Boundary Integral Equations Methods in Acoustic Scattering

A. Bendali Mathematical Institute, INSA, Toulouse and CERFACS, Toulouse M. Fares CERFACS, Toulouse

February 28, 2007

1 Abstract The main subject of this contribution is to present some recent methods, specially designed to be implemented on parallel platforms, to deal with acoustic scattering problems involving a bounded zone filled by a heterogeneous medium. The main approach is to couple a Finite Element Method, for handling this zone, with a Boundary Integral Equation, specially adapted to treat the unbounded part of the computational domain. After giving a short review of some alternative methods, we focus on the methods based on this approach and give a framework which makes it possible to construct almost all the standard Boundary Integral Equations. As well-known, each instance of this kind of scattering problems can be solved by a manifold of such equations. This framework allows one to have a good insight into the advantages and the drawbacks of each of them. It is seen next that the above coupling gives rise to non standard linear systems, with a matrix being partly sparse and partly dense. Serious difficulties then arise when the solution of such systems has to be tackled on a parallel platform. It is shown how techniques from domain decomposition methods can be used to efficiently overcome these difficulties.

Keywords: Acoustic Scattering, Helmholtz equation, Boundary Integral Equations, Finite Element Method, Coupling, Domain Decomposition Method, Cross-points.

1 Typical scattering problems in acoustics and brief review of some approaches to their numerical solution

The ﬁrst part of this section is devoted to state some examples of scattering problems in acoustics. They are used as a guiding line in this contribution to present some

2 numerical methods, based on Boundary Integral Equations (BIEs), for solving this class of problems. The second part gives a brief review of some alternative methods to the BIEs.

1.1 Some model scattering problems in acoustics

int RN Let Ω0 be a bounded domain of (N = 2, 3) with a sufﬁciently smooth boundary int 2 Σ. To make simpler the exposure, we assume that Ω0 is a C -domain (see, for example, [61, 59]). However, since we restrict ourselves here to the BIEs that are set as a variational equation, all the numerical procedures, hereafter described, remain valid int when Ω0 is a lipschitzian domain (see, e.g., [59] for a precise deﬁnition of such a domain) such that Σ is moreover piecewise C2. We denote by n the unit normal to Σ int int int pointing outwards to Ω0 . The complement Ω0 of the closure Ω0 of Ω0 is assumed to be also a domain. Note that Ω0 is an unbounded domain. It describes the air zone for which the corresponding wavenumber and sound speed are respectively k and c. It is implicitly assumed that the time dependence is in e−ikct. The incident wave uinc is always a plane wave propagating in Ω0 in the opposite direction indicated by θ0 ∈ S1 (the unit sphere of RN ) given by

inc N u (x) = exp(−ikθ0 · x) (x ∈ R ), (1) where a · b is the scalar (not Hermitian) product of two vectors a and b with N real or complex components.

1.1.1 Simple scattering problems

The ﬁrst of the two classes of acoustic scattering problems, hereafter considered, concerns cases where the determination of the total wave u0 in Ω0 is an outgoing solution of the Helmholtz equation

∆u + k2u = 0 in Ω , 0 0 0 (2) lim |x|(N−1)/2 ∂ (u − uinc)(x) − ik(u − uinc)(x) = 0, ½ |x|→∞ |x| 0 0 supplemented by a boundary condition¡ of one the following type ¢

• Neumann boundary condition

∂nu0 = 0 on Σ, (3)

• Dirichlet boundary condition

u0 = 0 on Σ, (4)

• Impedance boundary condition

−∂nu0 + βu0 = g0 on Σ. (5)

3 Figure 1: Schematic description of the scatterer

These conditions respectively correspond to a hard-sound obstacle, an elastic obstacle and a surface impedance condition. In (5), β and g0 are two given complex-valued functions deﬁned on Σ. The case g0 = 0 generally corresponds to a scattering problem with an imperfectly reﬂecting hard-sound surface. In this contribution, the con- sideration of g0 is connected with some coming procedures for solving more involved scattering problems. Standard notation in the theory of Partial Differential Equations (PDE) are used without further comment [67, 26].

1.1.2 Variable coefﬁcient Scattering problem

The second class of acoustic scattering problem that is considered concerns the cases where the wave penetrates an heterogeneous medium enclosed by Σ. More precisely, we assume that Ω1 is an annular domain limited by Σ and an interior boundary Γ, at least as smooth as Σ. The scatterer is schematically depicted in Fig 1.

The determination of u0 in Ω0 now requires the following coupling with the total wave in Ω1, denoted by u1,

u0 = u1 on Σ, (6)

∂nu0 = χ∂nu1 on Σ, (7) instead of a boundary condition on Σ. The determination of u1 in Ω1 is governed by the Helmholtz equation with variable coefﬁcients, χ and n, and a boundary condition on Γ, which for simplicity is supposed to be related to a hard-sound condition

∇ · χ∇u + k2χn2u = 0 in Ω , 1 1 1 (8) χ∂nu = 0 on Γ. ½ 1 Usual energy considerations and acoustic media properties imply the following

4 bounds on χ and n

<χ ≥ γ > 0,

−=β > 0. (11)

Under conditions (9–11), any of the above scattering problem admits one and only one solution in an appropriate functional setting (cf., e.g., [51]). It must be noticed that this contribution is not intended to be a discussion on the mathematical properties of the methods that are presented even if some insight is given on the existence and the uniqueness of the solution to the BIEs that are considered. We refer to [59, 27, 61, 22] for the mathematical analysis of this kind of BIEs and to [36, 30, 14] for the issues relative to the stability and convergence properties of the domain decomposition methods presented in this contribution.

1.2 Brief review of some numerical methods for solving acoustic scattering problems

A clear account of the available methods for solving the above kind of problems can be found in [46] (see also the preface of a special issue of the journal “Computer Methods in Applied Mechanics and Engineering” [43]). The more convenient way to deal with the equations posed in Ω1 is to use a Finite Element Method (FEM). The real difﬁculty of course concerns the unbounded character of Ω0. There exist two classes of such methods:

• Exact methods. These methods are called exact because they give rise to a solution where the only errors are coming from the discretization process. Among them, the BIE method is likely the most popular. Some particular versions of this solution procedure are detailed below. Another exact method can be developed when the solution in Ω0 can be performed by a separation of variables (see, e.g., [57]). Of course, this kind of solutions requires a very special geometry. However, by possibly making a deformation of Σ and extending the coefﬁcients χ and n by 1, the problem can be reduced to the case where the separation of variables applies. A very powerful extension of this method is provided by the Inﬁnite Element Method [42, 23, 34].

• Approximate methods. These methods are called approximate because they yield a solution which is generally not exact even without discretization. The most representative of them are those based on an absorbing (also called non reﬂecting) boundary condition. A detailed discussion of the performance of these methods, when applied to acoustic scattering problems, can be found in

5 [64] (See also [68] which presents a comprehensive example of the application of this technique). Another approximate method, which was extensively applied to time domain problems, is the Truncated Perfectly Matched Layer (PML) method [19, 47].

2 Boundary Integral Equations

A main feature of the BIEs is that several of them can be used for solving the same scattering problem. The use of one instead of another of these depends on several features: ease in the computation of the matrix coefﬁcients of the system to be solved, existence and uniqueness of the solution, stability of the solution process, direct or iterative method for solving the resulting linear system, etc. In this section, we introduce a general framework which makes it possible to obtain almost all the BIEs that are used in practical computations. We then present the most important of them including some particularly recent ones.

2.1 The basic framework

2.1.1 The integral representation

The starting point in the construction of almost every BIE is the following integral representation of u0 in terms of two unknown densities p and λ on Σ

inc u0(x)= u (x)+ Vp(x)+ Nλ(x) (x ∈ Ω0) (12) where Vp and Nλ are respectively the single- and the double-layer potentials deﬁned by int Vp(x)= GN (x, y)p(y) dΣy (x ∈ Ω0 ∪ Ω0), (13) ZΣ int Nλ(x)= − ∂ny GN (x, y)λ(y) dΣy (x ∈ Ω0 ∪ Ω0). (14) ZΣ The function GN (x, y), deﬁned for x =6 y, is the Green kernel related to the outgoing solutions of the Helmholtz equation

(i/4)H(1)(|x − y|), for N = 2, G (x, y)= 0 N exp(ik |x − y|)/4π |x − y| for N = 3, ½ (1) where H0 is the Hankel function of the ﬁrst kind and of order 0. (see, e.g., [61, 59, 27, 31]).

2.1.2 Properties of the potentials

As clearly indicated by the deﬁnition of the potentials, ﬁxing an integral representation int int for u0 yields an extension u0 of u0 in Ω0 obtained by allowing x in (12) to run

6 int through Ω0 . As seen below, the construction and the well-posedness of a BIE heavily relies on this extension. To go on with this task, we recall the following very classical properties of potentials (see, for example, the references just cited above). If w stands for either Vp or Nλ, it satisﬁes

2 int ∆w + k w = 0 in Ω0 ∪ Ω0, (15) (N−1)/2 lim |x| ∂|x|w(x) − ikw(x) = 0. (16) |x|→∞ ¡ ¢ An important feature of the representation of u0 in terms of potentials is that it directly yields an explicit expression for the radiation (also called far-ﬁeld) pattern of the scattered wave ik|x| inc e 1 u0(x)= u (x)+ ωN F0(x/ |x|)+ O( ) (17) |x|(N−1)/2 |x|(N+1)/2

1 2i 1 with ω2 := 4 πk , ω3 := 4π and q −iky·x/|x| −iky·x/|x| F0(x/ |x|)= p(y)e dΣy − λ(y)∂ny e (y)dΣy (18) ZΣ ZΣ (see, for instance, [59]). Another important properties of the potentials are provided by their ﬁrst and second traces on Σ (Cauchy data relatively to the Helmholtz equation). Using an exponent + int or −, according to the domain Ω0 or Ω0 from which they are taken, makes it possible to express these traces as follows (see, e.g., [61, 59, 27]) ± ± (Vp) |Σ = V p, (∂nNλ) |Σ = Dλ, (19) ± 1 T ± 1 n (∂ Vp) |Σ = ± 2 p − N p, (Nλ) |Σ = ± 2 λ + Nλ. (20) Now, Vp and Nλ are (singular) integral deﬁned as in (13) and (14) by then letting x run through Σ. The notation N T stands for the transpose of N relatively to the scalar (not Hermitian!) product of L2(Σ)

λ0N TpdΣ= pNλ0 dΣ, (21) ZΣ ZΣ and D is a hypersingular integral operator deﬁned in the distributional sense by ∇ · V (∇ λ × n) × n − k2n · V (λn) , N = 3, Dλ = Σ Σ (22) −∂ V (∂ λ) − k2n · V (λn) , N = 2, ½ s s where ∇Σ and ∇Σ· are respectively the surface gradient and surface divergence while ∂s indicates the derivative relatively to a unit-speed parameter s on Σ. The above trace formulas directly yield the following jump relations

int int u0 |Σ − u0|Σ = λ, ∂nu0 |Σ − ∂nu0|Σ = p (23) int which express the densities λ and p in terms of the traces of u0 and u0.

7 2.1.3 General principles governing the construction of a BIE

We start with the following observation. An easy calculation in the distributional sense int inc inc shows that if u0 − u and u0 − u satisfy the Helmholtz equation, and, for the latter, int the radiation condition, then u0 and u0 have the integral representation (12) in which the densities are deﬁned by the jump relations (23). The integral representation (12) int int hence appears as nothing else than a characterization of an extension u0 of u0 to Ω0 satisfying the Helmholtz equation. As a result, the construction of almost any BIE can be viewed as a way to design two equations linking the unknowns densities λ and p. For convenience, we will respectively refer to these equations as follows.

• Direct equation. This is a direct equation linking the densities. Direct here means that it is not set in terms of the potentials. Generally, this equation makes it possible to easily eliminate one of the densities. It directly results from either one of the boundary conditions (3), (4), (5), or the interface conditions (7), (8).

• Boundary Integral Equation. It is generally the actual equation to be solved. It is written as a relation linking the integral representations of the traces u0|Σ int int and ∂u0|Σ or u0 |Σ and ∂u0 |Σ in terms of the densities.

Actually, the above direct and boundary integral equations can be interpreted as an int equivalent expression of the extension of u0 by the solution u0 to a boundary-value int problem set in Ω0 . The question of existence and uniqueness of a solution to the BIE is hence brought back to an examination of this boundary-value problem. We show below how some of the most used BIEs can be obtained and analyzed following this general path. We ﬁrst deal with the direct BIEs based on the boundary equivalence principle. Then, we more rapidly present some popular BIEs as well as the extensions which have been recently carried out to improve them.

2.2 Direct formulations

2.2.1 Deﬁnition and general features

These BIEs are the most natural ones because they are based on the Helmholtz integral representation of u0 [62]. For this particular form of (12), obtained by the Green formula, the densities are the Cauchy data of u0 (really here the opposite of these ones: p = −∂nu0|Σ and λ = −u0|Σ) (cf. e.g., [27, 31]). It is important to notice that the conditions p = −∂nu0|Σ and λ = −u0|Σ are not really imposed in the formulation but will be the consequence of the solution procedure itself, even if one has to have them in mind when designing such a method. In engineering terminology, the densities are then sometimes termed the physical unknowns since they permit a direct interpretation of the result delivered by the solving procedure.

8 In view of the jump relations, such methods must therefore ensure that both the following two BIEs are satisﬁed

int inc 1 u0 |Σ = u |Σ + Vp + 2 λ + Nλ = 0 (24) int inc 1 T n n ∂ u0 |Σ = ∂ u |Σ + 2 p − N p + Dλ = 0. (25) According to the integral representation of the solution to Helmholtz equations or, more simply, by making use of the analytic continuation principle (cf., e.g. [51]), these conditions in turn are equivalent to

int int u0 = 0 in Ω0 . (26) For this reason, this kind of methods can be called “null ﬁeld method”. We present below the most important of them. The direct equation linking the densities is the same for all the methods. Since the densities are aimed to be equal to the (opposite) of the Cauchy data, the following conditions can be set a priory

• Neumann condition. In view of (3), only one of the two densities is unknown since p = 0. (27)

• Dirichlet condition. Taking account of (4), we have this once

λ = 0. (28)

• Impedance condition. For (5), the direct equation only yields the following relation linking the two densities

p − βλ = g0. (29)

• Coupling with a variable coefﬁcients problem. When the determination of u0 is coupled with the variable coefﬁcients problem (8), this relation is written in an implicit form through the interface conditions (6) and (7) which can be directly expressed as

p = −χ∂nu1|Σ, (30)

λ = −u1|Σ. (31)

Indeed, keeping (31) and performing a simple Green formula, we express (30) in the form 0 0 a(u1, u1)+ pu1dΣ = 0 (32) ZΣ 0 where u1 stands for any test function relative to the unknown u1 and

0 0 2 2 0 a(u1, u1) := χ(∇u1 · ∇u1 − k n u1u1)dΩ1. (33) ZΩ1

9 Relation (32), still implicit, more clearly brings out the direct equation linking the densities. More importantly, it is this way to write this equation which is used for coupling the FEM used to determine u1 with the BEM for which the main unknown will be the density p. Other coupling procedures have been developed mainly to get a ﬁnal coupled system which is symmetric [33]. However, since, for wave propagation problems, this way to proceed leads to a formulation which can involve spurious solutions, they will be not consider here.

2.2.2 BIEs constructed from an interior Dirichlet condition

Before stating these equations, we need to recall some well-known facts about the int eigenvalues of the Dirichlet problem for the laplacian in Ω0 (cf., e.g., [67]). There exits a non decreasing sequence {km,D}m≥0 of positive real numbers tending to infinity and such that if k =6 km,D, the following interior Dirichlet problem ∆w + k2w = 0 in Ωint, 0 (34) w = 0 on Σ, ½ has no other solution than w = 0. If k = km,D, problem (34) has non zero solutions. These solutions are the eigenfunctions of the interior Dirichlet problem. The eigenfunctions, associated with each exceptional value km,D, span a finite-dimensional space. It hence results from the analytic continuation principle, as already invoked above, that Sk,D := {∂nw|Σ; w is a solution to (34)} (35) is reduced to 0 for k =6 km,D, and is a finite-dimensional space otherwise. Therefore, by setting (24), we obtain a BIE which uniquely determines the densities for k =6 km,D. More particularly, we obtain the following BIEs for the above scattering problems.

• Neumann condition. Since p = 0, the equation reduces to

1 inc 2 λ + Nλ = −u |Σ. (36)

• Dirichlet condition. Now, it is λ which is equal to 0 Vp = −uinc. (37)

• Impedance condition. Since p − βλ = g0, we get

1 inc 2 λ + Nλ + βV λ = −u |Σ − Vg0. (38)

• Variable coefﬁcients. In view of (31) and (32), we obtain the system yielding the coupling method introduced in [49] a(u , u0 )+ pu0 dΣ = 0, ∀u0 , 1 1 Σ 1 1 (39) − 1 u − Nu + Vp = −uinc| . ½ 2 1 R1 Σ

10 In the terminology of BIEs, equation (37) is a ﬁrst kind integral equation while equation (36) is a second kind one. Beside other features, making this distinction is important mostly because ﬁrst kind formulations are valid for general boundaries, relative to a very thin or to a thick domain or even an open surface, while second kind ones are limited to thick obstacles only.

Let us now examine the case where k is equal to one of the exceptional values km,D. For such a value, equation (24) only guarantees that

int S ∂nu0 |Σ = sk,D ∈ k,D. (40)

We successively discuss each of the above equations.

• Neumann condition. Since p = 0, the jump relation yields

int int ∂nu0|Σ = ∂nu0|Σ − ∂nu0 |Σ + ∂nu0 |Σ =0+ sk,D = sk,D. (41) ¡ ¢ Hence, the boundary condition satisﬁed by u0 is not the actual homogeneous Neumann boundary condition but a Neumann boundary condition perturbed by the “spurious” values sk,D. From uniqueness of the Neumann problem, the obtained integral representation includes some spurious terms which corrupt the sought solution u0. In engineering terminology, it is thus said that “the spurious modes radiate”. Practically, this means that these formulations are not robust for solving scattering problems.

• Dirichlet condition. From the jump relation, the density p is no more equal to −∂nu0|Σ but is corrupted by sk,D

p = sk,D − ∂nu0|Σ. (42)

However, since u0|Σ = 0, u0 continues to satisfy the correct boundary condition and is not modiﬁed by the spurious value sk,D. Similarly, it is then said that “the spurious modes do not radiate”. In practice, the counterpart of this property is that these formulations result in robust solution procedures.

• Impedance condition. In this case, λ = −u0|Σ and p = sk,D − ∂nu0|Σ. In view of (29), u0 satisﬁes the following boundary condition

−∂nu0|Σ + βu0|Σ = g0 − sk,D. (43)

The boundary condition being modiﬁed, u0, λ and p are therefore all false.

• Variable coefﬁcients. In this case, it is the following transmission condition

χ∂nu1|Σ − ∂nu0|Σ = −sk,D (44)

that is satisﬁed instead of (7). Therefore, both u0 and u1 can be false.

11 The same constructions can be carried out using the interior Neumann boundary condition (25). We will see below that they can be obtained as particular cases of more general BIEs. We leave to the reader their analysis which then requires to consider the eigenvalues km,N of the interior Neumann problem and the associated ﬁnite- dimensional space Sk,N spanned by the traces w|Σ of the solutions w to the following Neumann problem ∆w + k2w = 0 in Ωint, 0 (45) ∂nw = 0 on Σ. ½ Instead, we will deal now with methods combining the two equations (24) and (25) in order to avoid any spurious mode.

2.2.3 Combined integral equations

Burton-Miller formulations [24] avoid spurious modes simply by writing the BIE in terms of a complex combination of the Neumann and the Dirichlet condition (see [31] where a comprehensive discussion concerning this approach can be found). The combined integral equation can be set in terms of two parameters as follows

int int (1 − α)∂nu0 |Σ + αηu0 |Σ = 0. (46)

Contrary to the two previous classes of BIEs, based on an interior Dirichlet or Neumann condition, the spurious modes are avoided if α and η are such that

0 <α< 1, =η =6 0. (47)

This can be checked simply from the following Green formula

int 2 2 int 2 int int int ∇u0 − k u0 dΩ0 = ∂nu0 u0 dΣ Ωint Σ Z 0 ³ ´ Z ¯ ¯ ¯ ¯ α ¯ ¯ ¯ ¯ = ηuintuintdΣ (48) α − 1 0 0 ZΣ which clearly shows that both (24) and (25) are satisﬁed. Particularizing (46), we get the following well-posed formulations for each of the above scattering problems

• Neumann condition

1 inc inc n (1 − α)D + αη 2 + N λ = − (1 − α)∂ u |Σ + αηu |Σ (49) ¡ ¡ ¢¢ ¡ ¢ • Dirichlet condition

1 T inc inc n (1 − α) 2 − N + αηV p = − (1 − α)∂ u |Σ + αηu |Σ (50) ¡ ¡ ¢ ¢ ¡ ¢ 12 • Impedance condition

1 T 1 (1 − α) D + β 2 − N + αη 2 + N + βV λ inc inc 1 T n = − (1 − α)∂ u |Σ + αηu |Σ − (1 − α) 2 − N + αηV g0 ¡ ¡ ¡ ¢¢ ¡ ¢¢ (51) ¡ ¢ ¡ ¡ ¢ ¢ • Variable coefﬁcients 0 0 0 a(u1, u1)+ Σ pu1dΣ = 0, ∀u1, 1 1 T − (1 − α)D + αη 2 + N u1 + (1 − α) 2 − N + αηV p (52)  R inc inc = − (1 − α)∂nu | + αηu | .  ¡ ¡ ¢¢ ¡ ¡ Σ ¢ Σ¢ In fact, in some recent formulations η may be¡ a boundary operator instead¢ of being simply a multiplicative constant. An insight into these approaches is given below when some coming alternative BIEs will be considered.

2.3 Indirect formulations

2.3.1 Brakhage-Werner formulations

We limit ourselves here to the most representative of the indirect formulations. It seeks u0 in the form of the following combined single- and double-layer potentials

inc u0(x)= u (x)+(iηV + N)λ(x) (x ∈ Ω0) (53) where η is a given real parameter. As reported in [31], this method has been indepen- dently devised by several authors even if it is generally known as the Brakhage-Werner integral equation [18]. It is called indirect because the integral representation is not based on a direct Helmholtz integral representation of u0 as above. Several advantages of direct formulations are hence lost as this can be brought out from the following ob- servations.

1. Solving the BIE does not directly yield the Cauchy data. 2. The existence of a solution is not guaranteed a priory. 3. There is no natural way to couple this representation with the FEM equations.

For the above scattering problems, related to a boundary condition on Σ, the BIEs can be obtained simply by enforcing this condition on Σ:

• Neumann condition

1 T inc n −iη 2 + N + D λ = −∂ u |Σ (54)

• Dirichlet condition ¡ ¡ ¢ ¢

1 inc iηV + − 2 + N λ = −u |Σ (55) ¡ ¡ ¢¢ 13 • Impedance condition

1 T 1 iη 2 + N − D + β iηV + − 2 + N λ = inc inc (56) ∂nu | − βu | + g . ¡¡ ¡ ¢ ¢ ¡ ¡ ¢¢¢ Σ Σ 0 To examine whether this equation has a solution or not, we ﬁrst note that it can be included in the above general framework. Indeed, it can be viewed as a special case of (12) corresponding to the following direct equation on the densities

p − iηλ = 0. (57)

int In turn, this equation implies that u0 satisﬁes the following interior problem

int 2 int int ∆u0 + k u0 = 0 in Ω0 , int int (58) ∂nu − iηu = ∂nu − iηu on Σ, ½ 0 0 0 0 which admits one and only one solution in an appropriate functional space. Since u0 is determined as a solution to a well-posed boundary-value problem, the integral representation of the solutions to the Helmholtz equation and the jump relations ensure that each of the above BIEs has one and only one solution.

2.3.2 Other recent formulations

The advent of Fast Multipole Method (FMM) (see for instance [40] for an application of this method to the BIEs considered in this contribution) has completely renewed the way to solve the BIEs involved in wave propagation problems at relatively high frequencies. The advantages of this method, in terms of reduction of memory storage and acceleration of the matrix-vector product, take a concrete shape through the use a Krylov iterative method, like GMRES [65]. However, the robustness and efﬁciency of an iterative method are highly depending on some restricted properties concerning the distribution of the eigenvalues of the coefﬁcients matrix of the linear system to be solved (cf., e.g., [65]). These properties can generally be improved by using a suitable preconditioning strategy. This task can be generally handled in an algebraic way through the construction of a Sparse Pattern Approximate Inverse (SPAI precon- ditionner) [2]. However, some recent studies have proved that the convergence of the Krylov iterative methods can be greatly improved by adequately using the analytical properties of the integral operators. In all these developments, the BIE is written as a perturbation of the identity by a compact operator. We start with the methods initiated in [53]. They are based on the following observation. The Helmholtz integral representation makes it possible to write any w satisfying ∆w + k2w = 0 in Ω , 0 (59) lim |x|(N−1)/2 ∂ w(x) − ikw(x) = 0, ½ |x|→∞ |x| in the following form ¡ ¢

w(x)= V (−∂nw|Σ)(x)+ N(−w|Σ)(x) (x ∈ Ω0). (60)

14 Introducing the Dirichlet-to-Neumann mapping S associated to (59) deﬁned by

Sϕ = −∂nw|Σ (61) where w is solution to (59) completed with the additional Dirichlet condition

w|Σ = −ϕ (62) and using (19) and (20), we get the following expression for the (opposite of the) identity operator 1 −ϕ = − 2 ϕ + V Sϕ + Nϕ. (63) Of course, S is not known in an explicit way. Otherwise the solution is given explicitly by the Helmholtz integral representation formula. But, if Sapp is an approximation of S, one can use the indirect method based on the integral representation

inc u0 = u + VSappϕ + Nϕ (64) which yields the following BIE for the Dirichlet problem

1 inc VSappϕ + Nϕ − 2 ϕ = −u |Σ. (65)

In [53], Sapp is obtained by computing the main symbol of S (in the terminology of pseudodifferential calculus) or from an explicit determination for simple shapes. In [7], Sapp is approximated by methods selecting one-way waves, also called On Surface Radiation Condition in this context (cf., [48, 25, 5]). As explained in [7], the crudest one-way approximation of the wave equation

∂nw − ikw ≈ 0 (66) yields Sappϕ = ikϕ and hence makes it possible to get the Brakhage-Werner formulation with the optimal parameter η = k found in [50]. Before ending this discussion, let us note that, by taking the transpose of the integral operator involved in the Brakhage- Werner BIE with optimal parameter, one gets the combined formulation with α = 1/2 and η = −ik. This is almost the formulation (α = 0.2, η = −ik) which has been obtained empirically in [58]. Starting with this remark, several extensions of the combined direct formulations (49) and (50) have been designed in [8] by allowing η to be an operator on Σ . The second class of methods is based on the so-called Calderon´ relations (cf., e.g., [61]) that can be obtained from the following observation. The above Helmholtz integral representation and the traces of single- and double layer potentials ensure that any function w satisfying (59) is such that

w| w| w| Σ = 1 Σ − H Σ (67) ∂nw| 2 ∂nw| ∂nw| · Σ ¸ · Σ ¸ · Σ ¸ N V with H = . D −N T · ¸

15 Since any function w in the form w = Nλ + Vp in Ω0 satisﬁes (59), in view of

w| λ Σ = − 1 + H , (68) ∂nw| 2 p · Σ ¸ · ¸ ¡ ¢ 1 1 1 2 we thus obtain − 2 +H = 2 − H − 2 + H which can be simpliﬁed to yield 4H = 1. Identifying each term in this matrix identity, we arrive to the desired Calderon´ relations ¡ ¢ ¡ ¢

2 4VD = 1 − 4N 2, DN = N TD, VN = N TV, 4DV = 1 − 4 N T (69)

In [28], the first and the last of these relations are used to define¡ left precondi-¢ tionners for respectively equation (49) with α = 1 and equation (37). Contrary to combined fields or Brakhage-Werner formulations, such kinds of approaches apply to a closed as well as to an open boundary (see [6] where these formulations are com- pared to other analytical preconditionners). In this part, we have not discussed specific methods for dealing with generalized impedance boundary conditions. A particular treatment of this class of problems by means of BIEs can be found in [13]. We will however show below how techniques from domain decomposition methods can be used to develop efficient solution procedures for this kind of problems.

3 Boundary element methods

We restrict ourselves here to a discretization of the above BIEs by Boundary Element Methods (BEMs). Other discretizations can of course be used (cf. e.g., [48, 62, 32]). However, we think that BEMs are the most robust methods and apply to the most general instances of scattering problems. After rapidly describing the basic steps of these methods, we present some numerical experiments.

3.1 General presentation

A BEM is closely related to a FEM. Indeed, it is a FE solution of a BIE. The ﬁrst step hence in the solution of a BIE by a BEM is to equivalently write it in a variational form. For instance, this way to proceed has an additional advantage. It makes it possible to write the hypersingular operator D through weakly singular integrals only

0 2 0 (V (∇Σλ × n) · ∇Σλ × n − k λ n · V (λn)) dΣ, N = 3, 0 Σ λ Dλ dΣ=  Z Σ 0 2 0n n Z  (∂sλ V (∂sλ) − k λ · V (λ )) dΣ, N = 2. Σ Z (70)  As for a FEM, the second step is to mesh Σ in the FEM terminology. We limit ourselves here to approximating Σ by straight segments in 2D and by planar triangles

16 in 3D, but more sophisticated approximations can be considered [60]. Either segments or triangles are called elements. These elements must fulfill the general overlapping requirements of FE geometrical decompositions (cf., e.g., [29, 48]). The vertices of the elements are generally compelled to be on Σ. The initial Σ is then discarded and the equations are directly set on the approximate boundary which is still denoted by Σ. The third step is to select the shape functions and a system of nodal values to be able to ensure the required continuity constraints. Although here we can choose functions that are constant within each element, either for the unknown or the trial densities, when none of their derivatives are involved, we limit the exposure to linear affine functions and characterize them through their nodal values at the vertices (cf., e.g., [48]). Each density λ or p is therefore characterized by a column-wise vector of complex numbers, still denoted by λ or p, characterizing the nodal values of respectively λ or p. We denote by XΣ the finite-dimensional space of such densities. Hence any boundary integral operator, as for instance V , is characterized by a matrix, denoted by the same symbol, and defined through the identification

0T 0 0 p Vp = p V pdΣ (p and p ∈ XΣ). (71) ZΣ To form matrix V , and likewise any other matrix associated to a boundary integral operator, one can proceed by an assembly process through the relation

0T 0T KL p Vp = K LpK V pL (72)

P P 0 where K and L are two generic elements, pL and pK the respective related nodal values of p and p0 on L and K, and V KL the associated elemental matrix. Since each couple of element K and L contributes to V , it is clear that this matrix is dense. Care must be taken in the computation of V KL since the integrals can be singular. More precisely, when K and L are adjacent or coincide, the singularity is extracted and exactly integrated. Otherwise, the integral is approximated by a Gaussian integration (see for instance [62] for the 2D and [44, 48] for the 3D case). It must be noticed that the corresponding matrix to the multiplicative operator 1/2, involved for instance in equation (36), is written in terms of the sparse mass matrix MΣ through the following identiﬁcation

1 0 0T 1 2 λλ dΣ= λ 2 MΣ λ. (73) ZΣ ¡ ¢ inc inc Let us finally note that, as for operators, we also denote by u or ∂nu the column- wise vectors defined through Gaussian integration on each element and a FE assembly process from the identification

0T inc inc 0 0T inc inc 0 λ MΣu = u λ dΣ and p MΣ∂nu = ∂nu p dΣ (74) ZΣ ZΣ

17 3.2 Some numerical results

To illustrate the above discussion, we present some error curves concerning the scattering of an acoustic wave by a hard-sound and an elastic unit sphere. The unit sphere is meshed so that there are at least 10 nodes per wavelength for all the considered frequencies. Each case is solved by the three direct methods. The results are depicted in FIG. 2 and FIG. 3. Beside the accuracy that is reached by these methods, these plots clearly show how the spurious modes can damage the computations and how to avoid these ﬂaws.

Errors on the densities Errors on the Scattering Cross Section 3 3 10 10

2nd kind BIE 2nd kind BIE 2 resonances resonances 10 2 10

1 10

Error in % Error in % 0 10

0 10 −1 10 Combined and 1st kind BIE 1st kind BIE Combined BIE Spurious modes do not radiate resonances −1 −2 10 10 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 Wavenumber Wavenumber

Figure 2: Errors on the density and the Scattering Cross Section for an elastic unit sphere.

Errors on the densities Error on the Scattering Cross Section 1 1 10 10 2nd kind BIE resonances 2nd kind BIE 0 resonances 10

−1 10

0 10

Error in % Error in % −2 10

−3 10 1st kind BIE Combined and 1st kind BIE Combined BIE resonances Spurious modes do not radiate

−1 −4 10 10 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 Wavenumber Wavenumber

Figure 3: Errors on the density and the Scattering Cross Section for an hard-sound unit sphere.

18 4 Finite Element-Boundary Element Methods

We first introduce the discretization of the above variable coefficient scattering problem by a FE-BE method. We then list the difficulties that then arise when solving the resulting matrix system by a direct solver. We next describe some standard algebraic approaches classically used to deal with them and bring out why they are not efficient for wave propagation problems. We finally present some alternative methods based on domain decomposition techniques which lead to suitable procedures for solving this kind of problems.

4.1 Standard solution of the FE-BE system

4.1.1 The FE-BE linear system

We have thus to solve either problem (39) or (52) where the BIE in both of these problems is ﬁrst equivalently written in the form of a variational equation. The ﬁrst step is to introduce a mesh of Ω1 in tetrahedra in 3D and in triangles in 2D. It is worth noting that this mesh automatically induces a mesh of the kind considered above on 0 Σ. In a standard way, we approach either the unknown function u1 or the test one u1 by functions which are polynomials of degree ≤ 1 over each element and globally continuous. Each of these functions, for instance u1, are characterized by a column- wise vector, still denoted by u1, whose components are the nodal values at the vertices.

Let us denote by XΩ1 the ﬁnite-dimensional space spanned by these functions.

It results from the deﬁnitions of these spaces that XΣ can be deﬁned as the space spanned by the traces u1|Σ for u1 ∈ XΩ1 . As a result, according to our convention to do not distinguish between functions and the column-wise vector formed by their nodal values, we can consider the following partitioning u u = 1,I (75) 1 u · 1,Σ ¸ where u1,I stands for the vector of the nodal values of u1 that are not on Σ and u1,Σ characterizes those of u1|Σ. Standard techniques of the FEM then yield the following sparse matrix characterizing the bilinear form (33) A A u u0T u0T II IΣ 1,I = a(u , u0 ). (76) 1,I 1,Σ A A u 1 1 · ΣI ΣΣ ¸ · 1,Σ ¸ As a result, either£ problem (39)¤ or (52) are reduced to the following linear system

AII AIΣ 0 u1,I 0 A A M u = 0 (77)  ΣI ΣΣ Σ   1,Σ    0 C −S p −b with       1 inc C = − 2 MΣ + N , S = −V, b = u (78) ¡ ¢ 19 for problem (39) and

1 1 T C = − (1 − α)D + αη 2 MΣ + N , S = − (1 − α) 2 MΣ − N + αηV inc inc b = (1 − α)∂nu |Σ + αηu |Σ ¡ ¡ ¢¢ ¡ ¡ ¢ (79)¢ for problem (52). ¡ ¢

4.1.2 A Schur complement procedure

The solution of linear system (77) cannot be tackled by a direct solver in general. The sparse and dense blocks, which are involved in its matrix, can considerably increase the amount of ﬁll-ins, hence generating not only huge memory storage but also an excessive cost in computing time. Furthermore, for large size problems, it is necessary to resort to high performance computing (HPC), that is, on parallel platforms, to be able to perform the actual computations. Unfortunately, an HPC library, that can deal with such a kind of linear system, does not exist as far as the authors knowledge is concerned. One way to avoid handling both sparse and dense blocks simultaneously is to use a block Gaussian elimination to express (77) as follows

A A u 0 II IΣ 1,I = (80) A A u −M p · ΣI ΣΣ ¸ · 1,Σ ¸ · Σ ¸ A A −1 0 S + 0 C II IΣ p = b (81) A A M Ã · ΣI ΣΣ ¸ · Σ ¸! £ ¤ Of course, the block in (81) is not inverted but instead the matrix product is computed by factorization and forward backward sweeps. The public domain library MUMPS for MUlti-frontal Massively Parallel Solver [3] is particularly suitable for this task. It computes the matrix of system (81), the so-called Schur complement matrix of system (77) associated with variable p, and adequately distributes it to a grid of processors so that this system can be solved by the parallel solver for dense matrices SCALAPACK [38]. Copies of the documentation and the package can be downloaded from the Web pages

1. http://mumps.enseeiht.fr/apo/MUMPS/

2. http://graal.ens-lyon.fr/MUMPS/

However, despite its attractive ability to deal with the dense and the sparse blocks separately, the Schur complement technique presents some drawbacks when used in this context. First, its construction can be excessively expansive in memory storage and computing time consuming. However, worst of all, equation (80) is the matrix form of variational equation (32), which, in turn, is related to a boundary-value problem consisting of Helmholtz-like equation (8) and a Neumann boundary condition on

20 Σ. As well-known, interior resonances can prevent such a kind of problem to be inverted in a stable way. The coming parts of this section will use techniques from the nonoverlapping domain decomposition methods (DDMs) and the adaptive radiation condition to avoid this drawback.

4.2 Domain Decomposition Methods

We ﬁrst present the nonoverlapping DDM as a tool to uncouple the FE and the BE solutions. We then show how it can deal also with the FE equations leading to a stable algorithm where only small size FE and a dense BE linear systems are solved at each iteration.

4.2.1 Uncoupling the FE and the BE solutions

We consider a nonoverlapping DDM as it was devised in [54] and extended later to Helmholtz equation in [35]. Note that this method coincides at the level of the FE equations with the so-called FETI method introduced in [39]. One way to introduce this DDM is to write the variable coefﬁcient scattering problem equivalently by con- sidering two auxiliary unknowns g0 and g1 on the interface Σ

0 0 0 0 a(u1, u1)+ βu1u1 dΣ= g1u1 dΣ, ∀u1, (82) ZΣ ZΣ 2 ∆u0 + k u0 = 0 in Ω0, lim |x|(N−1)/2 ∂ (u − uinc)(x) − ik(u − uinc)(x) = 0, (83)  |x|→∞ |x| 0 0 −∂nu + βu = g on Σ,  0 0 0 ¡ ¢ g = −g + 2βu | ,  0 1 1 Σ (84) g = −g + 2βu | . ½ 1 0 0 Σ The method depends on the complex parameter β which is used to improve its convergence properties. It is taken here in the form β = −ik(1 + iχ) with 0 <χ< 1. (85) More sophisticated choices in the form of a boundary operator can be developed to theoretically obtain more worthwhile rates of convergence [55, 56, 41, 30] (see [16] for a discussion about this choice). By setting

p = −g1 + βu1|Σ, (86) one can easily prove that the system (82), (83) and (84) is equivalent to the variable coefﬁcient scattering problem. Using the previous FE discretization and either BIE (38) or (51) as an equivalent formulation of (83), we get the following linear system as a discrete approximate version of the above equations A A u 0 II IΣ 1,I = (87) A A + βM u M g · ΣI ΣΣ Σ ¸ · 1,Σ ¸ · Σ 1 ¸

21 (C + βS)λ = b − Sg0 (88) to which must be joined (84) written in the form of column-wise vector equalities

g = −g + 2βu , 0 1 1,Σ (89) g = −g − 2βλ. ½ 1 0

The matrices C, S and b are respectively given by (78) or (79) according to the formulation which is used. It must be noticed that it is necessary to use the same formulation for both the impedance and the variable coefﬁcient problems to obtain that the DDM is simply an iterative procedure to solve the FE-BE equations. Here again, a simple veriﬁcation shows that (87), (88) and (89) yield a solution to (77) by setting

p = −g1 + βu1,Σ. (90)

Assuming that g1 is given, u1,Σ can be obtained by solving the plain FE linear system (87). In the same way, given g0, λ can be obtained by solving the linear system (88) relative to a pure BEM. As a result, g0 and g1 can be obtained through a successive approximation method. However, this way to proceed results in a slowly convergent method. Instead, the linear system (89) with an implicit matrix, associated with these two solutions can be solved more efﬁciently by the GMRES iterative method [65]. Indeed, this procedure is nothing else than an iterative solution by the GMRES algorithm of the linear system whose matrix is the Schur complement matrix associated with the variables g0 and g1. A detailed description of this procedure as well as some numerical experiments can be found in [15]. Indeed, the solution of the FE linear system can be completely avoided by tackling it by a DDM similar to the one given here for uncoupling the FE and BE solutions. However, a difﬁculty then stems from nodes that are shared by more than two subdomains or by nodes shared by two subdomains and which are on Γ or Σ. These nodes are called cross-points and dealt with by preserving there the FE continuity constraints on both the unknown and the test function (see [14] for a description of this method as well as its stability properties). It must be noticed that only cross-points related to the FEM are dealt with in this way. The matching at a cross-point shared by the FEM and the BEM continues to be treated through the iterative process associated to (89) (see [17] for details). The plots, depicted in FIG. 5, illustrate the remarkable convergence properties of the method, which seems to depend very little on the number of subdomains or the way the subdivision is being performed. The errors have been computed for a spherical geometry for which Ω1 is the annular domain between the two spheres of respective radii R0 = 0.75 and R1 = 1, and relatively to the following data n = 1.8, χ = 1.5 and k = 2π. The exact solution is known explicitly in this case in terms of a Mie’s series expansion (cf., e.g., [32]). In FIG. 4 are reported the type of decompositions that have been used. Particularly, it is depicted what has to be understood by a domain decomposition made up from 1 and 2 layers of subdomains.

22 (a) One layer DD (b) Three layers DD

Figure 4: Various kinds of Domain Decompositions.

Errors on the BIE densities Error on the Scattering Cross Section 3 2 10 10 Convergence is reached Convergence is reached after 45 iterations after 30 iterations 2 layer DD 72 subdomains

2 1 (dashed thick line)2 layers DD 10 10 2 layer DD 96 subdomains 72 subdomains (dashed thin line) (dashed thick line) 2 layers DD

Error in % 96 subdomains Error in % (dashed thin line) 1 0 10 10 1 layer DD 1 layer DD 72 subdomains 72 subdomains (solid thick line)1 layer DD (solid thick line)1 layer DD 96 subdomains 96 subdomains (solid thin line) (solid thin line) 0 −1 10 10 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 Iterations iterations

Figure 5: Errors on the densities and the Scattering Cross Section..

23 4.2.2 Treatment of a generalized impedance boundary condition

We then present an adaptation of the above nonoverlapping DDM to solve a boundary- value problem with an involved impedance boundary condition. We consider the following scattering problem related to a generalized boundary condition on Σ 2 ∆u0 + k u0 = 0 in Ω0, lim |x|(N−1)/2 ∂ (u − uinc)(x) − ik(u − uinc)(x) = 0, (91)  |x|→∞ |x| 0 0 −∂nu + Zu = 0 on Σ,  0 0 ¡ ¢ whereZ is polynomial or even a rational of boundary differential operators. This kind of impedance boundary condition incorporates the effect of Ω1 when this domain is a layer with a thin thickness (cf., e.g., [10]) or when Ω1 corresponds to a media with a high refractive index (cf., e.g., [4]). To avoid the treatment of such an involved boundary condition by a BIE formulation, we mimic the above approach and rewrite (91) in the form of (83) coupled with Zu1 + βu1 = g1 on Σ (92) and (84). Each iteration is performed by solving the BIE (88) and a plain differential problem on Σ. To illustrate this way to proceed, we consider the case when the effect of the thin layer is approximated by the second order Engquist-Ned´ elec´ impedance condition [37] which is posed in terms of the following boundary operator δκ δκ Z = −χδ ∂ 1+ ∂ + k2n2 1 − (93) s 2 s 2 µ µ ¶ µ ¶¶ where χ and n are the contrast coefﬁcients of Ω1, assumed to be constant here, δ is the thickness of Ω1,and s and κ are respectively the unit-speed parameter and the Gaussian curvature of Σ (cf., e.g., [10]). We have considered the following example: Σ is an ellipse with a majoraxis a = 1.5 and a minoraxis b = 0.5, δ = 0.1, χ = 1/n2 with n2 = 2(1+ i). InFIG. 6 are depicted the two ellipses Σ and Γ, that have been considered, as well as the exact scattering cross section , obtained by a direct computation, and the one corresponding to the approximated impedance boundary condition. This gives an idea on the thickness that can be correctly dealt with by means on this kind of approximate methods. The computation of the curvature is obtained directly from the mesh of the curve Σ (cf. [66] for a general formula expressing the curvature from three points of the curve and, for instance, [9] for its application in the present context). It is worth mentioning that the successive approximations procedure did not converge in this case. The problem in g0 and g1 has thus been solved using the GMRES iterative Krylov method as explained above, which required 16 iterations only to reduce the residual by a factor of 10−5.

24 Direct and 2nd order Engquist−Nedelec approximation

15 of the Scattering Cross Section of a coated hard−sound obstacle

Impedance boundary condition

0 thickness = 0.1 n2 = 2(1+i)

Scattering Cross Section in Db Direct solution χ = 1/n2

−5

−10 0 0.5 1 1.5 2 2.5 3 3.5 Observation direction (in radians)

Figure 6: The thin layer and its scattering cross section obtained by a direct computation and an approximate impedance boundary condition

4.2.3 Coupling a DDM with an adaptive radiation condition

To end this part, we now present an approach which was not published elsewhere for coupling a DDM with the so-called adaptive radiation condition. This method avoids solving any linear system unless small size FE linear ones. We first give some recalls on the adaptive radiation condition [1, 48]. In fact, the formulation underlying this iterative method has been devised and solved by means of a direct method in [52]. The first proof of the convergence of a method, close to the one considered here, has been given for the Laplace equation in [12, 11]. We first describe the method analytically and then give some new features, that have been added to extend its scope of application. The adaptive radiation condition requires the introduction of a fictitious boundary S enclosing Σ to truncate Ω0. We denote by Ω0,S the truncated domain. By using the integral representation (12) with λ = −u0|Σ and p = −∂nu0|Σ, the variable coefficient scattering problem can be replaced by the system consisting of (6), (7), (8), and

2 ∆u0 + k u0 = 0 in Ω0,S inc (94) ∂nu − iku =(∂nw − ikw)(u , u | ,∂nu | ) on S ½ 0 0 0 Σ 0 Σ with inc inc w(u , u0|Σ,∂nu0|Σ)= u − V (∂nu0|Σ) − N (u0|Σ) . (95) It is worth noting that no error is coming from this way to deal with the truncation of Ω0. The adaptive radiation condition simply consists in assuming that u0|Σ and ∂nu0|Σ are given before performing a new iteration, which is obtained by solving a standard transmission problem. The actual difﬁculty, when carrying out these iterations by means of a nodal FEM, is to compute the data ∂nu0|Σ. This can be done through several procedures. The ﬁrst, that has been considered, is to use an involved

25 integration by parts to reduce this term to a volume integral [52]. Another procedure is to use an approximation of the problem in Ω1 by the mixed FEM of Raviart and Thomas (cf., e.g., [20]). This way to proceed does not seem to have been used, but it can easily deduced from the treatment carried out in [48] for Maxwell’s equations with a magnetic field formulation in Ω1 and an electric field one in Ω0. However, mixed FEM are difficult to implement and to solve since they lead to problems of saddle-point type and considerably increase the size of the linear system to be solved. Another procedure, closely related to the introduction of a Lagrange multiplier to en- force the matching condition (6), amounts to performing a discrete Green formula. However, this way to proceed yields an additional computing cost and complicates the effective implementation. The method, we propose, makes use of the above nonoverlapping DDM to obtain ∂nu0|Σ. Indeed, with a suitable adaptation, it is possible to write the boundary condition in (94) in the form

inc ∂nu0 − iku0 =(∂nw − ikw)(u , u0|Σ, −g0 + βu0|Σ) on S. (96)

Now, all of the data, required to perform another iteration, become available when the solution procedure is carried out in terms of a nodal FEM. Furthermore, the FE problems in Ω1 and Ω0 are not solved. Instead, this solution is carried out in terms of the DDM with the special treatment for cross-point described above. Here again, continuity at the cross-points is enforced relatively to FE equations in Ω1 and in Ω0,S. Indeed, the treatment is analogous to the above coupling of a FEM with a BEM, the BEM being replaced by the FEM with an adaptive radiation condition in Ω0,S. In this way, only small size FE linear systems are solved during the iterative process. The numerical implementation of the method for the 3D case is still in progress. We just here validate the method in 2D by dealing with the above example of a hard-sound cylinder of unit diameter surrounded by a cylindrical annular domain of diameter 2 filled with an homogeneous medium. To place the fictitious boundary on which is prescribed the adaptive radiation condition, we choose another surrounding annular cylinder of diameter 3. We choose a decomposition so that only FE linear systems posed on approximately 1/8th of the resulting computational domain are solved at once (see FIG. 7). Two tests have been performed. The first one corresponds to slightly changing the contrast coefficients relatively to the air zone n = 1.1 and χ = 1. The second one deals with an absorbing medium with n =1+0.3i and again χ = 1. The results are slightly less good for the first case, probably due to the well-known dispersive flaws of the FEM with low degree local approximating functions. In FIG. 8, we report a comparison of the results, obtained for the absorbing case after 60 iterations, with the exact solution which, similarly to the above 3D case, can be expanded as a Fourier- Hankel series (see, for example, [62] for a similar calculation).

26 Figure 7: The decomposition of the computational domain in 8 subdomains

Real part of the computed and the exact Real part of the computed and the exact solution on the rigid obstacle solution on the fictitious boundary 0.6 1.5 Computed solution (solid line) 0.4 Exact value (bullet) 1 Exact solution (bullet) 0.2 0.5

−0.2 Computed (solid line) −0.5 −0.4

−1 −0.6

−0.8 −1.5 0 10 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 300

(a) Computed and exact values at the obstacle (b) Computed and Exact values at the ﬁctitious boundary

Figure 8: Exact and Computed values of the total wave at two locations.

27 5 Conclusion

The BIE method is by now extensively used in scientific and engineering studies. As, hopefully, this has been brought out from the above presentation of this method in the context of modeling time-harmonic acoustic waves phenomena, a tremendous progress due to an intensive research activity has been carried out in this field during the last years. Due to the lack of space, several important issues have been only evoked, as, for instance, for the FMM, or completely skipped, as for the matrix compression techniques for BEM equations (cf., e.g., [45, 63] ) or the high-order, high- frequency methods [21]. In the same way, only the FE-BE hybridization technique has been described. Important methods like the coupling of a BEM with an asymptotic high frequency computation have not been considered (see, for instance, [48] where a lot of hybridization techniques are described in the context of electromagnetism but which can be easily adapted to acoustics). Several research issues remain open to improve this kind of solution procedures despite the outstanding progress that has been done. One of the most urgent in our opinion is to design reliable stopping criteria for the iterative processes involved in the solution of such problems. As this can be observed from the numerical experiments given above, several expansive iterations can be performed improving only the solution of the discrete problem but without any impact on the real accuracy of the results. One approach to this problem will be probably to devise some efficient and reliable a posteriori error estimates.

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