PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Determination and optimization of sound insulation capabilities ofgeometrically complex walls
Elias PERRAS1; Chuanzeng ZHANG1; Jihao CHEN2; Zhijiang JI2
1
2

Chair of Structural Mechanics, University of Siegen, Siegen, Germany

State Key Laboratory of Green Building Materials, China Building Materials Academy, Beijing, China

ABSTRACT
Several methods have been developed in the past to calculate the sound transmission in building acoustics.
However, efficient determination and optimization of the sound insulation capabilities of geometricallycomplex walls like brick walls or layered wall structures remain a challenging task. In this work, a fully
coupled fluid-structure interaction model based on the frequency-domain spectral element method (FDSEM)
is proposed to determine the sound insulation characteristics of arbitrary wall structures. A virtual soundmeasurement laboratory following the guidelines of ISO 10140 is designed and used. The calculated
frequency responses agree well with the experimental results. Therefore, this method can be used to evaluate
the performance of the wall materials and structures in an efficient way. Furthermore, contrary to othermethods based on the plate theory, this method is more flexible and avoids many essential simplifications,which could restrict the applicability of the methods in the wall structure design. The present method isimplemented into a multi-objective optimization procedure based on a genetic algorithm to optimize various
wall structures (e.g. novel brick walls, multi-functional laminates), and the results are compared with that byother numerical and analytical methods.
Keywords: Frequency-domain spectral element method, Fluid-structure interaction, Layered structures

1. INTRODUCTION
Increased energy demand and consumption, especially in developing countries, lead to the use ofinnovative materials and structures. These structures have to fulfill special requirements on theirsound-insulation capabilities, as well as other demands such as structural and thermal properties.
Experimental determination of the acoustic behavior and heat insulation property is well established,but expensive and time-consuming. Therefore, a mathematical or numerical model for predicting theinsulation properties of such materials and structures is necessary to significantly reduce the costs of
their design and optimization process. Numerous studies have been reported in the past for the
analytical and numerical determination of the transmission loss of structures, such as sandwich panels,
honeycomb panels, and double-leafed walls, e.g. (12, 20, 21). These methods can usually deliver
reliable results with little computational effort, but they suffer from many restrictions which limit
their validity and flexibility in analyzing structures with complex geometry. The motivation for thiswork is to examine the suitability and applicability of a virtual sound insulation laboratory forcalculating the transmission loss of engineering structures in the design and optimization procedure.
The model follows the guidelines of the ISO 10140 (9) and solves a fully coupled fluid-structureinteraction (FSI) problem. To be efficient in an optimization procedure with many iterations, therequired numerical calculations must be sufficiently fast. For this purpose, the problem is solved byusing the spectral element method (SEM), an advanced and efficient variant of the finite element
method (FEM) to increase the efficiency and accuracy of the numerical calculations.
Several authors have used a fully coupled model to determine the transmission loss of structural
panels and walls. Since the computational resources necessary for the time-efficient numericalcalculations at high frequencies became available only in the last two decades, most previous studies
were usually only devoted to the low frequency range (5, 11). More recently, Arjunan et al. used a
FEM model to determine the transmission loss of double panels (1, 2). They reported that the1

perras@bau.uni-siegen.de257

computing time for their three-dimensional (3D) model was 30 hours by using a 2014 generation
computer. Del Coz Díaz et al. used a two-dimensional (2D) model to determine the transmission lossof brick walls (6). Arjunan et al. also reported a lack of references on the use of fully coupled
numerical models to determine the transmission loss of panels. Among many possible reasons for thislack is the computational effort necessary to calculate the wave propagation characteristics at highfrequencies. Although the SEM is a well-established method in the meantime, no previous studies
have been yet reported on a fully coupled SEM model for determining the transmission loss of wall
panels used in civil engineering, to the best knowledge of the authors. Originally, the SEM wasdeveloped by Patera to solve some problems in the field of computational fluid dynamics (15), but it
was also used later for a wide variety of other engineering problems such as wave propagation infunctionally graded materials (7), structural health monitoring (19), seismology (10) and many others.

2. PROBLEM FORMULATIONIn the determination of the transmission loss of a panel in a laboratory setting a source, usually aloudspeaker, emits a sound at varying frequencies in the source room. The resulting pressure wave
excites vibrations in the panel, which in turn results again in a pressure wave in the receiving room,
recorded by microphones (Figure 1, left). This process is simulated by a mathematical model. The
governing equation for the fluid in the source room and the receiving room (as well as air gaps in the
structure) is the Helmholtz equation for a compressible, inviscid fluid:

22
1   p x  ω 
x

p

Q,


 

ρ  x 2
c


(1)

where is the mass density of the fluid, c is the sound velocity, p(x) is the spatial pressure distribution,is the angular frequency, and Q is the source term. To simulate a loudspeaker, a point source is usedwhich satisfies (4) 2 p  x   ω 2
2ρω P
(2)
   p  x   4π eiφδ  x  x0  .
2
x
4π 2
c
where P and φ denote the power and the phase of the point source, and (x − x0 ) is the Dirac-delta
function. To take account of the absorption of the walls, the respective boundaries are described by
an impedance Z corresponding to the reflection index r. The impedance is determined by the boundary
condition:iω p  x 1n  p  x   
.
(3)
ρ
Z
In a laboratory setting, the absorption behaviour of the receiving room usually corresponds to anabsorption coefficient of ≈ 0.15. The reflection index r and the absorption coefficient
can becalculated byrZ  ρcand α  1  r 2 .
Z  ρc(4)

The considered structure is described by the equations of motion for a continuum in the frequencydomain asɶ T σ  ρω 2 u  x   f ,

(5)ɶ denotes a differential operator, σ is the stress tensor,
where small deformation u(x) is assumed, is the mass density of the material, and f is the body force vector. The interaction between the fluid
and the structure is described by the following coupling or boundary conditions on the interface:uS nF S ,F

 uF nF  and t   pnS  pnF
S ,F

(6)

with the normal vectors n being defined in Figure 1 and the Cauchy stress vector t. For further information
we refer to the literature, e.g. (18).

258

Figure 1 – Schematic representation of a virtual laboratory (left) and definition of normal vectors for the
coupled fluid-structure interaction problem (right)

3. FREQUENCY-DOMAIN SPECTRAL ELEMENT METHOD
3.1 Discretization
Using the weak-form or the Galerkin method, the boundary value problem under consideration canbe recast into a discretized form by approximating the pressure field with p(x) = Np and thedisplacement field with u(x) = Nu, where N is the vector of the shape functions and p and u are thenodal pressure vector and the nodal displacement vector (17). The resulting system of algebraic
equations, with neglected impedance matrix for simplicity, can be written asKF  ω2MF
1
 1

C1

0



 ρ F1 ω 2 C1T
KS  ω2MS
2

ρ F2 ω C 2T
  p1   Q1  u    f ,
C 2   
K F2  ω 2 M F2   p2  Q2 

0

(7)

where K and M denote the system stiffness matrices and mass matrices for the fluid domain (denotedby F) and the structural domain (denoted by S) respectively, and C is the coupling matrix (18).
3.2 Basics of the Spectral Element Method
The shape functions used in the developed SEM algorithm are the Lagrange polynomials LAp forthe interpolation order p given by:
p 1

ξ ξα
α 1 ξ β  ξ α

LAp , β ξ   

(8)

α β

which are constructed by using the nodal set i .
If evenly distributed nodes over the interval [0;1] ∶=ℝ|0 ≤ ≤ 1# are chosen, strong
oscillations with high orders occur, which is the so-called Runge-effect. This can be avoided by using
the zeros of the Lobatto polynomial LOp-1 as the locations for inner nodes to construct the Lagrange
polynomials or the GLL nodal set. The Lobatto polynomials can be calculated by n 2 dL ξ  2 n  2k  !
kLOn ξ   n 1with Ln ξ     1
ξ n2k .
(9)
n
dξ
n

2
k
!
n

k
!
k
!2




k 0
By using the GLL shape functions, some numerical difficulties can be overcome since the condition
number of the coefficient matrix in Eq. (7) is much lower and allows us to use higher-order shapefunctions. This, in turn, leads to a spectral convergence of the method, hence the name spectral
element method. It also offers the possibility of using the Lobatto quadrature for the numericalcalculation of the element mass matrices (17).
Figure 2 shows the set of the shape functions of a polynomial order p = 8 for an evenly distributed
nodal set and a GLL nodal set. The nodal set i is shown as red dots.

259

Figure 2 – Shape functions with evenly distributed nodes (left) and GLL distribution (right)
3.3 Advantages of using SEM for FSI Problems
One major advantage of the SEM is the reduced number of the degrees of freedoms (DOFs)
necessary to achieve a given error threshold. Here, a one-dimensional (1D) model problem is used fordemonstrating this advantage. The model is also used to a priori estimate the necessary order p andelement size for a given error threshold (3). The governing equation and boundary conditions (BCs)are given byp  x   k 2 p  x   0 , x   0; L  ,(10)
BCs: p  0   n  ikp  0   1 and p  L   n  ikp  L   0 ,where k = ⁄$ is the wave number, and n is the outward normal vector of the boundary. The error
can be calculated by using the analytical solution for the problem p(x,k) = − (ieikx )⁄(2k). The error 2
is defined as
2  100  n1 pnSEM  pnref N2

 n1 pnref 
N

2

(11)

with pref being the analytical solution. The boundary value problem described by Eq. (10) is solved
using the SEM. By varying the number of DOFs per wavelength for a given interpolation order p we
can obtain Figure 3, from which the following conclusions can be drawn:
 For a low error threshold, such as 2 = 0.05%, the necessary number of nodes per wavelength
decreases significantly with increasing order. For lower error thresholds, which are not shown in
the figure, the effect is even more pronounced.
 For a high error threshold, the use of higher order p gives rise to less advantages.

Figure 3 – Number of nodes per wavelength N⁄' needed to satisfy a given error threshold2

Since the computational effort for calculating the element matrices increases significantly, thehighest order used for achieving a low error threshold should be p = 10 to avoid overcompensation.
However, for calculating a large number of frequencies the element matrices do not need to beevaluated repeatedly, therefore the most time-consuming part usually is the solution of the linear

260

system of equations. In Figure 4, the number of DOFs and the time for solving the linear system of
equations tLS using the direct solver PARDISO (16) are shown.

Figure 4 – Comparison of the DOFs (left) and solution time (right)
After evaluating the pressure field in the source room and the receiving room by solving Eq. (7),the sound reduction index can be calculated by
R  20 lg

plp0 20 lgprp0

 10 lg

S
Ae

(12)

with the sound pressure pl in the source room and pr in the receiving room, p0 = 20 ∙ 10-6 Pa, the areaof the specimen S and the equivalent absorption area Ae (14). Eq. (12) is evaluated for a high numberof frequencies for every 1/3 octave between 100 Hz and 3150 Hz. To assess the speed of the developed
algorithm, a comparison with the commercial FEM software COMSOL Multiphysics 5.3 (4) was doneusing a layered panel and the 1D error estimator (Figure 3) to determine the necessary meshing
resolution. The results are shown in Figure 5, in which the time ttot for the calculation of thetransmission loss curve is given. Only for a low error threshold, such as 2 = 0.001%, the advantageof the SEM using high order shape functions can be clearly observed. The efficiency increases forhigher frequencies (higher number of DOFs) and lower error threshold. The reasons for the algorithm
being faster than the commercial software for a high error threshold could be: Pre-calculation and storage of the data for the fluid domains, since they have to be calculated
only once and can be reused for further calculations.
 Computation of the element mass matrices using the Lobatto integration introduces a negligible
error but significantly reduces the number of the matrix operations for the assembly process foreach frequency. The time-saving effect here is larger than expected.
 For high orders of the shape functions and high frequencies the parallelized calculation of several
frequencies instead of the parallelized solution of the linear system of equations is more timeefficient and was used where appropriate.
 Use of different orders of the shape functions in different directions for thin elements reduces thenumber of DOFs for the considered structure (8).

Figure 5 – Comparison of COMSOL and SEM for different orders p261

Figure 6 shows a good agreement between the predicted and the measured sound reduction index
R for an aluminum plate. For the layered wall, the deviations are rather large. The reasons could be
uncertainties in the material properties and mainly due to lacking knowledge about the measurement
conditions including the exact size of the laboratory, the absorption properties and the mounting
conditions of the specimen. The latter problem occurs not only in a virtual laboratory but also in theexperimental determination of R. Meier et al. found in their inter-laboratory-test that the deviations
for the same specimen could be of up to 20 dB in the low frequency range and 4-8 dB in the mid andhigh frequency range (13).

Figure 6 – Comparison between the measurement (data from Villot et al. (21)) and calculation results by the
SEM and Berger’s mass law (14) of a thin aluminum plate (left), and comparison between the measurement(data from CBMA) and calculation results for a layered panel (see Figure 7)

Figure 7 – Examples of some examined complex layered structures, E is Young’s modulus in [GPa],the mass density in [kg⁄m3 ]is

4. OPTIMIZATION SCHEME
An integer value for the sound insulation capabilities can be calculated using a procedure describedin the ISO 717 (14). Since an optimization algorithm requires a rather smooth function, the procedure
was adjusted by using a smaller step-size to obtain a single value Rw with several decimals. This
provides a smooth curve of Rw (di ), such as shown in Figs. 8 and 9, instead of the discontinuous
staircase function when the ISO 717 is directly applied.
The eigenfrequencies of the laboratory and the specimen have a significant effect on thetransmission loss curve. To assess the necessary number of frequencies per 1/3 octave nt and thenecessary accuracy, a test specimen with a varying thickness dwall was examined (Figure 8). The
necessary error threshold was found to be 2 <1% and the number of frequencies per 1/3 octave should
be nt >20.

262

Figure 8 – Effects of the error threshold

2 and the number of frequencies per 1/3 octave nt

The designs 1 and 2 as shown in Figure 7 were optimized regarding their acoustic insulation
capabilities. The objective function and constraints are defined byFDesign 1  max Rw  d1 , d3  , FDesign 2  max Rw d 2 , d ag
subject to di  0.07, 0.005  d1  0.04, 0.002  d3  0.04

 di  0.26, 0.005  d 2  0.06, 0.0025  d ag  0.08, d1  d3

(13)
design 1.
design 2

The numerical calculations of the objective function for a high number of functional values require a large
computational effort. For a large search space, an evolutionary algorithm can be used. It is possible to find agood solution for Eq. (13) in less than 100 iterations respectively, which require a few hours for the
optimization procedure. For the sake of brevity, more details, however, are not presented here.

Figure 9 – Objective function for Design 1 (left) and Design 2 (right)

5. CONCLUSIONSIn this paper, a SEM in the frequency-domain is presented for calculating the transmission loss oflayered wall structures by using a virtual laboratory and a fully coupled model for considering thefluid-structure interaction. For high frequencies and high accuracy, the SEM is an efficient method toevaluate the acoustic pressure field in the fluid and the structural deformations. However, forcomparison with many experimental measurements, the calculation of the sound reduction index witha low accuracy is usually sufficient, diminishing some of the advantages of the SEM. For an
optimization algorithm, the objective function should be evaluated with higher accuracy, regarding
the error threshold of the calculation and the number of frequencies, to avoid random oscillations of
the transmission loss curve. The developed SEM algorithm here reduces the time required for thenumerical evaluation by up to 80% for layered wall structures. This allows for a higher number of
iterations and therefore for a larger search space.
The efficiency of the algorithm decreases for a geometrically complex structure, such as Design 2(Figure 7). To approximate the complex geometry a high number of small elements is needed. The
order of the shape functions is taken as identical for both the fluid and the structural domains andtherefore unnecessary DOFs are introduced, since the requirement for a given error threshold isalready fulfilled by much larger elements. This problem can be avoided by using different orders for263

the shape functions for the fluid and the structural domains, which require special coupling algorithms.
This is a part of our further research.

ACKNOWLEDGEMENTS
This work was partially supported by the International Science & Technology Cooperation
Program of China (2014DFA71290). The experimental data have been provided by the China Building
Materials Academy, Beijing, People's Republic of China.

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