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A high order spectral algorithm for elastic obstacle scattering in three dimensions
Frédérique Le Louër
To cite this version:
Frédérique Le Louër. A high order spectral algorithm for elastic obstacle scattering in three dimen-
sions. Journal of Computational Physics, Elsevier, 2014, 279, pp.1-17. �hal-00700779v4�
A high order spectral algorithm for elastic obstacle scattering in three dimensions
Frédérique Le Louër
Abstract
In this paper we describe a high order spectral algorithm for solving the time-harmonic Navier equations in the exterior of a bounded obstacle in three space dimensions, with Dirichlet or Neumann boundary conditions. Our approach is based on combined-field boundary integral equation (CFIE) reformulations of the Navier equations. We extend the spectral method developed by Ganesh and Hawkins [20] - for solving second kind boundary integral equations in electromagnetism - to linear elasticity for solving CFIEs that commonly involve integral operators with a strongly singular or hypersingular kernel. The numerical scheme applies to boundaries which are globally parameterised by spherical coordinates. The algorithm has the interesting feature that it leads to solve linear systems with substantially fewer unknowns than with other existing fast methods. The computational performances of the proposed spectral algorithm are demonstrated using numerical examples for a variety of three-dimensional convex and non-convex smooth obstacles.
Keyword : Navier equations, Dirichlet condition, Neumann condition, boundary integral equations, spectral method
1 Introduction
In this paper we present a high order spectral algorithm for solving the time-harmonic Navier equations in the exterior of a three dimensional bounded obstacle, with a Dirichlet or a Neumann boundary condition. Efficient numerical solution of such scattering problems is of practical interest for various industrial applications related to inverse scattering problems such as non-destructive testing of materials [8].
The method of boundary integral equation (BIE) is a classical tool for solving scattering problems of time- harmonic waves in unbounded, homogeneous and isotropic media. Its efficiency has been widely demonstrated for low and middle frequency waves via boundary collocation methods or boundary element methods (BEM).
However, the boundary integral formulation has the drawback that it reduces the time-harmonic scattering problem to a fully-populated and, in general, non-Hermitian complex linear system. The complexity of direct solvers to invert the scattering matrix isO(N3)in time andO(N2)in memory ifN is the number of degrees of freedom. The complexity of iterative solvers such as GMRES, which iteratively build a sequence of approximate solutions, isO(nit×N2)in time and memory ifnitis the number of iteration. The boundary element mesh size depends on the frequency of the problem and the geometry of the obstacle. Therefore, the solution of realistic problems is limited by the number of degrees of freedom that can be handled by one or the other solver on any machine. In the scientific literature, there exist various methods to decrease the computation time and the memory size: preconditionning and fast methods. Preconditionning methods aim at reducing significantly the number of iteration of the iterative solver applied to a full system-matrix. Fast methods aim at reducing the size of the discretization space. One of them is the so-called fast multipole method (FMM) which speed up the matrix-vector multiplication used by iterative solvers with BEM discretization. Earlier works in acoustics and electronagnetism lead to a complexity of O(Nlog2N) in time and memory per iteration. It has recently been extended to linear elasticity by Bonnet, Chaillat and Semblat [9, 10, 12]. A different kind of compression for the BE matrices can be obtained by applying the adaptive cross approximation (ACA) algorithm to hierarchical matrices [3, 4, 5]. A comparison between these two techniques can be found in [7] in the low- and high-frequency
regimes for the Helmholtz equation. Other fast methods, as spectral methods, accelerate the convergence of the discrete solution so that the BIE can be reduced to linear systems with substantially fewer unknowns than with other BEM-based methods.
It is the purpose of this paper to use high order spectral methods to solve the BIEs in linear elasticity, posed on simply connected closed surfaces. It was first introduced by Ganesh and Graham for acoustic scattering [17]
and consists in transforming the integral equation on the surface Γto an integral equation on the unit sphere using a change of variable and then by expanding the integrand and looking for a scalar solution in terms of series of scalar spherical harmonics. The integral equation is finally reduced to a linear system whose matrix coefficients are the integral equation operator evaluated at each basis functions. Superalgebraic convergence of the discrete solution was proven for smooth obstacles and Fredholm integral equations of the second kind.
This technique has then been extended to electromagnetism by Ganesh and Hawkins for solving the perfect conductor problem [18] by expanding the integrand and looking for a solution in terms of (componentwise) series of scalar spherical harmonics. Spectral accuracy of the algorithm was proven for the magnetic field integral equation (MFIE) only. In [19] they exploited the vector spherical harmonics, but this approach does not yield spectrally accurate approximations : a stagnated convergence is observed (see [18, Remark 2]). However, as being eigenfunctions of integral operators on the sphere, vector spherical harmonics are a natural choice for surface formulation of three dimensional scattering problems. Expanding the integrand in terms of componentwise sum of scalar spherical harmonics, one obtains the hybrid high-order method [20] with a complexity ofO(n5), where nis the largest degree of the spherical harmonics. For smooth obstacles, the spectral convergence rate isO(n−s), for any positive integer s. The order constant in the spectral accuracy depends on the size, the geometry, and the smoothness of the obstacles.
Further efficient algorithms for non-spherical perfect conductors require construction of high-order basis functions that are also tangential on the surface of the non-spherical scatterers. This was achieved by Ganesh and Hawkins in [22], concluding the development of high-order algorithm for computation of tangential surface currents for electromagnetic scattering by a single perfect conductor. For elastic obstacle scattering the unknown field is not required to be tangential and hence in this work, we follow the approach developed in [20].
It is well known that BIEs in linear elasticity involve integral operators with a strongly singular or hypersin- gular kernel. To apply the hybrid spectral method of Ganesh and Hawkins we use a standard technique coming from the finite element theory, that is integration by parts. Indeed, the boundary integral operators with a strongly or hypersingular kernel - that is the double layer boundary integral operator and the Neumann trace of the single and double layer potentials - can be expressed in terms of integral operators with a weakly singular kernel and surface differential operators [25]. Surface derivatives of the vector spherical basis functions can be computed analytically, thus no more approximations is required. By this way, one expects that the spectral accuracy of the hybrid method developed for the MFIE naturally extends to the BIEs of the first kind in linear elasticity.
In the low and medium frequency region, the resulting full matrix system is sufficiently small (few thousands of unknowns) to be solved with a direct solver. To speed up the implementation of the arrays, one can use the Fast Fourier Transform. For higher frequencies and complex geometries (in terms of non-convexity and angularity) an iterative solver is required. Ganesh and Hawkins developed in [21] an algebraic preconditioner specifically for high order spectral algorithms using sparse approximations.
LetΩ⊂R3be a bounded domain with a simply connected closed boundaryΓof classC2at least and outward unit normal vectornand letΩcdenote the exterior domainR3\Ω. Throughout the paper we denote byHlocs (Ωc) andHs(Γ)the standard (local in the case of the exterior domain) complex valued, Hilbertian Sobolev space of orders∈Rdefined onΩc andΓrespectively (with the conventionH0=L2.) Spaces of vector functions will be denoted by boldface letters, thusHs= (Hs)3. We setH1loc(Ωc,∆∗) :=
u∈H1loc(Ωc) : ∆∗u∈L2loc(Ωc) . We use the notation of [2, section 2] for the formulation of the elastic scattering problems and quote some important properties of the boundary integral operators from [15, section 2]. We assume that the Lamé parametersµand λ+µand the density ρ are positive constants. The propagation of time-harmonic elastic waves in the three- dimensional isotropic and homogeneous elastic medium is described by the Navier equation
∆∗u+ρω2u=µ∆u+ (λ+µ)∇divu+ρω2u= 0, (1.1)
where ω >0 is the frequency. In this work we are interested in computing an approximation to the radiating solution and its far field pattern of the following exterior boundary value problems for elastic waves: Given vector densitiesf ∈H12(Γ)andg ∈H−12(Γ), find the solutionu∈H1loc(Ωc,∆∗)to the Navier equation (1.1) inΩc which satisfies either a Dirichlet boundary condition
u=f onΓ (1.2)
or a Neumann boundary condition
P u=g onΓ, (1.3)
where the traction operator is defined byP u= 2µ∂u
∂n+λ divu
n+µn×curlu.In addition the fielduhas to satisfy the Kupradze radiation condition
r→∞lim r ∂up
∂r −iκpup
= 0, lim
r→∞r ∂us
∂r −iκsus
= 0, r=|x|, (1.4)
uniformly in all directions. Here, the longitudinal wave is given byup=−κ−2p ∇divuand the transversal wave is given byus=u−up associated with the respective wave numbersκp andκsgiven byκ2p=ρω2(λ+ 2µ)−1 andκ2s=ρω2µ−1. The radiation condition implies that the solution has an asymptotic behavior of the form
u(x) = eiκp|x|
|x| u∞p (ˆx) +eiκs|x|
|x| u∞s (ˆx) +O 1
|x|
, |x| → ∞, uniformly in all directionsxˆ = x
|x|. The fieldsu∞p andu∞s are defined on the unit sphereS2 inR3 and known as the longitudinal and the transversal far-field pattern, respectively. Setting G(κ,x−y) = eiκ|x−y|
4π|x−y|, the fundamental solution of the Navier equation is given by
Φ(x,y) = 1 µ
G(κs,x−y)IR3+ 1
κ2s∇xT∇x
G(κs,x−y)−G(κp,x−y) .
It is a3×3matrix-valued function and we haveΦ(x,y) =TΦ(x,y) = Φ(y,x). For a solution to the Navier equa- tion (1.1) that satisfies the Kupraze radiation condition, one can derive the Somigliana integral representation formula forx∈Ωc:
u(x) = Z
Γ
T[PyΦ(x,y)]u(y)−Φ(x,y)Pyu(y)
ds(y), (1.5)
where Py =P(n(y), ∂y)and PyΦ(x,y)is the tensor obtained by applying the traction operatorPy to each column ofΦ(x,y). The elastic scattering problems can be reduced, via several different use of (1.5), to a single combined-field boundary integral equation (CFIE) [6, 13, 27]. In Section 2 we review the standard direct and indirect approaches. We also give variant representations of the boundary integral operators that are better suitable for their numerical evaluations. These formulations are obtained with the help of the tangential Günter derivative (see [25, 27] or Lemma 2.1 for a definition). Section 3 and 4 are concerned with the description of a high order spectral algorithm for solving boundary integral equations in linear elasticity. We globally use the same notations and steps as Ganesh and Hawkins in [20]. We start in Section 3 by transporting the boundary integral operators and the surface differential operators over the unit sphere and by splitting the kernels into a weakly singular part and a smooth part. The pull-back of surface differential operators on the unit sphere are evaluated using straightforward equalities that link them to the corresponding differential operators on the unit sphere. Then in Section 4, we summarize the procedure to obtain a fully discrete approximations of the CFIEs. Numerical results in the low and medium frequency region for a variety of three-dimensional convex and non-convex smooth obstacles are presented in Section 5 to show the efficiency of the method. And especially, we compare the fast spectral algorithm (FSM) with the fast multipole method (FMM). The results demonstrate that the FSM still is very competitive with the FMM in linear elaticity as was shown by Ganesh and Graham in [17, page 234] in acoustic scattering. Finally, we draw concluding remarks and we discuss possible research lines in Section 6.
2 The solution of elastic obstacle scattering problems
The scattering problem of time-harmonic elastic waves by a bounded obstacle Ωleads to special cases of the above boundary value problems (1.1)-(1.4) which can be reduced in several different ways to a single combined- field boundary integral equation. We review the standard direct and indirect approaches. For the Dirichlet problem we follow [13, Section 2] and for the Neumann problem we extend acoustic techniques [26, Chapter 5].
Let us start with the rigid body problem: the total displacement fieldu+uincis given by the superposition of the incident fielduinc, which we assume to be an entire solution of the Navier equation, and the scattered field u, which solves the Navier equation inΩc, and satisfies the Dirichlet boundary condition (1.2) withf =−uinc and the Kupradze radiation condition. In this case, the direct approach is based on the Somigliana integral representation formula (1.5) and the following result forx∈Ωc [15, formula (2.6)]
0 = Z
Γ
T[PyΦ(x,y)]uinc(y)−Φ(x,y)Pyuinc(y) ds(y).
Adding the last equation to (1.5) and using the boundary condition of u, we obtain the following integral representation:
u(x) = − Z
Γ
Φ(x,y)Py u(y) +uinc(y)
ds(y), x∈Ωc. (2.1)
Taking the Neumann and Dirichlet traces of the above identity we obtain [15, formulas (2.8)-(2.10)]
SP(u+uinc) = 2uinc and (I +D′)P(u+uinc) = 2P uinc onΓ,
where S is the single layer boundary integral operator and D′ is the traction derivative of the single layer potential, defined by
Sϕ(x) = 2 Z
Γ
Φ(x,y)ϕ(y)ds(y), D′ϕ(x) = 2
Z
Γ
[PxΦ(x,y)]ϕ(y)ds(y).
Setting ϕ=P(u+uinc), the fieldu given by (2.1) solves the Dirichlet boundary value problem (1.1)-(1.2) if the densityϕsolves the following integral equation
ϕ+D′ϕ+iηSϕ= 2 P uinc+iηuinc
onΓ. (2.2)
where η is a non-zero real constant. The operator S : H
1
2(Γ) → H
1
2(Γ) is compact while the operator D : H12(Γ) → H12(Γ) has a strongly singular kernel. It can be shown that the homogenous form of (2.2) only has the solution ϕ= 0, however we cannot use immediately Riesz theory to prove existence of a solution to (2.2). We use a left equivalent regularizer [27, Chapter IV, Section 5] to modify the integral equation as (I +K)ϕ=H 2P uinc+iηuinc
where (I +K) =H(I +D′+iηS), His an integral operator with a strongly singular kernel which is injective andK is an integral operator with a weakly singular kernel. Uniqueness of a solution to the regularized integral equation follows from the uniqueness of a solution to (2.2). We conclude with Riesz theory and the fact that the equation (2.2) and the regularized one have the same solutions.
The far-field pattern can be computed via the integral representation formula u∞=−F
DP(u+uinc), where the far-field operatorF
D is defined by (see [2, equations (2.12) and (2.13)]) FDϕ(ˆx) =
Z
Γ
1
µ[IR3−xˆ⊗x]ˆ e−iκsx·yˆ
4π + 1
λ+ 2µ[ˆx⊗x]ˆ e−iκpx·yˆ 4π
ϕ(y)ds(y). (2.3)
Now we turn to the cavity problem: the total displacement fieldu+uincis given by the superposition of the incident fielduinc, which we assume to be an entire solution of the Navier equation, and the scattered field u, which solves the Navier equation inΩc and satisfies the Neumann boundary condition (1.2) withg=−P uinc and the Kupradze radiation condition. A similar procedure as for the rigid body problem leads to the following integral representation of the solution
u(x) = Z
Γ
T[PyΦ(x,y)] u(y) +uinc(y)
ds(y), x∈Ωc. (2.4)
Taking the Neumann and Dirichlet traces of the above identity we obtain [15, formulas (2.8)-(2.10)]
(D−I)(u+uinc) =−2uinc and N(u+uinc) =−2P uinc onΓ,
where D is the double layer boundary integral operator and N is the traction derivative of the double layer potential, defined by
Dϕ(x)e = 2 Z
Γ
T[PyΦ(x,y)]ϕ(y)e ds(y), Nϕ(x)e = 2
Z
Γ
PxT[PyΦ(x,y)]ϕ(y)e ds(y).
Setting ϕe =u+uinc, the fieldugiven by (2.4) solves the Neumann boundary value problem (1.1)-(1.3) if the densityϕe solves the following integral equation
Nϕe+iηDϕe−iηeϕ=−2 P uinc+iηuinc
onΓ. (2.5)
Here again, existence and uniqueness can be shown by the use of left or right regularization techniques. Indeed, the operatorN :H12(Γ)→H−12(Γ)is bounded and the operatorD′−I, considered in these function spaces, is compact. By the identitiesSN=−I +D2andN S=−I +D′2, the single layer boundary integral operator for the Lamé system (ω= 0) denoted S0 can be considered as a regularizer. Note thatS0:H−12(Γ)→H12(Γ)is invertible [1, Proposition 2.4] andSis a compact perturbation ofS0. Therefore, the boundary integral equation operators became compact perturbations of the operators−I +D2or−I +D′2. Following [16, proof of Lemma 3.9] one can prove that these operators are Fredholm operators of index zero. Hence, we can apply Riesz theory.
The far-field pattern can be computed via the integral representation formula u∞=F
N(u+uinc),
where the far-field operatorFN is defined by (see [2, equations (2.12) and (2.13)]) FNϕ(ˆe x) =
Z
Γ
1 µ
T
Py[IR3−xˆ⊗x]ˆ e−iκsx·yˆ 4π
+ 1
λ+ 2µ
T
Py[ˆx⊗x]ˆ e−iκpx·yˆ 4π
! e
ϕ(y)ds(y).
For both of the Dirichlet and the Neumann boundary value problems, the indirect approach is based on the following ansatz for the integral representation of the scattered elastic field :
u(x) = Z
Γ
T[PyΦ(x,y)]ϕ(y)ds(y) +iη Z
Γ
Φ(x,y)ϕ(y)ds(y), (2.6) whereηis still a non-zero real constant andϕis a density inH
1
2(Γ). Taking the Dirichlet trace to (2.6) together with the boundary condition (1.2) we obtain the following CFIE for the Dirichlet boundary value problem
ϕ+Dϕ+iηSϕ= 2f onΓ. (2.7)
Taking the Neumann trace to (2.6) together with the boundary condition (1.3) we obtain the following CFIE for the Neumann boundary value problem
Nϕ+iηD′ϕ−iηϕ= 2g onΓ. (2.8)
The far field is then given byu∞=F
Nϕ+iηF
Dϕ.
To apply the spectral method of Ganesh and Hawkins [20], for solving one of the above CFIEs, we need to express the operators D, D′ and N in terms of boundary integral operators with a weakly singular kernel and surface differential operators. Such representations can be obtained with the help of the tangential Günter derivative whose properties are given in the following lemma (see [27, Equation (1.13) and Theorem 1.3, pages 282-284]) .
Lemma 2.1 Let Γ be a closed orientable surface in R3. The tangential Günter derivative denoted by M is defined for a vector functionv∈C1(Γ,C3)by
Mv= ∂
∂nv−(divv)n+n×curlv.
For any vector functions v,v˜ inC1(Γ,C3) there holds the Stokes formula Z
Γ
(Mv)·v˜ds= Z
Γ
v·(M˜v)ds. (2.9)
The tangential gradient∇Γ and the surface divergencedivΓ are defined for a scalar functionuand a vector functionv by the following equalities [30]:
∇u=∇Γu+ ∂u
∂nn, divv= divΓv+
n· ∂v
∂n
, and the Günter tangential derivative can then be rewritten as :
Mv= [∇Γv]−(divΓv)·IR3
n, (2.10)
where[∇Γv]is the matrix whose thej-th column is the tangential gradient of thej-th component ofv.
In the following two lemmas we give the integral representation of the operators D,D′ andN that we will consider in the remaining of the paper. We refer to [25, pages 47-49] for more details.
Lemma 2.2 The double layer boundary integral operator and the traction derivative of the single layer potential operator can be rewritten as :
(Dϕ)(x) = 2µ(SMϕ)(x) + 2
− Z
Γ
G(κs,x−y)Myϕ(y)ds(y) + Z
Γ
∂
∂n(y)G(κs,x−y)ϕ(y)ds(y) +
Z
Γ
∇y
G(κp,x−y)−G(κs,x−y)
n(y)·ϕ(y) ds(y)
,
(2.11)
and
(D′ϕ)(x) = 2µ(MSϕ)(x) + 2
− Mx
Z
Γ
G(κs,x−y)ϕ(y)ds(y) + Z
Γ
∂
∂n(x)G(κs,x−y)ϕ(y)ds(y) + n(x)
Z
Γ
ϕ(y)·∇x
G(κp,x−y)−G(κs,x−y) ds(y)
.
(2.12)
Proof. We use integration by part and the properties(i)and (ii)of Lemma 2.1. The procedure to obtain these formulations is already known (we follow [25, page 49] in the case of the stationary Navier equations). We recall the proof since we use this development to prove the next lemma. First of all we rewrite the operatorP as
P u= 2µMu+ (λ+ 2µ)(divu)n−µn×curlu. (2.13) Then we apply the operatorPy in the form (2.13) to the tensorΦ(x,y). It follows
T PyΦ(x,y)
= 2µT
MyΦ(x,y)
−T
n(y)×curlyG(κs,x−y)IR3
+(λ+ 2µ) µ
T[n(y)⊗divyΦ(x,y)],
divyΦ(x,y) =T
∇yG(κs,x−y) + 1
κ2s
T∇y∆y G(κs,x−y)−G(κp,x−y)
= κ2p κ2s
T
∇yG(κp,x−y) , n(y)×curlyG(κs,x−y)IR3 =
My− ∂
∂n(y)+n(y)·divy
G(κs,x−y)IR3. In virtue of the property(i)in Lemma 2.1 we can write
T
n(y)×curlyG(κs,x−y)IR3
=
−My− ∂
∂n(y)
G(κs,x−y)IR3+Th
n(y)⊗T∇yG(κs,x−y)i . Collecting the equalities we obtain
T
PyΦ(x,y)
= 2µT
MyΦ(κ,x−y) +
∂
∂n(y)+My
G(κs,x−y)IR3
+∇y
G(κp,x−y)−G(κs,x−y)
⊗n(y).
To obtain (2.11) we use the equalities (2.9) and to obtain (2.12) we take the adjoint form of (2.11).
Lemma 2.3 The traction derivative of the double layer potential operator can be rewritten as follows:
N = 2µD′M+ 2µM(D−2µSM) +V with
Vϕ(x) =ρω2 Z
Γ
n(x)G(κp,x−y) n(y)·ϕ(y) ds(y) +ρω2n(x)×
Z
Γ
G(κs,x−y)(ϕ(y)×n(y))ds(y)
−µcurlΓ Z
Γ
G(κs,x−y) curlΓϕ(y)ds(y), x∈Γ,
(2.14)
where we have curlΓ=−n×∇Γ andcurlΓ= divΓ(· ×n).
Proof. First, we writeD= 2µ SM+ (D−2µ SM). Applying the traction trace to the kernel of2µ SMwe obtain2µ D′M. Then, we apply the operatorPx, in the form (2.13), to the kernel of(D−2µ SM). Following the calculation of the previous proof, one can see that we have
(D−2µ SM)ϕ(x) =− Z
Γ
T[n(y)×curly{G(κs,x−y)I3}]ϕ(y)ds(y) + Z
Γ
∇yG(κp,x−y) n(y)·ϕ(y) ds(y).
We observe that
T[n(y)×curly{G(κs,x−y)I3}]ϕ(y) = T[curly{G(κs,x−y)I3}](ϕ(y)×n(y))
= −[curly{G(κs,x−y)I3}] (ϕ(y)×n(y))
= [curlx{G(κs,x−y)I3}](ϕ(y)×n(y))
= curlx{G(κs,x−y)(ϕ(y)×n(y))}.
The composition of the operator(λ+ 2µ)n(x) divx with this term vanishes since one hasdivcurl= 0and the composition withµn(x)×curlxyields
µn(x)×
(−∆x+∇xdivx){G(κs,x−y)(ϕ(y)×n(y))}
=µκ2sn(x)× {G(κs,x−y)(ϕ(y)×n(y))} −µcurlxΓdivx{G(κs,x−y)(ϕ(y)×n(y))}
=ρω2n(x)× {G(κs,x−y)(ϕ(y)×n(y))} −µcurlx
Γ{∇xG(κs,x−y)·(ϕ(y)×n(y))}
=ρω2n(x)× {G(κs,x−y)(ϕ(y)×n(y))} −µcurlxΓ{−∇yG(κs,x−y)·(ϕ(y)×n(y))}
=ρω2n(x)× {G(κs,x−y)(ϕ(y)×n(y))} −µcurlxΓ{−∇y
ΓG(κs,x−y)·(ϕ(y)×n(y))},
(2.15)
sincecurlxcurlx=−∆x+∇xdivxand ∇Γ·=n×(∇· ×n). In view of the following integral part formula for a scalar densityϕ1 and a tangential vector densityϕ2
− Z
Γ
(∇Γϕ1)·ϕ2ds= + Z
Γ
ϕ1divΓϕ2ds,
and the identitydivΓ(ϕ2×n) = curlΓϕ2, we obtain the last two terms of the right hand side in (2.14).
We observe that∇yG(κp,x−y) n(y)·ϕ(y)
=−∇xG(κp,x−y) n(y)·ϕ(y)
.The composition of the operator µn(x)×curlxwith this term vanishes since one has curl∇= 0and the composition with(λ+ 2µ)n(x) divx
yields
−(λ+ 2µ)n(x) divx∇xG(κp,x−y) n(y)·ϕ(y)
=−(λ+ 2µ)n(x)∆xG(κp,x−y) n(y)·ϕ(y)
=ρω2n(x)G(κp,x−y) n(y)·ϕ(y) .
We conclude by collecting all these formulas.
3 Spherical parametrization of the integral equations
The spectrally accurate algorithm is based on a spherical reformulation of the integral equations. We denote byS2 the unit sphere of R3. Letq : S2 →Γ be a parametrization of class C1 at least. We denote byτq the transformation ("pull-back") that maps a functionudefined on Γ onto the functionτq(u) =u◦q defined on S2. LetJq be the determinant of the Jacobian of the change of variablexˆ ∈S27→q(ˆx)∈Γ. We denote byθ, φ the spherical coordinates of any pointxˆ ∈S2, i.e.
ˆ
x=ψ(θ, φ) =
sinθcosφ sinθsinφ
cosθ
, (θ, φ)∈]0;π[×[0; 2π[∪{(0,0); (0, π)}.
The tangent and the cotangent planes at any pointxˆ =ψ(θ, φ)∈S2 are generated by the unit vectors :
eθ=∂ψ
∂θ(θ, φ) =
cosθcosφ cosθsinφ
−sinθ
andeφ= 1 sinθ
∂ψ
∂φ(θ, φ) =
−sinφ cosφ
0
.
The triplet (ˆx,eθ,eφ) forms a direct orthonormal system. The determinant of the Jacobian of the change of variable isJψ(θ, φ) = sinθ.
The matrix[DS2q(ˆx)] =T[∇S2q(ˆx)]maps the tangent planeTxˆ toS2at the pointxˆ onto the tangent plane Tq(ˆx) toΓat the pointq(ˆx). The latter is generated by the vectors
t1(ˆx) =∂q◦ψ
∂θ ◦ψ−1 and t2(ˆx) = 1
sinθ
∂q◦ψ
∂φ
◦ψ−1. (3.1)
We can write[DS2q(ˆx)] =t1⊗eθ+t2⊗eφ, where for two vectorsa, b∈R3 we seta⊗b=aTb.
The functions Jq andτq(n)can be computed via the formulas : Jq =t1×t2
andτq(n) = (t1×t2)/Jq . The parametrization q : S2 → Γ being a diffeomorphism, we set [DS2q(ˆx)]−1 = [DΓq−1]◦q(ˆx). The transposed matrixT[DS2q(ˆx)]−1 maps the cotangent planeT∗xˆ to S2 at the point xˆ onto the cotangent plane T∗q(ˆx) toΓat the pointq(ˆx). The latter is generated by the vectors
t1(ˆx) = t2(ˆx)×n(q(ˆx))
Jq(ˆx) andt2(ˆx) = n(q(ˆx))×t1(ˆx)
Jq(ˆx) . (3.2)
We can write[DS2q(ˆx)]−1=eθ⊗t1+eφ⊗t2.
In addition to the formulas [23, Eq. (3.2)] we will use, for non-tangential vector functionsv,w∈C1(Γ,C3), the following identities
τq(divΓv) = [DS2q]−1: [∇S2τq(v)], (3.3) τq(curlΓw) = −1
Jq
T[DS2q] :
curlS2τq(w)
, (3.4)
where for two(3×3)matricesAandBwhose columns are denoted by(a1, a2, a3)and(b1, b2, b3), respectively, we setA:B=a1·b1+a2·b2+a3·b3. These equalities are an extension of the formulas stated in [23, Equation (3.2) and Appendix A] by using the well known indentitiesdivΓv =T race([∇Γv])andcurlΓw=−T race([curlΓw]).
Spherical reformulation The key of the spectral method is to transport the boundary integral equation on the unit sphere. To do so, following [17, 18, 20], we use a change of variable in the integrand and we consider instead the following integral operators:
Sϕs=Jqτq
Sτq−1(ϕs)
= 2Jq
Z
S2
nΦ(q(·),q(ˆy)) ϕs(ˆy)o
Jq(ˆy)ds(ˆy), Dϕs=Jqτq
Dτq−1(ϕs)
= 2µSMqϕs+ 2Jq
− Z
S2
n
G κs,q(·)−q(ˆy)
Mq(ˆy)ϕs(ˆy)o
Jq(ˆy)ds(ˆy) +
Z
S2
∂
∂n(q(ˆy))G(κs,q(·)−q(ˆy))ϕs(ˆy)
Jq(ˆy)ds(ˆy) +
Z
S2
n
∇q(ˆy)
G(κp,q(·)−q(ˆy))−G(κs,q(·)−q(ˆy))
n(q(ˆy))·ϕs(ˆy)o
Jq(ˆy)ds(ˆy)
, D′ϕs=Jqτq
D′τq−1(ϕs)
= 2µMqSϕs+ 2Jq
− Mq
Z
S2
nG κs,q(·)−q(ˆy) ϕs(ˆy)o
Jq(ˆy)ds(ˆy) +
Z
S2
∂
∂n(q(·))G(κs,q(·)−q(ˆy))ϕs(ˆy)
Jq(ˆy)ds(ˆy) +n(q(·))
Z
S2
n
ϕs(q(ˆy))·∇q(·)
G(κp,q(·)−q(ˆy))−G(κs,q(·)−q(ˆy))o
Jq(ˆy)ds(ˆy)
, and Nϕs=Jqτq N τq−1 ϕs)
= 2µD′Mqϕs+ 2µMq(D −2µSMq)ϕs+Vϕswith Vϕs=Jqτq N2µτq−1 ϕs)
= 2Jqρω2
n(q(·)) Z
S2
G(κp,q(·)−q(ˆy))(n(q(ˆy))·ϕs(ˆy))Jq(ˆy)ds(ˆy) +n(q(·))×
Z
S2
G κs,q(·)−q(ˆy)
ϕs(ˆy)×n(q(ˆy))
Jq(ˆy)ds(ˆy)
+µ[DS2q]curlS2
Z
S2
n
G(κs,q(·)−q(ˆy)) T[DS2q(ˆy)] :
curlS2ϕs(ˆy)]o ds(ˆy), where
Mq(ˆy)ϕs(ˆy) =
T[DS2q(ˆy)]−1[∇S2ϕs(ˆy)]− [DS2q(ˆy)]−1: [∇S2ϕs(ˆy)]
IR3
n(q(ˆy)). (3.5)
Splitting of the kernels Following details in [17, 18, 20], to implement the integral operatorsS,D,D′ and N we split their kernels into a smooth and a weakly singular part. We introduce the functions :
G1(q;κ,x,ˆ y) =ˆ 1
2πcos(κ|q(ˆx)−q(ˆy)|), G2(q;κ,x,ˆ y) =ˆ 1
2π
sin(κ|q(ˆx)−q(ˆy)|)
|q(ˆx)−q(ˆy)| ifxˆ 6= ˆy,
κ ifxˆ = ˆy.
and
R(q; ˆx,y) =ˆ |ˆx−y|ˆ
|q(ˆx)−q(ˆy)|. ThenS can be rewritten as:
Sϕs(ˆx) = Z
S2
R(q; ˆx,y)ˆ
|ˆx−y|ˆ W1(q; ˆx,y)ϕˆ s(ˆy)ds(ˆy) +i Z
S2
W2(q; ˆx,y)ϕˆ s(ˆy)ds(ˆy) whereW1(q; ˆx,y)ˆ and W2(q; ˆx,y)ˆ are3×3 matrix-valued functions given by
W1(q; ˆx,y)ˆ = Jq(ˆx)Jq(ˆy) µ
G1(q;κs,x,ˆ y)ˆ − 1 κ2s
G1(q;κs,x,ˆ y)ˆ − G1(q;κp,x,ˆ y)ˆ
|q(ˆx)−q(ˆy)|2
IR3
−Jq(ˆx)Jq(ˆy) µκ2s
κsG2(q;κs,x,ˆ ˆy)−κpG2(q;κp,x,ˆ y)ˆ IR3
−Jq(ˆx)Jq(ˆy) µκ2s
κ2sG1(q;κs,x,ˆ y)ˆ −κ2pG1(q;κp,x,ˆ y)ˆ
|q(ˆx)−q(ˆy)|2 A(q,x,ˆ y)ˆ +3Jq(ˆx)Jq(ˆy)
µκ2s
G1(q;κs,x,ˆ ˆy)− G1(q;κp,x,ˆ y)ˆ
|q(ˆx)−q(ˆy)|4 A(q,x,ˆ y)ˆ +3Jq(ˆx)Jq(ˆy)
µκ2s
κsG2(q;κs,x,ˆ ˆy)−κpG2(q;κp,x,ˆ y)ˆ
|q(ˆx)−q(ˆy)|2 A(q,x,ˆ y),ˆ W2(q; ˆx,y)ˆ = Jq(ˆx)Jq(ˆy)
µ
G1(q;κs,x,ˆ y)ˆ − 1 κ2s
G2(q;κs,x,ˆ y)ˆ − G2(q;κp,x,ˆ y)ˆ
|q(ˆx)−q(ˆy)|2
IR3 +Jq(ˆx)Jq(ˆy)
µκ2s
κsG1(q;κs,x,ˆ y)ˆ −κpG1(q;κp,x,ˆ y)ˆ
|q(ˆx)−q(ˆy)|2 IR3
−3Jq(ˆx)Jq(ˆy) µκ2s
κsG1(q;κs,x,ˆ ˆy)−κpG1(q;κp,x,ˆ y)ˆ
|q(ˆx)−q(ˆy)|4 A(q; ˆx,y)ˆ
−Jq(ˆx)Jq(ˆy) µκ2s
κ2sG2(q;κs,x,ˆ y)ˆ −κ2pG2(q;κp,x,ˆ y)ˆ
|q(ˆx)−q(ˆy)|2 A(q; ˆx,y)ˆ +3Jq(ˆx)Jq(ˆy)
µκ2s
G2(q;κs,x,ˆ ˆy)− G2(q;κp,x,ˆ y)ˆ
|q(ˆx)−q(ˆy)|4 A(q; ˆx,y),ˆ with
A(q; ˆx,y) =ˆ q(ˆx)−q(ˆy)
⊗ q(ˆx)−q(ˆy) . The operatorDcan be rewritten as:
Dϕs(ˆx) = Z
S2
R(q; ˆx,ˆy)
|ˆx−y|ˆ
2µW1(q; ˆx,y)ˆ − G1(q;κs,x,ˆ y)Jˆ q(ˆx)Jq(ˆy)IR3
Mq(ˆy)ϕs(ˆy)ds(ˆy) + i
Z
S2
2µW2(q; ˆx,y)ˆ − G2(q;κs,x,ˆ y)Jˆ q(ˆx)Jq(ˆy)IR3
Mq(ˆy)ϕs(ˆy)ds(ˆy)
+ Z
S2
R(q; ˆx,y)ˆ
|xˆ−y|ˆ
D1(q; ˆx,y) +ˆ K1(q; ˆx,y) Iˆ R3
ϕs(ˆy)ds(ˆy) +i
Z
S2
D2(q; ˆx,y) +ˆ K2(q; ˆx,y) Iˆ R3
ϕs(ˆy)ds(ˆy)
whereK1(q; ˆx,y)ˆ andK2(q; ˆx,y)ˆ are scalar-valued functions given by K1(q; ˆx,y) =ˆ Jq(ˆx)Jq(ˆy)
G1(q;κs,x,ˆ y)ˆ
|q(ˆx)−q(ˆy)|2 +κsG2(q;κs,x,ˆ y)ˆ
n(q(ˆy))· q(ˆx)−q(ˆy) K2(q; ˆx,y) =ˆ Jq(ˆx)Jq(ˆy)G2(q;κs,x,ˆ y)ˆ −κsG1(q;κs,x,ˆ y)ˆ
|q(ˆx)−q(ˆy)|2 n(q(ˆy))· q(ˆx)−q(ˆy) andD1(q; ˆx,y)ˆ andD2(q; ˆx,y)ˆ are3×3matrix-valued functions given by
D1(q; ˆx,y) =ˆ Jq(ˆx)Jq(ˆy)G1(q;κp,x,ˆ y)ˆ − G1(q;κs,x,ˆ y)ˆ
|q(ˆx)−q(ˆy)|2 q(ˆx)−q(ˆy)
⊗n(q(ˆy)) +Jq(ˆx)Jq(ˆy)
κpG2(q;κp,x,ˆ y)ˆ −κsG2(q;κs,x,ˆ y)ˆ
q(ˆx)−q(ˆy)
⊗n(q(ˆy)), D2(q; ˆx,y) =ˆ Jq(ˆx)Jq(ˆy)G2(q;κp,x,ˆ y)ˆ − G2(q;κs,x,ˆ y)ˆ
|q(ˆx)−q(ˆy)|2 q(ˆx)−q(ˆy)
⊗n(q(ˆy))
−Jq(ˆx)Jq(ˆy)κpG1(q;κp,x,ˆ y)ˆ −κsG1(q;κs,x,ˆ y)ˆ
|q(ˆx)−q(ˆy)|2 q(ˆx)−q(ˆy)
⊗n(q(ˆy)).
The operatorD′ can be rewritten as:
D′ϕs(ˆx) =Mq(ˆx)
Z
S2
R(q; ˆx,y)ˆ
|xˆ−y|ˆ
2µW1(q; ˆx,y)ˆ − G1(q;κs,x,ˆ y)Jˆ q(ˆx)Jq(ˆy)IR3
ϕs(ˆy)ds(ˆy) +iMq(ˆx)
Z
S2
2µW2(q; ˆx,y)ˆ − G2(q;κs,x,ˆ y)Jˆ q(ˆx)Jq(ˆy)IR3
ϕs(ˆy)ds(ˆy) +
Z
S2
R(q; ˆx,y)ˆ
|xˆ−y|ˆ
D1′(q; ˆx,y) +ˆ K′1(q; ˆx,y) Iˆ R3
ϕs(ˆy)ds(ˆy) +i
Z
S2
D2′(q; ˆx,y) +ˆ K2′(q; ˆx,y) Iˆ R3
ϕs(ˆy)ds(ˆy) whereK′1(q; ˆx,y)ˆ andK′2(q; ˆx,y)ˆ are scalar-valued functions defined by
K1′(q; ˆx,y) =ˆ K1(q; ˆy,x),ˆ K′2(q; ˆx,y) =ˆ K2(q; ˆy,x).ˆ andD′1(q; ˆx,y)ˆ andD′2(q; ˆx,y)ˆ are3×3matrix-valued functions defined by
D′1(q; ˆx,y) =ˆ TD1(q; ˆy,x),ˆ D′2(q; ˆx,y) =ˆ TD2(q; ˆy,x).ˆ The operatorV can be rewritten as:
Vϕs(ˆx) = Z
S2
R(q; ˆx,y)ˆ
|ˆx−y|ˆ V1(q; ˆx,y)ϕˆ s(ˆy)ds(ˆy) +i Z
S2
V2(q; ˆx,y)ϕˆ s(ˆy)ds(ˆy) + µ[DS2q(ˆx)]curlS2
Z
S2
R(q; ˆx,y)ˆ
|xˆ−y|ˆ G1(q;κs,x,ˆ y)ˆ T[DS2q(ˆy)] : [curlS2ϕs(ˆy)]
ds(ˆy) + iµ[DS2q(ˆx)]curlS2
Z
S2
G2(q;κs,x,ˆ y)ˆ T[DS2q(ˆy)] : [curlS2ϕs(ˆy)]
ds(ˆy), where V1(q; ˆx,y)ˆ andV2(q; ˆx,y)ˆ are3×3matrix-valued functions defined by
V1(q; ˆx,y) =ˆ ρω2
G1(q;κp,x,ˆ y)Jˆ q(ˆx)Jq(ˆy)n(q(ˆx))⊗n(q(ˆy)) +G1(q;κs,x,ˆ y)ˆ
Jq(ˆx)Jq(ˆy)
n(q(ˆx))·n(q(ˆy))
IR3−n(q(ˆy))⊗n(q(ˆx)) ,
V2(q; ˆx,y) =ˆ ρω2
G2(q;κp,x,ˆ y)Jˆ q(ˆx)Jq(ˆy)n(q(ˆx))⊗n(q(ˆy)) + G2(q;κs,x,ˆ y)ˆ Jq(ˆx)Jq(ˆy)
n(q(ˆx))·n(q(ˆy))
IR3−n(q(ˆy))⊗n(q(ˆx)) . Finally, settingϕs=P(u+uinc)◦qand2fs=Jq(P uinc+iηuinc)◦q, the parametrized form of equation
(2.2) is
JqI
H
1
2(S2)+D′+iηS
ϕs=fs, onS2, (3.6)
and settingϕfs= (u+uinc)◦q andgs =−2Jq(P uinc+iηuinc)◦q the parametrized form of equation (2.5) is N +iηD −iη JqI
H
1 2(S2)
ϕfs=gs, onS2, (3.7)
The parametrized forms of the equations (2.7) and (2.8) are, respectively, the adjoint forms of the equations (3.6) and (3.7).
4 The high-order spectral algorithm
The spectral algorithm is based on that in [20]. In a first step, we deal with the singularities of the weakly singular parts of the integrands by introducing a change of coordinate system on S2. It will yield transformed operators with kernels which are singular only at one point on the sphere, namely the north pole. In a second step, we interpolate the integrands and project the integral equations on vector spherical function basis.
Treatment of the singularities For any xˆ ∈ S2, letTxˆ be the orthogonal transformation defined in [17, Subsection 3.1.1] that satisfies Txˆxˆ =T(0,0,1) = ˆη. For any yˆ ∈ S2, we setzˆ =Txˆy. We also introduce anˆ induced linear tranformation Txˆ defined byTxˆu(ˆz) =u(Txˆ−1z) =ˆ u(ˆy) and we still denote byTˆx its bivariate analogueTxˆv(ˆz1,ˆz2) =v(Txˆ−1ˆz1, Tˆx−1ˆz2). The boundary integral operator S can be rewritten in the form:
Sϕs(ˆx) = Z
S2
TˆxR(q; ˆη,z)ˆ
|ˆη−z|ˆ TxˆW1(q; ˆη,ˆz)Txˆϕs(ˆz)ds(ˆz) + i Z
S2
TxˆW2(q; ˆη,z)Tˆ xˆϕs(ˆz)ds(ˆz), (4.1) and it can be shown that (θ′, φ′)7→ TxˆR(q; ˆη,z(θˆ ′, φ′))TxˆW1(q; ˆη,z(θˆ ′, φ′))is smooth. An important point is that the singularity
1
|ηˆ−z(θˆ ′, φ′)| = 1 2 sinθ2′
is cancelled out by the surface element ds(ˆz) = sinθ′dθ′dφ′. We proceed in the same way for the integrand of the operatorsD,D′ andV.
Fully discrete approximations Let On and Ln be respectively the operators defined in [20, Equations (3.10) and (3.12)]. For practical purposes, we choose n′ = 2n+ 1 and n ≥ 5. (This satisfies the theoretical constraints required onnandn′ for convergence analysis based on that in [17, 18, 20].) The codomainHnof the fully discrete operatorOnis the3(n+ 1)2−2finite dimensional space spanned by orthonormal vector spherical harmonics of degree at mostn.
Following details in [20], the spectral algorithm for the rigid body problem is: find ϕn∈Hn such that Onϕn+OnD′n′ϕn+iηOnSn′ϕn= 2Onfs, (4.2) where, as in [20],Sn′ andDn′′ are defined as
Sn′ϕs(ˆx) = Z
S2
1
|ˆη−z|ˆ Ln′{TxˆR(q; ˆη,·)TxˆW1(q; ˆη,·)Txˆϕs(·)}(ˆz)ds(ˆz) + i
Z
S2
Ln′{TxˆW2(q; ˆη,·)Txˆϕs(·)}(ˆz)ds(ˆz),
and
D′n′ϕs(ˆx) = Z
S2
1
|ηˆ−z|ˆ Ln′
nTxˆR(q; ˆη,·)Txˆ
D′1(q; ˆη,·) +K′1(q; ˆη,·) IR3
Txˆϕs(·)o
(ˆz)ds(ˆz) +i
Z
S2
Ln′
nTxˆ
D′2(q; ˆη,·) +K′2(q; ˆη,·) IR3
Txˆϕs(·)o
(ˆz)ds(ˆz) +Mq(ˆx)
Z
S2
1
|ˆη−ˆz|Ln′
nTˆxR(q; ˆη,·)Txˆ
2µW1(q; ˆη,·)− G1(q;κs,η,ˆ ·)Jq(·)IR3
Txˆϕs(·)o
(ˆz)ds(ˆz) + iMq(ˆx)
Z
S2
Ln′
nTxˆ
2µW2(q; ˆη,·)− G2(q;κs,η,ˆ ·)Jq(·)IR3
Txˆϕs(·)o
(ˆz)ds(ˆz).
The spectral algorithm for the cavity problem is: find ϕn∈Hn such that
OnNn′ϕen+iηOnDn′ϕen−iηOnϕen = 2Ongs, (4.3) where
Dn′ϕfs(ˆx) = Z
S2
1
|ˆη−z|ˆ Ln′
nTxˆR(q; ˆη,·)Tˆx
D1(q; ˆη,·) +K1(q; ˆη,·) IR3
Tˆxfϕs(·)o
(ˆz)ds(ˆz) +i
Z
S2
Ln′
n Tˆx
D2(q; ˆη,·) +K2(q; ˆη,·) IR3
Tˆxfϕs(·)o
(ˆz)ds(ˆz) +
Z
S2
1
|ˆη−z|ˆ Ln′
n
TxˆR(q; ˆη,·)Txˆ
2µW1(q; ˆη,·)− G1(q;κs,η,ˆ ·)Jq(·)IR3
Txˆ(Mqϕfs)(·)o
(ˆz)ds(ˆz) +i
Z
S2
Ln′
nTˆx
2µW2(q; ˆη,·)− G2(q;κs,η,ˆ ·)Jq(·)IR3
Tˆx(Mqfϕs)(·)o
(ˆz)ds(ˆz),
andNn′ = 2µ(D′M)n′ + 2µM(D−2µSM)n′ +Vn′ denotes the corresponding approximation of the operator N with
Vn′fϕs(ˆx) = Z
S2
1
|ˆη−z|ˆ Ln′
TxˆR(q; ˆη,·)TxˆV1(q; ˆη,·)Txˆϕfs(·) (ˆz)ds(ˆz)
+i Z
S2
Ln′
TxˆV2(q; ˆη,·)Txˆϕfs(·) (ˆz)ds(ˆz)
+µ[DS2q(ˆx)]curlS2
Z
S2
1
|ˆη−z|ˆ Ln′
nTxˆR(q; ˆη,·)TxˆG1(q;κs,η,ˆ ·)Txˆ T[DS2q] : [curlS2fϕs] (·)o
(ˆz)ds(ˆz) +iµ[DS2q(ˆx)]curlS2
Z
S2
Ln′
nTxˆG2(q;κs,η,ˆ ·)Txˆ T[DS2q] : [curlS2ϕfs] (·)o
(ˆz)ds(ˆz).
As in [20], the above discrete operators can be written using summation form using the eigenfunction prop- erties of the spherical harmonics [14]. For details of convergence analysis of the method developped originally for electromagnetic scattering, we refer to [17, 20] where such bounds are established.
Remark 4.1 Contrary to the acoustic and the electromagnetic cases, the algorithm additionnally requires the evaluation of the tangential Günter derivative Mq (3.5) and the tangential vector curl (3.4) of both the approximate densities ϕn in the integrands (i.e. with respect to the variable y) and some densities resultingˆ from surface integrals (i.e. with respect to the variablex).ˆ
(i) A direct numerical evaluation of the tangential derivatives of ϕn in the integrands require, before all, the analytical computations of the Jacobian matrix of vector spherical harmonics and of the tangent and cotangent vectors (3.1)-(3.2) .
(ii) A numerical evaluation of the tangential derivatives with respect toxˆ is obtained through the application of the discrete projection operatorOn and the integration by parts (2.9) and
Z
Γ
curlΓu·v˜ds= Z
Γ
ucurlΓv˜ds, put in an other way
Z
S2
Mq(ˆx)v(ˆx)
·v(ˆ˜ x)Jq(ˆx)ds(ˆx) = Z
S2
v(ˆx)· Mq(ˆx)˜v(ˆx)
Jq(ˆx)ds(ˆx) Z
S2
[DS2q(ˆx)]curlS2u(ˆx)
·v(ˆ˜ x)ds(ˆx) =− Z
S2
u(ˆx) T[DS2q(ˆx)] : [curlS2v(ˆ˜ x)]
ds(ˆx).
Thus, we are lead to the evaluation of the tangential derivatives of the spherical vector harmonics. These integration by parts formulas justify why we multiply the boundary integral equations by Jq .
5 Numerical experiments
In the remaining of the paper we present numerical experiments for low and medium frequencies to highlight the fast convergence of our algorithm through the evaluation of the far-field patternu∞ and the boundary traces of the scattered field. We use the various smooth and non-smooth, convex and non convex obstacles whose the parametric representations and visualizations are given in [17]. The surfaces are characterized by their diameter denotedsize obj. The convex shapes are the sphere, denoted bysphere(size obj) and the ellipsoid with radius a,b andc denoted byellipsoid(a,b, c). The non convex shapes are the bean denoted bybean(size obj) and the peanut denoted bypea(size obj,α).
As a first test, using the CFIEs based on the indirect approach, we compute the far field, denoted by u∞ps, created by an off center point source located inside the elastic obstacle :
uinc(x) =−[Φ(x,s)]p, s∈Ωandp∈S2.
In this case the total exterior wave has to vanish so that the far-field pattern of the scattered waveus is the opposite of the far field pattern of the incident wave. The following far field representation is obtain by applying the kernel of the far field operatorFD to p(see (2.3)).
u∞exact(ˆx) = 1 µ
e−iκsx·sˆ
4π (ˆx×p)×xˆ+ 1 λ+ 2µ
e−iκpx·sˆ
4π (ˆx·p) ˆx.
We chooses=T(0,0.05,0.0866)andp=T(1,0,0). In the tabulated results we indicate the uniform-norm error (by taking the maximum of errors obtained over 1300 observed directions, i.en∞= 25) :
||[u∞ps]n−u∞exact||∞= max
ˆ x∈S2
[u∞ps]n−u∞exact.
As a second test, using CFIEs based on the direct approach, we compute the far field, denoted by u∞pw, created by the scattering of an incident plane elastic wave [2, Section 3] or [24, section 3]:
uinc(x) = 1
µeiκsx·d(d×p)×d+ 1
λ+ 2µeiκpx·d(d·p)d, whered,p∈S2. (5.1) When the polarization p is orthogonal to the propagation vector d, the incident plane waves oscillate in a direction orthogonal to the direction of propagation. They are called shearing waves. Whenp=d, the incident plane waves oscillate along the direction of propagation. They are called pressure waves. In the tabulated results (except for the sphere) we indicate the real part and the imaginary part of the polarization component of the far field evaluated at the incident direction : [u∞pw(d)]n·p. In the case of the sphere we know the analytical
