Accelerated vibro-acoustics of porous domains via a novel Coupled Multiscale Finite Element Method

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. Unfortunately , 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 simulation 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.

Full text

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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1

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˜
Qu
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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