Benoit Nennig

) Mabrouk

Ben Tahar

Emmanuel Perrey-Debain

A displacement-pressure finite element formulation for analyzing the sound transmission in ducted shear flows with finite poroelastic lining

Keywords: numbers: 43.20.Mv, 43.50.Gf, 43.28.Py

In the present work, the propagation of sound in a lined duct containing sheared mean flow is studied. Walls of the duct are acoustically treated with absorbent poroelastic foams. The propagation of elasto-acoustic waves in the liner is described by the Biot's model. In the fluid domain, the propagation of sound in a sheared mean flow is governed by the Galbrun's equation. The problem is solved using a mixed displacement-pressure finite element formulation in both domains. A 3D implementation of the model has been performed, and is illustrated on axisymmetric examples. Convergence and accuracy of the numerical model are shown for the particular case of the modal propagation in a infinite duct containing an uniform flow. Practical examples concerning the sound attenuation through dissipative silencers are discussed. In particular, effects of the refraction effects in the shear layer as well as the mounting conditions of the foam on the Transmission Loss are shown. The presence of a perforate screen at the air-porous interface is also considered and included in the model.

I. INTRODUCTION

Over the years, considerable effort has been directed toward the development of finite element (FE) models for predicting the propagation of sound waves in gas flow in the presence of acoustically treated walls [START_REF] Astley | A finite element scheme for attenuation in ducts lined with porous material: comparison with experiment[END_REF][START_REF] Cummings | Sound attenuation of a finite length dissipative flow duct silencer with internal mean flow in the absorbent[END_REF][START_REF] Eversman | Aft fan duct acoustic radiation[END_REF][START_REF] Peat | A finite element analysis of the convected acoustic wave motion in dissipative silencers[END_REF] . These treatments are generally multi-layered and composed of perforated plates backed by honey comb or backed with a porous blanket. Porous materials are generally used to broaden the absorption frequency range of the liner. The practical applications of such work range from noise transmission in vehicle exhaust systems, through ventilation and air conditioning ducts, to the prediction of more complex three-dimensional fields aircraft engine-duct systems.

In the early work dealing with absorption and FE, it was assumed that the liner was locally reacting, thus eliminating the need to discretize the absorbing material explicitly. Later, to illustrate the effects of a finite size bulk reacting lining, Craggs [START_REF] Craggs | A finite element model for rigid porous absorbing materials[END_REF] derived a finite element model for motionless porous materials. This model is well adapted for rigid materials. The solid frame can also be assumed to be infinitely soft and the so-called limp model [START_REF] Beranek | Acoustical properties of homogeneous, isotropic rigid tiles and flexible blankets[END_REF][START_REF] Doutres | Validity of the limp model for porous materials: a criterion based on the Biot theory[END_REF] must be used; the stiffness of the solid phase is neglected but its inertial effects are conserved. In both cases, only one compressional wave is allowed to propagate in the material; these are known as fluid equivalent models and can be characterized by a complex wave number and a complex characteristic impedance [START_REF] Beranek | Acoustical properties of homogeneous, isotropic rigid tiles and flexible blankets[END_REF][START_REF] Zwicker | Sound Absorbing Materials[END_REF][START_REF] Delany | Acoustical properties of fibrous absorbent materials[END_REF] . For a wide class of aborbent materials such as polymer foams, the solid structure has a finite stiffness. In this case the Biot's theory [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. I low-frequency range[END_REF][START_REF] Allard | Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials[END_REF] describing the propagation of elastic and pressure waves in poroelastic material must be a) Electronic address: benoit.nennig@utc.fr used. The numerical solution of Biot's equations using the finite element method with the h or p version have been extensively discussed in the literature and various FE formulations involving different variables have been proposed [START_REF] Kang | Finite element modeling of isotropic elastic porous materials coupled with acoustical finite elements[END_REF][START_REF] Atalla | Enhanced weak integral formulation for the mixed (u, p) poroelastic equations[END_REF][START_REF] Horlin | A 3-D hierarchical FE formulation of Biot's equations for elastoacoustic modeling of porous media[END_REF][START_REF] Rigobert | Investigation of the convergence of the mixed displacement-pressure formulation for three-dimensional poroelastic materials using hierarchical elements[END_REF] . In this regard, the Atalla et al.'s mixed (u, p p ) formulation has several advantages of reducing the number of degrees of freedom as well as easing the transmission conditions at the air-porous interface [START_REF] Atalla | Enhanced weak integral formulation for the mixed (u, p) poroelastic equations[END_REF][START_REF] Atalla | A mixed displacement-pressure formulation for poroelastic materials[END_REF] . It is remarkable that most of the studies related to sound attenuation through dissipative silencers are carried out using simplified fluid equivalent models and that no attempt was made to quantify the effects of the finite stiffness of the absorbent material. The first paper in this matter seems to be that of Kang 17 for circular ducts. The authors developed an axisymmetric finite element formulation for poroelastic liner and showed that mounting conditions can have great impact on the sound attenuation.

All references cited above do not consider the presence of a gas flow in the fluid domain. Ignoring the convection and refraction effects due to the mean flow can lead to erroneous results especially for relatively high speed flow. Indeed, the presence of a flow in the airway is known to have a non negligible effect on the upstream and downstream propagation. This has been investigated in a modal context for infinite lined circular duct [START_REF] Kirby | Simplified techniques for predicting the transmission loss of a circular dissipative silencer[END_REF] , slowly varying porous liner [START_REF] Rienstra | Contributions to the theory of sound propagation in ducts with bulk-reacting lining[END_REF] or using FE mode solvers for ducts with arbitrary cross section 1 . For lining of finite dimension, modal matching methods can sometimes be used for a certain class of problems [START_REF] Kirby | Simplified techniques for predicting the transmission loss of a circular dissipative silencer[END_REF][START_REF] Nennig | A mode matching method for modelling dissipative silencers lined with poroelastic materials and containing mean flow[END_REF] . In most work, the mean flow is considered uniform in the duct. This simplification is made to ease the numerical treatment of the wave equation, but it has the drawback to neglect refraction effects due to the rotational flow in the boundary layer which is inevitably present in the vicinity of the walls [START_REF] Eversman | Effect of Boundary Layer on the Transmission and Attenuation of Sound in an Acoustically Treated Circular Duct[END_REF] . In fact very little research have dealt with modeling the wave propagation inside the absorbent material Sound transmission in ducted shear flows 1 exposed to a grazing shear flow and the work is usually restricted to the modal analysis of infinite duct by solving the Pridmore-Brown equation [START_REF] Gogate | Sound propagation in ducts with bulk reacting lining in the presence of laminar mean flow[END_REF][START_REF] Pridmore-Brown | Sound propagation in a fluid flowing through an attenuating duct[END_REF] .

Because of these limitations, we think there is a need to develop more advanced numerical techniques based on the finite element methodology in order to incorporate these effects. For a general rotational mean flow, two equivalent physical models are proposed in the literature: the Linearized Euler's Equations (LEE) and the Galbrun's equation [START_REF] Galbrun | Propagation d'une onde sonore dans l'atmosphère terrestre et théorie des zones de[END_REF][START_REF] Godin | Reciprocity and energy theorems for waves in a compressible inhomogeneous moving fluid[END_REF] . The numerical solution of LEE can be carried out either in the frequency or in the time domain. For this latter, however, there remain some stability issues regarding propagation through shear layers and the correct implementation of time-domain impedance boundary conditions. Some interesting discussions can be found in a recent review of Astley [START_REF] Astley | Numerical methods for noise propagation in moving flows, with application to turbofan engines[END_REF] and in references therein. Galbrun's equation offers an interesting alternative to LEE. The equation describes exactly the same physical phenomenon as the LEE but is obtained using a different linearization process involving the displacement perturbation, instead of the acoustic velocity, as primitive variable. All details of the formulation as well as its FE discretization can be found in Refs. 24, 25, 27, 28 and some aspects will be discussed and reminded later.

In the context of poroelastic liners exposed to a grazing flow, Galbrun's equation offers several advantages: (i) it allows a direct treatment of the coupling condition at the interface as the continuity of normal acoustic displacement appears explicitly in the boundary terms [START_REF] Godin | Reciprocity and energy theorems for waves in a compressible inhomogeneous moving fluid[END_REF][START_REF] Gabard | A numerical method for vibro-acoustic problems with sheared mean flows[END_REF] , (ii) it can be written as an exact conservation law for homentropic rotational flow and a definition of the acoustic intensity is available [START_REF] Godin | Reciprocity and energy theorems for waves in a compressible inhomogeneous moving fluid[END_REF][START_REF] Poirée | Les équations de l'acoustique linéaire et non linéaire dans un écoulement de fluide parfait (equations of linear and non linear acoustics in a perfect fluid flow)[END_REF] , (iii) efficient and stable FE formulations in the frequency domain are available [START_REF] Treyssède | A mixed finite element method for acoustic wave propagation in moving fluids based on an eulerian lagrangian description[END_REF] . All this makes the method a candidate of choice for dealing with the presence of poroelastic materials which are defined via frequency dependent Biot's parameters.

Based on previous studies, the aim of the present work is to extend and propose a FE formulation for analyzing the sound transmission in ducted shear flows with finite poro-elastic lining. The propagation of waves in the liner is described via the Biot's model whereas the sound propagation in the fluid domain is governed by the Galbrun's equation. The problem is solved for a general 3D configuration using a mixed displacement-pressure finite element formulation in both domains allowing an efficient treatment of the coupling conditions at the air-porous interface. Details of physical models and their FE formulations are presented in Section 2 under the assumptions there is no induced flow in the liner. Convergence and accuracy of the numerical model are shown in Section 3 for the particular case of the modal propagation in a infinite duct containing an uniform flow. To illustrate the method, we compute the Transmission Loss for typical dissipative silencers encountered in the automotive industry and this is presented in Section 4. We show that the existence of multiple wave types in the porous material can have significative consequences on the sound attenuation and that the liner is very sensitive to the boundary conditions that exist at its surface. The presence of a perforate screen at the air-porous interface is also considered and included in the FE model. Compu-tational aspects of the method are discussed at the end of the paper.

II. PROBLEM FORMULATION A. Problem statement

The configuration considered here consists of a dissipative silencer comprising a expansion chamber of length L filled with a sound-absorbing poroelastic material. In the airway, a sheared mean flow with speed v 0 is present. In this work, we restrict the analysis to parallel shear flow where the density ρ 0 , the sound wave speed c 0 and the mean pressure p 0 are taken constant 30 (Eq. (1.12)). Furthermore, we consider problems with axial symmetry, so we can define the local Mach number as a function of the radial coordinate only: M (r) = v 0 (r)/c 0 . The inlet and outlet rigid wall pipes are identical, each having a circular cross section with radius r 1 . This simple configuration have been chosen for a sake of illustration, but the numerical scheme presented in this paper can be used for solving general 3D problems with non potential flow as in Ref. 31.

For a brief nomenclature, we call Ω a and Γ a , the airflow flow domain (r ≤ r 1 ) and its boundary. Similarly, we call Ω p and Γ p the poroelastic domain and its boundary. The acoustic field is generated by an incident field stemming from the left. On both ends Γ i and Γ o , appropriate modal conditions must be used to ensure that the reflected and transmitted fields are radiating away from the dissipative silencer. The mathematical formulation for this will be stated later. Finally we call Γ the air-porous interface and Γ w denotes the rigid walls.

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B. Galbrun's equation

Galbrun's equation derives from general fluid mechanic conservation equations, whereby the linearization process is carried out with a lagrangian perturbation of the displacement [START_REF] Galbrun | Propagation d'une onde sonore dans l'atmosphère terrestre et théorie des zones de[END_REF][START_REF] Godin | Reciprocity and energy theorems for waves in a compressible inhomogeneous moving fluid[END_REF][START_REF] Poirée | Les équations de l'acoustique linéaire et non linéaire dans un écoulement de fluide parfait (equations of linear and non linear acoustics in a perfect fluid flow)[END_REF][START_REF] Treyssède | A mixed finite element method for acoustic wave propagation in moving fluids based on an eulerian lagrangian description[END_REF] . Under the previous assumptions and without the presence of acoustic sources, the Galbrun's equation for the displacement perturbation w and the acoustic pressure p can be written in the frequency domain (e -iωt ) as follows :

ρ 0 d 2 0 w dt 2 + ∇p = 0, ( 1a
) p = -ρ 0 c 2 0 ∇ • w, (1b)
Sound transmission in ducted shear flows 2

where d0 dt ≡ -iω + v 0 • ∇ is the material derivative along the mean flow.

One may observe that it is possible to express (1) with the displacement perturbation only. However solving the Galbrun's equation in this form with conventional nodal finite element is known to give rise to corrupted results. Thus, to overcome the so-called locking effects in the FE formulation two ways have been investigated in the past (i) the regularization technique [START_REF] Bonnet-Bendhia | Time-harmonic acoustic propagation in the presence of a shear flow[END_REF] , which add a special treatment for the curl of the displacement. (ii) The pressure is kept as an independent variable as in Ref. 28: this is the displacement-pressure mixed formulation of Galbrun's equation. This approach will be used in the following. The interested reader may consults Ref. 28, 32 and the references therein.

After integration by parts, the associated weak formulation to the mixed form is

- Ωa ρ 0 d 0 w * dt • d 0 w dt dΩ + Ωa w • ∇p * dΩ + Ωa ∇p • w * dΩ - Ωa p * p ρ 0 c 2 0 dΩ + Γa ρ 0 (v 0 • n ′ ) d 0 w dt • w * dΓ - Γa p * w • n ′ dΓ = 0, (2)
for all the test functions p * , w * . Here, n ′ is the outward normal unit vector and the bar symbol on the material derivative means that we take the complex conjugate, i.e.

d0 dt ≡ iω + v 0 • ∇.

C. Biot's model

The wave propagation involving respectively the fluid and solid phases displacement U and u in the poroelastic medium is described by the Biot-Allard model which is well documented in the reference textbook [START_REF] Allard | Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials[END_REF] . For the numerical implementation, we use the Atalla's mixed (u, p p ) formulation [START_REF] Atalla | Enhanced weak integral formulation for the mixed (u, p) poroelastic equations[END_REF][START_REF] Atalla | A mixed displacement-pressure formulation for poroelastic materials[END_REF] . This has the advantage of reducing the number of degrees of freedom as well as easing the transmission conditions at the air-porous interface. This reduced formulation reads

∇ • σ s (u) + ω 2 ρ u + γ ∇p p = 0, ( 3a
)
∆p p + ω 2 ρ 22 R p p -ω 2 ρ 22 φ 2 γ ∇ • u = 0. (3b)
Here, p p is the pore pressure, φ is porosity of the porous

material, γ = φ ρ12 ρ22 -Q R and ρ = ρ 11 - ρ 2 12
ρ22 . The effective density coefficient ρ 11 , ρ 22 , respectively for the solid phase and the fluid phase, and the coupling density coefficient ρ 12 , are complex and the imaginary part takes into account viscous damping. The in vacuo stress tensor σ s reads

σ s (u) = I K b - 2 3 N ∇ • u + 2N ε s (u), (4)
with ε s the in vacuo strain tensor. Here, K b is the complex dynamic bulk modulus of the frame and takes into account the thermal dissipation. In the same way, the shear modulus N includes the structural damping. Lastly, R is the effective bulk modulus of the fluid phase and Q indicates the coupling of the two phases volumic dilatation. All these coefficients are related to the poroelastic structural parameters (see Table III

σ s (u) : ε s (u * ) dΩ -ω 2 Ωp ρu • u * dΩ + Ωp φ 2 ω 2 ρ 22 ∇p p • ∇p * p - φ 2 R p p p * p dΩ - Ωp (γ + φ ′ )(∇p * p • u + ∇p p • u * ) dΩ - Ωp φ ′ p * p ∇ • u + p p ∇ • u * dΩ - Γp σ t n • u * dΓ Iσ - Γp φ (U -u) • n p * p dΓ = 0 . ( 5
)
We put φ ′ = φ(1 + Q/R) and n denotes the outward normal unit vector to the poroelastic domain. Note that the presence of the boundary integral involves the fluid normal displacement U • n and the total stress tensor σ t . These new physical quantities do not need to be stated explicitly here as they will be replaced after applying boundary conditions.

D. Coupling conditions and global formulation

Coupling conditions between the acoustic domain and the poroelastic material are summarized by Debergue et al. [START_REF] Debergue | Boundary conditions for the weak formulation of the mixed (u, p) poroelasticity problem[END_REF] :

σ t n = -p n, (6a)
p p = p, (6b) w • n = φ(U -u) • n + u • n. (6c)
The first condition is the standard continuity requirement of the normal stress at the interface. The second condition ensures the continuity of the pressure between the acoustic domain and the pores. The last condition ensures the continuity of the displacement at the interface. Though these conditions have been specified in no-flow case [START_REF] Debergue | Boundary conditions for the weak formulation of the mixed (u, p) poroelasticity problem[END_REF] , they still hold when a flow is present in the airway [START_REF] Godin | Reciprocity and energy theorems for waves in a compressible inhomogeneous moving fluid[END_REF][START_REF] Gabard | Stability and accuracy of finite element methods for flow acoustics. II: Two-dimensional effects[END_REF] .

On the hard wall of the expansion chamber Γ w , two scenarios are investigated (i) the porous foam is considered clamped, that is the solid phase displacement vector is zero at the wall and (ii) the foam is sliding and only the normal component of the displacement is set to zero. In both cases, the boundary terms in (5) vanish on Γ w since u * is set to zero.

The global formulation is obtained by summing (2) and (5). Because the expressions are quite lengthy, we Sound transmission in ducted shear flows 3 shall detail only the boundary terms on the air-porous interface Γ. First, since there is no flow in the liner, v 0 and n ′ are orthogonal and the first boundary term in (2) (its restriction on Γ) vanishes. Both conditions (6a) and (6c) are directly substituted in the boundary integral over Γ. For the condition of continuity of pressure, we introduce an additional functional

Γ w * • n (p -p p ) dΓ = 0, (7)
where w * can be interpreted as a Lagrange multiplier. Now, (6b) is added thanks to (7) and we can switch p and p p in the boundary integral term I σ in (5). These two manipulations give a symmetric formulation of the coupling integrals. We finally arrive at the weak form for the coupling on Γ:

Γ p p u * •n dΓ+ Γ p * p u•n dΓ+ Γ pw * •n dΓ+ Γ p * w•n dΓ - Γ p * p w • n dΓ - Γ p p w * • n dΓ = 0. ( 8
)
The first line of ( 8) corresponds to a displacementpressure coupling within each separate domain whereas the second line represents the displacement-pressure coupling between the two domains. It can be noted the two displacement-pressure weak formulations ( 5) and ( 2) make easier the coupling between both domain and this is due to the continuity of the lagrangian perturbation of the displacement. In particular, this allows easy treatment of plug flows. In this latter case, the use of the LEE would have been more difficult as the Myers boundary condition [START_REF] Myers | On the acoustic boundary condition in the presence of flow[END_REF] is not taken into account naturally. The combination of ( 5), ( 2) and ( 8) yields the coupled fluid-poroelastic finite element system which can be written in the compact form

G C C t B    p w p p u    = F G F B (9)
. One can note that the discretization of the Galbrun's equation in its mixed form leads to the complex hermitian matrix block G. However the displacement-pressure formulation of Biot's equations gives a complex symmetric matrix B. The right hand sides vectors F G and F B arise from the boundary conditions. In this study, we have necessarily F B = 0 and F G stems from the pressuredisplacement condition at the inlet and outlet planes.

III. VALIDATION AND ASSESSMENT OF THE FE MODEL

In order to validate and assess the FE model, we shall consider the case of an infinite cylindrical silencer.On the outer radius r 2 , the foam is clamped and there is an uniform mean flow with Mach number M in the airway. In this scenario, the problem is separable in cylindrical coordinates (r,θ,z) and analytical modal solutions can be found. These solutions all have the e i(βz-ωt) dependence and finding these modes requires the computation of the axial wavenumbers β. In appendix A, the method for finding these modes is presented very briefly for the case of purely radial modes only. We refer the reader to Ref. 20 for more details. Now, for a chosen mode, the analytical solution for both the pressure and the displacement vector is imposed directly in the system (9) at nodes corresponding to the inlet and outlet sections Γ i and Γ o . A convergence rate study has been carried out for three different discretization schemes listed in Table I. Biot's equations are discretized with standard isoparametric tetrahedral elements called T4 and T10 using respectively a linear and a quadratic lagrangian polynomial interpolation basis for the pressure and for the solid displacement field. To ensure the convergence, the mixed formulation ( 2) is discretized with an appropriate choice for the interpolation function basis satisfying the inf-sup condition [START_REF] Bathe | Finite element procedures[END_REF] . For this purpose, the so-called T5-4c and T10-4c mixed elements are used [START_REF] Treyssède | A mixed finite element method for acoustic wave propagation in moving fluids based on an eulerian lagrangian description[END_REF][START_REF] Bathe | Finite element procedures[END_REF][START_REF] Bériot | A three dimensional finite element model for sound propagation in non potential mean flows[END_REF] . In the T5-4c element (also called "MINI") the pressure interpolation is linear while the displacement field is enriched with a bubble function. In the T10-4c, the pressure is linear whereas the displacement is interpolated with quadratic basis functions.

For a given physical quantity X, we define the relative error estimator E 2 in terms of the energy-norm as

E 2 (X) = 100 X fem -X ana L 2 (S) X ana L 2 (S) (10)
where S a surface of reference lying on the θ = 0 plane and corresponding either to Ω a or Ω p . A comparison between the analytical solution and the FE solution computed with the LQ scheme is shown in Fig. 2. The dimensions are r 1 = 0.06 m, r 2 = 0.1 m with a the total length of L = 0.6 m. In all cases the frequency is fixed at 600 Hz. For all physical quantities we can note a good agreement between the computed values and the modal solution. This is confirmed by the relative error E 2 given in Table II. In this example, approximately eight elements were used in the radial direction. In this regard, the FE mesh is shown on the reference surface in order to give an idea of the discretization level. Note that the pressure is continuous across the interface whereas the normal displacement presents a jump due to the separation of the total displacement field into the fluid and solid phase.

A convergence rate analysis has been performed for two configurations (i) a no-flow case with mode β = 17.135 + 4.7643i and (ii) an upstream propagation case Sound transmission in ducted shear flows 4 with M = -0.2 and mode β = 20.2572 + 7.1227i. Results for the pressure in both domains are conveniently plotted in Fig. 3. Errors in the displacement have been found to be higher than for the pressure but the convergence curves show similar trends. Clearly, best results have been obtained when using either LQ or QQ schemes. This indicates that quadratic interpolation for the poroelastic material discretization must be used in order to reduce computational costs and that errors of about one percent for the pressure in the fluid domain can be expected at a reasonable cost. Thereafter, the LQ scheme is used in all calculations.

Note that validations were also performed on silencer configurations with uniform mean flow and clamped boundary conditions. Comparisons with the recent mode matching technique proposed in Ref. 20 show a good agreement (see Figs. 8-12 therein).

IV. RESULTS

A. Test case configuration

As a test case we consider the geometry already studied by Kirby 18 so we take r 1 = 37 mm, r 2 = 76.2 mm and L = 315 mm. To illustrate the method we consider two foams which are standard polymer foams presenting a strong coupling between both fluid and solid phases: a 'soft' one called FM4 and the XFM foam. The material properties can be found in Refs. 16, 33 and are reminded in Table III for completeness. Here the diameter of the acoustic duct is relatively small and the first cut-off frequency is well above the frequency range of interest. Only the fundamental mode is allowed to propagate and this simplifies the radiating conditions at both ends of the duct. At the exit plane, the non-reflective condition is assured by the modal impedance of the first mode [START_REF] Treyssède | A mixed finite element method for acoustic wave propagation in moving fluids based on an eulerian lagrangian description[END_REF] of the rigid duct. In an uniform flow, this mode is given analytically in terms of Bessel functions. In a shear flow, the mode is computed by solving the Pridmore-Brown equation using a shooting method as in [START_REF] Treyssède | A mixed finite element method for acoustic wave propagation in moving fluids based on an eulerian lagrangian description[END_REF][START_REF] Mungur | Acoustic wave propagation in a sheared fluid contained in a duct[END_REF] . At the inlet plane, the pressure-displacement field is expressed as a sum of an incident and a reflected mode and the global matrix is rearranged so that the incident term appears on the right-hand side F G of the FE system. Note that the same procedure can be followed in a multimodal context as shown in Ref. 39.

On Fig. 4, the test case mesh is shown. Since the problem possesses axial symmetry, only one quarter of Sound transmission in ducted shear flows 5 the duct is meshed and normal displacement field are set to zero on the lateral sides (corresponding to θ = 0 and θ = π/2).

In the present work, we wish to identify the effects of the poroelastic material as well as the presence of the flow in the airway on the Transmission Loss (TL) of the dissipative silencer. By calling P i and P t the incident and transmitted power, the TL is defined by TL = -10 log

P t P i . ( 11
)
In the presence of a shear flow in the airway, the calculation for the transmitted and incident acoustic powers relies on the numerical evaluation of the intensity vector 40

I = ρ 0 ∂w ∂t • d 0 w dt v 0 + p ∂w ∂t . (12)
The powers are obtained after integration of the normal component (along the z-axis) of the intensity over the inlet and the outlet planes.

B. Mounting conditions and skeleton influence

It is known that the manner in which lining materials are fitted into the duct can have a great impact on the sound attenuation level [START_REF] Kang | Sound propagation in circular ducts lined with noise control foams[END_REF] , especially with poroelastic materials presenting a strong skeleton influence. This effect has been experimentally observed and discussed in a standing wave tube 41 and some criteria have been proposed to quantify the frame stiffness influence [START_REF] Doutres | Validity of the limp model for porous materials: a criterion based on the Biot theory[END_REF][START_REF] Pilon | Behavioral criterion quantifying the edge-constrained effects on foams in the standing wave tube[END_REF] .

To illustrate this, the clamped and sliding mounting conditions are compared and results on the TL are shown in Fig. 5. For the XFM foam for which the fluid and solid phases are strongly coupled, 'resonances' peaks are clearly identifiable. By comparing these results with a simplified bulk-reacting models such as the Limp model, it is concluded that these peaks stem from the elasticity of the frame [START_REF] Nennig | A mode matching method for modelling dissipative silencers lined with poroelastic materials and containing mean flow[END_REF] . In this context, the mounting conditions have a great impact on the dynamic response of the poroelastic liner and its absorbing properties.

The FM4 foam is softer; when considering sliding conditions on the hard wall, the TL curve shows a quite smooth behavior without marked skeleton resonances. Here, results are comparable with the bulkreacting model (Limp model) [START_REF] Nennig | Contribution à l'étude de matériaux absorbants acoustiques en présence d'écoulement (a contribution to acoustic absorbing materials understanding in the presence of a mean flow)[END_REF] and the solid frame influence is essentially due to the inertial coupling. However, when the foam is clamped, the coupling between phases is stronger and the skeleton resonances are clearly visible in the low frequency regime. These effects have been observed especially for small thickness liners.

C. Shear flow influence

In this section we shall investigate the effect of a shear flow in the airway on the upstream and downstream propagation for the dissipative silencer of the previous section. Sound transmission in ducted shear flows 6 [START_REF] Atalla | A mixed displacement-pressure formulation for poroelastic materials[END_REF][START_REF] Debergue | Boundary conditions for the weak formulation of the mixed (u, p) poroelasticity problem[END_REF] . With the flow resistivity σ, the tortuosity α inf , the viscous and thermal characteristic lengths Λ and Λ ′ , the poisson coefficient ν and the effective skeleton density ρ1. The effective skeleton density ρ1 = (1 -φ)ρs, where ρs is the density of the material of the frame. The flow profile 21

Foam φ σ [kNm -4 s] α inf Λ [µm] Λ ′ [µm] ρ1 [kgm -3 ] N [kPa] ν XFM 0.
M (r) = M * , if 0 ≤ r ≤ r 1 -δ M * cos π 2δ (r -(r 1 -δ)) , if r 1 -δ < r < r 1 (13)
is constant with a magnitude M * excepted in the shear layer, where a sinusoidal transition with a thickness δ is used. When using a uniform flow model with Mach number M 0 , we made sure that comparisons are made by keeping the same mean Mach number, that is M 0 = M (r). In the following example, the mean Mach number value is chosen equal to 0.2 and the lining material is sliding on the hard walls.

When the flow is uniform, the sound waves propagating upstream are generally more attenuated than when propagating downstream and this is in agreement with a modal analysis (it suffices to compare the imaginary part of the axial wavenumber). Results presented in Fig. 6 for the XFM foam confirm these facts. When the flow is sheared the same conclusion holds, but in addition, the refraction effects in the boundary layer are found to be more significant for the upstream propagation. Note that the convection of the waves by the mean flow changes the 'apparent' wave length which is shorter in the upstream propagation case. This has a noticeable effect on the elasto-acoustic excitation of the lining material. This explains the 'shift' of the resonance peaks on the TL curves.

Refractions effects due to the flow are conveniently illustrated in Fig. 7. Arrows corresponding to the intensity vector field calculated from (12) are shown. This clearly illustrates the noticeable differences between the shear flow and the uniform flow case. For the downstream propagation, the shear flow makes the acoustic intensity penetrate deeper in the poroelastic lining. On the contrary, the shear flow tends to make the intensity of upstream propagating acoustic waves more parallel to the duct axis. These observations are in line with results of Fig. 6, especially the upstream propagation case for which Losses are found to be higher for uniform flows in a large part of the spectrum.

D. Presence of a perforated screen

In many problems of practical interest, the presence of a perforated screen separating the absorbing material from the airway is sometimes inevitable. The reasons for this are either to avoid the flow to penetrate through the liner or for other purely practical reasons. The presence of the screen will in turn affect the performance of the dissipative silencer. Because of this, our FE model needs Sound transmission in ducted shear flows 7 to be modified in order to take into account the induced change of pressure δp over the perforated screen. Here, the difficulty lies in having appropriate values for the layer impedance defined as the ratio between the change of pressure and the normal velocity at the surface of the liner. These values are usually measured experimentally and various semi-empirical models have been proposed in the literature. A complete survey would be too lengthy to be inserted in this paper and we can refer to recent research works in this field [START_REF] Kirby | Simplified techniques for predicting the transmission loss of a circular dissipative silencer[END_REF] . In particular it was shown that the presence of the grazing flow in the airway and the porous backing behind the screen can have a substantial effect on the impedance [START_REF] Kirby | The impedance of perforated plates subjects to grazing gas flow and backed by porous media[END_REF] .

In the present work we assume in our modeling that the lining material is not directly in contact with the screen so that the normal stress at the surface of the poroelastic liner is due to the air pressure only. The transmission conditions at the interface Γ now read (note that the second equation is usually written in terms of the normal acoustic velocity v n = -iωw • n on the screen). The last condition stems from the continuity of the normal displacement at each side of the screen and (6c) remains unchanged. Finally, the presence of the perforated screen is taken into account in our FE model by simply adding the boundary integral

σ t n = -(p -iωZw • n) n , (14a) δp = p p -p = -iωZw • n. (14b)
-iω Γ w * • n Z w • n dΓ. ( 15
)
in the Galbrun's matrix G.

In the next example, we shall investigate the noflow case only. For a sake of illustration we chose the impedance model given in Ref. 43

Z ρ 0 c 0 = √ 8υω φ p c 0 1 + τ d + 1 8φ p (k 0 d) 2 -i ω φ p c 0 τ + 8 3π d 1 -0.7 φ p + 8υ ω 1 + τ d , (16)
with υ = 1.5 10 -5 m 2 s -1 the air kinematic viscosity, the screen thickness τ , the hole diameter d and the screen porosity φ p . Results are depicted in Fig. 8 for the two screens described in Table IV. The screen 1 is acoustically transparent below 500 Hz. In the mid frequency range (say 500 Hz up to 1400 Hz) the apparent resistivity increases and a gain of few decibels can be achieved. However, above 1400 Hz, performances drastically decrease because the perforated screen tends to behave like a rigid wall (high impedance value) and the air-porous coupling becomes less effective. For the screen 2, similar trends hold, however the porosity is relatively high and the coupling remains effective in a wider frequency range. Meshes are built using the free software Gmsh 44 , the FE implementation (integration and matrix assembly) is carried out on Matlab and the final system is solved with a Fortran-based solver. For homogenous porous material, all frequency dependent parameters of the Biot's model can be pulled out of the integrals. Thus, for a fixed mesh the numerical integration over the elements is performed only once. The system is solved with a direct parallel sparse solver: Mumps (with PORD ordering) [START_REF] Amestoy | Multifrontal parallel distributed symmetric and unsymmetric solvers[END_REF] on 8 processors on the High-performance computing platform Pilcam 46 . To give an idea of the computational cost, the sparse system (nnz = 6,589,999 and N dof =78,574) corresponding to the mesh of Fig. 4 is solved in 17 seconds and requires 250 Mbytes per working processor.

V. CONCLUSION

In this paper, a finite element model for analyzing the sound transmission in ducted shear flows with finite poroelastic lining was presented. The numerical model has been developed for general 3D configurations. The use of a displacement-pressure formalism for both propagative domains (absorbing material and the airway) allows an efficient and natural coupling at the air-porous interface. Comparisons with analytical solutions show the efficiency of the method. It is demonstrated that the numerical model presented in this work gives a complete computational tool for tackling complex configurations of practical interest.

To illustrate the method, the case of a circular dissipative silencer is discussed. In particular, the effects of the mounting conditions of the porous foam on the elasticity of the skeleton are shown. In some cases, elastic resonances are clearly identified and this can have significant consequences on the sound attenuation.

Using the acoustic intensity defined in (12) for arbitrarily mean flow, refraction effects in the boundary layer are shown for the downstream and the upstream propagation. In the case of the fundamental mode, these effects are noticeable only in the upstream case. Work is ongoing by the authors to extend the results when higher order propagative modes are also present in the airway.

Comparisons with experimental data would also be interesting and very instructive, in particular to identify the effect of an internal flow in the absorbent on the porous material properties 2 . This aspect was ignored in the present work but we think that this could be an interesting subject for further investigation.

Sound transmission in ducted shear flows 9

FIG. 1 .

 1 FIG.1. Geometry of the silencer benchmark.

FIG. 2 .

 2 FIG.2. Numerical results (fe) computed with the LQ scheme and analytical solutions (ana) for the real part of the pressure, the axial and the radial displacement (quantities are scaled for the sake of clarity). The displacement 'Disp.' (resp. the pressure 'Pres.') signifies the fluid displacement perturbation in the airway w (resp. pressure p) for r ≤ r1, and the solid phase displacement u for r1 < r ≤ r2 (resp. pore pressure pp). Computed with β = 9.7112 + 2.8826i, M = 0.3 and f = 600 Hz.

FIG. 3 .

 3 FIG. 3. Convergence curves for the pressure. The discretization scheme is given by the line style : LL scheme, . . . LQ scheme and QQ scheme. The line thickness indicates the domain under consideration: bold line for the poroelastic domain and thin line for the fluid domain. (a) No flow case with β = 17.135 + 4.7643i. (b) M = -0.2 flow with β = 20.2572 + 7.1227i.

FIG. 5 .

 5 FIG. 5. Effects of the mounting conditions, sliding, clamped, for two foams: (a) XFM, (b) FM4.

FIG. 6 .

 6 FIG. 6. Effects of the shear layer with a mean Mach number M0 = 0.2, no flow, δ = 0.2, uniform, for XFM foam. (a) Downstream propagation, (b) Upstream propagation.

FIG. 7 .FIG. 8 .

 78 FIG. 7. Acoustic intensity field at the entrance of the liner at 1021 Hz, uniform flow; shear flow. (a) Downstream case, (b) Upstream case.Sound transmission in ducted shear flows 8

  ) by the Johnson-Champoux-Allard model and can be found in Ref.11.Given tests functions p * p , u * , the associated weak formulation is[START_REF] Atalla | Enhanced weak integral formulation for the mixed (u, p) poroelastic equations[END_REF]

	Ωp

TABLE I .

 I Three discretisation schemes.

	Scheme	Ωa	Ωp
	LL	T5-4c	T4
	LQ	T5-4c	T10
	QQ	T10-4	T10

TABLE II .

 II Error E2 for M = 0.3, f = 600 Hz, β = 9.7112 + 2.8826i computed with the LQ scheme. Displacement 'Disp.' (resp. 'Pressure') signifies the fluid displacement perturbation in the airway w (resp. the pressure p), and the solid phase displacement u in the poroelastic material (resp. the pore pressure pp).

	Error E2 (%)	Pressure	Radial Disp.	Axial Disp.
	Poroelastic mat.	0.586	0.785	0.525
	Airway	1.372	2.718	0.847

TABLE III .

 III Materials properties used in numerical tests

TABLE IV .

 IV Perforated screen characteristics.

	Screen	d [mm]	τ [mm]	φp [-]
	1	1	1	.05
	2	3.5	1	.263
	E. Notes on computational aspects

Acknowledgement

The authors acknowledge the Projet Pluri-Formations PILCAM2 at the Université de Technologie de Compiègne for providing HPC resources that have contributed to the research results reported within this paper 46 .

APPENDIX A: MODE CALCULATION FOR AN UNIFORM FLOW

In this section, the silencer chamber is assumed to be infinite in length and a uniform mean flow is present in the airway. The eigenvalue analysis is performed by assuming that all perturbative quantities have the e i(βz-ωt) dependence. Within these assumptions, the displacement perturbation w satisfies the convected wave equation in the airflow domain (see (1))

with Ω = M β -k 0 , k 0 = ω/c 0 . Since the flow is uniform, we can look for purely acoustic mode by putting w = ∇ϕ 0 and the acoustic pressure is simply obtained from p = ρ 0 c 2 0 Ω 2 ϕ 0 . In the axisymmetric case, we find that ϕ 0 (r) = A 0 J 0 (α 0 r) where the transverse wavenumber satisfies the dispersion relation α 2 0 + β 2 = Ω 2 . The fluid and solid phase displacements in the porous materials are governed by the Biot's model (3). When the medium is homogeneous, both displacements admit the Helmholtz decomposition u = ∇(ϕ 1 + ϕ 2 ) + ∇ × Ψ and U = ∇(µ 1 ϕ 1 + µ 2 ϕ 2 ) + µ 3 ∇ × Ψ where potentials ϕ 1 , ϕ 2 and Ψ are solutions of the Helmholtz equation with wavenumbers k 1 , k 2 and k 3 , (see [START_REF] Allard | Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials[END_REF] ). So we put ϕ e (r) = A e J 0 (α e r) + B e Y 0 (α e r), e = 1, 2, (A2a)

where α 2 e + β 2 = k 2 e (e = 1, 2, 3). Note that, for the sake of simplicity, the shear wave potential ϕ 3 simply stands for the azimuthal component of Ψ. The modal vector V = [A 2 , B 2 , A 3 , B 3 , A 1 , B 1 , A 0 ] T must be found so that the transmission conditions (6a),(6b) and (6c) at the fluid-porous interface (r = r 1 ) are satisfied. For this, the pore pressure p p and the total stress tensor σ t must be expressed in terms of the potentials as shown in [START_REF] Allard | Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials[END_REF] . Similarly, at the wall (r = r 2 ), the foam is clamped, i.e. u = 0 and U • n = 0. All these conditions leads to the nonlinear eigenvalue problem

where M(β) is a 7×7 matrix. We can now exploit the fact that det M(β) is a meromorphic function in the complex β-plane and solve det M(β) = 0 using the argument principle. All simple zeros β ± k (k = 1, . . . , K) are classified with the convention that superscript ± stands for the sign of the imaginary part (+ refers to rightgoing modes and -to leftgoing modes).


B Nennig

E Perrey-Debain

J.-D Chazot

The method of fundamental solutions for acoustic wave scattering by a single and a periodic array of poroelastic scatterers

Keywords: Method of fundamental solutions, Biot's equations, poroelastic, porous material, scattering

The Method of Fundamental Solutions (MFS) is now a well-established technique that has proved to be reliable for a specific range of wave problems such as the scattering of acoustic and elastic waves by obstacles and inclusions of regular shapes. The goal of this study is to show that the technique can be extended to solve transmission problems whereby an incident acoustic pressure wave impinges on a poroelastic material of finite dimension. For homogeneous and isotropic materials, the wave equations for the fluid phase and solid phase displacements can be decoupled thanks to the Helmholtz decomposition. This allows for a simple and systematic way to construct fundamental solutions for describing the wave displacement field in the material. The efficiency of the technique relies on choosing an appropriate set of fundamental solutions as well as properly imposing the transmission conditions at the air-porous interface. In this paper, we address this issue showing results involving bidimensional scatterers of various shapes. In particular, it is shown that reliable error indicators can be used to assess the quality of the results. Comparisons with results computed using a mixed pressure-displacement finite element formulation illustrate the great advantages of the MFS both in terms of computational resources and mesh preparation. The extension of the method for dealing with the scattering by an infinite array of periodic scatterers is also presented.

Introduction

Poroelastic materials are often used for their good sound absorbing capabilities in the middle and high frequency range. Typical applications can be found in the context of the transport noise reduction or to enhance the quality of room acoustics. The description of wave propagation in porous media is not limited to audible acoustics as Biot's model [START_REF] Biot | Theory of propagation of elastic waves in a fluidsaturated porous solid. I. low-frequency range[END_REF] was originally developed for geological applications. Because of their inherent diphasic features and the strong contrasts that may exist between the solid and the fluid phases, wave propagation modeling remains a difficult task often leading to heavy computational costs. In the context of the Finite Element Method (FEM), some developments have been proposed using Lagrange or hierarchical finite elements [START_REF] Atalla | Enhanced weak integral formulation for the mixed (u, p) poroelastic equations[END_REF][START_REF] Horlin | A 3-D hierarchical fe formulation of Biot's equations for elasto-acoustic modeling of porous media[END_REF][START_REF] Rigobert | Investigation of the convergence of the mixed displacement-pressure formulation for threedimensional poroelastic materials using hierarchical elements[END_REF]. Because of the scale disparity, the so-called poroelastic elements have a slower convergence rate than purely elastic or fluid elements [START_REF] Rigobert | Investigation of the convergence of the mixed displacement-pressure formulation for threedimensional poroelastic materials using hierarchical elements[END_REF]. To make matters worse, Biot's equations are frequency dependent and large FEM system matrices have to be recalculated for each frequency. For homogeneous and isotropic materials, the Boundary Element Method (BEM) offers an alternative [START_REF] Tanneau | A boundary element method for porous media[END_REF]. The method has the advantage of reducing the entire problem to one with only unknowns on the boundaries. However, the system matrix is full and there is still the need to discretize the boundary surface as well as performing regular and singular integrations over each boundary element.

In the past decade, several researchers have focused their work on meshless methods in order to avoid the time-consuming mesh generation process for complicated geometries. In this regard, the Method of Fundamental Solution (MFS) has been shown to be efficient for solving a large variety of physical problems as long as a fundamental solution of the underlying differential equation(s) is known. In particular, the MFS is suitable for scattering problems by choosing appropriate fundamental solutions satisfying the radiation condition at infinity. The method shares the same advantages as the BEM over domain discretization methods because there is no need to create a mesh over the entire domain. Furthermore, as no integration is needed, some numerical difficulties encountered with the BEM are avoided. For comprehensive reviews on applications of the MFS for scattering and radiation problems one can refer to Fairweather et al. [START_REF] Fairweather | The method of fundamental solutions for elliptic boundary value problems[END_REF][START_REF] Fairweather | The method of fundamental solutions for scattering and radiation problems[END_REF].

In this work, we are interested in applying as well as assessing the MFS for the numerical simulation of a bidimensional incident acoustic wave scattered by a poroelastic material. To the authors' knowledge, such problems have never been addressed using the MFS and although analytical solutions are available for canonical geometries such as cylindrical and spherical scatterers [START_REF] Hasheminejad | Acoustic scattering by a pair of poroelastic spheres[END_REF][START_REF] Hasheminejad | Acoustic scattering by an elliptic cylindrical absorber[END_REF], there is a need for fast and accurate methods taking into account scatterers of arbitrary shape. The applications we have in mind range from multiple scattering, the modeling of double porosity materials [START_REF] Atalla | Acoustic absorption of macro-perforated porous materials[END_REF][START_REF] Olny | Acoustic wave propagation in double porosity media[END_REF] to shape foam optimization modeling [START_REF] Lee | Two-dimensional poroelastic acoustical foam shape design for absorption coefficient maximization by topology optimization method[END_REF][START_REF] Sgard | On the use of perforations to improve the sound absorption of porous materials[END_REF].

In a preliminary note presented by the authors [START_REF] Nennig | On the efficiency of the method of fundamental solutions for acoustic scattering by a poroelastic material[END_REF], the application of the MFS to such problems was briefly presented. In this work, all the details of the implementation of the method and the investigation of its accuracy and numerical characteristics, including the condition number of the resulting matrices are given. The present paper is organized as follows. After presenting the MFS formulation in Section 2, the method's performance is measured with respect to an academic problem, the circular-shaped scatterer, for which analytical solutions are available. In particular, it is shown that reliable error indicators can be used to assess the quality of the results. In Section 4, various numerical examples of increasing difficulty are presented with a comparison to FEM results. The last section shows applications concerning the scattering of an incident plane wave by an infinite periodic array of porous structures.

Formulation of the method

Consider a time-harmonic acoustic plane wave ϕ inc 0 = A inc exp ik 0 ζ • x (with the convention e -iωt ) in an unbounded exterior domain Ω e incident upon one (or more) poroelastic acoustic foam(s) denoted by Ω i with boundary Γ as shown in Fig. 1. We denote by n, the unit outward normal vector to Ω i . In the surrounding acoustic domain Ω e , the fluid is inviscid and the acoustic displacement potential ϕ 0 obeys the wave equation

∆ϕ 0 + k 2 0 ϕ 0 = 0. (1)
Here, k 0 = ω/c 0 is the classical wavenumber defined as the ratio of the angular frequency ω and the sound speed c 0 . To express the transmission conditions at the interface Γ, it is convenient to introduce the particle displacement perturbation w = ∇ϕ 0 . With this definition, the acoustic pressure is obtained from the linearized momentum equation as p = ρ 0 ω 2 ϕ 0 . In [START_REF] Abramowitz | Handbook of Mathematical functions[END_REF], it is natural to split the potential into an incident and a scattered part as ϕ 0 = ϕ inc 0 +ϕ sc 0 (and similarly for the pressure and displacement). Here we require the scattered field to satisfy the usual Sommerfeld radiation condition at infinity. In the poroelastic medium the acoustic waves propagation is described by Biot's model [START_REF] Biot | Theory of propagation of elastic waves in a fluidsaturated porous solid. I. low-frequency range[END_REF]. This latter is based on the superposition of a fluid phase and a solid phase which are coupled together and respectively described by the fluid phase displacement U and the solid phase displacement u. For the time-harmonic representation, we have the following coupled system [8]

∇ • σ s + ω 2 (ρ 11 u + ρ 12 U) = 0, (2)
∇ • σ f + ω 2 (ρ 12 u + ρ 22 U) = 0. ( 3
)
The solid and fluid phase stress tensors are given by

σ s = (A ∇ • u + Q ∇ • U) I + 2N ε s , (4a)
σ f = (Q ∇ • u + R ∇ • U) I, (4b)
where ε s = 1/2(∇u + (∇u) t ) is the usual strain tensor and I is the identity matrix. The total stress tensor σ t is, by definition, the sum of σ f and σ s . Biot's coefficients A, N, Q, R are related to the material properties by the Allard-Johnson model. Their expressions can be found in the literature or in the reference textbook [START_REF] Allard | Propagation of sound in porous media: modeling sound absorbing materials[END_REF] as well as the other quantities introduced in this section. These quantities are all complex and frequency-dependent, A and N correspond to the Lamé coefficients, R is the effective bulk modulus of the fluid phase and Q indicates the coupling of the two phases volumic dilatation. The imaginary parts of A and N include the structural damping and, in Q and R these parts include the thermal dissipation. The imaginary parts of the effective density coefficients ρ 11 , ρ 22 and ρ 12 take into account viscous damping. The complete solution to the problem is found after applying the classical air-porous transmission conditions [START_REF] Debergue | Boundary conditions for the weak formulation of the mixed (u, p) poroelasticity problem[END_REF][START_REF] Lovera | Boundary conditions for a fluid-saturated porous solid[END_REF] on the interface Γ, i.e.

p p -p sc = p inc , ( 5a
) φU • n + (1 -φ)u • n -w sc • n = w inc • n , (5b)
σ t n + p sc n = -p inc n . ( 5c
)
Here φ is the porosity and the pore pressure p p is obtained from the fluid phase tensor as -Iφp p = σ f . For homogeneous and isotropic materials, the wave equation for the fluid phase and solid phase displacements can be decoupled thanks to the Helmholtz decomposition. Both solid and fluid displacement fields are then written as

u = ∇ϕ + ∇ ∧ (ϕ 3 • e 3 ) and U = ∇χ + ∇ ∧ (Θ • e 3 ). (6)
After decoupling the equations, we have [START_REF] Allard | Propagation of sound in porous media: modeling sound absorbing materials[END_REF]: ϕ = ϕ 1 + ϕ 2 and χ = µ 1 ϕ 1 + µ 2 ϕ 2 where

µ i = P k 2 i -ω 2 ρ 11 ω 2 ρ 12 -Qk 2 i , i = 1, 2, (7)
are the wave amplitude ratios between the two phases in the porous material (here, P = A + 2N ). Similarly, the potential Θ is simply obtained as Θ = µ 3 ϕ 3 with µ 3 = ρ 12 /ρ 22 . Under this form, each potential ϕ i (i = 1, 2, 3) satisfies the Helmholtz equation

∆ϕ i + k 2 i ϕ i = 0, (8)
and the associated complex wavenumbers are

k 2 1 = ω 2 2(P R -Q 2 ) (P ρ 22 + Rρ 11 -2Qρ 12 + √ D), (9)
k 2 2 = ω 2 2(P R -Q 2 ) (P ρ 22 + Rρ 11 -2Qρ 12 - √ D), ( 10
)
k 2 3 = ω 2 N ρ 11 ρ 22 -ρ 2 12 ρ 22 . (11)
Here, D stands for the discriminant of a quadratic equation and D = (P [START_REF] Doutres | Validity of the limp model for porous materials: a criterion based on the Biot theory[END_REF] ). Physically, there are two compressional waves associated with ϕ 1 , ϕ 2 and one rotational (shear) wave associated with ϕ 3 . They all propagate in the two phases and their relative contributions are given by the coefficients µ i . If such a decomposition holds in elastodynamics, the coexistence of two phases in the poroelastic media adds another fluid-borne compressional wave which is not present is elastic solids.

ρ 22 + Rρ 11 -2Qρ 12 ) 2 -4(P R -Q 2 )(ρ 11 ρ 22 - ρ 2
The MFS implementation starts by choosing an appropriate set of fundamental solutions for both propagative domains Ω e and Ω i . In the acoustic domain, a natural choice is to choose these solutions using the well-known free field Green's function, i.e. G 0 (x, y) = i/4H 0 (k 0 |x -y|). So, we seek the scattered field as a distribution of monopoles

p sc (x) = ρ 0 ω 2 Q0 q0=1 A 0 q0 G 0 (x, y 0 q0 ), (12)
where the source points y 0 q0 are chosen to be located on a curve Γ0 in the interior domain Ω i . The displacement vector is obtained by simply taking the gradient of [START_REF] Doutres | Validity of the limp model for porous materials: a criterion based on the Biot theory[END_REF]. To simulate the wave field in the foam, a possible choice is to use fundamental solutions for poroelastic media for which an explicit form can be found, for instance, in [START_REF] Bonnet | Basic singular solutions for a poroelastic medium in the dynamic range[END_REF]. An easier option is to construct these solutions by simply expanding each potential in the form

ϕ i (x) = Qi qi=1 A i qi G i (x, y i qi ), for i = 1, 2, 3, (13)
where

G i (x, y) = i/4H 0 (k i |x -y|)
is the fundamental solution of the Helmholtz equation. Here, Q i is the number of source points y i qi associated with the i th potential and the A i qi 's are unknown amplitudes. These points are located on a fictitious boundary Γi in the exterior domain Ω e as shown in Fig. 1. The influence of the location of the source points on the quality of the solution will be discussed in the next section. Now, using these expansions in [START_REF] Atalla | Acoustic absorption of macro-perforated porous materials[END_REF], we find the expression for the different physical quantities involved:

φp p (x) = 2 i=1 k 2 i (Q + µ i R) Qi qi=1 A i qi G i (x, y i qi ) (14a) u(x) = 2 i=1 Qi qi=1 A i qi ∇G i (x, y i qi ) + Q3 q3=1 A 3 q3 ∇ ⊥ G 3 (x, y 3 q3 ) (14b) U(x) = 2 i=1 Qi qi=1 A i qi µ i ∇G i (x, y i qi ) + Q3 q3=1 A 3 q3 µ 3 ∇ ⊥ G 3 (x, y 3 q3 ) (14c)
where

∇ ⊥ ≡ (∂ x2 , -∂ x1 )
t stands for the orthogonal gradient operator. From these expressions and using (4), we finally obtain the normal total stress tensor σ t . The explicit forms for the tensor coefficients are too cumbersome to be inserted in this paper. Note that other type of solutions could have been considered (Green's functions and/or their derivatives or plane waves for instance) and the present choice was mainly motivated by simplicity. Furthermore, it has the advantage of allowing us to specify independently the number of source points Q i for each kind of wave, which is not the case when using Green's functions. In this respect, our method is not strictly speaking an MFS since it does not rely on Biot's equations fundamental tensor. Now, given a set of N col collocation points x l (l = 1, . . . , N col ) on Γ, substituting [START_REF] Fairweather | The method of fundamental solutions for scattering and radiation problems[END_REF] in the transmission conditions (5) at each collocation point yields three linear systems of the form

M α A = F inc α , α = a, b, c. (15)
Here, the subscript α refers to the type of condition involved (there is one system for each condition (5a), (5b), (5c)). The right hand side vector F inc α stems from the incident wave (pressure and displacement). The unknown vector A contains the amplitudes of all sources. Since the acoustic displacement is expected to behave like |w| ∼ k 0 φ 0 , it is judicious to rescale the kinematic conditions (5b) by multiplying the associated lines by ρ 0 c 0 ω. This yields the matrix system

MA = F inc . ( 16
)
In this work, early results showed that it is preferable for reasons of stability to consider more collocation points than the number of unknowns N dof = scatterers (the circular scatterer for instance), the MFS has been observed to perform slightly better with interpolation schemes (i.e. N col = N dof ) than least-squares schemes (N col > N dof ). However, for the sake of robustness and generality the second approach was preferred. After multiplying the system ( 16) by the Hermitian transpose M † we finally get the Hermitian square system

M † MA = M † F inc . (17)
A important advantage of performing this operation is that a simple error estimator is available at very low cost. Indeed, going back to the original collocation problem [START_REF] Hasheminejad | Acoustic scattering by a pair of poroelastic spheres[END_REF], we can define the a posteriori algebraic error estimator for each of the conditions at the interface (α = a, b, c). We thus define

E α = 100 M α A -F inc α 2 F inc α 2 , ( 18
)
where A is solution of [START_REF] Hennion | Analysis of the scattering of a plane acoustic wave by a periodic elastic structure using the finite element method : application to compliant tube gratings[END_REF]. The numerical evaluation of ( 18) is computationally cheap since the matrices M α have already been calculated. Of course, taking N col = N dof will automatically produce errors with zero value (except from when interpolation matrices are very ill-conditioned).

By increasing the number of collocation points, errors will grow until it stabilises.

As for the choice of positions of the source points, we follow Alves [START_REF] Alves | On the choice of source points in the method of fundamental solutions[END_REF] and take these points along the discrete normal direction, so we put y i qi = ŷi qi + s i ñi qi where the points ŷi qi (q i = 1, . . . , Q i and i = 0, . . . , 3) are distributed on Γ and ñi qi is the approximate normal vector defined by

ñi qi = (ŷ i qi +1 -ŷi qi-1 ) ⊥ max qi |ŷ i qi +1 -ŷi qi-1 | . ( 19
)
Here, the symbol ⊥ signifies that we take the orthogonal vector pointing outward. The normalization is chosen here for convenience to ensure that the normal vector amplitudes never exceed unity and that coefficients s i correspond to the farthest distance from the interface Γ. Before we end this section, we should point out that, as far as engineering applications are concerned, the physical variables of primary interest are the pressure and the normal velocity at the interface. In practical terms, the scattering characteristics of the porous obstacle are conveniently quantified by the scattering cross section (SCS) defined as the ratio between the scattered power and the incident power flux [START_REF] Morse | Theoretical acoustics[END_REF]

Σ = W sc I inc = Γ Re (p sc v sc n ) dΓ |A inc | 2 ω 4 ρ 0 /c 0 , (20)
where v sc n = -iωw sc • n is the normal acoustic velocity for the scattered field. Similarly, we can define the absorption cross section as the ratio between the total power and the incident power flux. These quantities combine the reflection, transmission and the absorption properties of the porous scatterer. By construction, the SCS is a far field estimator and numerical errors at the interface only have a mild impact on its numerical evaluation. This will be commented further when necessary.

Validations for the circular-shaped scatterer

We shall validate and assess the efficiency of the method in the specific case of the scattering of a horizontal acoustic plane wave by a circular-shaped poroelastic scatterer. In polar co-ordinates, the inner and outer wave fields can be represented by separable solutions. Each potential is given by infinite series which are well behaved allowing us to produce very accurate results without deterioration at high frequency. The derivation of the exact form for these solutions is quite lengthy and this is inserted in Appendix. For a proper assessment of the method's performance, it is convenient to define the relative error on the boundary for each physical quantity (call it X):

Ê(X) = 100 X -X 2 X 2 , (21)
where X represents either the acoustic pressure, the fluid phase or solid phase normal displacement or the normal stress at the interface and X is the associated vector containing the value of X at the collocation points. Here, X stands for a reference solution vector, either obtained analytically in the present case or computed numerically with another method when analytical solutions are not available. The aim of this section is to identify the effects of the main parameters of the problem, namely the locations and the number of sources. For this latter, we are still left with the problem of finding a quasi-optimal relationship between the Q i 's. Through intensive calculations, not presented here, we found that choosing the same number for each wave type was probably the best option offering the best trade-off between accuracy, conditioning and simplicity. This simplifies the analysis significantly as we can now put

Q = Q i for i = 0, . . . , 3, (22)
and perform the analysis with a single parameter Q. Similarly, we take the points ŷi qi all equal and put s i = s with s 0 = -s. Note that this choice seems to be contrary to the results given in [START_REF] Perrey-Debain | Plane wave decomposition in the unit disc: Convergence estimates and computational aspects[END_REF] where it is advocated that the best option, when using plane waves, is precisely not to take the same number of wave directions for each type of waves (that is shear and pressure wave types in the elasticity case) and this is particularly relevant when the ratio between the wavenumbers is large. The reasons for this probably lie in that, plane waves and singular sources behave differently in terms of their approximation properties and it would be interesting to explore this further via numerical experiments in the spirit of [START_REF] Alves | Numerical comparison of two meshfree methods for acoustic wave scattering[END_REF] With the resistivity σ, the tortuosity α inf , the viscous and thermal characteristic lengths Λ and Λ ′ , the poisson coefficient ν and the effective skeleton density ρ 1 . The effective skeleton density ρ 1 = (1 -φ)ρs, where ρs is the density of the material of the frame. As an exemple we chose a polymer foam (XFM) commonly used in the transport industry. The material properties are reported in Table 1. To give an idea of the complex wavenumbers for the wave potentials in the material, we find that k 1 ≈ 17.6 + 1.21i, k 2 ≈ 21.4 + 14.1i and k 3 ≈ 38.9 + 2.1i (these values correspond to the frequency 500 Hz). The corresponding acoustic wavenumber is k 0 ≈ 9.2. On Fig. 2 (a) we show the influence of the source locations as well as the number of sources (i.e. Q) on the normal solid displacement error u n = u • n, i.e. Ê(u n ) and the a posteriori error E b . The evolution of the conditioning (in the 2-norm) of the associated system is also shown (Fig. 2 (b)). Before we comment on these results, we should point out that we chose to measure the normal displacement error rather than the pressure error for the simple reason that it was observed, that a small error on the displacement normally guarantees a smaller error on the pressure. The convergence curves together with the conditioning curves typically illustrate the MFS paradox which many authors have already observed and discussed: despite the ill-conditioning of the MFS systems, the method produces accurate results. In the present situation, it is concluded that the further away from the boundary the source points are (i.e. the larger s), the better the results as long as the conditioning of the system does not exceed a certain value above which results are likely to be corrupted by round off errors. This value, say cond 2 ≈ 10 16 , is in line with standard double precision arithmetic. Note that, in our algorithm, matrices are inverted using the pinv function from MATLAB so that for very ill-conditioned matrices (cond 2 > 10 16 ) the SVD is automatically used with some thresholding in order to dampen the effects of the round-off errors. In this respect, we believe that better results could perhaps be obtained by properly filtering the small singular values. Another approach would be to apply Tikhonov regularization in order to speed up the computation. In fact, the most striking feature of these convergence curves is the almost perfect agreement between the true error Ê(u n ) and its algebraic counterpart, the a posteriori error E b and through many numerical experiments, we always observed very good correlations. The mathematical reasons for this goes beyond the scope of this paper but we believe that in most cases the algebraic a posteriori errors E α should serve as reliable error indicators especially when reference solutions are not available. This section comprises three examples of increasing difficulty illustrating the efficiency of the MFS. The first shows the MFS performance for foams presenting geometrical singularities such as corners. In the second example, we tackle circular shaped corrugated foams. The last example concerns scatterers of arbitrary shapes. Because there are no analytical solutions to these problems, all reference solutions are computed using Finite Element models. These computations are carried using Lagrange quadratic finite elements in both fluid and poroelastic domains. The (u, p p ) formulation of Biot's equations [START_REF] Atalla | Enhanced weak integral formulation for the mixed (u, p) poroelastic equations[END_REF][START_REF] Debergue | Boundary conditions for the weak formulation of the mixed (u, p) poroelasticity problem[END_REF] is used and the non reflecting boundary conditions are implemented using Bermudez's perfectly matched layer (PML) formulation [START_REF] Bermudez | An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems[END_REF]. As for the accuracy of these FE reference solutions, numerical tests carried out on the circular shaped scattering problem shows that these results are reliable up to around 1 percent of error on the solid phase normal displacement which is acceptable if engineering accuracy is sought.

Square scatterer

The first example concerns that of a square-shaped poroelastic scatterer of side length a = 0.2 m. The incident plane wave is horizontal, travelling in the ζ = (1, 0) direction and the frequency is 1500 Hz. In this scenario, the boundary curve is not regular and source points around the square are placed along the normal to the boundary avoiding the corners, see Fig. 3. In this respect, we found that putting more sources in the vicinity of the corners did not show any improvement. Comparisons between the MFS and the FE results are conveniently shown in Fig. 4 for the pressure (Fig. 4a) and the normal displacement (Fig. 4b). Here, the abscissae θ refers the usual polar coordinate of the collocation points on the boundary (the square is centered at the origin). One can note the associated global errors are in good agreements with the a posteriori error estimator as shown in Table 2.

As expected, the discrepancies reach a maximum near the corners and this is particularly noticeable for the displacement curve. The pressure remains more stable even near the corners and is free of spurious oscillations. Fortunately these spurious oscillations, present on the normal displacement curve have a low radiation efficiency and their effect on the SCS is negligible.

We should point out that, the displacement and pressure magnitudes do not correspond to any realistic case as we took A inc = 1 in our calculations.

Scattering by a corrugated cylinder

Here, we consider the scattering of a horizontal plane wave by a circular shaped corrugated foam with boundary The shape of the boundary as well as the location of the collocation points and the source points are displayed in Fig. 5. The perfectly circular shape of radius a = 0.1 m is also illustrated. For this specific example, a multifrequency analysis is performed in the range [0,5000 Hz]. In Fig. 6 we show the evolution of the SCS for different materials: the XFM foam and the B wool whose properties are reported in Table 1. The perfectly circular shape is also considered for the sake of illustration. The results are compared with the perfectly reflecting wall case (rigid wall). These are computed using the MFS with acoustic sources only and by applying the normal displacement condition at the interface: w • n = 0.

In the mid-frequency range, the presence of peaks for the XFM foam can be observed. An analysis of the poroelastic wave field in the absorbing material reveals that these are due to some resonance effects of the elastic foam's skeleton. For the wool, these peaks are highly attenuated and the use of an equivalent fluid model [START_REF] Doutres | Validity of the limp model for porous materials: a criterion based on the Biot theory[END_REF] would have been sufficient for the analysis. The effects of the corrugations on the SCS are noticeable above 3000 Hz for the rigid case. This frequency corresponds to an acoustic wavelength which is comparable to the length of the obstacle. For porous scatterers, the effects of the corrugations are dampened due to the absorbing properties of the materials, especially in the high-frequency range.

All the results have been computed with an accuracy that does not exceed 1.2% error. A closer analysis, however, is instructive as it shows behaviors which are inherent to the method. In Fig. 7, the associated a posteriori errors E b are plotted in logarithmic scale. For the rigid circular cylinder case, clearly identifiable peaks can be observed at the frequencies: 1626, 2592 and 3733 Hz. These peaks correspond to a loss of accuracy due to some non-uniqueness problems of the MFS formulation. This point is discussed in length in [START_REF] Wilton | A clarification of nonexistence problems with the superposition method[END_REF] and in [START_REF] Chen | Using the method of fundamental solutions in conjunction with the degenerate kernel in cylindrical acoustic problems[END_REF] for the circular cylinder case. It emerges that these 'critical' frequencies are in fact the eigenvalues of an interior problem corresponding to the internal surface on which are located the source points. Because the source points are placed on a perfectly circular ring, it was easy to check that the frequencies correspond to the interior Dirichlet problem [START_REF] Chen | Using the method of fundamental solutions in conjunction with the degenerate kernel in cylindrical acoustic problems[END_REF]. In general, XFM foam, B wool, rigid wall. Computed using Q = 60.

these irregular frequencies are hard (if not impossible) to predict though their effects are somewhat noticeable even with porous scatterers (except with the B wool where it is harder to conclude).

In this work, the MFS algorithm was implemented using MATLAB and, for the specific examples illustrated here, around 0.73 second (CPU time) is needed per frequency (Intel Core 2 Duo T7100 @ 1.80 GHz, 2 GB RAM). To the authors' knowledge these performances cannot be met with the FEM. Using a compiled language and avoiding some redundant evaluations of Hankel functions, it is anticipated that the computational time could be reduced further by a factor of 10. For the sake of illustration, the total pressure field is shown in Fig. 8. This corresponds to a high frequency calculation (15000 Hz). Around 1.6 seconds are required to compute the solution with less than 1.2 % of a posteriori error and only 400 degrees of freedom (dof) are needed (Q = 100 and we took s = 0.25a for the sources locations). The computation of the pressure on a refined grid containing about 150,000 points takes approximately 30 seconds. Note the performance of the method could be further enhanced for this example and those of sec. 3 and 4.1 if symmetries of the domains were taken into account.

Multiple scatterers of arbitrary shapes

The last numerical experiment concerns a multiple scattering problem with non-convex objects having the shape of the three letters 'M', 'F' and 'S'. The boundaries have been drawn using a graphical software and interpolated with Bezier curves. Fig. 9 shows the locations of the source points as well as the collocation points on the letter 'M'.

The example considered here is that of an incident plane wave in the ζ = (cos π/4, -sin π/4) direction and the frequency is 6000 Hz. In our calculation, we took Q = 80 for each scatterer; this yields N dof = 3×4×Q = 960 variables. The a posteriori errors for the pressure and the displacement are E a = 1.78% and E b = 2.77%. The total pressure in the acoustic domain and in the porous domain is shown in Fig. 10. The absorbing properties of the porous material can be visually identified. Note that the CPU time needed for computing the matrix coefficients and inverting the system does not exceed 20 s on a simple PC. The use of the FEM would be far more demanding in order to achieve the same level of accuracy.

Periodic scattering

Let us now consider an infinite periodic array of identical porous structures. The problem is depicted in Fig. 11. Here the geometry is assumed d-periodic in the e 1 direction. When the array is illuminated by a plane wave of angle of incidence θ inc (defined about the vertical axis as shown), the solution of the scattering problem is pseudoperiodic, that is d-periodic with a phase shift, and each physical variable (call it X) must fulfill the condition

X(x + md e 1 ) = X(x)e imβd . ( 24
)
Here m is a relative integer and β = k 0 sin θ inc is the projection of the incident wave-number vector on the e 1 direction. This problem can be easily solved with the MFS by simply modifying the Green's function for the acoustic domain. The simplest form is a representation as a sum of images For convenience we shall write the field point x = (x 1 , x 2 ), the source point y = (y 1 , y 2 ) and introduce the notation δ 1 = x 1 -y 1 and δ 2 = x 2 -y 2 . In [START_REF] Nennig | On the efficiency of the method of fundamental solutions for acoustic scattering by a poroelastic material[END_REF], r m = δ 2 2 + (δ 1 -md) 2 is the distance between the field point and the source point of the m th scatterer. The series is unfortunately very slowly convergent and is therefore not appropriate for numerical computations. In [START_REF] Linton | The Green's function for the two dimensional Helmholtz equation in periodic domains[END_REF], the author has compared several methods to efficiently compute the 2D periodic Green's function G d β . Through extensive numerical tests it was shown that the Ewald method is probably the best technique giving a good trade-off between CPU time and robustness. This is based on the alternative form for Green's function:

G d β (x, y) = - i 4 ∞ m=-∞ H 0 (k 0 r m )e imβd . (25)
G d β (x, y) = 1 4d ∞ m=-∞ e iβmδ1 γ m e γmδ2 erfc γ m d 2e + eδ 2 d + e -γmδ2 erfc γ m d 2e - eδ 2 d + 1 4π ∞ m=-∞ e imβd ∞ n=0 1 n! k 0 d 2e 2n E n+1 e 2 r 2 m d 2 , (26)
Here, β m = β + 2πm/d and

γ m = (β 2 m -k 2 0 ) 1 2 = -i(k 2 0 -β 2 m ) 1 2 . ( 27
)
The definition for the square root is chosen in order to ensure that the scattered acoustic waves are radiating away from the array or decay exponentially. The advantage of such a representation is that the series is quickly convergent and only a small number of terms is needed to achieve extremely accurate results (to double standard precision if necessary). Furthermore, the complementary Error function erfc and the exponential integral can be computed quickly [START_REF] Zhang | Computation of Special Functions[END_REF]. In [START_REF] Olny | Acoustic wave propagation in double porosity media[END_REF], the positive constant e is arbitrary though its value will affect the convergence of the two summations, this is discussed in [START_REF] Linton | The Green's function for the two dimensional Helmholtz equation in periodic domains[END_REF] and references therein. As shown in Table 3 in [START_REF] Linton | The Green's function for the two dimensional Helmholtz equation in periodic domains[END_REF], e = 4 seems to be a good choice allowing fast convergence and it was observed that less than twenty terms are required in the two summations over m and around ten in the summation over n to nearly achieve machine precision. The spatial derivatives involved in the computation of the displacement vector and the stress tensor are obtained by simply differentiating the truncated version of [START_REF] Olny | Acoustic wave propagation in double porosity media[END_REF].

E n (x) = ∞ 1 u -n e -xu du (28)
As an example, we consider an infinite periodic array of 'M'-shaped obstacles filled with XFM foam (Fig. 9). The results for the pressure field around the scatterer located near the origin (m = 0) are plotted in polar coordinates in Fig. 12. In our calculations, we took f =1500 Hz, d = 0.3 m, θ inc = π/6, Q = 80 and the location for the source points is chosen with s = 0.2a (a is the characteristic length of the obstacle as shown Fig. 9). The MFS results are compared with those computed using pseudo-periodic finite elements as described in [START_REF] Hennion | Analysis of the scattering of a plane acoustic wave by a periodic elastic structure using the finite element method : application to compliant tube gratings[END_REF]. The a posteriori error estimator is E b = 2% which is in accordance with the 3% of relative error measured against the FEM solution. Once the MFS coefficients have been found, quantities such as the reflection and transmission coefficients or the acoustic absorption can be easily computed in a straight forward manner using the plane wave decomposition for the periodic Green's function and we refer to [START_REF] Linton | Resonant effects in scattering by periodic arrays[END_REF] for more details. The effects of the periodic array on the scattering can be identified in Fig. 13 where the real part of the total pressure is shown around and inside the porous scatterers.

Conclusions

The application of the MFS for the acoustic wave scattering by porous scatterers has been presented. The wave equations for the fluid phase and solid phase displacements can be decoupled using the Helmholtz decomposition which allows for a systematic and simple way to construct fundamental solutions describing the wave displacement field in the poroelastic material. An assessment of the MFS efficiency with respect to the number and position of the virtual sources and of the collocation points is presented in detail for the specific case of a circular- shaped scatterer. In particular, it is shown that reliable error indicators can be used to assess the quality of the results. Through many illustrative examples of increasing difficulty, comparisons with results computed using a mixed pressure-displacement finite element formulation illustrate the considerable advantages of the MFS both in terms of computational resources and mesh preparation. Finally, the extension of the method to dealing with the acoustic wave scattering by an infinite array of periodic porous scatterers is discussed and presented. It is believed that the method show promise in solving a wide range of noise control problems ranging from multiple scattering to shape foam optimization. The development of the MFS for 3D problems is also of particular interest to us and this could be the subject of futur work.

Figure 1 :

 1 Figure 1: Scattering geometry.

Figure 2 :

 2 Figure 2: Effects of the sources location on the error and the conditioning ( s = 0.1a, s = 0.2a, s = 0.3a). (a): error on the solid phase normal displacement for the XFM foam at 1500Hz; line with markers '•' refers to Ê(un) error and the simple line to the a posteriori error E b . (b): conditioning (2-norm) of M † M.

Figure 3 :

 3 Figure 3: Square-shaped scatterer. Here symbols +, ×, • indicate the location of the source points in Ω e and ♦ those in Ω i . The collocation points on Γ are identified by the markers '•'.

Figure 4 :

 4 Figure 4: (a): Real part of the pressure field on Γ at 1500 Hz for the square-shaped scatterer; solid line: FEM solution and dots refers to the MFS solution. (b): Real part of solid phase normal displacement.

aFigure 5 :Figure 6 :

 56 Figure 5: Corrugated cylinder shape. Here +, ×, • are the source points in Ω e and ♦ those in Ω i . The collocation points on Γ are given by markers '•'.

Figure 7 :

 7 Figure 7: a posteriori error estimator for the straight (black) and corrugated (gray) cylinder. XFM foam, B wool, rigid wall. Computed using Q = 60.

Figure 8 :

 8 Figure 8: Real part of the pressure fields at 15000 Hz for a XFM foam obstacle impinged by plane wave with angle of incidence θ inc = 0.2 rad.

  a

Figure 9 :

 9 Figure 9: M-shaped scatterer with a = 0.1 m. Here the symbols +, ×, • indicate the location of the source points in Ω e and ♦ those in Ω i . The collocation points on Γ are identified by the markers '•' .

Figure 10 :

 10 Figure 10: Real part of the total pressure in the acoustic domain and in the porous material (XFM foam).

Figure 11 :

 11 Figure 11: Periodic scattering geometry.

2 πFigure 12 :

 212 Figure 12: Real part of the pressure field around a M-shaped scatterer (m = 0) at 1500 Hz. The marker '•' refers to the FEM solution and '+' to the MFS solution.

Figure 13 :

 13 Figure 13: Real part of the MFS pressure field at 1500 Hz for a periodic 'M'-shaped porous scatterer.

Table 1 :

 1 for instance. Materials properties.

	Material	φ	σ (kNm -4 s) α inf Λ (µm) Λ ′ (µm) ρ 1 (kgm -3 )	N (kPa)	ν	Ref.
	XFM foam 0.98	13.5	1.7	80	160	30	200(1 -0.05i) 0.35 [11]
	B wool	0.95	23	1	54.1	162.3	58	8.5(1 -0.1i)	0	[12]

Table 2 :

 2 Relative errors on a square scatterer (a=0.2 m, s=0.2a) at 500 Hz and 1500 Hz for XFM foam with Q=60 and 2 × 4 Q = 480 collocation points.

			500 Hz	1500 Hz
	Fields p (α =a)		E α (%) Ê (%) 5.56 0.75	E α (%) Ê (%) 3.95 1.03
	u • n (α =b)		4.42	1.88	4.95	4.31
	given parametrically by
	x(t) = 0.1	cos t sin t	+ 0.005	cos 8t sin 8t	, t ∈ [0, 2π[. (23)

i=0 Q i . To be more specific, it was found that taking N col = 2N dof guarantees that results have converged, i.e. the numerical solution becomes insensitive to the number of collocation points, as discussed in[START_REF] Alves | On the choice of source points in the method of fundamental solutions[END_REF]. All calculations were performed using this ratio. Note that when dealing with 'nice' shaped

Acknowledgement

The authors wish to thank the Projet Pluri-Formations Pilcam2 [28] at the Université de Technologie de Compiègne for providing the high performance computing resources that have contributed to the research results reported within this paper as well as the reviewers for their helpful comments.

Appendix A. Scattering of a horizontal acoustic plane wave by a circular-shaped poroelastic scatterer.

Take an incident plane wave along the e 1 direction with amplitude A inc . Following [START_REF] Abramowitz | Handbook of Mathematical functions[END_REF][START_REF] Morse | Theoretical acoustics[END_REF], the outer wave field in the acoustic domain can be represented by separable solutions in polar coordinates (r, θ) as

and, similarly for the total field:

For the wave field in the poroelastic medium we can follow a recent paper [START_REF] Nennig | A mode matching method for modelling dissipative silencers lined with poroelastic materials and containing mean flow[END_REF] and expand each potential as the Fourier series:

Âi,m J m (k i r) cos mθ, with i = 1, 2, (A.4a)

Now, applying the boundary conditions ( 5) at the airporous interface r = a yields a linear system of the form

where all the quantities involved depend on the azimuthal mode number m = 0, 1, 2.... The resolution of this 4 × 4 matrix system gives the coefficients Âi,m (i = 0, . . . , 3) and the analytical solution can thus be computed. For completeness, the analytical expressions for the coefficients are explicitly written:

with the constants

and φ i = φµ i +1-φ . The right hand side vector contains the contributions from the incident plane wave:

and these coefficients have to be divided by 2 for the special case m = 0. The conditioning of the system can be improved by multiplying the line of M corresponding to the normal displacement condition by the factor ρ 0 c 0 ω (and similarly for the corresponding line of F). In practice, the summation of the series is carried out until the relative difference between two consecutive steps is lower than 10 -10 percent.