case mentioned in Figure 2.
The absolute value of the coupling coefficient decays with increased inter-resonator distance l. Fischer et
al. [13] show, that the coupling strength is supposed to be a function of the distance between the pipes, with asound attenuation of 1/r2 in the near field and 1/r in the far field matching our results. Additionally, the zerosof the real and imaginary parts occur periodically. The former concur with the maximum frequency mismatchof mode 1 and mode 2 and the latter with the modal (anti-)crossings, represented by the dotted and dashed lines
in Figs. 2(c) and 2(e). A noteworthy finding taken from Figs. 2 and 3 is the imaginary part of the coupling
coefficient turning zero at modal (anti)crossings and thus we fulfill the criterion that the real part of two modes
become degenerate for a purely imaginary coupling. The analytical model also predicts the reciprocal behaviorof ℜ(κ) turning zero, where the real parts of mode 1 and mode 2 show a maximum deviation and the imaginaryparts cross. These results correlate with the findings by Haus et al. [28]. Since we increase the aperture widthin Figure 2(e), κ also increases and hence, the modal crossings of the real part are restored. Furthermore,
the model also replicates the decreased periodicity of modal crossings in the imaginary part and the additional
degeneracies of the imaginary parts disappear.
2.2 Relative orientation
We now examine the relative orientation of the C-shapes’ apertures. Powell et al. [12] show that it is relevant inelectromagnetic metamaterials. Additionally, Powell et al. [11] and Hesmer et al. [29] suggest that the relativeorientation plays a major role for the coupling of split ring resonators, hence it may also be of importance forthe acoustic meta-atoms. The apertures face each other for zero twist angle α = 0◦ , the two C-shapes are rotatedand reach the configuration in Figure 1(a) for α = 90◦ . The apertures look away from each other for α = 180◦ .
Figure 4(a) displays the absolute difference between the cavity resonant frequencies ∆ f = |ℜ(ω̄1 − ω̄2 )| in theparameter plane of inter-resonator distance l and twist angle α. The dark blue regions indicate the course ofdegenerate modes with ∆ f = 0. The modal crossings provide the locations of possible transmission minima, that

1,040mode 180

mode 2 CMT

mode 1mode 2 CMT

γ [Hz]

f [Hz]

1,020
1,000

60

40
980
0

200

400
l [mm]

20600

0(a) real part.

200

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l [mm]

600(b) imaginary part.

1,080
mode 180

mode 2 CMT

mode 1mode 2 CMT

1,040

γ [Hz]

f [Hz]

1,060

1,020

60

40
1,000
980

0

200

400
l [mm]

20600(c) real part, frequency detuning ∆ = 5 %.

0

200

400
l [mm]

600(d) imaginary part, frequency detuning ∆ = 5 %.

1,100

120
mode 1mode 1mode 2 CMT

mode 2 CMT

1,050

γ [Hz]

f [Hz]

100
80
60

1,000
40
0

200

400
l [mm]

600(e) real part, frequency detuning ∆ = 5 %, increased aperture width
wA = 6 mm (initially wA = 4 mm).

20

0

200

400
l [mm]

600(f) imaginary part, frequency detuning ∆ = 5 %, increased aperture
width wA = 6 mm (initially wA = 4 mm).

Figure 2. Real and imaginary part of the cavity resonant frequencies ω̄ [Hz] as a function of the inter-resonatordistance l [mm]. The dashed black lines indicate the zeros of ℜ(κ) and the dotted lines the zeros of ℑ(κ).

ℜ(κ)

40

ℑ(κ)

|κ|

κ [−]

κ [−]

ℑ(κ)

|κ|

20

20
0

0
−20

−20
−40

ℜ(κ)

40

0

200

400
l [mm](a) 2nd case.

600

−40

0

200

400
l [mm]

600(b) 3rd case.

Figure 3. κ is displayed for the second and third case mentioned in Figure 2. The absolute value is plotted as
black line, the real part as red line and the imaginary part as blue line.
is attenuation maxima.
The normalized total transmitted sound power Pout /Pinc in the α-l-plane is presented in Figure 4(b). The
incident wave with f = 1000 Hz is matching the C-shapes’ resonant frequency. As a result, transmission dips andmodal crossings occur at the same distances and twist angles. For comparison, the crossings are indicated with
red crosses in both plots in Figure 4. The transmission dips only occur where the symmetric mode dominates,
because of the relative orientation of resonators and incident wave.
The dark blue regions with ∆ f = 0 in Figure 4(a) cover a certain range regarding the inter resonator distance. Furthermore, we can shift the point of degenerate modes to certain distances by adjusting the orientationof the C-shapes. Thus, the period of degenerate points is maintained. The transmission study shows that theattenuation maxima can be found at the points of modal degeneracy. This leads to the conclusion that distanceand orientation are tuning parameters for metamaterials. Surprisingly, the attenuation maxima for this configuration is not found at α = 90◦ but at α = 72◦ and l = 204mm. Similar results are observed by varying frequenciesof the incoming wave.

3

CONCLUSIONS

Two cavity resonant modes of a pair of identical coupled Helmholtz resonators were found. With varying separation distance, the corresponding eigenfrequencies oscillate around the uncoupled resonant frequency. A period
equal to the uncoupled resonant wavelength λ̄ means that the two modes cross every λ̄ /2. One mode is symmetric, the other one is antisymmetric, implying in- and anti-phase oscillation of the two resonators, respectively.
The identity of resonant frequencies at the modal crossings holds within a certain range of frequency detuning,
the region of modal degeneracy. Consequently, certain separation distances might enable larger manufacturing
tolerances than others. C-shapes with larger aperture width and a separation distance corresponding to a modal
crossing could be used to mitigate the effect of fabrication inaccuracies. This allows for higher manufacturing
tolerances in exchange for increased losses. Coupling induces the possibility of tuning via the resonators’ relative arrangement, that is distance and orientation. Starting with facing apertures, the two resonators were twisted
in opposite directions. Thereby, the modal synchronization points can be shifted to arbitrary distances, while theperiod is maintained. This means that the coupling strength depends on both relative orientation and distance.
Accordingly, these two influence parameters offer additional degrees of freedom for tuning the metamaterial
response with unchanged components. Additionally, a new surrogate model based on the CMT was derived
that predicts the modal behavior of the system and, hence, replaces the time-consuming numerical analysis. In

150
0.95

50

50

0.9
α [◦ ]α [◦ ]

100
100

100
0.85
50

Pout /Pinc [-]150∆ f [Hz]

150

0.8
0

100

200
l [mm]

300(a) cavity resonant frequency difference.

0

0

0

100

200
l [mm]

300(b) normalized total transmitted sound power.

Figure 4. (a) Difference between the real parts of the cavity resonant frequencies ∆ f = |ℜ(ω̄1 − ω̄2 )| [Hz]of mode 1 and 2 in the plane of inter-resonator distance l [mm] and angle α [◦ ]. The modal crossings are
represented by the dark blue lines with zero frequency difference ∆ f = 0. The black line indicates a constantinter-aperture distance la starting at the second crossing point for α = 0◦ . (b) Normalized total transmitted
sound power Pout /Pinc [−] for a plane wave with f = 1000 Hz in the plane of inter-resonator distance l [mm]and angle α [◦ ]. (a, b) For comparison, the second modal crossing is indicated by red crosses in both figures.
conclusion, one can state that the interaction between the resonators in a metamaterial structure significantly
affects its characteristics. The better the understanding, the larger the benefits of including synchronization andcoupling effects in the design of acoustic metamaterials.

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ABS-0959

Broadband acoustic lens design using gradient-based optimization
Feruza Amirkulova(1) , Samer Gerges(2) , Andrew N. Norris(3)(1) Mechanical Engineering Department, San José State University, San José, 95192, USA, feruza.amirkulova@sjsu.edu(2) Mechanical Engineering Department, San José State University, San José, 95192, USA, samer.gerges@sjsu.edu(3) Mechanical and Aerospace Engineering Department, Rutgers University, Piscataway, 08854, USA, norris@rutgers.edu

ABSTRACTIn this work, we present a gradient-based optimization (GBO) method [1, 2] for the design of a broadband
acoustic lens. This design is based on an optimization process using the semi-analytical optimization approachby means of multiple scattering theory. The GBO algorithm maximizes the sound amplification at the focal
point by evaluating pressure derivative with respect to the cylinder positions and then perturbatively optimizing
the position of each cylinder in the lens. The results of the GBO of the uni- and multi-directional broadband
acoustic lens designs are presented including performance measures for the frequency and the incidence angle
dependency. Examples of sound localization and focusing effects produced by planar nonuniform and uniform
configurations of cylindrical structures such as rigid or void cylinders as well as thin elastic shells are shown.

Keywords: Sound localization, Gradient-based optimization, Acoustic metamaterials

1

INTRODUCTION

The manipulation and control of sound stimulated the development of critical technologies such as acoustic
cloaking [3], sound focusing [4], ultrasound imaging, wavefront manipulation [5], beam splitting [6], absorption [7], vibration attenuation [8], and sound diffusion [9]. Adapting ideas from optics, acoustic lenses arecapable of focusing sound in like manner as optical lenses apt to focus light. The ability to focus sound facilitate enhancements in biomedical imaging [10], structural health monitoring, and enable radar reflection to
make stealth aircraft visible on radar screens [11]. Sound super-focusing effect can be achieved using acoustic metamaterials with the acoustic anisotropy required by transformation acoustics [12–15]. Here, we consider
more limited focusing for a single incidence and a range of incidence directions at a single frequency and bandof frequencies by maximizing the absolute pressure amplitude at the focal point produced by a given configuration of scatterers, and develop uni- and multi-directional broadband acoustic lens designs. We specificallydesign the multi-directional broadband acoustic lens to localize the sound and to focus acoustic incident waves
received from multiple directions onto a prearranged localization region or focal point. We achieve this with
an appropriate arrangement of the target metastructures made of conventional simple isotropic materials that areavailable in nature and obtain useful and interesting effects.
Optimization methods [16, 17] such as stochastic algorithms [18–20], topology optimization [21–23], shape
optimization [24], and gradient-based optimization (GBO) [24–26] have proven to be a powerful tool for designing metamaterials and metadevices. In this work, we present the implementation of a semi-direct method
for broadband acoustic lens design using GBO strategy [1, 26, 27] by means of multiple scattering theory [28]which mimics the behavior and performance of acoustic Luneburg lens that adepts to focus the incoming planesound wave at the focal point on the other side of the lens. Instead of using glass and mirrors, our acoustic
lens designs consist of arbitrary planar configurations of cylindrical structures such as rigid or void cylinders aswell as thin elastic shells embedded in water environment. We obtained an analytical formula for the gradientsof the absolute pressure amplitude at the focal point with respect to positions of a set of cylinders in closed
form rather than relying on finite difference schemes. This is of preeminent importance for efficient calculations in the optimization process. The closed analytical form of the gradients enhances modeling capability andcomputational efficiency when combined with GBO algorithms and parallel computing via use of f mincon andMultiStart Global optimization solvers on MATLAB.

2

PROBLEM FORMULATION

Consider acoustic multiple scattering [28] by an arbitrary planar configuration of M cylindrical structures in
the context of the 2D acoustic time harmonic wave equation. The total acoustic pressure p(x) is defined as:
p = pinc + psc , where pinc is the incident and psc is the scattered pressure fields which satisfy the Helmholtz
equation [26, 28]. The incident wave pinc is the plane wave of unit amplitude in the direction ψ. We denotethe total acoustic pressure at the focal point x f by p f ≡ p(x f ) for a given set of scatterer positions {xm },m = 1, . . . . , M. see Fig. 1. The evaluation of p f ≡ p(x f ) using the forward and reciprocal formulations of
inverse design of acoustic lens are provided in [1]. Our goal here is to rearrange the target cylinders to find an
optimal set {xm } that maximizes the absolute value |p f | = (p f p∗f )1/2 where ∗ denotes the complex conjugate.

Figure 1. The schematic of an acoustic lens in the near field. The incident wave is a plane wave of unit
amplitude in the direction k shown under the angle of incidence ψ. The cylinders are constrained to staywithin the circular region of radius Rout , the artificial boundary, denoted by black dotted line.
2.1 Broadband multi-directional lens design
The broadband multi-directional acoustic lens model maximizes the root mean square (RMS) of a set of theabsolute pressure at the focal point |p f | over a range of wavenumbers and angles of incidence, by iteratively
rearranging the position of each cylinder starting from a random configuration. The procedure for a broadband
multi-directional GBO strategy is as follows. We define the objective function as the RMS of a set of the absolute pressure amplitudes |p f | at the focal point over some range of normalized wavenumbers ki a (i = 1, 2, ..., Nk )and over a range of incident angles ψl (l = 1, 2, ..., Nl ):|p f ({xm })|RMS = 1 1 Nk Nψ
1/2|p f (ki a, ψl , {xm })|2.
∑∑
Nk Nψ i=1 l=1(1)This objective function is non-convex and highly non-linear; it has many local maxima and global maximum at
infinity. We solve the non-convex optimization problems with non-linear constraints. To avoid cylinder overlapping the distance between the centers of cylinders are constrained to be greater than minimal allowable distancethat is chosen greater than the diameters of the cylinders. In addition, the cylinders are constrained to move
inside a fixed circular region.
To enhance the GBO process, we define the broadband gradient vectors qRMS
with respect to positions r jjwhich can be found in terms of the individual single frequency gradients q j (ki a) as:
N

qRMS
=
j

∂ |p f ({xm })|RMS
1
1 1 Nk ψ=∑ ∑ |p f (ki a, ψl , {xm })| q j (ki a, ψl ),∂xj|p f |RMS Nk Nψ i=1
l=1

j = 1, 2, . . . M,

(2)

where |p f |RMS and q j (ki a) are defined at the normalized wavenumbers ki a (i = 1, 2, ..., Nk ) and angles of incidence ψl (l = 1, 2, ..., Nl ), and the derivation of gradients q j is provided in [1].

3

NUMERICAL RESULTS(a) ka = 0.75 : prel = 6.43(b) ka = 1.5 : prel = 4.13(c) ka = 2 : prel = 4.62

Figure 2. The illustration of absolute total acoustic pressure field p (a) at normalized wavenumbers ka = 0.75 :|p ||p f |rel = pi (xf ) = 6.43, (b) ka = 1.5 with prel = 4.13, and (c) ka = 2 with prel = 4.62 for the optimized uniformfconfigurations of M = 50 rigid cylinders. The focal point is located at x f = (0.0865, 0)m. For drawing purposesvalues higher than 2 are denoted by yellow color.
The numerical results of the GBO of the uni- and multi- directional acoustic lens designs are demonstrated.
Computations are performed on MATLAB using MultiStart and f mincon solvers along with sequential quadratic
programming (SQP) algorithms, while supplying the gradient of the pressure at the focal point and the gradientof the nonlinear geometrical constraints. Several configurations of randomly placed scatterers, via MultiStart,
are initialized. MultiStart has efficient local solvers, and can search a wide variety of start points in parallel.
Providing the analytical gradients of the pressure amplitude enhances the optimized solution accuracy, reduces
run time, and enables parallel computing using MultiStart. The multiple scattering solver [1, 26] evaluates theabsolute RMS pressure at the focal point as well as it gradients with respect to positions for each configuration,and f mincon combined with MultiStart solver continue to perturb the scatterer positions toward minimized
gradients until the optimal configurations are found. We investigate the sound localization effect for a plane
wave incident on a configuration of cylindrical scatterers considering various values of normalized wavenumber,ka and the total number of scatterers in acoustic lens, M.
3.1 Sound localization at single frequency and normal plane wave incidence
In this section, we present the numerical results for the unidirectional lens designs at single frequency andnormal plane wave incidence on uniform and non-uniform configurations of rigid cylinders. 100 configurationsof randomly placed scatterers, via MultiStart, are initialized.
3.1.1 Sound localization using uniform configuration of rigid cylinders
Here we present the localization effect for a plane wave incident on uniform configuration of 50 rigid cylinders
at various discrete values of normalized wavenumber: ka = 0.75, 1.5, and 2. The optimized lens configuration isa uniform configuration of identical rigid cylinders of radius a = 0.0075m located inside of circular region ofradius Rout = 0.0715 m. The focus of the lens is located at x f = (0.0865, 0)m. Figure 2 illustrates the absolute
total acoustic pressure |p| computed at normal plane wave incidence for a uniform configuration of M = 50 rigid
cylinders at fixed normalized wavenumbers ka = 0.75, 1.5, and 2. We computed the absolute pressure |p f |r el|p |relative to the incident background pressure at the focal points specified as: |p f |rel = pi (xf ) . The |p f |rel valuesf
are given in Figure 2 for each optimized configuration at each ka = 0.75, 1.5, and 2. As expected, the soundfocusing and localization effects are enhanced with the increase of the wavenumber.

3.1.2 Sound localization using non-uniform configuration of rigid cylinders
Now we consider a normal plane wave incidence on non-uniform planar configurations of rigid cylinders ofradii a ∈ [amin , amax ] where amin = 0.005m and amax = 0.01m. Figure 3 illustrates the absolute total acousticpressure |p| computed for the optimized nonuniform configurations of M = 10, 20, 25, and M = 30 and kamax = 2
with amax = 0.01m. The value of |prel |, the absolute pressure relative to incident background, are provided
in Figure 3 for each optimized configuration. The focusing and localization effect becomes more noticeable
and improves with the increase of the number of scatterers. At kamax = 2, the highest amplitude is achieved
producing prel = 4.5769 for M = 30 scatterer nonuniform configuration. These results are comparable to thoseshown previously in section 3.1.1 for the optimized uniform configuration with much larger number of identicalrigid scatterers, i.e. M = 50.

(a) M = 10 : prel = 2.7036(b) M = 20 : prel = 3.7112(c) M = 25 : prel = 4.1243(d) M = 30 : prel = 4.5769

Figure 3. The illustration of absolute total acoustic pressure field p at normalized wavenumbers kamax = 2 forthe optimized non-uniform configurations of M = 10 with p f |rel = (a), M = 20 : prel = 3.7112 (b), M = 25 : prel =4.1243(c), and M = 30 with prel = 4.5769 (d) rigid cylinders. The focal point is located at x f = (0.102, 0)mwhere radii of cylinders vary between amin = 0.005m and amax = 0.01m. For drawing purposes values higher
than 2 are denoted by yellow color.
3.2 Broadband sound localization using uniform configuration of cylindrical scatterers
3.2.1 Sound localization implementing uni-directional lensHere we study the focusing and localization frequency over the range of wavenumbers. Numerical results arepresented for uniform cylindrical void configurations and uniform sets of thin elastic shells for various numberof scatterers M = 10, 20 and 30 and ranges of wavenumbers: ka ∈ [0.6, 1.0], ka ∈ [1.0, 1.4], ka ∈ [1.4, 1.8] andka ∈ [1.6, 1.8]. 50 uniform configurations of randomly placed scatterers, via MultiStart, are initialized. Figure
4 illustrates the localization effect for a normal plane wave incident on uniform configurations of cylindricalvoids at various discrete values of normalized wavenumber, ka ∈ [0.6, 1] depicted at top row figures (a)-(c)and ka ∈ [1, 1.4] given at middle row figures (d)-(f), and ka ∈ [1.6, 1.8] shown in bottom row figures (g)-(i)varying the total number of scatterers M = 10, 20 and 30. Figure 5 illustrates similar localization effects foruniform configurations of elastic thin cylindrical shells at various discrete values of normalized wavenumber,ka ∈ [0.6, 1] depicted at top row figures (a)-(c) and ka ∈ [1, 1.4] shown in bottom row figures (d)-(f) varying thetotal number of scatterers M = 10, 20 and 30. Figure 4 and 5 show 9 black dots the horizontal axis indicating
considered discrete values of ka and results for unoptimized initial random configuration denoted by blue color
curves that are included for a comparison. Generally, as the number of optimization wavenumbers raises and as
the number of scatterers increases, the distribution of the focal absolute pressure becomes more uniform under
the optimization process. Specifically, for |p f |RMS computed at 9 values of ka shown by green curves, a more
uniform broad focusing performance is achieved within the considered range of wavenumbers.
|pf |

3

M = 10M = 20

3

2

2

2

1

1

1

0

0
1

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1.15

(a)

1.2

1.25

1.3

1.35

1.4

1

3

3

2

2

1

1

1

1.6

1.7

1.8

0
1.4

1.45

1.5

(d)

1.55

1.6

1.65

1.7

1.75

1.8

0
1.4

3

3

2

2

1

1

1

0
1.6

0
1.6

1.7

1.75

(g)

1.8

1.65

1.7

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

(f)

2

ka

1.15

ka 2 values
ka 3 values
ka 5 values
ka 9 values
Unoptimized

(e)

3

1.65

1.1

(c)

2

1.5

1.05

(b)

3

0
1.4

M = 30

3

1.75

1.8

0
1.6

1.65

1.7

ka

ka

(h)

(i)

1.75

1.8

Figure 4. The absolute pressure at focal point, |p f |, optimized at 2, 3, 5, and 9 equally spaced values ofwavenumber ka ∈ [1.0, 1.4] depicted at the top row Figures 4(a)-(c), ka ∈ [1.4, 1.8] shown at the middle row
Figures 4(d)-(f), and ka ∈ [1.6, 1.8] given at the bottom row Figures 4(g)-(i) for uniform configurations of M =10 (left column), M = 20 (middle column) and M = 30 (right column) cylindrical voids. The focal point is
x f = (0.0865, 0)m. The plots for unoptimized initial random uniform configuration are denoted by blue color
curves and are included for a comparison.

3.2.2 Sound localization developing multi-directional lens
Now we present the numerical results illustrating the performance of multi- directional acoustic lens designs initializing 100 uniform configurations of randomly placed scatterers, via MultiStart. Figure 6 illustrates |p f |RMS ,as a function of angle of incidence ψ and the normalized wavenumber ka performing concurrently a multifrequency and multi-angle optimization of function |p f |RMS at a broad range of the incidence angle ψ ∈[−π/4, π/4] and 3 values of wavenumber ka ∈ [0.35, 0.55] for configurations of M = 50 rigid cylinders. The
optimized configuration on left top corner of Figure 6 is the result of GBO for 3 values of wave number ka
and 9 incidence angles. The results for unoptimized initial random uniform configuration is depicted by blue
color curves.

4

CONCLUSIONSWe presented the GBO strategy which combines the multiple scattering theory with the principle of acoustic
reciprocity and leads to an effective method to optimally position scatterers so as to produce acoustic focusing.
The GBO technique uses semi-analytic expressions for the gradient of the focal pressure with respect to the po-M = 10|pf |M = 30M = 20

2.5

4

5

2

4

3
1.5

3
2

1

2
1

0.5
0
0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1

0
0.6

0.65

0.7

0.75

(a)

0.8

0.85

0.9

0.95

1

0
0.6

2.5

2

2

2

1.5

1.5

1.5

1

1

1

0.5

0.5

0.5

0

0
1.05

1.1

1.15

1.2

ka
(d)

0.7

0.75

1.25

1.3

1.35

1.4

0.8

0.85

0.9

0.95

1

1.25

1.3

1.35

1.4

(c)

2.5

1

0.65

(b)
2.5

ka 2 values
ka 3 values
ka 5 values
ka 9 values
Unoptimized

0
1

1.05

1.1

1.15

1.2

ka
(e)

1.25

1.3

1.35

1.4

1

1.05

1.1

1.15

1.2

ka
(f)

Figure 5. The absolute pressure at focal point, |p f |, optimized at 2, 3, 5, and 9 equally spaced values ofwavenumber ka ∈ [0.6, 1.0] depicted at the top row Figures 5(a)-(c), and ka ∈ [1.0, 1.4]] illustrated at the bottom
row Figures 5(d)-(f) for uniform configurations of M = 10 (left column), M = 20 (middle column) and M = 30
(right column) empty elastic cylindrical thin shells with material properties given in [2]. The focal point is
x f = (0.0865, 0)m.

Figure 6. The absolute pressure at focal point, |p f |, optimized at 1, 3, 5, and 9 equally spaced values of angle
of incidence ψ and three values of wavenumber ka = 0.35, 0.45, and 0.55 for the optimized uniform configurations of M = 50 rigid cylinders. The focal point is x f = (0.0865, 0)m. The results for unoptimized initial
random uniform configuration are depicted by blue color curves. The performance of lens is shown at normalized wavenumber ka = 0.55.
sitions of each scatterer. The results presented for the uni- and multi-directional broadband acoustic lens designs
display that a considerable sound pressure amplification can be achieved by optimal scatterer replacement using this analytical-numerical technique. Providing the gradient information, the GBO strategy can be integrated
with deep reinforcement learning algorithms [29] and generative neural networks [30] which have potentials tosearch for the globally optimized devices over a broad range of parameters and can provide better solutions
than ones produced by traditional optimization methods. Optimized lens models obtained by using non-uniform

configurations allowed for the reduction of compute time achieving a similar performance as produced by uniform configurations but employing lesser number of scatterers. The presented GBO assisted acoustic lens model
can be further improved by considering the cylinder radii as an additional design parameter which will require
deriving analytical formulas for the gradients of absolute pressure with respect to cylinder radii. This study isunder the development and the preliminary results for an implementation of lens design via radii and position
adjustments will be presented at the ICA2022 meeting.

ACKNOWLEDGEMENTS
FA acknowledges Small Grant Project grant support from San Jose State University.

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[2] Feruza Amirkulova, Linwei Zhou, Anam Abbas, Peter Lai, Cheng Qiu, and Tristan A. Shah. Acoustic
metamaterial design framework using deep learning and generative modeling. The Journal of the AcousticalSociety of America, 151(4):A253–A253, apr 2022.
[3] A. N. Norris. Faxen relations in solids - a generalized approach to particle motion in elasticity and viscoelasticity. Journal of the Acoustical Society of America, 123(1):99–108, 2008.
[4] A.-C. Hladky-Hennion, J. O. Vasseur, G. Haw, C. Croënne, L. Haumesser, and A. N. Norris. Negativerefraction of acoustic waves using a foam-like metallic structure. Appl. Phys. Lett., 102(14):144103, 2013.
[5] Hongchen Chu, Qi Li, Bingbing Liu, Jie Luo, Shulin Sun, Zhi Hong Hang, Lei Zhou, and Yun Lai. A
hybrid invisibility cloak based on integration of transparent metasurfaces and zero-index materials. Light:
Science & Applications, 7(1), aug 2018.
[6] Jierong Cheng, Sandeep Inampudi, and Hossein Mosallaei. Optimization-based dielectric metasurfaces forangle-selective multifunctional beam deflection. Scientific Reports, 7(1), sep 2017.
[7] Clément Lagarrigue and Damien Lecoq. Acoustic metamaterials for industrial applications. wiley online
library. Wiley Online Library, aug 2019.
[8] Xinyue Wu, Yabin Jin, Abdelkrim Khelif, Xiaoying Zhuang, Timon Rabczuk, and Bahram Djafari-Rouhani.
Topological surface wave metamaterials for robust vibration attenuation and energy harvesting. Mechanics
of Advanced Materials and Structures, pages 1–9, jun 2021.
[9] Noé Jiménez, Jean-Philippe Groby, and Vicent Romero-García. Spiral sound-diffusing metasurfaces basedon holographic vortices. Scientific Reports, 11(1), may 2021.
[10] A. Spadoni and C. Daraio. Generation and control of sound bullets with a nonlinear acoustic lens. Proceedings of the National Academy of Sciences, 107(16):7230–7234, Apr 2010.
[11] Dario Leone. How luneburg lens radar reflectors are used to make stealth aircraft visible on radar screens.
https://theaviationgeekclub.com/these-devices-make-stealth-aircraft-visible-on-radar-screens/, June 2021.
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[15] Xiaoshi Su, Andrew N Norris, Colby W Cushing, Michael R Haberman, and Preston S Wilson. Broadband
focusing of underwater sound using a transparent pentamode lens. Journal of the Acoustical Society ofAmerica, 141(6):4408–4417, 2017.
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3522 https://doi.org/10.3397/IN–2021–2431, aug 2021.

ABS-0054

Identification and assessment of flow-induced sound sources inside
encapsulated flow domains - Investigating a clinically applied hydraulicprosthesis
Andreas WURZINGER(1) , Bernhard MAYR-MITTERMÜLLER(2) , Harald SIMA(2) , Manfred KALTENBACHER(1) , Stefan
SCHODER(1)(1) Graz University of Technology, IGTE, A-8010 Graz, Austria, andreas.wurzinger@tugraz.at(2) Otto Bock Healthcare Products GmbH, A-1110 Wien, Austria

ABSTRACT
Noise emissions of hydraulic prostheses can have significant negative impacts on users and their environment. In
this context, the sound production mechanisms inside the hydraulics of a clinically-applied knee prosthesis were
investigated with the goal to identify tonal and broadband source regions and assess counter measures. Therefore, scale resolving incompressible and weakly compressible flow simulations are used to calculate acoustic
sources based on common aero-acoustic formulations inside the encapsulated flow domain. These sources arevisualized both in time and frequency domain to obtain qualitative information about their location and underlying source mechanisms. This analysis allows to distinguish between wall excitation based on turbulent
incompressible flow phenomena on the one hand and compressible wave propagation effects on the other. Finally, validation experiments show a reduction of noise emissions obtained by adaptation of the flow guidance
based on elimination of one source mechanism that is identified during the analysis.

Keywords: Computational Acoustics

1

INTRODUCTION

The human knee joint allows people a functional, self-healing, low-noise [11] and impact-tolerant [1] force
transmission during walking. Unfortunately, accidents and maladies can lead to the fatal situations in which
over-knee amputation of the leg is a medical necessity. In 2014 only, 57.637 amputations of the lower extremities were performed in Germany [3]. Technical aids like a leg-prosthesis can help in these difficult circumstances to significantly improve the life quality. Providers of medical equipment in the field of prosthetics
as Otto Bock Healthcare Products GmbH are therefore interested in the development of prostheses that allow
for safe, natural human walking. Therefore, the knee-prosthesis under investigation in this work is based on amechatronic, controlled hydraulic damping system. Apart from mechanical and functional requirements soundemissions in several use-cases are of major interest, as the generated noise can be a stress factor for the user
and also unpleasant for both user and the user’s environment [4]. Figure 1a shows the sound power density
spectrum emitted by the prosthesis under investigation (design 1) in a certain use-case. The sound emissions
consist of broad-band noise between 500 Hz and 1.5 kHz as well as a tonal whistling sound at approximately
4.5 kHz. Especially, the tonal components of the emitted sound should be avoided, which most often requires
minor changes to the overall design only (eg. design 2). To detect acoustic hot-spots as early as possible in thedevelopment process, a simulation workflow for identification and systematic analysis and prevention of acoustic
source hot-spots is developed in the funded project VirtualProsthesis1 .
Figure 1b depicts the schematic of the hydraulic system applied in the prosthesis under investigation. The
system is used to control the damping of the artificial knee joint and consists of a double-chamber piston,1 https://projekte.ffg.at/projekt/4163257

(b) Schematic of the hydraulic system. Separated
flow guides for extension and flexion movement.
(a) Power spectrum of the sound emissions of the investigated prosthesis (design 1) and an improved design 2. Frequency ranges of interest for broadband
noise and tonal noise are marked.

Figure 1. Microphone measurement of 2 design variants of the prosthesis and a schematic of the hydraulicsystem.
which is moved in extension direction, when no force is applied due to the static pressure. For each directionof piston movement, flexion and extension, the hydraulic fluid flows through separate servo and check valves.
These servo valves can be controlled from fully closed to fully open to meet the target resistance for each gatephase.

2

METHODOLOGY

2.1 Flow simulation
To reduce complexity and numerical effort of the problem, the hydraulic system is approximated by a simplified stationary system (see Fig. 2a) that models the flexion motion. This assumption is based on the phaserelation and the tones observed by experimental investigation. Therefore, the piston is replaced by velocity inlet
and mass flow outlet and the expansion chamber by constant pressure boundary conditions respectively.
The computational domain is discretized by a combination of hexahedral, polyhedral and wedge elementswith edge sizes between 5 · 10−5 and 4 · 10−4 m. All walls are refined with 8 prism layers to achieve full
resolution (y+ < 1) of the boundary layer. The hydraulic fluid is modeled by either an incompressible or aweakly compressible fluid based on the Tait equation with nominal density ρ = 860 kg/m3 , dynamic viscosityη = 0.02752 Pa s, isothermal compression modulus K = 1.5 · 109 Pa, and adiabatic exponent n = 7. A scale resolving Stress-Blended Eddy simulation (SBES) [6], similar to hybrid approaches used eg. in [7], is used toefficiently resolve sufficient turbulent scales. It uses a k-ω-SST turbulence model and a Wall-Adapting Local
Eddy-viscosity (WALE) sub-grid-scale model. The incompressible continuity equation is fulfilled by the direct
coupling of the pressure correction equation. Spatial discretization is of order 2 and time integration is done by
an implicit 2nd order scheme with a fixed time step of ∆t = 5 · 10−6 s. The described setup was computed with
Ansys Fluent and performed on the Vienna Scientific Cluster Vienna Scientific Cluster2 (VSC). The overall computational effort for simulating 8000 time steps was about 14500 CPU h and 24000 CPU h for the incompressible
and weakly compressible setups respectively.
Figures 2b and 2c show the average of 5 point spectra (center and 4 points close to walls) of the staticpressure at 4 monitoring cross-sections indicated in Fig. 2a for both the weakly compressible and the incompressible fluid model (as investigated in [12]). The spectra in monitor 2 and 3 show dominant broad bandcharacter due to turbulent pressure fluctuations while the spectra in monitor 1 and 4 differ significantly to the
2 vsc.ac.atincompressible spectra that show broad band character mostly. Further investigation shows, that the spectra aresignificantly modulated by compressible modes. The frequencies of the amplitude peaks coincide roughly withthose obtained solving the Helmholtz equation for a point source at monitor 2 with the finite element-based
multi-physics solver openCFS3 , as done in [12].

(a) Outline of the simplified flow domain. Inlet andoutlet boundary conditions on the piston surfaces,
and a pressure boundary condition towards the expansion chamber. Servo and check valves closed
during extension neglected.

(b) Averaged spectra of the static pressure of 5 points in the monitoring crosssections indicated in Fig. 2a. Compressible modes obtained from the frequency response function (FRF) of a point source located in M2 [12], Fig. 8
are indicated.

(c) Averaged spectra of the incompressible static pressure of 5 points in themonitoring cross-sections indicated in Fig. 2a (as depicted in [12], Fig. 5).

(d) Iso-contours at ±108 Pa
s of the acoustic source
term based on the PCWE formulation.

Figure 2. Computational Fluid Dynamics (CFD) simulation setup and resulting pressure spectra and acoustic
sources based on incompressible CFD.

2.2 Acoustic sources
To obtain information about the acoustic source hot-spots, several formulations of hybrid aero-acoustic workflows [8] can be used to calculate an acoustic source term. Visualizing the source terms can help understandthe source characteristics during a detailed cause-effect analysis [9]. In Fig. 2d iso-contours of the RHS sourceof the Perturbed Convective Wave Equation [2]3 opencfs.org

(a) Real part of the fieldFFT of the incompressible static pres- (b) Real part of the fieldFFT of the static pressure at f = 4654 Hz
based on the weakly compressible CFD.
sure at f = 4500 Hz.

Figure 3. Visualizations of the fieldFFT computed from incompressible and compressible flow pressure fields.

1 D2 ψ a
1 Dpic− ∇ · ∇ψ a = − 222ρ̄c Dt
c Dt

(1)

∂•
ic
with D•Dt = ∂t + v · ∇• are depicted. Thereby, p is the incompressible flow pressure, and v the mean flow
velocity. Therein, the main regions of acoustic sources can be observed: At the inlet and outlet of the servo
valve, at the check valve, and at 2 bends of the side channel in the vicinity of M3 and M4 according to Fig. 2a.
To get further insight in the structure of the acoustic sources, the flow pressure time data was transformed
into the frequency domain by using a field fast Fourier transform (fieldFFT). Figure 3a shows the real part ofthe incompressible static pressure obtained by the fieldFFT in a plane at the inlet of the servo valve, for thefrequency of the tonal component of the measurement f = 4500 Hz.
Most apparent, one can observe structures of pressure fluctuations, that convect with the flow velocity. Thisbehavior indicates vortex generation by a mixing layer instability phenomenon, which explains the peaks in theamplitude spectrum of Fig. 2c at approximately 4500 Hz. However, further investigations have to be made tounderstand this mechanism, as it seems that the instability is triggered by compressible effects even though theanalyzed data set stems from an incompressible CFD. Counterintuitively, such global pressure fluctuations canbe observed in Fig. 3a. These can be related to artificial compressible phenomena introduced by numericalinaccuracy as the pressure correction equation
∇2 p′ = −∇ · ∇ · ρ vic ⊗ vic − ∇2 p0 + ∇ · ∇ · τ(2)with the viscous stress tensor τ, the pressure correction p′ = pic − p0 , and the initial guess for the pressurep0 allows for a pressure correction |p′ | > 0. It is noted at this point that Eq. 2 is equivalent to a degenerated form of Lighthill’s inhomogeneous wave equation [5] for the case if speed of sound tends towards infinity
and consequently the wave operator turning into a Laplace operator. Consequently, these artificial compressible
effects vanish with increased convergence of the CFD solver for the incompressible material model. Interestingly enough, this artificial compressibility yields similar results in triggering the mixing layer instability to theweakly compressible approach, as the Mach number doesn’t exceed Ma < 0.02. This leads to insignificant errors, introduced by the violation of the phase relation by neglecting wave traveling effects for low Mach number
flows[10]. Figure 3b shows the real part of the Fourier transformed flow pressure from the weakly compressible CFD for the slightly higher frequency f = 4654 Hz. Again the vortices are visible, however the extensive
pressure fluctuations change phase at the valve inlet and are of much lower magnitude at this frequency.

3

CONCLUSIONS

A simplified stationary model of the fluid dynamics during the flexion motion of a hydraulic knee prosthesis
was established that has been solved both for the incompressible and the compressible pressure fields. Comparison to compressible modes obtained from FRFs shows the influence of compressibility on the flow pressure
field. Furthermore visualization of both the pressure fields and acoustic source terms based on aero-acoustic
formulations yield insight in the spatial distribution of flow-induced sound source hot-spots and source characteristics like flow instability triggered by coupled compressible effects, that cannot be predicted reliably byan incompressible flow simulation. Therefore, special care has to be taken for flows that are prone to become
unstable like mixing layers, which will be the focus of further investigations.

ACKNOWLEDGEMENTS
This project has received funding from the Austrian Research Promotion Agency (FFG) under the Bridge projectNo. [71694]. The computational results presented have been achieved in part using the Vienna Scientific Cluster(VSC).

REFERENCES[1] A. Hoshino and W. Wallace. Impact-absorbing properties of the human knee. The Journal of bone andjoint surgery. British volume, (5), 1987.
[2] M. Kaltenbacher. Computational Acoustics. CISM International Centre for Mechanical Sciences. Springer
International Publishing, 2017.
[3] K. Kröger, C. Berg, F. Santosa, N. Malyar, and H. Reinecke. Amputationen der unteren extremität indeutschland. Dtsch Arztebl Int, 2017.
[4] S. Kumar, H. M. Forster, P. Bailey, and T. D. Griffiths. Mapping unpleasantness of sounds to their auditory
representation. The Journal of the Acoustical Society of America, (6), 2008.
[5] M. J. Lighthill. On sound generated aerodynamically I. General theory. Proceedings of the Royal Societyof London, 1951.
[6] F. Menter. Stress-blended eddy simulation (SBES)—A new paradigm in hybrid RANS-LES modeling. In
Symposium on Hybrid RANS-LES methods, 2016.
[7] S. Schoder, C. Junger, and M. Kaltenbacher. Computational aeroacoustics of the eaa benchmark case ofan axial fan. Acta Acustica, (5), 2020.
[8] S. Schoder and M. Kaltenbacher. Hybrid aeroacoustic computations: state of art and new achievements.
Journal of Theoretical and Computational Acoustics, (04), 2019.
[9] S. Schoder, P. Maurerlehner, A. Wurzinger, A. Hauser, S. Falk, S. Kniesburges, M. Döllinger, and M.
Kaltenbacher. Aeroacoustic sound source characterization of the human voice production-perturbed convective wave equation. Applied Sciences, (6), 2021.
[10] S. J. Schoder. Aeroacoustic analogies based on compressible flow data. PhD thesis, Wien, 2018.
[11] L.-K. Shark, H. Chen, and J. Goodacre. Knee acoustic emission: a potential biomarker for quantitative
assessment of joint ageing and degeneration. Medical engineering & physics, (5), 2011.
[12] A. Wurzinger, B. Mayr-Mittermüller, M. Kaltenbacher, and S. Schoder. Untersuchung strömungsakustischer
quellen einer klinisch angewandten hydraulischen knieprothese. In DAGA 2022-48. Jahrestagung für Akustik,2022.

ABS-0071

Aeroacoustic formulations for coupling of confined flows with the flow-guiding
structure
Paul MAURERLEHNER(1) , Johannes TIEBER(2) , Stefan SCHODER(1) , Helfried STEINER(2) , Günter BRENN(2) , Florian
TOTH(3) , Manfred KALTENBACHER(1)(1) Institute of Fundamentals and Theory in Electrical Engineering, Graz University of Technology, Austria,

paul.maurerlehner@tugraz.at(2) Institute of Fluid Mechanics and Heat Transfer, Graz University of Technology, Austria, helfried.steiner@tugraz.at(3) Institute of Mechanics and Mechatronics, TU Wien, Austria, florian.toth@tuwien.ac.at

ABSTRACTTwo aeroacoustic formulations, namely Lighthill’s wave equation and the Perturbed Convective Wave Equation(PCWE), are applied within the hybrid aeroacoustic approach to compute the propagation of flow-induced soundin a pipe flow. A low Mach number flow (Ma=0.06) through a diverging and remerging plastic branch pipe
is investigated. An analysis of the source terms computed by Large-Eddy Simulation (LES) reveals that the
primary acoustic sources occur at flow detachments downstream of the sharp edges of the branch. Spectral
results of the wall pressure estimated by the PCWE show good agreement with measurements. In contrast, thespectral peaks due to acoustic resonances are overestimated by Lighthill’s wave equation by at least an orderof magnitude, while correctly reproducing the incompressible pressure field. Overall, the results indicate that
the PCWE is better suited for the present flow configuration. By coupling the internal acoustic domain withthe surrounding structural domain through appropriate interface conditions, the interaction of the fluid and thestructure is simulated. The comparison with results of the non-coupled simulations and the measurement showsthat Fluid-Structure Interaction (FSI) must be modeled for accurate results, because the damping of the structure
mitigates the occurring acoustic resonances.

Keywords: Computational Aeroacoustics, Fluid-Structure Interaction, Finite-Element Method, Duct Acoustics

1

INTRODUCTION

The hybrid aeroacoustic approach is an efficient way to address the issue of the disparity of scales in Computational Aero-Acoustics (CAA) at low Mach number [1]. In literature, the application to various kinds offlow configurations, such as fans [2, 3], jets [4], other external flows [5, 6], and internal flows [7, 8, 9] canbe found. Furthermore, the PCWE, in particular, has been applied to compute human voice production using a
hybrid approach [10, 11].
When applying this two-step approach for the computation of flow-induced sound in ducts, the sound radiationof the flow-guiding structure can be additionally obtained by coupling the fluid and the solid domains via
appropriate coupling conditions. Furthermore, a strong coupling of the internal fluid domain and the confining
structural domain must be considered when FSI is involved, which typically appears for flexible structures. For
the particular case of small deformations, where the fluid flow is not affected by the deformation, a couplingof the internal acoustics and the structure is sufficient. Consequently, the strong coupling can be realized in the
acoustic propagation simulation within the hybrid aeroacoustic approach avoiding FSI in the relatively costly
Computational Fluid Dynamics (CFD) simulation.

In this work, the hybrid (two-step) approach is applied to predict the sound generation and propagation ofa confined low Mach number flow. A diverging and remerging pipe flow (simply called pipe branch in thefollowing), obtained by combing two T-junctions, is investigated. Dedicated measurements were performed ina flow through a geometry manufactured in a single part using 3D printing. This plastic part is modeledin the simulations when FSI is considered. The first step of the simulation workflow consists of computing
the aeroacoustic source terms by performing an incompressible LES based on the Finite-Volume (FV) method.
For the subsequent Computational Acoustics (CA) simulation, two aeroacoustic formulations, namely Lighthill’swave equation and the PCWE, are applied. The governing equations are solved using the Finite-Element (FE)multiphysics solver openCFS [12]. Comparing the resulting Power Spectral Densitiy (PSD) of the wall pressure
with measurements shows that the PCWE yields more accurate results than Lighthill’s wave equation. The
Lighthill source term involves spatial derivatives and, thus, strongly depends on the spatial discretization andthe numerical approximation of the derivative. In contrast, the relevant source term of the PCWE consists ofthe substantial time derivative of the incompressible pressure. Considering FSI within the (coupled) PCWEbased CA simulation improves the agreement with the measurements by mitigating acoustic resonances by thedamped structure, revealing the relevance of FSI in the investigated flow configuration.

2

AEROACOUSTIC FORMULATIONSIn the following, two aeroacoustic formulations are considered within the hybrid aeroacoustic workflow to solve
for the propagation of flow-induced sound. The PCWE is based on a perturbation ansatz, which decomposes
the flow quantities, namely the pressure p, the flow velocity v , and the density ρ into a mean part ( p̄, v̄v, ρ̄)and fluctuating incompressible (pic0 , v ic0 ), and acoustic components (pa , v a , ρ a ). For the pressure p and velocity
v , this splitting approach reads
p = p̄ + p0 = p̄ + pic0 +pa and| {z }(1)v = v + v 0 = v + v ic0 +vva .
| {z }

(2)

pic

v ic

Application of this perturbation approach to the compressible flow equations, yields the inhomogeneous wave
equation
1 D2 ψ a
1 Dpic0
a
−
∇
·
∇ψ
=
−
= Qic
(3)
PCWE ,
ρ0 c2 Dt
c20 Dt 2
which is solved for the scalar acoustic potential ψ a with v a = −∇ψ a , where v a denotes the irrotational particle
velocity [13]. In (3), c0 denotes the constant speed of sound and ρ0 the density of the uniform medium. TheD∂= ∂t+ v · ∇ in the wave operator and in the source term considers the convection ofsubstantial derivative Dtaic
acoustic waves. The acoustic pressure is obtained by the relation pa = ρ0 Dψ
Dt . The source term QPCWE and themean velocity v in the wave operator are provided by the incompressible CFD simulation.
While the PCWE solves for a purely acoustic quantity, Lighthill’s wave equation in the pressure formulation
reading
1 ∂2 0p − ∇ · ∇p0 = ∇ · ∇ · [ T ] = QLH(4)
c20 ∂t 2
solves for the pressure fluctuation p0 . The right hand side (RHS) of (4) defines the acoustic source term denoted
T ] containing non-linear convective
as QLH consisting of the double divergence of the Lighthill stress tensor [Tfluxes, an excess term and a viscous stress term. For low Mach number flows, where the flow can be assumed
as incompressible, the latter can be neglected, reducing the stress tensor to solely the non-linear velocity product[ T ] ic = ρ0 v ic ⊗ v ic

(5)and the Lighthill source term becomes
ic
Qic
LH = ∇ · ∇ · [ T ] .

(6)

The source term (6) is again obtained by an incompressible flow simulation.
For a strong coupling of acoustics and structural dynamics, dynamic (continuity of forces) and kinematic (continuity of normal velocity) coupling condition must be fulfilled at the coupling interface ΓC [14]. For the PCWE,
the dynamic coupling condition reads
pa n = ρ0Dψ aσ mech ] · n ,n = −[σDt(7)where n denotes the normal unit vector on ΓC . Additionally, the kinematic coupling condition is enforced by∇ψ a · n = −∂ u mech
· n,
∂t

(8)

with umech being the mechanical displacement. For the non-coupled case, ΓC is a rigid boundary, where u mech =0, reducing (8) to zero, which corresponds to a soundhard boundary condition.

3

WORKFLOW

The presented simulation workflow is based on the hybrid approach, which divides the task of sound computation into a CFD simulation and a Computational Acoustic (CA) simulation (propagation simulation), as
displayed in Fig. 1. As a part of this concept, the computed source terms are interpolated onto the CA mesh
Hybrid aeroacoustic approach
Coupled CA simulationinc. CFD
simulation
AVL FIRETM

ST (raw)

CFD/CA
grid

Pre-Processing(Interpolation, Filtering)

ST

CA
grid

Aeroacoustics

Mechanics
openCFS

Figure 1. Simulation workflow for strong coupling of fluid and structure based on the hybrid aeroacoustic
approach.
during the source data export in AVL FIRETM , where conservation of energy is enforced by considering theintersection of the CFD grid and the CA mesh [15]. Furthermore, mean values of the source term, which do
not contribute to sound generation, are removed prior to the propagation simulation [16]. For the considerationof FSI, strong coupling of the fluid region and the structural mechanical region is considered in the propagation simulation. The structural mechanical domain is governed classically by the balance of momentum anda linear stress-strain relationship. The weak forms of the coupled partial differential equations governing the
aeroacoustics and structural dynamics are solved using the FE-based solver openCFS.

4

APPLICATION

The presented workflow is applied to a fully developed turbulent airflow with bulk inflow velocity U = 20.8 m/s(Ma=0.06), passing through a diverging and remerging branch pipe (see Fig. 2). The inner diameter is D =50mm, the wall thickness of the branch pipe is 2mm. For the CFD domain, the inlet and outlet section of alength of Lin = 4.2 D and Lout = 10.3 D is considered, respectively. In the validation experiment, the internal wallpressure is captured by pressure transducers (Kulite XCS-093) at selected positions P1 and P2 (see Fig. 2b).
More information on the experimental setup can be found in [17].

(a) Picture of the experimental setup with insertedpressure transducers (Kulite).

(b) Geometry of the fluid and structural domain including sensor positionsof the wall pressure probes P1 and P2 (all measures in mm).

Figure 2. Experimental setup and corresponding definition of the simulation domains. The inflow Lin andoutflow length Lout define the fluid domain of the CFD and CA simulation.
The fluid density ρ0 = 1.176 kg/m3 , the dynamic viscosity µ0 = 18.3 µPa s, and the speed of sound c0 =345.2 m/s are chosen as the material parameters required for the simulations. The plastic mounts at both endsof the pipe are considered in the simulation, where the bottom surface is assumed to be fixed according to theexperimental setup. The material parameters (Young’s modulus E, Poisson ratio ν, and density ρ) of the 3D
printed branch and mounts are listed in Tab. 1. The mass-proportional damping coefficient αR and stiffnessproportional damping coefficient βR for Rayleigh damping of the structure are fitted for the damping ratio (ofselected structural modes) obtained from experimental modal analysis in the range of 0 to 5 kHz.
Table 1. Structural mechanical material parameters (Young’s modulus E, Poisson ratio ν, and density ρ).
E (MPa)νρ (kg/m3 )αRβR

Branch pipe

2 000

0.43

900

777

1.387e-5

Mounts

2 200

0.40

840

777

1.387e-5Part

4.1 INCOMPRESSIBLE FLOW SIMULATION
The incompressible Navier-Stokes equations are solved using LES in AVL FIRETM based on the FV method. At
the inlet face, a turbulent fluctuating velocity distribution was prescribed, which was generated in a precursor
LES, considering fully developed, axially periodic, turbulent flow through a 5D long pipe. At the outflow
boundary, an averaged pressure boundary condition (BC) was used. The computational grid of the 25D longdomain consisted of about 8.14 million FV cells. The time step was set to 10 µs, and the simulation time was
0.62 s where convergence was reached, while additional 0.2 s were simulated for the export of the source terms.
More information about the simulation setup is provided in [18], where a LES of a similar flow configuration
was carried out and validated. These predicted source terms were further interpolated onto the CA mesh for thesubsequent propagation simulation and the mean value was removed in a filtering process.
Figure 3 shows the instantaneous source terms for both considered aeroacoustic formulations. Large source
terms occur downstream of the sharp edges at the diverging and converging branches, respectively. The Lighthillsource terms indicate the detachment of vortices at the sharp-edges, especially at the converging branch (Fig. 3b).

4.2 PROPAGATION SIMULATION
The propagation simulations are carried out by the multiphysics solver openCFS, applying the FE method [12].
A time step size of ∆t = 10 µs was chosen, and a zero initial condition was used. The simulation time was 0.2 s,corresponding to the export time of the CFD simulation. At the first 80 time steps, a temporal blending function

(a) PCWE source term Qic
PCWE .

(b) Lighthill source term Qic
LH .

Figure 3. Instantaneous contours of the source terms obtained by incompressible LES.
was used to smoothly activate the acoustic sources to avoid impulsive excitation. In Fig. 4a, the computational
domains and in Fig. 4b a detail of the multiphysics mesh are displayed. The mesh consists of the internal(a) Full setup with inlet left and outlet right.

(b) Clipped view of the mesh in horizontal plane at the inlet.

Figure 4. FE simulation setup for the coupled propagation simulation.
aeroacoustic domain (orange) including the aeroacoustic sources and the structural mechanical domains of themounts (green) and the branch pipe (pink). Adjacent Perfectly Matched Layer (PML) regions (red) at the inlet
and outlet avoid reflections at the open boundaries [19]. At the bottom surfaces of the mounts, all three degrees
of freedom are locked for the coupled simulation. The total number of elements sums up to approximately
1.2 million. The maximum element size in the fluid domain is 3 mm (see Tab. 2), resolving acoustic waves
up to fmax = 11 kHz by at least ten finite elements. Linear elements are used for the aeroacoustic domain,
while quadratic FE basis functions are applied to the structural mechanical domain. More information about themultiphysics mesh is provided in Tab. 2.
Figure 5 shows the Power Spectral Densities (PSDs) of the wall pressure estimated by the incompressible LES
and the PCWE-based propagation simulation at the positions P1 and P2 (see Fig. 2b). The pressure fromthe incompressible CFD simulation must be added to the pressure estimated by the PCWE according to (1)to be comparable with the wall pressure measurements. At P2, various peaks are visible in the measured
spectrum, which is caused by the excitation of acoustic resonances, where the length of the centerline loop
Llp = 0.63 m of the branch pipe coincides with a multiple of the acoustic wavelength λi . This results in aresonance frequency fi = cL0 i , where i denotes the order of the mode. In Figs. 6a and 6b, the real part of thelp
Fourier transformed acoustic pressure predicted by the PCWE is depicted at frequencies f3 = 1667.2 Hz and
f8 = 4134.7 Hz, respectively. The contours clearly exhibit the acoustic resonances, where the latter is above

Table 2. Details of the multiphysics mesh for the coupled propagation simulation.
Regiondomainno. regions

no. elements

max el. size

el. order

Fluid

acoustic

1

1 002 1713mm

1

tetrahedron

PML

acoustic

2

2 016

3mm

1

wedge

Branch pipe

struct. mech.

1

163 9382mm

2tetrahedron

Mount

struct. mech.

2

14 8173mm

2tetrahedron

6

1 199 775

2-6mm

Total
PSD (Pa2 /Hz)(a)el. type

PSD (Pa2 /Hz)(b)

Figure 5. PSDs of the internal wall pressure at the two sensor positions (a) P1 and (b) P2.
the cut-off frequency for plane wave propagation fc = 4046 Hz. Due to the assumption of incompressible flow,
the LES does not capture these acoustic modes. When adding the acoustic pressure predicted by the PCWE,
the spectrum of the measurement is reproduced well (Fig. 5a). The peaks of the acoustic resonances are notvisible in the PSD of P1 due to the local dominating superimposed turbulent pressure fluctuations. Overall, thepressure fluctuations are underestimated in P1, especially in the higher frequency range. For P1 (Fig. 5a), theresult of Lighthill’s wave equation is also shown (for better visibility, it is not shown for P2 in Fig. 5b.). The
spectra resulting from Lighthill’s wave equation partly overestimate the acoustic pressure, especially above thecut-off frequency. Comparing the contour of the pressure fluctuation p0 at an instantaneous time step in Fig. 6dreveals that the pressure from the incompressible CFD (of the LES in Fig. 6c) is reproduced well.
Additionally, strong coupling is considered for the PCWE-based propagation simulation (which yielded moreaccurate results) to evaluate the impact of FSI. The coupling does not have a significant impact on the PSD in
P1 (Fig. 5a). In P2, the strong coupling results in reduced amplitudes at the higher acoustic resonances (above
3000 Hz), showing that these resonances are mitigated by the (Rayleigh) damped structure.

5

CONCLUSIONS

Two aeroacoustic formulations, namely the PCWE and Lighthill’s wave equation, were applied to compute theflow-induced sound in a diverging and remerging branch pipe flow based on the hybrid aeroacoustic approach.
Strong coupling of the acoustic and the structural mechanical domains allows for considering FSI within the
FE framework of openCFS. Incompressible LES is used to compute the acoustic source terms of the low Mach
number flow. Contours of the source terms showed that the dominating acoustic sources occur in the detachedflow region downstream of the sharp edges of the branch. The result of the PCWE yielded good agreementwith wall pressure measurements, while the propagation simulation based on Lighthill’s wave equation overesti-

(a) CA (PCWE): Real part of the Fourier transformed acousticpressure ℜ (pa ) at f3 = 1667.2 Hz.

(b) CA (PCWE): Real part of the Fourier transformed acousticpressure at ℜ (pa ) at f8 = 4134.7 Hz.

(c) CFD: Instantaneous incompressible pressure pic0 .

(d) CA (Lighthill): Instantaneous pressure fluctuation p0 .

Figure 6. Field visualizations of the non-coupled simulation results.
mates the amplitudes of the acoustic resonances at least by an order of magnitude, especially above the cut-off
frequency. Consideration of strong coupling in the PCWE-based propagation simulation showed, that for accurate results, FSI must be considered to model the mitigation of the acoustic resonances in the higher frequency
range.

ACKNOWLEDGEMENT
The support from the Austrian Research Promotion Agency (FFG) under no. 874784 and from AVL List GmbH
Graz are gratefully acknowledged.

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[4] M. Tautz, K. Besserer, S. Becker, and M. Kaltenbacher, “Source formulations and boundary treatments forLighthill’s analogy applied to incompressible flows,” AIAA Journal, vol. 56, pp. 2769–2781, 2018.
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M. Kaltenbacher, “Hybrid aeroacoustic approach for the efficient numerical simulation of human phonation,” The Journal of the Acoustical Society of America, vol. 147, pp. 1179–1194, 2020.
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M. Kaltenbacher, “Aeroacoustic sound source characterization of the human voice production-perturbed
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Journal of Computational Physics, vol. 235, pp. 407–422, 2013.

ABS-0075

A non-conforming and overlapping DG formulation for the acoustic
conservation equations
Johannes HEINZ(1) ; Manfred KALTENBACHER(2)(1) TU Wien, Institute of Mechanics and Mechatronics, 1060 Vienna, Austria, Email: johannes.heinz@tuwien.ac.at(2) TU Graz, Institute of Fundamentals and Theory in Electrical Engineering, 8010 Graz, Austria, Email:manfred.kaltenbacher@tugraz.at

ABSTRACT
High-order discontinuous Galerkin (DG) methods are ideally suited to solve the acoustic conservation equationsin the time domain. To compute, e.g., the acoustic field originating from a ventilator, it is common to use twomesh regions. One of the regions rotates while the other one stays fixed. To solve this kind of problems, weneed implementations that can deal with non-conformities at mesh interfaces. If elements do not precisely represent the geometry, i.e., by using non-uniform rational B-splines, it is not sufficient to utilize non-conforminginterfaces alone. Within this work, we show that Nitsches’ idea — which is commonly used at non-conforminginterfaces — can be applied for overlapping elements straightforward. Instead of using complex element cut
algorithms, we directly use element faces at non-conforming transitions as integration domains. To counteract
aliasing, we use over-integration to generate a stable method.

Keywords: Discontinuous Galerkin, Non-matching methods, Moving meshes

1

INTRODUCTION

While nonconforming interfaces (NCIs) have the nice property of easy modular mesh generation, overlapping
meshes enhance this property. This is because boundaries of mesh regions do not have to coincide anymore.
This becomes especially relevant in simulations in which elements move relative to each other. Imagine thecomputation of the aeroacoustic sound field around a rotating fan. In this case the computational mesh typicallyconsists of two regions, a rectangular outer region with a cylindrical hole and a cylinder as inner region. The
outer region stays fixed, while the inner domain — containing the rotor blades — rotates, see e.g. [13, 23].
Solving this kind of problems using NCIs would require the surface of the cylinder to exactly coincide withthe surface of the hole in the outer domain. This can only be achieved using isogeometric analysis, i.e. use
non-uniform rational B-splines, see, e.g. [20]. Using overlaps between the region, it is possible to compute theemerging sound field without the use of isogeometric analysis.
To account for the rotating inner domain typically arbitrary Lagrangian–Eulerian (ALE) methods are used. The
derivation of ALE methods for mass, momentum, and energy conservation equations is given in [5]. This approach has been succesfully applied to the acoustic conservation equations in [8] in a FEM setting.
Nitsche presented the idea to incorporate Dirichlet boundary conditions in the weak form in [19]. Nitsches’
method lead to non-conforming discretizations using the same principle — see amongst others [14, 21, 10] —and naturally extends from non-conforming interfaces to overlapping meshes. Typically element cut algorithms
are used to ensure integration in the same integration points on the non-conformingly connected or overlapping
meshes, see e.g. [17, 13, 23, 22]. The implementation for those cut cell algorithms is a challenging endover.
Element cut algorithms in the overlapping case are espescially complicated and thus it is desirable to avoid
them. A different approach for NCIs is to consequently evaluate all quantities in the integration points dictatedby the master element, see [10, 16]. This methodology is also referred to as point-to-point interpolation and has
been compared to the mortar approach in [16]. To supress aliasing an over-integration at non-conforming cell

faces is necessary [16]. To the best of the authors knowledge this approach has not been applied to overlapping
elements to this date.
Following the point-to-point interpolation method we proposed a non-conforming discontinuous Galerkin discretization for the acoustic conservation equations in [9]. If elements on a NCI do not contain hanging nodes,
the method collapses to the standard DG discretization. With this work we extend mentioned methodology torotating overlapping mesh regions and provide a test case which highlights that the method is well suited forthe computation of aeroacoustic borne sound around a fan as described above.

2

GOVERNING EQUATIONS

The governing equations are the well known acoustic conservation equations and are stated in Eulerian description amongst others in [18, 12, 15]. To be able to consider moving meshes we have to transferre the equations
into the arbitrary Lagrangian–Eulerian description. For the derivation of the equations we referre to [5]. On atime dependent domain Ω(t) ⊂ Rd of dimension d and in a time interval t ∈ [0, T ] they read1∂u
− ∇uuu g + ∇p = 0∂t χρon Ω(t)(1)∂p− ug · ∇p + ρc2 ∇ · u = c2 F∂t χ

on Ω(t)(2)

on ∂ ΩDp (t),(3)

p = gpYρcuu · n = Y p

on ∂ Ω (t).

(4)

In the equations, the material parameters c and ρ are the speed of sound, and the density of the underlying
material respectively. The acoustic pressure is denoted as p, the acoustic particle velocity is u = (u1 , ..., ud )T ,and the velocity of an ALE observer at position χ = (χ1 , ..., χd )T is u g = (ug1 , ..., ugd )T . We can apply pressure Dirichlet boundary conditions (BCs) Eq. (3), and admittance BCs Eq. (4) at the corresponding domainboundaries ∂ ΩDp (t), or ∂ ΩY (t). g p is the acoustic pressure prescribed at Dirichlet boundaries. The first-orderabsorbing BC [6] is obtained with admittance Y = 1, while a reflecting boundary requires Y = 0.

3

NUMERICAL METHOD

3.1 Notation
The physical domain Ω(t) depends on time t if domain boundaries move ∂ Ω(t). The computational domain Ωh (t) represents the physical domain and also dependents on time if its boundaries ∂ Ωh do not movesince the mesh might also move in its internal. The computational domain consists of Nel possibly overlapping
rectangular/hexahedal finite elements. Each finite element has a volume of Ωe (t) and is restricted by ∂ Ωe (t).
The acoustic pressure und acoustic particle velocity are approximated as ph and u h by means of shape functions Nik of polynomial degree k which are constructed from Lagrange polynomials
n kpp̃eh (ξξ ) =

N

∑

i=1

k
Ni p (ξξ )pei ;ũueh (ξξ ) =nN ku∑ Niku (ξξ )uuei .

(5)

i=1

In Eq. (5), nN represents the number of shape functions used for a volume element. Throughout the work we
use the same polynomial degree for pressure and velocity ku = k p . Corresponding broken polynomial spaces Vhfor test and trial functions are those of [9]onVhu = u h ∈ [L2 (Ωh (t))]d : u h (xx)|Ωe (t) = ũu(ξξ )|Ω̃e (t) ∈ [Pku (Ω̃e )]d , ∀e ∈ [1, Nel ] ,(6)noVhp = ph ∈ L2 (Ωh (t)) : ph |Ωe (t) = p̃h |Ω̃e (t) ∈ Pk p (Ω̃e ), ∀e ∈ [1, Nel ] ,(7)where Pk is the space of shape functions of polynomial degree k.

Ω̃1−

ξ−

ϕ −1 (ξξ − , Ω1− )Ω1−x

ϕ (xx, Ω2+ )
Ω2+

Ω̃2+

ξ+

ϕ −1 (ξξ − , Ω1− ), Ω2+ ]ϕ [ϕFigure 1. Picture adapted from [9]. The integration point ξ − on the reference element Ω̃1− is associated to theedge of the master element Ω1− which is connected to the other mesh region via NCI or overlap. The mappingto the physical space followed by the mapping to the slave reference element Ω̃2+ yields an integration point ξ +which can be located inside the slave reference element.
The bidirectional mapping used in [9] now depends on time Ωe (t) → Ω̃e Ω̃e → Ωe (t)ϕ (t) :
; ϕ −1 (t) :.
 x (t) 7→ ξ = ϕ (xx(t), Ωe (t))
 ξ 7→ x (t) = ϕ −1 (ξξ , Ωe (t))(8)

It transfers reference coordinates ξ = (ξ1 , ..., ξd )T to the corresponding physical coordinates x (t) = (x1 (t), ..., xd (t))Tand vice versa, see Fig. 1. As in [9] the implementations of the mapping is realized by means of a search algorithm. Since the mapping depends on time it is now required to search for the integration points each time
step.
Interior informations of a specific element Ωe (t) are denoted with (·)− ; exterior informations are denotedwith (·)+ . The element under consideration Ωe− (t) is also called master element within the following. Exterior
information appears on slave elements which are those connected to the master element Ωe+ ; this connection
can be achieved via coinciding element boundaries or if an element overlaps an element boundary of the masterelement. Each facet of the master element has an outward pointing normal vector n − (t). If element boundaries
coincide n (t) = n − (t) = −nn+ (t) with the normal vector of the slave element n + (t). If the slave element is an
overlapping one we imagine the element boundary of the master element as an artificial truncation of the slave
element and require the same identity. An arbitrary scalar or vectorial quantity b is always defined on the masterelement if no superscript occurs, i.e. b = b− . We use the notation of Bassi et al. [3, 4] for the averaging operator {{·}}, jump operator [·], and normal jump operator J·K; i.e. {{b}} = (b− +b+ )/2, [b] = b− −R b+ , and JbK =b− ⊗ n− + b+ ⊗ n+ . Integrals are expressed in the usual compressed notation (a, b)Ωe (t) = Ωe (t) a · b dΩ(t)R
and (a, b)∂ Ωe (t) = ∂ Ωe (t) a · b dΓ(t). Here, · implys an inner product and a is a different arbitrary scalar orvectorial quantity. As in [10, 16, 9] faces of the master elements serve as integration domains directly, independent how master and slave elements are connected.
Standard Gaussian quadrature is used for numerical integration. Volume integrals are computed using nq =(k + 1)d quadrature points, where d is the spatial dimension. Integrals over element facets inside each meshregion use nq = (k + 1)d−1 quadrature points. To suppress aliasing we use nq = (2k + 1)d−1 quadrature points onelement facets which are on a non-conforming or overlapping interface. We apply boundary conditions using
the mirror principle, cf. [11]. Hence, outer information for acoustic pressure and acoustic particle velocity at
domain boundaries readp+ = −p− + 2g p ; u + = u −
2Y −p+ = p−u+ =p − u− · n nρcon ∂ ΩDp (t),(9)

on ∂ ΩY (t).

(10)

3.2 SPATIAL DISCRETIZATIONThe ALE observer is thought to move with the grid. We are evaluating the integration points at the deforemed
elements instead of computing on a reference domain. This approach automatically fulfills the discrete geometric(a) Computational domain at refinement ref = 1(b) Overlap δ = 0.03
2refFigure 2. The computational domain consists of two mesh regions. The outer (rectangular) one stays fixed,
while the inner (circular) region rotates counter-clockwise with ω = 2π rads .
conservation law [7]. For the spatial discretization we multiply the governing equations by the test functions w
and q. Resulting system of equations is integrated by parts and numerical fluxes, denoted with superscript ∗ ,are introduced to arrive at the weak formulation. We are integrating the formulation by parts once more. This
gives us surface integrals which have to vanish and is commonly referred to as the strong formulation cf. [11].
The resulting semi-discrete system of equations reads
!


1
∂ uh
g
wh ,
w
u
u
w
e
− (w h , ∇u h h )Ω (t) +h , ∇ph∂t χρΩe (t)Ωe (t)wh ∈ Vhu , (11)
∀w



1
g
− w h (uuh · n ), u ∗h,c − u h ∂ Ωe (t) +w h · n , p∗h − ph=0ρ∂ Ωe (t)



∂ ph
g
qh ,
Ωe (t) − (qh , u h · ∇p)Ωe (t) + ρc2 qh , ∇ · u h Ωe (t)∂t χ∀qh ∈ Vhp . (12)



g
∗
2
∗
2
− qh u h · n , ph,c − ph ∂ Ωe (t) + ρc qh · n , u h − u h ∂ Ωe (t) = c qh , F Ωe (t)As fluxes in the pressure gradient and velocity divergence term we use the Lax–Friedrichs fluxesp∗h = {{ph }} +τJu h K ;2γu ∗h = {{uuh }} + J ph K ,
2

(13)

1
, cf. [18, 12, 15, 24]. As fluxes for the convective terms, centralwith penalty parameters τ = ρc and γ = ρcfluxes are sufficientp∗h,c = {{ph }} ;u ∗h,c = {{uuh }} .

(14)

Overlapping and non-conforming element edges are treated by the consistent evaluation of integration points
and over-integration as de-aliasing routine as described before or more precicely in [16].

4NUMERICAL RESULTS

Our implementations build on the open source software project EXADG [2] which internally uses the software
library deal.II [1]. Hereinafter, all spatial values are given in m.
4.1 PLANE WAVE IN DUCTThe density is ρ = 1 mkg3 and the speed of sound is c = 1 ms . To verify the non-conforming rotating implementation we compare the results of a computation with and without a non-conforming rotating interface.
For the rotational computation the setup is as follows: The computational domain is a channel Ωh,outer =(−1, −0.25) × (1, 0.25) with a hole of radius r = 0.125, centered around x c = (0, 0)T . A circle is placed above
the hole and rotates counter-clockwise with an angular velocity ω = 2π rad
s . To steer the spatial resolution thereference

rotating

reference

1.0

rotating

1.00

p in Pa

p in Pa

0.98

0.0

−1.0

1.85

1.95
t in s(a) Full pressure wave

2.05

0.94

0.90

1.92

1.93
t in s(b) Detail view

Figure 3. Acoustic pressure p over time t at x = (0.9, 0.0)T .
elements of the mesh can be refined by splitting. Given a certain spatial refinement ref, the element size h com−1 s with mesh refinement ref = 1.
putes as h = h2initialref . Figure 2a shows the computational domain at t = 1.5 · 10
To ensure all integration points can be found at any time, the circle is of radius r = 0.125+δ . Here, δ describes
the overlap between inner and outer domain and is defined as δ = 0.03
, cf. Figure 2b. For the conforming com2refputation without any rotation we choose ω = 0 and δ = 0.
At the left side of the channel Γleft we excite a single plane pressure wave by means of a pressure Dirichletboundary condition
 sin 2π t  if t < 0.1 s
0.1
gp =on Γleft .
(15)
 0
else
At the top and botton we apply reflecting boundary conditions and at the right side of the channel we apply
a first-order absorbing boundary condition. All computations are performed with a polynomial degree of k = 4
and a spatial refinement of ref = 4. For temporal discretization we apply an explicit Euler scheme with a time
step size of ∆t = 6 · 10−7 .
The pressure wave traversing the rotating domain is depicted in Fig. 4. For comprehensibly the grid in Fig 4 isobtained with ref = 1 while the result is obtained with ref = 4. The pressure results over time at x = (0.9, 0.0)T
can be found in Fig. 3. Especially in the detail view in Fig. 3b we can see that the results are in good
agreement.

5

CONCLUSIONS

We used Nitsches’ idea to generate a DG discretization for the acoustic conservation laws which can deal withoverlapping elements and moving meshes. Instead of using element cut algorithms we evaluate quantities onslave elements in integration points dictated by the master element. To avoid aliasing the approach uses overintegration at domain regions [16]. A test case using this strategy and a test case without non-conformities
and without any mesh movement yield the same results. This highlights that the method is well suited for
aeroacoustic applications such as a rotating fan.

(a) t = 0.9 s(b) t = 1.05 s(c) t = 1.2 s(d) t = 1.35 s

Figure 4. Pressure pulse traversing rotating domain. Note, that displayed result is achieved with ref = 4 but themesh is the one of ref = 1 to keep the plots comprehensible.

ACKNOWLEDGEMENTS
This project has received funding from the European Union’s Framework Programme for Research and Innovation Horizon 2020 (2014-2020) under the Marie Skłodowska–Curie Grant Agreement No. [812719].
The computational results presented have been achieved in part using the Vienna Scientific Cluster (VSC).

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Maier, P. Munch, J.-P. Pelteret, S. Sticko, B. Turcksin, and D. Wells. The deal.II library, version 9.4.
Journal of Numerical Mathematics, 2022.
[2] D. Arndt, N. Fehn, G. Kanschat, K. Kormann, M. Kronbichler, P. Munch, W. A. Wall, and J. Witte.
ExaDG: high-order discontinuous Galerkin for the exa-scale. In Software for Exascale Computing - SPPEXA
2016-2019. Springer International Publishing, 2020.
[3] F. Bassi, A. Crivellini, D. D. Pietro, and S. Rebay. An artificial compressibility flux for the discontinuous
Galerkin solution of the incompressible Navier-Stokes equations. Journal of Computational Physics, (2),2006.
[4] F. Bassi, A. Crivellini, D. A. D. Pietro, and S. Rebay. An implicit high-order discontinuous Galerkinmethod for steady and unsteady incompressible flows. Computers & Fluids, (10), 2007.
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Encyclopedia of Computational Mechanics. American Cancer Society, 2004. Chapter 14.
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on deforming domains. International Journal for Numerical Methods in Fluids, (12), 2006.
[8] O. Guasch, M. Arnela, R. Codina, and H. Espinoza. A stabilized finite element method for the mixed
wave equation in an ALE framework with application to diphthong production. Acta Acustica united with
Acustica, (1), 2016.
[9] J. Heinz and M. Kaltenbacher. A nonconforming DG formulation for the acoustic conservation laws. In
Fortschritte der Akustik - DAGA 2022. Deutsche Gesellschaft für Akustik e.V. (DEGA), 2022.
[10] V. Hermann, M. Käser, and C. E. Castro. Non-conforming hybrid meshes for efficient 2-D wave propagation using the discontinuous Galerkin method. Geophysical Journal International, (2), 2010.
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[12] M. Hochbruck, T. Pažur, A. Schulz, E. Thawinan, and C. Wieners. Efficient time integration for discontinuous Galerkin approximations of linear wave equations. ZAMM - Journal of Applied Mathematics andMechanics / Zeitschrift für Angewandte Mathematik und Mechanik, (3), 2014.
[13] A. Hüppe, J. Grabinger, M. Kaltenbacher, A. Reppenhagen, G. Dutzler, and W. Kühnel. A non-conforming
finite element method for computational aeroacoustics in rotating systems. In 20th AIAA/CEAS Aeroacoustics Conference. American Institute of Aeronautics and Astronautics, 2014.
[14] M. Kaltenbacher and S. Floss. Nonconforming finite elements based on nitsche-type mortaring for inhomogeneous wave equation. Journal of Theoretical and Computational Acoustics, (03), 2018.
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[17] A. Massing, M. G. Larson, and A. Logg. Efficient implementation of finite element methods on nonmatching and overlapping meshes in three dimensions. SIAM Journal on Scientific Computing, (1), 2013.
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for acoustics and elastodynamics. Journal of Computational Physics, (10), 2011.
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ABS-0136

Adjoint-based analyzes in acoustic time-domain simulations
Mathias LEMKE
Technische Universität Berlin, Germany, mathias.lemke@tnt.tu-berlin.de

ABSTRACTThis manuscript presents an adjoint-based approach for analyzing acoustic configurations in the time domain.
The technique allows the optimization and the identification of sound sources considering effects like diffraction. As an example, the identification of rotating sound sources based on microphone array measurements isdemonstrated. All parameters sought, including rotation speed, relative positions and amplitudes of the sources,
are identified using a measurement time corresponding to one rotation of the sources. In addition, preliminary
results concerning the consideration of boundary conditions within the approach are presented. It is shown thatreflections and diffraction can support the identification of sound sources.

Keywords: time domain, adjoint approach, source identification

1

INTRODUCTION

Wave-based acoustic time-domain simulations enable the description of effects that are difficult to account forin geometric acoustics, for example, diffraction. In addition, they often offer higher accuracy compared tofrequency-based methods. These advantages often justify the higher numerical effort compared to classic, geometric or frequency-based methods.
A large number of acoustical problems, such as the search for optimal driving functions or geometric arrangements of loudspeakers as well as the optimal design of walls using acoustic materials, can be describedas high-dimensional optimization tasks in the time domain. An adjoint-based approach has proven suitable forthese problems since the computation time is mainly independent of the number of parameters examined.
This contribution presents the application of the adjoint approach for identifying rotating sound sources using
microphone array measurements. In addition, it is shown that reflecting boundary conditions do not only hinder
the identification of sound sources but instead can help to provide extra information for the identification.
Sound source identification is a typical problem in acoustics. Various parameters of one or more soundsources are sought, including signal and position. Usually, microphone array measurement techniques are used,in particular the so-called beamforming. Depending on the acoustic setup considered, different algorithms areused, operating in the time or frequency domain. A recent overview of algorithms is given by [1]. An adjointbased method for the identification of stationary and moving sound sources in the time domain is presented in[2] and [3] respectively.

2

FRAMEWORK

2.1 Adjoint in generalIn general, adjoint equations arise from a scalar-valued objective function J, defined by the user. The function
encodes the goal of the analysis, for example, the localization of sound sources. It is given by the scalar productbetween a weight vector g and a system state vector q
J = g · q = gT q

g, q ∈ Rn .

(1)

The system state q is the solution of the system under consideration
Mq = sM ∈ Rn×n , s ∈ Rn ,(2)

with M as the governing operator and s as (sound) sources on the right-hand side. To optimize J using the nentries of s, the governing equation must be solved n + 1 times.
To reduce the computational effort the adjoint equation can be used instead
M T q∗ = gq ∗ ∈ Rn ,(3)

with the adjoint variable q∗ . Using
J = gT q = M T q∗

T

q = q∗ T Mq = q∗ T s = q∗ · s,(4)

a formulation is found, which allows the objective J to be computed without solving equation (2) for various s.
By solving the adjoint equation (3) once, the objective can be computed by a scalar product. Thus, the adjoint
approach allows efficient computation of gradients for J as a function of s.
2.2 Adjoint Euler equations
In the following, the definition of the used variables is adjusted in the context of a Cartesian tensor notation.
The scalar objective function J is defined in space and time with dΩ = dx1 dx2 dx3 dt throughout the computational
domain:Z
21 q − qmic dΩ.
(5)J=2
The field variable q(x j ,t) contains the complete state q = [ρ, u j , p] of the system governed by the Euler equations.
   
ρ
ρui
0
0




      ∂t ρu j  + ∂xi ρui u j + pδi j  − ui ∂xi  0  =  0 
(6)




   
ui pγ
p
p
sp
γ−1
γ−1
Therein, ρ denotes the density, u j the velocity in the x j direction, γ the isentropic exponent and p the pressure.
The source terms s(x j ,t) = [0, 0, s p ] describe monopole sound sources that allow controlling the system state andthe solution of the equations, respectively. For details on the formulation, in particular, the reformulation of theenergy equation in terms of pressure, see [4].
For the identification of sound sources by means of microphone measurements, the objective function depends only on the pressure and the pressure fluctuation:
J=

1
2

Z 

p − pmic

2

σ dΩ,

(7)

where pmic contains the microphone measurements. The additional weight σ (xi ,t) defines where and when thetarget is evaluated, i.e. the microphone positions and sampling rate.
To simplify the derivation of the adjoint, the above system of partial differential equations is abbreviated by
E (q, s) := E(q) − s = 0.

(8)The overall goal is the acquisition of a solution of the Euler equations that reduces the objective functional(7) by adjusting s. Thus, an optimization of s corresponds to an identification of the sound sources sought.
In order to optimize s with the adjoint approach, the governing system (6) and the objective function (7)must be linearized. This leads toZq − qtarget σ δ qdΩ
δJ =
(9)
|
{z
}
=g

and
Elin (q)δ q − δ s = 0,

(10)

with δ q(x j ,t) = [δ ρ, δ u j , δ p] and δ s(x j ,t) = [0, 0, δ s p ]. The weight g(x j ,t) = (q − qtarget )σ corresponds to thedifference between the numerical solution of (8) and the measured target field.
Z

(11)
δJ =
p − ptarget σ δ pdΩ
|
{z
}
=g

By combining the linearized system and the objective function in a Lagrangian manner
δJ= gT δ q − q∗ T (Elin (q)δ q − δ s)
|
{z
}
∗T

T

= q δs+δq=0Tg − Elin
(q)q∗

(12)
(13)

results. Note that the spatial and temporal integrals are not shown for simplicity.
The adjoint equation sought results from the requirementTE ∗ (q, q∗ , g) := g − Elin(q)q∗ = 0,

(14)

with q∗ (x j ,t) = [ρ ∗ , u∗j , p∗ ] as adjoint state variable. See [5] for a detailed derivation of the adjoint Euler equations.
Finally, the change in the objective function is given byδ J = q∗ T δ s.

(15)

The solution of the adjoint equation can again be interpreted as the gradient of J with respect to the source
terms s∇s J = q∗ .
(16)
The adjoint-based gradient is typically used iteratively: First, the Euler equations are solved forward intime. Then, the adjoint equations are calculated using the direct solution and g backward in time. Based on
the adjoint solution, the gradient ∇s J is determined and employed to update the source distribution sl using a
gradient approach:sl+1 = sl + α∇s J,(17)where α denotes a suitable step size and l denotes the iteration number. The gradient is calculated for theentire computational region and the entire simulation time.
For the determination of sound sources with a known position, the gradient is evaluated exclusively at theselocations. The procedure is repeated with the current sl until a suitable convergence criterion is reached. For
acoustic problems, convergence is typically achieved within 20 loops or less.
The identification of a global optimum is not guaranteed since the method optimizes towards local extrema.
The computational costs of the approach are independent of the number of sources and their arrangement.
However, they depend on the size and the spatial and temporal resolution of the computational domain definedby the considered frequency range.
2.3 Moving sound sources
In particular, when s0 = 0 holds, the first adjoint solution already contains information about the position of thesources. By summing the absolute adjoint sensitivity p∗ point-wise over time
tn=endp̄(x j ) = ∑ |p∗ (x j ,t)|,

(18)

tn=0

the positions that have the highest impact on the objective function can be identified.
For tracking of moving sources, time moving averages of the sensitivities over 2N + 1 samples can be usedNp̄(x j ,t) =∑ |p∗ (x j ,t − δtn )|,n=−N

as well as methods of image processing, such as shift or rotation and convolution filters.

(19)

Figure 1. (Left) Signals of the three rotating sound sources si . The amplitude ratios compared to the first sources1 correspond to -3 dB and -6 dB, respectively. (Right) Considered configuration to identify three rotating
incoherent sound sources ◦ based on 64 microphones circularly arranged in a plane •.

Figure 2. (Left) Reference sound field in the plane of the microphones (x3 = 0.5 m, •) for a selected time step.
(Right) Reference measurement signal over the whole time of 2000 samples considered, measured at microphone
position 1, highlighted by • in the figure above.

3

EXAMPLE: THREE ROTATING SOURCES

As an example, a synthetic system similar to [6] is studied. Three incoherent sound sources ◦ rotating at 1500
rpm in a quiet and anechoic environment are considered. The signals are bandlimited to 1250 - 4500 Hz withthe following amplitude ratios: 1.00, 0.71 (-3 dB) and 0.50 (-6 dB), see Fig. 1 (left). For the measurement,
64 microphones • are centered 0.5 m above the plane of rotation arranged in a circle with equal distances,
see Figs. 1 (right) and 2 (left). For the analysis, 2000 samples at a sampling rate of 48 kHz are used, whichcorresponds to approximately one rotation of the sources. The goal is to identify the three sound sources, inparticular their rotation speed, the position of the sound sources and their relative amplitudes based on the
microphone data, exemplified for a microphone shown in Fig. 2 (right).
The adjoint solution q∗ (x,t) and p∗ (x,t), respectively, is calculated and used analogous to [2, 5, 7] for theidentification of sound sources. Figure 3 (left) shows a selected time step.
A simple image processing method was used to evaluate the adjoint solution. The sensitivities p∗ (x,t) arerotated in the plane of the sources with different angles corresponding to different rotation speeds, see Fig. 3(right). Integration of such rotated absolute sensitivities corresponding to equation (18) over time and subsequent
determination of the maximum values results in the curve in Fig. 4 (left). Consistent with expectations, themaximum is reached at the reference rotation speed of 1500 rpm. The integrated absolute sensitivity for thesame plane considering the previously determined rotation speed is shown in Fig. 4 (right). The positions of
the sensitivity maxima correspond to the relative positions of the sound sources. The amplitude ratios of thesensitivities were found to be 1.00, 0.76 and 0.54, i.e. in good agreement with the reference values. To increase
the accuracy, further iterative optimization is possible.

Figure 3. (Left) Normalized adjoint-based sensitivities p∗ for a selected time step in the plane of the rotating
sources. Current positions of the reference sources are marked by ◦. (Right) Rotation of the adjoint-based
sensitivities over time for subsequent integration, see text for details.

Figure 4. (Left) Maximum values after time integration of the absolute sensitivities to determine the rotation
speed (reference: 1500 rpm). (Right) Normalized absolute adjoint-based sensitivities considering the previously
identified rotation speed of 1500 rpm. Reference positions are marked by ◦.
Thus, the rotational velocity was determined from 2000 samples (corresponding to 42 ms) of the synthetic
microphone data without additional information. Furthermore, the low-frequency sources (relative to the dimensions of the microphone array) were spatially separated and the amplitude ratios were determined in good
agreement with the reference.

4

CONSIDERATION OF BOUNDARY CONDITIONS

The localization of sound sources typically involves the presumption of an anechoic environment. The soundpropagation between source and microphone is usually modeled by free-field Green’s functions.
Many practical problems, however, involve impedance-impaired boundaries. To consider such boundary conditions tailored transfer functions can be used [8]. The functions are determined either analytically, numericallyor experimentally. The first is restricted to simple conditions like a straight fully reflecting wall. The latter
requires a high effort.
The presented adjoint-based approach allows a simple consideration of boundaries by including them in thegoverning and the corresponding adjoint equations. Here, the boundaries are modeled using a volume penalization approach, enabling the consideration of complex impedance conditions [3].
As a representative example, a simple synthetic system is studied. A single sound source ◦ is considered,
see Fig. 5. In detail, the source is defined by white noise, band-limited in the range from 750 to 4000 Hz. For
measurement, 8 microphones • located at x1 = 0.75 m are employed. For the analysis, 7500 samples are used
at a sampling rate of 96 kHz, which corresponds to a measurement time of less than 0.1 s.
Three cases are examined in detail: (a) localization of the source in an anechoic environment, (b) source

(a)

(b)

(c)

Figure 5. Normalized absolute adjoint-based sensitivities: (a) source in anechoic environment, (b) source infront of a rigid edge (c) source located behind a wedge. The reference position of the sound source is markedby ◦. The position of the microphones by •.
located in front of a rigid edge and (c) source located behind a wedge.
The objective is to identify the source position by means of the adjoint-based sensitivities (18). Corresponding results are shown in Fig. 5. In all three cases, the localization of the reference sound source is successful.
In case (a), however, the uncertainty with regard to the depth position, which is typical for a planar microphonearray, becomes apparent. The consideration of existing ridig walls in case (b) improves the result significantly.
The reflections increase the information density or rather the observability [5] of the sound source.
By directly considering existing walls, configurations become accessible in which there is no line of sight
between the sound source and the microphone array. In case (c), the source is covered by a rigid wedge, seeFig. 5. Nevertheless, the maximum of the absolute adjoint-based sensitivity is found at the reference position.
Thus, the identification of the sound source employing microphone array measurement succeeds here only byaccounting reflections and diffraction.

5

CONCLUSION

An adjoint-based method for identifying moving sound sources in the time domain was presented. The method
was successfully validated using a synthetic example of three rotating incoherent sources.
The approach allows the determination of the parameters sought based on a few measurement samples. In
perspective, the integration of complex boundary conditions into the approach [3], the consideration of baseflows [9] as well as the identification of directional characteristics [10] are possible.

ACKNOWLEDGEMENTS
The author acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG) within the projectLE 3888/2.

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PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0508

Prediction of turbulent flow-induced noise in the cavity
Bon-Heon Ku1; Young-Woo Yi1; Dong-Sub Lee2; Chin-Suk Hong3; Hee-Chang Lim1, a
1School of Mechanical Engineering, Pusan National University, Busandaehak-ro 63beon-gil 2, Geumjgeong-gu, Busan, Korea
2
3

Agency for Defense Development, Korea

School of Mechanical Engineering, Ulsan College, Bongsu-ro 101, Dong-gu, Ulsan, Korea

ABSTRACTAs the contribution of sonar self-noise by flow has recently increased, a numerical prediction method forthe noise, generated by the wall pressure fluctuations in the turbulent boundary layer is of interest. In thisstudy, the prediction method of the flow induced noise of a cuboid cavity, simplified sonar model, werestudied. The random vibroacoustics is adapted to reflect the temporal/spatial randomness of the wallpressure fluctuations. The structural excitation by the wall pressure fluctuation was modelled by the Corcos
model and Smol’yakov-Tkachenko model. The acoustics-structural reciprocity technique was applied toeffectively calculate the noise transfer functions, required for the random vibroacoustic model. Thisapplication technique leads to effectively calculate the noise transfer functions for many points, and to apply
random vibroacoustics. In order to verify the acoustic-structural reciprocity technique, the noise transfer
functions in the direct and reciprocal directions were compared and showed a good agreement. The effectsof flow velocity on noise in the cuboid cavity were studied and showed that the noise level increases as theflow velocity increases. We developed a method for predicting noise by flow in structures for potential use
for internal sound field estimation.
Keywords: wall pressure fluctuation, flow induced noise, random vibroacoustics, acoustics-structural
reciprocity, noise transfer functions

1. Introduction
Although there have been many relevant studies during last several decades, the flow noise
phenomenon caused by the wall fluctuating pressure under the turbulent boundary layer has beenof the most great importance in various industrial fields such as watercrafts, aviation vehicles, andspacecrafts [1-3]. In addition, structural noises are mainly caused by those structural and acoustic
noises in fluids. Recently, as the running speed of ship and aircraft objects has increased, a turbulent
boundary layer is formed over the surface of the structure due to the fluid viscosity. As the operating
speed increases, therefore, the thickness of the turbulent boundary layer becomes thinner, and thewall fluctuating pressure is generated close to the surface of the bodies, which would be due to thecomplex behavior of the eddy in the turbulent boundary layer. Finally, the wall pressure fluctuation
within the turbulent boundary layer causes a random load to occur on the surface of the structure,causing vibration and flow noise inside the structure.
In this study, since last several decades there have been actively studied in the area of flow noise
phenomena in relatively complex shapes in the mainly industrial sectors, but this study aims to findthe details of flow noise in a simplified shape such as a cuboid cavity. The turbulent boundary layer
has been thoroughly analyzed to understand the effect of turbulent flow on the wall pressure
fluctuation close to the elastic plate under the turbulent boundary layer. The mathematical model
proposed by Corcos[4] has been used for modelling the wall fluctuation pressure caused by the
turbulent flow over the flat plate. The finite element method was applied for obtaining the transfer
function of acoustic-structural noise, and the sound field level inside the cuboid cavity model wasa

Corresponding author : hclim@pusan.ac.kr

estimated by the free-stream velocity.

2. Finite element modeling of wall pressure fluctuation
In this study, we used a finite element method suggested by Hong et al.[5] to implement turbulent boundary
layer to a simple supported beam such as acting wall fluctuation pressure on the flat plate. In addition,
depending on the free-stream velocity, the level of the internal sound pressure was also predicted by thecalculation of the turbulent-induced fluctuating pressure and the appropriate noise transfer function.
2.1. Modeling of Wall pressure fluctuation
In this study, the Corcos model was used to model the wall pressure fluctuation, and it is mathematically
expressed as follows.

𝑆𝑝𝑞 (𝜉, 𝜂, 𝜔) = 𝑆𝑝𝑝 (𝜔)𝑒

−𝛼|

𝜔𝜉
|
𝑈𝑐

𝑒

−𝛽|

𝜔𝜂
|
𝑈𝑐

𝑒

−𝑖(

𝜔𝜉
)
𝑈𝑐

(1)

where the 𝑆𝑝𝑞 expresses the cross spectral density function of the wall pressure fluctuation, the 𝑆𝑝𝑝 is
the auto spectral density function, and the symbol 𝛼, 𝛽 are the longitudinal and lateral decay rates of thecoherences direction, and 𝑈𝑐 is the convection velocity necessarily required for obtaining the Corcos
model numerically. Based on the extensive study, Eq.(1) can be expressed as a function of the similarity
variables

𝜔𝜉
𝑈𝑐

𝜔𝜂

and 𝑈 , respectively.
𝑐Fig.1. Mean streamwise velocity profiles through DNS numerical analysis

Fig. 2 Cross-correlation distribution of fluctuating pressure near the wall by using DNS calculation

Fig.3 Coherence of fluctuating pressure near the wall by using DNS simulation
2.2. Modeling of flow induced noise
The characteristics of wall pressure fluctuation due to turbulent boundary layers appears in the form ofrandom loads on thick elastic structures. Based on the random vibroacoustics theory, when an ergodic
random load is applied to an elastic structure targeting a linear system or a deterministic system, theresponse appears to be a form of power spectral density or auto-correlation function and the RMS values
such as the terms of displacement, velocity, acceleration, and a sound pressure. Finite element method wasused to obtain the noise transfer function caused by a random force under the turbulent boundary layer.
The structural response must be assessed by applying a unit force of 1𝑁 to all nodes modelled in the finite
element structure. However, when a unit force is applied to all nodes, the calculation becomes costly.

Fig. 4 FEM Cuboid cavity model
In order to overcome this problem, the acoustics-structural reciprocity technique[6] was used to verify andapply the calculated results.
The idea of acoustics-structural reciprocity is expressed as follows.
𝑃𝐹𝑣= 𝑄𝑛 (2)where the symbols 𝐹, 𝑃 are the force [𝑁] and the pressure [𝑁/𝑚2 ], and the others 𝑄, 𝑣𝑛 are a kindof sound source (monopole source) [𝑚3/𝑠] and the wall-normal velocity [𝑚/𝑠] on the wall.

Fig. 5 Verify of the acoustic-structural reciprocity techniqueThis study performs the prediction of the internal sound field at any point inside the Cuboid cavity model interms of the various free-stream velocity based on these verification methods.

Fig.6 The effects of flow velocity on noise in the Cuboid cavity model

3. RESULTS
In this study, the wall pressure fluctuation was modeled by the calculation of the DNS simulation. As shownin Fig. 1, the characteristics of the turbulent boundary layer were precisely compared our current data withother existing data, and reference DNS data. It was found that the characteristics of the distribution of the
turbulent boundary layer showed well the well-known typical turbulent boundary layer according to various
Reynolds numbers(𝑅𝑒𝜏 ) and analysis techniques. In Fig. 2, it is confirmed that the convection velocity (𝑈𝑐 )agrees well with almost 73% of the free-stream velocity(𝑈∞ ). In this study, the convection velocity (𝑈𝑐 )was assumed to be 70% of the free-stream velocity. As shown in Fig. 3, as the frequency increases, it tends
to be exponentially decreased, and the turbulence coherence (𝛼) along the streamwise direction isassumed to be 0.1.
As shown in Fig. 5, the noise transfer functions to which the acoustic-structure reciprocity is applied are
expressed in 𝑀𝑎𝑔[1/𝑚2] and 𝑃ℎ𝑎𝑠𝑒[𝑅𝑎𝑑], confirming that the results agree well. Fig. 6 predicted thesound level inside the cuboid cavity model according to various flow rates. This study conducted to predict
the internal sound field of the cuboid cavity model in terms of various free-stream velocity by comparing
three scaled models - the Corcos, the non-correlated, and perfectly-correlated models. When the frequency(𝑓) is large, it was confirmed that the Corcos and non-correlated models have similar internal sound fields.
On the other hand, regarding the perfectly-correlated model, it can be seen that the internal sound fieldtends to be relatively higher than that of the two models. In addition, as the free-stream velocity increased,
it is confirmed that the internal sound field also increased.

4. CONCLUSIONSIn this study, using direct numerical analysis (DNS) and finite element method (FEM) techniques, we
performed to predict the internal sound field of the cuboid cavity model by modeling the wall fluctuationpressure of the turbulent boundary layer and developing a noise transfer function calculation technique.
The results can be summarized as follows.
1. The characteristics of wall pressure fluctuation has been modeled using the Corcos model by
the results of DNS simulation.
2. The noise transfer function was also calculated and well applied by using the finite element
method.
3. The internal sound field caused by the free-stream velocity was confirmed through various
flow-induced force models.

ACKNOWLEDGEMENTS
The authors gratefully acknowledge the financial support provided by the Agency for Defense
Development under the contract UD190002DD.

REFERENCES
1. Zhao, X., Lei, J., 2020. Improved Technique for Evaluation of Wall Pressure Fluctuations from Turbulent
Boundary Layer. AIAA Journal 58, 3320–3331.
2. Magionesi, F., Di Mascio, A., 2016. Investigation and modelling of the turbulent wall pressure fluctuations
on the bulbous bow of a ship. Journal of Fluids and Structures 67, 219–240.
3. Karthik, K., Jeyakumar, S., Sebastin, J.S., 2021. Numerical prediction of flow noise levels on towed sonararray. Proceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the
Maritime Environment 235, 600–606.
4. Corcos, G. M., 1963, Resolution of Pressure in Turbulence, JASA, 35(2), 192-199
5. Hong. C. S., Shin K. K., and Kim, S. Y. A Modelling of Structural Excitation Forces Due to Wall Pressure
Fluctuations in a Turbulent Boundary Layer. 2001;11(2):226-233.
6. Fahy, F.J., 2003. Some applications of the reciprocity principle in experimental vibroacoustics. Acoust. Phys.
49, 217–229.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0510

Modelling of spectral properties of wall pressure fluctuations beneathturbulent boundary layers
Young-Woo Yi1; Dong-Sub Lee2; Chin-Suk Hong3; Hee-Chang Lim1, a
1School of Mechanical Engineering, Pusan National University, Busandaehak-ro 63beon-gil 2, Geumjgeong-gu, Busan, Korea
2
3

Agency for Defense Development, Korea

School of Mechanical Engineering, Ulsan College, Bongsu-ro 101, Dong-gu, Ulsan, Korea

ABSTRACT
Large Eddy Simulation(LES) has been popularly and used in the last several decades to simulate turbulent
boundary layer in the numerical domain. Synthetic Inflow Generation(SIG), which is based on exponential
correlation function both in space and in time is also widely used for a fully developed turbulent boundary
layer flow simulation. Recent results show that the flow characteristics of the results of LES-SIG are wellmatched with the results of Direct Numerical Simulation(DNS) and theoretical results. In this study, we
compared the fluctuating pressure near the wall of LES-SIG with those of DNS results and suggested theproper model for the modeling the fluctuating pressure near the wall. As a results, the fluctuating pressurenear the wall of LES-SIG are very close to those of DNS and new model related to the Smol’yakov-Tkachenko
model are suggested.
Keywords: Large Eddy Simulation(LES), Synthetic Inflow Generation(SIG), Smol’yakov-Tkachenko model

1. INTRODUCTION
The noise generated by the turbulent boundary layer over various bluff bodies has been major
issue for a wide range of engineering applications. Examples include the noise generated bypropellers, rotors, wind turbines, fans, wings, and hydrofoils. Even having no disturbances in theoncoming flow, bluff bodies can generate noise due to an unsteady turbulent f low and wake. This socalled self-noise or leading edge noise would be a major contributor to the overall noise generated
by an induced noise by bluff bodies [1-3]. Reducing self-noise is challenging and requires an efficientmethod for describing and understanding the sources of noise near the bluff bodies.
On the other hand, turbulent boundary layer over bluff bodies has also been taken great attention
over a long period of time regarding the understanding of heat and mass transfer mechanisms [4]. In
order to understand the source noise from the turbulent boundary layer, it is requisite to generate
the turbulent flow over the bluff bodies as well as flat plates. There have been several techniques togenerate turbulent flow in a calculation domain. The most straightforward approach to simulate aspatially developing turbulent boundary layer, which start the calculation far upstream with a laminar
profile having some random disturbances and then allow a natural transition to generate realistic
turbulence [5]. In order to reduce the cost of simulation for this transition process, most of spatiallyevolving simulations start with a synthetic inflow generator(SIG) having a short distance domain. Thismethod generates inlet velocity conditions for the simulation from the correlation based on thelength scales of the turbulent flow [6].
In this study, we analyze the fluctuating pressure as a sound source in turbulent boundary layer
come from both direct numerical simulation using periodic boundary conditions (PBC-DNS) and large
eddy simulation using synthetic inflow generator (SIG-LES) and find out the common characteristicsof the fluctuating pressure in turbulent boundary layer regardless of the method of simulation andchanging Reynolds number.
aCorresponding author : hclim@pusan.ac.kr

2. EVALUATION OF WALL TURBULENT BOUNDARY LAYER
We firstly aimed to generate the fluctuating pressure near the wall, allowing the turbulent
boundary layer develop in the domain. Figure 1 shows the mean streamwise velocity profiles among
different approaches and turbulent Reynolds number (Re τ). The turbulent Reynolds number (Re τ) was
drawn by friction velocity (u τ) at the flat plate and a half height of the channel (d). In the simulation,
no-slip condition was used at the top and bottom flat plates and periodic condition at side the flow
properties should not be diffused.
In SIG-LES, the size of the domain was 20πd×2d×πd (here d is channel height), which stands for
streamwise, vertical, and spanwise direction. The free-stream velocity (U∞) was almost 20 andkinematic viscosity (ν) was 2.5e-2. The turbulent Reynolds number in this case (Re τ=395) was sameas DNS case [5]. In case of the PBC-DNS, the size of the domain was 12d×2d×6d, which has the same
domain height as the SIG-LES. The free-stream velocity (U∞) was set to 1.14 and kinematic viscosity(ν) was 3.1e-4. Turbulent Reynolds number in this case (Re τ=180) was nearly half of the SIG-LES case.
In order to evaluate the mean statistics of turbulent boundary layer, we compare the current
results (i.e., SIG-LES and PBC-DNS) with reference DNS data [5] and the existing theory [7]. As can beseen in the figure, the mean streamwise velocity profile for all two cases of different approaches
shows excellent agreement with the DNS data obtained within the wall distance range, i.e., the linear

Fig. 1 Mean streamwise velocity profiles among SIGLES(Reτ=395), PBC-DNS(Reτ=180), DNS [5], and theoreticalresults

Fig. 2 Temporal variation of non-dimensional fluctuating pressure near the wall by using (a) SIGLES, (b) PBC-DNS

Fig. 3 Cross-correlation distribution of fluctuating pressure near the wall by using (a)SIG-LES, (b)PBC-DNS.

Fig. 4 Coherence of fluctuating pressure near the wall by using (a) SIG-LES, (b) PBC-DNS

viscous sublayer, buffer layer, and logarithmic region.

3. RESULTS
Figure 2 show the temporal variation of fluctuating pressure in turbulent boundary layer at theclosest grid of the wall. The fluctuating pressure is non-dimensionalized by p 0 (=(ϕ p(0)× Δω) 1/2 ), which
was calculated by the auto spectrum of pressure at the frequency 0 (ϕ p(0)) and the gradient ofangular frequency (Δω). As shown in the figure, the variation was mostly spanning in the range of -5
and 5 regardless of the simulation method. Even the simulation has been made in different Reynolds
number over the range of 180 -395, the variation was limited in this range.

As shown in Fig.3, the cross-correlation of fluctuating pressure is a good indicator for analyzing
and modelling the pressure spectrum near the wall. The convection velocity (UC ) can be computedby using the difference between two points having the duration time of first peak of cross -correlation
profile. Based on this result, it turns out that the convection velocity of the t urbulent boundary layer
would be estimated by the 70% of the freestream velocity (U∞).
Figure 4 shows the coherence distribution computed from fluctuating pressure close to the wallby using SIG-LES and PBC-DNS. The coherence describes the correlation among waves or noises atdifferent points in space, either streamwise or spanwise direction [8]. In this study, the coherence of
streamwise direction can be calcualated. As shown in Fig.4, the quantity is subject to exponential
decay having the coefficient of α=0.06 for both cases.

4. CONCLUSIONSThis study has tackled one of properties in predicting sound and self-noise sources - fluctuatingpressure in turbulent boundary layer by using direct numerical simulation and large eddy simulationand found the common characteristics of the fluctuating pressure in turbulent boundary layer. The
work is summarized as follows.
1. The method of SIG-LES and PBC-DNS have found to be suitable for generating a turbulent
boundary layer in the domain.
2. The variation of temporal fluctuating pressure fell in a certain range regardless of simulation
method and changing Reynolds number.
3. It turns out that the characteristics of fluctuating pressure shows an interesting feature suchas the ratio of convection velocity (UC) to freestream velocity (U∞) and the coherence (α)
structure.

ACKNOWLEDGEMENTS
The authors gratefully acknowledge the financial support provided by the Agency for Defense
Development under the contract UD190002DD.

REFERENCES
1. Kim, J. N. and Sung, H. J. Wall pressure fluctuations and flow-induced noise in a turbulent boundary layerover a bump. Journal of fluid dynamics. 2006;558(10):79-102.
2. Shin, K. K., Hong C. S. and Lee, H. G., A study of Flow Induced Noise for Multilayered Cylinder due to
Turbulent Boundary Layer. 1996;6(5):671-677.
3. Hong. C. S., Shin K. K., and Kim, S. Y. A Modelling of Structural Excitation Forces Due to Wall Pressure
Fluctuations in a Turbulent Boundary Layer. 2001;11(2):226-233.
4. Kovasznay, L. S. Turbulent Boundary Layer. Annual Review of Fluid Mechanics. 1970;2(1):95-112.
5. Moser, R. D. and Kim, J. Direct numerical simulation of turbulent channel flow up to Reτ=590. Physics of
fluids. 1999;11(4):943-945.
6. Xie, Z. T. and Castro, I. P. Efficient generation of inflow conditions for large eddy simulation of street-scale
flows. turbulence and combustion. 2008;81(3):449-470.
7. Pope, S. B. Turbulent Flows. 1st revised ed . Cambridge, UK: Cambridge University Press & Assessment;
2000. p. 272-274.
8. Hecht, E. Optics. 3rd ed. New York, USA: Eddison Wesley Longman Inc. 1998. p. 554-574

ABS-0744

Linearized perturbed compressible equation (LPCE) for low Mach number
aeroacoustics: a review of theory and application
Young J. Moon(1) ,Jung-Hee Seo(2)(1) Korea University, Korea, Rep. of,yjmoon@korea.ac.kr(2) Johns Hopkins University, USA,jhseo@jhu.edu

ABSTRACTThe linearized perturbed compressible equation (LPCE) proposed for various low Mach number aeroacoustic
problems is reviewed. In this method, the acoustic field is predicted by acquiring the noise source, DP/Dt,
from the hydrodynamic flow field computed by large-eddy simulation. We present theoretical backgrounds of
LES/LPCE hybrid approach and its application to turbulent edge-scattering noise.

Keywords: Linearized perturbed compressible equation (LPCE), Low Mach number, Aeroacoustics

1

INTRODUCTION

Many industrial flow applications such as automobiles, fans, and etc. operate at low Mach numbers, e.g. M 
0.3. Direct numerical simulation (DNS) of low Mach number aeroacoustics still remains a challenging problem
because of scale disparities between the hydrodynamic vortical motions and the acoustic waves. In this regard,a hydrodynamic/acoustic splitting method is considered a good alternative for prediction of flow noise at low
Mach numbers. Since a flow at low Mach number is obtained much faster by solving the incompressible NavierStokes equations, the hydrodynamic/acoustic splitting method is considered computationally more efficient thanthe direct numerical simulation as well as the other hybrid methods, based on the compressible Navier-Stokes
equations (see Figure 1). Moreover, computational efficiency can be further enhanced, if grid systems for flow
and acoustics are treated separately.
In previous hydrodynamic/acoustic splitting methods, a flow field is obtained by solving the incompressible
Navier-Stokes equations, while the acoustic field is predicted by the perturbed Euler equations (PEE) which
are derived by subtracting the incompressible Navier-Stokes equations from the compressible ones, neglecting
the ’viscous terms’. It is, however, to be noted that the perturbed Euler equations are mixed-scales, non-linear
equations, in which a coupling effect is allowed between the incompressible flow variables and the perturbed
quantities. Through this coupling effect, a ‘perturbed vorticity’ is generated in the perturbed system, which isacoustically a non-radiating vortical component. Although role of perturbed vorticity on sound generation is
negligible at low Mach numbers, this perturbed quantity can easily become unstable for various reasons andeventually becomes self-excited. As a result, inconsistent acoustic solutions may result from the perturbed Euler
equations.
The perturbed compressible equations (PCE) proposed by Seo and Moon (1) remedy the false generationof perturbed vorticity by handling its generation and diffusion properly, especially when noise source is in theshear flow. In PCE, the perturbed Euler equations are modified by introducing the ‘perturbed viscous stresses’ inthe momentum equations and the perturbed energy equation is formally derived from the compressible thermalenergy equation. The PCE has been tested for various applications, including laminar dipole tone from a circularcylinder at ReD = 200 and M• = 0.3. For this problem, the PCE solution was shown to be accurate as the DNS
solution. In parallel, computational efficiency and accuracy of the PCE in conjunction with the ‘grid-splitting’technique has been investigated. It has been shown that perturbed vorticity could become unstable, if acoustic
grid resolution is not properly used. This is because the length scale of the perturbed vorticity is not comparable

Figure 1. LES/LPCE hybrid method for prediction of turbulent flow noise at low Mach numbers
to that of acoustic waves but closer to the hydrodynamic vortical scale. Thereby, in some cases, grid-dependent
acoustic solutions may result from the falsely resolved perturbed vorticity.
In order to resolve aforementioned matters, the linearized perturbed compressible equations (LPCE), a modified version of the original PCE, are proposed by Seo and Moon (2). In LPCE, generation of perturbed vorticity
is suppressed by dropping the non-linear coupling terms that contribute to generate vortical components in theperturbed system. This will exclude errors related to the perturbed vorticity, deliberating the fact that it is not
an important acoustic source at low Mach numbers. Through this modification, LPCE solutions can be independent of grid resolution and computational efficiency with the grid-splitting technique can be fully achieved. The
present LPCE ensures grid-independent acoustic solutions by suppressing the generation of perturbed vorticity
in the formulation.
The present paper is organized as follows. In Section 2, the perturbed compressible equations (PCE) andperturbed vorticity transport equations are revisited, and the linearized perturbed compressible equations (LPCE)
are derived. In Section 3, the present method is applied to the prediction of turbulent edge-scattering noise tovalidate its accuracy. Finally, concluding remarks are summarized in Section 4.

2

THEORY OF LOW MACH NUMBER AERACOUSTICS

2.1 Perturbed Compressible Equations (PCE)
In the hydrodynamic/acoustic splitting method, the instantaneous total flow variables are decomposed into incompressible and perturbed compressible variables asr(~x,t) = r0 + r 0 (~x,t)~ x,t) + ~u0 (~x,t)~u(~x,t) = U(~(1)

p(~x,t) = P(~x,t) + p0 (~x,t)
The incompressible flow variables describe unsteady viscous flow, while acoustic fluctuations and other compressibility effects are represented by perturbed compressible variables. A hydrodynamic flow field is solved by
the incompressible Navier-Stokes equations, while its compressibly perturbed field is calculated by the perturbed
compressible equations (PCE).
The perturbed compressible equations are written as∂ r0+ (~u · —)r 0 + r(— · ~u0 ) = 0
∂t

(2)

(a)

(b)

1 DP
Figure 2. (a) Instantaneous r1 Dr
Dt contours (compressible N–S eqs., ReD = 150, M = 0.2), (b) instantaneous P Dt
contours (incompressible N–S eqs., ReD = 150)

0 ~
∂ ~u0
~ + 1 —p0 + r DU = 1 f~0+ (~u · —)~u0 + (~u0 · —)U
∂t
r
r Dt
r vis

∂ p0
+ (~u · —)p0 + g p(— ·~u0 ) + (~u0 · —)P =
∂t

DP
+ (g
Dt

1) (F

(3)
— ·~q)(4)0 is the perturbed viscous force vector, F and ~~ · —), ~fvis
where D/Dt = ∂ /∂t + (Uq represent thermal viscous
dissipation and heat flux vector, respectively. At low Mach numbers, the perturbed viscous forces can be approximated as!
0∂∂ u0i ∂ u j 2 ∂ u0k
0fvis,i = µ0
+
di j
(5)
∂ x j ∂ x j ∂ xi 3 ∂ xk

by assuming viscosity µ ⇠= µ0 (= constant), and F and ~q are expressed as✓◆∂ u j ∂ uk 2 ∂ ul
∂ uk
F=µ
+
d jk
,
∂ xk ∂ x j 3 ∂ xl
∂ xk∂
qj = k
∂xj

✓

gp
r

◆

(6)

(7)

More details of PCE derivation can be found in reference (1).
For low Mach number and non thermally-driven flows, the total derivative of the hydrodynamic pressure in
the right hand side of Eq. (4) can be regarded as the only explicit acoustic source term. It is also interesting
to note that Eq. (4) can be re-written as a ‘dilatation rate equation’,
✓
◆
1
dP
d p0
D=
+
+T
(8)
rc2 dtdt
where D = — ·~u0 is the dilation rate, d/dt = ∂ /∂t + (~u · —), T = (g 1)(F — ·~q)/(rc2 ) is the thermal effectterm, and relation of g p = rc2 is used. From Eq. (8), one can see that compressibility (or dilation rate) effectis related to a total derivative of the pressure (hydrodynamic and perturbed combined) (see Figure 2) (3).
2.2 Perturbed Voticity Transport Equations
Since perturbed variables in PCE are residuals of the total variables with incompressible components subtracted,
they represent not only the acoustic fluctuations but also the other compressibility effects such as coupling
effects between the hydrodynamic flow and the perturbed field. One particular component of the perturbed~ 0 = — ⇥~u0 , a ‘non-radiating’variables that concerns consistency of the acoustic solution is perturbed vorticity, w
vortical component generated in the PCE system. This fluctuating quantity becomes unstable for various reasons

and generates unwanted errors in acoustic calculations. Here, attention is given to identify terms associated withproduction and diffusion of perturbed vorticity in its transport processes.
In order to derive the perturbed vorticity transport equation, the perturbed momentum equation, Eq. (3) is
re-written as∂~u0
1~ + —( 1~u0 ·~u0 ) + (w
~ + (~W ⇥~u0 ) + (w~ 0 ⇥ U)~ 0 ⇥~u0 ) + —p0 =+ —(~u0 · U)∂t2r

r 0 DU 1 ~ 0+ fvisr Dtr(9)Taking a curl on the left hand side of Eq. (9) yields~0∂w~ 0 +w~ 0 (— ·~u0 ) + (~u0 · —)~W + ~W(— ·~u0 )+ (~u · —)w∂t(~W · —)~u01(—r ⇥ —p0 ).
r2~ 0 · —)~u(w

With the incompressible Navier-Stokes equations, the right hand side of Eq. (9) can be re-written as
r0
—P
r0 r

r0
0
~ + 1 ~fvis
n0 —2U,rr(10)

and the curl of Eq. (10) yields
✓
1
1 ~0
(—r
⇥
—P)
+
—
⇥
f
2
r
r vis

◆
r0
~ .
n0 —2Ur(11)

Combining Eq. (9) and (11) and dividing both sides by r, one obtains the perturbed vorticity transport equations.
∂
∂t

✓

~0
w
r

◆

✓

~0
w
+ (~u · —)
r

◆

=

1 ~
~ 0 · —)~u]
W · —)~u0 + (wr [(| {z } | {z }

|

I a

{z
I

+

I b

}

1
u0 · —)~W + ~W(— ·~u0 )]r [(~| {z } | {z }

|

II a

1
1
(—r ⇥ —p) + — ⇥ ~Fvis ,
3
r
r
|
{z
} | {z }
III

{z
II

II b

}

(12)

IV

0
~
where ~Fvis = (~fvis
r 0 n0 —2U)/r.
In Eq. (12), one can clearly see that perturbed vorticity is generated and diffused by source terms on theright hand side through: (i) coupling effects between the hydrodynamic vorticity and the perturbed velocities(terms I and II), (ii) entropy field (term III), and (iii) viscous force (term IV). Term I is related to a threedimensional effect of vortex stretching: stretching of hydrodynamic vorticity by perturbed velocities (term I-a)and stretching of perturbed vorticity by total velocities (term I-b). Term II represents a more direct coupling between the hydrodynamic vorticity and the perturbed velocities. The convective effect of hydrodynamic vorticity
by perturbed velocity is represented by term II-a, whereas term II-b is related to the dilatation rate effect. Term
III is not so important for low Mach number, non thermally-driven flows and term IV only provides physicaldiffusion to the perturbed vorticity. In our previous study (1), it was shown that term II-a is the most dominantsource term that generates perturbed vorticity and term II-b is considered less important at low Mach numbers.
It is interesting to note that perturbed vorticity is not a ‘radiating’ acoustic quantity but a ‘convecting’ hydrodynamic vortical. Its physical meaning represents modification of hydrodynamic vorticity through interactions
between the hydrodynamic vorticity and the velocity fluctuations. At low Mach numbers, the magnitude of theperturbed vorticity is small but, if falsely resolved, it becomes self-excited and grows to affect the acoustic

solution. Since term II-a is related to the gradient of hydrodynamic vorticity W, perturbed vorticity w 0 usually appears at the edge of the hydrodynamic vorticity and its length scale is similar to (or sometimes smallerthan) the hydrodynamic vortical scale. Therefore, acoustic grid resolution must carefully be handled in PCE
calculation. More details can be found in reference (1).
2.3 Linearized Perturbed Compressible Equations (LPCE)
The linearized perturbed compressible equations (LPCE) are proposed to secure a stable and grid-independent
acoustic solution. Neglecting the second-order, non-linear terms such as (~u0 · —)~u0 , the original PCE, Eqs. (2)-(4)can be re-written as∂ r0~ · —)r 0 + r0 (— ·~u0 ) = 0
+ (U
∂t
∂~u0
~ + 1 —p0 = (~W ⇥~u0 )+ —(~u0 · U)∂tr0~
~ 0 ⇥ U)(w

∂ p0
~ · —)p0 + gP(— ·~u0 ) + (~u0 · —)P =
+ (U
∂t

DP
+ (g
Dt

(13)
~
r 0 DU
1 0+ ~fvis
r0 Dt
r0

(14)

— ·~q)

(15)

1) (F

Since the left hand side of Eq. (14) does not generate any vortical component, only the right hand side terms~ correspond to~ 0 ⇥Uare responsible for the generation of perturbed vorticity. The first two terms, ~W ⇥~u0 and w
the dominant source terms (terms I and II) in the perturbed vorticity transport equations and the last two terms
are associated with the entropy and viscous effects (terms III and IV).
With the thermal terms neglected in the perturbed energy equation, a set of linearized perturbed compressible
equations (LPCE) is written as∂ r0~ · —)r 0 + r0 (— ·~u0 ) = 0
+ (U
(16)
∂t
∂~u0
~ + 1 —p0 = 0+ —(~u0 · U)
(17)
∂t
r0
∂ p0
~ · —)p0 + gP(— ·~u0 ) + (~u0 · —)P = DP .
+ (U
∂t
Dt
Since the curl of the linearized perturbed momentum equations, Eq. (17) yields
~0
∂w
= 0,
∂t

(18)

(19)

Thus, generation of perturbed vorticity is suppressed in the LPCE formulation: i.e. any further changes (generation, convection, and decaying) of perturbed vorticity in time are prevented. Now, one can use larger normal
grid spacings near the wall because LPCE is an inviscid form. More details can be found in reference (2).

3

TURBULENT EDGE-SCATTERING NOISE

A flow (Uo = 20m/s) over the flat plate at an angle of attack a = 0 is considered. The plate has a chord lengthc = 10cm, thickness h = 0.03c, span L = 3c, and the plate edges have four corners cut with a radius of 0.5mm
in order to smoothen the geometrical singularities. The Reynolds number based on the chord length, Rec is
1.3 ⇥ 105 and the flow Mach number, Mo is 0.06. An incompressible large eddy simulation is performed forthe computational domain shown in Figure 5. Flow periodicity is assumed in the spanwise direction for thesimulated span, LS = 0.03c. An O-type grid is used with 657⇥201⇥21 (about 2.8 millions) points in x, y, and+z. The minimum grid spacing for x and y is 0.0005c (or Dxmin= Dy+
min = 3) and a uniform grid spacing of+0.0015c (or Dz = 15) is used in the spanwise direction. More details can be found in reference (4).
Figure 3(a) shows the iso-surfaces of the second invariant property of the velocity gradients, Q(= 200) overthe plate. A separation occurs at the leading-edge and the shear layer approximately reattaches at x = 0.8c.

(a)

(b)

Figure 3. Instantaneous hydrodynamic flow field at Rec = 1.3 ⇥ 105 (Uo = 20m/s); (a) iso-surfaces of Q (=200)
non-dimensionalized by (Uo /h)2 , (b) turbulent boundary layer characteristics; boundary layer profiles (timeaveraged) at x = 0.2c, 0.4c, and 0.6c
Then, it breaks off into eddies that are convected downstream along the plate surface. It also exhibits the
expected von Kármán vortex shedding in the wake region. Figure 3(b) shows the boundary layer profiles overthe plate, which are expected to be turbulent but with a pressure gradient effect imposed upon it. A boundary
layer thickness, d at x = 0.2c (upstream from the trailing-edge) is 1.12h and the turbulent Reynolds number,
Ret is 231.
Temporal variations of the wall pressure fluctuations at various stations over the plate (from A to I) are
exemplified in Figure 4(a). First, one can identify that at station A, pressure fluctuations with a range ofscales are monitored because it is close to the reattachment point of the shear layer (x = 0.8c). Small scaleeddies are rapidly decaying but large-scale structures generated at very low frequency (St = f h/Uo = 0.01 or
67Hz) seem to survive to the trailing-edge and influence the region close to the stations at H and I. Station
H clearly shows the pressure fluctuations at the von Kármán vortex shedding frequency (St = 0.2 or 1333Hz).
At station I, pressure fluctuations at St=0.01 are only visible. A x t plot of the wall C p presented in Figure4(b) clearly indicates that large-scale pressure bursts ‘originated’ from the shear layer reattachment point havea non-dimensional time scale close to 100 (= tUO /h), which corresponds to the frequency close to St = 0.01. It
is also indicated that pressure bubbles are convected downstream with UC = 0.83UO .
The flat plate self-noise at ReC = 1.3 ⇥ 105 and Mo = 0.06 (for a = 0) is computed by solving the linearized
perturbed compressible equations (LPCE). The LPCE calculation is conducted in a two-dimensional compu-

(a)

(b)

Figure 4. (a) Time history of surface pressure fluctuations at streamwise stations A, B, C, D, E, F, G, H,
I; curves are separated by a vertical off set of 0.1 with the corresponding zero lines located at -0.1, 0, 0.1,
0.2,. . . ,0.7, (b) x t plot of C p (plate wall): exhibiting large-scale pressure fluctuations at St = 0.01 with aconvection velocity, Uc = 0.83Uo

(a)

(b)

(c)

Figure 5. (a) Instantaneous DP/Dt field (spanwise-averaged) projected onto the acoustic grid (347⇥247:top) from the hydrodynamic grid (657⇥201⇥21: bottom), (b) instantaneous pressure fluctuation field (Dp0 =(P + p0 ) (P + p0 )), (c) time history of pressure fluctuations at r = 70h (or 2.1c) vertically away from thetrailing-edge; non-dimensionalized by ro co 2tational domain, where it is coupled with the ‘spanwise-averaged’ incompressible LES solution. An O-type
acoustic grid is used with 347⇥247 points and a minimal normal spacing at the wall is five times larger thanthe hydrodynamic one. Figure 5(a) shows the grid details near the trailing-edge and the noise sources, i.e. thevon Karman vortex shedding and the convected eddies in the boundary layer. The noise source, DP/Dt field is
acquired from the LES solution and projected onto the coarse acoustic grid.
The instantaneous pressure fluctuation field (Dp0 = (P + p0 ) (P + p0 )) computed by LPCE is presented in
Figure 5(b). First, a dipole tone generated by the von Karman vortex shedding at the trailing-edge is clearly
noticeable. The acoustic wavelength, l /c is close to 2.5, which corresponds to the frequency, St = 0.2. Besides,
one may note that the other higher frequency components superimposed in the near-field are being generated
from the trailing-edge and the reattachment point of the shear layer. Due to the grid coarseness, however,acoustic waves with short wavelengths are expected to be numerically dissipated as propagating away fromthe plate. Figure 5(c) also shows the time history of the pressure fluctuations monitored at r = 70h (or 2.1c)
vertically away from the trailing-edge.
Since the microphone is located 20c from the plate, the 2D acoustic field computed by LPCE within thedomain of 10c needs to be extrapolated by the Kirchhoff method and also to be corrected to 3D spectral
pressure by Oberai’s method (*). Finally, the 3D spectral pressure in the far-field, which is radiated from span(h), needs to be corrected for the total span (100h or 3c) used in the experiment. In this procedure, we need
information on the spanwise coherence function of the surface pressure , g(z) in the most dominant noise source
region.
Figure 6(a) shows the g(z) function, which is computed by LES for h as well as measured for 3c (or100h). The spanwise coherence function rapidly drops in most frequencies, except for the tone at St = 0.2. The

(a)

(b)

(c)

Figure 6. (a) Spanwise coherence; o: computations, (o, o): wind-tunnel measurements, (b) spanwise coherence
length; data interpolated by a Gaussian law, g(z) = exp( (z/Lc )2 ), (c) comparison of SPL spectrum at r = 20c
vertically away from the mid-chord of the plate; computation (blue), experiment (black)

(a)

(b)

(c)

(d)

Figure 7. Directivity patterns of Dp0rms at r = 20c; (a) St = 0.2 (c/l = 0.4), (b) St = 0.4 (c/l = 0.8), (c) St =1 (c/l = 2), (d) St = 2 (c/l = 4)
spanwise coherence length of the surface pressure, LC calculated by the Gaussian law, g(z) = exp( (z/Lc )2 isshown in Figure 6(b). The largest value of LC is approximately 7h at St=0.2 but LC is mostly under 2h atother frequencies. The far-field SPL predicted for the actual span is now compared in Figure 6(c) with themeasured data at Ecole Centrale Lyon. The comparison is very good, except at low frequency (St < 0.03). The
measured SPL spectrum indicates an increase of SPL as the Strouhal number decreases towards St = 0.01 but theprediction fails to show it. The discrepancy comes from the fact that for the tone at St = 0.2, the microphone
location (at 20c vertically away from the mid-chord of the plate) is 8 times larger than the corresponding
acoustic wavelength (2.5c) and one can say that the far-field approximation is quite valid. On the other hand, at
St = 0.01, the acoustic wavelength is approximately 50c and the far-field assumption is not valid any more. As
a reference, the acoustic wave generated at St = 0.025 has a wavelength comparable to the microphone location(i.e. 20c).
Figure 7 shows the directivity patterns of Dp0rms at r = 20c for various Strouhal numbers (or ratios of theplate chord length to the acoustic wavelength). At the von Kármán vortex shedding frequency (St = 0.2 or
c/l = 0.4), it clearly shows a dipolar pattern. The figure also shows that as Strouhal number increases (oracoustic wavelength becomes shorter than the chord length, the directivity pattern changes to a finger-like shapedue to wave diffractions around the leading and trailing edges of the plate. Finally, one can see that the present
computational methodologies for flow and acoustics reasonably capture the physical aspects of flat plate selfnoise, i.e. the noise generation mechanism and its interactions with the flow and structure.

4

CONCLUSIONSThis paper reviews the perturbed compressible equations (PCE), the perturbed vorticity transport equation, andthe linearized perturbed compressible equations (LPCE). The concept of perturbed vorticity is introduced andits importance in hydrodynamics/acoustics splitting methods is fully discussed. This paper successfully demonstrates that the present hybrid method using LES and LPCE was found to be accurate in predicting turbulent
edge-scattering noise.

REFERENCES[1] Seo, JH and Moon YJ. Perurbed compressible equations for aeroacoustic noise prediction at low Mach
numbers. AIAA Journal. 2005;43(8).
[2] Seo, JH, and Moon, YJ. Linearized perturbed compressible equations for low Mach number aeroacoustics.
Journal of Computational Physics. 2006;218.
[3] Moon, YJ. Sound of fluids at low Mach numbers. European Journal of Mechanics B/Fluids. 2013;40.
[4] Chang, KW, Seo, JH, Moon, YJ, and Roger, M. Prediction of Flat Plate Self-Noise. AIAA Paper 2006-2513.
2006.

PROCEEDINGS of the24th International Congress on Acoustics
October 24 to 28, 2022, in Gyeongju, Korea
Extended abstractABS-0141

Novel formulation of semi-analytical finite element method foracoustic problems
Timur ZHARNIKOV1; Roman PONOMARENKO2; Denis SABITOV2; Christopher AYADIUNO3
1

Aramco Research Center, Russia

2

Aramco Research Center, Russia
3

Saudi Aramco, Saudi Arabia

ABSTRACTIn this work, novel formulation of the SAFE approach is presented. It is based on the spectral element method(SEM) variant of the finite element method and offers high order of accuracy. Also, model geometry
description with the curvilinear elements is developed and implemented. Proposed SEM-SAFE formulation
has the potential to improve the spectrum estimation and the dispersion curve computation by offering moreaccurate description of the interfaces with less elements and higher degree of approximation. Mass andstiffness matrix computation is simplified by using SEM formulation. This approach is exemplified bycomputing the spectrum of fluid-filled waveguides in anisotropic medium. Better quality of theeigenfunctions approximation is observed.
Keywords: SAFE, dispersion, anisotropy

1. INTRODUCTION
The semi-analytical finite element method (SAFE) is a powerful tool to compute the spectrum andthe dispersion curves of complex waveguide structures. Such problems attract attention and have many
applications, especially in the situations when no analytical solution is available. Of note are the nondestructive testing and geophysics (references (1,2)). The latter deals with the acoustics of boreholes
in rock formations, which are often anisotropic. In addition, the computations have to be fast andaccurate. It stimulates the interest to evaluate SAFE techniques with improved characteristics.
Spectral element method (SEM) is a version of the finite element method with specially designed
nodal basis functions (references (3,4)). Their hallmark is the property of discrete orthogonality,
which results in block diagonal mass matrix. In addition, it allows for high orders of approximationand for the possibility to use the curvilinear mesh elements. Therefore, the number of elements
required for the description of the borehole, can be significantly reduced.
In this work we present the formulation of SAFE approach using SEM version of the finite element
method. The methodology and the examples are outlined in below Sections.

2. METHOD
SAFE method formulation is well developed (see references (1,2) and references therein). It isbased on the finite element discretization of the virtual work principle. Assuming translational
invariance of the waveguide and the absence of cross-terms with respect to the frequency, thedisplacement field can be discretized as:)(1)
𝒖(𝑥, 𝑦, 𝑘, 𝜔) = 𝑵(𝑥, 𝑦)𝒖(𝑘, 𝜔)𝑒 (where 𝑵(𝑥, 𝑦) is the matrix of the basis functions and 𝒖(𝑘, 𝜔) are the displacement field values in
the nodes in the (𝑘 − 𝜔) domain. Substituting such an approximation into the variational principle
results in the generalized eigenvalue problem of the form
(𝐊 + 𝑖𝑘𝐊 + 𝑘 𝐊 − 𝜔 𝐌 + 𝑖𝜔𝑷)𝐔 = 𝟎(2)

1

timur.zharnikov@aramcoinnovations.com

Saudi Aramco: Public

Here, the matrices are computed as the combinations of blocks of the matrix integrals of the form
(references (1,2))
( )

𝐊 =

( )

𝐁 ∗ 𝐂𝐁 𝑑𝑉𝐌=

𝐍 ∗ 𝜌𝐍𝑑𝑉(3)

and the like. The integrands in these expressions involve the density of the medium, stiffness tensor,
basis functions and their derivatives, etc.
The key difference of the developed approach from the standard formulation of the SAFE method
is that here spectral element basis functions 𝑁 (𝑥, 𝑦) are used (references (3,4)). They are taken as
Lagrange functions: each function equals to 1 at the specific node and to zero at all other nodes asshown in Figure 1. The nodes are distributed inside the cell as the nodes of Gauss-Lobatto-Lagrange(GLL) quadrature. Such a choice is made to simplify numerical integration described further.

Figure 1 – Example of basis function and distribution of nodes in the physical cell (left) and reference
coordinate system (right)
The calculation of mass and stiffness matrices is a cumbersome operation. Integration with GLL
quadrature for structured grid is easy. In the unstructured curvilinear grid, each element is deformed
in a unique way, transformation from the reference coordinate system to the physical element is nonlinear. Hence, the matrices should be computed for each cell separately. Transformation from thereference coordinates to the coordinates in physical space can be obtained by considering the mapping
function in the following form:
𝑥 = 𝑎 + 𝑏 𝜉 + 𝑐 𝜂 + 𝑑 𝜉𝜂 + 𝑒 𝜉 + 𝑓 𝜂 + 𝑔 𝜉 𝜂 + ℎ 𝜂 𝜉
𝑦 = 𝑎 + 𝑏 𝜉 + 𝑐 𝜂 + 𝑑 𝜉𝜂 + 𝑒 𝜉 + 𝑓 𝜂 + 𝑔 𝜉 𝜂 + ℎ 𝜂 𝜉(4)

The mass, stiffness, and other matrices appearing in SAFE, can be calculated by integration over thereference element as
𝑀 =𝑁 (𝑥, 𝑦)𝑁 (𝑥, 𝑦)𝑑𝑥𝑑𝑦 =𝑁 (𝜉, 𝜂)𝑁 (𝜉, 𝜂)𝐽𝑑𝜉𝑑𝜂(5)using the Jacobi matrix of the mapping function (4):
𝐽=

𝜕𝑥⁄𝜕𝜉
𝜕𝑦⁄𝜕𝜉

𝜕𝑥⁄𝜕𝜂
𝜕𝑦⁄𝜕𝜂

(6)

Once the matrix coefficients of generalized eigenvalue problem (2) are computed and assembled,
the latter can be solved by standard algorithms. Doing so for a series of frequencies, the dispersion
curves of the modes can be obtained.

3. RESULTS AND DISCUSSION
To illustrate the capabilities of the developed approach, we considered the waveguide in thehomogeneous horizontal transverse isotropy (HTI) medium. The parameters of the corresponding
vertical transverse isotropy (VTI) medium were as follows: density 𝜌 = 2230 𝑘𝑔⁄𝑚 ; elastic moduli

Saudi Aramco: Public

of corresponding VTI medium are 𝐶 = 40.9 𝐺𝑃𝑎, 𝐶 = 8.5 𝐺𝑃𝑎, 𝐶 = 26.9 𝐺𝑃𝑎, 𝐶 = 10.5 𝐺𝑃𝑎,𝐶 = 15.3 𝐺𝑃𝑎. The waveguide is filled with water with the density 𝜌 = 1000 𝑘𝑔⁄𝑚 and the bulk
modulus 𝐾 = 2.25 𝐺𝑃𝑎 , its radius is R=0.1016 m. The method was tested by calculating the
eigenfunctions for several frequencies and constructing the dispersion curves.
In line with the SAFE-SEM formulation introduced above, the computational domain was
discretized using the curvilinear elements (see Figure 2). Implemented algorithm supports high orderbasis functions. For simplicity’s sake, the second order basis functions were used in this case. As aresult, coarser grid can be used to approximate the problem.

Figure 2 – Example of curvilinear mesh used for the computation
Solving the generalized eigenvalue problem (2) allowed the construction of the dispersion curves.
The examples of the latter for the ground mode (known as the Stoneley mode in geophysicalapplications) and for the dipole modes are presented in Figure 3. Black dots correspond to the Stoneleymode, red and blue dots denote fast and slow dipole modes, respectively. The examples of theeigenfunctions’ shape for another model are shown in Figure 4. Good quality of the eigenfunction
approximation can be observed.

Figure 3 – Example of computed dispersion curves

Figure 4 – Example of eigenfunctions computed for another model
In addition, the comparison of the eigenfunctions and the dispersion curves with those obtained by

Saudi Aramco: Publicthe standard SAFE approach and previously developed pseudospectral method demonstrated good
agreement. It confirms that developed approach can be considered as viable alternative to the
traditional SAFE formulation.

CONCLUSION
The formulation of the SAFE approach using the spectral element (SEM) version of the finite
element technique is presented. The advantage of developed approach is the possibility to use highorder basis functions and the curvilinear elements. It results in significant reduction of the elements
without sacrificing the accuracy. The numerical example of computing the dispersion curves for the
anisotropic medium demonstrates and confirms the capabilities of SAFE-SEM formulation.

REFERENCES
1. Ellefsen KJ, Cheng CH, Toksoz MN. Effects of anisotropy upon the normal modes in a borehole. J Acoust
Soc Am. 1991;89:2597–2616.
2. Zharnikov T, Syresin D. Repulsion of dispersion curves of quasidipole modes of anisotropic waveguides
studied by finite element method. J Acoust Soc Am. 2015;137(6):EL402.
3. Komatitsch D, Vilotte J-P. The spectral element method: An efficient tool to simulate the seismic responseof 2D and 3D geological structures. Bulletin of the Seismological Society of America. 1998;88(2):368–
392.
4. Charara M, Sabitov D. Efficient and accurate 3D sonic logging modeling in anisotropic viscoelastic media.
Society of Exploration Geophysicists. Global Meeting Abstracts. 2020;33–36.

Saudi Aramco: Public

Extended abstractABS-0229

Spectral estimation for nonstationary audio signals:
Beyond the uncertainty principle
Adrien MEYNARD1 ; Bruno TORRÉSANI2
1 ENSL,CNRS, Laboratoire de physique, Lyon, France
2 Aix Marseille Univ, CNRS, I2M, Marseille, France

ABSTRACTSpectral analysis aims to determine the frequency content of stationary signals. However, the stationarity assumption is too strong a restriction for modeling audio signals, whose statistical properties intrinsically varyover time. To address this challenge, we have proposed a spectral estimation method that takes into accountseveral physical sources of nonstationarity in acoustics, such as intensity variations, or the Doppler effect. The
signal’s model is formed of a stationary signal deformed by a stationarity-breaking operator (namely, a combination of amplitude modulation and time warping). The resulting algorithm provides simultaneous estimates
for the underlying spectrum and the deformation operator. The approximation relation the algorithm is basedon is only valid for slowly varying deformations. To overcome these limitations, we developed a Bayesian approach that allows us to include sharpness priors on the spectrum’s shape, and hence concurrently improve theestimation quality and the time-scale signal’s representation.

Keywords: Nonstationarity, Wavelet transform, Time-scale synthesis, Time warping

1

INTRODUCTION

Spectral analysis consists in the determination of the frequency content of stationary signals. Among othertechniques, Welch’s method [13] has been widely used to perform spectral estimation of stationary signals.
Whatever the method used, the stationary assumption is necessary to give sense to the estimated quantity, called
power spectrum [11].
Stationarity is a mathematical notion that guarantees the independence in time of the statistical properties ofa random process. A signal is said to be stationary when it is modeled as a realization of a stationary random
process. Most audio signals cannot be satisfactorily modeled as stationary. Indeed, the physical sources thatgenerate these sounds are inherently varying over time. For example, when a rotating engine accelerates, thepitch of the sound produced increases with the speed of the engine. Other phenomena, such as the Doppler
effect or intensity variations are sources of non-stationarity.
Several extensions of spectral analysis to classes of nonstationary signals have been proposed [10]. In ourwork, we have considered nonstationary signals modeled as stemming from stationary signals deformed bystationarity-breaking operators. We considered a physically relevant deformation, namely a combination of amplitude modulation and time warping. We then developed JEFAS [6], an algorithm for the simultaneous estimation of the stationarity-breaking operator and the power spectrum of the underlying stationary signal. Based on
an idea introduced by Clerc and Mallat [2] and developed by Omer [8], our approach approximates the effectof the deformation in the wavelet domain as a mere displacement of coefficients in the time-scale plane. The
estimation procedure proposed in JEFAS is thus performed in the wavelet domain.
We have also developed and studied an alternative approach, called the synthesis approach, inspired by amodel developed by Turner & Sahani [12], which interprets time-frequency transform in terms of probabilistic
inference. The Turner-Sahani was applied to various time-frequency problems. Our model expresses nonstationary signals as the result of a time-scale synthesis from a random time-scale representation, governed by aprescribed prior distribution, which involves the time warping model. We study the corresponding model, andcompute the posterior distribution which is used to construct an estimator for the model parameters. This approach, called JEFAS-S (where the “S” stands for “synthesis”), also outputs an adapted time-scale representationfrom which signal synthesis is possible. Remarkably enough, JEFAS-S is able to handle time warping models
in situations where the time warping function can have much faster variations. In addition, JEFAS-S can also
yield time-frequency representations which are much sharper than conventional transforms.
In this article, we describe the nonstationary signals involved. We then successively present the analysis
approach that gave rise to JEFAS and the synthesis approach used to develop JEFAS-S. Finally, we present
numerical results where we put a particular emphasis on the pros and cons of both approaches. This paper isan extended abstract of our full paper [7], currently under peer review.

2

NON-STATIONARITY MODEL

Amplitude modulated and time warped (AM-TW) signals are defined as realizations of nonstationary processes
taking the form
Y = Mα Dγ X,(1)where Mα is the amplitude modulation operator (pointwise multiplication with the positive valued function α),
and Dγ is the time warping operator (composition with the monotone function γ, assumed to be differentiable
and strictly increasing):p(2)
Mα x(t) = α(t)x(t) ,Dγ x(t) = γ 0 (t)x(γ(t)) .
Here, X is a second order wide-sense stationary random process, with zero mean. Its power spectrum is denoted
by SX .
In analogy with the spectral estimation of stationary random processes (estimate the power spectrum fromone or several realizations of the process), spectral analysis of AM-TW signals is defined as the problem of
the joint estimation of the functions α, γ and the underlying power spectrum SX from one realization of thenonstationary process Y .

3

JEFAS: AN ANALYSIS APPROACH

3.1 Wavelet transform
Under certain existence conditions [4], the wavelet transform Wx of a signal x is defined by:Wx (s, τ) =Z

x(t)ψs (t − τ) dt ,(3)

R

where ψ is the analysis wavelet, and ψs are the scaled wavelets defined byψs (t) = 2−s/2 ψ 2−st , s ∈ R.
We will focus here on the so-called sharp wavelet, introduced in [6]. Using such wavelet, the synthesis formula
reads (for a suitably chosen normalization of ψ)Z ∞ Z ∞dsx(t) = ℜψs (t − τ)Wx (s, τ)dτ
.
(4)
s
−∞ −∞
3.2 Approximation result
In that framework, it can be shown [6] that the respective wavelet transforms Wy and Wx of y and x (realizationsof the processes defined in (1)) are approximately related by
Wy (s, τ) ≈ Wx (s + log2 (γ 0 (τ)), γ(τ)) .

(5)

In the setting of random processes, under Gaussian assumption, it can be shown [8, 6] that the approximation
error is unbiased and its variance can be controlled thanks to the decay properties of the wavelet ψ, and thevariations of γ 0 .

1.5

1.5

1.25

1.25

1

Frequency (kHz)Frequency (kHz)

1

0.75

0.5

0.75

0.5

0.25

0.25
0

2

4

6

8

0

10

2

4

6

8

10

Time (s)Time (s)
10 1

1
10

0.5

2

4

6

8

10

Estimated α2 (t)

2
1.5
1

Estimated spectrum

Estimated log( '(t))

1.5

0

10 -1

10 -2

10

-3

0.5
0
0

2

4

6

8

10

Time (s)

10 -4

0

500

1000

1500

2000

2500

3000

Frequency (Hz)Figure 1. Spectral analysis of a wind sound with JEFAS. Top left: wavelet transform (absolute value) of theoriginal signal. Bottom left: estimates of TW an AM functions. Top right: wavelet transform (absolute value)of the estimated underlying stationary signal. Bottom right: estimated power spectrum.
3.3 JEFASJEFAS relies on the approximated behavior (5). Indeed, we proposed an algorithm which performs alternate
estimations of the non-stationarity functions α and γ, and the underlying power spectrum SX . The estimationsof α and γ are performed in the wavelet domain, and consist in resolving a maximum likelihood problem.
JEFAS has been implemented and tested on various audio signals such as: recording of a racing car passingby (non-stationarity caused by the Doppler effect), wind blowing through a window, or dolphins vocalizations.
The output of JEFAS applied to a sound recording produced by wind blowing is displayed in Fig. 1. One
must note that nonstationarity caused by speed variations (i.e. time warping) is well estimated, while amplitude
modulation is badly handled. The theoretical study of the performance of JEFAS—in particular the Cramér-Rao
bound on the variance of the estimators—explains that behavior [6].
JEFAS is a highly effective method for estimating TW deformation of nonstationary signals. It is however
limited by the intrinsic properties of the wavelet transform. In particular the uncertainty principle limits thecapability to estimate fast varying time warping functions.

4

JEFAS-S: A SYNTHESIS APPROACH

4.1 Synthesis model
Instead of starting from the wavelet transform, the approach is based on a synthesis model. The nonstationary
random discretized signal y is modeled as stemming from a random time-scale representation, in the form of
the inversion formula (4), namely
y = ℜ (DW) + ε ,(6)where ε is a zero mean, Gaussian white noise, D is a wavelet dictionary. W is a random time-scale matrix, withprescribed prior probability distribution, which will implement the time warping model. Based on the latter, thegoal is to estimate W from the realization y.

We now introduce the time-scale warping prior model on W. This prior model is designed so as to mimic
the behavior of wavelet transform of time warped stationary processes, and involve the following elements
1. Columns of W are assumed independently distributed, according to a circular complex Gaussian law (thiswas an approximation in JEFAS, and is now a prior model in JEFAS-S). We refer to e.g. [9] for an
extended description of multivariate complex Gaussian distributions.
2. The natural choice for the covariance matrix of a given column of W is given by JEFAS, and the approximation result (5). However, we will also investigate other choices that provide sharper time-scale
representations.
Unlike JEFAS, amplitude modulation is not included in JEFAS-S. Nevertheless, there is no theoretical obstruction to the inclusion of amplitude modulation in the model if necessary.
4.2 JEFAS-S
In a Bayesian framework, the posterior distribution of the time-scale representation is derived in [7]. It is azero-mean complex Gaussian multivariate random variable, whose covariance and relation matrices depend on
the nonstationarity parameters, i.e. SX and the TW function.
in this paradigm, the Expectation-Maximization (EM) strategy [3] leads to an estimation algorithm for thesimultaneous estimation of the nonstationarity parameters and the time-scale representation. The update rules
are detailed in [5, 7], together with some numerical considerations (e.g. initialization, dimension reduction).

5

Numerical results

The numerical results presented in the current section are reproducible. The MATLAB code is available online
at https://github.com/AdMeynard/JEFAS.
We illustrate here the ability of JEFAS-S to handle non-stationary TW signals with fast varying spectral
content. To this end, a nonstationary signal is synthesized from a stationary signal x comprised of two sine
waves with close frequencies. We have




n
n
+ ϕ1 + 2 cos 2π f2+ ϕ2 ,(7)x[n] = cos 2π f1
NτNτ
where Nτ = 1024, f1 = 184 and f2 = 154. The phases ϕ1 and ϕ2 are independent random scalars drawn from a
uniform distribution in the interval [0, 2π). We then apply the time warping operator. The warping function γ
is derived from the electrocardiogram of a patient with irregular heartbeat. Its derivative is obtained evaluating
the R-peak to R-peak intervals, followed by a cubic spline interpolation. This fast varying function is shown in
Fig. 2 (dotted line). Eventually, a Gaussian white noise of variance σ 2 = 2.5 × 10−3 is added to the observations. The wavelet transform of the TW signal, shown in the top left panel of Fig. 3, is highly disturbed byinterference between the two components. Indeed, on the one hand, the ability to discriminate two close sine
waves requires the analysis wavelet to be sharply localized in frequency, but then interference appears along
the time axis. On the other hand, the fast variations of the warping function are only discernible if the wavelet
is sharp in time, but interference then appears along the scale axis. The impossibility to avoid interference isa consequence of the uncertainty principle. Hence, the approximated behavior of the wavelet transform JEFAS
is based on is not valid. On this example JEFAS turns out to converge in 3 iterations, but the algorithm does
not converge satisfactorily towards the ground truth warping function. This is shown in the left panel of Fig. 2.
The rapid variations of γ 0 are strongly smoothed by the JEFAS estimate.
JEFAS-S converges in 111 iterations. The estimated γ 0 function is represented in the right panel of Fig. 2.
The rapid variations are now accurately reproduced. The accuracy of the estimate is markedly improved: themean square error goes from 0.0261 for JEFAS to 0.0069 for JEFAS-S. The time-scale representation estimatedwith JEFAS-S is depicted in the top right panel of Fig. 3. Although the presence of two components is not
visible, we clearly visualize the rapid variations of the time warping function. Unlike the wavelet transform,
interference is absent here.

1

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Frequency (Hz)Frequency (Hz)Figure 2. Derivative of the warping function, derived from real heartbeat measurements (dotted lines). The
resulting JEFAS (left) and JEFAS-S (right) estimates are superimposed in red.

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Time (s)Figure 3. Wavelet transform of the TW signal (top left). JEFAS-S time-scale representation (top right), andits sharpened version (bottom right). The synchrosqueezed wavelet transform is shown for comparison (bottom
left).
Once the parameters of the model have been estimated, we can take advantage of the Bayesian frameworkto adapt the form of the time-scale representation to new a priori constraints. To do this, we simply modify theexpression of the covariance template, i.e. the shape of the prior covariance matrix. We choose here to take
into account the knowledge of the sparse structure of the power spectrum to modify the covariance prior so thatthe estimated power spectrum is of the formK∆
S˜X,sp (ξ ) = ∑ ã2k δ (ξ − ξ˜k ) ,k=1

where ãk and ξ˜k are the estimates of the amplitudes and frequencies of the stationary signal. These estimates
are obtained using the local maxima of the spectrum estimated by JEFAS-S. In this example w find ξ˜1 = 155
and ξ˜2 = 184. The resulting time-scale transform is shown in the middle Fig. 3, and as expected turns out tobe much sharper than the time-scale representation given on the top image. In addition, the two components
forming the signal appear quite distinctly, the synthesis was indeed able to separate these two components.
Besides, the comparison with the synchrosqueezed wavelet transform [1], depicted in the bottom left ofFig. 3, shows that our sharpened representation presents a much better concentration of the time-scale coefficients along the instantaneous frequencies, and does not suffer from the smoothing effect which is visible on
the synchrosqueezed scalogram.

6

CONCLUSION

We have presented two approaches for spectral estimation of a class of nonstationary signals.
JEFAS is an analysis approach that relies of the approximate behavior of the wavelet transform of AM-TW
signals. It is a fast algorithm, which has been applied to diverse audio signals. The main limitation of JEFAS
is its inaccuracy when the TW function varies rapidly.

JEFAS-S is a new synthesis approach for time-scale representation, which is able to handle fast varying nonstationarities. In comparison with existing approaches, a main originality is that JEFAS-S is based upon explicit
probabilistic modeling of a time-scale synthesis representation, which allows to bypass (to a certain extent)
constraints due to uncertainty principles. JEFAS-S can handle both locally broadband and narrowband signals.
In the latter case, i.e. for signals generated by time warping of sums of sine waves, JEFAS-S is also able
to yield extremely sharp time-scale representations. Since JEFAS-S is a synthesis approach, we also point out
that the signal can be synthesized from the sharpened representation (as well as a stationary signal constructedfrom a corrected time-scale transform). The approach is limited so far to non-stationarities generated by time
warping, but could be extended to other types, for example by adding amplitude modulation, or introducing
frequency modulation or more general frequency nonlinear transformation. JEFAS-S also suffers from a longer
computation time compared to JEFAS.

REFERENCES[1] F. Auger, P. Flandrin, Y.-T. Lin, S. Mclaughlin, S. Meignen, T. Oberlin, and H.-T. Wu. Time-frequency
reassignment and synchrosqueezing: An overview. IEEE Signal Processing Magazine, 30(6):32–41, 2013.
[2] M. Clerc and S. Mallat. Estimating deformations of stationary processes. Ann. Statist., 31(6):1772–1821,
Dec. 2003.
[3] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM
algorithm. Journal of the Royal Statistical Society. Series B (Methodological), 39(1):1–38, 1977.
[4] S. Mallat. A Wavelet tour of signal processing. Elsevier/Academic Press, 1998.
[5] A. Meynard. Stationnarités brisées : approches à l’analyse et à la synthèse. Theses, Aix-Marseille Université,Oct. 2019.
[6] A. Meynard and B. Torrésani. Spectral Analysis for Nonstationary Audio. IEEE/ACM Transactions on
Audio, Speech and Language Processing, 26(12):2371 – 2380, Dec. 2018.
[7] A. Meynard and B. Torrésani. Synthesis-based time-scale transforms for non-stationary signals. preprint,
May 2022.
[8] H. Omer and B. Torrésani. Time-frequency and time-scale analysis of deformed stationary processes, withapplication to non-stationary sound modeling. Applied and Computational Harmonic Analysis, 43(1):1 – 22,
2017.
[9] B. Picinbono. Second-order complex random vectors and normal distributions. IEEE Transactions on Signal
Processing, 44(10):2637–2640, 1996.
[10] M. B. Priestley. Non-linear and non-stationary time series analysis. Academic Press, 1988.
[11] P. Stoica and R. L. Moses. Introduction to Spectral Analysis. Prentice Hall, 1997.
[12] R. E. Turner and M. Sahani. Time-frequency analysis as probabilistic inference. IEEE Transactions on
Signal Processing, 62(23):6171–6183, Dec. 2014.
[13] P. D. Welch. The use of fast Fourier transform for the estimation of power spectra: A method basedon time averaging over short, modified periodograms. IEEE Transactions on Audio and Electroacoustics,
15(2):70–73, June 1967.

ABS-0234

Local time-frequency fading
Marina KRÉMÉ1 ; Bruno TORRÉSANI2 ; Antoine DELEFORGE3
1,3 Université de Lorraine, CNRS, Inria, Loria, F-54000 Nancy, France
2 Aix-Marseille Univ, CNRS, I2M, Marseille, France

ABSTRACTFor efficient analysis of non-stationary signals, such as radar, sonar, speech, music, or general audio, timefrequency (TF) representations are required. They allow performing non-stationary filtering, ie extracting nonstationary "components" in signals. However, ad-hoc procedures are frequently used, whose accuracy is difficult
to assess and control. In this contribution, we are interested in such time-frequency filtering, which we define
as follows: the attenuation of a given, small region Ω in the time-frequency plane. This problem, addressed in a
recent publication, is referred to as time-frequency fading (TFF).
We address a local version of TFF, called LTFF, that only exploits a neighborhood Ω of the time-frequency
region of interest Ω, which is supposed to be much smaller than the full time-frequency domain. This problem
will be referred to as local time-frequency fading. We propose a method for determining such a neighborhood Ω.
The local TFF problem is then solved using a variational approach, that minimizes a suitable quadratic objectivefunction. Thus, there is a closed-form solution involving Gabor multipliers.

Keywords: Time-Frequency Fading, Local time-frequency fading, Gabor transforms, Gabor multipliers

1

INTRODUCTION

Filtering is one of the fundamental techniques in signal processing. By filtering one often means linear timeinvariant (LTI) filtering, which has a simple characterization in time and/or frequency domains. We refer to [1]for an in-depth description, in view of audio applications. Sticking to linear filters, time varying (LTV) filteringis also fundamental, as most devices actually feature time variations. LTV filters form a much wider class than
LTI filters, and may be represented in several different ways, for example as matrices as described in [1] for
digital signals, or pseudo-differential operators in the mathematical literature [2] in the analog case. Among
LTV filters, filters that can be efficiently represented in a joint time-frequency domain have received significant
interest in the last two decades. These include filters whose spectral characteristics (frequency response) vary
slowly as a function of time, or filters designed to enhance, attenuate or cancel out components which possess
specific localization properties in joint time-frequency domain (e.g., chirps with prescribed or parameterized frequency modulation laws, transient oscillatory waves). Several formulations for non-stationary Wiener filtering
have been proposed, that rely on time-frequency representations such as the spectrogram [3] or Wigner distribution [4, 5, 6]. These generally rely on pointwise multiplication in the time-frequency domain prior to inversion,very much in the spirit of classical LTI filtering. Filtering in the time-frequency domain has been used for awide range of applications including decision, detection, or time-frequency segmentation.
Time-frequency filtering exploits the ability of some time-frequency representations to efficiently represent
signals of interest. Limited to linear transforms, usual choices are Gabor transform [7] or Short Time Fourier
Transform [8], on which we will focus in the present paper, or wavelet transform, constant Q transform or S
transform, the choice often depending on the application domain.
In this paper, we discuss a local version of time-frequency fading (TFF) [9], which is a type of timefrequency filter design. Time-frequency fading is the process of restoring a signal that has been degraded byan additive disturbance, in the case where the latter is well localized in a specific region of the time-frequency
domain. We previously investigated and proposed a rigorous formulation of TFF in our previous paper [9]. We
1

ama-marina.kreme@inria.fr
bruno.torresani@univ-amu.fr
3 antoine.deleforge@inria.fr
2

solved it and proposed several algorithms that make use of random projection techniques to enable numericalscaling. TFF, on the other hand, formulates the problem in the signal dimension. As a result, despite the factthat dimension reduction methods have been used to approximate the solution, TFF remains difficult to apply
to long signals, such as those sampled at 44100 Hz. Here, we rely on the formulation proposed in and takeadvantage of the support reduction. In section 2 we briefly recall the time-frequency fading problem, and we
introduce and formulate local TFF. In section 3, we illustrate LTFF on an example of signal mixing. Section 4
presents the ongoing work. An extended version of this work is under preparation.

2

GLOBAL AND LOCAL TIME-FREQUENCY FADING

2.1 TFF and notations
The Time-Frequency Fading (TFF) problem introduced in [9] may be seen as a time-frequency analogue of astop-band filter design problem, where one tries to attenuate a set of prescribed regions Ω p ⊂ Λ of the timefrequency plane Λ, with prescribed gains. This is achieved via a variational formulation, of the formP2
+ ∑ λ p ∥Vg x∥2Ω p ,x∗ = arg min ∥Vg x − Vg z ∥Ω
x∈CL

(1)

p=1

where z ∈ CL is the input signal, Vg : CL → Λ is a linear time-frequency transform (see below for notations),Ω p ⊂ Λ, p = 1, . . . P are the regions of interest in the time-frequency plane Λ, and the λ p ∈ R∗+ are parametersthat control the tradeoff between the various terms in (1). ∥·∥ is the usual Frobenius norm on the time-frequency
space Λ, and ∥ · ∥Ω denotes its restriction to a given region Ω.
The quadratic nature of the objective function naturally leads to a linear system, whose closed-form solution
reads (see [9] for details)!−1Px∗ = z − I + ∑ (1 − λ p )MΩ p
p=1p∑ λ p MΩ p z ,

(2)

p=1

which can involve high dimensional linear system for large signals. In [9], dimension reduction techniques were
proposed to handle the problem and use a spectral approach that exploits time-frequency localization properties.
However, the formulation of [9] involves a L-dimensional problem, which may involve regions of the timefrequency plane quite far away from Ω and do not significantly influence the solution.
The approach can be developed for any linear time-frequency transform, but in the present paper we will
limit to the discrete Gabor transform (DGT) defined as follows (we refer to [9] for more details and notations,and to [10, 11] for numerical implementations we use in this work). Let a, b be two divisors of L, and set
M = L/b and N = L/a. The discrete time-frequency space is denoted by Λ = ZM × ZN . For any x ∈ CL , its DGT
Vg ∈ ℓ2 (Λ) = CM×N is defined by
L−1

Vg x[m, n] = ⟨x, g mn ⟩ = ∑ x[ℓ]gg[ℓ − na]e−2iπmℓ/M ,(m, n) ∈ Λ ,

(3)

ℓ=0

where the g mn ∈ CL , called Gabor atoms, are obtained from g by time-frequency translations, i.e. g mn [ℓ] =e2iπmℓ/M g [ℓ − na]. For convenience we introduce the time-frequency shift operators πmn such that g mn = πmn g .
′These operators satisfy the composition rule πmn πm′ n′ = e−2iπm na/M πm+m′ n+n′ , and the adjoint operator reads∗ = e−2iπmna/M ππmn−m −n .
Given M ∈ CM×N , the corresponding Gabor multiplier is the linear operator MM : CL → CL given by
MM = Vg∗ MVg ,(4)where by abuse of notation we have denoted by M the operator of pointwise multiplication with M. M iscalled time-frequency mask, or upper symbol, of MM . When M is the indicator function of a subset Ω ⊂ Λ
we will use the notation MΩ . Elementary properties of Gabor multipliers can be found in [12]. Among these,
the approach of [9] relies strongly on the fact that for real valued masks, corresponding Gabor multipliers are
self-adjoint, and their spectrum can be controlled under suitable assumptions on the window g .

The closed form solution of the problem (1) can be expressed in terms of Gabor multipliers, and will appear
as a special case of the solution given below, we thus refrain to reproduce it in this section.
2.2 Local TFF
In this section, we are interested in the estimation of a part of a locally degraded signal. We define an
envelope around the masked region, and formulate LTFF from the newly defined regions.
2.2.1 Determination of the time-frequency regions of interest
Let ε > 0 be a fixed tolerance parameter. For each sub-region Ω p , we define the influence envelope Ωεp of Ω p
as follows:noΩεp = (m, n) ∈ Λ : ∃ (m′ , n′ ) ∈ Ω p , |⟨ggmn , g m′ n′ ⟩|2 ≥ ε(5)The set Ωεp consists in the time-frequency points where the corresponding atoms "communicate" with those in
the masked region Ω p , and therefore depends on the window g and the sampling parameters. Notice that
′

′

′

′

⟨ggmn , g m′ n′ ⟩ = ⟨πm∗ ′ n′ πmn g , g ⟩ = e2iπ(m−m )n a/M ⟨πm−m′ n−n′ g , g ⟩ = e2iπ(m−m )n a/M Ag [m − m′ , n − n′ ] ,(6)Ag being the ambiguity function of the window g : Ag [m, n] = ⟨πmn g , g ⟩. Therefore the ambiguity function of
the analysis window g is used as the selection criterion. We also denote by∆ p = Ωεp \Ω p ,

∆=

[

∆p ,

Γ = Ω\∆(7)p

respectively the complementary region of Ω p in Ωεp , the union of these complementary regions and the comple
S
mentary region of ∆ in Ω, and Ω = Λ \ p Ω p .
We will also need the projection of the envelope Ωεp onto the time domain, which is the interval obtained
as follows:
I p = [anmin (p), anmax (p)],
(8)
n
o
n
o
where nmin (p) = min n | ∃m , (m, n) ∈ Ωεp and nmax (p) = max n | ∃m , (m, n) ∈ Ωεp are respectively the minimum and maximum time indices in the time-frequency envelope Ωεp (we recall that a is the time sampling step,i.e. the hop size).
2.2.2 Local TFF formulation
Similarly to TFF, and using the above notations, we formulate local TFF as follows:x∗ = arg min
x∈CL

P

P

p=1

p=1

∑ ∥Vg x − Vg z ∥2∆ p + ∑ λ p ∥Vg x∥2Ω p + ∥Vg x − Vg z ∥2Γ ,(9)In this formulation, we keep the part of the signal that is not masked (terms 1 and 3) and control the energy
within each sub-region Ω p (term 2) Note that, at this point, the reconstructed signal has length L. The following
resolution will make the concept of "local" more clear. By setting the gradient of the objective function in (1)equal to zero, we obtain the following linear system:"PMΓ + ∑ M∆ p + λ p MΩ p
p=1

#


"

P

#

x = MΓ + ∑ M∆ p z ,

(10)

p=1

which we now approximate as a family of smaller dimensional problems.
For all q = 1, . . . P, denote by 1q the indicator function of Iq . Assuming that the intervals Iq are disjoint∩Pq=1 Iq = 0,
/ we haveP

x = ∑ 1q x + 1J x,
q=1

(11)

where 1J the complementary of ∪Pq=1 Iq . The linear system becomes (10):
#

"

"

#



∑ ∑ M∆ p + λ p MΩ p 1q + ∑ MΓ 1q x = ∑ ∑ MΩ p 1q + ∑ MΓ 1q z
p q

q

p q

q

Assuming that the non-diagonal terms in the sums can be neglected, i.e. thatMΩ p 1q
M∆ p + λ p MΩ p 1 q ,are small enough for all q ̸= p, andMΓ 1q
are small enough for all q, we end up withM∆ p + λ p M Ω p x p = MΩ p z p ,(12)where we have set x p = 1 p x. The operator M∆ p + λ p MΩ p is in fact a Gabor multiplier associated to a constantmask M = 1∆ p + λ p 1Ω p , where 1∆ p (respectively 1Ω p ) is the indicator function of the subregion Ω p (respectively∆ p ). If λ p > 0 for all p, then the smallest value of the mask M is strictly positive, so the mask is said tobe semi-normalized 1 , which implies invertibility (see Proposition 3.7 in [13]). From now on, we assume that
λ p > 0 for all p. The solution (12) is then written :−1
x p = M∆ p + λ p MΩ p
MΩ p z p(13)

Thus, for each sub-region p, the masked time domain signal is reconstructed. In this formula we have reduced the dimension of the problem. From a Gabor multiplier, initially of dimension L, we pass to a multiplierin the dimension of the masked region which is much smaller.

3

Experiments

We illustrate the approach on a (single) synthetic mixture of two real audio signals.We first describe the signals,
but we do not recall the actions of Gabor multipliers already widely discussed in [9]. We also consider the casewhere P = 1, i.e. a single masked region.
3.1 Experimental setting
We consider two audio signals sampled at 8 kHz of length 16384 (about 2 seconds). The signal of interest is
a car engine noise and the perturbation signal is the song of a bird. As mentioned in the previous sections, thesignal of interest has a broadband spectrogram (i.e., with energy spread all over the time-frequency plane), while
the perturbation signal has its energy well localized in the time-frequency plane. The Gabor transform for eachof these signals is calculated with a Gauss window of length 256, the time-frequency lattice parameters are set
to a = 32 and b = 512, generating a 513 × 256 time-frequency matrix. The observation is a linear combinationsof these two signals, as shown in Fig 1 (left). The corresponding binary mask is on the right. The goal is tofilter the bird song from the mixture. We construct the Ωεp region as described in section 2.2.1. The value of
ε here is 10−6 . The Fig 2 shows the masked Ω p region (in dark yellow) as well as the Ωεp region (in lightyellow). The corresponding time interval is I p = [47 127].
We then restrict the frequency time plane of dimension of size 513 × 256 to 513 × 80. We solve TFF
from this reduced time-frequency plane. The time domain signal to be reconstructed corresponds only to thetime domain signal which is in the Ω p region and and is of length 1600 here. The closed-form solution (12)depends on the regularization parameter λ p = λ > 0 which is adjusted using the strategy presented in [14]: theoptimal value of λ is chosen as the one for which the energy of the reconstructed signal in the region Ω pcorresponds to a given target energy E. Here, the energy E was chosen roughly by hand. In this example, theoptimal value of λ is 10−3 .
1 A sequence M = (M[n])n>1 is said to be semi-normalized if there exist a and b such that 0 < a ≤ |M[n]| ≤ b < ∞ for all n ∈ N.

-40

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Time (s)Figure 1. Left: Spectrogram of the mixture of the engine noise of a car (energy spread everywhere in thetime-frequency plane) and of a bird "cuicui" (energy well localized - dark yellow area). Right: binary mask
constructed from the spectrogram of two sources as in [9]

3500

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Time (s)Figure 2. Left: Illustration of Ω p and Ωεp regions in the case of a single region. The value of e is 10−6 . Right:
a zoom on reconstruction of the part of the signal of interest masked by the bird song.
We quantitatively evaluate the quality of the reconstruction, via the source-to-distortion ratio (SDR) [15]expressed in decibel by
!
f

∥xre
p ∥
re f
,
(14)
SDR x p , x p = 20log10f∥x p − xrep ∥fwhere xre
p is the reference signal. The right panel of Figure 2 shows the reconstruction of the part of the car
engine noise signal masked by the bird song in red. We can see that on this simple example, LTFF reconstructs
almost perfectly the missing part of the signal with SDR = 40 dB. This experiment shows that LTFF could
lead to satisfactory reconstruction results in the case of several regions P >1. Moreover, LTFF could solve suchproblems in high dimension since it has fast computation time.

4

Ongoing work

We present a novel method for performing local time-frequency fading in this paper. Because it is limited to aneighborhood of the masked region, this method can handle longer signals. However, the obtained theoreticalresults are based on assumptions concerning a control on the norm of the sum of the two Gabor multipliers
that we have not presented here. As part of this work, we will establish more rigorous proof results. From anexperimental point of view, we have presented one example to illustrate the idea of our work. In the rest of ourwork, we will extend these experiments to the case where we have P > 1 regions and also to signals sampled
at 44.1 kHz. We will also compare the results of reconstructions obtained with TFF as well as the computation
times. The experiments will then be extended to a larger class of signals by automating the code. Finally, thecombination of TFF and LTFF may result in a novel source separation method.

References[1] J. O. Smith, Introduction to Digital Filters: With Audio Applications, ser. Music signal processing series.
W3K, 2007. [Online]. Available: https://books.google.fr/books?id=pC1iCQUAsHEC[2] G. F. Margrave, P. C. Gibson, J. P. Grossman, D. C. Henley, V. Iliescu, and M. P. Lamoureux, “The Gabortransform, pseudodifferential operators, and seismic deconvolution,” vol. 12, pp. 43–55, 2005.
[3] P. Lander and E. J. Berbari, “Enhanced ensemble averaging using the time-frequency plane,” in IEEE-SP
Proc. Int. Symp. on Time-Freq. and Time-Scale Analysis, 1994, pp. 241–243.
[4] G. F. Boudreaux-Bartels and T. W. Parks, “Time-varying filtering and signal estimation using Wigner distribution synthesis techniques,” vol. 34, no. 3, pp. 442–451, 1986.
[5] F. Hlawatsch, G. Matz, H. Kirchauer, and W. Kozek, “Time-frequency formulation, design, and implementation of time-varying optimal filters for signal estimation,” vol. 48, no. 5, pp. 1417–1432, 2000.
[6] L. Stankovic, “On the time-frequency analysis based filtering,” vol. 55, no. 5, pp. 216–225, May 2000.
[Online]. Available: https://doi.org/10.1007/BF02994785[7] D. Gabor, “Theory of communication. part 1: The analysis of information,” Journal of the Institution ofElectrical Engineers - Part III: Radio and Communication Engineering, vol. 93, no. 26, pp. 429–441, Nov.
1946.
[8] M. R. Portnoff, “Time-frequency representation of digital signals and systems based on short-time Fourier
analysis,” vol. 28, no. 1, pp. 55–69, 1980.
[9] A. M. Krémé, V. Emiya, C. Chaux, and B. Torrésani, “Time-frequency fading algorithms based on Gabormultipliers,” IEEE Journal of Selected Topics in Signal Processing, vol. 15, no. 1, pp. 65–77, Jan 2021.
[10] P. Soendergaard, B. Torrésani, and P. Balazs, “The linear time frequency analysis toolbox,” International
Journal of Wavelets and Multiresolution Information Processing, vol. 10, no. 4, pp. 1 250 032–1 – 1 250 032–
27, 2012.
[11] Z. Průša, P. L. Søndergaard, N. Holighaus, C. Wiesmeyr, and P. Balazs, “The Large Time-Frequency
Analysis Toolbox 2.0,” in Sound, Music, and Motion, ser. LNCS. Springer International Publishing, 2014,
pp. 419–442.
[12] H. G. Feichtinger and K. Nowak, “A first survey of Gabor multipliers,” in Advances in Gabor Analysis,
H. G. Feichtinger and T. Strohmer, Eds. Birkhaüser, 2002, pp. 99–128.
[13] D. T. Stoeva and P. Balazs, “A survey on the unconditional convergence and the invertibility of multiplierswith implementation,” in Sampling – Theory and Applications (A Centennial Celebration of Claude Shannon),
M. R. S. Casey, K. Okoudjou and B. Sadler, Eds. Springer, 2020, to appear, preprint arXiv:1803.00415.
[14] A. M. Krémé, V. Emiya, C. Chaux, and B. Torrésani, “Filtering out time-frequency areas using Gabormultipliers,” 2020, pp. 5790–5794.
[15] E. Vincent, R. Gribonval, and C. Févotte, “Performance measurement in blind audio source separation,”vol. 14, no. 4, pp. 1462–1469, Jul. 2006.

ABS-0380

Scattering at the angles of polyhedral rooms: Application to stress-energy
tensor conservation in Riemannian spaces
Jean-Dominique POLACK(1) , Aidan MEACHAM(2) , Roland BADEAU(3) , Jean-Christophe VALIERE(4)(1) Institut D’Alembert, Sorbonne Université, France, jean-dominique.polack@sorbonne-universite.fr(2) Institut D’Alembert, Sorbonne Université, France, aidan@lam.jussieu.fr(3) LTCI, TelecomParis, France, roland.badeau@telecom-paris.fr(4) Institut PPrime, Université de Poitiers, France, jean.christophe.valiere@univ-poitiers.fr

ABSTRACT
Riemannian spaces with negative curvature constitute the proper setting for the distribution of images createdby irregular polyhedral rooms with obtuse angles. The crucial parameter is the excess angle that arises around
specific edges, called hinges, when first and second order images are considered, as it pilots the metric tensor of
the space and all its geometrical properties. With the use of these geometrical properties, and complementing
them with the uncertainty principle, we describe the scattering of wave packets around dihedral angles: it is
proportional to the excess angle, and is best described in terms of the conservation of the stress-energy tensor.
The basic elements for computing the scattering are given.

Keywords: Riemannian geometry, Polyhedral rooms, Scattering, Stress-energy tensor

1

INTRODUCTION

In a previous paper [1], we showed that the same mirroring technique as known for rectangular rooms can beapplied to polygonal and polyhedral rooms of arbitrary shapes, leading to the tessellation of a Riemannian spacewith negative curvature. We were then able to show that the number of image sources increases exponentiallywith layer order, building a bridge between rooms with flat walls and generic mixing rooms with partiallycurved walls.
In the present paper, we take a further look at the implications of this tessellation, introducing the wave
nature of the sound field in a room and the conservation of its energy and intensity, in order to explain scatteringat corner and edges, while keeping mathematical complexity at minimum.

2

SCATTERING ON NON RECTANGULAR DIHEDRAL ANGLES

The proper setting for computing scattering on non rectangular dihedral angles is Riemannian geometry. We
therefore first expose the basic principles of this geometry.
2.1 Riemannian geometry
We consider an n-dimensional space with its positive-definite metric tensor gi j and the volume element dV =√ 1
gdx . . . dxn , where g = det gi j is positive [2]. The infinitesimal distance element is given by:
ds2 = gi j dxi dx j

(1)

and we note gi j the inverse matrix of gi j . Oi is the covariant derivation with respect to xi , which differs fromthe usual partial derivation ∂i in a way that depends on the tensor rank. For example, for a function Φ:
O jΦ = ∂ jΦ = Φ j

(2)

ε

ε

ε

ε

Figure 1. Reflection and scattering of sound rays on obtuse angle β . ε is the excess angle, since second order
image angles overlap by ε.
but O j Φi = ∂ j Φi − Γkji Φk and O j X i = ∂ j X i + Γijk X k , where X i are the (contravariant) components of vector X
and Γkji the Christoffel symbols linked to the derivatives of the elements of the metric tensor gi j :1Γkji = gkl ∂ j gil + ∂i gl j − ∂l g ji
2

(3)

Note that, unlike ordinary differentiations, covariant derivations do not commute.
2.2 Images rooms and obtuse angles
We consider the obtuse angle β of Figure 1 in a 2-dimensional Euclidean space. A sound source S (blackstar) emitting inside the angle emits rays in all directions. Some rays impinge on the left arm Ox of the angle
and are reflected (red arrows). As the position of impact moves clockwise toward the apex, the reflected ray
gradually moves upwards and eventually hits the right arm on which it is reflected once more. Finally, the ray
impacts the apex of the angle (black upwards broken-line arrow).
In a similar fashion, some rays will impinge on the right arm Oy and be reflected (green arrows). As
the position of impact moves anticlockwise toward the apex, the reflected ray gradually moves upwards andeventually hits the left arm on which it is reflected once more. Despite the continuity of the impinging rays
around the apex, there is no continuity of the reflected rays, and this creates scattering.
In order to visualise the scattering, one needs to consider the images of the sound source by reflection on
the two arms of the angle. Let’s call Sl0 the image of the source on the left arm (red star), and Sr0 the image
on the right arm (green star); Sl00 the left-most second order image (dim red star), and Sr00 the right-most second
order image (dim green star). Reflected rays on the arms of the angle are first emitted from the first order
image sources Sl0 and Sr0 , then from the second order sources Sl00 and Sr00 when the reflected rays hit the oppositearm, until the rays emitted from the first-order image sources reach the apex (red and green dotted-line arrows).
In that position, the rays emitted from the second order image sources do not coincide in direction, since they
are emitted from two different image sources at angle ε with respect to the apex. Scattering therefore comesin to fill the gap between these two directions, and in fact beyond them. In other words, one must consider
a continuum of image sources along the sector between Sl00 and Sr00 (grey sector), that is, one must rotate thesecond order image angle by ε from position y00 0x0 to position y0 0x00 .
In fact, when rays rotate clockwise around the original source S, the reflected rays rotate anticlockwisearound the first order source Sl00 , and clockwise around the second order source Sr00 . So, when the secondary
source moves clockwise from Sl00 to Sr00 on Figure 1, the diffracted rays rotate clockwise around the apex, thus
filling the grey sector with continuity of rotations at its boundaries.
In order to make the second order images coincide, one needs to embed Figure 1 in a 3-dimensional space

α
α

α
α

Figure 2. Embedding scattering on obtuse angle in 3-dimensional space. Angles α are vertical and measuredeviations of angle arms from horizontal plane. Note that the two second order image sources coalesce in one
single image source.
as in Figure 2. We call x1 , y1 , and z1 the three Cartesian coordinates, and keep notations Ox and Oy for thearms of the original flat angular sector, which is now slanted so that angle β exactly projects on a right angle.
As a consequence, Ox is elevated by angle α above Ox1 and Oy is lowered by angle α below Oy1 ; similarly,O(−x) is elevated by angle α above O(−x1 ) and O(−y) is lowered by angle α below O(−y1 ), so that the apex
angle remains equal to β . In such a way, we obtain a locally flat space where rays are free to cross the borders
between subsequent angular sectors without changing their direction.
2.3 Metric for obtuse angles
When the angle α is not null in Figure 2, that is, when β > π/2, the position of any point M in the system
is given by its coordinates x and y, that are no longer orthogonal. In order to compute the corresponding
infinitesimal distance element, one must project coordinates x and y on the Ox1 y1 plane, then compute thevertical coordinate with equation:
z1 = tan α(| x1 | − | y1 |)(4)valid for the four sectors of the system. With the convention sgn(0) = 0, one obtains everywhere, even on thearms of the angles:ds2= dx12 + dy21 + dz21

= (1 + tan2 α)dx12 + (1 + tan2 α)dy21 − 2sgn(x)sgn(y) tan2 αdx1 dy1√√with dx = 1 + tan2 αdx1 and dy = 1 + tan2 αdy1 , that is:
ds2

(5)

2

tan α
= dx2 + dy2 − 2sgn(x)sgn(y) 1+tan
2 α dxdy= dx2 + dy2 − 2sgn(x)sgn(y) sin2 α dxdy

(6)

Simple projection on the plane Ox1 y1 shows that − sin2 α = cos β . Note that cos β < 0 since π/2 < β < π for
obtuse angles. The last equation can therefore be written as:
ds2 = dx2 + dy2 + 2sgn(x)sgn(y) cos β dxdy

(7)

Using the embedding in the 3-dimensional space, distance elements ds are obviously continuous along a linethat crosses one of the arms, implying that dx and dy are modified when the line crosses the arms. From theexpression of ds2 , it is easy to deduce the metric tensor, its inverse, and its determinant:1
sgn(x)sgn(y) cos βgi j = (8)sgn(x)sgn(y) cos β
1


1
1

gi j =2sin β−sgn(x)sgn(y) cos β−sgn(x)sgn(y) cos β
1

g = sin2 β




(9)
(10)

and gi j

Note that gi j
reduce to the identity matrix on the arms (x = 0 or y = 0) because of the convention
sgn(0) = 0, with g = 1.
Equation (3) then leads to the Christoffel symbols, that are all equal to 0, but for:Γyxx

= gyy ∂x gxy = 2δ (x)sgn(y) cos βΓxyy= gxx ∂y gxy = 2sgn(x)δ (y) cos β(11)where δ is the Dirac distribution, and where we have used the form for gxy valid on the arms because δ (x),resp. δ (y), is null everywhere except on the arms x = 0, resp. y = 0 ;It can further be shown [3] that the space has negative curvature at the apex O where it tends toward4(π/2 − β ) for small values of π/2 − β . Indeed, Regge [4] has shown that the curvature is equal to the total
deficit angle at the apex −ε = 2π − 4β .
For a 3-dimensional obtuse dihedral angle, Fig 1 represents a projection of the rays on a plane perpendicular
to the apical edge. But no equivalent of Fig 2 can be drawn, as the embedding takes place in a 4-dimensional
space. Full computation shows that the local curvature remains null everywhere - flat space - except on theapical edge that Regge [4] calls hinge. Note that hinges are subspaces of co-dimension 2 where excess angle isnon null.

3

SCATTERING

3.1 Time-space metricIn thispSection, we now consider a 4-dimensional time-space with its metric tensor gi j and the volume element
dV = |g|dx0 . . . dx3 , where g = det gi j [2]. Here, g is negative as the first eigenvalue of the metric tensor is
negative and equal to −c2 , where c is the speed of sound: its eigenvector corresponds to the time direction dx0 .
The infinitesimal distance element is still given by equation (1):ds2 = gi j dxi dx j

(12)

Therefore, covariant derivations and Christoffel symbols are still defined as in Sec. 2.3. More precisely, we
have:


−c2
0
0
0




 0
1
sgn(x1 )sgn(x2 ) cos β 0 
gi j = 
(13)


 0
sgn(x1 )sgn(x2 ) cos β
1
0 


0
0
0
1


−c−2
0
0
0




sgn(x1 )sgn(x2 ) cos β
1
0
−
0


sin2 βsin2 βgi j = 
(14)
1
2


) cos β1 0− sgn(x )sgn(x
0
2
2
sin β
sin β


0
0
0
1
where β is one fourth of the total dihedral angle around the hinge, and with determinant g = −c2 sin2 β < 0. gi j
and gi j reduce to diagonal matrices on the arms, now denoted (x1 = 0 and x2 = 0), with all diagonal elements
equal to 1 except g00 = −c2 and g00 = −c−2 , and with g = −c2 . As for the Christoffel symbols, they are allequal to 0, but for Γ211 = 2δ (x1 )sgn(x2 ) cos β and Γ122 = 2sgn(x1 )δ (x2 ) cos β (eq 11). It can be shown [3] thatthe time-space remains flat everywhere, except on the hinges, now given by the apical edges extended by thetime laps.
3.2 Conservation of stress-energy tensor
The stress-energy tensor formalism was introduced into Acoustics by Morse and Ingard [5] in order to extend
the wave equation by a set of conservation equations that are sum-upped in the conservation of the stress-energy
tensor T i j :
T tt T tx T ty T tz

 tx
xx T xy T xz 


T
T

(15)
Tij = 

 ty
xy
yy
yz
T
T
T
T 


T tz T xz T yz T zz
where T tt is the total energy density; T tx , T ty and T tz the sound intensity vector; and the remnant terms
represent the spatial distribution of kinetic energy. Note that the stress-energy tensor is symmetrical.
The conservation of the stress-energy tensor can be written in terms of the covariant derivative of the stressenergy tensor as:(16)
Oi T i j = ∂i T i j + Γiik T k j + Γikj T ik = 0
that is ([2] p.54):p1
p ∂i ( |g|T i j ) + Γikj T ik = 0
|g|

(17)

or, after integration on a small 4-dimensional volume V with border ∂V :Z

ni T i j dS +

∂V

Z
V

Γikj T ik dV = 0(18)

In the last equation, ni is the outgoing normal covector to the boundary, normalized by ni gi j n j = ±1, with anegative sign for time boundaries.
With the form chosen for the metric tensor, the conservation of the stress-energy tensor reduces to:R∂V ni T

i0 dS

=0

i1
∂V ni T dS

= −2 cos β V sgn(x1 )δ (x2 )T 22

p|g|dx0 dx1 dx2 dx3

R∂V ni Ti2 dS

= −2 cos β V δ (x1 )sgn(x2 )T 11

p|g|dx0 dx1 dx2 dx3

R

i3 dS

=0

R

∂V ni T

R

R

(19)

Note that the first line simply corresponds to the conservation of the total energy. For the two next lines, sincecos β < 0, when the stress-energy tensor crosses the arm Ox2 of the angle, it receives a positive acceleration
proportional to T 22 in the x1 direction; and when it crosses the arm Ox1 of the angle, it receives a positive
acceleration proportional to T 11 in the x2 direction; no acceleration occurs in the x3 direction. The accelerations
along the x1 and x2 directions, however, do not correspond to deviations of the direction of propagation since
the space is flat everywhere but on the hinge. It only takes into account the change of coordinates across thearms of the angle.
3.3 Wave packets
Any ray issued from a sound source will have some thickness. This is simply due to the uncertainty principle(see for example [6]), which states that positions and directions cannot be both determined with infinite precision. Note that this is not the case with the geometrical constructions of the previous Sections, where positions
and directions are simultaneously defined with infinite precision.

Figure 3. Wave packets crossing angle arms. Packet on the left crosses arm Ox1 between x10 and x20 ; packet in
the middle symmetrically crosses the two arms. Green lines mark the areas where deviations occur.
We now consider a wave packet, that is, a wave of finite extension, both along the direction of propagationand laterally, and we apply the stress-energy conservation equations (19) to it. We choose for integration atime interval over which the wave packet moves from one side of the angle arms to the other. Two cases are
depicted in Figure 3.
The first wave packet is located in sector (−x2 )Ox1 at time x10 . It is here defined by the projection of its
distribution on the plane of Figure 3, where it is represented by a rectangle area with levels of grey proportional
to the probability of presence of the wave packet. The size of the rectangle is chosen such that the componentsof the stress-energy tensor are negligible on its boundaries. As time increases, the wave packet moves at angle
θ with respect to direction x2 toward sector x1 Ox2 in which it is located at time x20 . It crosses arm Ox1 (x2 = 0)
in-between times x10 and x20 , where it is deviated according to the second line of equation (19). In other words,
the components of T i j do not change, but for component T 01 . Complete computation of the field [3] shows thatthe corresponding impulse exactly compensates for the change of coordinates, so that the wave packet keeps on
travelling in its initial direction.
This is not the case for the second wave packet, located in sector (−x1 )O(−x2 ) at time x30 and moving tosector x1 Ox2 at time x40 . It symmetrically crosses the arms x1 = 0 and x2 = 0 on its way. Since the function sgnchanges sign around the origin O, the negative deviation compensates for the positive one on each axis, andno deviations occurs. However, if a larger part of the packet passes on one side of the origin, some deviation
occurs in proportion to the offset. This case is not represented in Figure 3.
In both cases, direct calculation shows that T 00 does not change when crossing arm Ox1 , as expected forenergy conservation.
3.4 Ray scattering
Instead of thickening the rays by considering wave packets, an alternative is to thicken the hinges by low-passfiltering the Christoffel symbols on the arms (equation 3), and to look at the geodesics in the vicinity of thehinges. Indeed, differential geometry is a high-frequency theory, that does not account for wave-like phenomena.
As a consequence, rays have infinitesimal thickness and are not deviated by hinges, unless they cross them.
We therefore propose to replace, in the Christoffel symbols of equations (11), δ (x) by the approximation
sin kx/πx, the weak limit of which is known to be the DiracR delta function when the wave number k tends
x(sin ku/πu)du. We thus obtain:
towards infinity; and the function sgn(x) by the integral −1 + −∞
Γyxx
Γxyy


R x sin ku = sinπxkx −1 + 2 −∞πu du cos βR x sin ku  sin ky= −1 + 2 −∞ πu du πy cos β(20)

The crucial change introduced by these low-pass versions of the Christoffel symbols is the smooth and gradualtransition from −1 to +1 that occurs across a hinge.

Figure 4. Ergodesics crossing angle arms. Geodesic x10 -x20 crosses arm Ox1 away from the apex and remains
straight; geodesic x30 -x40 crosses arm Ox1 near the apex and is deflected. Green curve corresponds to low-passsign function; red curves correspond to low-pass delta function.
In a Riemannian time-space, the equations of the geodesics are given by [2]:dvk
= −Γkij vi v j
dτ

(21)

where τ is the proper time, vi = dxi /dτ the velocity vector in direction xi , and where dvk /dτ is the accelerationin direction xk In the present case, the only non-zero accelerations are therefore along x1 when crossing the arm
x2 = 0, and along x2 when crossing the arm x1 = 0. We therefore obtain:
dv1
dτ
dv2
dτ


R x sin ku   2 2
= − cos β sinπxkx −1 + 2 −∞πu du v
R x sin ku  sin ky  1 2
= − cos β −1 + 2 −∞ πu du πy v

(22)

As a consequence, a ray that crosses one of the arms far from the hinge carries on in its original direction - ray x10 -x20 in Figure 4 - since the integral of the sinc-function is equal to 1: the total impulse in theneighbourhood of the arm just compensates for the coordinate changes. But a ray that crosses one of the arms
near the hinge, at a distance less than π/k - ray x30 -x40 in Figure 4 -, undergoes an impulse that does not fullycompensate for the coordinate change, with less compensation the nearer it comes to the hinge. In the limit
case where the ray crosses at the hinge, the impulse vanishes and the ray is deviated by half the excess angle.

4

CONCLUSIONS

We have presented a geometrical theory that naturally accounts for scattering on the boundaries of a room. It
introduces Riemannian spaces with negative curvature, which constitute the proper setting for the distributionof images created by non-rectangular rooms with obtuse angles, that is, created by irregular polyhedra with alarge enough number of faces. The crucial factor is the excess angle that arises around specific edges, called
hinges, when first and second order images are considered, as it pilots the metric tensor of the space and all its
geometrical properties, including its curvature.
Using the uncertainty principle, we were able to describe the scattering of wave packets around dihedral
angles. The scattering is proportional to the excess angle, and can be described in terms of the stress-energy
tensor, that is, in terms of energy conservation.
The present theory must now be developed to derive scattering coefficients from the distribution of hinges
around a room. The excess angles are the main factors, complemented by the lengths of the hinges. But asshown in Sections 3.4, wave length also plays a rôle. Proper definition of scattering coefficients has long been
missing for the application of the diffusion equation in Room Acoustics [7], despite some recent attempts [8].

There remains to introduce absorption in the present theory. Due to its geometrical nature, it cannot accountfor losses in its present form. The classical way to introduce absorption, which consider intensity flows inside
the boundaries, cannot be used here without some adaptation. But accounting for losses should not proove
difficult since absorption is easily described with the stress-energy tensor used in Sections 3.2 and 3.3, asshown in [9, 10].

REFERENCES[1] Polack JD, Meacham A, Badeau R, Valière JC. Riemannian space tessellation with polyhedral room images.
Proc 16th CFA; 11-15 April 2022; Marseille, France, 2022.
[2] Linet B. Notes de cours de relativité générale. Paris; 2004-5.
[3] Polack JD, Meacham A, Badeau R, Valière JC. Riemannian space tessellation with polyhedral room images.
hal-03598258; 2022.
[4] Regge T. General Relativity without Coordinates. Nuovo Cimento. 1961;XIX:558-571.
[5] Morse PM, Ingard KU. Theoretical Acoustics. New York, USA; McGraw-Hill; 1968.
[6] Stephenson UM. An energetic Approach for the Simulation of Diffraction within Ray Tracing Based on the
Uncertainty Relation. Acta Acust united Ac. 2010;96:516-535.
[7] Picaut J, Simon L, Polack JD. A mathematical model of diffuse sound field based on a diffusion equation.
Acta Acust united Ac. 1997;83(4):614-621.
[8] Gül ZS. Exploration of room acoustics coupling in Hagia Sophia of Istanbul for its different states. J Acoust
Soc Am., 2021;149(1):320-339.
[9] Dujourdy H, Pialot B, Toulemonde T, Polack JD. Energetic wave equation for modelling diffuse sound field- Applications to corridors. Acta Acust united Ac. 2017;103(3):480-491.
[10] Dujourdy H, Pialot B, Toulemonde T, Polack JD. Energetic wave equation for modelling diffuse soundfield - Applications to open offices. Wave Motion. 2019;87:193-212.

ABS-0565

Room shape reconstruction using acoustic super-resolution
Tom SPRUNCK(1,2) , Yannick PRiVAT(1) , Cédric FOY(3) and Antoine DELEFORGE(2)(1) Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France (firstname.name@inria.fr)(2) IRMA, Université de Strasbourg, CNRS UMR 7501, 67084 Strasbourg, France (yannick.privat@unistra.fr)(3) UMRAE, Cerema, Univ. Gustave Eiffel, Ifsttar, Strasbourg, 67035, France (cedric.foy@cerema.fr)

ABSTRACT
Room geometry reconstruction methods based on the image source model and room impulse response (RIR)
measurements are still underdeveloped. Indeed, determining the 3D locations of the image sources is a difficult
non-linear inverse problem in acoustics. The method presented here relies on super-resolution, a frameworkfrom the signal processing community. The fundamental objective is to reconstruct the locations and amplitudes
of a linear combination of Dirac masses by using linear measurements. We consider here the linear operator
mapping a weighted sum of 3D Dirac measures – modeling the image sources – to the corresponding discrete
RIR. The reconstruction can be achieved by solving a relaxed convex problem in the space of Radon measurescalled BLASSO. Usual reconstruction guarantees do not directly apply here due to the singularity of the operator
at each microphone position. We adapt and apply the Sliding Frank-Wolfe algorithm, which allows for an
efficient numerical resolution of this problem. Experimental results on noisy synthetic data demonstrate theability of the method to find the locations of hundreds of image sources in a variety of room configurations.

Keywords: Acoustic reflectors, room shape, sound field, super-resolution, convex optimization

1

INTRODUCTION

The pressure field p : R3 × [0, ∞) → R resulting from a perfect omni-directional point source placed at a location
3r src
0 ∈ R inside an empty room and emitting a perfect impulse obeys an inhomogeneous wave equation withcomplex boundary conditions. However, Allen and Berkley [1] showed that in the case of a rectangular roomwith rigid, perfectly specular walls this pair of partial differential equations is equivalent to the following freefield wave equation:
+∞
1 ∂2p
r
r
(r
,t)
−
∆p(r
,t)
=
(1)
∑ ak δr srck (rr ) δ0 (t).
c2 ∂t 2k=0
The boundary conditions are removed by considering instead a source term composed by a sum of image
sources δr src
that simultaneously emit an impulse at the initial time t = 0. Each image source models a specularkreflection path of the original impulse with regards to the walls of the room. Technically this equivalence only
holds for perfectly reflective surfaces and ak = 1, although it can be generalized by weighing images sources
with coefficients ak ∈ [0, 1] to account for a proportion of sound energy absorbed by the walls.
The image sources are constructed iteratively by applying to the original source a sequence of symmetries
with respect to the walls of the room, thus the 3D locations of image sources contain the information necessary
to find the dimensions of the room. However, reconstructing the locations r src
k from punctual and discrete-timemeasurements of the sound field p (i.e a Room Impulse Response or RIR) remains a difficult inverse problem.

sesd sources10(arbitrary unit)

1.5

target RIR
reconstructed RIR

5

1
0

0.5
-5

0

-15
-10
-5

−0.5
10

20
t (ms)

30

-10

40

-15
-10

(a) Part of a RIR at one microphone

-5

0

5

10

15

(b) 3D plot of the target and reconstructed sources

Figure 1. Simulated experiment: 32-channel spherical microphone array with diameter 16.8 cm, fs = 16 kHz,
Tmax = 50 ms and no noise.

2

INVERSE PROBLEM

Equation (1) can be further generalized to an arbitrary source mass distribution ψ, yielding
1 ∂2p(rr ,t) − ∆p(rr ,t) = ψ(rr )δ0 (t),c2 ∂t 2(2)where ψ belongs to the space M (R3 ) of Radon measures. Following a classical development, the generalsolution of equation (1) is given by a spatial convolution with a well known Green function:p(rr ,t) = (G(·,t) ∗ ψ)(rr ) =δ (t − ∥rr − r ′ ∥2 /c)ψ(rr ′ )drr ′ .
4π∥rr − r ′ ∥2r ′ ∈R3Z(3)

The discrete multichannel RIR measured by M microphones placed at locations {rr micm }m∈J1,MK inside theroom can then be expressed asxm [n] = (κm ∗ p(rr mic
m , ·))(n/ f s ), n ∈ J0, N − 1K.

(4)

Here ∗ denotes time-domain convolution, fs is the microphones’ sampling frequency and κm : R → R aretime-domain filters that model the response of the source and the microphones. By combining this relation withequation (3) we obtain the complete expression of the discrete observation vector x ∈ RMN resulting from asource distribution ψ:Zmic
def κ n/ f s − r m − r 2 /cxm [n] =γm,n (rr )dψ(rr ) = ⟨γm,n , ψ⟩ where γm,n (rr ) =.
(5)
4π ∥rr micr ∈R3
m − r ∥2This allows us to define a linear observation operator Γ : M (R3 ) → RMN that maps a source distribution ψ
to its corresponding observation vector x . We now can particularize (5) to the discrete image sources distribution
def(rr ) to get the measured RIR:ψa ,rr src (rr ) = ∑Kk=0 ak δr src
k


src
κm (n/ fs −∥r mic
m −rr k ∥2 /c)
KKsrc
x = Γ ψa ,rr src = ∑k=0 ak γ (rr k ) = ∑k=0 ak∈ RMN .
(6)src
4π ∥r micm −rr k ∥2m,nRecall (%)

100
80
60

8 kHz
16 kHz24 kHz32 kHz

40
20
0

1

2

3

4
Source order

5

6

7

8

Figure 2. Recall in function of source order for varying sampling frequency for a 32-channel spherical microphone array with diameter 16.8 cm and no noise
Note that the infinite image source distribution can be accurately approximated by a finite number K ofimage sources as higher order image sources have a negligible impact on the measured RIR, due to repeatedabsorptions and increasing source-microphone distances.
Let M∗ (R3 ) ⊂ M (R3 ) denote the subset of sparse Radon measures of the form ψa ,rr src with K ∈ N, a ∈K+1
R+
and r src ∈ (R3 )K+1 , i.e. measures that are finite positive combinations of spikes. The inverse problem
of recovering the amplitudes and positions of these spikes given noisy observations x belongs to a generalclass of problems where the goal is to recover the continuous locations of a set of spikes given discrete linear
observations over their measure [2, 4, 3, 5].
Rather than solving a non-convex optimization problem on the amplitudes and positions of the image
sources, we follow the approach in [3] and consider a convex relaxation to the whole space of Radon measures (BLASSO):
1(7)
argmin ∥xx − Γ ψ∥22 + λ ∥ψ∥TV ,3
ψ∈M (R ) 2
where λ > 0 is a regularization parameter.
Strictly speaking, Γ is not well defined because γ is singular at each microphone position. These singulardef
micities can be ignored by changing the integration domain in (5) to R3ε = R3 \ ∪M
m=1 B(rr m , ε) for a fixed ε > 0and by only considering measures ψ ∈ M (R3ε ), without altering the convexity of the optimization problem.

3

NUMERICAL RESOLUTION AND RESULTS

In order to solve (7) numerically, we adapt the Sliding Frank-Wolfe algorithm proposed in [3], which is briefly
reviewed below.
(i) ∈ (R3 )Qi the lists of Q spike amplitudes and positions estimated ati
Let us denote by a (i) ∈ RQi
+ and r(0)(0)
iteration i, with a = r = 0.
/ At iteration i + 1, the following four steps are performed:• Step 1: A new spike location r Qi +1 is first added to r (i+1) by maximizing the following dual, non-convex
def

objective based on the current residual y (i) = x − Γ ψa (i) ,rr (i) :
h
i
def
(i)
max η (i) (rr ) = Γ ∗ (yy(i) ) (rr ) = ∑ ym [n]γm,n (rr )
r ∈R3

(8)

m,n

where Γ ∗ denotes the Hermitian adjoint of Γ .
• Step 2: The whole list of amplitudes a (i+1) is updated by minimizing the cost function in (7) over a only.
This amounts to a classical non-negative LASSO convex optimization problem for which efficient solvers areavailable.
• Step 3 (Sliding): The value of the cost function in (7) is further decreased by jointly refining all the values
in a (i+1) and r (i+1) through non-convex local search.

• Step 4: The spikes whose amplitudes are lower than a threshold αmin are removed from a (i+1) and r (i+1) .
The main modification applied to the original algorithm concerns the first step, spike finding. Solving thismaximization problem requires choosing an accurate initial guess for the optimization algorithm (BFGS). Thisis usually achieved by first solving the problem over a discrete grid covering the entire 3D search space, whichis computationally intractable here. We instead use the time signals to compute an approximate time of arrivalat each microphone. This allows us to reduce the search space to spheres centered around the microphones,
which are consequently meshed with a finite grid. Step 3 is also only applied once at the end of the algorithm,
as the large number of spikes to recover makes it computationally intensive.
We apply this algorithm to a set of 200 randomly generated rectangular rooms. We consider here a compact
spherical array of 32 omni-directional microphones that all carry the same response, an ideal low-pass filter
with cutting frequency fs /2, i.e κm = κ : t 7→ sinc(π fst). The source and array are randomly located in eachroom, with a minimum separating distance of 1 m from each other and to the walls. We apply operator Γ
to the theoretical image sources distribution to generate the target RIR and add white Gaussian noise. All the
RIRs are finally cut to last 50 ms. The code used to run the experiments is available at https://github.
com/Sprunckt/acoustic-sfw.
Figure 2 shows an example of simulated RIR, the image sources found by the algorithm and the corresponding time signal for a given microphone. We perform an experimental study of the impact of noise, microphonearray diameter and sampling frequency on the accuracy of the reconstruction. We observe that for a great
enough sampling frequency and array diameter we recover over 90% of the observable image sources with anangular error lower than 2 °. All of the 200 original sources were correctly located, and nearly every first ordersource.

4

CONCLUSIONS

We introduced a new method to recover the continuous 3D positions and amplitudes of all audible image
sources given the early part of a discrete-time multichannel room impulse response from a compact microphonearray. Although there is room for improvement on the algorithmic side, simulated experiments suggest that under idealized conditions, namely, no noise, omni-directional point source and microphones with known responses
and perfectly specular room boundaries the inverse problem is essentially solved. Further theoretical investigations are needed to understand the conditions under which existence, uniqueness and stability guarantees can beestablished for problem (7).

ACKNOWLEDGEMENTS
This work was made with the support of the French National Research Agency through project DENISE (ANR20-CE48-0013).

REFERENCES[1] J. B. Allen and D. A. Berkley. Image method for efficiently simulating small-room acoustics. Journal of the
Acoustical Society of America, 65:943–950, 1976.
[2] E. J. Candès and C. Fernandez-Granda. Towards a mathematical theory of super-resolution. Communications
on pure and applied Mathematics, 67(6):906–956, 2014.
[3] Q. Denoyelle, V. Duval, G. Peyré, and E. Soubies. The Sliding Frank-Wolfe Algorithm and its Application
to Super-Resolution Microscopy. Inverse Problems, 2019.
[4] V. Duval and G. Peyré. Exact Support Recovery for Sparse Spikes Deconvolution. Foundations of Computational Mathematics, 15(5):1315–1355, 2015.
[5] Y. Traonmilin and J.-F. Aujol. The basins of attraction of the global minimizers of the non-convex sparse
spike estimation problem. Inverse Problems, 36(4):045003, 2020.

ABS-0567

Audio InpaintingPeter BALAZS(1) , Georg TAUBÖCK(2) , Shristi RAJBAMSHI, Nicki HOLIGHAUS(1)(1) Acoustics Research Institute, Austrian Academy of Sciences, Austria(2) Institute of Telecommunications, TU Wien, Austria

ABSTRACTThe objective of audio inpainting is to conceal missing or heavily distorted segments in an audio signal. Thisis ideally done by reconstructing the original signal or, at least, by inferring a meaningful surrogate signal. We
present here a novel approach applying sparse modeling in the time-frequency (TF) domain which was recently
published.
In particular, we devise a dictionary learning technique which learns the dictionary from reliable parts around thegap with the goal to obtain a signal representation with improved TF sparsity. The presented method is based ona basis optimization technique that deforms a given Gabor frame such that the sparsity of the analysis coefficientsof the resulting frame is maximized (with respect to a certain sparsity measure). Furthermore, we modify theSParse Audio INpainter (SPAIN) such that it is able to exploit the improved TF sparsity and—in turn—benefits
from dictionary learning. As an alternative, we combine dictionary learning with weighted ℓ1 -minimization
adapted for audio inpainting to compensate for the loss of energy within the gap after restoration. Our experiments demonstrate that the developed methods achieve significant gains in terms of signal-to-distortion
ratio (SDR) and objective difference grade (ODG) compared with several state-of-the-art audio inpainting techniques.

Keywords: Audio inpainting, time-frequcny, Gabor dichtionary, compressive sensing, basis optimization,
convex optimization, Alternating Direction Method of Multipliers (ADMM)

1

INTRODUCTION

Audio Inpainting [1] is a technique that fills missing or disturbed information in an audio signal. Usually,
consecutive samples are considered - a "hole" or gap. Such techniques were first used in image processing- hence the name - and later were also applied to audio signals. Early methods exploiting sparsity in thetime-frequency domain were only able to cover short holes, approximately up to 10 ms. Here, due to the quasistationary nature of sound, one can expect that an "interpolation" of the conditions before and after the gap will
give very satisfying results. Hence, it is reasonable that the original signal can be recovered. We note that the
LPC method from [2] has been applied successfully to this scenario.
For long gaps, above 100 ms, reconstruction of the true original signal is by no means feasible. The
distribution of audio parameters will, most likely, change drastically in this period. Thus, one has to aim ata "decent" inpainting - meaning a perceptual or information-related reasonable fitting. Here let us mention theapproach in [3], where a similarity graph - comparing the parts around the gaps to other parts of the signal is used, or [4], based on generative neural networks.
In this contribution we focus on the intermediate case, i.e gaps between 10 and 100 ms, not covered by themethods mentioned above. We present the methodology introduced in [5] and [6]. The key ideas are the following: First, a time-frequency dictionary - a Gabor frame - is used with an optimization approach. Here, sparsity
is not promoted in both directions, but rather only in the frequency domain, while simultaneously controllingthe maximal values in the time domain. Optimization is performed using an Alternating Direction Method ofMultipliers (ADMM) [7] solver. Alternatively, we perform weighted ℓ1 -minimization applying the ChambollePock (CP) algorithm [8]. Both approaches avoid or at least reduce the energy loss problem. Secondly, we use

Gabor Parameters

Used Value

window g

Hann window

window length wg

2800 samples (∼64 ms)

hop size a

700 samples

number of modulations M

2800

Table 1. Gabor Parameters
a basis optimization technique to adapt the dictionary to our signal. Discussions with other researchers lead us
to believe that the usage of perturbed Gabor atoms - instead of a completely "free", i.e unstructured, learned
dictionary - seems to be the crucial step that facilitates the mehotd’s success.
We will present the results of our algorithms. But, in the spirit of the special session “mathematical methods
in acoustics”, we further elaborate on some of the mathematical details.

2

SIMULATION AND RESULTS

In this section, we compare the audio inpainting performance of modified SPAIN combined with dictionary
learning, (A-SPAIN-LEARNED. [5]), and weighted ℓ1 -minimization combined with dictionary learning, (CPLEARNED, [6]), with the original methods A-SPAIN and CP, presented in [9] and [10], respectively. We
also compare our approaches with other audio inpainting techniques such as A-SPAIN-MOD [5], i.e., modified
SPAIN without dictionary learning, and JANSSEN [2]. The letter ‘A-’ indicates that an algorithms optimizes
analysis sparsity[11, 12]. Note that A-SPAIN uses a frame-wise DFT dictionary, whereas A-SPAIN-LEARNEDand A-SPAIN-MOD use a (deformed) Gabor dictionary. Following common practice in time-frequency audio
processing, the dictionary is four times overcomplete in either case. JANSSEN uses an auto-regressive modelwith the same parameters as in [5, 6, 10].
As a means to compare the performance, we use the signal-to-distortion ratio (SDR) [1] defined as,
SDR(xorig , xinp ) = 10 log10∥xorig ∥22∥xorig − xinp ∥22[dB],

where xorig and xinp denote original and inpainted signal within the gaps, respectively. Note that higher SDR
values imply better signal restoration.
For our experiments, we used the same ten music signals from the EBU SQAM dataset [13] as in [5, 6, 10],in order to allow for a valid and comparable assessment. All considered signals were sampled at 44.1 kHz andrepresent different levels of sparsity with respect to the original Gabor dictionary. In each test instance, theinput was a signal with 5 gaps at random positions. The lengths of these gaps ranged from 5 ms up to 85 ms.
For fixed lengths, the results over all ten signals containing the 5 gaps were averaged.
Furthermore, we also calculated the PEMO-Q values [14], which utilize a human auditory system model
to assess the quality of the restored signal. Thus, the PEMO-Q criterion is closer to the subjective evaluation
than the SDR. The PEMO-Q model provides two quantities quantifying overall perceptual similarity: perceptual
similarity measure (PSM) and the more commonly used objective difference grade (ODG). For simplicity, we
restrict to ODG in this contribution. ODG quantifies the perceptual impact of audio artifacts in the restored
signal and its value ranges from −4 (very annoying) up to 0 (imperceptible). Hence, ODG can be interpreted
as the degree of perceptual similarity of xorig and xinp .
Throughout our experiments, we used the fast implementation of Gabor transforms available in the LTFAT
toolbox [15, 16], and adopted its time-frequency conventions. The Gabor parameters used in our experiment aresummarized in Table 1. For the remaining parameters, we refer the reader to [6].

0
20
-0.5
-1

15

ODG

SDR [dB]

-1.5
10

-2
-2.5

5
CP
CP-LEARNEDA-SPAIN-MOD
A-SPAIN-LEARNEDA-SPAIN
JANSSEN

0CP
CP-LEARNEDA-SPAIN-MOD
A-SPAIN-LEARNEDA-SPAIN
JANSSEN

-3
-3.5

-5

-4
5

15

25

35

45

55

65

75

85

5

15

25

35

45

55

gap length [ms]gap length [ms]

(a)

(b)

65

75

85

Figure 1. Performance comparison of different audio inpainting algorithms in terms of (a) SDR and (b) ODG
for varying gap lengths.

Fig. 1(a) shows the inpainting performance in terms of SDR of all tested algorithms. We can observe that
CP-LEARNED performs significantly better than CP, i.e., the original weighted ℓ1 -minimization method. Even
more, it beats all other methods except A-SPAIN-LEARNED, which is slightly superior for gap lengths below 45 ms. Remarkably, CP-LEARNED even outperforms the otherwise best performing A-SPAIN-LEARNED
algorithm for longer gaps (≥ 45 ms) although the observed improvements are minor.
Fig. 1(b) shows the overall inpainting performance in terms of ODG. For smaller gap lengths (≤ 35 ms),
the performance of CP-LEARNED is superior to CP and a bit worse compared to other competing methods,
whereas for longer gap lengths (≥ 45 ms), CP-LEARNED’s performance relative to the other methods improves,
exceeding all the other methods except A-SPAIN-LEARNED for gap lengths above 65 ms.

ACKNOWLEDGEMENTS
This work was supported by Vienna Science and Technology Fund (WWTF) grant MA16-53 (INSIGHT) as
well as by the Austrian Science Fund (FWF) projects I 3067-N30 (MERLIN), P 34624 (LoFT), and P 34922(NoMASP).

REFERENCES[1] A. Adler, V. Emiya, M. Jafari, M. Elad, R. Gribonval, and M. Plumbley, “Audio inpainting,” IEEE Trans.
Audio, Speech, and Language Processing, vol. 20, pp. 922–932, March 2012.
[2] A. Janssen, R. Veldhuis, and L. Vries, “Adaptive interpolation of discrete-time signals that can be modeled
as autoregressive processes,” IEEE Trans. Ac., Sp., Sign. Proc., vol. 34, no. 2, pp. 317–330, 1986.
[3] N. Perraudin, N. Holighaus, P. Majdak, and P. Balazs, “Inpainting of long audio segments with similarity
graphs,” IEEE Transactions on Audio, Speech and Language Processing, vol. 26, no. 6, pp. 1079–1090, 2018.
[4] A. Marafioti, P. Majdak, N. Holighaus, and N. Perraudin, “Gacela: A generative adversarial context encoder for long audio inpainting of music,” IEEE Journal of Selected Topics in Signal Processing, vol. 15,
no. 1, pp. 120–131, 2020.

[5] G. Tauböck, S. Rajbamshi, and P. Balazs, “Dictionary learning for sparse audio inpainting,” Journal of
Selected Topics in Signal Processing, vol. 15, no. 1, pp. 104–119.
[6] S. Rajbamshi, G. Tauböck, N. Holighaus, and P. Balazs, “Audio inpainting via ℓ1 -minimization and dictionary learning,” in 2021 29th European Signal Processing Conference (EUSIPCO), pp. 2149–2153, 2021.
[7] S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein, et al., “Distributed optimization and statistical learning
via the alternating direction method of multipliers,” Foundations and Trends® in Machine learning, vol. 3,
no. 1, pp. 1–122, 2011.
[8] A. Chambolle and T. Pock, “A first-order primal-dual algorithm for convex problems with applications toimaging,” Journal of mathematical imaging and vision, vol. 40, no. 1, pp. 120–145, 2011.
[9] O. Mokrỳ, P. Záviška, P. Rajmic, and V. Veselỳ, “Introducing SPAIN (SParse Audio INpainter),” in 201927th European Signal Processing Conference (EUSIPCO), pp. 1–5, IEEE, 2019.
[10] O. Mokrý and P. Rajmic, “Audio inpainting: Revisited and reweighted,” IEEE/ACM Transactions on Audio,
Speech, and Language Processing, vol. 28, pp. 2906–2918, 2020.
[11] M. Elad, P. Milanfar, and R. Rubinstein, “Analysis versus synthesis in signal priors,” Inverse problems,
vol. 23, no. 3, p. 947, 2007.
[12] S. Kitić, N. Bertin, and R. Gribonval, “Sparsity and cosparsity for audio declipping: a flexible non-convex
approach,” in International Conference on Latent Variable Analysis and Signal Separation, pp. 243–250,
Springer, 2015.
[13] EBU, “EBU SQAM CD: Sound quality assessment material recordings for subjective tests.”[14] R. Huber and B. Kollmeier, “PEMO-Q–A new method for objective audio quality assessment using amodel of auditory perception,” IEEE Transactions on Audio, Speech, and Language Processing, vol. 14,
no. 6, pp. 1902–1911, 2006.
[15] P. L. Søndergaard, B. Torrésani, and P. Balazs, “The linear time frequency analysis toolbox,” Int. J. Wavel.,
Multires. a. Inf. Process., vol. 10, no. 04, p. 1250032, 2012.
[16] Z. Průša, P. L. Søndergaard, N. Holighaus, C. Wiesmeyr, and P. Balazs, “The Large Time-Frequency
Analysis Toolbox 2.0,” in Sound, Music, and Motion (M. Aramaki, O. Derrien, R. Kronland-Martinet, andS. Ystad, eds.), Lecture Notes in Computer Science, pp. 419–442, Springer International Publishing, 2014.

ABS-0587

Active subspace realization for accelerated training of parametric
reduced-order models in vibroacoustics
Harikrishnan K. SREEKUMAR(1) , Yannik HÜPEL(1) , Sabine C. LANGER(1)(1) Technische Universität Braunschweig, Institute for Acoustics, Germany, hk.sreekumar@tu-braunschweig.de

ABSTRACTParametric model order reduction techniques have seen significant applications in the field of multi-query problems like uncertainty quantification, optimization and sensitivity analysis to yield faster evaluation of large-scalemodels. However, capturing parametric effects of a large number of parameters is still challenging owing to thecurse of dimensionality. This is the case for practical vibroacoustic models where for instance a range of material parameters have to be considered. As a consequence, the offline or the training phase in a parametric modelorder reduction algorithm becomes computationally demanding due to the large training set. In this contribution,
we investigate and present the results of reducing the high-dimensional parameter domain from vibroacoustic
models to a low-dimensional subspace by applying the method of active subspaces. The method identifies a setof parameters in the low-dimensional subspace which primarily influence the system’s response. We present theexistence of such active subspaces for classical vibroacoustic problems reduced in a desired parametric domain.
Finally, the training in such active subspaces has shown to yield a significant reduction in search space andtraining cost.

Keywords: Parametric model order reduction, Active subspaces, Vibroacoustics

1

INTRODUCTION

At an early design phase, simulation of vibroacoustic systems delivers insights to optimizing design parameterswith the aim of improving their acoustic characteristics among other properties. However, every simulation ata new parameter setting is computationally demanding. One way to circumvent the high computational cost is
to approximate the frequency response functions (FRF) of such large-scale models using model order reduction(MOR) methods to yield a frequency-domain reduced-order model (fROM) valid for a desired frequency region.
Krylov-subspace projection techniques have shown an efficient reduction of such large-scale models in the frequency domain. In addition, one can include parametric effects to the reduction framework with the help ofparametric model order reduction (PMOR) methods [2]. As a result, parametric computations can be efficiently
performed with a PMOR generated surrogate model.
The offline or training phase of PMOR utilizes adaptive algorithms, such as the greedy algorithm, to obtainan accurate parametric reduced-order model (pROM). In the context of projection-based MOR techniques, arange of options are available to finally yield an accurate pROM spanning a desired parametric domain. The
global basis approach resulting in a single projection basis is less reliable for high-dimensional problems. Consequently, local basis approaches became popular where the system’s response at a desired parametric point isobtained using information from a number of local fROMs with interpolation or regression techniques. The
most popular interpolation techniques include interpolating underlying subspaces [1], reduced-order model matrices, output responses or transfer functions and pole-matching [13, 10]. In this contribution, we choose thislocal approach by interpolating multiple FRFs obtained from locally generated fROMs.
The cost of training is an important factor when it comes to practical high-dimensional parametric problems.
In the context of parametric computations and surrogate modelling, the problem of the curse of dimensionality
is infamous, which makes it infeasible to consider a large number of parametric variations thereby increasing thetraining cost exponentially. A classical full grid computation is possible only for a small number of parameters

(n p < 3). For a higher number of parameters (n p ≥ 3), there are various approaches to circumvent the curseof dimensionality; with the method of sparse grids [4, 15] being one popular among them. A sub-samplingstrategy to reduce the computational cost for large parameter time-dependent problems can be referred to [5].
Altogether, the requirement to deal with high-dimensional problems is of great demand and is an active field of
research.
Another approach is to perform dimensionality reduction, where the number of parameters is reduced by
exploiting the hidden anisotropy of a high dimensional parametric problem. With the sensitivity analysis one
can identify parameters which have the strongest influence on the system’s response. As a result, it is possible
to train a pROM by finding the most important and contributing fROMs along the direction of highly ranked
parameters based on their sensitivity measures. A disadvantage of such an approach is that they reveal theprominent parametric (univariate-)directions which are only aligned with the natural parameter coordinate direction. On the other hand, the active subspace method (ASM) [7, 6] identifies the prominent directions which
may not be aligned with the parametric dimension or in other words oriented multi-dimensionally. The mainidea is to establish a rotated coordinate system, where the parametric surface variation is now aligned to (1)vary the most along certain new rotated coordinates called the active parameter coordinates and (2) vary theleast or invariant along the inactive parameter coordinates. The rotated coordinates enable one to define a lowdimensional subspace called the active subspaces constructed with only the active parameters. As a result, it
is now possible to investigate the parametric computations within this low-dimensional parameter space. The
method so far has shown significant advantages in many applications for instance [17, 18].
In the context of PMOR, this means that the reduced dimensionality with active subspaces has greater potential to lower the cost of PMOR training and narrow the search space to this low-dimensional active subspace
to find the most contributing fROMs characterizing the entire parametric domain of interest; thereby offering apromising method to approach the curse of dimensionality. In this contribution, we present results and insights
which imply the existence of a low-dimensional active subspace for vibroacoustic problems and their advantages
in performing a PMOR training phase with reduced computational cost.
The paper is structured to firstly introduce the considered vibroacoustic problem in Section 2 and the essential mathematical concepts of PMOR in Section 3. The theory of active subspaces based on gradient evaluations
is detailed in Section 4. Postliminary, Section 5 elaborates on the methodology and algorithms for performing
PMOR training in active subspaces which are the main contribution of the paper. The advantages and robustness of the method are then examined by application to a generic vibroacoustic model of a coupled plate-fluid
cavity presented in Section 6. Finally in Section 7, we present our conclusions and final remarks.

2

VIBROACOUSTIC PROBLEM DEFINITIONIn this paper, we deal with typical vibroacoustic problems for modelling structure-borne and air-borne soundwave propagation. The basic system of equations is obtained from the finite element method (FEM), approximating the system behaviour of large and complex domains. The core problem in consideration showcases aclassical vibroacoustic problem, a strongly coupled fluid-structure system. The system represents a range ofpractical examples where one is interested to estimate the sound radiating from a vibrating structure such as theestimation of noise in a room due to vibration of surrounding walls. A more complex example is an aircraft
cabin subjected to external loading on the fuselage exterior in order to assess the cabin noise [3]. The resulting
FEM system of equations of the coupled problem can be expressed as:     



D
0
K
−K
x
F
M
0
s
s
c
s
 + iω +  s  =  s  ⇒ −ω 2 M + iωD + K x = F,
−ω 2 
(1)
0 Df
0
Kf
xf
0
KcT M fwhere M, D, K ∈ Cn×n are the mass, damping and stiffness matrix respectively. The system matrices and vectors
with subscripts s and f denote the structural and fluid domain. Excitation is considered for the vibrating structure with the load Fs . In our case, the corresponding state or modelled degrees of freedom in xs and x f accountto the deflection and rotation of the structural nodes and acoustic pressure for the nodes in the fluid medium. In
addition, the term Kc denotes the coupling matrix accounting for the strong coupling between the two domains.
Finally, ω = 2π f denote the angular frequency in radians per second with f the frequency in Hertz (Hz).

3

PARAMETRIC MODEL ORDER REDUCTION

The basic system of equations presented in (1) is often computationally demanding to solve especially forsystem evaluations at higher frequencies owing to the wave resolving nature of FEM. This system of equations
is considered as the large-scale model in upcoming discussions, which is subjected to MOR procedure to yield
faster system evaluations.
Consider the essential parameter variables represented as p ∈ P n p , where p = [p1 , p2 , · · · , pn p ] contains thevarious parameters and the parameter space P is, for our problems, a bounded domain normalized to [−1, 1].
The parametric system of equations for the dynamic steady-state analysis in the frequency domain can be thenexpressed as:−ω 2 M(p) + iωD(p) + K(p) x(ω, p) = F(2)y(ω, p) = Cx(ω, p).
Equation (2) is similar to the dynamic system of equations presented in (1) with the parameter list p containing the underlying material and design parameters. For the numerical examples considered in this paper, thematerial parameters include the variables occurring from a viscoelastic isotropic material behaviour assignment
for the structural domain and a damped Helmholtz domain for the fluid-cavity. We discuss the parameters indetail within Section 6. For simplicity, we consider a single-input-single-output configuration where F ∈ Cn×1 is
the input or loaded excitation to the system, y(ω, p) = Cx(ω, p) is the output FRF with C ∈ C1×n and x(ω, p) is
the state vector. The superscript T denotes the hermitian transpose. An extension towards the practical multipleinput-multiple-output configuration for the presented work in this paper is straight forward. The output FRF of
the above parametric system reads:y(ω, p) = C(−ω 2 M(p) + iωD(p) + K(p))−1 F.
With projection-based techniques, dynamic systems in the frequency domain can be approximated with aprojection basis V (p) ∈ Cn×r and the resulting FRF yielded from the reduced system reads:yR (ω, p) = CR (−ω 2 MR (p) + iωDR (p) + KR (p))−1 FR
where the reduced-order matrices obtained after Galerkin projection can be expressed as [·]R = V (p)T [·]V (p)with [·] = {M(p), D(p), K(p)}, CR = CV (p), FR = V (p)T F(p). The projection basis V is orthogonal as a resultof iterative Arnoldi algorithm with Gram-Schmidt orthogonalization [16]. Consequently, the computations are
performed faster with reduced systems MR (p), DR (p), KR (p) ∈ Cr×r , FR ,CR ∈ C1×r where r  n.
Using global PMOR approaches, one tends to find a single V (p) to span the desired parametric domain. On
the contrary, with local PMOR approaches, there are multiple local projection basis V (p1 ),V (p2 ), · · · ,V (pn ) at
various parameter points and computes V (p) at any desired parameter point by interpolation or regression. We
refer to [2] for more background on PMOR and [8] for projection-based MOR with Krylov subspaces.
In this contribution, we use the state-of-the-art second-order Krylov subspaces expressed as:V ∈ colspan −K̃ −1 D̃, −K̃ −1 M̃, −K̃ −1 Fand multi-point Padé approximation to yield accurate fROMs within the desired frequency range where K̃ =−ω 2 M + iωD + K, D̃ = 2iωM + D and M̃ = M. A detailed discussion on the same for vibroacoustics can befound in [16].
A range of adaptive algorithms are available to yield an accurate pROM. Following the conventional termsof offline and online phases in PMOR, in this paper, we refer to the offline phase also as the training procedure. During the training procedure, we apply suitable adaptive algorithms like the greedy algorithm to actively
identify a small set of local fROMs which, in turn, can be used to describe the parametric response. On theother hand, the online phase refers to the faster response evaluation procedure at any desired parametric pointwith the trained pROM using interpolation or regression.
The adaptive algorithms use a suitable error estimator or an error indicator to observe the convergence of theyielded pROM and to choose new local fROM(s) in further iterations. We use a relatively computationally cheap

error estimate of normalized relative frequency difference (NRFD) to access the convergence of interpolated
pROM by comparing the natural frequencies of the respective FRFs. The maximum NRFD error is chosen for
the frequency domain of interest, expressed as:
εNRFD = maxi

ωr,i − ω̂r,i
ωr,i

(3)

where ωr,i represents the reference natural frequencies obtained from a direct fROM evaluated FRF yR andω̂r,i represents the compared natural frequencies obtained from the pROM online evaluated FRF ŷR for a given
parametric point. The success of using this error estimate is presented later in Section 6. While NRFD checks
the quality of a pROM with the occurrence of resonances in the FRFs, we use absolute error norms to evaluate
the quality of generated fROMs and consequently the FRF itself.

4

ACTIVE SUBSPACE METHOD USING THE GRADIENT BASED APPROACH

The ASM, introduced in [7], is an efficient mathematical tool to perform dimensionality reduction of highdimensional parametric problems. The method is highly suitable for problems where the output varies predominantly in certain parametric directions. Moreover, we present later in Section 6, that the sensitivity indicesfor the considered vibroacoustic models support the existence of strong anisotropy from the parameters. However, strong variability along multi-direction is hard to realize when using sensitivity indices to rank the input
parameters according to the strongest influence. The ASM reorients coordinate directions by rotation so as toalign the strongest and the weakest variations to the new rotated coordinates. The rotated subspace can bethen approximated with a set of prominent active subspace parameters thereby constructing the low-dimensional
subspace called the active subspace.
A gradient-based approach is used to construct the active subspace which can then reveal a set of active andinactive parameters. The active parameters are now the coordinates in the active subspace capturing the strongest
influence. On the other hand, the inactive parameters constitute coordinate directions where the output responseis almost invariant. Such a separation of parameters into active and inactive is enabled with mainly threesteps: (1) computing the gradient response of the system, (2) construction and factorization of a covariance-stylematrix using the computed gradients and (3) segregation of active parameters and computing the approximatedlow-dimensional or active subspace. Below we briefly describe these three steps.
The gradient can be used to capture the variability of theh model response. iLet f (p) represent the output
response function for input parameters p ∈ P n p and ∇ p f (p) = ∂∂pf , ∂∂pf , · · · , ∂∂pnf is the corresponding gradient.
12p
To compute the gradient for the whole parametric domain, a Monte-Carlo (MC) integration is performed assuggested in [7], which reads for m MC samples in the parameter space P n p :∇ p f j = ∇ p f (p j ) ∀p j ∈ P n p , j = 1, · · · , m.
A suitable method can be used to compute the gradient at each sampled point. In this contribution, we use
the computationally expensive method of finite differences for gradient computation. The number of samples
considered for the MC integration is commented on later in Section 5.
As a second step, the computed gradient enables the building of a covariance-style matrix C ∈ Cn p ×n p or
termed as the active subspace matrix reads:

 Z
1
(4)
C = E (∇ p f )(∇ p f )T = (∇ p f )(∇ p f )T ρ(p) dp ⇒ C ≈ C̃ = GGT = Σmj=1 (∇ p f j )(∇ p f j )T ,m
where G = √1m ∇ p f1 , ∇ p f2 , · · · , ∇ p fn p . The active subspace matrix in (4) is expressed first in a probabilistic
setting with an underlying density function ρ(p) and then expressed using evaluation with MC integration.
Now, an eigenvalue decomposition is performed on the matrix C̃, such that C̃ = W̃ Λ̃W̃ T , to yield a set of
eigenvalues Λ and corresponding eigenvectors W , which are then partitioned suitably to form:


h
i
Λ1
 , and W = W1 W2 .
Λ=
Λ2

√
Alternatively, a singular value decomposition (SVD) can be performed on the matrix G yielding G = W̃ Λ̃V T .
For the algorithm used in this paper, we follow this procedure with a SVD.
Thirdly, the huge gaps between the eigenvalues obtained from an SVD can be used to perform dimensionality reduction where a partition is done based on arranging the eigenvalues in descending order of theirmagnitudes and observing the most dominant eigenvalues. This strongly relates to the principal component analysis [11] which can be used to discover active subspaces by preserving the components with high variability.
Or in other words, eigenvalues which are almost of zero magnitude can be discarded or be considered inactive.
As a result, diag (Λ1 ) ∈ Cl p and W1 ∈ Cn p ×l p contain the most important eigenvalues and eigenvectors where thenumber of active parameters l p < n p . The eigenvectors W1 define the rotation of the coordinates corresponding
to the most prominent variation. The respective transformation of original parameter coordinates p to the rotatedcoordinates can be achieved with:
φ = W1T p, ψ = W2T p.

(5)

As a result, we have now obtained a set of active and inactive parameters. Using these active parameters
and the transformations in (5), one can perform the parametric training procedure within this low-dimensional
active subspace with less numbers of parameters. Finally, the goal of approximating the system response isto approximate an n p -variate function with an l p -variate function. Hence, the response function in the original
parametric domain is approximated with the dominant l p number of variables. For this approximation in theoriginal parametric domain, firstly we perform k functional evaluations in the active subspace expressed as:
G(φi ) ≈ f (pi ) for pi = W1 φi +W2 ψi ≈ W1 φi and i = 1, · · · , k.

(6)Then, we compute the interpolated or regression response over the original parametric domain of interest using
the above k functional evaluations expressed as:
f (p) ≈ Ĝ(φ )where Ĝ(φ ) is the interpolated or regression surface constructed with k data points in active subspace (φi , f (pi )).
The preference for interpolation and regression techniques in context of PMOR is commented in Section 5.
In (6), the grid transformation from the active subspace to the physical space is approximated with contribution from only the active parameters. Also, the transformation from φ to p is ill-posed [6]. The approximation
introduces an error when the eigenvalues corresponding to the inactive parameters Λ2 are not zero. Hence, theinfluence of the approximation on the overall subspace approximation is fully dependent on the nature of the
problem. In our investigations, this influence is negligible as the eigenvalues λl p +1 , · · · , λn p are small.

5

PROPOSED TRAINING STRATEGY WITH ACTIVE SUBSPACES

The major computational cost in PMOR accounts for the training procedure. A suitable test grid, which isoften a fine grid, is chosen to evaluate the error evolving in each iteration of an adaptive algorithm. For a large
number of parameters, we, therefore, need a very large test grid adding to the curse of dimensionality, making
it impossible to train for high-dimensional parametric problems. In this section, we describe the combined approach of ASM presented in Section 4 within a PMOR framework presented in Section 3. Firstly, an outline of
the computational pipeline is presented. This is followed by the algorithms for computing the active subspaces
and the overall PMOR training using ASM. Finally, we comment on the computational cost.
5.1 Computational pipeline
To illustrate the role of active subspaces in the PMOR training phase, a typical pipeline comparing the conventional training and training in the active subspace is depicted in Figure 1.
A conventional training pipeline with greedy algorithm, marked in Figure 1, involves an iterative procedureof interpolating a set of fROMs, also known as the interpolated grid Ξinterpolated , which essentially forms thepROM. A test or training grid Ξtrain is chosen to check the convergence of the produced pROM in each greedy
iteration. An error measure is computed for Ξtrain and the convergence of pROM is then checked against aGreedy algorithm
Add new
fROM fromtraining grid tointerpolation gridConventional pipeline

Parametricproblem

Active subspace
pipeline

Generationof initial
Generationofinterpolation andinitialinterpolation
training grid for all
andparameters
training gridfor all parameters

Discover active
Discoveractivesubspacesubspace

Interpolation ofInterpolation
of
fROMs
fROMs

Evaluation of
Evaluation
oferrorerror
estimate for training
estimate
grid fortraining grid

No
Yes
Converged?

Converged
pROM

Generationof initial
Generationof initialinterpolation andinterpolation andtraining grid for active
training
grid for active
parametersparameters

Active subspace realization

Figure 1. Flow chart representing both the conventional PMOR training pipeline and PMOR training pipeline
with active subspacesuitable error tolerance. The interpolated grid continues to grow and contain more fROMs from the training
grid until the pROM converges. This training procedure is marked as the conventional pipeline in Figure 1.
In general, the interpolation grid is initially a coarse grid and the training grid accounts for a finer grid.
Using suitable sampling techniques, one can generate a fitting number of parameter points for Ξinterpolated andΞtrain . However, Ξinterpolated and Ξtrain can be huge when the number of parameters are high owing to the curseof dimensionality. Next, we present the pipeline for training with active subspaces to circumvent this problem.
For a PMOR training pipeline with active subspaces, also marked in Figure 1, the aim is to find a lowdimensional active subspace where the parametric response variations are dominant. As a result, training insuch an active subspace reduces the number of parameters and also reduces the search space. As a first step inthe pipeline, we construct the active subspace using the gradient-based approach mentioned in Section 4 so as
to yield a low-dimensional subspace of active parameters. Consequently, Ξinterpolated and Ξtrain are defined on theactive subspace domain with reduced parameters. It is often observed that for problems with strong anisotropyof considered parameters, the active subspace is constructed with very few (one or two) parameters. We observe
this later for the considered vibroacoustic example in Section 6. Considering such a small number of parameters,
it is now affordable to just use the full-grid sampling to proceed with the training. Therefore, in this paper, weconsider full-grid sampling which delivers converging pROMs. However, the usage of sparse grids on the active
parameter grid is still possible for better performance and is interesting for future research. As depicted in
Figure 1, the consequent part of the pipeline is the greedy procedure to finally yield a converged pROM trainedin the active subspace. As a result, one can integrate the active subspace training into a conventional PMOR
algorithm with less effort. The algorithms for this active subspace pipeline are detailed further in Section 5.2.
Commenting on the online evaluations in the parametric domain, we choose two different approaches for (1)while in training procedure and (2) while in post-training evaluations. While in training procedure, we preferinterpolation techniques over regression for better convergence of the greedy algorithm. As already mentioned in
Section 4, the ASM rotates the parametric domain and aligns to a reduced set of active parameters. Training insuch an active subspace evaluates a narrow parametric domain where the response variations are dominant. The
resulting pROM is still valid and accurate for the entire parameter domain of interest, which includes the lessdominant regions. As a result, we require a fitted response surface using regression for a global evaluation of
the response functions outside the active subspace interpolated grid Ξinterpolated . Hence after a successful training
procedure with the interpolation grid Ξinterpolated , a regression surface is computed with the converged Ξinterpolated
as the data points. In this paper, a response surface for global evaluation is constructed using Gaussian process
regression (GPR) using an exponential kernel. For the response functions presented in this contribution, we usethe GaussianProcessRegressor from scikit-learn [14]. Details on the construction of response surfaces
with kriging in the context of ASM can be referred to [7].
5.2 Algorithm
With an understanding of the theory of active subspaces from Section 4 and the pipeline explained in theprevious section, we present the underlying major algorithms in the form of pseudocodes in Algorithm 1 andAlgorithm 2.
Algorithm 1 Algorithm to perform PMOR training procedure based on a greedy approach with active subspacefunction PERFORM P MORT RAINING G REEDYA SM(Parametrized full system Σ, frequency range ω =[ωmin : ωmax ], parameter domain P ∈ [−1, 1], error tolerance σtol )Φ = CONSTRUCTACTIVE S UBSPACE(Σ, m, P, σtol,AS ). See Algorithm 2
Ξtrain = GENERATE S AMPLES(ntrain , Φ, ‘full-grid’). Generates ntrain samples
Ξinterpolation = GENERATE S AMPLES(ninterpolation , Φ, ‘full-grid’). Generates ninterpolation samples
do. Greedy iteration/* Compute interpolated function with current Ξinterpolation */yR (ω, p) = COMPUTE L OCAL ROMS A ND I NTERPOLATE R ESPONSE F UNCTIONS(Σ, Ξinterpolation )ε = []
for all pi ∈ Ξtrain do. Computing error over test grid/* Compute FRF using the interpolated function */ŷR,i (ω, pi ) = yR (ω, pi )/* Compute FRF by building new local ROM */ỹR,i (ω, pi ) = COMPUTE L OCAL ROM A ND G ENERATE R ESPONSE F UNCTIONS(Σ, pi , ω)εi = COMPUTE E RROR NRFD(ỹR,i (ω, pi ), ŷR,i (ω, pi ))if εi < σtol then
. Check for local convergenceΞtrain = REMOVE P OINT F ROM G RID(Ξtrain , pi ). Converged points are removed from gridend ifend forεmax , pnew = argmax(ε). Maximum error point is chosen for interpolation in next greedy iterationΞinterpolation = ADD M AXIMUM E RROR P OINT T O G RID(Ξinterpolation , pnew )while εmax > σtol
. Check for convergence in entire training grid
return Converged interpolated grid Ξinterpolatedend function
Algorithm 1 highlights the major steps in a PMOR training procedure for the proposed approach using
ASM. The ASM training routine includes the construction of the covariance matrix with function CONSTRUC TACTIVE S UBSPACE detailed in Algorithm 2. The inputs to the algorithm are (1) the parametric problem Σ(p)- which are the parametrized FEM system evaluations from an in-house FEM software in our case, (2) the
frequency domain of interest ω = [ωmin : ωmax ], (3) the parameter domain P normalized to scale [−1, 1], and(4) the error tolerance for the greedy algorithm σtol . In every greedy iteration, new local fROMs are added toΞinterpolated from Ξtrain that correspond to the maximum error estimate expressed in (3). Each local fROM inΞinterpolated and Ξtrain , computed with function COMPUTE L OCAL ROMS A ND I NTERPOLATE R ESPONSE F UNCTIONS,
is produced using Krylov-based MOR techniques. For sake of brevity, the algorithm for fROM generation is notelaborated. The authors advise to refer to the classical MOR and PMOR literature for more details. While in
ASM training, it is worth mentioning that the samples are drawn from the constructed active subspace Φ. Alsowith regards to the interpolation of FRFs in Ξinterpolated , we use the linear n-dimensional interpolation capabilityof interpolate.griddata function from scipy in Python.
The function CONSTRUCTACTIVE S UBSPACE to construct the active subspace is further detailed in Algorithm
2. The algorithm performs an evaluation of the parametric functional response gradient using finite differences
and builds the covariance matrix G. Using SVD, the matrix is factorized into eigenvalues and eigenvectors
which are then sorted accordingly to construct the active subspace Φ which are essentially the dominant eigen-values Λ1 and eigenvectors W1 .
Algorithm 2 Algorithm to construct the low-dimensional active subspacefunction CONSTRUCTACTIVE S UBSPACE(Parametrized full system Σ, number of gradient evaluations m, parameter domain P ∈ [−1, 1], tolerance for grouping eigenvalues σtol,AS )ΞMC = GENERATE S AMPLES(m, P, ‘random-uniform’). Generates m samples for MC integration
for all pi ∈ ΞMC do. Computing gradient∇ p fi = COMPUTE G RADIENT(Σ, pi ). Here with a finite difference scheme
end for . Compute covariance matrix
G = √1m ∇ p f1 , ∇ p f2 , · · · , ∇ p fn pW, Σ,V T = SINGULARVALUE D ECOMPOSITION(G)/* Grouping into active and inactive */Λ = Σ2
Λ,W = SORT PAIR D ESCENDING(Λ, W )Λ1 , Λ2 ,W1 ,W1 = [ ]for all Λi ∈ Λ and W:,i ∈ W do
if Λi /Λ0 >h σtol,AS ithen hiΛ1 = Λ1 Λi ; W1 = W1
else
h
i
h
Λ2 = Λ2 Λi ; W2 = W2
end ifend forreturn Φ : {Λ1 ,W1 }end function. Factorization

W:,i
W:,i

i

5.3 Computational costCommenting on the computational cost, the extra cost for the proposed algorithm 1 is the construction of theessential active subspace. The major costs incur from the number of MC samples m used to establish thecovariance matrix in (4), which includes the evaluation of gradients of the original function f (p). We use thefinite-differences method, which adds to the computational cost. However, more affordable gradient computation
methods, such as the adjoint method, are suggested when available. A multi-fidelity approach is presented in[12] to approximate the covariance matrix with both low and high-fidelity gradient functions, thereby reducing
the cost of realizing the active subspaces. However, for the considered example in Section 6, we observe a
requirement of a few sample evaluations - around 100 samples - to sufficiently capture the dominant eigenpairs.
The usage of the above mentioned efficient methods for gradient computation and further optimization regarding
the required number of samples is proposed for further investigations. Regardless, the cost of training with theoriginal grid outweighs the cost incurred with active subspaces.

6

NUMERICAL EXAMPLE

In this section, we present the results of PMOR training with ASM discussed in Section 5 applied to a genericvibroacoustic model. The study is focused on an evaluation in the frequency region of 100 − 400 Hz for every
1 Hz frequency step and to a single-input-single-output configuration for simplicity. The response evaluations
for generated fROMs and gradient computations correspond to the fROMs yielded from respective FEM models
using an in-house vibroacoustic solver coded in C++ for high-performance parallel computing.
The coupled plate-cavity example represents a typical vibroacoustic problem with the respective FEM discretised system mentioned in Section 2. The discretised FE model is illustrated in Figure 2. The model essentially
includes a two-dimensional plate with dimensions 1m × 0.7m strongly coupled to a three-dimensional acoustic
cavity with dimensions 1m × 0.7m × 1.3m. The plate is modelled with the Reissner-Mindlin plate formulationand viscoelastic material properties. Whereas, the acoustic medium is a Helmholtz domain modelling acoustic

wave propagation through a stationary, ideal and compressible fluid. The plate and fluid domain are discretisedwith respective quadratic elements and strongly coupled with conforming meshes yielding a full-order systemdimension of 10395. In total, the parametric response can be described using 9 parameters which are: (a) 5parameters from the plate domain - Young’s modulus (E), Poisson’s ratio (ν), density (ρ), thickness (t) anddamping loss factor (η), and (b) 4 parameters from the fluid domain - the real and imaginary part of speedof sound cf,real , cf,imag and fluid density ρf,real , ρf,imag . The imaginary part for the fluid parameters is accounted
to consider numerical damping in the fluid domain. The nominal values and considered variations of eachparameter are described in Table 1.

Figure 2. Discretised plate-cavity model with plate domain shaded in cyan and fluid-cavity domain shaded ingrey. Also, the input node is marked as red • and output node marked as blue • points.
Table 1. Parameter variations considered for the coupled plate-cavity model. The individual parameter bounds
and nominal values presented in the table are later normalized to scale [−1, 1] for the active-subspace algorithms
dealing with general domains.
DomainPlate

CavityMaterial

Aluminium

Air

Parameter

Units

Nominal value

Variation

Bound

NRFD

E

N/m2

7 · 1010

10%[6.65, 7.35] · 1010

6.26 · 10−3

η

.

0.05

10%

[0.0475, 0.0525]

3.98 · 10−3

ν

·

0.3

10%

[0.285, 0.315]

5.45 · 10−3

ρ

kg/m3

2700

10%

[2565, 2835]

9.79 · 10−3

t

m

0.003

10%

[0.00285, 0.00315]

8.45 · 10−3

cf,realm/s

343.1110%[325.95, 360.26]

7.31 · 10−3

cf,imag

m/s

8.572

10%

[8.144, 9.001]

8.07 · 10−3

ρf,real

kg/m3

1.21

10%

[1.15, 1.27]

6.25 · 10−3

ρf,imagkg/m3

0.030210%[0.029, 0.032]

3.98 · 10−3

The input to the model is a node at (0., 0., 0.) in plate for excitation perpendicular to the plate surface
and output is observed at a node at (1., 0.7, 1.3) at the end of the fluid domain also depicted in Figure 2.
Therefore, the MOR procedure for this model yields a FRF consisting of the acoustic pressure response forthe output node. The aim is to perform accurate PMOR training by interpolating the least number of FRFs
obtained from the respective local fROMs. For the nominal parametric setting, an accurate local fROM of sizer = 90 and absolute error norm 1.3 · 10−10 is generated with expansion points ωEP = 2π {100, 400} rad·s−1 using
Krylov-based MOR techniques presented in Section 3.
For the current parametric problem involving 9 parameters, we deploy Algorithm 1 and thereby Algorithm
2 to perform PMOR training using ASM. In Figure 3, the eigenvalues corresponding to the covariance matrix
evaluated using 100 Monte Carlo integration samples are plotted. It is evident from the figure that for thecurrent model, there is a single parameter direction in the active subspace where the system response is largely
varying. To further support this, we also performed a Sobol sensitivity analysis with 12000 Saltelli samples
and respective fROM evaluations using the SALib python library [9]. The first order Sobol indices are plottedfor each of the parameters over frequency in Figure 4 and are in close accordance with the results in [15].
It is evident that the speed of sound cf,real has a significant influence on the system response for the entire
frequency domain. In summary, the few dominant active parameters exposed with ASM and the sensitivity
measures from the parameter study support hidden anisotropy within vibroacoustic parametric problems which
can be leveraged to achieve significant dimensionality reduction of high-dimensional problems. As a result, we
expect ASM to perform efficiently with the found reduced low-dimensional active subspace. For further training,
we build the active subspace with the first two eigenpairs or active parameters. Considering full-grid samplingin the constructed active subspace using (5), the initial interpolation grid is chosen to contain 8 × 8 = 64 fROMs
and the training grid with 30 × 30 = 900 fROMs. For a given NRFD error tolerance σtol = 10−2 , the greedy
algorithm converges with 303 fROMs in Ξinterpolated for a final maximum NRFD error of 7.99 · 10−3 .
20

Sobol sensitivity indices

1

Eigenvalues, λi

15

10

5

0.8
E
η
ν
ρ
t
c f ,realc f ,imag
ρ f ,realρ f ,imag

0.6
0.4
0.2
0

0
2

4
6
Parameter indices, i

8Figure 3. Plate-cavity model eigenvalue decay plot
evaluated for 100 gradient evaluations based on highfidelity solution response obtained from respective
fROMs. Dominant eigenvalues are visible.

100

150

200 250 300
Frequency (Hz)

350

400

Figure 4. First order Sobol sensitivity indices obtained from Monte-Carlo simulation and pROM online computation for each parameters computed with
12000 Saltelli samples.

We present the efficiency of the proposed training method using the parametric surface plots for the mostdominant and least dominant parameters along with their NRFD error values. The response plot obtained fromPMOR online phase for pROM obtained with training in ASM is compared with the response plot obtainedwith direct full-order model expensive computation. The response plots are computed by varying the parameterof interest for the entire desired parametric domain and other parameter values corresponding to their nominal
values as per Table 1. As a result, the response plots for the PMOR online phase are essentially the FRFs
which are interpolated from the respective pROM. We obtain a maximum NRFD error of 7.31 · 10−3 for thedominant parameter cf,real with respective response plots in Figure 5. Similarly, Figure 6 shows the responseplots obtained for the least dominant parameter ν with a maximum NRFD error of 5.45 · 10−3 . Observing bothof the response plots along with their NRFD error, we see the response obtained from PMOR online phaseaccurately approximating the direct response. The maximum NRDF error for other parameters, see Table 1, arealso within tolerable limits. Hence, using NRFD error for our PMOR-ASM framework has shown to provide
accurate and converging pROM thereby sufficiently capturing the dynamics of the system response.

0.8

0.6

0.6

|y| (Pa)|y| (Pa)

0.8

0.4

0.4

0.2

0.2

0
360

0
360
400
340

400
340

300

300

200

cf,real (m/s)

320

100

200

cf,real (m/s)

Frequency (Hz)

320

100

Frequency (Hz)

0.8

0.8

0.6

0.6

|y| (Pa)|y| (Pa)

Figure 5. Pressure response of the coupled plate-cavity system by varying the most dominant parameter - realpart of speed of sound cf,real , obtained from PMOR online phase (left) and full-order model computation (right)
with corresponding maximum NRFD error 7.31 · 10−3 .

0.4

0.4

0.2

0.2

0
0.32

0
0.32
400
0.3

400

300

0.3

Frequency (Hz)

()

300

200

()

0.28

100

200
0.28

100

Frequency (Hz)Figure 6. Pressure response of the coupled plate-cavity system by varying the least dominant parameter the Poisson’s ratio ν, obtained from PMOR online phase (left) and full-order model computation (right) withcorresponding maximum NRFD error 5.45 · 10−3 .

7CONCLUSIONSIn this paper, we presented a strategy along with algorithms for accelerating a PMOR training procedure using
low-dimensional active subspaces. Using such an approach, we have shown the potential of such a method toaddress the curse of dimensionality by dimensionality reduction using gradient-based active subspace realization.
For the representative vibroacoustic problem presented in this paper, the active subspace method reduces thenumber of parameters to a few number of active parameters - just two in our case. The advantage of using
an active subspace for the training phase of a PMOR algorithm is significant and has yielded an accurate
approximation of parametric response for the entire desired space. Furthermore, we see the requirement for
efficient gradient evaluations of systems involving high dynamics, especially in the higher frequency domain
with high modal density. Also, efficient optimization of presented PMOR algorithms with suitable interpolation
or regression techniques to support large-scale practical applications are interesting for future research.

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[5] S. Chellappa, L. Feng, and P. Benner. A training set subsampling strategy for the reduced basis method.
Journal of Scientific Computing, 89(3):63, 2021.
[6] P. G. Constantine. Active subspaces: Emerging ideas for dimension reduction in parameter studies, volume 2of SIAM spotlights. Society for Industrial and Applied Mathematics, Philadelphia, 2015.
[7] P. G. Constantine, E. Dow, and Q. Wang. Active subspace methods in theory and practice: applications tokriging surfaces. SIAM Journal on Scientific Computing, 36(4):A1500–A1524, 2014.
[8] E. J. Grimme. Krylov Projection Methods for Model Reduction. PhD thesis, 1997.
[9] J. Herman and W. Usher. SALib: An open-source python library for sensitivity analysis. The Journal ofOpen Source Software, 2(9), 2017.
[10] Y. Hüpel, H. K. Sreekumar, and S. C. Langer. Pole-matching technique for parametric model order reduction of second order vibroacoustic systems. In Proceedings of the DAGA Conference, Vienna, Austria,
2021.
[11] I. T. Jolliffe and J. Cadima. Principal component analysis: a review and recent developments. PhilosophicalTransactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 374(2065):20150202,
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[12] R. R. Lam, O. Zahm, Y. M. Marzouk, and K. E. Willcox. Multifidelity dimension reduction via active
subspaces. SIAM Journal on Scientific Computing, 42(2):A929–A956, 2020.
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ABS-0637

Random time-varying filtering with subsampling
Georg TAUBÖCK(1) , Shristi RAJBAMSHI, Peter BALAZS(2)(1) Institute of Telecommunications, TU Wien, Austria, georg.tauboeck@tuwien.ac.at(2) Acoustics Research Institute, Austrian Academy of Sciences, Austria, peter.balazs@oeaw.ac.at

ABSTRACTTime-varying filtering is of crucial importance in many areas of signal processing and, in particular, in audio
processing. From a mathematical point of view, time-varying filters correspond to so-called “frame multipliers”.
Here, the audio signal is first transformed into another signal domain, typically by means of a time-frequency
transformation; then, the resulting transformed signal coefficients are multiplied with a “symbol”; and, finally,an audio signal is re-synthesized. Depending on the choice of the symbol, a frame multiplier allows to amplify certain audio components and/or to suppress others. In this contribution, we consider frame multipliers,
whose symbols are chosen at random with an additional subsampling step of the output signal. We show thatthe corresponding subsampled random time-varying filters satisfy remarkable mathematical properties. Moreover, we demonstrate that they are highly efficient for signal compression. Our analysis is based on the recent
mathematical methodology “compressive sensing”.

Keywords: Frames, Compressive Sensing, Frame Multiplier, Random Symbol Vector

1

INTRODUCTION

In audio and acoustics, time-dependent signal representation and processing techniques are highly relevant forvarious applications. A mathematical theory which is particularly suited to serve as framework for the development and analysis of such methods is frame theory. Frames provide the following general properties: perfectreconstruction, stability, redundancy, and a signal-independent, linear inversion procedure.
1.1 Frames and frame multipliers
A (finite) frame Ψ over CN with frame bounds 0 < A ≤ B < ∞ is a family of Q vectors Ψ = {ψ0 , . . . , ψQ−1 },ψq ∈ CN , such thatQ−12

A∥x∥22 ≤ ∑ ⟨x, ψq ⟩ ≤ B∥x∥22 ,∀x ∈ CN .

q=0

The synthesis operator of a frame Ψ is defined as the operator1 TΨ : CQ → CN , a = [a0 , . . . , aQ−1 ]T 7→ TΨ a =Q−1∑q=0 aq ψq , and is (equivalently) represented via the N × Q synthesis operator matrixTΨ = ψ0 ψ1 · · · ψQ−1 .
A frame Ψ = {ψ0 , . . . , ψQ−1 } over CN is denoted as full spark frame, if any subfamily of N vectors {ψq0 ,. . . , ψqN−1 } is linearly independent [1]. Accordingly, a matrix is called full spark, if the set of its column vectors
constitutes a full spark frame.
1 The superscripts T and H represent transposition and Hermitian transposition, respectively.

Suppose next that we are given two frames Ψ = {ψ0 , . . . , ψQ−1 } over CN and Φ = {φ0 , . . . , φQ−1 } over CL
with the same cardinality Q, respectively. The operator Mm,Φ,Ψ : CN → CL defined as2
Q−1

x 7→ Mm,Φ,Ψ x = ∑ mq ⟨x, ψq ⟩φq

(1)

q=0

is called frame multiplier with associated symbol (vector) m = [m0 , . . . , mQ−1 ]T ∈ CQ [2]. Note that the corresponding frame multiplier matrix can be represented by the matrix product



m0
0
ψ0H





  .. ..
(2)
Mm,Φ,Ψ = TΦ diag(m) TH . ,.
Ψ = φ0 · · · φQ−1 



H
ψQ−1
0
mQ−1
where diag(m) denotes the diagonal matrix with symbol vector m on its main diagonal.
1.2 Gabor systems
Let us define the following two linear operators on CN : the cyclic shift or translation operator T : CN → CN ,which is given by
x = [x0 , x1 , . . . , xN−1 ]T 7→ Tx = [xN−1 , x0 , x1 , . . . , xN−2 ]Tand the modulation or frequency shift operator E : CN → CN , which is given by
x = [x0 , x1 , . . . , xN−1 ]T 7→ Ex = [x0 , ωx1 , . . . , ω N−1 xN−1 ]T ,where ω = e2πi/N . The operators π(λ ) = E j Tk , λ = (k, j), are called time-frequency shift operators and thecollection of vectors(g, Λ) = {π(λ )g : λ ∈ Λ},
where Λ ⊆ ZN × ZN
with ZN = {0, 1, . . . , N − 1}, is denoted as Gabor system with window g. If Λ = ZN × ZN , (g, Λ) is called fullGabor system with window g. The number R = card(Λ)/N is referred to as redundancy of the Gabor system.
Gabor systems exhibit very rich structural properties. The following theorem was first proven in [3] for N
prime and later generlized to all positve integers N ∈ N in [4].
Theorem 1. For every positive integer N, there is some window g ∈ CN , such that for all Λ ⊆ ZN ×ZN with card(Λ) ≥N the Gabor system Ψ = (g, Λ) is a full spark frame. Morever, the set of such windows g is of full measure, i.e., its
complement has N-dimensional Lebesgue measure zero.
Obviously, one can consider multipliers for which Ψ and(/or) Φ are Gabor systems. A typical application
scenario of such Gabor multipliers is to enhance, reduce, suppress, or even completely remove time-frequency(TF) components of a given signal according to a predefined TF mask (represented by the symbol m) [5, 6].
1.3 Linear time-invariant (LTI) and time-varying (LTV) filters
Let us consider a finite-dimensional LTI filter. It is described by a circulant matrix H, for whichh0hN−1 · · · h1




h0
h2 
 h1
H


H= .
.  = FN diag(m) FN ,..
 ..
. .. 


hN−1

hN−2

···

h0

where FN denotes the unitary N×N discrete Fourier transform (DFT) matrix and [h0 , . . . , hN−1 ]T = √1N FN m.
2 ⟨·, ·⟩ denotes the conventional Euclidean inner product.

(3)

Comparing (3) with (2), we see that an LTI filter is a special case of frame multiplier with frame vectors givenby the columns of FN .
Furthermore, let us consider a finite-dimensional LTV filter. It can be represented by a general matrix H,
for which the singular value decomposition is given by
H = U diag(m) VH .

(4)

Again, cf. (4) and (2), we conclude that any LTV filter can be rewritten as a frame multiplier.

2

COMPRESSIVE SENSING

Compressive sensing (CS) has been one of the major developments in applied mathematics in the past 15
years [7–9]. It considers the recovery of (approximately) sparse vectors from incomplete and possibly perturbed
linear measurements via efficient algorithms such as ℓ1 -minimization.
In general, reconstruction of x = [x0 , . . . , xN−1 ]T ∈ CN from measurementsy = Ax,(5)where A ∈ CM×N and M ≪ N, is impossible without substantial a priori information about x. In compressive
sensing (CS), the assumption that x is S-sparse, that is, ∥x∥0 = card(supp(x)) ≤ S for some S ≪ N is introduced
to guarantee uniqueness and efficient recoverability of x. An estimate x̂ of the sparse vector x can be obtainedby means of solving the ℓ0 -minimization problem,(P0)x̂ = arg min ∥z∥0

subject toy = Az.

z

In words, we search for the sparsest vector consistent with the measured data y = Ax. Unfortunately, ℓ0 minimization is NP-hard in general. Consequently, a number of alternatives, for example, greedy algorithms[10], have been proposed in the literature. The most popular approach utilizes ℓ1 –minimization [7, 8, 11], thatis, the convex program(P1)x̂ = arg min ∥z∥1

subject toy = Azz

is solved, where ∥z∥1 = |z0 | + |z1 | + . . . + |zN−1 | denotes the usual ℓ1 vector norm.
Whether the estimate x̂ obtained by programs (P0) or (P1) coincides with the true solution x in (5), depends
on the properties of the so-called measurement matrix A. It is rather elementary to show [9] that x̂ obtainedby (P0) satisfies x̂ = x for all S-sparse x, if every set of 2S columns of A is linearly independent. Hence, it isdesirable if A is full spark, since then any S-sparse signal x with S ≤ M/2 is necessarily the unique sparsest
solution to y = Ax.
To guarantee recoverability of the sparse vector x in (5) by means of program (P1) and greedy algorithms,
it suffices to establish the restricted isometry property (RIP) of A [9]: define the restricted isometry constant δSof an M × N matrix A to be the smallest positive number that satisfies(1 − δS )∥x∥22 ≤ ∥Ax∥22 ≤ (1 + δS )∥x∥22

for all x with ∥x∥0 ≤ S.

(6)

In words, the statement (6) requires that all column submatrices of A with at most S columns are wellconditioned. Informally, A is said to satisfy the RIP with order S when δS is “small”. Now, if the matrix
A obeys (6) with δκS < δ ∗ for suitable constants κ ≥ 1 and δ ∗ < 1, then many algorithms precisely recover any
S-sparse vectors x from the measurements y = Ax. Moreover, the RIP usually implies robustness w.r.t. noiseand ensures that the algorithms produce reliable results not only for perfectly sparse but also for approximately
sparse (compressible) vectors x [9].
It is worth mentioning the difference between matrices satisfying the RIP and full spark matrices. While thesubmatrices of full spark matrices are invertible, they may not be well-conditioned as required for the RIP tohold. Unfortunately, for deterministic matrices, it is difficult to determine the conditioning of every submatrix;
to date, no deterministic RIP matrix is known to perform optimally [12], whereas deterministic constructions offull spark matrices are known [1].
A natural generalization of the CS framework mentioned above is the situation when x is assumed to besparse w.r.t. a given dictionary D, i.e., x = Ds with ∥s∥0 ≤ S ≪ N, where D is allowed to differ from theidentity matrix. A straightforward (though maybe not necessarily optimum) way to address this generalization
is to consider the composite measurement matrix A′ ≜ AD and solve programs (P0) or (P1) using A′ instead of
A to obtain an estimate ŝ of the coefficient vector s; an estimate of x is then obtained via x̂ = Dŝ.

3

TIME-VARYING FILTERS AS CS MEASUREMENTS

We now investigate the applicability of time-varying filters or, equivalently, of frame multipliers, cf. Subsection
1.3, as CS measurements (see also [13, 14] for an approach without subsampling). Let us first recall the key
assumption in CS that the number of measurements is (much) smaller than the ambient dimension of the unknown vector x, i.e., M ≪ N. Although our definition of a frame multiplier (1) allows for different dimensions
N and L, we note that many frame multipliers used in practice restrict to the special case N = L. Therefore,
to ensure “compatibility” with the CS framework for these practically relevant frame multipliers as well, we
incorporate the necessary dimensionality reduction via an additional subsampling step. More formally, for asubset M ⊆ {0, . . . , L − 1} of cardinality M, let PM : CL → CM , x 7→ xM be the operator that restricts a vector
to the entries indexed by M . The main question we would like to address is if measurement matrices of theform
A = PM Mm,Φ,Ψ ∈ CM×N

(7)

are suitable for CS reconstruction.
In the following, we assume that Ψ and Φ are full spark frames. This encompasses the important cases,
where Ψ and Φ are harmonic frames [15], i.e., frames whose synthesis operator matrix consists of the first
N rows of the Q × Q DFT matrix, as well as where Ψ and Φ are Gabor systems for almost all choices ofwindows according to Theorem 1. Thus, in fact, both subsampled Fourier and Gabor multipliers are covered by
our framework, or, equivalently stated (see Subsection 1.3), subsampled time-invariant and time-varying systems,
respectively, are investigated as measurement procedures.
Furthermore, let Q = card(Ψ) = card(Φ) ≥ N + L − 1. We have (see next theorem for its formal statement)that there exists a symbol m ∈ CQ such that the card(M ) × N measurement matrix A defined in (7) is full spark
for all M ⊆ {0, . . . , L − 1}. Moreover, this holds for almost all symbols.
Theorem 2. Let Ψ and Φ are full spark frames over CN and CL with same cardinality Q = card (Ψ) = card (Φ).
Provided that Q ≥ N + L − 1, the symbol m can be chosen in a way such that the subsampled frame multiplier matrix
PM Mm,Φ,Ψ is full spark for all subsets M ⊆ {0, . . . , L − 1}, M ̸= 0.
/ Moreover, the set of such symbols is of fullmeasure, i.e., its complement has Q-dimensional Lebesgue measure zero.
Note that this theorem can be generalized to the situation when x is assumed to be sparse w.r.t. a given
dictionary D (cf. Section 2).
Theorem 3. Let Ψ and Φ are full spark frames over CN and CL with same cardinality Q = card (Ψ) = card (Φ),
respectively. Provided that Q ≥ N + L − 1, the symbol m can be chosen in a way such that a vector x which is Ssparse (S ≤ L/2) w.r.t. a given dictionary D is necessarily the unique sparsest solution (w.r.t. D) to the reconstruction
problemy = PM Mm,Φ,Ψ x
for all subsets M ⊆ {0, . . . , L − 1} with card(M ) ≥ 2S. Moreover, the set of such symbols is of full measure, i.e., its
complement has Q-dimensional Lebesgue measure zero.
Therefore, time-varying filtering with subsampling constitutes an effective CS measurement procedure provided that the assumptions of the Theorems 2 and 3 are met. The proofs of Theorems 2 and 3 can be foundin [16].

4

AUDIO SIGNAL COMPRESSION

We now assess the compression capabilities of time-varying filtering with subsampling for real-world audio
signals. To that end, we choose the signal “a16_clarinet” from the EBU SQAM dataset [17] as our input audio
signal x. This signal is sampled at 22.05 kHz and consists of 131072 samples. Since it is real-valued, we set
x = Ds, where D corresponds to a Gabor synthesis matrix that only takes into account positive frequencies (realdiscrete Gabor transform). More specifically, we use a Gabor system with rectangular grid Λ = {0, a, . . . , (N/a−1)a}×{0, b, . . . , (N/b−1)b}, where a = 512 and b = 64, and with a periodized and normalized Gaussian window.
Hence, the dimension of D is 131072 × 262400. As shown in Fig. 1, this signal is approximately sparse in the
TF domain. The measurement vector y is obtained via y = PM Mm,Φ,Ψ x, where Φ = Ψ are again Gabor systems
with rectangular grid Λ = {0, a, . . . , (N/a−1)a} × {0, b, . . . , (N/(b)−1)b}; however, with parameters a = 16 andb = 4096 (redundancy R = 2). We choose the Gabor window at random (standard normal distributed; canonicaltight window), as this will yield a full spark frame with probability 1 according to Theorem 1. Similarly, thesymbol is a realization of a zero-mean proper complex Gaussian random vector with identity covariance matrix,
so that the assumptions of Theorems 2 and 3 are met with probability 1. Moreover, we subsample every
third element, i.e., M = {0, 3, 6 . . . , 131070}, achieving a compression ratio of 1:3. Since this is a large scale
problem, we employ the SPGL1 software [18, 19] to recover TF coefficients ŝ and apply x̂ = Dŝ to reconstruct
the original audio signal. It can be seen from Fig. 1 that the difference between original and reconstructedsignal is relatively small.

Figure 1. Real discrete Gabor transform magnitude of the audio signal “a16_clarinet”: original signal (left),reconstructed signal (right).

5

CONCLUSIONS

We demonstrated that random time-varying filtering with subsampling can be efficiently used for compressive
sampling of signals that are sparse with respect to a given dictionary. We showed excellent compression performance not only for ideally sparse but also for approximately sparse (compressible) signals. This is importantbecause real-world signals such as audio signals are rarely ideally sparse. Our experiments indicated that foraudio signals a compression ratio of 1 : 3 is easily possible without notable differences. Therefore, we believethat random time-varying filters, or equivalently, random frame multipliers, will be a very useful tool for present
and future audio applications.

ACKNOWLEDGEMENTS
This work was supported by WWTF grant INSIGHT (MA16-53).

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[2] P. Balazs, “Basic definition and properties of Bessel multipliers,” Journal of Mathematical Analysis andApplications, vol. 325, no. 1, pp. 571–585, 2007.
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vector spaces,” Journal of Fourier Analysis and Applications, vol. 11, no. 6, pp. 715–726, 2005.
[4] R.-D. Malikiosis, “A note on Gabor frames in finite dimensions,” Applied and Computational Harmonic
Analysis, vol. 38, no. 2, pp. 318–330, 2015.
[5] H. G. Feichtinger and K. Nowak, “A first survey of Gabor multipliers,” in Advances in Gabor analysis,
pp. 99–128, Springer, 2003.
[6] P. Balazs, B. Laback, G. Eckel, and W. A. Deutsch, “Time–frequency sparsity by removing perceptuallyirrelevant components using a simple model of simultaneous masking,” IEEE Transactions on Audio, Speech,
and Language Processing, vol. 18, no. 1, pp. 34–49, 2010.
[7] E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction fromhighly incomplete frequency information,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, 2006.
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[9] S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing. Applied and Numerical
Harmonic Analysis, Basel: Birkhäuser, 2013.
[10] J. A. Tropp, “Greed is good: Algorithmic results for sparse approximation,” IEEE Trans. Inf. Theory,
vol. 50, pp. 2231–2242, Oct. 2004.
[11] S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM review,
vol. 43, no. 1, pp. 129–159, 2001.
[12] A. S. Bandeira, M. Fickus, D. G. Mixon, and P. Wong, “The road to deterministic matrices with therestricted isometry property,” Journal of Fourier Analysis and Applications, vol. 19, no. 6, pp. 1123–1149,
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[13] G. Tauböck, S. Rajbamshi, P. Balazs, and L. D. Abreu, “Random Gabor multipliers and compressive
sensing,” in 13th International conference on Sampling Theory and Applications (SampTA), pp. 1–4, 2019.
[14] S. Rajbamshi, G. Tauböck, P. Balazs, and L. D. Abreu, “Random Gabor multipliers for compressive sensing: A simulation study,” in 27th European Signal Processing Conference (EUSIPCO), pp. 1–5, 2019.
[15] O. Christensen et al., An introduction to frames and Riesz bases. Springer, 2016.
[16] G. Tauböck, S. Rajbamshi, and P. Balazs, “Frame multipliers and compressive sensing,” preprint, 2022.
[17] EBU, “EBU SQAM CD: Sound quality assessment material recordings for subjective tests.”[18] E. van den Berg and M. P. Friedlander, “SPGL1: A solver for large-scale sparse reconstruction,” Dec.
2019. https://friedlander.io/spgl1.
[19] E. van den Berg and M. P. Friedlander, “Probing the pareto frontier for basis pursuit solutions,” SIAM
Journal on Scientific Computing, vol. 31, no. 2, pp. 890–912, 2008.

ABS-0794

Grid-like wavelet sampling for audio processing applications
Clara HOLLOMEY∗ , Nicki HOLIGHAUS, Günther KOLIANDER
Acoustics Research Institute, Academy of Sciences, Austria*clara.hollomey@oeaw.ac.at

ABSTRACT
We present a novel sampling strategy that enables the use of wavelet transforms with audio processing algorithms that rely on a time-frequency matrix structure as present in the short-time Fourier transform (STFT).
We apply the proposed scheme to a blind source separation task based on non-negative matrix factorizationand give practical guidance on the usage of the transform for audio applications. Comparable results to thoseachieved via the STFT demonstrate the potential of the method. Wavelet transforms exhibit a constant center
frequency to bandwidth ratio (Q-factor) and are thus particularly useful for representing audio data, but prior,
non-uniform sampling strategies do not lend themselves well to audio processing. Specifically, the samplinginterval classically decreases with the wavelets’ center frequency, and the resulting coefficients have no timefrequency matrix structure. By introducing quasi-random delays, we construct invertible wavelet transforms
with linearly spaced center frequencies and a fixed sampling interval. Interpreting the resulting coefficients as atime-frequency matrix provides access to processing schemes that are normally not readily usable with wavelet
transforms.

Keywords: Wavelets, Sampling, Audio analysis

1

INTRODUCTION

Constructed by scaling a fixed mother wavelet, wavelet filters are functions that exhibit a fixed number of oscillations, usually designed for simultaneously localizing a signal in time and frequency [2]. Due to the scaling,
their bandwidth increases linearly with frequency, such that wavelets favor signal representations on logarithmic frequency axes that give rise to efficient invertible filter bank implementations, most notably [11]. As aconsequence, wavelet transforms can offer a worthwhile alternative to the Fourier transform, particularly for theprocessing of signals with frequencies spanning several orders of magnitude, such as audio.
However, precisely their constant center frequency to bandwidth ratio forms an impediment to the application of the wavelet transform in practice. The efficient approaches for constructing invertible wavelet bases
mentioned above rely on the dyadic splitting of the time-frequency plane, which restricts the bandwidth of
the frequency channels, potentially yielding undesirably low frequency resolution [4]. Approaches that provide
more flexibility [3], on the other hand, usually yield highly redundant time-frequency coefficients [14], and arethus less efficient. Moreover, in addition to flexibility in the choice of frequency scaling and the invertibilityof the time-frequency coefficients, many audio processing algorithms (e.g. [6, 15, 9]) require the alignment oftime-frequency coefficients over frequency. For the short-time Fourier transform (STFT), those requirements areeasily fulfilled when its downsampling factor which is usually the same for all frequency channels, is chosensuch that the invertibility of the resulting coefficients is ensured. For wavelets, choosing the same downsampling
factor for all frequency channels and imposing invertibility of the transform implies excessive redundancy andtherefore often impairs its practical usefulness.
In [8], we introduced a sampling scheme mitigating those trade-offs, and exemplified its suitability for audio
processing applications. Here, we summarize that approach and describe the implementation and usage of theresulting filter bank.

(a) Mallat’s algorithm, from [1](b) Resulting sampling grid(c) Resulting time-frequency geometry, from [8]Figure 1. A typical filter bank structure for implementing the fast dyadic wavelet transform, also termed Mallat’salgorithm. The resulting sampling grid yields a time-frequency geometry that covers the time-frequency plane
evenly.

(a) Uniform sampling grid(b) Time-frequency geometry resulting
from equidistant sampling, from [8](c) Time-frequency geometry resulting
from grid-like sampling, from [8]Figure 2. A uniform time-frequency sampling grid; the resulting time-frequency geometry does not cover thetime-frequency plane well. The proposed grid-like sampling provides even coverage.

2

GRID-LIKE WAVELET SAMPLINGClassical approaches to the wavelet transform, such as Mallat’s algorithm depicted in Figure 1(a) achieve spectral coverage by tieing the downsampling factor to the channel number per tree level. The consequential timefrequency sampling grid is not equidistantly spaced, as schematically depicted in Figure 1(b), but achieves evencoverage of the time-frequency plane, see Figure 1(c). Uniform sampling on a grid, on the other hand, does
not lead to similarly even coverage of the time-frequency plane, as depicted in Figure 2(b).
Intuitively, to obtain a matrix-like time-frequency structure of samples and invertibility at the same time, asampling scheme for wavelets should thus, on the one hand, be equidistributed on the time-frequency plane toensure that one sampling value lies within each tile of the grid depicted in Figure 2(a). On the other hand,
it should permit sufficient variation in the spacing of sampling points to achieve even coverage of the timefrequency plane.
A sequence in the domain [0, 1)d is called equidistributed, if, asymptotically, the number of points in every subset of the domain is approximately proportional to the size of the subset. Random and pseudorandom
processes yield such equidistributed sequences, but do not satisfy a counterpart of equidistribution for finite
sequences. Thus, although in discrete scenarios, equidistribution of a sequence may arise empirically, it is
not guaranteed, and may only do so for subsequences that are prohibitively long in a sampling context. Low
discrepancy sequences [5] exhibit characteristics of random and pseudorandom sequences, but are constructedaccording to deterministic principles to achieve a specific type of equidistribution on specified intervals evenfor finite subsequences of almost arbitrary length. Many low discrepancy sequences are designed such that theyguarantee a certain order of convergence in their prime application, numeric integration via the quasi-MonteCarlo method [12]. As we demonstrate, they can be used equally well to ensure even coverage of the timefrequency plane by wavelet atoms.
We use low discrepancy sequences for generating a small delay that is applied to each filter in the wavelet
transform to achieve the even coverage of the time-frequency plane depicted in Figure 2(c). More specifically,−1for a given mother wavelet ψ, wavelet filters are generated via dilation by s j = 1b + qj
> 0 and translationby d(l + δ j )n − d(l + δ j )1(1)
ψl, j [n] = √ ψsjsj
where d, b, q > 0 and δ j corresponds to the delay and is generated by a low discrepancy sequence (δ0 , δ1 , δ2 , . . .).

3

IMPLEMENTATION AND USAGE

The grid-like sampled wavelet transform is implemented in MATLAB/GNU Octave as part of the filter bankframework of the Large Time-Frequency Analysis Toolbox (LTFAT) 2.5 [13]. It comprises waveletfilters, afunction that derives the wavelets along with their associated downsampling factors and center frequencies, andthe actual filterbank function, that, based on the output of waveletfilters, computes the time-frequency
coefficients for a specific input signal.
waveletfilters is a general purpose tool for generating wavelet functions that cover a specified filterbank length and frequency range. In its default configuration, besides the filter bank length to be covered,waveletfilters accepts a vector of nearly arbitrary scales as input arguments. A mother wavelet is then generated, and the downsampling factors are derived therefrom according to the input scales. The mother wavelet is
specified in the frequency domain, with a Cauchy wavelet as the default and centered at a relative frequency of
fCW = 0.1, i.e., a scale of 1 always corresponds to a wavelet with relative center frequency 0.1. In compliancewith LTFAT’s conventions for relative frequency that standardize the Nyquist frequency fNL to one, the minimum scale sN (and corresponding relative center frequency fN ) accepted by waveletfilters is thus limited
by its internal Nyquist frequency
fNL
1
fN =
=
= 10,
(2)
fCW
0.1
to sN ≥ f1N = 0.1. Alternatively to providing the scales directly, linear frequency spacing across the range ofinterest can be achieved by specifying a desired minimal and maximal frequency, as well as the desired numberof wavelet channels M. For our proposed grid-like sampling scheme, the desired sequence of delays is additionally passed to waveletfilters as an optional parameter, either as a vector or as an anonymous function,
describing e.g. a low discrepancy sequence, such as a Kronecker [10] or the van der Corput sequence [16].
Since the bandwidth of a wavelet increases proportional to its center frequency, the lower frequency regionsof the transform may, as depicted in Figures 1(c) and 2(c), be insufficiently covered for allowing stable inversion. To mitigate this issue the waveletfilters implementation adds one lowpass filter by default to cover
the frequency range from zero to the lowest wavelet center frequency specified. For greater flexibility, it is also
possible to parametrize waveletfilters such that this frequency range is covered by several frequency-shifted
copies of that wavelet.
An example for the usage of a wavelet filter bank in a uniform filter bank setting is given in Listing 1.
The synthesis of the time domain signal can be achieved efficiently by calculating the dual window via thefunction filterbankrealdual and making use of the filter bank inversion routine ifilterbank, which arereadily available in LTFAT. ifilterbank generally outputs complex signals. Energy-preserving recovery of real
signals, such as audio, thus necessitates the multiplication by a factor of two, as indicated in lines 33 and 34
of the listing.

Listing 1: GNU Octave usage example for grid-like wavelet sampling in a uniform filter bank setting% access the audio file and its sampling frequency[x,fs] = audioread(’inputsignal.wav’);%the filter bank length corresponds to the length of the audio file
Ls = length(x);5% delays via a golden Kronecker sequencealpha = 1-2/(1+sqrt(5)); % 1−1/(goldenratio)delays = @(n,a) a*(mod(n*alpha+.5,1)-.5);
10% parameters of the wavelet transformredundancy = 2;
M = 1021; %number of frequency channels minus the lowpass filter
start_index = 3; % number of channels to be covered by copies% of the lowest frequency wavelet

1520% derive the scales
fmax = 10; % 10=Nyquist frequency
freq_step = fmax/M; % steps in frequency
fmin = freq_step*start_index; % center frequency of the minimum wavelet channel
scales = 1./linspace(fmin,fmax,M-start_index+1);

25% generate grid−like sampled wavelet filters with a redundancy of two% for the usage in a filter bank with uniform downsampling factors[g, a, fc, L, info] = waveletfilters(Ls,scales,’delay’,delays, ’redtar’, redundancy,...
’uniform’, ’repeat’);% compute the time−frequency coefficients via a uniform filter bank
X = ufilterbank(x,g,a);30% synthesize the time−domain signal via an inverse filter bank% calculate the dual windows
gd = filterbankrealdual(g,a,L);%set flag ’real’ to implicitly multiply by two and take the real part of the output
y=ifilterbank(X, gd, a, Ls,’real’);4

AUDIO PROCESSING APPLICATIONS

Grid-like wavelet sampling can be used as a plug-and-play alternative to the STFT. In [6], Févotte et al. usednonnegative matrix factorization (NMF)—a dimension-reduction technique for non-negative data, optimized withregards to the Itakura-Saito divergence—for the decomposition of audio signals. They obtained a meaningful
factorization of spectrograms S as S = W · H, with component and activation matrices W and H, respectively.
Adapting the code from [6], we achieved similar results in [8] when replacing the spectrogram with the timefrequency coefficients obtained from the grid-like sampled wavelet transform. In [8], we chose the parametersof the wavelet transform according to frame theoretic considerations. Here, we take a more practical angle,
using the STFT as a baseline to outline necessary considerations when applying the grid-like wavelet transform
as its substitute.
As an input signal, we consider a piano recording, a sequence of six chords and use the NMF algorithm
to split them into eight components. The smoothing prior introduced in [6] was not used, i.e., its weight was
set to 0. For both transforms, M = 1024 frequency channels were used, with M = 1018 wavelet channels and 6
lowpass channels of the wavelet transform matching the M = 1024 uniform channels of the STFT used in [6].
Similarly, the total oversampling rate of the wavelet transform was set to 2, matching [6]. The delays δi weregenerated via a Kronecker sequence [10]. A Cauchy wavelet with hyperparameter α = 1200 and a sinebell window were used, respectively. This resulted, after truncation for numeric reasons, in a wavelet transform where
the frequency support of the middle scales (i.e. those around 5.5kHz for the audio signal used in this setup)
had a similar frequency support to the window of the STFT. Nonetheless, the bandwidth of the STFT filters
remains fixed across frequency, whereas the wavelet filters have a constant center frequency to bandwidth ratio,such that there is no direct correspondence between both representations.
The results, depicted in Figures 3 and 4, were sorted in decreasing order of variance of their temporalcomponents and analyzed via a constant-Q transform [7] for better display of the characteristic patterns. Overall,
the audio tracks are similar. In both cases, tracks 1-4 (counted from highest to lowest variance of the temporalcomponents) represent the various notes of the chord, with the wavelets providing slightly less separation acrossfrequency, while the last two components appear to capture low-intensity, residual sound. The main differences
lie in tracks 5 and 6. From visual inspection and informal listening, it appears that the audio captured in track
5 by the STFT-NMF has been split into two tracks, track 5 and 6, by the wavelet-NMF.

5

FURTHER RESOURCES

The code for the grid-like sampled wavelet transform can be freely downloaded as part of the Large TimeFrequency Analysis Toolbox (LTFAT) 2.5 under the General Public License version 3. LTFAT, supplemental
code and additional material comprising experimental results and audio examples are available from http://ltfat.org.

ACKNOWLEDGEMENTS
We would like to thank Cédric Févotte for kindly providing their code for Itakura-Saito NMF and some guidance on its use. We further thank Georg Tauböck for fruitful discussion on potential applications of the proposed
method.

Figure 3. Audio components obtained from the NMF based on the grid-like sampled wavelet transform, sorted
in order of descending variance of their temporal components, plots were normalized to equal magnitude forbetter display

Figure 4. Audio components obtained from the NMF based on the STFT, sorted in order of descending varianceof their temporal components; plots were normalized to equal magnitude for better display

References[1] Homepage of the Large Time-Frequency Analysis Toolbox. ltfat.org. Accessed: 2022-08-04.
[2] C. Burrus, R. Gopinath, and H. Guo.
Recherche, 67, 01 1998.

Introduction to Wavelets and Wavelet Transform—A Primer.

[3] R. Coifman and M. Wickerhauser. Entropy-based algorithms for best basis selection. IEEE Transactions on
Information Theory, 38(2):713–718, 1992.
[4] I. Daubechies. Orthonormal bases of compactly supported wavelets. Communications on Pure and Applied
Mathematics, 41(7):909–996, 1988.
[5] J. Dick and F. Pillichshammer. Digital Nets and Sequences: Discrepancy Theory and Quasi–Monte Carlo
Integration. Cambridge University Press, 2010.
[6] C. Févotte. Majorization-minimization algorithm for smooth Itakura-Saito nonnegative matrix factorization.
In 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 1980–
1983, 2011.
[7] N. Holighaus, M. Dörfler, G. A. Velasco, and T. Grill. A framework for invertible, real-time constant-Q
transforms. IEEE Transactions on Audio, Speech, and Language Processing, 21(4):775–785, 2012.
[8] N. Holighaus, G. Koliander, C. Hollomey, and F. Pillichshammer. Grid-based Decimation for Invertible
Wavelet Transforms in Audio Processing. submitted, 2022. https://ltfat.org/notes/ltfatnote057.
pdf.
[9] F. Jaillet and B. Torrésani. Time-Frequency Jigsaw Puzzle: adaptive multiwindow and multilayered Gaborexpansions. International Journal of Wavelets, Multiresolution and Information Processing, 05, 03 2007.
[10] L. Kuipers and H. Niederreiter. Uniform Distribution of Sequences. John Wiley, 1974.
[11] S. Mallat. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions
on Pattern Analysis and Machine Intelligence, 11(7):674–693, 1989.
[12] H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, 1992.
[13] Z. Průša, P. L. Søndergaard, N. Holighaus, C. Wiesmeyr, and P. Balazs. The Large Time-Frequency Analysis Toolbox 2.0. In M. Aramaki, O. Derrien, R. Kronland-Martinet, and S. Ystad, editors, Sound, Music,
and Motion, pages 419–442. Springer, Cham, Switzerland, 2014.
[14] Z. Průša, P. L. Søndergaard, and P. Rajmic. Discrete Wavelet Transforms in the Large Time-Frequency
Analysis Toolbox for Matlab/GNU Octave. ACM Trans. Math. Softw., 42(4):32:1–32:23, June 2016.
[15] G. Tauböck, S. Rajbamshi, and P. Balazs. Dictionary Learning for Sparse Audio Inpainting. IEEE Journalof Selected Topics in Signal Processing, 15(1):104–119, 2021.
[16] J. G. van der Corput. Verteilungsfunktionen I-II. Proc. Akad. Wet. Amsterdam, 38:813–821, 1058–1066,
1935.

ABS-0930

Differentiable Dictionary Search: Integrating Linear Mixing with Deep
Non-linear Modelling for Audio Source Separation
Lukáš Samuel Marták, Rainer Kelz, Gerhard Widmer
1 Institute of Computational Perception & LIT AI Lab, Johannes Kepler University Linz, Austria, lukas.martak@jku.at

ABSTRACTThis paper describes several improvements to a new method for signal decomposition that we recently formulated under the name of Differentiable Dictionary Search (DDS). The fundamental idea of DDS is to exploit aclass of powerful deep invertible density estimators called normalizing flows, to model the dictionary in a linear
decomposition method such as NMF, effectively creating a bijection between the space of dictionary elements
and the associated probability space, allowing a differentiable search through the dictionary space, guided by theestimated densities. As the initial formulation was a proof of concept with some practical limitations, we will
present several steps towards making it scalable, hoping to improve both the computational complexity of themethod and its signal decomposition capabilities. As a testbed for experimental evaluation, we choose the taskof frame-level piano transcription, where the signal is to be decomposed into sources whose activity is attributed
to individual piano notes. To highlight the impact of improved non-linear modelling of sources, we compare
variants of our method to a linear overcomplete NMF baseline. Experimental results will show that even in theabsence of additional constraints, our models produce increasingly sparse and precise decompositions, according to two pertinent evaluation measures.

Keywords: differentiable dictionary search, non-negative matrix factorization, normalizing flows, invertible
neural networks, audio signal decomposition.

1

INTRODUCTION

Decomposing a signal that was produced as a mixture of source signals is an essential task in signal analysis,
with applications in audio source separation, automatic speech recognition, audio de-noising, musical instrument recognition, musical audio de-mixing, harmonic-percussive source separation, musical melody extraction,
polyphonic music transcription, and many more. In different domains, various assumptions are made about thenature of what is to be considered an elementary source signal, and different artifacts of decomposition aredesirable, dictated by the respective objectives.
This has led to a multitude of conceptually different approaches to signal mixture decomposition. Linear
methods such as Non-negative Matrix Factorization (NMF) have been widely extended, adjusted and customized
with various constraints, to incorporate assumptions and inductive biases accommodating specificities of particular problems. In contrast, more general and highly flexible non-linear models such as, specifically, deep
end-to-end neural networks have been successfully trained to perform mixture decomposition. While the linear
methods have inherent performance limitations relating to their modeling capacity, they seem to have an edgeat both the interpretability of inference, and robustness of performance in the face of distribution shift between
training and inference, when compared to these deep non-linear models.
In this work, we aim to advance an ongoing effort in integrating the strengths of these contrasting approaches. The assumption of approximately linear mixing of sources is reflected in our use of additive source
re-composition of NMF. This can be seen as a strong inductive bias on the algorithm searching for optimal
signal decomposition, since it significantly constrains the size of available hypothesis space, as opposed to a
non-linear mixing model. However, the constituent sources often arise as a result of some complex non-linearphysics. To address this, we assume non-linear sound producing processes for constituent sources, and train
deep non-linear models to represent them.
We present a new method for signal decomposition based on the concept of Differentiable Dictionary Search(DDS), which was originally introduced in [15]. The general idea of DDS is to use a trained explicit-likelihooddensity model to impose a distance measure on the semantically structured space of spectral activity that therelevant sound sources exhibit, and use it to constrain the optimization process guiding the decomposition of an
audio mixture. This combines the efficiency of dictionary sharing across time inherited from the non-negative
matrix factorization framework, with improved modelling capacity of deep normalizing flows used to express
the acoustic diversity of individual sound sources.
The initial DDS formulation [15] was mainly a proof of concept, still exhibiting a number of severe limitations. In particular, lack of dictionary sharing and parameter reuse in source modelling resulted in scalability
issues with respect to growing input size and number of sources. In this paper, we present several extensionsof the original model with the goal to further increase parameter reuse and data efficiency. We address this bychoosing to model all sources together with a single conditional density model, with the source identity expressedby the variable being conditioned on, which we hope also leads to better generalization. These adjustments are
aimed at reducing time and space complexity of the decomposition algorithm, while improving performance atthe task of source attribution.
We use audio recordings of polyphonic piano music with temporally aligned labels to evaluate the decomposition quality by assessing the precision of spectral activity attribution to the sources. We compare variantsof our method to an overcomplete NMF baseline, providing all of the compared methods with equivalent data
for training (i.e. initialization), and iteration budget for inference (i.e. decomposition), ensuring that the conditions for comparison are fair, and that differences in performance can be attributed purely to differences in themethods’ mechanics. To highlight these differences, we assess the memory and computation requirements as afunction of the size of training data. Our experiments will show systematic improvement, in terms of source
attribution, along the axis of method adjustments introduced by our advanced formulations. We will also seethat the adjustments lead to increasingly sparse decompositions, according to a standard sparsity measure.

2

RELATED WORKThe model class used for modeling the spectral content of individual sources is called “normalizing flows”,
which have been introduced in [6]. We used deep parametric density model architectures RealNVP [3] and Glow[11]. Adaptive dictionaries, or generative models, have been used for audio decomposition before. Examples
using Generative Adverserial Networks include [18, 13], examples using (variational) autoencoders are [17, 9].
All these approaches have a shortcoming in their lack of explicit access to the likelihood of their samples. It istherefore somewhat difficult to prevent the generative models from producing samples “too far away” from thetraining data manifold. Although Variational Autoencoders can provide the likelihood of samples they generate,
Monte Carlo methods need to be used to approximate an expectation of a lower bound on the likelihood [1].
A very interesting line of research is discussed in [16, 14, 5], where Gaussian Processes are used for source
separation. These models are adaptive, well grounded in probability theory, and even have an accessible, explicit
likelihood, but are unfortunately extremely computationally demanding.

3

METHOD

The idea at the core of the DDS approach is to take the NMF framework S ≈ W · H and replace the dictionary
W in a specific way. A dictionary entry corresponding to a particular source represents the spectral profile ofthat source by using a deep generative density model. The spectrogram S is decomposed into linear combinations of samples (likely spectral profiles) that are generated by these non-linear source density models. Those
samples are allowed to adapt to the input that is being decomposed. The normalizing flow establishes a bijective
differentiable mapping between the data space and a latent space, and allows explicit access to the likelihoodof a sample as well. This enables to (softly) constrain the likelihoods of samples in a natural way, which inturn (softly) constrains how far the samples can deviate from the training distribution. The direction in which
the samples should adapt is indirectly given by the gradient of the reconstruction error with respect to the latentcode that produced a sample. We are able to do this because the mapping between data space and latent space
is fully differentiable.
To put it differently: by fitting generative density models to the training samples of isolated sources, we fit aparametrized function to estimate the data generating distribution. Since the model class we chose is a smooth,differentiable and invertible mapping between two manifolds (a so called diffeomorphism), we can travel on thedata manifold by taking small steps in arbitrary directions of the high-likelihood region of the latent space towhich the data manifold has been mapped. This enables us to search through this differentiable dictionary inany direction in data space we desire — thus the name “Differentiable Dictionary Search”. Several benefits ofthis approach have been demonstrated in [15], through a set of highly controlled experiments.
In the following, we describe three different formulations of the DDS method, along with a semi-supervised
NMF baseline, all of which will be evaluated in Section 5. These three variants of DDS represent the iterative
process towards improving both scalability and performance of the method.
3.1 The Linear Baseline: NMF
We will denote the log-magnitude spectrogram of an audio snippet as S ∈ RD×T
+ , with D the spectral resolution(number of frequency bins) and T the length of the input (number of time frames). The NMF approximation is
M×Tgiven as Ŝ = W · H, where W ∈ RD×M
is the dictionary matrix, and H ∈ R+
is the activation matrix.
+
The inner dimension M of the NMF matrix multiplication corresponds to the number of components ofthe decomposition, and gives the number of columns in W (“dictionary entries”) as well as the number ofrows in H (“component activations”). In an unsupervised NMF, M would be a free parameter of the method.
In our experiments, M will correspond to the total number of all samples (spectral frames of isolated notes)
from both training and validation sets combined, that are otherwise used for training (and selection) of thedictionary model(s) in our variants of the DDS method. We simply store all of these samples in the dictionary
W to initialize the method, and keep it fixed during the decomposition. This is what makes our baseline asemi-supervised and over-complete variant of the NMF method. The motivation is to enable fair comparison
conditions by keeping the access to raw training data equal among the methods.
We will denote the number of sources (e.g. notes in the case of piano recordings) modeled by each method
as K. After decomposition, the M “component activations” (rows in H) are collapsed into the K rows of “source
activations” by summing components according to their source identity.
3.2 The Unconstrained Formulation: DDS-1
This is the original formulation of the DDS method, as proposed in [15]. The primary deviation of the DDS-1
formulation from NMF (as well as the other two formulations) stems from the independent treatment of timeframes in the decomposed spectrogram S. Each time frame is modeled separately as a linear combination of K
different dictionary entries. Each of the K possible sources is modeled with a separate density model using the
RealNVP architecture, structure and configuration following [15].
The dictionary for the k-th source Wk ∈ RD×T
is generated by the corresponding density model Fθk . Please+
note, that each Fθk has its own set of parameters θk . Transforming the corresponding set of latent codes Zk ∈⊤RD×T as Fθ−1(Z⊤k ) = Wk for each source makes the full dictionary W of shape (D × T × K) large. Thekactivation matrix H ∈ RK×T
then contains one scalar for scaling each dictionary entry. With the row vector hk+
denoting k-th source activations in H, the input approximation is given as Ŝ = ∑Kk=1 Wk ⊙ hk .
This specific setup was meaningful for the evaluation of the proof of concept implementation, where wetreated the monophonic test samples independently in the highly controlled experiment of [15], and had only
12 sources (1 octave) in the exemplary semantic decomposition at the end. However, it is also the source ofcomputational bottleneck when applying this variant to larger scale problems.
3.3 Constraining the Dictionary Size: DDS-2
As a first step to improve scalability, we re-introduce the sharing of dictionary entries across time from the
NMF framework, such that the input approximation is given by the product of the dictionary and activation
matrices Ŝ = W · H.

We introduce a new free parameter N denoting “number of components per source”, which specifies how
many dictionary entries (columns in W) will be used to explain activity of each source. This yields a new
structure of the dictionary W ∈ RD×KN
and activation H ∈ RKN×T
matrices, with each of the K sources rep++resented by N components. After decomposition, the N “component activations” (rows in H) of each source
are combined by summation, to make the K × T activation matrix H. Dictionary entries for each source are
generated by a separate, source-specific density model Fθk using the RealNVP architecture, following DDS-1.
⊤
With the N dictionary latent codes Zk ∈ RD×N generating a subset of the dictionary as Wk = Fθ−1
(Z⊤k ) forkeach of the K sources, the full dictionary is constructed by concatenating the generated entries of all source
models: W = [W1 , W2 , · · · , WK ].
This imposes another regularization on the adaptation of dictionary entries (other than the likelihood penalty
that is fundamental to the general DDS approach), as the entries now need to be re-usable across multiple
spectrogram frames of the polyphonic mixture. As a result, computational efficiency is increased along withperformance, due to the reduction in potential for "overfitting" any particular frame.
3.4 Unifying the Dictionary Model: DDS-3
This formulation builds on the previous one (DDS-2) by additionally regularizing the source modeling for thedifferentiable dictionary. Instead of a set of multiple source models, a single model Fθ (with a unique set ofparameters θ ), is now trained as a conditional density model — allowing sampling x ∼ pmodel (x|y) conditionedon the source class label y — used to model the full set of dictionary entries for all K sources. To accomplish
this, we choose to use an adaptation of the architecture described in [8], which allows for the class conditioning
to be expressed explicitly in the latent mapping z by splitting it into a semantic part zs ∈ RK , and a nuisance partzn ∈ RD−K . The source identity is expressed by a 1-hot vector in the semantic part zs , while most of the intraclass variance of the source should manifest in the nuisance part zn . To train the conditional generative density
model, we follow the training procedure from [8] up to a small change in the independence cross-entropy(iCE) objective: we replace the softmax cross-entropy on the semantic dimensions zs with a mean squared
error against the 1-hot class labels. Because the softmax has no explicit inverse, the conditional generationof dictionary entries using our model’s explicit inverse Fθ−1 would become difficult. The full dictionary W ∈D×KN
R+is then generated form the corresponding set of latent codes Z ∈ RD×KN by the multi-source dictionarymodel as W = Fθ−1 (Z⊤ )⊤ , and the decomposition is given by Ŝ = W · H, following DDS-2.
By having a single model jointly express probability densities of all sources, the set of parameters θ , albeit
larger than that of any individual source model in the prior DDS variants, is now optimized using all of theparameters
thus goes down drastically, increasing bothavailable training samples combined. The ratio of # trainable# training data
data efficiency, and utilization of parameters.
All of the DDS variants obtain decomposition by minimizing the reconstruction error ∥S − Ŝ∥2 jointly with− log p(z) — the negative log-likelihoods on the dictionary entries — amounting to a maximum likelihood
estimation (MLE) term, which is further weighted by relative contributions of dictionary entries to the full
activation matrix. This objective is minimized via gradient descent by alternating updates of dictionary latentcodes Z and the activations H. In the specific case of DDS-3, only the nuisance parts of the latent code zn are
updated, as semantics zs are held fixed.
3.5 Computational Complexity
To highlight how the methods that we will evaluate compare in terms of asymptotic computational complexity,
we provide a rough assessment of their respective costs in an analysis with several simplifying assumptions.
Since training of the NMF method corresponds to simple storage of data samples into the dictionary, it has
virtually no computational cost, as opposed to the DDS method variants, where one or many instances of asource model need to be trained.
All compared decomposition methods proceed iteratively, and the cost of one iteration stays constant, hence
giving the runtime complexity of one iteration is sufficient to compare the four different approaches. Furthermore, since the gradient-based parameter update has a cost that is a constant factor of the forward pass cost
— the inference of Ŝ from these parameters — we limit our assessment in Table 1 to reporting on two major
components of single iteration: (i) the reconstruction of input Ŝ from W and H, and (ii) the construction ofTable 1. Comparing Methods by Computational Complexity of Single Iteration
type

component
NMF

DDS-1

DDS-2

DDS-3

Θ(DT M)

Θ(DT K)Θ(DT KN)

Θ(DT KN)

W ← f (Z, θ )—Θ(T D2 ) · KUVΘ(ND2 ) · KUVΘ(KND2 ) ·UVŜ ← f (W, H)

Θ(DM + MT + DT )Θ(DT K + T K + DT )Θ(DKN + KNT + DT )Θ(DKN + KNT + DT )—Θ(T D + D2 ) · KUVΘ(ND + D2 ) · KUVΘ(KND + D2 ) ·UVŜ ← f (W, H)

time

space

methodW ← f (Z, θ )dictionary W from Z, using a set of parameters θ ; computed by each method as described above, omitting theassociated costs of the gradient computation and gradient-based parameter update.
As inference of the dictionary W corresponds to passing a specific batch of data (Z) through a (set of)normalizing flow(s), we assess this by recounting the number of matrix multiplications involved, leaving out
other operations of cost proportional to the input size via constant factor, which is negligible. We denote thenumber of flow steps in a normalizing flow model by U, and the number of matrix multiplication operations
involved in a single flow step1 by V . Further assumption that those matrix multiplications are on the orderof the data dimension D, which approximately holds for all of our DDS variants parametrizations among our
experiments, yields us the estimates in Table 1.
First, the NMF baseline has no cost of dictionary inference, as that is fixed to the training data, but this advantage comes at the cost of linear time and space complexity of NMF with respect to the size of training dataM, as opposed to the constant complexity of the DDS variants. This is one of the many benefits of dictionarymodeling, as typically M ≫ KN for most practical scenarios, such as decomposition of piano recordings. On
the other hand, the cost of dictionary modeling grows quadratically with the spectral resolution D in both timeand space for all DDS methods, as opposed to merely linear growth in NMF.
The computational bottleneck of the initial DDS-1 formulation can now easily be pinpointed to the obviously
linear growth of already D-quadratic dictionary modeling costs in terms of temporal length T of the input,
which is the ultimate, fastest growing parameter in any practical scenario. The dictionary sharing with an
adjustable parameter N introduced in DDS-2 (and DDS-3) purposefully eliminates this linear cost. Ultimately,
the step from DDS-2 to DDS-3 – consolidating the dictionary into a single model – does not save much timein the big picture, but improves the space complexity by eliminating the K-factor from the D-quadratic cost2 ,which definitely makes a difference with higher spectral resolutions D and larger sets of sources K.

4

EXPERIMENTAL SETUP

4.1 Data and Basic Audio Signal Processing
To assess the relevant aspects of performance with respect to varying degrees of task complexity (such as
number of concurrently sounding notes, or variance in note intensities), we use the MAPS (MIDI-Aligned
Piano Sounds) dataset [4], as its large variety of systematically structured examples allows us to study certain
aspects of our method formulations in better isolation. We will use the ISOL subset of MAPS — samples
of isolated notes — for training. To evaluate the decomposition performance, we will use the RAND subsetof MAPS, which contains samples of randomly generated note combinations (chords) drawn from a specificrange of notes, with varying degrees of polyphony and intensity (loudness). The MAPS dataset also allows tocontrol for timbral variance in terms of the used instrument model and its recording conditions. We evaluate
1 Such as the dimension shuffling operation, or the densely connected layers of the Multi-Layer Perceptron (MLP) networks used to learn the

coupling functions.
2 After multiplying by K in DDS-2, the inference of W costs Θ(KND + KD2 ) · UV as opposed to Θ(KND + D2 ) · UV in DDS-3, making it adifference of (K − 1)D2 ·UV .

behavior of the tested systems in presence of timbral distribution shift, by carefully selecting subsets of samples
for training and evaluation with no overlap along the timbre-determining axes. In the following experiments,
we train all systems on samples of synthetic, sample-based instruments, excluding only the Disklavier(ENSTDk*)
samples from training. We randomly split this set into 80% training and 20% validation samples. Evaluation isdone on samples from the Disklavier instrument in “Close” recording setting — code (ENSTDkCl) in [4].
The MAPS audio samples are encoded as 44.1 [kHz] stereo WAV files with the temporally aligned groundtruth in the form of MIDI files. Before computing spectrograms, we downsample the recordings to 16 [kHz]mono waveforms, preserving spectral activity in the frequency range of [0; 8] [kHz]. We analyze the audio at
spectral resolution of 512 frequency bins, given by DFT window size of 1024, using the Hann window function,
and a hop size of 512 samples. The temporal resolution, as determined by the combination of sampling rateand hop size, is then used to quantize the MIDI ground truth into a piano-roll with a shape matching that of theactivation matrix produced by the decomposition. After spectral analysis of different sample audio files, silent
frames are discarded based on the ground truth annotations.
4.2 Parametrization of Methods (1): Training
We choose to evaluate on a portion of RAND subset of MAPS that uses the pitch range of the centered 5
octaves, marked by M36-95 in [4], corresponding to 60 notes between and including C2 and B6, that theevaluated methods need to model as individual sources. This range is commonly used to evaluate multi-pitch
algorithms. Note that this is a 5-fold increase from the single octave (12 notes) modelled by DDS-1 in [15].
NMF. The dictionary W in NMF is initialized by loading all of the samples, which are otherwise also used
for both training and validating the normalizing flows used in our DDS method formulations, and storing them
directly in the dictionary, giving this method a slight edge in terms of training data volume.
DDS. The differentiable dictionaries in both the DDS-1 and DDS-2 formulations, are initialized by training
a set of unconditional density models (parametrized by a modified single-scale 1-dimensional RealNVP architecture, as per Section 3), one for each note. We follow [15] closely in regard to the model architecture andhyperparameters. In particular, the 16 flow steps are realized via affine coupling blocks [2] with their coupling
functions approximated using MLPs with 4 dense layers of 128 units with SELU [12] activations. The shufflingof dimensions between the coupling blocks is realized via randomly initialized, fixed permutation layers, which
are easily invertible via transposition.
The differentiable dictionary in DDS-3 formulation is initialized by training a single conditional density
model (parameterized by a modified single-scale 1-dimensional Glow architecture with semantic conditioning,as per Section 3). Specifically, following the Glow architecture [11], each of the 32 flow steps has (i) an
ActNorm layer, (ii) a trainable mixing layer parameterized in its LU-decomposition, and (iii) an affine coupling
parameterized as MLP with 3 dense layers of 1024 units and Leaky ReLU [19] activations.
All density models were trained with the Adam optimizer [10] for up to 2500 epochs using mini-batch sizeof 512, and learning rate of 1 · 10−3 with the single-source models (DDS-1 and DDS-2) and 1 · 10−5 with themulti-source model (DDS-3). After each training epoch, the model was evaluated on the validation data forsubsequent model selection.
4.3 Parametrization of Methods (2): Inference
For all methods, the decomposition is run for a maximum of 1000 update steps, each of which allows for two
independent updates of the dictionary, and activation matrix, respectively, following the procedure of alternating
updates. NMF uses a step size of 1 · 10−2 , the DDS variants use 5 · 10−3 . The remaining parameters and aspects
of decomposition, such as learning rate reduction upon loss plateau detection or the early stopping condition,
are set equally for all methods, and follow the experimental setup outlined in [15] identically.
Initial Conditions for the NMF method. Let M denote total number of samples used for training and validation, and thus also the number of entries forming the dictionary W in NMF. Then, the activation matrix H isinitialized with M1 everywhere.3 .

3 This by no means implies that NMF is allowed twice as many updates of H (due to having W fixed) as the DDS methods. The maximum
number of updates for each parameter is equal for all methods. A single update step allows for exactly one update of H and one update of W.

Initial Conditions for DDS variants. In all variants of the DDS method, the latent space mapping vectors Z ofthe differentiable dictionaries are initialized to 0 vectors. The activation matrix H is initialized to small, randomEnumbers drawn from a uniform distribution hi ∼ U (0, 1), re-scaled to a factor of √EK for DDS-1 and √K·N
for DDS-2 and DDS-3, where E is the average energy of the input magnitude spectrogram S, K is number ofsources, and N is number of components per source.
4.4 Post-processing
Once the inference of activations H via the decomposition of input spectrogram S is finished, yet before calculating any performance quantifiers, we re-scale each scalar value in H by the norm of its associated dictionary
vector from W, the one that is “activated” by it. This works slightly differently for each of the methods, andfor NMF, DDS-2 and DDS-3, it is followed by aggregating the activity of sources over the components that
represent them. Nonetheless, the principle is the same, and this step has an equally positive effect on the performance of all methods. The basic idea is that normalizing the dictionary entries to equal-length vectors will
put the values of H on a rather more “comparable” scale of activity contribution, without burdening the optimization process of decomposition with enforcing equal norms of vectors that are being searched over. As aresult, this is a cheap way to express the activation values in H on a consistent, W-independent scale.
4.5 Evaluation Metrics
To quantify performance of the inspected methods for a relative comparison of their properties, we calculate
and report two pertinent evaluation measures, defined as follows.
Decomposition Correctness. In order to measure the quality of decomposition, we define a metric that wecall Precision of Source Attribution (PSA). As per the Eq. 1, both the activation matrix H and the thresholded,
binary labels Y have the shape [K × T ]. The numerator in Eq. 1 quantifies the amount of activity attributed
correctly, while the denominator quantifies all of the activity attributed in total.
PSA =∑k,t H ⊙ Y∑k,t H(1)

Intuitively, PSA quantifies the fraction of activity in the activation matrix H that is attributed to those sourcesthat are active according to the ground truth annotations. Combined with the post-processing of H describedabove, it helps capture how well the activity in magnitude spectra is explained by the dictionary elements
associated with those sources that are truly active.
Decomposition Sparsity. To quantify the sparsity of the activation matrices produced, we will report a simple
measure of sparsity based on Lε0 [7]. The standard L0 -sparsity simply computes fraction of the count of zeros
to the total number of elements in the matrix. The Lε0 -sparsity also includes small, non-zero elements into thatcount, if they are below some ε threshold. We use ε = 5 · 10−2 to correct for noisy activity in H, which our
NMF baseline is more prone to generate due to its excessive dictionary size.

5

RESULTSTo examine the general trends of task difficulty and method performance on different problem configurations, we
executed multiple audio decomposition experiments using audio snippets created using various parametrizations.
To highlight the relative performance of the tested methods, we first report a detailed result on a single suchconfiguration, picked at random. Nonetheless, the observed trends in relative performance of the comparedmethods seem to hold consistently across varying test configurations.
5.1 Method Comparison
The numbers reported in Table 2 were obtained by running decompositions on a test snippet constructed byconcatenating only segments with actual note content, excluding all silence. The segments were selected tofulfill the following criteria: (i) notes have an intensity range I32-96 representing the more complex scenario
of higher dynamic range, (ii) there are only two notes sounding at any given time. The number of individual
dictionary components used for each source was set to N = 64 for both DDS-2 and DDS-3.

Table 2. Improving attribution correctness.
NMF

DDS-1

DDS-2

DDS-3

PSA

0.35680.27600.43230.6475Lε00.68240.70390.81210.9367

As our measure of decomposition quality PSA (Eq. 1) indicates, each of the adjustments introduced to
our formulation of the DDS approach led to an improvement in performance, ultimately outperforming thelinear baseline by a large margin. In particular, the introduction of dictionary sharing into DDS-2 seems to
be responsible for a 56.7% relative improvement against DDS-1, while the additional introduction of parametersharing in the parameterization of source model in DDS-3 seems to have improved the decomposition quality
by another 49.8% with respect to DDS-2. The under-performance of DDS-1 compared to the NMF baseline
points towards its inefficient scaling, as discussed in 3.5.
Moreover, all of the considered variants of DDS appear to be producing increasingly sparse decompositions,demonstrating increasing suitability to the problem domain of audio decomposition, in which mixtures are typically composed of small subsets of all potential sources. However, we observed this as an unintended positiveside effect of the adjustments introduced, having designed none of them with the specific goal of encouraging sparse decompositions in mind. Using any of these methods, sparsity can still be additionally incentivizedthrough an explicit regularization term in the decomposition objective.
5.2 General Trends
In broader set of experiments, we varied different problem parameters and method arguments, such as spectral
resolution of the input, polyphony level and dynamics range of the test samples, and the method flexibility for
DDS-2 and DDS-3 controlled via the dictionary size argument N (components per source). While the trend of
relative performance gains reported in 5.1 appeared quite consistently, we observed several additional trends.
In general, all methods — including the baseline — usually achieve slightly better decomposition metrics at
a larger spectral resolution of 1024, albeit at higher computational cost. Most of the trends we observe, appear
to be more pronounced at the higher spectral resolution of 1024. In terms of both the decomposition andreconstruction quality measures, the samples with a higher dynamic range (I32-96) were generally representing
a harder task than those of lower dynamic range (I60-68), as all methods had consistently better metrics on
the samples with low dynamic range. The problem difficulty also seems to grow almost linearly with the levelof polyphony, at least between P2 and P7, for which we have measurements from the MAPS dataset. When
tweaking the method capacity parameter N, higher values in DDS-2 produced generally smaller reconstruction
error, but not necessarily more precise decomposition. On the contrary, lower N seemed to produce moreaccurate decompositions. In contrast, with growing N, DDS-3 seemed to be producing either equally good,or better decompositions than for lower values of N, in both decomposition quality and reconstruction error.
This can be interpreted as evidence for a difference between properties of the “bag of models” approach todictionary modeling of DDS-2, and the “one conditional model for all sources” approach of DDS-3, considering
all other aspects being equal. We conjecture that the extra regularization of the single dictionary model in DDS3 markedly reduced the potential to “misuse” the spare capacity at higher N, as contrasted with DDS-2 and its
many less regularized models.

6

CONCLUSION

We presented several adjustments to a recently proposed method for audio decomposition, improving its computation and memory costs, as well as quality of the decompositions in terms of precision of source attribution.
As a positive side effect, these improvements also incentivize sparser decompositions. An investigation into
these dictionary models’ dynamics could reveal promising directions for further improvements.

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