Acoustic topology optimization of the material distribution on a simply
supported plate
Jan VAN DEN WYNGAERT(1) , Mattias SCHEVENELS(2) , Edwin REYNDERS(1)(1) KU Leuven, Faculty of Engineering Science, Department of Civil Engineering, Kasteelpark Arenberg 40, B-3001 Leuven,Belgium. jan.vandenwyngaert@kuleuven.be, edwin.reynders@kuleuven.be(2) KU Leuven, Faculty of Engineering Science, Department of Architecture, Kasteelpark Arenberg 1, B-3001 Leuven, Belgium.

mattias.schevenels@kuleuven.beAbstract
Lightweight plates typically have a low sound reduction index (SRI) due to their low mass. Their SRI can beincreased by roughly 6 dB by doubling the thickness of the plate. The distribution of additional material in anon-uniform way may result in a higher increase in SRI. A good material distribution strategy is therefore needed
to reduce the transmitted sound in a certain frequency range. In this paper, the optimal material distribution ata single frequency and in a frequency band for a simple system was considered: a simply supported plate. TheSRI is modelled within the hybrid Finite Element-Statistical Energy Analysis (FE-SEA) framework to reduce
computational cost while maintaining a high prediction accuracy. The thicknesses of the elements comprising
the plate are used as optimization variables. The sensitivity of the SRI to the element thicknesses is derived
explicitly because a gradient based optimization algorithm is used. It is observed that the optimization algorithm
increases the SRI by shifting the eigenfrequencies of the plate and creating stop bands by locally adding massthat disturbed the mode shapes. In the case of a PMMA-plate, an increase up to 32.1 dB in the SRI is achieved
for the single frequency case and up to 29.90 for the frequency band case compared to 6 dB for the uniform
material distribution.
Keywords: topology optimization, airborne sound insulation, optimal material distribution, simply supported plate, PMMA plate

1INTRODUCTION

Lightweight plates typically have a low sound reduction index (SRI) due to their low mass. Following the masslaw, an increase of 6 dB can be obtained by doubling the mass of the plate for inﬁnitely large plates under thecoincidence frequency. By distributing this added mass onto the plate in a non-uniform way, a higher increase
in SRI can be attained.
Up until recently, the prediction of the airborne sound insulation of a simply supported plate had been either
too inaccurate or computationally too expensive to allow for numerical design optimization. To model the soundinsulation of the plates, the hybrid FE-SEA modelling framework [1, 2] is used. The transmission rooms aremodelled to carry a diffuse sound ﬁeld and are coupled to a ﬁnite element model of the plate to reduce thecomputational cost while retaining a high prediction accuracy.
In this paper, the optimal material distribution for a maximal sound insulation is considered. The sound insulation at a single frequency or the minimum of the sound insulation in a certain frequency band are used as
objective functions. A volume constraint is used to limit the total material used. The optimization problem issolved using the standard Optimality Criteria method [3].
The rest of this paper is structured as follows: in Section 2 the prediction of the transmission loss of a simply
supported plate is elaborated. The optimization problem is formulated in Section 3. The results of the optimization are shown in Section 4 and discussed in Section 5. Some conclusions and ﬁnal remarks are given inSection 6.

1208

2Transmission loss prediction

The plate is modelled deterministically to capture its vibration behavior in full detail while the sound ﬁelds inthe sending and receiving rooms are modelled as diffuse. The hybrid FE-SEA modelling framework [1, 2] isused to rigorously couple the diffuse sound ﬁelds in the transmission rooms to the deterministic wall model byemploying the diffuse ﬁeld reciprocity relationship [4]. In what follows, the deterministic model of the wall
system is explained ﬁrst. Subsequently, the interaction between the wall and the sound ﬁelds in the adjoining
rooms is described.
2.1 Plate model
The plate is a polymethyl methacrylate (PMMA) panel with density ρ = 1275 kg/m3 , modulus of elasticity
E = 4.5 GPa, Poisson’s ratio v = 0.35, a minimal thickness of 15 mm and dimensions 1x1 m2 . In the presentwork, the plate is modeled numerically, using the ﬁnite element method [5]. As the plate thickness is small
compared to its lateral dimensions, it can be adequately modelled with structural shell elements. The particular element type that is employed here is a four node DKT element with six degrees of freedom – threedisplacements and three rotations – at each node. Simply supported boundary conditions are assumed.
The model is then employed for computing the natural frequencies and mass-normalized displacement modeshapes of the simply supported plate. These numerically computed mode shapes are taken to be basis functions
for the computation of the transmission loss. The corresponding generalized coordinates are then the modalamplitudes of the plate. The equation of motion of the plate therefore reads
Dpl qpl = f̃pl ,(1)2Dpl,kk = −ω 2 + ωpl,k(1 + iηpl,k ),(2)where Dpl is a diagonal matrix with entries

with ωpl,k the numerically computed angular natural frequency of mode k the plate, with corresponding modeshape φ pl,k , and ηpl,k the corresponding structural damping loss factor.
2.2 Transmission room modelWithin the hybrid DET-SEA framework, a transmission suite (room-wall-room) model has been developed byReynders et al. [2, 6, 7]. In this model, which is adopted also in the present work, the rooms are taken to
carry a diffuse ﬁeld, while the wall is modeled deterministically. The hybrid model contains two diffuse (SEA)
subsystems - the sending and receiving rooms - and, in the context of a sound transmission analysis, the quantityof interest is the so-called coupling loss factor, η12 , between both rooms. Although the coupling loss factor is arandom quantity (because the sound ﬁelds in the rooms are random, diffuse ﬁelds), only its mean value will beof interest in the present analysis as the intention is to predict the mean sound insulation of the wall across theensemble of random sound ﬁelds. The coupling loss factor relates directly to the sound transmission coefﬁcient,τ, which is deﬁned as the ratio between the power ﬂow from room 1 to room 2 through the wall, and theincident sound power on the wall in room 1. The relationship at frequency ω reads [8]τ=4V1 ω
η12 ,Lx Ly c(3)where V1 denotes the volume of the sending room, Lx and Ly are the width and the height of the wall and cis the speed of sound in air. The sound insulation of the wall then immediately follows fromR := 10 log

Lx Ly c1= 10 log.
τ4V1 ωη12(4)1209The mean value of the coupling loss factor is obtained from [1]η12 =where    −H 2Im Ddir2 rs D−1∑tot Im Ddir1 Dtot rsπωn1 r,sDtot := Dpl + Ddir1 + Ddir2 .

(5)

(6)

Ddir1 and Ddir2 are the direct ﬁeld acoustic dynamic stiffness matrices expressed in terms of the generalized
wall coordinate vector qpl , and n1 is the modal density in the ﬁrst transmission room.
The direct ﬁeld response of a room is the sound ﬁeld that would occur if the room would be of inﬁnite extent,
in other words, if the room would behave as an acoustic half-space as seen from the room-wall interface whenthat interface is embedded in an inﬁnite planar bafﬂe. The related acoustic dynamic stiffness matrix is thentermed the direct ﬁeld dynamic stiffness matrix Ddir of the room. For room 1 for example, the direct ﬁeld
dynamic stiffness matrix Ddir1 describes the relationship between the displacements and forces at the interface
with the plate:(7)Ddir1 q1 = f̃dir1 .
where the components of f̃dir1 denote the forces acting on the (generalized) degrees of freedom of the plate q1
due to the pressure ﬁeld in the acoustic half-space.

3

Optimization methodology

The sound insulation of a plate, computed according to (4), is a property that depends on the frequency, theplate setup and the rooms separated by the element. The airborne sound insulation is modelled in laboratory
conditions that comply with the ISO 10140-2:2010 standard [9]. This standard deﬁnes among others the volumeof the transmission rooms and the size of the plate section. In this paper the volume and measurement openingof the KU Leuven Laboratory of Acoustics are considered.
3.1 Parametrization
In this paper, the plate thickness of a simply supported PMMA plate is optimized for an optimal airborne
sound insulation. The PMMA-plate is divided into shell elements using a square grid. The individual element
thicknesses, x, are used as design variables.
3.2 Optimization problem and constraints
The sound insulation can be evaluated at a single number frequency of in a frequency band. When the soundinsulation is evaluated in a frequency band either the sum of the individual sound insulations can be optimizedor the minimum sound insulation in the frequency band can be maximized. In this paper, the minimum soundinsulation is optimized in the case of a frequency band.
maxx

min(R(x, f), f ∈ D)s.t.

∑ xi Ai ≤ 2Atot tmin

(8)

itmin ≤ xi ≤ 4tmin
with Atot the total surface area of the plate and D the frequency band of interest.
The sound insulation of the simply supported plate is optimized under a volume constraint. The minimal thickness of the simply supported plate is tmin = 15 mm and ensures there is always a minimum amount of material
between the transmission rooms. The total volume of the plate cannot exceed the volume of a plate withhomogeneous thickness 2tmin . The maximal thickness of the plate cannot exceed 4tmin , as shown in Eq. (8).

1210

Table 1. Comparison of the Sound Reduction Index (SRI) shows an increase up to 29.9 dB in the case of afrequency band and 32.1 dB for a single frequency with respect to the reference plate.
Case123456Frequency (band) [Hz]
53134
210
50-100100-150
50-300

Reference SRI [dB]
0.00
16.03
33.28
0.00
15.90
0.00

Optimized SRI [dB]
32.12
45.90
45.67
29.90
34.05
17.35

The optimization problem is solved using the standard Optimality Criteria method [3]. As a starting point ofthe optimization, a plate with a uniform thickness of 2tmin is used. The optimization is performed on a 21x21grid.
In what follows, six setups are considered: the optimization of the sound insulation at a single frequency being
either the ﬁrst, second or third eigenfrequency of the plate with a uniform thickness 2tmin , and the optimization
in a frequency band of 50-100 Hz, 100-150 Hz and 50-300 Hz.
3.3 Sensitivity analysis
By direct analytical derivation, it can be shown that, after some algebra, the sensitivity of the coupling loss
factor can be written as   ∂ η12∂ Dtot
∂φ4−TT=Re ∑(−Dtot AC )rs+ 2 Im ∑(E Re(C))rs,(9)
∂ xi
πωn1
∂ xi rs∂ xi rsrsrswith   −H
A := Im Ddir2 , C := D−1
tot Im Ddir1 Dtot and E := Ddir2 φ pl .

(10)with Ddir2 the direct ﬁeld dynamic stiffness matrix in room 2 in terms of the physical degrees of freedom ofthe plate.
The derivative of the total dynamic stiffness matrix can be elaborated as:∂ Dpl ∂ Ddir1 ∂ Ddir2∂ Dtot=++∂ xi∂ xi∂ xi∂ xiT∂φ∂ Dpl∂ φplpl=+2
Ddir1 φpl + 2φplT Ddir1,
∂ xi
∂ xi
∂ xi
with

∂ ωpl,k∂ Dpl,kk= 2ωpl,k (1 + iηpl,k ).
∂ xi∂ xi(11)
(12)

(13)

The gradients of the angular natural frequencies and mode shapes are computed according to [10].

1211Transmission loss [dB]5040302010063a)

125

250

500

Frequency [Hz]Transmission loss [dB]50403020100b)

63

125

250500Frequency [Hz]Transmission loss [dB]

50403020100c)

63125

250500Frequency [Hz]Figure 1. Thickness and sound insulation curve of of the plate optimized at a) 53, b) 132 Hz and c) 210 Hz.
The sound insulation of the optimized plate is shown in red and the sound insulation of the reference plate isshown in blue. The frequency at which the optimization is performed is indicated with a dot.

4Results

The sound insulation of the optimized plates is compared to a reference case where the plate has a uniform
thickness of 2tmin , which is the maximal mass that still satisﬁes the volume constraint. The sound insulation isrespectively given in blue for the reference case and in red for the optimized plates in Figs. 1-2.

5Discussion

In the ﬁrst case, the sound insulation is optimized at 53.5 Hz. This single frequency corresponds to the ﬁrst
eigenfrequency of the plate in the reference case. The optimized shape, shown in Fig. 1.a, exhibits stiffening in
the corners and the middle of the plate. This increases the ﬁrst eigenfrequency, resulting in a stiffness controlledzone.
For the second case, the sound insulation is optimized at the second and third eigenfrequency of the reference
plate at 134 Hz in the reference case. The optimized shape, shown in Fig. 1.b, displays a single centralizedmass. This results in an antiresonance at 134 Hz.

1212  a)







Transmission loss [dB]

5040302010063b)

125

250

500

Frequency [Hz]Transmission loss [dB]

50403020100c)

63

125

250

500

Frequency [Hz]Figure 2. Thickness and sound insulation curve of of the plate optimized in a frequency band of a) 50-100 Hz,b) 100-150 Hz and c) 50-300 Hz. The sound insulation of the optimized plate is shown in red and the soundinsulation of the reference plate is shown in blue. The frequencies at which the optimization is performed areindicated with a dot.
In the third case, the sound insulation is optimized at the fourth eigenfrequency of the reference plate whichcorresponds to the 2-2-mode in the reference case. The optimized shape, shown in Fig. 1.c. The eigenfrequencyof the 2-2-mode of the optimized plate is shifted from 210 to 174 Hz. This results in an antiresonance at
210 Hz.
In the fourth case, the sound insulation is optimized in the frequency range of 50-100 Hz which comprises only
the ﬁrst mode in the reference case. The minimum of the sound insulation in this band is maximized. The
optimized shape, shown in Fig. 2.a, shows four regions where the material is arranged around a central part.
The two horizontal gaps between the bottom, three middle and top masses, and the two vertical gaps between
the left, three middle and right masses result in a low plate bending stiffness in both directions. This results ina ﬁrst eigenfrequency of 26.4 Hz. The second and third eigenfrequencies, at respectively 117.4 and 117.6 Hz,
stay relatively close to the original eigenfrequencies of the homogeneous plate and out of the frequency bandof interest. This results in a stop band between 27 and 117 Hz with a high sound insulation.
In the ﬁfth case, the sound insulation is optimized in the frequency range of 100-150 Hz, comprising two
coincident modes in the reference case. The minimum of the sound insulation in this band is maximized. The
optimized shape, shown in Fig. 2.b, shows two regions of concentrated material and a central line. By shaping
the added material as ellipses, the eigenfrequencies of the 2-1- and 1-2-bending mode of the plate no longer
coincide since the bending stiffness in the two orthogonal directions is different. The large dip in the sound1213

insulation at 132 Hz for the homogeneous plate, shown in Fig. 2.b, gets shifted to 170 Hz. This dip corresponds
to the 2-1-bending mode of the plate. The eigenfrequency of the 1-2-mode of the plate is shifted to 100 Hz.
This optimal shape also shows good results in the 50-100 Hz range since the ﬁrst eigenfrequency is also shiftedto the low frequency range, though the sound insulation is lower than the optimal distribution of the ﬁrst case.
In the sixth case, the sound insulation is optimized in the frequency range of 50-300 Hz, comprising six modesin the reference case, of which two are coincident modes. The minimum of the sound insulation in this bandis maximized. The optimized shape, shown in Fig. 2.c, shows an irregular distribution of the material. The ﬁrst
mode is shifted down out of the relevant frequency range. The eigenfrequencies of the 2-1- and 1-2-bending
mode of the plate no longer coincide since the bending stiffness in the two orthogonal directions is different.
This results in a dip in the sound insulation that is less shallow compared to the reference case. The irregularpattern also disturbs the 2-2-bending mode resulting in only a small dip in the sound insulation at 153 Hz.

6Conclusions

In this paper, the airborne sound insulation of a simply supported plate is maximized by optimizing the plate
thickness under a maximal material constraint. The sound insulation of the plate is modelled in the hybrid
FE-SEA framework to lower the computational cost while retaining a high accuracy.
By shaping the added material to either increase or decrease the plate bending stiffness in different ways toshift modes or split up coinciding modes, the sound insulation is effectively maximized.The sound insulationshows an increase up to 32.1 dB for the single frequency optimization and 29.9 dB for the frequency band
optimization.

Acknowledgments
The research presented in this paper has been performed within the framework of theresearch project G0A7717N AcOpt: Numerical optimization of the airborne sound insulation of single anddouble walls, funded by the Research Foundation - Flanders (FWO), Belgium, and the research project ERC2016-STG: Virtual building acoustics: a robust and efﬁcient analysis and optimization framework for noisetransmission reduction, funded by the European Research Council (ERC). The ﬁnancial support by FWO andERC is gratefully acknowledged.

1214

References[1] P.J. Shorter and R.S. Langley. Vibro-acoustic analysis of complex systems. Journal of Sound and Vibration,288(3):669–699, 2005.
[2] E. Reynders, R.S. Langley, A. Dijckmans, and G. Vermeir. A hybrid ﬁnite element - statistical energy
analysis approach to robust sound transmission modelling. Journal of Sound and Vibration, 333(19):4621–4636, 2014.
[3] M. P. Bendsøe and O. Sigmund. Optimization of structural topology, shape, and material, volume 414.
Springer, 1995.
[4] P.J. Shorter and R.S. Langley. On the reciprocity relationship between direct ﬁeld radiation and diffuse
reverberant loading. Journal of the Acoustical Society of America, 117(1):85–95, 2005.
[5] K.J. Bathe. Finite Element Procedures. Prentice-Hall, Englewood Cliffs, NJ, second edition, 1996.
[6] C. Decraene, A. Dijckmans, and E.P.B. Reynders. Fast mean and variance computation of the diffuse
sound transmission through ﬁnite-sized thick and layered wall and ﬂoor systems. Journal of Sound andVibration, 422:131–145, 2018.
[7] E. Reynders, C. Van hoorickx, and A. Dijckmans. Sound transmission through ﬁnite rib-stiffened andorthotropic plates. Acta Acustica united with Acustica, 102(6):999–1010, 2016.
[8] M.J. Crocker and A.J. Price. Sound transmission using statistical energy analysis. Journal of Sound andVibration, 9(3):469–486, 1969.
[9] International Organization for Standardization. ISO 10140-2: Acoustics – Laboratory measurement of soundinsulation of building elements – Part 2: Measurement of airborne sound insulation, 2010.
[10] Siddhi Vijendra. A generalized approach for calculation of the eigenvector sensitivity for various eigenvector
normalizations. PhD thesis, University of Missouri – Columbia, 2005.

1215