Generalized Impedance Boundary Conditions and Shape Derivatives for 3D Helmholtz Problems

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Generalized Impedance Boundary Conditions and Shape Derivatives for 3D Helmholtz Problems

Djalil Kateb, Frédérique Le Louër

To cite this version:

Djalil Kateb, Frédérique Le Louër. Generalized Impedance Boundary Conditions and Shape Deriva-

tives for 3D Helmholtz Problems. Mathematical Models and Methods in Applied Sciences, World

Scientific Publishing, 2016, �10.1142/S0218202516500500�. �hal-01267024v2�

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Generalized Impedance Boundary Conditions and Shape Derivatives for 3D Helmholtz Problems

Djalil Kateb

^∗1

and Frédérique Le Louër

^†1

1Sorbonne Universités, Université de technologie de Compiègne, LMAC EA2222 Laboratoire de Mathématiques Appliquées de Compiègne - CS 60 319 - 60 203 Compiègne cedex, France

Abstract

This paper is concerned with the shape sensitivity analysis of the solution to the Helmholtz transmission problem for three dimensional sound-soft or sound-hard obstacles coated by a thin layer. This problem can be asymptotically approached by exterior problems with an improved condition on the exterior boundary of the coated obstacle, called Generalised Impedance Boundary Condition (GIBC). Using a series expansion of the Laplacian operator in the neighborhood of the exterior boundary, we retrieve the first order GIBCs characterizing the presence of an interior thin layer with either a constant or a variable thickness. The first shape derivative of the solution to the original Helmholtz transmission problem solves a new thin layer transmission problem with non vanishing jumps across the exterior and the interior boundary of the thin layer. In the special case of thin layers with a constant thickness, we show that we can interchange the first order differentiation with respect to the shape of the exterior boundary and the asymptotic approximation of the solution. Numerical experiments are presented to highlight the various theoretical results.

1 Introduction

This paper is devoted to the shape sensitivity analysis of the solution to time-harmonic acoustic scattering problems in the special case where the scattering object is a three-dimensional sound-soft or a sound-hard obstacle coated by a thin layer whose width ε tends to zero. It is well known that the use of boundary and finite elements methods for solving this scattering problems fail since some numerical instabilities arise.

Indeed, we face two kind of scalings : a big scale for the exterior of the obstacle and a very small one which corresponds to the thin layer. To avoid the phenomenon, we are led to approximate the original model by a new exterior boundary value problem with high order boundary conditions in terms of surface derivatives, called generalized impedance boundary conditions (GIBC). The exact solution is given through an asymptotic expansion in terms of the thickness parameterεwhere each coefficient function is the solution of a boundary value problem set on a geometry independent onε. In practice, we are only interested by a finite number of terms in the asymptotic expansion. The GIBC satisfied by the approximate solution leads to an error estimate up toO(ε^N+1), whereN is the order of truncation in the asymptotic expansion of the exact solution. These conditions have been first derived by Bendali and Lemrabet in [4] in the case of thin layer with a constant thickness and more recently they were generalised to the 2D case of thin layer with a variable thickness in [3].

The work finds its motivation in the recent study of inverse scattering problems (see [7, 8, 9, 11]) or shape optimization problems (see [15]). The authors take the approximation of order 1 of the original problem and present a theoretical analysis based on the shape derivative of the approximate solution. Our natural question is the following : what happens if we compute first the shape derivative of the original problem (with the coated context) and then take the corresponding GIBC of order1. The purpose of the paper is to give here a general result about the norm of the difference of the shape derivatives for an approximation of orderN. We show that the error is up toO(ε^N+1).

Let consider a simply connected bounded domain Ω in R³, with a closed orientable boundary Γ, as smooth as we need, representing a sound-soft or a sound-hard scatterrerΩ^ε coated by a thin layer denoted

∗djalil.kateb@utc.fr

†frederique.le-louer@utc.fr

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Ω^ε_int. Letε >0and hbe a positive smooth function defined onΓ. The thin layer with variable thickness surrounding the acoustic object is defined by:

Ω^ε_int={x+δh(x)n(x)|x∈Γand −ε < δ <0}.

Ω^εint

Ωext

Γ Γ^ε κi

κe

n nε εh

Ω^ε

We setΓ^ε=∂Ω^εso that we haveΩ = Ω^ε_int∪Γ^ε∪Ω^ε. We denote bynandnε the outward unit normal vectors to ΓandΓ^ε, respectively, and by Ωext =R³\Ω the unbounded exterior domain. Throughout the paper we denote byH^t(Ω^ε_int),H_loc^t (Ωext)andH^t(Γ)the standard (local in the case of the exterior domain) complex valued, Hilbertian Sobolev space of ordert∈Rdefined onΩ^ε_int,Ωext,ΓandΓ^εrespectively (with the conventionH⁰ =L².) The exterior wavenumberκe, the interior wavenumberκi and the density ratio ρ are given positive constants. We are concerned with the following transmission problem : Given any densitiesfext∈H¹²(Γ)andgext∈H⁻¹²(Γ), find the solution(u^ε_int, u^ε_ext)∈H¹(Ω^ε_int)×H_loc¹ (Ωext)satisfying











∆u^εint+κ²iu^εint = 0 inΩ^εint

∆u^ε_ext+κ²_eu^ε_ext = 0 inΩext

u^ε_int−u^ε_ext = fext onΓ ρ∂nu^ε_int−∂nu^εext = gext onΓ,

(1.1)

and either a Dirichlet boundary condition onΓ^ε

u^ε_int=f_int^ε , (1.2)

or a Neumann boundary condition onΓ^ε

∂n_εu^εint=nε·(∇u^εint)_|Γε=gint^ε . (1.3) To ensure the uniqueness of the solution to either the problem (1.1)-(1.2) or (1.1)-(1.3), the scattered field u^ε_extis assumed to solve the Sommerfield radiation condition lim

|x|→+∞|x| |∂nu(x)−iκu(x)|= 0uniformly in all directionsx/|x|. Following the proof of Theorem 2.1 in [25], one can prove that the thin-layer transmission problem has at most one solution. Existence of a solution can be proved using boundary integral equation methods [25, 34]. More details can be found in the Appendix. The radiation condition implies that the scattered fielduext has an asymptotic behavior of the formu^εext(x) = ^e^iκ|x|_|x| u^ε∞(ˆx) +O

1

|x|

, |x| → ∞, uniformly in all directionsxˆ=_|x|^x. The far-field patternu^ε∞is a scalar function defined on the unit sphere S² ofR³ and is always analytic.

The scattering problem of time-harmonic waves by the coated obstacleΩleads to special cases of the above transmission problems where the given densitiesfext andgextare the boundary data of an incident plane waveuînc(x) =eîκê^d·x,d∈S². The total displacement fieldu^εext+uîncis then given by the superposition of the incident fielduînc, which is an entire solution of the Helmholtz equation, and the scattered fieldu^ε_ext, which solves the Helmholtz equation in Ωext and the Sommerfield radiation condition. In this cases, we assumef_int^ε = 0andg^ε_int= 0. In Section 3, for small positive real values ofε, we approach the solutionu^εext

of (1.1) by the solutionv^ε_[N]of some exterior boundary value problems of the form







∆v^ε_[N]+κ²ev^ε_[N] = 0 inΩext

C ε, ∂n(v_[N]^ε +u^inc),(v^ε_[N]+u^inc)

= 0 onΓ lim

|x|→+∞|x|

∂nv_[N]^ε (x)−iκv_[N]^ε (x)

= 0,

(1.4)

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where the linear condition on the boundaryC ε, ∂n(v_[N]^ε +u^inc),(v_[N]^ε +u^inc)

= 0is the so-called Generalised Impedance Boundary Condition (GIBC). To construct the GIBCs, we extend toR³ the approach developed in [3] to model the 2D case of thin layers with variable thickness. The technique originates from the PhD thesis [33] and is based on an asymptotic expansion of the Helmholtz equation in Ω^εint in terms of ε and surface derivatives onΓpresented in Section 2. We introduce a rapid variable S and thanks to the transformation of the thin layer into a band of thicknesshwe getu^εintas a solution to a Sturm-Liouville type problem of variable S . Once the differential equation is solved, then thanks to boundary conditions and jump condition, we get the corresponding boundary condition corresponding to the exterior domain. The existence and uniqueness of a solution to these problems can be found in [12] forN = 1,2. This approach leads first to estimate||u^εext−v^ε_[N]||

H¹2(Γ) =O(ε^N+1)and we deduce||u^εext−v_[N]^ε ||H¹(Ω_ext∩B_R) =O(ε^N+1) for every ballBR of radiusR and||u^ε∞−v^ε_∞,[N]||_L2(S²) =O(ε^N+1) wherev^ε_∞,[N] is the far-field pattern of the approximate solutionv^ε_[N].

Then, assuming the thin layer having a constant thickness, we analyze the dependence of the solution, or equivalently its far-field pattern, to the transmission problem (1.1) with respect to the shape of the exterior boundaryΓ. The first shape derivativeu˙^εextsolve the transmission problem (1.1) with non vanishing jumps accross the exterior and the interior boundaries. On one hand, the shape derivativeu˙^ε_ext is approached in Section 4 by the solutionw^ε_[N]of some exterior boundary value problems of the form











∆w^ε_[N]+κ²_ew^ε_[N] = 0 inΩext

C ε, ∂nw_[N]^ε , w^ε_[N]

= F_[N]^ε,1 onΓ lim

|x|→+∞|x|

∂nw^ε_[N](x)−iκw^ε_[N](x)

= 0,

where the right-hand side F_[N]^ε,1 can be expressed in terms of the boundary data of the exterior total field v^ε_[N]+u^inc. In this case we naturally obtain||u˙^ε∞−w_∞,[N]^ε ||L²(S²)=O(ε^N+1)where w_∞,[N]^ε is the far-field pattern of the approximate derivativew^ε_[N]. On the other hand, we provide in Section 5 the characterisation of the first shape derivativev˙^ε_[N]of the solutionv^ε_[N]to the exterior problem (1.4) of the form











∆ ˙v^ε_[N]+κ²ev˙_[N]^ε = 0 inΩext

C ε, ∂nv˙_[N]^ε ,v˙^ε_[N]

= F_[N]^ε,2 onΓ

|x|→+∞lim |x|

∂nv˙^ε_[N](x)−iκv˙^ε_[N](x)

= 0,

where the right-hand side F_[N]^ε,2 can be expressed in terms of the boundary data of the exterior total field v^ε_[N]+u^inc. In Section 6, we prove for N = 0,1,2that the two approaches are equivalent, which means

||v˙_[N]^ε −w^ε_[N]||

H¹2(Γ)=O(ε^N+1)and||v˙_∞,[N]^ε −w∞,[N]^ε ||L²(S²)=O(ε^N+1)wherev˙^ε_∞,[N]is the far-field pattern of the derivative v˙^ε_[N]. The various theoretical results are illustrated by some numerical experiments in Section 7. The transmission problem and the exterior boundary value problems are solved using boundary integral equation methods [13, 34] (see the Appendix) and the high order spectral method [18]. Finally, we draw concluding remarks and we discuss possible research lines in Section 8.

2 Elementary differential geometry and asymptotic expansions

In this section, we derive the asymptotic expansion of the Laplacian operator in the neighborhood ofΓusing the high-order material derivatives of some surface differential operators and Taylor-Young expansions. We use the surface differential operators: The tangential gradient∇Γ, the surface divergencedivΓand the scalar Laplace-Beltrami operator∆Γ. For their definitions we refer to Nedelec’s book [27] (pp. 68-75). We use the notations of [27] and quote some usefull results from [27] (pp. 67-78) and [14].

SinceΓis a smooth closed orientable boundary, there exists a tubular neighbourhood˚Γs₀ ofΓin which any pointyadmits the unique expansion

y=x+sn(x), withx∈Γ, ands∈]−s0;s0[ withs0>0.

For any s ∈]−s0;s0[, we set Γs :={y =x+sn(x) |x ∈Γ}. We denote ∇Γ_s and divΓ_s the tangential gradient and the surface divergence onΓs, respectively, and we denote bynsthe outward unit normal vector toΓs. For any scalar functionuand vector functionwdefined in˚Γs₀, the following expansions hold onΓs:

∇u=∇Γsu+ns∂su ,

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and

divw= divΓsw+ (ns·∂sw).

We denote by τs the transformation that maps the restrictionu|Γ_s of uto Γs to the function defined on Γby(τsu|Γ_s)(x) =u|Γ_s(x+sn(x)). Setting (τsu|Γ_s)(x) = ¯u(x, s), we define an isomorphism between˚Γs₀

andΓ×]−s0;s0[. The outward unit normal vectorns to the boundaryΓssatisfiesns=τ_s⁻¹n. Using this change of coordinate system we can write fory∈Γs:

(∇u)(y) = (τs∇u)(x) =τs∇Γ_sτ_s⁻¹u(x, s) +¯ n∂su(x, s)¯ , and

(divw)(y) =τs(divw)(x) =τsdivΓ_sτs⁻¹w(x, s) +¯ n·∂sw(x, s)¯ .

The material derivatives of the surface differential operators has been analysed in [14, Section 5] and we obtain the following result.

Proposition 2.1. The functions defined by s ∈]−s0;s0[7→ τs∇Γ_sτs⁻¹ ∈ L C¹(Γ),C⁰(Γ,R³)

and s ∈ ]−s0;s0[7→τsdivΓsτs⁻¹∈L C¹(Γ,R³),C⁰(Γ)

are infinitely differentiable and we have for anyu0∈C¹(Γ) andw0∈C¹(Γ,R³):

∂s τs∇Γ_sτs⁻¹

u0=−τsRs∇Γ_sτs⁻¹u0 (2.1) and

∂s τsdivΓ_sτs⁻¹

w0=−τsdivΓ_sRsτs⁻¹w0+τs∇Γ_sHs·w0−Trace[τsR²s](n·w0) (2.2) whereRs=∇Γ_snandHs= Trace[Rs].

The first order material derivatives corresponds to the commutators given in [27, Eqs. (2.5.228) and (2.5.229)]. To obtain the high order derivatives, it suffices to use the chain rule since we have [27, Eq (2.5.154) and (2.5.155)]

∂s(τsRs) =−τsR²sand∂s(τsHs) =−Trace[τsR²s]. Further, we will use the gaussian curvature denoted byGs which satisfies

Trace[R²s] + 2Gs=H²s. (2.3)

and if we setΠ3= I3−n⊗n, then the Cayley Hamilton’s theorem implies

R²s− HsRs+GsΠ3= 0. (2.4)

Using the Taylor-Young formula in the neighbourhood of s = 0 and (2.1), we can expand the gradient operator in the coordinate system(x, s)∈Γ×]−s0;s0[and we obtain for anyN∈N

(∇u)(x+sn(x)) = n(x)∂su(x, s) +¯ ∇Γu(x, s)¯ +

N

X

`=0

s^`1

`!∂_s^`(τs∇Γsτ_s⁻¹)|s=0u(x, s) +¯ O(s^N+1), (2.5) with 1

`!∂s^`(τs∇Γsτs⁻¹)|s=0= (−1)^`R^`∇Γ. In the same way, we write (divw)(x+sn(x)) = n(x)·∂sw(x, s) + div¯ Γw(x, s)¯

+

N

X

`=1

s^`1

`!∂s^`(τsdivΓ_sτs⁻¹)|s=0w(x, s) +¯ O(s^N+1), (2.6) with

∂s(τsdivΓsτs⁻¹)|s=0w¯ =−divΓRw¯ +∇ΓH ·w¯−Trace[R²](n·w)¯

=− divΓ(R − H)Π3w¯ +HdivΓΠ3w¯+ (H²−2G)(n·w)¯ , whereΠ3= I3−n⊗nand using the chain rules we obtain the following high order terms

1

2!∂_s²(τsdivΓ_sτ_s⁻¹)|s=0w¯ =HdivΓ(R − H)Π3w¯+ (H²− G) divΓΠ3w¯ + (H³−3HG)(n·w)¯ ,

and 1

3!∂s³(τsdivΓ_sτs⁻¹)|s=0w¯ =−

(H²− G) divΓ(R − H)Π3w¯ + (H³−2GH) divΓΠ3w¯

−Trace[R⁴](n·w).¯

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The formula (2.5) is in accordance with the formula (2.5.182) in [27]

τs∇Γ_sτs⁻¹u(x, s) = (I +¯ sR(x))⁻¹∇Γu(x, s)¯ .

Indeed, the Neumann series of (I +sR(x))⁻¹ yields the infinite series given in (2.5). We deduce that the gradient operator is equal to its Taylor series in the tubular˚Γs₀. Since we havedivw= Trace[∇w], we also deduce that the divergence operator is equal to its Taylor series in˚Γs₀. However, the high-order terms are easier to obtain by computing the material derivatives than taking the trace of[R^`∇Γw]for any`∈N.

Assuming∀x∈Γ,0< εh(x)< s0, then we use the change of variables=−εSwithS ∈[0 ;h(x)]. We set¯u(x, s) = ¯u(x,−εS) =U(x, S)and we have

∂su(x, s) =¯ −1

ε∂SU(x, S).

Combining (2.5) and (2.6), we obtain the asymptotic expansion of the Laplacian∆ = div∇

∆ = 1 ε² ∂_S²+

N

X

`=1

ε^`Λ`+O(ε^N+1)

! , where

Λ1=−H∂S, Λ2= ∆Γ−S(H²−2G)∂S, Λ3=S divΓ(2R − H)∇Γ+H∆Γ

−S²(H³−3HG)∂S, Λ4=S² divΓ(2R²− HR)∇Γ+HdivΓ(2R − H)∇Γ+ (H²− G)∆Γ

−S³Trace[R⁴]∂S.

The following proposition gives an expression of the outward unit normal vector to the interior boundary Γ^ε={y=x−εh(x)n(x)|x∈Γ}for any functionh.

Proposition 2.2. The outward unit normal vector toΓ^ε is given by

nε= (1−εhH+ε²h²G)n+ε(I3+εh(R − H))∇Γh p(1−εhH+ε²h²G)²+ε²k(I3+εh(R − H))∇Γhk² .

Proof. Assume that the tangent plane toΓat the pointxis generated by the unit vectorse1(x)ande2(x) such that the outward unit normal vector toΓis defined byn=e1×e2. The cotangent vectors are given bye¹=e2×nande²=n×e1. We haveei·e^j=δ_i^jwhereδ_i^jis the kronecker symbol. The tangent plane to Γ^ε at the pointy =x−εh(x)n(x) is generated by the vectors e1(x, ε) = D[I−εh(x)n(x)]e1(x) and D[I−εh(x)n(x)]e2(x)and the outward unit normal vector toΓ^ε is given by

nε(y) = e1(x, ε)×e2(x, ε) ke1(x, ε)×e2(x, ε)k. It remains to computeN_h^ε(x) =e1(x, ε)×e2(x, ε). We have

Nh^ε=e1×e2−ε [D(hn)]e1×e2+e1×[D(hn)]e2

+ε² [D(hn)]e1×[D(hn)]e2

=n−εh(Re1×e2+e1× Re2)−ε((∇Γh·e1)n×e2+ (∇Γh·e2)e1×n) +ε²h²Re1× Re2+ε²h((∇Γh·e1)n× Re2+ (∇Γh·e2)Re1×n) . To conclude we use the following equalities

Re1×e2+e1× Re2= (H − R)e1×e2= (H − R)n=Hn,

(∇Γh·e1)n×e2+ (∇Γh·e2)e1×n=−(∇Γh·e1)e¹−(∇Γh·e2)e²=−∇Γh , Re1× Re2= cof[R](e1×e2) =Gn,

n× Re2=n× Re2+Rn×e2=−(H − R)e¹, Re1×n=Re1×n+e1× Rn=−(H − R)e².

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3 Construction of the GIBCs

The construction of the GIBC is based on the assumption that the interior and exterior fields admits the following expansion whenεtends zero :

u^εint(y) =U^εint(x, S) =X

`≥0

εⁿU^`int(x, S)inΓ×[0, h(x)],

u^ε_ext(y) =X

`≥0

εⁿu^`_ext(y)inΩext. The problem (1.1) can be rewritten as follows:









 P

`≥0

ε^` ∆u^`_ext+κ²_eu^`_ext

= 0 inΩext

P

`≥0

ε^`∂_S²U^`int = −P

`≥1

ε^`Λ1U^`−1_int −P

`≥2

ε^`(Λ2+κ²ih²I)U^`−2_int

−P

k≥3

P

`≥k

ε^`Λ`U^`−k_int inΓ×(0, h) P

`≥0

εⁿU^`_int(x,0) = u^inc(x) +P

`≥0

ε^`u^`_ext(x) onΓ× {0}

P

`≥0

ε^`∂SU^`_int(x,0) = −¹_ρ ε∂nu^inc(x) +P

`≥1

ε^`∂nu^`−1_ext(x)

!

onΓ× {0},

(3.1)

and the interior field satisfies either a Dirichlet boundary condition onΓ× {h}

X

`≥0

ε^`U^`int= 0,

or a Neumann boundary condition onΓ× {h}that can be rewritten using Proposition 2.2 as follows X

`≥0

ε^`∂SU^`int=hHX

`≥1

ε^`∂SU^`−1_int

−h²GX

`≥2

ε^`∂SU^`−2_int +X

`≥2

ε^`∇Γh· ∇ΓU^`−2_int

+X

k≥3

h^k−2X

`≥k

ε^`∇Γh·(2R − H)R^k−3∇ΓU^`−k_int .

We identify the right and left hand sides of each equations in (3.1) according to the power`≥0ofεand we solve iteratively the new systems - that can be split into two systems of unknownsU^`intandu^`extrespectively - to compute first U^`_int and then recover the boundary condition satisfied by u^`_ext. From these results we deduce the GIBC satisfied byv_[N]^ε , which is an approximation of P^N

`=0

ε^`u^`extup toO(ε^N+1). The final results are stated in the following two propositions. We obtain similar results than in the 2D case [3].

Proposition 3.1. The GIBCs modeling sound-soft obstacles coated by thin layers with a variable thickness are given forN = 0,1,2,3by

(v^εh_[N]+u^inc) +B^εh,N∂n(v_[N]^εh +u^inc) = 0 where

B^εh,0= 0, B^εh,1=−¹_ρ(εh)I, B^εh,2=−¹_ρ(εh)

1 +(εh) 2 H

I and B^εh,3=−¹_ρεh

I +εh

2 HI−(εh)² 6 ∆Γ+

εh

2∆Γ+εh

3(κ²_i+H²− G)I

εh

. Proof. Collecting the equations when`= 0, we obtain the two systems







∂²_SU⁰_int = 0 inΓ×(0, h(x))

∂SU⁰int = 0 onΓ× {0}

U⁰_int = 0 onΓ× {h(x)}

and

∆u⁰_ext+κ²_eu⁰_ext = 0 inΩext

u⁰ext+u^inc = U⁰_int onΓ.

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The first equation implies thatU⁰_int(x, S)is a polynomial function of degre 1 in the variableS. The second equation implies that the leading coefficient is0and the third equation gives the constant term. We conclude

U⁰int(x, S) = 0.

In this case we approachu^ε_extby the functionv_[0]^ε =u^ext₀ and thenu^ε_ext−v^ε_[0]=O(ε). When`= 1, we obtain the two systems







∂S²U¹int = −Λ1U⁰int inΓ×(0, h(x))

∂SU¹_int = −¹_ρ∂n u⁰_ext+u^inc

onΓ× {0}

U¹int = 0 onΓ× {h(x)}

and

∆u¹ext+κ²eu¹ext = 0 inΩext

u¹_ext = U¹_int onΓ, We conclude with similar arguments that

U_int¹ (·, S) =−(S−h(x))¹_ρ∂n

u⁰_ext+u^inc . We compute u^inc+

1

P

`=0

ε^ù^èxt = ¹_ρεh∂n u⁰ext+uînc on Γ. In this case we approach the solution u^εext

by the function v_[1]^ε that satisfies the Helmholtz equation and the boundary condition (u^inc+v_[1]^ε ) = εh¹_ρ∂n u^inc+v^ε_[1]

and we getu^εext−v_[1]^ε =O(ε²). When`= 2, we obtain the two systems







∂S²U²int = −Λ1U¹int−(Λ2+κ²i)U⁰int inΓ×(0, h(x))

∂SU_int² = −¹_ρ∂nu¹_ext onΓ× {0}

U_int² = 0 onΓ× {h(x)}. and

∆u²ext+κ²eu²ext = 0 inΩext

u²_ext = U²_int onΓ. We compute

∂S²U²int=−H¹_ρ∂n

u⁰ext+u^inc , and we conclude

U_int² (·, S) =−

S²−h²(x) 2

Hρ⁻¹∂n

u⁰_ext+u^inc

−(S−h(x))ρ⁻¹∂nu¹_ext. We computeu^inc+

2

P

`=0

ε^ù^èxt= ¹₂(εh)²H¹_ρ∂n u⁰ext+uînc

+εh¹_ρ∂n

u^inc+

1

P

`=0

ε^`u^`ext

onΓ. In this case we approach the solutionu^εext by the functionv^ε_[2] that satisfies the Helmholtz equation and the boundary conditionu^inc+v^ε_[2]= (εh)

1 +εh

2 H

1

ρ∂n u^inc+v_[2]^ε

and we get u^εext−v_[2]^ε =O(ε³). When`= 3, we obtain the two systems







∂_S²U³_int = −Λ1U²_int−(Λ2+κ²_i)U¹_int inΓ×(0, h(x))

∂SUint³ = −¹_ρ∂nu²ext onΓ× {0}

Uint³ = 0 onΓ× {h(x)}

and

∆u³_ext+κ²_eu³_ext = 0 inΩext

u³ext = U³int onΓ. We compute

∂_S²U³_int=−S(2H²−2G)¹_ρ∂n

u⁰_ext+u^inc

− H¹_ρ∂nu¹_ext+ (∆Γ+κ²_i)(S−h(x))¹_ρ∂n

u⁰_ext+u^inc , and we conclude

U_int³ (·, S) =−

S³−h³(x) 6

h

2H²−2G ρ⁻¹∂n

u⁰_ext+u^inc

−(∆Γ+κ²_i)ρ⁻¹∂n

u⁰_ext+u^inci

−

S²−h²(x) 2

h

Hρ⁻¹∂nu¹_ext|Γ+ (∆Γ+κ²_i)hρ⁻¹∂n

u⁰_ext+u^inci

−(S−h(x))ρ⁻¹∂nu²_ext. We computeu^inc+

3

P

`=0

ε^`u^`ext= (εh)²1

3(εh)(κ²i+H²− G)I +¹₂(εh)∆Γ−¹₆∆Γ(εh)1

ρ∂n u⁰ext+u^inc +

1

2(εh)²H¹_ρ∂n

u^inc+

1

P

`=0

ε^`u^`ext

+εh_ρ¹∂n

u^inc+

2

P

`=0

ε^`u^`ext

onΓ. In this case we approach the solution u^ε_ext by the function v^ε_[3] that satisfies the Helmholtz equation and the boundary condition u^inc+v^ε_[3] =

εh

1 +εh

2 H+^(εh)₃²(κ²i+H²− G)

I +^(εh)₂³∆Γ−^(εh)₆²∆Γ(εh)

1

ρ∂n u^inc+v_[3]^ε and we getu^εext−v^ε_[3]= O(ε⁴).

(9)

Proposition 3.2. The GIBCs modeling sound-hard obstacles coated by thin layers with a variable thikness are given forN = 0,1,2by

∂n(v^εh_[N]+u^inc) +B^εh,N(v_[N]^εh +u^inc) = 0 where

B^εh,0= 0, B^εh,1 = ρ

divΓ(εh)∇Γ+ (εh)κ²_iI , B^εh,2= ρ

divΓ(εh) 1 + (εh)(R −¹₂H)

∇Γ+ (εh) 1−¹₂(εh)H κ²_iI

, B^εh,3= divΓεh

1 +εh

2(2R − H) +(εh)²

3 (2R²− HR)

∇Γ+εh 1−εh

2 H+(εh)² 3 G

κ²_iI

−(εh)³

6 [∆Γ+κ²iI]²+(εh)²

2 [∆Γ+κ²iI]

divΓ(εh)∇Γ+κ²i(εh)I +1

2∇Γ(εh)· ∇Γ

divΓ(εh)²∇Γ+κ²_i(εh)²I .

Proof. The rank`= 0allows us to computeU⁰_intonly. We obtain the system







∂S²Uint⁰ = 0 inΓ×(0, h(x))

∂SU_int⁰ = 0 onΓ× {h(x)}

U_int⁰ = (u^inc+u⁰_ext) onΓ× {0}.

The first equation implies thatUext⁰ (x, S)is a polynomial function of degre 1 in the variableS. The second equation implies that the leading coefficient is0and the third equation gives the constant term. We conclude that

U_int⁰ (·, S) = (u^inc+u⁰_ext).

When`= 1, we obtain the two systems







∂_S²Uint¹ = −Λ1Uint⁰ = 0 inΓ×(0, h(x))

∂SUint¹ = H∂SUint⁰ = 0 onΓ× {h(x)}

U_int¹ = u¹_ext onΓ× {0}.

and

∆u⁰_ext+κ²_eu⁰_ext = 0 inΩext

∂n(u^inc+u⁰ext) = −ρ∂SUint¹ onΓ. We conclude with similar arguments that

Uint¹ (·, S) =u¹ext.

We compute∂n(u^inc+u⁰ext) = 0onΓ. In this case we approachu^εextby the functionv^ε_[0]=u^ext0 and then

∂n(u^ε_ext−v^ε_[0]) =O(ε). When`= 2, we obtain the two systems







∂S²Uint² = −Λ1Uint¹ −(Λ2+κ²i)Uint⁰ inΓ×(0, h(x))

∂SU_int² = ∇Γh· ∇Γ(u^inc(x) +u⁰_ext(x)) onΓ× {h(x)}

U_int² = u²_ext onΓ× {0}.

and

∆u¹_ext+κ²_eu¹_ext = 0 inΩext

∂nu¹ext = −ρ∂SUint² onΓ. We compute

∂S²U²int=−(∆Γ+κ²i)(u^inc+u⁰ext), and we conclude

Uint² (·, S) =− S²

2 −Sh(x)

(∆Γ+κ²i)(u^inc+u⁰ext) +S∇Γh· ∇Γ(u^inc+u⁰ext) +u²ext, We compute∂n

u^inc+

1

P

`=0

ε^`u^`ext

=−ρ divΓ(εh)∇Γ+ (εh)κ²i

(u^inc+u⁰ext)onΓ. In this case we approach the solution u^ε_ext by the function v_[1]^ε that satisfies the Helmholtz equation and the boundary condition

∂n(u^inc+v_[1]^ε ) =−ρ divΓ(εh)∇Γ+ (εh)κ²i

u^inc+v^ε_[1]and we get∂n(u^εext−v^ε_[1]) =O(ε²). When`= 3, we obtain the two systems











∂S²Uint³ = −Λ1Uint² −(Λ2+κ²i)Uint¹ −Λ3Uint⁰ inΓ×(0, h(x))

∂SU_int³ = hH∂SU_int² −h²G∂SU_int¹ +∇Γh· ∇ΓU_int¹

+h∇Γh·(2R − H)∇ΓUint⁰ onΓ× {h(x)}

U_int³ = u³_ext onΓ× {0}.

(10)

and

∆u²_ext+κ²_eu²_ext = 0 inΩext

∂nu²ext = −ρ∂SU_int³ onΓ. We compute

∂S²Uint³ (·, S) =−(S−h(x))H(∆Γ+κ²i)(u^inc+u⁰ext) +H∇Γh· ∇Γ(u^inc+u⁰ext)

− ∆Γ+κ²_i

u¹_ext|Γ−S divΓ(2R − H)∇Γ+H∆Γ

(u^inc+u⁰_ext), and

∂SU_int³ (·, h(x)) =hH∇Γh· ∇Γ(u^inc+u⁰_ext) +∇Γh· ∇Γu¹_ext|Γ+h∇Γh·(2R − H)∇Γ(u^inc+u⁰_ext). We conclude

Uint³ (·, S) =− S³

6 −Sh²(x) 2

h

H(∆Γ+κ²i)(u^inc+u⁰ext) + divΓ(2R − H)∇Γ+H∆Γ

(u^inc+u⁰ext) i

+ S²

2 −Sh(x) h

hH(∆Γ+κ²i)(u^inc+u⁰ext) +H∇Γh· ∇Γ(u^inc+u⁰ext)− ∆Γ+κ²i

u¹ext|Γ

i

+Sh

∇Γh· ∇Γu¹ext|Γ+h∇Γh·2R∇Γ(u^inc+u⁰ext)i +u³ext. We compute∂n

u^inc+

2

P

`=0

ε^`u^`ext

=−¹₂ρ

divΓ(εh)²(2R − HI )∇Γ−(εh)²Hκ²iI

u⁰ext+u^inc

−ρ divΓ(εh)∇Γ+ (εh)κ²i

u^inc+

1

P

`=0

ε^`u^`ext

. In this case we approach the solutionu^εextby the function v^ε_[2]that satisfies the Helmholtz equation and the boundary condition

∂n(u^inc+v_[2]^ε ) =−ρ ε

∆Γ+κ²iI

−¹₂ε²

divΓ(2R − HI )∇Γ− Hκ²iI

u^inc+v^ε_[2] and we get ∂n(u^εext− v^ε_[2]) =O(ε³).

When`= 4, we obtain the two systems











∂_S²U_int⁴ = −Λ1U_int³ −(Λ2+κ²_i)U_int² −Λ3U_int¹ −Λ4U_int⁰ inΓ×(0, h)

∂SUint⁴ = hH∂SUint³ −h²G∂SUint² +∇Γh· ∇ΓUint²

+h∇Γh·(2R − H)∇ΓU_int¹ +h²∇Γh·(2R²− HR)∇ΓU_int⁰ onΓ× {h}

Uint⁴ = u⁴ext onΓ× {0}.

and

∆u³_ext+κ²_eu³_ext = 0 inΩext

∂nu³ext = −ρ∂SUint⁴ onΓ. We obtain

∂SUint⁴ (·, S)

=− H ¹₆(S³−h³)−¹₂(S−h)h²h

H(∆Γ+κ²_i)(u^inc+u⁰_ext) + divΓ(2R − H)∇Γ+H∆Γ

(u^inc+u⁰_ext)i +H¹₂(S−h)²h

hH(∆Γ+κ²_i)(u^inc+u⁰_ext) +H∇Γh· ∇Γ(u^inc+u⁰_ext)− ∆Γ+κ²_i u¹_ext|Γi + (S−h)Hh

∇Γh· ∇Γu¹_ext|Γ+h∇Γh·2R∇Γ(u^inc+u⁰_ext)i

−(S−h)(∆Γ+κ²_i)u²_ext +(S³−h³)

6 (∆Γ+κ²_i)(∆Γ+κ²_i)(u^inc+u⁰_ext)−¹₂(S²−h²) divΓ(2R − H)∇Γ+H∆Γ

u¹_ext|Γ

−(S²−h²)

2 (∆Γ+κ²i)h

h(∆Γ+κ²i)(u^inc+u⁰ext) +∇Γh· ∇Γ(u^inc+u⁰ext)i

−(H²−2G)h

1

3(S³−h³)−¹₂(S²−h²)h

(∆Γ+κ²_i)(u^inc+u⁰_ext)−¹₂(S²−h²)∇Γh· ∇Γ(u^inc+u⁰_ext)i

−¹₃(S³−h³) divΓ(2R²− HR)∇Γ+HdivΓ(2R − H)∇Γ+ (H²− G)∆Γ

(u^inc+u⁰_ext) +hHh

∇Γh· ∇Γu¹ext|Γ+h∇Γh·2R∇Γ(u^inc+u⁰ext)i

−h²G∇Γh· ∇Γ(u^inc+u⁰ext) +∇Γh· ∇Γ

h1

2h²(∆Γ+κ²i)(u^inc+u⁰ext) +h∇Γh· ∇Γ(u^inc+u⁰ext) +u²ext

i

+h∇Γh·(2R − H)∇Γu¹_ext+h²∇Γh·(2R²− HR)∇Γ(u^inc+u⁰_ext).