Prediction of the sound transmission loss of two-dimensional periodic
structures with a hybrid framework
Carolina DECRAENE(1) , Edwin P.B. REYNDERS(2) , Geert LOMBAERT(3)(1) Structural Mechanics Section, Dept. of Civil Engineering, KU Leuven, Belgium, carolina.decraene@kuleuven.be(2) Structural Mechanics Section, Dept. of Civil Engineering, KU Leuven, Belgium, edwin.reynders@kuleuven.be(3) Structural Mechanics Section, Dept. of Civil Engineering, KU Leuven, Belgium, geert.lombaert@kuleuven.be

Abstract
The diffuse sound transmission through a complex ﬁnite-sized wall or ﬂoor can be efﬁciently computed witha hybrid approach. The wall is then modelled deterministically as ﬁnite size effects and modal behaviour canbe important in this frequency range, while the rooms carry a diffuse ﬁeld and are modelled as stochastic
subsystems. The ﬁnite element method is usually employed to compute the modal behaviour of the wall. At
higher frequencies, the computational cost then increases signiﬁcantly as a ﬁner mesh is required due to theshorter wavelength of structural deformation. For this reason, an alternative approach was recently developed,
which allows to replace the ﬁnite element model by an analytical model for ﬁnite-sized thick and layered walls.
However, the application of this approach is limited to layered structures such as sandwich panels or double
glazing. In the present work, the elaboration towards more complicated building elements which exhibit spatial
periodicity, will be considered by invoking periodic structure theory. As a ﬁrst validation example, a steel panel
is considered, in which the predictions are compared to other numerical and experimental data.
Keywords: sound transmission loss, periodic, hybrid

1

INTRODUCTION

Lightweight systems are more frequently used in buildings to achieve a high thermal performance and to savematerial and transportation costs. Because of their relatively low weight and complex vibro-acoustic behaviour,
achieving a sufﬁcient level of sound insulation with such systems is, however, challenging. The goal in thiswork is to create an accurate sound transmission loss prediction method, which is efﬁcient enough to be used
as a design tool and can replace the need of excessive experimental prototype testing.
The framework of the hybrid ﬁnite element-statistical energy analysis (FE-SEA) approach [1, 2, 3] serves as astarting point in this article. Using the diffuse ﬁeld reciprocity relationship [4, 5], the hybrid framework allows
modelling the rooms of the overall room-wall-room system as stochastic subsystems (like in statistical energy
analysis, SEA), while treating the wall in a deterministic way with ﬁnite element (FE) analysis. A diffuse ﬁeld
model is by deﬁnition a stochastic model. The uncertainty of the sound transmission loss concerning the diffuse
ﬁeld assumption in the SEA subsystems can be assessed [3]. This allows computing the mean of the soundtransmission loss as well as its variance. Note that the computation time of the FE-SEA method will increaseat higher frequencies as a ﬁner mesh is needed to capture the small wavelength deformations of the structure.
To overcome this limitation, two alternative approaches were already explored. First, the ﬁnite element analysis
could be replaced with a semi-analytical transfer matrix method (TMM) [6, 7, 8], which allows to predict the
behaviour of inﬁnite walls and ﬂoors, consisting of homogeneous solid, ﬂuid and/or poroelastic layers. By
projecting the wall displacements onto a set of sinusoidal lateral basis functions [9, 10], the modal behaviourof the structure is approximately taken into account. This recently developed approximate modal TMM-SEA
approach (mTMM-SEA) is however limited to layered systems [11].
Secondly, a hybridization between periodic ﬁnite element modelling (peFE) and statistical energy analysis wasachieved in [12]. The periodic lay out of most building elements could therefore be exploited. The latterpeFE-SEA approach has two challenges that must be tackled: (1) modal behaviour, which inﬂuences the sound1200

Figure 1. A room-wall-room system, in which the wall is modelled in a deterministic way while the rooms aremodelled as SEA subsystems.
transmission loss in the considered frequency range (50-3150 Hz), is disregarded; (2) computing the soundtransmission loss results in an integration over each phase constant surface in the phase lag domain and isthus computationally inefﬁcient. The two problems are solved by discretizing the phase domain into the phasecombinations causing standing waves in the structure. Ideas in this direction were already mentioned in [12].
Whereas the authors in [12] focus mainly on Born-von Kármán boundary conditions, the wall or ﬂoor in thepresent work is assumed to have simply supported boundary conditions, because modal testing has revealed that
simply supported boundary conditions occur in the small transmission opening of the KU Leuven Laboratoryof Acoustics [13]. Contributions of the present work include also a theoretical expansion of the method withspecial attention to implementation issues and a detailed validation example.
The remainder of this article is organized as follows. The theory is summarized in section 2. Section 2.1 introduces the hybrid deterministic-SEA approach to sound transmission modelling in general. Section 2.2 elaborates
periodic structure theory with emphasis on obtaining the necessary dynamic stiffness matrices. In section 3 themethod is validated for a steel plate by comparing the predicted sound transmission loss with other prediction
tools and with experimental data. The conclusions are to be found in section 4.

2

A HYBRID MODAL BASED PERIODIC FINITE ELEMENT-SEA APPROACH

2.1 The hybrid framework
Throughout this article, a room-wall-room system is considered, where the rooms carry a diffuse wave ﬁeld
and the wall a deterministic wave ﬁeld (cfr. Fig. 1). The out-of-plane displacement uz of the partition wall at
ϕ andposition x = (x, y) and at frequency ω is decomposed using a ﬁnite set of Ndof global basis functions ϕ̂corresponding generalized coordinates a:Ndof

ϕ p (x)a p (ω)
uz (x, ω) ≈ ∑ ϕ̂

(1)

p=1

All generalized response degrees of freedom (DOFs) are collected in the vector a(ω) ∈ CNdof , so that the timedomain response is given by Re(aeiωt ), with i the imaginary unit. Similarly, the corresponding generalizedharmonic loads are collected in the load amplitude vector f(ω) ∈ CNdof . Since a contains all (generalized)
interface degrees of freedom between the wall and the rooms, the equations of motion of the whole system(room-wall-room) can be written as
Da = f,(2)
with D ∈ CNdof ×Ndof the dynamic stiffness matrix at frequency ω. D may be decomposed into the dynamic
stiffness matrix of the wall, denoted as Dd (subscript d stands for deterministic), and the dynamic stiffness1201

matrices of the rooms, denoted as D1 and D2 :D = D d + D1 + D2

(3)

Since the rooms are assumed to carry a diffuse ﬁeld in this work, they are modelled as random subsystems in
the overall room-wall-room system. The dynamic stiffness matrix of each acoustic room volume k is decomposed as(k)(k)
Dk = Ddir + Drev ,k = 1, 2
(4)
(k)

(k)

where Ddir denotes the mean of the subsystem’s dynamic stiffness matrix Ddir := E [Dk ]. This term describes the
response of subsystem k to the outgoing wave ﬁeld caused by the displacements of the deterministic boundary[4, 14]. Boundaries and objects in the room induce however random wave scattering. The inﬂuence of this(k)reverberant ﬁeld on subsystem k is captured by Drev . With this decomposition, the equations of motion for arandom subsystem are(k)(k)
Ddir a = −fk − frev ,
(5)
(k)

(k)

where the reverberant forces are deﬁned as frev := −Drev a, and fk denotes the sum of the loads applied tosubsystem k at its DOFs. The overall equations of motion (2) become
(1)

(2)

Dtot q = f − frev − frev ,

(6)

(k)

where Dtot := Dd + ∑2k=1 Ddir is a purely deterministic matrix. When a diffuse ﬁeld acts in both rooms, the(k)(k)
reverberant forces frev are related to Ddir through the diffuse ﬁeld reciprocity relationship [4]:(k) (s)E[frev frev ] = δks

 (k) 4Êk
Im Ddir ,
ωπnk

(7)

with nk the modal density of subsystem k [15] and δks the Kronecker delta. Note that the hat symbol is
employed here as shorthand notation for ensemble mean. Using this equation, it is possible to obtain the meantime-averaged total energy Êk of room k from a stationary power balance which involves the other random
subsystems as well as the deterministic master system, assuming the ﬁelds are statistically independent of eachother. For the case where the external loading acts solely on the random subsystems (rooms), the power balanceis formulated as [1]:2ω ηk + η̂d,k Êk + ∑ ω η̂k j nk
j=1Êk Ê j−nk n j= P̂k ,

k = 1, 2.

(8)

In this expression, ηk is the damping loss factor of subsystem k, Pk the power input from external loading
injected directly into the diffuse ﬁeld of this subsystem, and




2
(k)
−H
Im (Dd,rs ) D−1∑
tot Im Ddir Dtot
ωπnk r,s
rs






2
(k)
( j)
−H
η̂k j =Im Ddir,rs D−1∑
tot Im Ddir Dtot
ωπnk r,srs

η̂d,k =

(9)(10)

where the superscript H denotes Hermitian transpose and the integer subscripts r, s select an element in row
r and column s of a matrix. When the sound insulation provided by the wall is sufﬁciently high, P̂k can beapproximated as the ensemble mean of the power input into a hard-walled room. It can be noted that the power
balance equation (8) has formally the same structure as in conventional SEA. Therefore the factors η̂k j canbe interpreted as coupling loss factors, and (10) provides a rigorous way to compute their ensemble mean. In

1202

a similar manner, (9) enables to rigorously estimate the power that is lost by energy dissipation in the wall.
By solving for the mean coupling loss factor η̂12 between the two subsystems, the mean sound transmission
coefﬁcient τ̂12 can be obtained from [15]
4ωV1
,
(11)
τ̂12 = η̂12ca Swith S the surface area of the wall, V1 the volume of the sending room, and ca the speed of sound. Note thatthe dynamic stiffness matrices are diagonal when cross modal coupling is neglected, in which case equation(11) can be rewritten as:

(2)
(1)
Im{Ddir,p }Im{Ddir,p }16π(12)
τ̂12 = 2 ∑Ska p|Dtot,p |2
where ka is the acoustic wavenumber.

Figure 2. Indication of the edge and corner degrees of freedom of a periodic unit cell. The red arrows show
the path of a travelling wave, reﬂected at the edges. Combination of these four waves result in a standing wave.

2.2 Determining Dd and Ddir using periodic structure theory
If a two-dimensional periodic structure with dimensions Lx × Ly is considered, it is possible to extract a periodic
unit cell which is repeated Nx and Ny times within the structure in respectively the x- and y-direction. A ﬁnite
element model of such a unit cell with dimensions lx and ly is shown in ﬁgure 2 where the local coordinate
system is denoted by x = (x , y ). The degrees of freedom q of the unit cell are reorganized into interior(I=interior), edge (T=Top, B=Bottom, R=Right, L=Left) and corner DOFs (LB, RB, LT, RT). The dynamic
response of the unit cell is then investigated using Bloch’s theorem, in which periodic boundary conditions are
imposed onto the edge and corner DOFs [16, 17, 18, 19, 20]. Deﬁning the phase lag combination (εxp , εyq ),1203

the relations between the boundary DOFs can be written in matrix format:
⎤ ⎡
⎤
⎡
I
0
0
0
qI
⎥
⎢ qB ⎥ ⎢0
I
0
0
⎥ ⎢
⎥
⎢
⎥⎡
⎢ qT ⎥ ⎢0 Ie−iεyq
⎤
0
0
⎥ ⎢
⎥ qI
⎢
⎥
⎢ qL ⎥ ⎢00
I0⎥ ⎢ qB ⎥
⎥ ⎢
⎢

−iεxp
⎥ = ⎢0
⎥⎢
⎥
0
0
Ie
q
q=⎢
R
⎥ ⎢
⎥ ⎣ qL ⎦ = Rq⎢⎥
⎢qLB ⎥ ⎢0
0
0
I
⎥
⎥ ⎢
⎢
−iεxp ⎥ qLB⎢qRB ⎥ ⎢0
0
0
Ie
⎥ ⎢
⎥
⎢
⎣ qLT ⎦ ⎣000
Ie−iεyq ⎦qRT000
Ie−iεxp −iεyq(13)with q = [qTI qTB qTL qTLB ]T the reduced DOFs. This reduction is then applied to the equations of motion of theunit cell as well, so that the following reduced eigenvalue problem is obtained:[K − ω 2 M ]q = 0(14)

where K = RH (εxp , εyq )KR(εxp , εyq ) and M = RH (εxp , εyq )MR(εxp , εyq ) are respectively the reduced mass andstiffness matrix of the periodic unit cell. As modal behaviour is investigated, only phase constants that allow
standing waves in the structure need to be considered. For a mode of the system, the summation of the phase
constants of a back and forward travelling wave in the global structure adds up to a multiplication of 2π, sothat [22]:εxp = pπ/Nx(p = 0, . . . , Nx )andεyq = qπ/Ny ,(q = 0, . . . , Ny )For each phase constant combination (εxp , εyq ), the eigenvalue problem in equation (14) is solved for a numberof possible corresponding eigenfrequencies Ω pq,n and corresponding eigenvectors ϕ pq,n (x ) as multiple waves
can propagate under an imposed periodic boundary condition. Note that the mode shapes of the unit cellare mass normalized: ϕ Hpq,n M ϕ pq,n = 1. Post-processing the results includes (1) selecting only the normal
displacements; (2) converting standing waves into travelling waves at the edges and corners of the phase domain.
This is necessary to guarantee the correct use of the proposed general method, which accounts for a twodimensional travelling wave; (3) eliminating waves with zero wavenumber in x- or y- direction as they do not
add up to a standing wave in one direction and (4) eliminating the phase shift of the travelling waves to ensure
zero displacements at the boundary of the global structure. The obtained dynamic properties of the unit cell are
now used to determine the direct ﬁeld response of the subsystems and the deterministic response of the wall,
which in their turn will be inserted into the hybrid framework.
First, the direct ﬁeld dynamic stiffness matrix Ddir,pq,n of standing wave pq, n is determined as follows:
Ddir,pq,n (ω) =in which G(ω, k, z = 0) = √ iω2 ρa 2
2

ka −||k ||1
4π 2

 ∞ ∞−∞ −∞ψ pq,n (k)|2 dk,G∗ (ω, k, z = 0)|ψ(15)

is the Green’s function with ρa the density of air. To avoid singular-

ity at coincidence, a Green’s function averaged in the wavenumber domain was used [21]. Assuming simply supported boundary conditions, the displacement ﬁeld ψ pq,n (x) consists of four travelling waves with theϕ (−p)q,n (x), ϕ̂ϕ p(−q),n (x) and ϕ̂ϕ (−p)(−q),n (x). Because
ϕ pq,n (x), ϕ̂same eigenfrequency as illustrated in ﬁgure 2: ϕ̂∗∗
ϕ pq,n and ϕ̂ϕ (−p)q,n = ϕ̂ϕ p(−q),n with ∗ indicating the complex conjugate, the standing wave in theϕ (−p)(−q),n = ϕ̂ϕ̂wavenumber domain k = (kx , ky ) is found by:
ϕ pq,n (k) + ϕ̂
ϕ ∗pq,n (−k) − ϕ̂ϕ p(−q),n (k) − ϕ̂ϕ ∗p(−q),n (−k)
ψ pq,n (k) = ϕ̂(16)

ϕ pq,n (k) is derived from the obtained eigenvectors of the unit cell
The travelling wave of the total structure ϕ̂
ϕ pq,n (k) using the following relation [12]: 
1 − e−iNx (εx −kx lx )1 − e−iNy (εy −ky ly )ϕ pq,n (k) = ϕ pq,n (k)
·
(17)
ϕ̂
1 − e−i(εx −kx lx )1 − e−i(εy −ky ly )

1204

50

Transmission loss [dB]

Transmission loss [dB]

50

40

30

20

10

30

20

10
16

(a)

40

31.5

63

125

250

500 1000 2000

Frequency [Hz]

63

(b)

125

250

500

1000

2000

Frequency [Hz]Figure 3. Harmonic sound transmission loss of a steel plate (a) as predicted with the hybrid modal peFE-SEA
approach (black continuous line), the FTMM (blue dashed line), the hybrid mTMM-SEA method for whichcross modal coupling is neglected (red line), the hybrid mTMM-SEA with cross modal coupling (black dottedline); and (b) as measured in the Acoustic Laboratory of the KU Leuven (blue continuous line).
ϕ pq,n (k) and ϕ p(−q),n (k) follow from a Fourier transform of ϕ pq,n (x ) and ϕ p(−q),n (x ). As ϕ p(−q),n (x ) can becomputed from ϕ pq,n (x ) by mirroring the displacement ﬁeld around the x-axis, the eigenvalue problem for thephase constant combination (p, q) only needs to be performed once. Note that the mesh in the x-domain needs
to be more reﬁned for the computation of the direct ﬁeld response than required for the FE analysis of theperiodic unit cell. Decoupling this so-called acoustic mesh and the FE mesh is therefore recommended.
Secondly, the deterministic dynamic stiffness matrix Dd,pq,n is determined by:


2
(18)
Dd,pq,n = 4Nx Ny (1 + iη)Ω pq,n
− ω2
with η the loss factor of the structure.

3

VALIDATION EXAMPLE: STEEL PLATE

The modal peFE-SEA is applied to a thin, simply supported steel plate. The steel plate has a density ρ =7750 mkg3 , Young’s modulus E = 200 GPa, Poisson’s ratio ν = 0.28 and dimensions 1.25 m×1.50 m×0.002 m. The
experimentally determined damping loss factor of the mounted plate can be found in [11]. In the simulations,the reverberation time T , the air density ρa and the sound speed ca in the rooms are taken to be 1.5 s, 1.2 mkg3
and 343 ms , respectively. Speciﬁcally for the considered peFE-SEA method, the steel panel is divided into squareperiodic unit cells with length 0.125 m. The ﬁnite element model of the periodic unit cell consists of 15
divisions in both directions. 8-nodes shell elements are used for the free wave propagation analysis. In order todetermine the direct ﬁeld response with sufﬁcient accuracy, the acoustic mesh has to be ﬁner for standing waves
with higher natural frequencies. The spatial gridspacing is therefore gradually reduced from 3.1 to 1.6 mm forincreasing natural frequencies. Contrarily, the wavenumber domain should be meshed ﬁner at lower frequencies
to capture the peak of the Green’s function and the gridspacing ranges from 0.1 to 1 rad/m over the frequency
range of interest.
Figure 3 (a) shows the narrow-band (1/48 octave bands) predictions of the modal peFE-SEA approach anddifferent alternative prediction tools, such as the TMM with a simpliﬁed spatial windowing technique (FTMM)[23] and the hybrid mTMM-SEA approach with and without cross modal coupling [11]. The FTMM only takes

1205diffraction effects into account, while the other approaches also include modal behaviour. Clearly, the modal
behaviour of the steel panel affects the sound transmission loss up to about 500 Hz as the FTMM prediction
coincides with the other methods from that frequency onwards. When comparing the modal peFE-SEA approachwith the mTMM-SEA method, a good agreement is found when cross modal coupling is neglected in themTMM-SEA method as well. The shift (less than 1 dB) between both methods is due to the acoustic meshreﬁnement in both methods and adopting a ﬁner acoustic mesh would reduce the discrepancy. Comparison
with the mTMM-SEA, that accounts for cross modal coupling, shows that the peaks at anti-resonances in thepeFE-SEA predictions do not occur, while the dips in the predicted sound transmission loss appear at the same
frequencies [11]. Figure 3 (b) compares the predictions with experimental data for which at lower frequencies
the modal behaviour of the rooms result in oscillations in the sound transmission loss. Above 250 Hz the
diffuse ﬁeld assumption in the SEA subsystems is justiﬁed and there is an overall good agreement.

4

CONCLUSIONSIn this work, a hybridization between two-dimensional periodic ﬁnite element modelling and statistical energy
analysis has been investigated for simply supported structures. In order to decompose the displacements of thestructure into standing waves, the dynamic properties of the unit cell only need to be computed for a speciﬁc
number of imposed propagation constants. The resulting travelling waves are combined next into standing waves
to compute the direct ﬁeld response, taking special care for zero displacements at the boundaries of the global
structure. As a validation example, the sound insulation predictions of a steel plate are compared with otherprediction tools and experimental data. The predicted sound transmission loss of the steel plate are in close
agreement with the measured sound insulation.

ACKNOWLEDGEMENTS
This research was supported by a PhD Fellowship (C. Decraene) from the Research Foundation - Flanders. The
ﬁnancial support of FWO is gratefuly acknowledged. The ﬁrst author also thanks Prof. Brian Mace for their
discussions at the University of Auckland during her research stay, funded by a travel grant of the FWO.

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