Antoine Legay

The extended finite element method combined with a modal synthesis approach for vibro-acoustic problems

Keywords: XFEM, modal synthesis method, reduced order model, vibro-acoustics

The optimization of a vibro-acoustic problem is of main importance for passengers' comfort in transportation vehicles in terms of interior noise. Engineers use numerical tools to predict the response of this coupled problem, but it may lead to a prohibitive computational time. Based on FEM, this work aims at reducing the computational time. The first idea is to keep the same mesh of the acoustic cavity for all the structure configurations and to enrich the pressure approximation by using the extended FEM (XFEM). The enrichment is based on a Heaviside function completed at the structure tip by a continuous ramp function. The second idea is to build reduced basis. The structure basis is composed of its eigenmodes, whereas a modal synthesis method with a fixed interface is used to build the fluid basis. The interface DOFs are the enriched DOF of the XFEM, whereas the internal domain corresponds to the acoustic cavity with no structure. These two combined ideas enable to minimize the computational time in the study of the influence of the structure positions in an acoustic cavity. The method is implemented for shell structures embedded in a 3D acoustic domain.

INTRODUCTION

Noise reduction of passenger acoustic compartment in transport industry becomes a more and more important factor to be taken into account during the design process [START_REF]Boeing uses LMS SYSNOISE to predict acoustics of aircraft cabins[END_REF]. It is shown in [START_REF] Cherng | Characterization and validation of acoustic cavities of automotive vehicles[END_REF] that flexible and/or absorbing structures in an acoustic cavity may have significant influences on both the amplitude of the sound pressure level and the resonant frequency shifts. These flexible and absorbing structures have to be included in the model in order to predict a realistic noise mapping in the cavity. The acoustic characteristics of such a coupled problem can be predicted using numerical tools: for instance, the FEM for the acoustic part as well as for the structural part. Engineers can study several configurations of the equipments in the passenger acoustic compartment, and they can optimize the proportion and the localization of absorbing materials in order to reach the expected noise level: the total computational time increases drastically, which may become impossible to perform in practice. In order to catch the complexity of the 3D geometry of both the acoustic domain and flexible structures, the meshes may be very fine, involving a large number of DOFS.

Noise reduction is performed in industrial applications by using additional treatments such as porous material on the walls [START_REF] Allard | Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials[END_REF] or viscoelastic material on (or inside) the structure [START_REF] Rao | Recent applications of viscoelastic damping for noise control in automobiles and commercial airplanes[END_REF]. The computational time may be reduced by the use of modal basis [START_REF] Ohayon | Reduced models for fluid-structure interaction problems[END_REF][START_REF] Bobillot | Iterative techniques for eigenvalue solutions of damped structures coupled with fluids[END_REF][START_REF] Rouleau | Vibro-acoustic study of a viscoelastic sandwich ring immersed in water[END_REF], subdomain decomposition [START_REF] Rumpler | Performance of a restrained-interface substructuring FE model for reduction of structural-acoustic problems with poroelastic damping[END_REF][START_REF] Davidsson | A reduction method for structure-acoustic and poroelastic-acoustic problems using interface-dependent Lanczos vectors[END_REF][START_REF] Junge | Interface-reduction for the Craig-Bampton and Rubin method applied to FE-BE coupling with a large fluid-structure interface[END_REF][START_REF] Tran | A robust component mode synthesis method for stochastic damped vibroacoustics[END_REF], or simplification of the absorbing material as rheological devices [START_REF] Deü | Vibration and transient response of structural acoustic interior coupled systems with dissipative interface[END_REF][START_REF] Bermúdez | Modelling and numerical solution of elastoacoustic vibrations with interface damping[END_REF][START_REF] Legay | A rheological interface model and its space-time finite element formulation for fluid-structure interaction[END_REF]. Moreover, the accuracy of the numerical solutions may be controlled to ensure the quality of the results [START_REF] Ihlenburg | Finite element solution of the Helmholtz equation with high wave number. Part 1: the h-version of the FEM[END_REF][START_REF] Ph | Error estimation and adaptivity for the finite element method in acoustics: 2D and 3D applications[END_REF]. However, all the numerical strategies developed to reduce the computational time need a remeshing step when a new configuration of the structures in the cavity is studied.

In this work, the acoustic cavity is considered with no variation in the overall optimization process. Because the flexible structures are generally thin compared with the size of the cavity, the structures are modeled as shells. For simplicity reasons, the dissipation is only introduced in this work as a modal structural damping. These structures are localized arbitrarily in the acoustic cavity, which enables to study several configurations (different number of structures, change of the size, and the positions of the structures) with no fluid remeshing. In order to catch the pressure discontinuity around the structures in the context of incompatible meshes, the extended FEM (XFEM [START_REF] Moės | A finite element method for crack growth without remeshing[END_REF]) is used to enrich the acoustic pressure field around the structure. Because the structures are considered as shells, the normal to the structure, and so the pressure, is not defined at the structure tip: it varies continuously from one side of the structure to the other one. This effect is taken into account in the enrichment function while it is harder to model with a compatible mesh.

Based on this approach, a reduction of the structural part using a standard modal basis is proposed. Concerning the fluid part, the reduction approach is based on a modal synthesis method [START_REF] Rumpler | Performance of a restrained-interface substructuring FE model for reduction of structural-acoustic problems with poroelastic damping[END_REF]. The XFEM discretization of the fluid part provides a natural separation between the internal and the interface DOFS: the interface DOFS are the enriched additional unknowns, whereas the internal DOFS correspond to the cavity with no structure inside. The advantage is that the modal basis of the fluid part is the same for all the configurations and then it can be computed only once. This modal basis is enriched by static modes that correspond to the response of the fluid to unit pressure imposed successively to the interface DOFS. These static modes have to be computed for each configuration. The size of the problem is finally reduced to the number of enriched fluid nodes plus the number of both structure and fluid modes, leading to a significant reduction of the number of DOFS.

The second section describes the problem and the discretized spaces and finally gives the system of equations to be solved. The details of the XFEM formulation and implementation of the structural-acoustic coupled problem can be found in [START_REF] Legay | An extented finite element method approach for structural-acoustic problems involving immersed structures at arbitrary positions[END_REF]. The third section details the modal reduced structure basis. The fourth section gives the key idea of this work which is the original use of the XFEM DOFS separation in order to condense the fluid domain on the interface. The last section explores academic and industrial-like applications so that the convergence and the computational time reduction is shown. Finally, an influence study of the positions of 2 structures in an acoustic cavity is performed.

DISCRETIZATION OF THE COUPLED PROBLEM USING XFEM

Description of the problem

The acoustic fluid domain and the thin flexible structures are, respectively, denoted by F and S (Figure 1). The problem is studied in the frequency domain for a permanent harmonic response at angular frequency !. The fluid is assumed to be compressible and inviscid, it satisfies the classical Helmholtz equation, the pressure fluctuation is used as variable. It is characterized by the constant speed of sound c 0 and by the density F . The structural domain is assumed satisfying the elastodynamic linearized equation, the displacement field is used as variable. The thin structures are modeled as shell structures meaning that they are seen by the fluid as surfaces. These surfaces are described in the fluid domain by the zero contour of a level set .M /, which is the signed distance to the closest structure (Figure 1). By denoting d the distance of a point M in the fluid domain to the structure, the level set is defined as

.M / D ˙d; (1)
where the sign is determined using a conventional unit normal vector E n pointing outward of one side of the structure to the arbitrarily chosen positive domain.

Discretized fluid pressure fluctuation and structural displacement fields

2.2.1. Fluid pressure fluctuation discretized space. A thin structure, modeled as a shell, placed in an acoustic domain separates locally the fluid into two parts, meaning that the pressure is discontinuous from one side of the structure to the other one. Moreover, the real structure tip is curve and the pressure field is continuous from one side to the other at the structure tip (Figure 3(a)). The classical discretization strategy is to make the different meshes compatible at the interface (Figure 2(a)) where three nodes are localized at the same geometrical point: two nodes for the two fluid domains and one node for the structure. The classical discretization, having one fluid node and one structure node at the same place, cannot represent correctly the local pressure field at the structure tip: in this case, the pressure has a unique value at the structure tip while it should vary. Another drawback of the classical discretization is that a remeshing is needed for each position of the structure in the fluid domain.

The alternative strategy used in this work is to embed the structure arbitrarily in the fluid mesh of the empty cavity (Figure 2(b)). The XFEM has been used in the context of fluid-structure interaction for structures embedded in a fluid flow [START_REF] Legay | An Eulerian-Lagrangian method for fluid-structure interaction based on level sets[END_REF][START_REF] Zilian | The enriched space-time finite element method (EST) for simultaneous solution of fluid-structure interaction[END_REF][START_REF] Gerstenberger | An eXtended finite element method/Lagrange multiplier based approach for fluidstructure interaction[END_REF] and for moving interface problems [START_REF] Duddu | A combined extended finite element and level set method for biofilm growth[END_REF][START_REF] Chessa | An enriched finite element method and level sets for axisymmetric two-phase flow with surface tension[END_REF], whereas it was first devoted to structural mechanics [START_REF] Sukumar | Modeling holes and inclusions by level sets in the extended finiteelement method[END_REF][START_REF] Legay | Strong and weak arbitrary discontinuities in spectral finite elements[END_REF]. Other techniques such as immersed element method-like are developed to deal with non-compatible meshes for a fluid-structure interaction problem [START_REF] Wang | The immersed/fictitious element method for fluid-structure interaction: volumetric consistency, compressibility and thin members[END_REF][START_REF] Wang | Extended immersed boundary method using FEM and RKPM[END_REF][START_REF] Zhang | Immersed finite element method[END_REF][START_REF] Laure | Collision strategy for the direct simulation of moving fibers in viscous fluid[END_REF][START_REF] Rabczuk | Immersed particle method for fluid-structure interaction[END_REF].

In order to correctly catch the pressure jump from one side of the structure to the other, the pressure field approximation is enriched by a Heaviside function using a partition of unity strategy [START_REF] Melenk | The partition of unity finite element method: basic theory and applications[END_REF], namely, XFEM [START_REF] Moės | A finite element method for crack growth without remeshing[END_REF]. The Heaviside function is completed at the structure tip by a continuous ramp function (Figure 3(c)). The pressure approximation becomes

p.M / D X i 2F N i F .M /P i C X i 2A .M /N i F .M /A i ; ( 2
)
where F is the set of nodes of the whole fluid mesh, N i F .M / is the standard shape function associated to the fluid node i, P i is the standard nodal pressure value, A is the set of enriched nodes, .M / is the enrichment function, and A i is the additional nodal unknown coming from the enrichment. The set of enriched nodes are those that are connected to the elements cut by the structures (Figure 2(b)). The enrichment function is defined by .M / D sign. .M // if jÂ.M /j > 2 ;

.M / D 2 Â.M / if jÂ.M /j < 2 ;

(

) 3
where  is defined in Figure 3(b).  has the same sign as the level set, it is equal to zero in the continuation of the structure. This enrichment function enables to have a continuous variation of pressure at the structure tip because 1 6 6 C1 if r D 0.

For tridimensional applications, the structure edge is curve and the  angle is measured in the local normal plane to this curve. In this work, the structure edge is supposed to be smooth enough such that it does not contain corners.

The full approximation containing both the standard part and the enrichment can be written from Equation (2) as

p.M / D OEN F .M / .M /N F .M /ˇF A Ä P F P A ; (4)
where N F is the shape function matrix of the fluid, P F contains the nodal pressure of the standard part, and P A contains the nodal additional unknowns coming from the enrichment. The Boolean localization matrix ˇFA is used to extract enriched set of nodes A from the set of nodes F. The tridimensional acoustic domain is discretized by four-node tetrahedral elements in this work. There are blending elements in the discretization, these are the elements that are connected to both enriched and non-enriched nodes [START_REF] Chessa | On the construction of blending elements for local partition of unity enriched finite elements[END_REF]. These elements have spurious effects only for the elements close to the structure tip enrichment. For the others, the Heaviside enrichment does not introduce errors in the solution. Because these spurious effects affect only a few elements, no correction has been implemented and no significant error has been found in the various numerical tests.

Structural displacement discretized space.

The thin structures are discretized using shell elements, three-node discrete kirchhoff triangle (DKT) in this work (Kirchhoff-Love theory) [START_REF] Stricklin | A rapidly converging triangular plate element[END_REF][START_REF] Dhatt | An efficient triangular shell element[END_REF][START_REF] Batoz | A study of three-node triangular plate bending elements[END_REF][START_REF] Batoz | Modélisation Des Structures Par éléments Finis[END_REF]. The structure DOFS are collected in U S , which is linked to the structural displacement by

u S D N S U S ; (5)
where N S is the shape function matrix. The stiffness and mass matrices of the structure are, respectively, denoted by K SS and M SS . A structural damping matrix D SS is introduced in the following using a modal damping.

Discretized weak form of the coupled problem

The continuous weak form of the equations of the structural-acoustic coupled problem at a given angular frequency ! ¤ 0 gives a linear system of equations in the discretized fluid and structure spaces. Because a structural damping is considered, the solution is complex and the imaginary unit number j is introduced (j 2 D 1). The following system is obtained:

0 B @ 2 6 4 K FF K FA 0 K T FA K AA 0 0 C T AS K SS C !j D SS 3 7 5 ! 2 2 6 4 M FF M FA 0 M T FA M AA C AS 0 0 M SS 3 7 5 1 C A 2 6 4 P F P A U S 3 7 5 D 2 6 4 ! 2 U F 0 F S 3 7 5 ; (6)
where F S is the discretized external load on the structure, U F is the discretized particle displacement, and the other matrices are defined as

K FF D 1 f Z F B T F B F dV; (7a) M FF D 1 f c 2 0 Z F N T F N F dV; (7b)
K AA D ˇT FA 1 f Z F B T A B A dV ˇFA ; (7c)
M AA D ˇT FA 1 f c 2 0 Z F 2 N T F N F dV ˇFA ; (7d)
K FA D 1 f Z F B T F B A dV ˇFA ; (7e)
M FA D 1 f c 2 0 Z F N T F N F dV ˇFA ; (7f) C AS D 2ˇF A T Z N T F n T N S d †; (7g) with B F D Ä @N T F @x @N T F @y @N T F @´ T ; (7h)
and

B A D Ä @N T F @x C @ @x N T F ˇ @N T F @y C @ @y N T F ˇ @N T F @´C @ @´N T F T : (7i)
The coupling between the acoustic fluid and the structures is only realized between the structural displacement U S and the pressure enrichment P A through the coupling matrix C AS . The details of the numerical implementations of the aforementioned matrices are given in [START_REF] Legay | An extented finite element method approach for structural-acoustic problems involving immersed structures at arbitrary positions[END_REF].

MODAL REDUCTION OF THE STRUCTURE

The structure reduced basis is chosen as the modal basis is enriched with the static solution. Each vector i S of the modal basis ˆSm , composed of the m first eigenmodes, is the solution of the following structural eigenvalue problem:

K SS ! i S 2 M SS Á i S D 0; (8)
where the boundary conditions on the structure are taken into account. The modes are normalized according to the mass matrix

ˆT Sm M SS ˆSm D 1 mm ; (9)
and the diagonal generalized stiffness matrix mm is obtained:

mm D ˆSm T K SS ˆSm : (10)
The static solution ‰ S is the solution of the following system of equations:

K SS ‰ S D F S ; (11)
where ‰ S satisfies the boundary conditions. In the case of a singular stiffness matrix, an arbitrary shift using the mass matrix can be introduced. This vector is made orthogonal to the modal basis ˆSm by using a Gram-Schmidt procedure. It is finally normalized according to the mass matrix

‰ T S M SS ‰ S D 1: (12)
Using the combination of the modal basis and the static mode, the structural reduced basis is denoted by ˆSn

ˆSn D OEˆS m ‰ S ; (13)
where n D m C 1 is the number of columns of ˆSn . The structural displacement vector expressed in the reduced basis ˆSn is denoted by ˛n. The projected stiffness matrix of the structure in this reduced basis is

K nn D " mm 0 m 0 T m ‰ T S K SS ‰ S # ; (14)
where 0 m is a column of zeros of size m.

Because a modal damping is introduced, the projected structural damping matrix D nn is diagonal. Its i th term corresponds to the modal damping 2 i ! i S , where i is the modal damping factor. The coupling system of Equation ( 6) becomes 0 B @ 2 6 4

K FF K FA 0 K T FA K AA 0 0 C T An K nn C !j D nn 3 7 5 ! 2 2 6 4 M FF M FA 0 M T FA M AA C An 0 0 1 nn 3 7 5 1 C A 2 6 4
P F P A ˛n 3 7 5 D 2 6 4 ! 2 U F 0 F n 3 7 5 ; (15)
where 1 nn is the identity matrix of size n. The coupling matrix between the acoustic fluid and the pressure enrichment is defined by

C An D C AS ˆSn ; (16)
and the generalized structural force is given by

F n D ˆT Sn F S : (17)

MODAL SYNTHESIS REDUCTION OF THE ACOUSTIC DOMAIN

A restrained-interface component mode synthesis method is chosen in order to condense the acoustic domain on the fluid-structure interface following the classical strategy used in structural mechanics, known as Craig-Bampton method [START_REF] Craig | Coupling of substructures for dynamic analysis[END_REF]. This strategy is applied in [START_REF] Rumpler | Performance of a restrained-interface substructuring FE model for reduction of structural-acoustic problems with poroelastic damping[END_REF] for a structural-acoustic problem using compatible meshes between the fluid and the structures, it is shown that the computational time is significantly reduced. The structural-acoustic problem written as the system of Equation ( 15) provides a natural separation between the fluid DOFS coupled to the structure (interface DOFS) and the uncoupled ones (internal DOFS): in the present work, the interface DOFS are those denoted by the subscript A, whereas the internal DOFS are those denoted by the subscript F (Figure 4). An acoustic reduction basis of the internal DOFS is built using a truncated low frequency modal basis. It is completed by static modes taking into account mobility of the interface DOFS, contribution of higher order modes to the response, and separations of the acoustic cavity by the structures. Static modes are computed with pressure DOFS 'restrained' at interface (P A D 0) and are solution of the following eigenvalue problem

K FF ! i F 2 M FF Á i F D 0: (18)
The p first modes are collected in the ˆFp matrix. The modes are normalized according to the mass matrix

ˆT Fp M FF ˆFp D 1 pp ; (19)
and the diagonal generalized stiffness matrix pp is obtained,

pp D ˆT Fp K FF ˆFp : (20)
It is noteworthy that the internal DOFS are the ones corresponding to the empty acoustic cavity with no immersed structure inside. The interface static modes are computed as successive responses of the empty cavity to a unit prescribed enriched pressure value. For the i th enriched node, the corresponding static mode ‰ i F is the solution of the following system

K FF K FA " ‰ i F 1 i A # D 0 F ; (21)
where 1 i A denotes a column vector of 0 with 1 located at the i th row. The obtained static modes are collected in the matrix ‰ FA , which can also be written from a mathematical point of view as

‰ FA D K 1 FF K FA : (22)
In the case of a singular stiffness matrix, an arbitrary shift using the mass matrix can be introduced. The discretized fluid pressure field is written using both the modal basis and the static modes as

Ä P F P A D Ä ˆFp ‰ FA 0 Ap 1 AA Ä ˛p P A ; (23)
where

h ˛T p P T A i T
is the pressure field expressed in the reduced basis " defined as (using Equation ( 4) and ( 23))

" D OEN F .M /ˆF p N F .M /‰ FA C .M /N F .M /ˇF A : (24)
The reduced coupled system of equations can be written by replacing Equation ( 23) into Equation [START_REF] Ihlenburg | Finite element solution of the Helmholtz equation with high wave number. Part 1: the h-version of the FEM[END_REF] as

0 B @ 2 6 4 pp 0 0 0 O K AA 0 0 C T An K nn C !j D nn 3 7 5 ! 2 2 6 4
1 pp M pA 0 M T pA O M AA C An 0 0 1 nn 3 7 5 1 C A 2 6 4 ˛p P A ˛n 3 7 5 D 2 6 4 ! 2 ˆT Fp U F 0 F n 3 7 5 ; (25)
where the matrices are defined by

O K AA D K AA C ‰ T FA K FA ; (26)
O M AA D ‰ T FA ? M FA C M T FA ‰ FA C M AA ; (27)
M pA D ˆT Fp ? M FA ; (28)
?

M FA D M FF ‰ FA C M FA : (29)
This system of equations can be solved for each frequency step and is referred to as the noncondensed version in the following. However, the ˛p unknowns can be eliminated using the first line of the system

˛p D ! 2 pp ! 2 1 pp 1 ˆT Fp U F C M pA P A Á (30)
After replacing in the second line, the reduced system is

" O K AA 0 C T An K nn C !j D nn # ! 2 " Q M AA .!/ C An 0 1 nn #! Ä P A ˛n D " Q F A .!/ F n # (31)
with

Q F A .!/ D ! 4 M T pA pp ! 2 1 pp 1 ˆT Fp U F ; (32)
and

Q M AA .!/ D O M AA C ! 2 M T pA pp ! 2 1 pp 1 M pA : (33)
This system of equations is referred to as the condensed version in the following. After solving this system, the pressure field in the cavity can be deduced by using successively Equations ( 30) and [START_REF] Sukumar | Modeling holes and inclusions by level sets in the extended finiteelement method[END_REF]. Note that the inverse matrix in Equation ( 33) is diagonal and is trivial to solve at each frequency step.

PRACTICAL USE OF THE PROPOSED METHOD

In order to clarify the practical implementation and to highlight the advantages of the presently proposed reduced approaches, the algorithms are detailed in the following. In the context of an optimization, uncertainty or reliability study of the structural-acoustic problem, several configurations have to be tested. It is supposed that the varying parameters are those related to the number of structures and to their positions. In the following, a configuration is a chosen set of parameters.

The classical approach, as used in a commercial software, has a conforming mesh at the fluidstructure interface and does not perform any reduction. As presented in the Algorithm 1, the fluid mesh is built for each configuration. The matrices associated to the fluid and to the fluidstructure coupling are computed for each configuration. The generic notations K, D, and M denote, respectively, the stiffness, the damping, and the mass matrices of the structural-acoustic coupled problem.

The algorithm of the proposed reduced method, in its non-condensed version, is presented in Algorithm 2. The empty cavity fluid mesh and the structure meshes are built once before the loop over the configurations. The corresponding uncoupled matrices K SS , M SS , K FF , and M FF as well as the fluid and structure modal basis ˆSn and ˆFp are computed at this step. A geometry variation of the structures is possible but the structure matrices K SS , M SS , and ˆSn have then to be computed for each configuration. In this paper, only variations of the structure positions are considered. For each configuration, the level set .M / and the enriched set of nodes A have to be updated. This allows to perform the computation of the static modes ‰ FA and the coupling matrix C An . The matrices O K AA and O M AA are then computed. For each frequency step, the system of Equation ( 25) is solved; the fluid pressure and the structure displacement are deduced using, respectively, the reduced basis " and ˆSn .

The algorithm of the proposed reduced method, in its condensed version, is detailed in Algorithm 3. The difference with the non-condensed version lies in the frequency loop. For each frequency, the condensed matrix Q M AA .!/ has to be computed, involving the trivial inversion of a diagonal matrix. After solving the system of Equation ( 31), the modal fluid contribution ˛p is computed using Equation [START_REF] Rabczuk | Immersed particle method for fluid-structure interaction[END_REF]. The fluid pressure and the structure displacement are computed as previously. This condensed version is very powerful when the number of interface DOFS is small compared with the number of fluid modes. However, because this is not usually the case in this work, the condensed version may be less efficient than the non-condensed version.

All the numerical developments are done within Python at the global level, coupled to Fortran routines for elemental computations. The convergence study of the proposed method using a tip enrichment is realized using six nonconforming meshes from meshes 1 to 6 (Table II). These meshes are unstructured and have a uniform element size through the cavity.

The frequency range is from 0 to 300 Hz. The sound level indicator is the mean quadratic pressure, expressed in dB, and computed according to the following definition:

L p D 10 log 10 Z F p 2 dV p 2 0 Z F dV ; ( 34
)
where p 0 D 20 Pa is the reference sound pressure in the air. The frequency response functions in terms of the mean quadratic pressure in the cavity are plotted in Figure 6(a) for a few meshes above the six different non-conforming ones; they are compared with the reference solution. Because no damping is introduced for this problem, the plot is arbitrarily cut to 130 dB because the pressure level at resonances is not defined. For mesh 2, the frequency response function is also plotted when the tip enrichment is not use. The tip enrichment function for this mesh improves the accuracy of the solution. Mesh 3 gives a good accuracy compared with the reference solution.

A mean dB-difference (denoted by ) is computed for each case using the following simple formulae

D P jdB-differencej Number of frequency steps : ( 35
)
The mean dB-difference between the reference frequency response function and the six different frequency response functions of the six non-conforming meshes is shown in Figure 6(b) versus the number of DOFS. The mean dB-difference is also computed when the tip enrichment is not use. For a coarse mesh (mesh 1), the mean dB-difference is quite high and the use of the tip enrichment does not improve the accuracy. For a fine mesh (meshes 5 and 6), the tip enrichment does not improve the accuracy because this enrichment is local to the structure tip. For intermediate size of mesh, the tip enrichment improves the accuracy of the solution.

The pressure field of the reference solution at f D 210 Hz is shown in Figure 7(a). Because there is no damping in this application, this frequency is arbitrarily chosen to be far enough from an eigenfrequency. The solution obtained by the proposed approach and for the fine mesh 6 is not plotted because it is visually the same. However, in order to compare the two solutions, the pressure at the structure tip is compared. It is plotted at the frequency of 210 Hz in Figure 7(b) for the different meshes and compared with the one obtained along the semicircle for the reference solution. The pressure (in dB) is plotted in terms of the angle  defined in Figure 3 varying from 90 o to C90 o . The convergence of the proposed method with a tip enrichment can be clearly seen. However, the solution does not converge (red curve, Mesh 6) to the reference one (black curve). This is probably because the structure has a non-zero thickness in the reference model, whereas it has a zero thickness in the non-conforming model. This academic application shows that the tip enrichment improves the solution of the proposed method. This tip enrichment enables to extend the scope of the applications of this XFEM-based method for structural-acoustic coupled problems; compared with the first work proposed by the author [START_REF] Legay | An extented finite element method approach for structural-acoustic problems involving immersed structures at arbitrary positions[END_REF], it is now possible to embed structures with free edges in the acoustic cavity.

Convergence study of the reduced basis on a circular ring structure in an acoustic cavity

One-fourth of a circular ring structure in a thin square acoustic cavity bounded by rigid walls is considered (Figure 8 and Table III). Boundary conditions as well as load are chosen such that the solution of the problem is two-dimensional. The structure is simply supported on its upper left boundary, the displacement along the cavity thickness direction is fixed for the whole structure. A harmonic load of amplitude 1N is applied on the other side. Because the acoustic cavity is separated into two different cavities by the structure, no tip enrichment is used in this application. The chosen mesh is shown in [START_REF] Legay | An extented finite element method approach for structural-acoustic problems involving immersed structures at arbitrary positions[END_REF] to be fine enough to give accurate results in the frequency range of interest (Table III and Figure 8). The studied frequency range is 1-600 Hz, involving seven structure modes and 12 fluid modes of the empty cavity. The structural modal damping factor is chosen to be equal to 0.02 for all the modes. The first six structural modes are presented in Figure 9. The meshes and the post-processing are performed with Gmsh [START_REF] Geuzaine | Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities[END_REF].

The convergence of the XFEM non-reduced method (system of Equation ( 6)) has been proven by comparisons with a standard compatible mesh approach in [START_REF] Legay | An extented finite element method approach for structural-acoustic problems involving immersed structures at arbitrary positions[END_REF]. The reference solution in the following is then based on the XFEM non-reduced approach to only get the error coming from the reduction and not the one coming from the discretization. This reference solution is thus obtained by solving the system of Equation ( 15) using the same mesh with 100 structure modes enriched by the static solution. The eigenfrequency of the 100 th mode is around 25,100 Hz, which is far more than 600 Hz, meaning that taking 100 structure modes gives a converged solution.

Two static modes ‰ i F are plotted in Figure 10. These vectors can be seen as a global XFEM enrichment of the fluid basis in order to take into account the structure in the coupled problem, there are 228 static modes in this application (228 enriched nodes).

Convergence of the fluid basis.

In order to study the convergence of the reduction method in terms of fluid reduced basis, the number of structure modes is kept as a constant, 100 modes are chosen. Five cases are studied, containing 5, 10, 20, 40, and 80 fluid modes in the basis ˆFp , they are named, respectively, in the following as 5F, 10F, 20F, 40F, and 80F.

The first eight modes of the fluid basis ˆFp of the cavity with no structure inside are presented in Figure 11.

The frequency response functions are compared in Figure 12(a) with the reference solution for cases 5F, 20F, and 80F. In order to quantify the influence of the number of fluid modes, an absolute value of the dB-difference with the reference solution is computed for each case and for each frequency. This means that dB-difference is compared with the ratio between the highest eigenfrequency in the fluid basis (depending on the number of fluid modes) over the highest observed frequency (600 Hz in this case) in Figure 12(b). For this application, choosing the fluid modes with an eigenfrequency up to twice the observed frequency gives accurate results with a mean dB-difference less than 1 dB (case 40F).

Convergence of the structure basis.

In order to study the convergence of the reduction method in terms of structure reduced basis, the fluid is not reduced and the system of Equation ( 15) is solved. Four cases are studied, containing 2, 4, 8, and 16 structure modes in the basis ˆSm , they are named, respectively, in the following as 2S, 4S, 8S, and 16S. The static solution is added to each case in order to form the reduced basis ˆSn .

The frequency response functions for the cases 2S, 8S, and 16S are plotted in Figure 13(a) and compared with the reference solution. The convergence can be clearly seen because only the two first resonances are correctly represented with the case 2S, whereas the case 8S gives accurate results up to about 500 Hz. The mean dB-difference is compared with the ratio between the highest eigenfrequency in the structure basis (depending on the number of structure modes) over the highest observed frequency (600 Hz in this case) in Figure 13(b). For the case 16S, which gives very good results, the highest eigenfrequency of the structural basis is more than twice larger than the observed frequency.

Convergence of both fluid and structure basis.

In order to study the convergence of the solution to both the fluid and the structure basis, three cases are studied: 14F-9S, 44F-14S, and 68F-17S. The frequency response functions are compared with the reference in Figure 14(a). The fluid pressure field and the structural displacement are presented in Figure 15(a) at 336 Hz for the case 44F-14S. The structural displacement is plotted with an amplification factor of 1000. The XFEM results show a discontinuous jump at the interface because the plot routine does not take into account the Heaviside enrichment in the enriched elements, whereas the enrichment is used to compute the real pressure at an enriched node according to Equation [START_REF] Cherng | Characterization and validation of acoustic cavities of automotive vehicles[END_REF].

The mean dB-difference is compared with the ratio between the highest eigenfrequency of the fluid basis and of the structure basis over the highest observed frequency (600 Hz in this case) in The computational time versus the number of frequency steps is presented in Figure 15(b), it is normalized to one of the reference solutions. A fixed time is needed at the beginning in order to form the reduced system. The computational time for the optimal case 44F-14S represents only 25% of the reference time for 1000 computed frequency steps when using the non-condensed version (gain factor of 4), whereas it is about 40% with the condensed version. The computational time is not really affected by the increasing number of modes.

Study of the influence of the positions of two structures in an acoustic cavity

6.3.1. Definition of the geometry and the meshes. The acoustic cavity is described in Figure 16, and the numerical values are given in Table IV. Two identical structures are immersed in this acoustic cavity. Their geometries are defined in Figure 17 in a local coordinate system. The two structures are arbitrarily localized in the cavity according to the a and b parameters, which give the positions of point number 1 for each structure in the cavity. A harmonic load is applied in ´-direction on point number 10 of the structure localized by the parameter a. The fluid mesh (22,473 nodes; 125,327 tetrahedral elements) is shown in Figure 18(a), whereas the structure mesh (1894 nodes; 3613 DKT elements) is plotted in Figure 18(b). These two meshes have been chosen after a convergence study, 44S. The eigenfrequency of the last mode on the basis corresponds approximately, for each case, to 1, 1.5, and 2 times the observed frequency (200 Hz in this application).

The frequency response functions in terms of the mean quadratic pressure expressed in dB in the acoustic cavity for the three basis are plotted in Figure 20(a). They are compared with a reference solution obtained with no reduction in the fluid, whereas 44 structure modes are taken into account in the structure basis. The differences are more important for large frequencies: basis 38F-24S shows differences from 100 to 200 Hz, basis 102F-32S gives good results up to 150 Hz, whereas basis 250F-44S is accurate up to the highest observed frequency (200 Hz). The mean dB-difference is shown in Figure 20(b) for the three basis, which confirm the fact that the more modes are taken into account in the basis, the more accurate are the results. The computational time is plotted in Figure 20(c), this time is normalized to the reference time to compute 500 frequency steps. The computational time is divided by a factor of 10 for basis 250F-44S, which gives very accurate results in the frequency range of interest.

Variation of the parameters.

In order to show the ability of the method to perform a parametric study of the structure positions, the two structures are placed arbitrarily in the fluid cavity. Parameter a varies from 2 to 3, whereas parameter b varies from 3 to 4. Eleven values of each parameter are chosen (a 2 OE2:0; 2:1; : : : ; 3:0 and b 2 OE3:0; 3:1; : : : ; 4:0), which leads to a total of 121 studied configurations. The fluid mesh as well as the structure mesh are the same as in the previous section. The reduced basis 250F-44S is chosen for all the computations, it is computed only once. A parallel computing strategy is used: eight processors computes eight different configurations at the same time, leading to an additional computational time saving. The envelope curves of the 121 frequency response functions are presented in Figure 21.

CONCLUSIONS

A reduced method based on XFEM is presented in this work in order to efficiently compute a structural-acoustic problem involving variations of the structures' positions and/or geometries. The XFEM enables to immerse the thin structures at arbitrary positions in the acoustic fluid without remeshing the whole domain. A structure tip enrichment is introduced in order to take into account the pressure variation at one geometric point. This effect cannot be modeled by a standard finite element approach if the structure is a shell.

The reduced proposed approach is based on a modal synthesis method. The enriched fluid nodes give a natural interface between the fluid and the structures. The internal fluid domain corresponds to the empty cavity with no structures. The advantage is that the modal basis of the fixed interface problem is the modal basis of the acoustic cavity with no structure, which can be thus computed independently of the immersed structures. The structural DOFS are projected on the structure modal basis decoupled from the fluid. A structural modal damping is added in this work. The size of the reduced system is the number of enriched fluid nodes increased by the number of fluid and structure modes taken into account.

The different test cases show that the optimal choice of fluid and structure bases corresponds to the last eigenfrequencies of both bases taken as, at least, twice the observed frequency. The computational time is reduced by a factor from 2 to 10 depending on the problem. Indeed, the reduced system is full in part corresponding to the enriched DOFS (the larger part of it), whereas the not-reduced initial large system is sparse. Therefore, the computational time is not proportional to the size of the system. The numerical tests show that for large applications, it is better to not condense the internal fluid modes on the interface in order to save computational time. However, the computational time comparisons done in this work do not take into account the needed remeshing time of a standard compatible mesh approach: the real gain factor is larger.

Finally, a parametric study of the positions of two structures in an acoustic cavity demonstrates the full potential of the proposed reduced method.

Figure 1 .

 1 Figure 1. Description and notations of the vibro-acoustic problem.

Figure 2 .

 2 Figure 2. Classical approach and XFEM approach.

Figure 3 .

 3 Figure 3. XFEM approximation at the interface.

Figure 4 .

 4 Figure 4. Separation of the DOFS into internal (F ) and interface (A).

Algorithm 1 Figure 5 .

 15 Figure 5. Geometry of the separated cavity.

Figure 6 .

 6 Figure 6. Results for the separated cavity.

Figure 7 .

 7 Figure 7. Pressure field for the separated cavity at f D 210 Hz.

Figure 8 .

 8 Figure 8. One-fourth of a circular structure ring in an acoustic cavity.

Figure 9 .

 9 Figure 9. Structure modal basis of the circular ring.

Figure 10 .

 10 Figure 10. Example of two static modes for the circular ring in an acoustic cavity among 228 static modes.

Figure 11 .

 11 Figure 11. Fluid modal basis of the empty cavity for the circular ring problem.

Figure 12 .

 12 Figure 12. Fluid basis convergence for the circular ring problem.

Figure 13 .

 13 Figure 13. Structure basis convergence for the circular ring problem, different structure bases.

Figure 14 .

 14 Figure 14. Convergence for the circular ring problem.

Figure 15 .

 15 Figure 15. Results for the circular ring problem.

Figure 16 .

 16 Figure 16. Two structures in an acoustic cavity.

Figure 14 (

 14 Figure 14(b). These three cases are in good agreement with the previous conclusion concerning the choice of the number of modes in the reduced bases, namely, twice the observed frequency for the reduced basis gives accurate results (case 44F-14S).The computational time versus the number of frequency steps is presented in Figure15(b), it is normalized to one of the reference solutions. A fixed time is needed at the beginning in order to form the reduced system. The computational time for the optimal case 44F-14S represents only 25% of the reference time for 1000 computed frequency steps when using the non-condensed version (gain factor of 4), whereas it is about 40% with the condensed version. The computational time is not really affected by the increasing number of modes.

Figure 17 .

 17 Figure 17. Description of one structure.

Figure 18 .

 18 Figure 18. Details of the meshes: fluid mesh (22,473 nodes/125,327 tetrahedral elements); structure mesh (1894 nodes/3613 DKT elements); enriched mesh (771 nodes/1917 tetrahedral elements).

Figure 19 .

 19 Figure 19. Pressure field in the acoustic cavity for two specific frequencies.

Figure 20 .

 20 Figure 20. Results.

Figure 21 .

 21 Figure 21. Frequency response function envelope of the parametric study.

Table I .

 I Data of the separated cavity.

			L (m)		1
			a (m)		0.6
			b (m)		0.65
			e (m)		0.0004
			c 0 (m s 1 )	340
		Mesh 1 Mesh 2 Mesh 3 Mesh 4 Mesh 5 Mesh 6
	nb. nodes	13	123	963	8554	77,653 161,196

F (kg m 3 ) 1.

2

Table

II

. Different non-compatible used meshes of the separated cavity.

elements. The semicircle structure tip is discretized by 47 nodes, which give an accurate solution of the local pressure at this point.

Table III .

 III Data of the circular structure ring in an acoustic cavity.

	Fluid		Structure
	L (m)	1	E (Pa)	70 10 9
	h (m)	0.03		0.27
	c 0 (m s 1 ) F (kg m 3 )	340 1.2	S (kg m 3 ) e, thickness (m)	2700 6 10 3
	Fluid elements	10086	R, radius (m)	0.8
	Fluid nodes	3528	F (N)	1
	Enriched elements	336	Structure elements	650
	Enriched nodes	228	Structure nodes	440

Table IV .

 IV Data of the acoustic cavity with two structures.

		Fluid		Structure
	L 1 (m)		5	E (Pa)	75; 000 10 6
	L 2 (m) H (m) h (m)	q	2 2 1.6	S (kg m 3 ) e, thickness (m)	0.33 2700 15 10 3
	R	.H h/ 2 C L 2 2 =4	l (m)	0.7
	c 0 (m s 1 ) F (kg m 3 )		340 1.2
	a, b (m)		Variable

 30) Get fluid pressure using the reduced basis " Get structure displacement using the reduced basis ˆSn 6. APPLICATIONS

Validation of the tip enrichment

One of the originalities of the present work compared with the previous enrichment proposed by the author [START_REF] Legay | An extented finite element method approach for structural-acoustic problems involving immersed structures at arbitrary positions[END_REF] is to enable free structure edges in the acoustic fluid. In order to validate this edge enrichment, a rigid wall is immersed in a thin square cavity bounded by rigid walls (Figure 5 and Table I). A harmonic unit pressure is imposed on the left side of the cavity. Boundary conditions as well as load are chosen such that the solution of the problem is two-dimensional.

In order to catch correctly the local pressure variation around the top of the structure tip, the structure has a small thickness e for the classical approach using a conforming mesh. This reference solution is computed using a fine mesh consisting of 153,637 nodes and 305,311 three-node triangle


Antoine Legay

An extended finite element method approach for structural-acoustic problems involving immersed structures at arbitrary positions

Keywords: fluid-structure interaction, structural-acoustic, XFEM, parametric study

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Noise reduction of passenger acoustic compartment in transport industry is taking more and more importance in the final characteristics of a vehicle. Undeniably, passengers expect to travel with an increasing comfort whether it is by car, train, or plane. Therefore, this constraint has to be taken into account by the engineers in the design process to reach the set noise level in the passenger compartment.

It is shown in [START_REF] Cherng | Characterization and validation of acoustic cavities of automotive vehicles[END_REF] that the seats in an aircraft cabin have significant influences on both the amplitude of the sound pressure level and the resonant frequency shifts. To predict a realistic noise mapping and level in the cavity, the seats and all the significant flexible and absorbing structures have to be taken into account.

This elasto-acoustic problem can be modeled and numerically computed. The finite element method can be used, and a mesh has then to be produced. Classically, this mesh is compatible at the interfaces between the fluid and the structures. During the design step, several configurations are tested to optimize the design, to perform an uncertainty study, or to compute a reliability level. Moreover, a same passenger compartment could be developed into several uses: the same car can have back seats or not, and an aircraft can have small seats or large seats depending on the class. For each configuration, a compatible mesh at the fluid-structure interfaces is needed. This remeshing step may become a waste of time for the engineers if a parametric meshing software is not available and if a large number of configurations have to be tested. Moreover, the engineer in charge of the study may possess only the fluid mesh of the empty cavity and so may not be able to remesh it according to the structures. A possible alternative is the use of the boundary element method; however, it suffers the same drawback. Indeed, a new surface mesh would have to be built for each configuration leading to an update of the corresponding full matrices.

The aim of this work is to be able to efficiently analyze different configurations of immersed structures in the acoustic domain and their influences on the noise level in the cavity. The basic idea of the proposed method is to always use the same fluid mesh of the empty cavity for all the possible configurations, assuming that the cavity's geometry does not change. The structures are then arbitrarily immersed within the acoustic fluid mesh. The embedded structures in the fluid are considered to be thin shells. Therefore, the pressure is discontinuous from one side of the structure to the other one. Although this discontinuity is naturally described in the case of a compatible mesh, it is introduced in this work for noncompatible meshes by using a partition of unity based approach [START_REF] Melenk | The partition of unity finite element method: basic theory and applications[END_REF] where the enrichment is a Heaviside function (XFEM, extended finite element method).

The XFEM was first developed to model cracks [START_REF] Moës | A finite element method for crack growth without remeshing[END_REF] and inclusions [START_REF] Sukumar | Modeling holes and inclusions by level sets in the extended finite-element method[END_REF][START_REF] Legay | Strong and weak arbitrary discontinuities in spectral finite elements[END_REF] in solid mechanics. It has been applied to several kind of problems where an evolving discontinuity has to be taken into account as in [START_REF] Duddu | A combined extended finite element and level set method for biofilm growth[END_REF] for biofilm growth for instance. The XFEM has already been used in the context of nonlinear fluid-structure interaction for structures embedded in a fluid flow [START_REF] Legay | An Eulerian-Lagrangian method for fluid-structure interaction based on level sets[END_REF][START_REF] Zilian | The enriched space-time finite element method (EST) for simultaneous solution of fluid-structure interaction[END_REF][START_REF] Gerstenberger | An extended finite element method/Lagrange multiplier based approach for fluid-structure interaction[END_REF].

Other techniques such as immersed element method-like are developed to deal with noncompatible meshes for a fluid-structure interaction problem [START_REF] Wang | The immersed/fictitious element method for fluid-structure interaction: volumetric consistency, compressibility and thin members[END_REF][START_REF] Wang | Extended immersed boundary method using FEM and RKPM[END_REF][START_REF] Zhang | Immersed finite element method[END_REF][START_REF] Laure | Collision strategy for the direct simulation of moving fibers in viscous fluid[END_REF]. However, these approaches need either to have a thick solid structure or to refine the mesh around the interface to have good results.

The structure is considered to have small displacements, so the discontinuity surface does not evolve with time in the fluid domain. However, for a parametric study of the structure positions in the cavity, and eventually their geometries, the discontinuities move according to the chosen parameters, whereas the acoustic cavity keeps the same geometry. It is assumed that there is fluid on both sides of the structure. The case where the fluid is only on one side of the structure can be performed using a compatible mesh with no change of the fluid mesh. If such a structure is supposed to change its configuration, the fluid mesh has to be build a priori such that all configurations are compatible to avoid remeshing during the parametric study.

No damping is considered in this work for simplicity reasons of implementation. Nevertheless, a structural-acoustic mechanical problem is closely linked to it if the final aim is to reduce noise level in the passenger cavity. This noise level reduction can be performed by several different techniques:

A porous material can be added on the walls [START_REF] Allard | Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials[END_REF], and the resulting numerical problem may be reduced by subdomain decomposition [START_REF] Rumpler | Performance of a restrained-interface substructuring FE model for reduction of structural-acoustic problems with poroelastic damping[END_REF][START_REF] Davidsson | A reduction method for Structure-acoustic and poroelastic-acoustic problems using interface-dependent Lanczos vectors[END_REF]. The absorbing material can be modeled as rheological devices [START_REF] Deü | Vibration and transient response of structural acoustic interior coupled systems with dissipative interface[END_REF][START_REF] Bermúdez | Modelling and numerical solution of elastoacoustic vibrations with interface damping[END_REF][START_REF] Legay | A rheological interface model and its space-time finite element formulation for fluid-structure interaction[END_REF] to simplify its numerical treatment. A viscoelastic material may be put on (or inside) the structure [START_REF] Rao | Recent applications of viscoelastic damping for noise control in automobiles and commercial airplanes[END_REF], and special numerical strategies can be used to efficiently compute the complex coupled problem [START_REF] Bobillot | Iterative techniques for eigenvalue solutions of damped structures coupled with Fluids[END_REF][START_REF] Rouleau | Vibro-acoustic study of a viscoelastic sandwich ring immersed in water[END_REF].

These different techniques can be used in addition to the approach developed in this work.

The second section of the paper states the equations of the coupled fluid-structure problem. The third section focuses on the discretization, especially the XFEM approximation. The fourth section gives details of the algorithm as well as the numerical implementation of the method. The last section shows the accuracy of the method by comparing the results to reference solutions and demonstrates the efficiency of the approach to compute different configurations of an industriallike problem.

STRONG AND WEAK FORMS OF AN ELASTO-ACOUSTIC PROBLEM

Description of the problem

A fluid-structure interaction problem is considered (Figure 1). The acoustic fluid domain and the thin flexible structures are respectively denoted by F and S . The problem is studied in the frequency domain for a permanent harmonic response at angular frequency !. The compressible fluid domain is described using pressure (p) as primary variable, whereas displacement (u S ) is used for the structural part. Their boundaries are separated into contours of imposed Dirichlet boundary conditions denoted @ 1 F and @ 1 S , prescribed Neumann boundary conditions denoted @ 2 F and @ 2 S , and coupling interface between acoustic fluid and structures denoted .

The thin structures are considered with no thickness in the fluid domain meaning that the structures are seen by the fluid as surfaces. These surfaces are described in the fluid domain by the zero contour of a level-set .M / that is the signed distance to the closest structure (Figure 2). By denoting d the distance of a point M to the structure, the level-set can be defined as

.M / D ˙d , ( 1
)
where d is the minimal distance to the structure; the sign is determined using a conventional unit normal vector E n pointing arbitrarily outward of one side of the structure. For simplicity reasons, only one structure is considered in the following of this section.

Strong form of the equations

Structure (u S ).

The structural domain is assumed satisfying the elastodynamic linearized equation at angular frequency !:

div S C ! 2 S u S D 0 in S , ( 2 )
where S is the constant density per unit volume and S is the Cauchy stress tensor, given by where D S is the classical Hooke's linear elasticity tensor and ".u S / is the strain tensor associated to the displacement vector field u S , defined by

S D D S ".u S / in S , ( 3 )
".u S / D 1 2 grad u S C grad T u S Á . ( 4
)
Prescribed displacements and forces are imposed on the external structural boundaries @ 1 S and @ 2 S , respectively:

u S D 0 on @ 1 S , ( 5 )
S n S D f Sb on @ 2 S , ( 6 )
where n S is the unit normal vector pointing outward S .

Compressible fluid (p).

The fluid is assumed compressible and inviscid, and satisfies the classical Helmholtz equation derived from the motion, continuity, and constitutive equations:

p C ! 2 c 2 0 p D 0 in F , ( 7 )
where c 0 is the constant speed of sound in the fluid and p the pressure fluctuation scalar field. The pressure fluctuation is named pressure in the following for simplicity reasons. The limit case where ! D 0 (static solution) is not given by Equation ( 7); this case is not considered in this work; more details are given in [START_REF] Morand | Fluid Structure Interaction[END_REF].

The fluid displacement vector, denoted by u F , can be related to the pressure field using the linearized Euler equation expressed for a harmonic solution:

grad p D ! 2 F u F in F , ( 8
)
where F is the fluid density. At external boundary of the acoustic fluid domain, rigid cavity conditions are imposed by setting a free pressure field. An harmonic excitation is prescribed via an acoustic source:

u F n F D u Fb on @ 2 F , ( 9 )
where n F is the unit normal vector pointing outward the fluid domain and u Fb is set to zero out of the acoustic source included in @ 2 F .

2.2.3.

Fluid-structure coupling. At interface between the structural domain and the acoustic fluid, normal stress and normal displacement continuity conditions are given by

S n S p n F D 0 on , (10)
.u F u S / n F D 0 on .

The fluid displacement u F is related to the pressure field using Equation (8).

Weak form

The test-function method is used to derive the variational formulation of the coupled problem. For this purpose, the spaces of sufficiently smooth functions C u S and C p are introduced, associated to the field variables u S and p, respectively.

Let ıu S be the frequency-independent test function, associated to u S , belonging to the admissible space C u S D ¹ıu S 2 C u S jıu S D 0 on @ 1 S º. Multiplying Equation (2) combined with Equation (3) by ıu S 2 C u S , applying a Green's formula, and taking Equations ( 6) and [START_REF] Wang | The immersed/fictitious element method for fluid-structure interaction: volumetric consistency, compressibility and thin members[END_REF] into account lead to

Z S D S ".ıu S / W ".u S / dV ! 2 Z S S ıu S u S dV Z ıu S n F p d † D Z @ 2 S ıu S f Sb d †.
(12) Let ıp be the frequency-independent test function, associated to p, belonging to the admissible space C p . Multiplying Equation ( 7) by ıp 2 C p , applying a Green's formula, and taking Equations ( 11) and ( 9) into account lead to

1 F Z F gradıp gradp dV ! 2 c 2 0 F Z F ıp p dV ! 2 Z ıp u S n F d † D ! 2 Z @ 2 F ıp u Fb d †. (13)
3. DISCRETIZATION OF THE COUPLED PROBLEM

Structure

The discrete Kirchhoff triangle element [START_REF] Stricklin | A rapidly converging triangular plate element[END_REF][START_REF] Dhatt | An efficient triangular shell element[END_REF][START_REF] Batoz | A study of three-node triangular plate bending elements[END_REF][START_REF] Batoz | Modélisation des structures par éléments finis[END_REF] is used to model and discretize the thin shell structure (Figure 3(a)). This element is based on the Kirchhoff-Love theory. There are six DOFs, three translations, and three rotations in the global coordinate system. The structure DOFs are collected in a vector U S that is linked to the structural displacement u S by

u S D N S U S , (14)
where N S is the shape function matrix at the global level including a transformation matrix from local to global coordinates. The structure stiffness matrix and mass matrix are respectively denoted by K SS and M SS .

where N F is the shape function matrix whereas the pressure gradient is given by where B F is the discretized gradient operator. For this element, the constant discretized gradient operator makes easier the integration through the element.

grad p D B F P, (16)

Enrichment of the pressure field.

A thin structure placed in a fluid separates locally the fluid into two parts, meaning that the pressure is discontinuous from one side of the structure to the other one (Figure 4(a)). The classical discretization strategy is to make the different meshes compatible at the interface (Figure 4(b)) where three nodes are placed at the same geometrical point: two nodes for each fluid domain and one node for the structure. The discontinuous pressure can be represented naturally by the fluid discretization but needs a remeshing or a mesh-moving technique for each configuration of the structures in the fluid domain. This method is used in this work to compute reference solutions in the application section. In the context of a parametric study of the positions of the structures, this remeshing step can become heavy and difficult to realize. The alternative strategy used in this work is to have the structure arbitrarily placed in the fluid mesh of the empty cavity (Figure 4(c)): there is no need to remesh the fluid for each structure configuration. To correctly catch the pressure jump from one side of the structure to the other one, the pressure field approximation is enriched by a Heaviside function using a partition of unity strategy [START_REF] Melenk | The partition of unity finite element method: basic theory and applications[END_REF], namely XFEM [START_REF] Moës | A finite element method for crack growth without remeshing[END_REF]. The pressure approximation becomes

p.M / D X i 2F N i F .M /P i C X i 2A N i F .M / i .M /A i , ( 17
)
where F is the set of nodes of the whole mesh, N i F .M / is the shape function associated to the fluid node i , P i is the standard nodal pressure value, A is the set of enriched nodes, i .M / is the enrichment function, and A i is the additional nodal unknown coming from the enrichment. The set of enriched nodes are those that are connected to the elements cut by the structures (Figure 5) or, in other words, cut by the zero contour of the level-set. The enrichment function is chosen such as where H is the Heaviside function and .M / is the signed distance function. The Heaviside function is modified to be the sign of the level-set. The pressure approximation can be written:

i .M / D H. .M //, (18)
p.M / D X i 2F N i F .M /P i C H. .M // X i 2A N i F .M /A i (19)
because the enrichment function is not node dependant. The full approximation containing both the standard part and the enrichment can be written from Equation [START_REF] Legay | A rheological interface model and its space-time finite element formulation for fluid-structure interaction[END_REF] as

p D N F P C N A A, (20)
where

N A .M / D H. .M //N F .M /ˇF A . ( 21
)
The Boolean localization matrix ˇFA is used to extract enriched set of nodes A from the set of nodes F . The pressure value at an enriched node i 2 A is then

p.M i / D P i C H. .M i //A i . ( 22
)
To keep the property of having p.M i / D P i , the enrichment may be chosen as i .M / D H. .M // H. .M i //. However, this shift is not used in this work to make the integration of an enriched element easier.

The level-set function is discretized using the fluid mesh, so .M / is defined as

.M / D X i 2F N i F .M / i , ( 23
)
where i is the nodal value of .M /.

3.2.3.

Fluid-structure coupling. The coupling term in the structural weak form ( 12) is given by Z

ıu S n F p d †. ( 24
)
It is recalled that in this work, it is assumed that there is fluid on both sides of the structure. Therefore, the interface is decomposed into C and , and a point M belonging to the interface is split into M C and M (Figure 6):

C is the structure surface in contact with the fluid domain where .M C / > 0:

ı the unit normal vector is denoted by n C F (with n C S D n C F ) ı the Heaviside function H. .M C // is C1
is the structure surface in contact with the fluid domain where .M / < 0: With the use of these notations, the coupling term is developed as

ı the unit normal vector is denoted by n F with n F D n C
Z ıu S n F p d † D ıU T S Z N T S n F N F d † P C ıU T S Z H.M /N T S n F N F d † ˇFA A D ıU T S Z C N T S n C F N F d † P C ıU T S Z C N T S n C F N F d † ˇFA A (25) C ıU T S Z N T S n F N F d † P ıU T S Z N T S n F N F d † ˇFA A. Because n F D n C F , the coupling term becomes Z ıu S n F p d † D ıU T S 2 Z N T S n C F N F d † ˇFA A. (26)
The coupling between fluid and structure is only carried out by the fluid pressure enrichment; the coupling matrix C SA is defined by

C SA D 2 Z N T S n C F N F d † ˇFA . ( 27
)
The coupling term in the fluid weak form is similar and gives the transpose matrix multiplied by ! 2 .

Discretized weak form

By replacing the continuous pressure field as well as the structure displacements by their approximations in Equations ( 12) and ( 13), the following discretized system is obtained:

0 @ 2 4 K FF K FA 0 K T FA K AA 0 0 C SA K SS 3 5 ! 2 2 4 M FF M FA 0 M T FA M AA C T SA 0 0 M SS 3 5 1 A 2 4 P A U S 3 5 D 2 4 ! 2 U Fb 0 F S 3 5 , ( 28
)
where K SS and M SS are respectively the stiffness and mass matrices of the structure, F S is the discretized external load on the structure, U Fb is the discretized acoustic source, and the other matrices are defined as

K FF D 1 f Z F B T F B F dV , (29)
M FF D 1 f c 2 0 Z F N T F N F dV , (30)
K AA D ˇT FA K FF ˇFA , (31)
M AA D ˇT FA M FF ˇFA , ( 32
)
K FA D 1 f Z F H. .M //B T F B F dV ˇFA , (33)
M FA D 1 f c 2 0 Z F H. .M //N T F N F dV ˇFA , ( 34
)
C SA D 2 Z N T S n C N F d † ˇFA . ( 35
)
This nonsymmetric formulation can be made symmetric for a solving in the frequency domain by dividing the acoustic equations (first and second lines of the system) by ! 2 (! ¤ 0).

Integration of the enriched elements

To compute the elemental enriched matrices K FA and M FA , an analytical solution is used following the formulas given in [START_REF] Ventura | On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method[END_REF] for a Heaviside enrichment combined with four-node tetrahedral elements.

Because B F is constant in a four-node tetrahedral element, the elemental expression of K FA in an enriched element e becomes

K e FA D 1 f B F T B F Z e F H. .M //dV . ( 36
)
The integral of the Heaviside function gives the following simple expression:

Z e F H. .M //dV D v C e v e V e v C e C v e D V e 12v C e 1 , ( 37
)
where the volumes of the positive and negative parent elemental domains are respectively v C e and v e (Figure 7) and V e is the total physical volume of the element. The expression is simplified because the total parent elemental volume is v e D v C e C v e D 1=6. The elemental expression of M FA in an enriched element e is

M e FA D 1 f c 2 0 Z e F H. .M //N F T N F dV . ( 38
)
The integration is carried out by using two quadrature points in the parent element, one at the center of each domain noted G C and G :

M e FA D 1 f c 2 0 V e 6v C e N T F j G C N F j G C 1 6v C e N T F j G N F j G Á . (39)
A tetrahedron can be cut by the zero contour of the level-set in two ways as mentioned in Figure 7 in the parent element with coordinate system .r, s, t/: case 1-3 where one node is one on side and three nodes are on the other side; case 2-2 where two nodes are on each side. Ventura [START_REF] Ventura | On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method[END_REF] gives only the formulas for the first case (1-3), whereas nothing is mentioned for the second case (2-2). The analytical formulas are recalled for the first case and developed for the second in the following.

The first case is supposed to have node 4 in domain C where > 0 and nodes 1, 2 and 3 in domain where < 0. The opposite situation according to the signed of is obtained by switching v C e and v e . With the notations defined in Figure 7(a), the parent volume of the positive side v C e is simply the volume of a subtetrahedron domain bounded by the four following points: .0, 0, 0/ , .r 0 , 0, 0/ , .0, s 0 , 0/ , .0, 0, t 0 / , Table I. Subdecomposition of the positive domain for case 2-2 into three subtetrahedra.

Four tetrahedron vertices Tetrahedron volume .0, 0, 0/ , .1, 0, 0/ , .0, s 0 , 0/ , .0, 0, t 0 / v 1 D s 0 t 0

6

.1, 0, 0/ , .0, s 0 , 0/ , .0, 0, t 0 / , 1

p 2 2 u 0 , 0, p 2 2 u 0 Á v 2 D p 2 12 s 0 u 0 .1 t 0 / .1, 0, 0/ , .0, s 0 , 0/ , 1 p 2 2 u 0 , 0, p 2 2 u 0 Á , 1 p 2 2 v 0 , p 2 2 v 0 , 0 Á v 3 D u 0 v 0 .1 s 0 / 12
where r 0 , s 0 , and t 0 are defined such that .r 0 , 0, 0/ D .0, s 0 , 0/ D .0, 0, t 0 / D 0. For this situation, as mentioned in [START_REF] Ventura | On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method[END_REF], the v C e volume is

v C e D r 0 s 0 t 0 6 . ( 40
)
Coordinates in the parent element of the center points G C and G are given by

G C W Â r 0 4 , s 0 4 , t 0 4 Ã and G W Â 1 r 2 0 s 0 t 0 4.1 r 0 s 0 t 0 / , 1 r 0 s 2 0 t 0 4.1 r 0 s 0 t 0 / , 1 r 0 s 0 t 2 0 4.1 r 0 s 0 t 0 / Ã .
The second case is supposed to have nodes 1 and 4 in domain C where > 0 and nodes 2 and 3 in domain where < 0. With the notations defined in Figure 7(a), the parent volume of the positive side v C e is the volume of three assembled subtetrahedra domains. The two variables s 0 and t 0 go from 0 to 1 and are defined such that .0, s 0 , 0/ D .0, 0, t 0 / D 0. The two variables u 0 and v 0 go from 0 to p 2 and are defined such that

1 p 2=2u 0 , 0, p 2=2u 0 Á D 1 p 2=2v 0 , p 2=2v 0 , 0 Á D 0.
The coordinates of the four vertices of each tetrahedron as well as their volumes are given in Table I.

The parent volume of the positive side v C e is given by

v C e D v 1 C v 2 C v 3 D 1 12 2s 0 t 0 C p 2s 0 u 0 p 2s 0 u 0 t 0 C u 0 v 0 u 0 v 0 s 0 Á . ( 41
)
The center points G C and G are given by their coordinate vectors using the following barycentric equations:

OG C D 1 v C e .v 1 OG 1 C v 2 OG 2 C v 3 OG 3 /, (42)
OG D 1 1 6v C e .OG 6v C e OG C /, (43)
where OG i denotes the coordinate vector of the center of the subtetrahedron number i assuming that point O is localized at .0, 0, 0/.

STRATEGY FOR A PARAMETRIC STUDY OF SEVERAL IMMERSED STRUCTURES

The discretized system to be solved for a configuration of the coupled fluid-structure system is given by Equation (28). This system can be extended to n immersed structures as

.K w 2 M /X D F (
K FF K FA 1 0 K FA 2 0 . . . K FA n 0 K T FA 1 K A 1 A 1 0 0 0 . . . 0 0 0 C S 1 A 1 K S 1 S 1 0 0 . . . 0 0 K T FA 2 0 0 K A 2 A 2 0 . . . 0 0 0 0 0 C S 2 A 2 K S 2 S 2
. . . 0 0 : : : : : : : : : : : : : : : : : : : : : : : :

K T FA n 0 0 0 0 . . . K A n A n 0 0 0 0 0 0 . . . C S n A n K S n S n 3
M FF M FA 1 0 M FA 2 0 . . . M FA n 0 M T FA 1 M A 1 A 1 C T S 1 A 1 0 0 . . . 0 0 0 0 M S 1 S 1 0 0 . . . 0 0 M T FA 2 0 0 M A 2 A 2 C T S 2 A 2 . . . 0 0 0 0 0 0 M S 2 S 2
. . . 0 0 : : : : : : : : : : : : : : : : : : : : : : : :

M T FA n 0 0 0 0 . . . M A n A n C T S n A n 0 0 0 0 0 . . . 0 M S n S n 3
where subscripts 1 to n denote the n different structures. It is supposed in this work that the structures are far enough from each other to not share enriched nodes. With this assumption, only one level-set can be used to localize several structures in the fluid domain.

In system (44), matrices K FF and M FF are those corresponding to the empty fluid cavity, meaning that they do not need to be recomputed when the structures are placed differently. Matrices K A i A i and M A i A i do not need to be recomputed because they can be computed from K FF and M FF by using the localization matrix ˇFA i (Equations 31 and 32). Matrices K S i S i and M S i S i are those corresponding to the structures with no fluid; again, they do not need to be recomputed for each configuration unless the geometry of the structure itself is changed. Finally, when many structures are placed arbitrarily in the fluid, the only matrices to be computed in the parametric study are the coupling matrices between the standard part and the enrichment of the pressure (K FA i , M FA i ) and the coupling matrices between the fluid enrichment and the structures C S i A i .

The algorithm used to compute frequency response functions during a parametric study is Compute K FF and M FF from the given empty cavity mesh Compute K S i S i and M S i S i for each structure Loop over the parameters:

ı Loop over the structures that have been modified:

-Update structure position.

-If the structure geometry has changed, compute K S i S i and M S i S i .

-Compute the level-set .M /.

-Compute ˇFA i matrix; in other words, get the set of enriched nodes A i .

-Compute the fluid-enrichment coupling matrices K FA i and M FA i .

-Compute the structure-enrichment coupling matrix C S i A i .

ı Form the system as in Equation (44). ı Compute the frequency response function for this configuration.

The algorithm could be simplified by considering all structures as one, avoiding the structure loop. However, it may become difficult to modify one specific structure.

The implementation has been carried out with Python for the main program at the global level and using Fortran functions at the elemental level. This Python-Fortran coupling takes benefit of the two different languages: speed of Fortran compiled functions and ease of Python to manipulate large sparse matrices. Geometry, meshes, and results viewing are performed with Gmsh [START_REF] Geuzaine | Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities[END_REF]. This implementation strategy allows to deal with quite large number of degrees of freedom.

VALIDATION AND APPLICATIONS

Validation on a parallelipipedic cavity

A parallelipipedic cavity divided into two parts by a plane flexible structure is considered (Figure 8). The data of the problem are given in Table II. The structure is simply supported on its four edges; a harmonic load of amplitude 1 N is applied to point A in the normal direction of the plane structure. The position of point A is chosen such that it cannot be a vibration node of the structure (Figure 8). The studied frequency range is 10 to 600 Hz.

Analysis of the problem.

Because the air is a light fluid compared with the structure, the modes are almost decoupled; therefore, a separate modal analysis of the fluid and the structure is carried out to verify the convergence of the numerical results obtained by the coupled system. In the case of a rigid structure, the two cavities are decoupled. The analytical eigenmodes p OEij k and eigenfrequencies f OEij k of the fluid cavities are given by

p OEij k .x, y, ´/ D cos i x L x cos j y l cos k h , ( 48
)
f OEij k D c 2 Â i 2 L 2 x C j 2 l 2 C k 2 h 2 Ã 1 2 , ( 49
)
where i , j , and k are integers and L x is the length of the cavity along x direction (L x D a for the large cavity and L x D L a for the small one). Table III gives the eigenfrequencies of the two cavities up to 600 Hz. The analytical eigenmodes and eigenfrequencies of the four-edge simply supported structure are given by

w OEij .x, y/ D sin i y l sin j h , ( 50
)
f OEij D s Ee 2 12.1 2 / S Â i 2 2 l 2 C j 2 2 h 2 Ã , ( 51
)
where i and j are integers and w OEij is the transverse displacement. Table IV gives the analytical eigenfrequencies of the structure up to 600 Hz.

Frequency analysis.

The frequency response function is computed using the present approach based on XFEM as well as using a conforming mesh between fluid and structure at the interface to validate the results. The two meshes are presented in Figure 9. Table V shows the number of nodes and elements used for each computation. The number of enriched nodes (set of nodes A ) is 1129, and there are 3067 enriched elements for the XFEM computation. The sound level indicator is the mean quadrature pressure, expressed in dB, and computed according to the following definition:

L p D 10 log 10 R F p 2 dV p 2 0 R F dV , ( 52
)
where p 0 D 20 Pa is the reference sound pressure in the air. The comparison of the frequency response functions is shown in Figure 10. On this figure, the decoupled analytical eigenfrequencies of the two cavities as well as those of the structure are mentioned as vertical lines. The coupled problem eigenfrequencies corresponding to the resonances of the FRF are close to these vertical lines. The results are compared for four frequencies close to the four first resonances of the frequency response function in Figure 11; the structure is placed above the fluid cavity to visualize it. The XFEM results show a discontinuous jump at the interface because the plot routine does not take into account the Heaviside enrichment in an element, whereas the enrichment is used to compute the real pressure at an enriched node according to Equation [START_REF] Rouleau | Vibro-acoustic study of a viscoelastic sandwich ring immersed in water[END_REF]. The plots show the good ability of the method to correctly catch the pressure jump through the structure.

(a) Reference, f = 92Hz (b) XFEM, f = 92Hz (c) Reference, f = 164Hz (d) XFEM, f = 164Hz (e) Reference, f = 228Hz (f) XFEM, f = 228Hz (g) Reference, f = 263Hz (h) XFEM, f = 263Hz
A dB difference error between the two results obtained by the two methods is computed in Figure 12. This error is more a qualitative tool than a quantitative tool for two main reasons:

The compatible mesh has been made smaller to have a good reference solution, so the error contains the discretization error as well as an error coming from the XFEM method itself. There is no damping in this problem, so the solution becomes highly singular at resonances; the value has thus no signification close to a resonance.

For these reasons, the error is not a relative error (it would add more uncertainty in the value), and the error is arbitrarily cut up to a value of 5 dB. However, the difference is less than 1 dB up to 225 Hz and globally increases to less than 2 dB away from the resonances, which represents approximately 2% in terms of relative error.

Parametric study.

To show the ability of the proposed approach to perform a parametric study according to the structure position, the plane flexible structure is arbitrarily placed at four different positions (a D 0.5, a D 0.55, a D 0.65, and a D 0.75), whereas the fluid mesh remains to be the same. The fluid and structure meshes used for this parametric study are the same as those used in the previous section for the XFEM noncompatible case. The number of enriched nodes vary from one configuration to an other because the zero-contour level-set is placed differently in the fluid mesh: 1062 enriched nodes for a D 0.5, 1047 enriched nodes for a D 0.55, 1082 enriched nodes for a D 0.65, and 1129 enriched nodes for a D 0.75. The different frequency response functions are plotted in Figure 13. Because the first two resonances correspond to structure eigenfrequencies, these are not shifted whereas all the cavity resonances are shifted according to the a value. The limit case a D 0.5 is where the two cavities have exactly the same size, meaning that their eigenfrequencies overlap.

Validation of the proposed approach.

This application shows that the method can reproduce a pressure discontinuity in a noncompatible mesh. The dB difference error between a reference solution and the XFEM approach is small, even if the mesh used with the XFEM method contains less elements. A parametric study of the position of a structure in the fixed fluid mesh of the empty cavity is easy to perform and gives accurate results.

Convergence study on an immersed circular ring structure

The accuracy of the method is principally based on the interface discretization. This discretization depends on the fluid element length compared with the structure curvature. To study this effect on the convergence, one-fourth of a circular ring structure is immersed in a thin square cavity of thickness denoted by h (Figure 14 and Table VI). Boundary conditions as well as load are chosen such that the solution of the problem is two dimensional. One side of the structure is clamped; a harmonic load of amplitude 1 N is applied on the other side. A reference solution is obtained by using a compatible mesh: 1018 nodes and 1676 elements for the structure; 123,748 nodes and 631,012 elements for the fluid.

The structure mesh is chosen to be the same as for the reference solution. The frequency response function is computed for these five fluid meshes and are compared with the reference solution (Figure 17(a) and 17(b)). The frequency response function for mesh e is not plotted because the difference with the reference solution cannot be observed visually. A dB difference with the reference solution is computed for each mesh and for each frequency. This dB difference is plotted in Figure 17(c it is arbitrarily cut up to a value of 5 dB. To quantify the influence of the mesh size, a mean dB difference (denoted by ) is computed for each mesh using the following simple formulae;

D P dB difference Number of frequency steps . ( 53
)
This mean dB difference (Table VII) is compared with the relative characteristic fluid mesh size to the radius of the circular structure in Figure 17(d). This last plot is more qualitative than quantitative; however, it gives a trend on how the fluid mesh size has to be compared with the curvature to give enough accurate results:

The fluid element length has to be less than 1=20 of the radius to have less than an average of 2 dB difference (c, d , and e meshes). A fluid element length larger than around 1=5 gives inaccurate results (a mesh).

The fluid mesh has thus to be prepared before the parametric study such that it is fine enough in the area where large structure curvatures are supposed to be.

Application to an industrial-like problem: an aircraft cabin

A cavity containing one or two separate flexible structures is considered. The geometry and the dimensions of the cavity and the structures are described in Figure 18 and in Table VIII. This case The frequency response for each configuration as well as the dB difference with a reference solution obtained with a compatible mesh are plotted in Figure 21 For each configuration, a representation of the pressure field and the structural displacement is plotted in Figure 22 for frequencies close to resonances. For configuration c at the 86.3 Hz resonance (Figure 22(f)), whereas only one structure is loaded by an external force, the second structure vibrates at the same displacement level as the first one: the coupling between the two structures is realized by the acoustic fluid. This last case demonstrates the ability and the flexibility of the method to deal with more complex problems where one or several structures are placed arbitrarily in an acoustic cavity.

CONCLUSIONS

The proposed method aims at avoiding remeshing process when performing a parametric study of different structure positions in an acoustic cavity. The structures are considered with no thickness in the fluid domain, and the fluid is supposed to be present on both sides of the structures. Therefore, the structures introduce pressure discontinuities through the fluid-structure interfaces. The XFEM is used because it is well suited to model such discontinuities inside elements, allowing to keep the same fluid mesh for all the structure positions in the cavity. The pressure enrichment is chosen as the Heaviside function on the basis of the signed distance to the structures.

The implementation is carried out in three dimension using four-node tetrahedral elements for the fluid and three-node discrete Kirchhoff triangle shell elements for the thin structures. A first application involving a plane structure in a parallelipipedic cavity shows the good ability of the method to deal with a plane surface of discontinuity. Moreover, several positions of the structures are computed, keeping the same fluid mesh for the whole parametric study.

A second case, made of a circular structure immersed in a thin squared fluid domain, enables studying the convergence of the method in terms of number of elements needed to represent correctly the fluid-structure interface. It turns out that the length of a fluid element has to be smaller than 1=10 of the curvature radius of the structure around the interface. The fluid mesh has then to be built a priori with this condition according to the possible positions of the structures in the parametric study. A classical compatible mesh method would have the same restriction, but a mesh would have to be made for each configuration. This restriction could be less severe if the used fluid element approximation was be quadratic (10-node tetrahedral elements for instance), but for simplicity reasons, it has not been tested in this first work.

Finally, an industrial-like problem is studied, taking care of the previous recommendations for the fluid mesh. Three structure configurations are computed involving one or two structures. Results are in good agreement with reference solutions, showing that the developed approach is well suited for complex problems close to industrial applications.

ΩFigure 1 .

 1 Figure 1. Description and notations of the elasto-acoustic interaction problem.

Figure 2 .

 2 Figure 2. Signed distance to the thin structure.

3. 2 . Fluid 3 . 2 . 1 .

 2321 Standard part of the approximation. The tridimensional fluid domain is discretized by fournode tetrahedral elements (Figure 3(b)). The pressure fluctuation p is linked to the node pressure vector P by p D N F P,

4 -node tetrahedral fluid element 3 Figure 3 .

 433 Figure 3. Fluid and structure finite elements. (a) Three-node discrete Kirchhoff triangle shell element; (b) four-node tetrahedral fluid element.

Figure 4 .Figure 5 .

 45 Figure 4. Different ways of meshing the interface. (a) Fluid divided into two parts by the structure; (b) compatible mesh; (c) incompatible mesh.

Fı 1 Figure 6 .

 16 Figure 6. Interface coupling details.

2 Figure 7 .

 27 Figure 7. Tetrahedral finite element in the parent domain cut by the zero contour of the level-set. (a) Case 1-3; (b) case 2-2.

Figure 8 .

 8 Figure 8. Parallelipipedic cavity divided into two parts by a flexible structure.

Figure 9 .

 9 Figure 9. The two meshes of the parallelipipedic cavity: left, compatible; right, noncompatible for XFEM.

Figure 10 .

 10 Figure 10. Frequency response function of the parallelipipedic cavity separated by a plane flexible structure.

Figure 11 .Figure 12 .

 1112 Figure 11. Results at different frequencies close to resonances of the coupled system. (a) Reference, f D 92 Hz; (b) XFEM, f D 92 Hz; (c) Reference, f D 164 Hz; (d) XFEM, f D 164 Hz; (e) Reference, f D 228 Hz; (f) XFEM, f D 228 Hz; (g) Reference, f D 263 Hz; (h) XFEM, f D 263 Hz.

Figure 13 .Figure 14 .

 1314 Figure 13. Parametric study results for the parallelipipedic cavity separated by a structure.

Figure 15 .

 15 Figure 15. Iso-contours of the level-set for the immersed circular ring. (a) Mesh a; (b) mesh b; (c) mesh c; (d) mesh d.

  Five different structured fluid meshes are used; their characteristics are shown in Table VII. The fluid mesh as well as the level-set projections are shown for the first four meshes in Figure 15. It can be visually observed that meshes a and b gives a quite inaccurate representation of the level-set, specially around the circle center. The enriched elements are plotted in Figure 16 for the first four meshes: meshes a and b have large partition of unity supports around the interface, leading to a poor approximation of the Heaviside function enrichment.

Figure 16 .Figure 17 .

 1617 Figure 16. Enriched elements for the immersed circular ring. (a) Mesh a; (b) mesh b; (c) mesh c; (d) mesh d.

Figure 18 .

 18 Figure 18. Aircraft cabin geometry.

Figure 19 . 3 )Figure 20 .Figure 21 .

 1932021 Figure 19. Fluid and structure meshes of the aircraft cabin. (a) Fluid mesh, 121,789 tetrahedra and 20,832 nodes; (b) structure mesh, 2228 triangles and 1168 nodes.

  (a), 21(b), and 21(c) for configurations a, b, and c, respectively. The dB difference is less than 0.5 dB in this frequency range for the three configurations apart around the resonances.

Figure 22 .

 22 Figure 22. Pressure field and structure displacement for the three configurations close to a chosen resonance. (a) Configuration a, fluid pressure, 50.4 Hz; (b) configuration a, structure displacement, 50.4 Hz; (c) configuration b, fluid pressure, 88.7 Hz; (d) configuration b, structure displacement, 88.7 Hz; (e) configuration c, fluid pressure, 86.3 Hz; (f) configuration c, structure displacement, 86.3 Hz.

Table II .

 II Data of the parallelipipedic cavity cut by a plane structure.

	Geometry	Fluid		Structure
	L (m) l (m) h (m) a (m)	1 0.6 0.4 0.75	c 0 (m s 1 ) F (kg m 3 )	340 1.2	E (Pa) S (kg m 3 ) e, thickness (m)	70,000 10 6 0.27 2700 4 10 3

Table III .

 III Eigenfrequencies of the two parallelipipedic fluid cavities.

	i	j	k	f (Hz)
	Large cavity, L x D a
	1	0	0	226.7
	0	1	0	283.3
	1	1	0	362.8
	0	0	1	325
	2	0	0	353.3
	1	0	1	381.7
	0	1	1	510.8
	2	1	0	533.6
	1	1	1	558.8
	0	2	0	566.7
	Small cavity, L x D L a
	0	1	0	283.3
	0	0	1	325
	0	1	1	510.8
	0	2	0	566.7

Table IV .

 IV Eigenfrequencies of the plane structure.

	i	j	f(Hz)
	1	1	86.6
	2	1	166.5
	1	2	266.3
	3	1	299.7
	2	2	336.3
	3	2	379.6
	3	1	386.2
	1	3	566.2

Table V .

 V Mesh characteristics of the parallelipipedic cavity.

		Compatible mesh					XFEM mesh
	Fluid mesh	Structure mesh	Fluid mesh	Structure mesh
	Nodes	Elements	Nodes	Elements	Nodes		Elements	Nodes	Elements
	44,643	263,274	1232		2334	21,089	120,306	1243	2356
			150			Compatible mesh
		Mean quadratic pressure (dB)	90 100 110 120 130 140			XFEM approach small cavity eigen freq. stucture eigen freq. large cavity eigen freq.
			0 80	100	200	300	400	500	600
					Frequency (Hz)

Table VI .

 VI Data of the circular structure ring immersed in a thin cavity.

	Fluid		Structure
	L (m)	1	E (Pa)	70,000 10 6
	h (m)	0.03		0.27
	340 F (kg m 3 ) 1.2 c 0 (m s 1 )	S (kg m 3 ) e, thickness (m) R, radius (m)	2700 6 10 3 0.8
			F , harmonic load (N)	1

Table VII .

 VII Parameters and results of the convergence study for the immersed circular ring.

	Mesh	a	b	c	d	e
	Fluid elements	216	726	2646	10,086	196,830
	Fluid nodes	98	288	968	3528	40,344
	Element length	0.16	0.09	0.05	0.025	0.012
	Mean dB difference	7.73	3.41	1.37	0.45	0.21
				(d) Mesh d

Table VIII .

 VIII Data of the aircraft cabin.