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Accelerated vibro-acoustics of porous domains via a

novel coupled multiscale ﬁnite element method

A. Sreekumar 1, S. P. Triantafyllou 2, F.-X. B´

ecot 3, F. Chevillotte 3, L. Jaouen 3

1University of Nottingham, Faculty of Engineering,

Centre for Structural Engineering and Informatics, UK

2National Technical University of Athens, School of Civil Engineering,

Institute for Structural Analysis and Aseismic Research, Greece

e-mail: savtri@mail.ntua.gr

3Matelys - Research Lab

7 rue des Maraˆ

ıchers Bˆ

atiment B, Vaulx-en-Velin, 69120, France

Abstract

The use of foam materials as a means of improving the sound absorption and transmission properties of

structures, especially at low frequencies, has been receiving considerable attention over the past years. Un-

fortunately, the underlying complex material domain that may include solid inclusions, renders numerical

simulation using the standard FEM a computationally taxing task. Within the taxonomy of multiscale sim-

ulation methods, the Coupling Multiscale Finite Element Method (CMsFEM) has been recently introduced

as a means of reducing computational costs for the consolidation analysis of highly heterogeneous domains.

In this work, we originally apply the CMsFEM framework to the Biot theory of elastic wave propagation

in air-saturated porous media. The resulting numerical scheme allows inclusions of air and other potential

sources of heterogeneity at the mesoscopic scale. The performance and accuracy of the method, along with

applications to materials with inclusions are explored using an in-house MATLAB code.

1 Introduction

1.1 Overview

The vibro-acoustic performance of porous materials is controlled by structural and visco-inertial-thermal

dissipation effects of the solid skeleton and pore-ﬂuid, respectively. In several cases the solid skeleton is

quite stiff and the elastic effects may be neglected under the rigid skeleton assumption. In such cases, the

classical Helmholtz theory [1] can be applied to compute pore-ﬂuid pressures. This theory however cannot

capture the signiﬁcant resonance effects that are manifested due to the deformability of the solid skeleton. In

such cases the modiﬁed Biot equations [2] have to be employed.

The Biot theory provides a phenomenological model that predicts the behaviour of waves propagating inside

a fully-saturated porous material at the meso-scale. Considering that the wavelengths under examination are

much larger than the average pore diameter, a periodically repeating unit cell, representative of the entire

domain (also known as an RVE) is constructed. Effective material parameters are obtained over this RVE

through homogenization schemes [3].

In vibro-acoustics, all dissipative effects are captured using complex parameters. Structural dissipation is

accounted for by the loss factor ˜ηs(ω). Fluid dissipation effects can be either viscous or thermal in character

at the microscopic level. The dynamic density ˜ρeq(ω)and dynamic bulk-modulus ˜

Keq(ω)capture these

effects.

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The model, as originally formulated by Biot [4, 5] is a u−Uformulation, i.e., the primary ﬁeld variables

are the solid-skeleton and pore-ﬂuid displacements. Hence, using a standard ﬁnite element approach to

discretize and solve the governing equations results in 4 degrees of freedom (DOFs) per node in 2-D and 6

DOFs per node in 3-D. An alternate mixed u−p formulation was proposed in [6], where the primary ﬁeld

variables are solid-skeleton displacements and pore-ﬂuid pressures, respectively. In this case, only 3 DOFs

per node in 2-D and 4 DOFs per node in 3-D are required hence resulting in a signiﬁcant reduction to the

corresponding computational costs. However this alternate formulation requires a reworking of the original

Biot parameters.

The formalism developed in [7], overcomes this limitation by providing a robust methodology to decouple

the computation of the dissipation parameters from the rest of the terms. This allows one to insert any

chosen dissipation model into the equations. This is desirable, as estimation of macroscopic parameters for

porous materials is a challenging task. For instance, when one does not have all the parameters required in

a six parameter semi-phenomenological dissipation model (the Johnson, Champoux, Allard, Lafarge model

(JCAL)) [8, 9, 10], one can instead choose a model for which all required parameters are indeed available,

e.g. the one parameter Delaney-Bazley-Miki model (DBM) [11, 12].

Upscaling techniques are used in numerical methods to drive down computational costs incurred when mod-

elling physical phenomena exhibited by porous and composite materials.

Figure 1: Motivation for multiscale approaches

Such materials have observably different scales as illustrated in Fig. 1. They can include complicated pore

geometries created by ﬂuid-solid skeleton interfaces, and varying material properties, e.g, elastic moduli,

porosity, permeability. These variations can have a measurable impact on the behaviour of the material

under consideration, at all scales. It is evident that numerical approaches that can account for these vari-

abilities in micro-structural conﬁgurations can provide valuable insights into mechanical and vibro-acoustic

characterizations.

The transfer matrix method (TMM) [13] is used to model acoustic wave propagation in layered media. These

can include solid, porous and ﬂuid layers. Matrices are used to describe the wave propagation through each

medium and the coupling constraints at layer interfaces. Semi-inﬁnite ﬂat surfaces are assumed for all layers.

A signiﬁcant shortcoming of this method is its inability to account for heterogeneous non-planar conﬁgu-

rations. Alternative methods, relying on double porosity theory [14, 15] were proposed to model anechoic

wedges and heterogeneous porous composites, respectively. It is however, still insufﬁcient to model complex

shapes with varied heterogeneous meso-scale inclusions. Such involved material layouts necessitate the use

of numerical techniques such as the Finite Element Method (FEM).

Classical FE approaches require an explicit resolution of all complex ﬁne-scale morphologies. This can

be prohibitively expensive even when studying static behaviour. Dynamic or spectral approaches used for

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Table 1: Material parameters used in governing equations

Parameter Description

E elastic Young’s modulus

νPoisson’s ration

ηs(ω)structural loss factor

˜ρ(ω)modiﬁed Biot density

˜ρeq(ω)dynamic mass density

˜

Keq(ω)dynamic bulk modulus

˜γ(ω)coupling factor

studying evolving phenomena, e.g., consolidation and sound absorption, further exacerbate this problem,

especially in the ﬁeld of acoustic topology and shape optimization problems [16, 17].

Heterogeneous multiscale methods seek to address this by deﬁning coarse-elements that cluster their own

set of ﬁne elements. The coarse-element is understood as a generalization of the notion of RVEs. The

more typical notion of being representative of the entire domain is recovered in the periodically repeating

limit. The ﬁne-scale heterogeneities are directly mapped to the coarse-scale through a set of multiscale basis

functions.

1.2 Governing Equations

In the following, we employ the mixed u−p formulation adopted by [2] and [7]. The momentum balance

equations for both phases are expressed as:

div(σs) + ω2˜ρ(ω)u=−˜γ(ω)∇p (1a)

∆p

˜ρeq(ω)+ω2p

˜

Keq(ω)=ω2˜γ(ω)div(u),(1b)

where σs=˜

D(ω)ε(u)denotes the in-vacuo stress tensor.

The elastic constitutive tensor ˜

D(ω)depends on the Young’s modulus E, the Poisson’s ratio ν, and the

structural loss factor ηs(ω), i.e., ˜

D(ω)≡˜

D(E, ν, ηs(ω)).(2)

The terms ˜ρ(ω)and ˜γ(ω)represent a modiﬁed Biot density and coupling factor, respectively.

These equations are efﬁciently solved in the frequency domain. The material parameters used are sum-

marized in Table 1. The (˜

·)symbol denotes the complex-valued nature of those parameters. There is a

parametric dependence of the complex-valued parameters involved, on the angular frequency ω. This is

consistent with experiments, as dissipation effects are frequency dependent.

1.3 Finite Element Formulation

Following the weak formulation proposed in [2] the following coupled system of equations is obtained, i.e.,

˜

K−ω2˜

M−˜

C

−ω2˜

CT˜

H−ω2˜

Qu

p=fu

fp(3)

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Figure 2: Schematic diagram of a ﬂat poroelastic layer in an impedance tube with rigid wall backing. The

material is subject to a normal incidence plane acoustic wave.

where the state matrices have the following forms:

˜

K=ZΩ

BT˜

D(ω)BdΩlinear elastic stiffness matrix

˜

M=ZΩ

NT˜ρ(ω)NdΩmass matrix

˜

H=ZΩ

(∇Np)T1

˜ρeq(ω)(∇Np)dΩpermeability matrix

˜

Q=ZΩ

NT

p

1

˜

Keq(ω)NpdΩcompressibility matrix

˜

CT=ZΩ

NT˜γ(ω)∇NpdΩcoupling matrix.

Eq. 3 reduces to the simpler Helmholtz equivalent ﬂuid representation in the limit of rigid skeletons or

inﬁnitely soft skeletons (in the latter case however, one would end up with a different expression of the mass

density). The boundary conditions are ﬁrst speciﬁed for a ﬂat layer subject to impedance tube conditions, as

shown in Fig. 2:

u= 0,on Γ3:bonded conditions (4a)

uy= 0,on Γ2∪Γ4:sliding conditions (4b)

p=p0,on Γ1:free end conditions (4c)

The constraints will change with the nature of the medium the poroelastic is coupled with (e.g., elastic solid,

ﬂuid, poroelastic, rigid-wall etc.). For the study of absorption in poroelastic materials, there is no external

load. As a result, it is sufﬁcient to have a zero vector in place of the load vectors fuand fp. The free-end

pressure constraint is a simpliﬁcation of more complicated impedance-type Robin boundaries.

2 A heterogeneous multiscale method for vibro-acoustics

The Coupled Multiscale Finite Element Method [18], has been developed to drive down computational costs

when dealing with mechanical phenomena in highly heterogeneous two-phase porous media. In principle, to

accurately resolve heterogeneities, a very ﬁne mesh discretization is required, often leading to prohibitively

expensive computations. The CMsFEM alleviates this problem by using a second, coarser, mesh; each coarse

element clusters its own portion of the underlying ﬁne mesh. Fine-scale details are mapped onto the coarse

mesh through an upscaling procedure using numerically derived multiscale basis functions. The solution

procedure is ﬁnally performed at the coarse scale, thereby signiﬁcantly reducing computational complexity.

In this work, the Biot (u−p) formulation in the ω-domain, as described in Section 1.2 is originally introduced

within the multiscale framework to achieve these objectives.

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Figure 3: Visualizing multiscale basis functions for a heterogeneous domain

2.1 Evaluation of multiscale basis functions

The heterogeneous domain shown in Fig. 3 comprises nM= 36 coarse nodes and nMel = 25 quadrilateral

coarse elements. Each coarse element contains nm= 81 ﬁne nodes and nmel = 64 ﬁne elements. In each

coarse element a random distribution of elastic properties is assumed sampled from a normal distribution as

shown in Fig. 3. We restrict our discussion to the 2-D case, however, generalisation to 3-D is straightforward.

Assuming a generalized representation for the abstract weak form of the governing equations:

Find (u,p)∈ V × W :

a(u,v) = F(v),∀v∈ V,

b(p,w) = G(w),∀w∈ W,

(5)

the multiscale basis functions are evaluated through the solution of the piece-wise continuous homogeneous

version of Eq. 5 over each coarse element domain ΩM(α), α = 1, . . . nMel for the solid and ﬂuid phase,

respectively, as follows:

Solid Phase

Find u∈ V(ΩM(α)) :

a(u,v)=0,∀v∈ V(ΩM(α))

u|∂Ω(M(α)) =¯

u,

(6)

Fluid Phase

Find v ∈ W(ΩM(α)) :

b(p,w)=0,∀w∈ W(ΩM(α))

p|∂Ω(M(α)) =¯

p,

(7)

where Vand Wrepresent the space of all trial and test functions for the solid and ﬂuid phases, respectively.

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The terms ¯

uand ¯

p denote the prescribed displacement and pressure ﬁelds over the Dirichlet boundary. The

solutions uand p are the static modes of the coarse element and are equivalent to the multiscale basis

functions Nu

iand Np

i,i= 1, . . . 4.

The prescribed boundaries ¯

uand ¯

p can inﬂuence the shape of the basis and are therefore critical to the

accuracy of the method. Assuming the barycentric property Nu,p

i(xj) = δij, where xj,j= 1, . . . 4denote

coarse-nodal coordinates, these prescribed kinematical constraints are broadly classiﬁed into (a) linear and

(b) periodic boundary conditions.

Linear boundary conditions often result in coarse elements that over-estimate the stiffness of the underlying

micro-structure particularly in the case of highly oscillatory material coefﬁcients. Periodic boundary con-

ditions relax this restriction although necessitating the periodicity of the underlying mesh hence restricting

their applicability to periodic domains. Such an approach is not valid when encountering randomly deﬁned

material distributions or dealing with more ﬂexible polygonal/polyhedral RVE deﬁnitions.

Oscillating boundary conditions have been shown to alleviate the aforementioned issues [19, 20]. In these,

a reduced version of the governing Eqs. 6 and 7 is solved over the boundary under consideration hence ac-

counting for the heterogeneities along the boundary. Furthermore, it can be easily shown that linear bound-

aries arise naturally as a limit case when dealing with homogeneous material deﬁnition along the boundary.

In the case where, the coarse and ﬁne length scales approach each other, resonance errors start to propagate

[20]. This problem is alleviated by evaluating the basis over a larger domain that encapsulates ΩM(α); this

approach is known as an oversampling strategy.

The discretized matrix forms of Eqs. 6 and 7 are now expressed:

(˜

Kα

m(ωk)uα

m={∅} , on KM(α)

uS=¯

uIJ , on ∂KM(α)

, I = 1 . . . nM, J = 1,...ndim, k = 1 . . . nfreq (8)

(˜

Hα

m(ωk)pα

m={∅} , on KM(α)

pS=¯

pIJ , on ∂KM(α)

, I = 1 . . . nM, J = 1,...ndim, k = 1 . . . nfreq (9)

where ndim =2 or 3, depending on the dimension of the problem and nfreq denotes the number of frequency

steps. The multiscale basis functions are iteratively evaluated for each frequency.

2.2 Upscaling procedure

The RVE speciﬁc vectors of nodal displacements uα

m=umx, umy Tand pressures pα

mare associated

with the corresponding coarse-element ﬁeld variables through the following Eqs.:

uα

m(i)=Nu

m(i)uM(α)(10a)

pα

m(i)=Np

m(i)pM(α),(10b)

where uα

m(i)and pα

m(i)denote the displacement and pressure vectors for the ith ﬁne-element in the αth

element. The arrays Nu

m(i)and Np

m(i)represent the multiscale basis functions mapping the αth coarse-element

nodal displacements uM(α)and pressures pM(α)to the ﬁne-scale, respectively.

Collecting the contributions from each ﬁne-element, Eq. (10) can be expressed over the entire RVE:

uα

m=Nu

muM(α)(11a)

pα

m=Np

mpM(α),(11b)

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Using Eq. 3, the governing equations at each frequency ωkare expressed for each micro-element as

"˜

Kel,α

m(i)(ωk)−ω2

k˜

Mel,α

m(i)(ωk)−˜

Cel,α

m(i)(ωk)

−ω2

k˜

Cel,α

m(i)(ωk)T˜

Hel,α

m(i)(ωk)−ω2

k˜

Qel,α

m(i)(ωk)#(uα

m(i)

pα

m(i))="fu el,α

m(i)

fp el,α

m(i)#(12)

Substituting the micro to macro mapping relations from Eq. (10) into Eq. (12), and multiplying the ﬁrst

row-set of equations by Nu T and second row-set of equations by Np T , the following equation is obtained:

"˜

Kel

M(α),m(i)−ω2

k˜

Mel

M(α),m(i)−˜

Cel

M(α),m(i)

−ω2

k˜

Cel T

M(α),m(i)˜

Hel

M(α),m(i)−ω2

k˜

Qel

M(α),m(i)#uM(α),m(i)

pM(α),m(i)="fu el

M(α),m(i)

fp el

M(α),m(i)#,(13)

where ˜

Kel

M(α),m(i),˜

Mel

M(α),m(i),˜

Hel

M(α),m(i),˜

Qel

M(α),m(i), and ˜

Cel

M(α),m(i)correspond to the ﬁne-element state

matrices mapped onto the coarse element nodes and assume the following form

˜

Kel

M(α),m(i)=NuT

m(i)˜

Kel,α

m(i)Nu

m(i)(14a)

˜

Mel

M(α),m(i)=NuT

m(i)˜

Mel,α

m(i)Nu

m(i)(14b)

˜

Hel

M(α),m(i)=NpT

m(i)˜

Hel,α

m(i)Np

m(i)(14c)

˜

Qel

M(α),m(i)=NpT

m(i)˜

Qel,α

m(i)Np

m(i)(14d)

˜

Cel

M(α),m(i)=NuT

m(i)˜

Cel,α

m(i)Np

m(i)(14e)

In Eqs. (13) and (14) the dependence of the state matrices on (ωk)is omitted for brevity. Similarly, the

forcing terms assume the following form

fu el

M(α),m(i)=NuT

m(i)fu el,α

m(i)(15)

fp el

M(α),m(i)=NpT

m(i)fp el,α

m(i)(16)

for the nodal forces and outﬂows, respectively.

On the basis of the principle of energy equivalence between the coarse element and its ﬁne-scale discretiza-

tion, the following relations between the micro-macro transition state matrices and vectors and their local

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coarse element representations are established:

˜

Kel

M(α)=

nmel

X

i=1

˜

Kel

M(α),m(i)(17a)

˜

Mel

M(α)=

nmel

X

i=1

˜

Mel

M(α),m(i)(17b)

˜

Hel

M(α)=

nmel

X

i=1

˜

Hel

M(α),m(i)(17c)

˜

Qel

M(α)=

nmel

X

i=1

˜

Qel

M(α),m(i)(17d)

˜

Cel

M(α)=

nmel

X

i=1

˜

Cel

M(α),m(i)(17e)

fu el

M(α)=

nmel

X

i=1

fu el

M(α),m(i)(17f)

fp el

M(α)=

nmel

X

i=1

fp el

M(α),m(i)(17g)

The local coarse element state matrices and vectors shown in Eqs. (17) can be assembled over the coarse

domain using standard assembly operations, i.e.,

KM=

nMel

A

α=1 Kel

M(α),(18a)

MM=

nMel

A

α=1 Mel

M(α),(18b)

HM=

nMel

A

α=1 Hel

M(α),(18c)

QM=

nMel

A

α=1 Qel

M(α),(18d)

CM=

nMel

A

α=1 Cel

M(α),(18e)

fu

M=

nMel

A

α=1 fu,el

M(α),(18f)

fp

M=

nMel

A

α=1 fp,el

M(α).(18g)

Hence, the upscaled global governing equations assume the following form:

˜

KM−ω2

k˜

MM−˜

CM

−ω2

k˜

CT

M˜

HM−ω2

k˜

QM

| {z }

˜

ZM

uM

pM

| {z }

XM

=fu

M

fp

M,

|{z }

FM

(19)

where the unknown ﬁeld vectors uMand pMdenote the coarse-nodal displacements and pressures, respec-

tively.

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2.3 Solution procedure at the coarse scale

The coarse-scale governing equations Eq. (19) are solved repeatedly over each frequency step.

XM(ωk) = ˜

ZM(ωk)−1FM,for k=1:nfreq (20)

The complex impedance at each frequency can now be computed at the incident surface:

˜

Zn(ωk) = −p|Γi

vn|Γi

,(21)

where p|Γiand vn|Γidenote the average pressure and average normal ﬂuid velocity at the incident surface

Γi, respectively. The average normal ﬂuid velocity is computed as follows:

vn|Γi=jωkUn,(22a)

Un=PFg|Γi

Γi

,(22b)

Fg=˜

ZM(ωk)XM(ωk)/ω2

k,(22c)

where Zair represents the impedance of air. The sound absorption coefﬁcient α(ωk)is ﬁnally computed in

terms of the complex frequency dependent impedance ˜

Zn(ωk)as follows:

α(ωk) = 1 −

˜

Zn(ωk)−Zair

˜

Zn(ωk) + Zair

2

.(23)

2.4 Downscaling

While the SAC is computed at the coarse scale in Eq. (23). It is also possible to compute it from the ﬁne-scale

information, to offer a more detailed account of the underlying heterogeneities.

The ﬁne-scale displacements and pressures at each time-step can be evaluated from the solution of Eq. (20) by

employing the following down-scaling procedure. The coarse element-wise displacements and pressures are

ﬁrst extracted at the desired frequency steps from uM(ωk)and pM(ωk), respectively. These values are now

stored in the vectors of coarse-nodal displacements uM(α)and pressures pM(α), α = 1 . . . nMel , respectively.

The displacements and pressures associated with the ith ﬁne-element in the αth coarse-element / RVE can

be evaluated using Eq. (10). The ith ﬁne-element Fginformation is also recovered with the same mapping.

Now the SAC can be computed from all the ﬁne-scale information.

3 Numerical Examples

Three benchmarks have been examined to investigate the validity of the proposed multiscaling solution

procedure for a) equivalent ﬂuid rigid motionless skeleton models and b) elastically deformable Biot solid

skeleton models, when subjected to plane wave normal incidence acoustical excitation. Melamine foam

and non-dissipative air are used in all cases. The macroscopic material parameters of melamine foam are

summarized in Table 2. The Sound Absorption Coefﬁcient (SAC) of these conﬁgurations are computed

through the MsFEM over the frequency range 20Hz ≤f≤5500Hz. These results are compared to the

corresponding TMM computations.

The efﬁciency of the MsFEM is measured by comparing its performance against the corresponding FEM.

The FEM operates over the associated global ﬁne mesh. Computational times taken to (A) evaluate the

multiscale basis functions and (B) perform the solution procedure are recorded and averaged over three runs.

Assembly and upscaling of state matrices are included within (A). Similarly, the downscaling operations are

included within (B).

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Table 2: Melamine foam macroscopic parameters

Parameter Name Value

σstatic air ﬂow

resistivity 104N·s·m−4

φopen porosity 0.99

α∞

high frequency limit of

dynamic tortuosity 1.01

Λviscous characteristic

length 9.8×10−5m

Λ0thermal characteristic

length 1.96 ×10−4m

k0

0

static thermal

permeability 4.75 ×10−9m2

E Young’s modulus 1.6×105N·m2

νPoisson’s ration 0.44

ηs

structural damping

coefﬁcient,(loss factor) 0.1

ρ1mass-density 8kg ·m−3

The acceleration provided by the MsFEM is demonstrated through computing speedup:

speedup =tFEM

tMsFEM

,(24)

where tFEM and tMsFEM denote the total times taken by each method, respectively.

3.1 Foam layer with rigid backing

This example is intended to verify the accuracy of the proposed method against the equivalent ﬂuid and Biot

models. An 80 mm thick foam layer is placed inside an impedance tube of side 80 mm. The conﬁguration

is subjected to sliding boundaries on the sides and a rigid wall backing at the rear. The 2-D geometry is

discretized with [8 ×8] coarse quadrilateral elements with each coarse element clustering an underlying

[2 ×2] ﬁne quadrilateral mesh. The 3-D geometry is similarly discretized with [8 ×8×8] coarse hexahedral

elements with each coarse element clustering an underlying [2×2×2] ﬁne hexahedral mesh. The coarse-scale

discretizations are illustrated in Fig. 4 .

The sound absorption coefﬁcients for the rigid skeleton and elastic skeleton assumptions as evaluated by the

MsFEM and the TMM are shown in Figs. 5a and 5b, respectively. Near exact correspondence between the

MsFEM and the TMM is observed for all cases.

The averaged computational times and speedup offered by the MsFEM over the FEM in the case of the

elastic skeleton assumption is provided in Table 3. A signiﬁcant speedup of 15.66 is obtained for the 3-D

discretizations.

3.2 Foam layer with air backing

This example is provided to examine the accuracy of porous-ﬂuid coupling. A 40 mm melamine foam layer

with a 40 mm air backing is placed in an impedance tube of side 80 mm. The sides are subjected to sliding

boundaries. Both a 2-D and a 3-D model are examined.

The 2-D geometry is discretized with [10 ×10] coarse quadrilateral elements with each coarse element

clustering an underlying [2 ×2] ﬁne quadrilateral mesh. The 3-D geometry is similarly discretized with

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(a) 2-D mesh (b) 3-D mesh

Figure 4: Coarse-scale meshes (a) 8×8quadrilateral coarse-elements with 2×2quadrilateral ﬁne-elements

per coarse-element and (b) 8×8×8hexahedral coarse-elements with 2×2×2hexahedral ﬁne-elements

per coarse-element.

(a) Rigid skeleton model (b) Elastic skeleton model

Figure 5: Sound absorption coefﬁcients computed for a 80mm single melamine foam layer with rigid back-

ing, through 2-D and 3-D MsFEM models and TMM models.

Table 3: Computation time taken (in seconds) and speedup offered by the MsFEM over FEM for a elastic

skeleton description of a melamine foam with rigid backing

MS Basis

Computation times

Solving and

Downscaling times Total time Speedup

2-D FEM 0.20 16.82 17.01 3.19

MsFEM 0.87 4.45 5.32

3-D FEM 4.08 803.67 807.73 15.66

MsFEM 8.93 42.66 51.59

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(a) 2-D mesh (b) 3-D mesh

Figure 6: Coarse-scale meshes (a) 10×10 quadrilateral coarse-elements with 2×2quadrilateral ﬁne-elements

per coarse-element and (b) 10×10 ×10 hexahedral coarse-elements with 2×2×2hexahedral ﬁne-elements

per coarse-element.

(a) Rigid skeleton model (b) Elastic skeleton model

Figure 7: Sound absorption coefﬁcients computed for a 40mm melamine foam layer with 40mm air backing

through 2-D and 3-D MsFEM models and TMM models.

[10 ×10 ×10] coarse hexahedral elements with each coarse element clustering an underlying [2 ×2×2] ﬁne

hexahedral mesh. The coarse-scale discretizations are illustrated in Fig. 6. The sound absorption coefﬁcients

for the rigid skeleton and the elastic skeleton assumptions as evaluated by MsFEM and TMM are shown in

Figs. 7a and 7b, respectively.

Once again, near exact correspondence between the MsFEM and TMM is observed in all cases. The averaged

computational times and speedup offered by the MsFEM over the FEM in the case of the elastic skeleton

assumption is provided in Table 4. A signiﬁcant speedup of 19.04 is obtained for the 3-D discretizations.

3.3 Melamine foam layer with meso-scale perforation

This example is provided to illustrate the ability of the MsFEM to adequately handle mesoscale inclusions.

An 80 mm foam layer with a square perforation of side 40 mm is placed in an impedance tube of side

80mm and length 400mm. The conﬁguration is subjected to rigid wall backing and sliding boundaries on

the sides. In this case, to account for reﬂection, scattering, dispersion and edge effects introduced by the

mesoscale perforation, the entire impedance tube geometry is modelled with impedance mismatch values

being computed at the extreme left end.

The 2-D geometry is discretized with [40×8] coarse quadrilateral elements with each coarse element cluster-

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Table 4: Computation time taken (in seconds) and speedup offered by the MsFEM over FEM for a elastic

skeleton description of a melamine foam with air backing

MS Basis

Computation times

Solving and

Downscaling times Total time Speedup

2-D FEM 0.35 16.53 16.89 2.34

MsFEM 1.82 5.40 7.23

3-D FEM 14.46 1592.07 1606.52 19.04

MsFEM 27.46 56.89 84.36

(a) 2-D mesh (b) 3-D mesh

Figure 8: Coarse-scale meshes (a) 40 ×8quadrilateral coarse-elements with 2×2quadrilateral ﬁne-elements

per coarse-element and (b) 40 ×8×8hexahedral coarse-elements with 2×2×2hexahedral ﬁne-elements

per coarse-element.

ing an underlying [2 ×2] ﬁne quadrilateral mesh. The 3-D geometry is similarly discretized with [40 ×8×8]

coarse hexahedral elements with each coarse element clustering an underlying [2 ×2×2] ﬁne hexahedral

mesh. The coarse-scale discretizations are illustrated in Fig. 8 .

The sound absorption coefﬁcients for the rigid skeleton and elastic skeleton assumptions as evaluated by

MsFEM and TMM are shown in Figs. 9a and 9b, respectively.

The 3-D MsFEM computations for both the rigid and elastic skeleton models agree with the TMM porous

composite model [15] , accounting for dynamic radiation correction and neglecting the pressure diffusion

effect. The SAC at the quarter wavelength resonance for the elastic model is slightly overestimated in relation

to the TMM model.

The 2-D MsFEM computations are compared against the TMM mixing law (ML) [15]. While the trend

is replicated correctly, the MsFEM is found to slightly overestimate the SAC. This can be attributed to the

differences in the cross-sectional geometry. The 3-D MsFEM and TMM result can be recovered from the

ML procedure when the mesoscale porosity is computed for a 3-D cross-section, i.e., φm=402

802= 0.25

. However, to facilitate ML comparisons with the 2-D MsFEM model, a 2-D cross-section is considered,

resulting in a mesoscale porosity φm=40

80 = 0.5. The comparison proves meaningful only in the case of

low resistive materials, like Melamine foam.

The averaged computational times and speedup offered by the MsFEM over the FEM in the case of the

elastic skeleton assumption is provided in Table 5. The FEM procedure in case of 3-D discretizations in

the elastic skeleton assumption, proves very expensive, as illustrated by the total times. The MsFEM proves

invaluable in accelerating computations here, offering a speedup of 28.94.

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(a) Rigid skeleton model (b) Elastic skeleton model

Figure 9: Sound absorption coefﬁcients computed for a 80mm melamine foam layer with a square mesoscale

air perforation of side 40mm through 2-D and 3-D MsFEM models and TMM models.

Table 5: Computation time taken (in seconds) and speedup offered by the MsFEM over FEM for a elastic

skeleton description of a melamine foam with mesoscale perforation

MS Basis

Computation times

Solving and

Downscaling times Total time Speedup

2-D FEM 1.29 51.81 53.10 3.22

MsFEM 5.20 13.96 19.16

3-D FEM 28.28 14,118.37 14,146.65 28.94

MsFEM 106.88 381.89 488.77

4 Concluding remarks

We discuss the particulars of the application of the Multiscale Finite Element Method (MsFEM) to vibro-

acoustic problems in porous media. This method is shown to signiﬁcantly drive down computational costs

while retaining the ﬁdelity of the standard FEM. Three benchmark examples are examined to establish the

validity of the proposed methodology in its application to coupled multilayered porous segments and porous

composite materials. Transmission loss problems as well as the case of oblique incidence and diffused ﬁeld

excitations are further extensions to be investigated.

Acknowledgements

This work has been carried out under the auspices of the grant: “European industrial doctorate for advanced,

lightweight and silent, multi-functional composite structures—N2N.” The N2N project is funded under the

European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie

Actions Grant: 765472.

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