Shape sensitivity analysis for elastic structures with generalized impedance boundary conditions of the Wentzell type -Application to compliance minimization

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Shape sensitivity analysis for elastic structures with generalized impedance boundary conditions of the Wentzell type -Application to compliance minimization

Fabien Caubet, Djalil Kateb, Frédérique Le Louër

To cite this version:

Fabien Caubet, Djalil Kateb, Frédérique Le Louër. Shape sensitivity analysis for elastic structures

with generalized impedance boundary conditions of the Wentzell type -Application to compliance

minimization. Journal of Elasticity, Springer Verlag, In press, Journal of Elasticity. �hal-01525249�

Shape sensitivity analysis for elastic structures with generalized impedance boundary conditions of the Wentzell

type - Application to compliance minimization

Fabien Caubet ^∗1 , Djalil Kateb ^†2 , and Frédérique Le Louër ^‡2

1 Institut de Mathématiques de Toulouse, Université de Toulouse, F-31062 Toulouse Cedex 9, France.

2

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

May 19, 2017

Abstract

This paper focuses on Generalized Impedance Boundary Conditions (GIBC) with second order derivatives in the context of linear elasticity and general curved interfaces. A condition of the Wentzell type modeling thin layer coatings on some elastic structure is obtained through an asymptotic analysis of order one of the transmission problem at the thin layer interfaces with respect to the thickness parameter. We prove the well-posedness of the approximate problem and the theoretical quadratic accuracy of the boundary conditions. Then we perform a shape sensitivity analysis of the GIBC model in order to study a shape optimization/optimal design problem. We prove the existence and characterize the first shape derivative of this model. A comparison with the asymptotic expansion of the first shape derivative associated to the original thin layer transmission problem shows that we can interchange the asymptotic and shape derivative analysis. Finally we apply these results to the compliance minimization problem. We compute the shape derivative of the compliance in this context and present some numerical simulations.

Keywords: asymptotic analysis, generalized impedance boundary conditions, Wentzell conditions, shape calculus, shape sensitivity analysis, compliance minimization, linear elasticity.

AMS Classification: 35C20, 49Q10, 49Q12, 74B05, 74P05

1 Introduction and main results

1.1 Introduction and general notations

This paper deals with the mathematical analysis and the shape optimization of some elastic struc- tures coated with a very thin layer of constant thickness. Our objective is first to construct a model problem where the thin layer effects are characterized by perturbed boundary conditions with second order derivatives called boundary condition of the Wentzell type. Then we aim to make a shape sensitivity analysis for this reduced problem using shape calculus tools. These subjects find applications in various area such as elastographic imaging and structural optimal design.

On the one hand, the theoretical and numerical analysis of a thin structure could be a very complex work. Hence, in order to overcome this difficulty, a classical strategy is to approximate an original structure with a thin layer (see, e.g., [10, 30, 46, 16]) or a rough boundary (see, e.g., [36, 1])

∗

fabien.caubet@math.univ-toulouse.fr

†

djalil.kateb@utc.fr

‡

frederique.le-louer@utc.fr

by another domain with new boundary conditions, called Generalized Impedance Boundary Condi- tions (GIBC). The GIBC contain some informations on the geometry and the material characteris- tics of the thin structure. For exterior problems, artificial (or approximate) conditions are another example (see, e.g., [25, 29]). These boundary conditions are generally simple differential conditions obtained from an asymptotic analysis of the full elliptic equations in the thin layer with respect to a characteristic length: the thickness of the layer, the scale of the roughness, the diameter of the artificial boundary, for the previous three examples, respectively. In linear elasticity, it deserves to mention the book of Ciarlet [21], where a local representation of the GIBC is proposed, and others works [26, 11] in the context of thin elastic plates or shells. The construction of GIBC is also treated for example by Antoine et al [9], by Poignard [44] or by Haddar et al [31, 32, 18] in acoustics and electromagnetism for general three-dimensional surfaces.

On the other hand, the problem of finding an optimal shape for physical issues described by elliptic boundary value problems is widely studied from many years. In linear elasticity, the idea is to study the influence of the shape of a structure on its behavior as for example its rigidity. To this end, several methods can be employed as the classical geometrical shape optimization approach (see, e.g., the work of Murat et al. [39]), the homogenization method (see, e.g., the work of Al- laire et al. [5]), the topological shape optimization approach (see, e.g., the work of Garreau et al. [28]) or the so-called level-set method (see, e.g., the works of Allaire et al. [7]). Several works deal about the problem of minimizing the compliance of the structure where standard boundary conditions are imposed on the free boundary. We can here mention the recent works of Allaire et al. [6], Amstutz et al. [8], Novotny et al. [41] and Dambrine et al. [24]. This list of references is far from being exhaustive and we also refer to the books of Sokolowski et al. [45], of Henrot et al. [35], of Allaire [3, 4] and of Haslinger et al. [33] for background notions about shape optimization methods.

Finally, more recently, one can find new advances on the computation of shape derivatives and/or the solution of shape optimization and inverse obstacle scattering problems when a GIBC of the first and second orders are imposed on the unknown boundary. For example, we can men- tion the works of Cakoni et al. [15, 14] and Caubet et al. [17] for the Laplace’s equation, of Bourgeois et al. [12] and Kateb et al. [37] for the Helmholtz equation and of Chaulet et al. [19] for the Maxwell’s equations.

The present paper is dedicated to the shape sensitivity analysis when first-order GIBC are imposed on the free boundary of some elastic structures.

General notations. For a smooth bounded open set ω of R ^d (d ≥ 2) with a boundary Γ, we denote by H ^s (ω) and H ^s (Γ) the standard complex valued, Hilbert-Sobolev spaces of order s ∈ R defined on ω and Γ respectively (with the convention H ⁰ = L ² ). Spaces of vector functions will be denoted by boldface letters, thus H ^s = (H ^s ) ^d . Moreover the tangential gradient is denoted by ∇ Γ

and the surface divergence is denoted by div Γ . For any vector field u, ∇ Γ u is the matrix whose the j-th column is the tangential gradient of the j-th component of u. For any matrix-valued function F , div Γ F is the vector whose the j-th component is the surface divergence of the j-th column of F. We denote by I d the d × d identity matrix. Finally, notice that for two matrices A and B, we denote by A : B the matrix scalar product between A and B.

1.2 Setting of the problem, main results and organization of the paper

Introduction of the two considered problems. Let Ω be a Lipschitz bounded open set of R ^d , where d ≥ 2 is an integer representing the dimension. We assume that the solid Ω consists of an isotropic material with a linear behavior. The boundary of Ω is such that ∂Ω =: Γ D ∪Γ N where Γ D

and Γ N are two non-empty open sets of ∂Ω and |Γ D | > 0. Let δ > 0 be a fixed (small) real number and let us define the following set of admissible shapes:

O δ := n

ω ⊂⊂ Ω be a smooth (at least C ² ) open set such that d(x, ∂Ω) > δ for all x ∈ ω and such that Ω\ω is connected o

. (1.1)

Let us consider a (nonempty) inclusion ω ∈ O _δ with boundary ∂ω =: Γ. Then Ω\ω represents a

reference configuration of an elastic solid assumed to be built on Γ _D . We denote by n the unit

normal vector to ∂Ω and Γ directed outward to Ω\ω. Moreover, H represents the mean curvature of Γ and b is the signed distance.

We denote by A _ext the Hooke’s law defined, for any symmetric matrix ξ, by A ext ξ := 2 µ ext ξ + λ ext Tr(ξ) I d ,

where µ ext > 0 and λ ext > 0 are two positive constants which represent the Lamé coefficients of the material making up the solid and we introduce, for any u ∈ H ¹ (Ω), the symmetrized gradient

e(u) := 1

2 ∇u + ^t ∇u .

Let ε > 0. We consider that Γ has an interior thin layer with thickness ε surrounding ω defined by ω ^ε _int := {x + sn(x) | x ∈ Γ and 0 < s < ε} .

We recall that the normal vector n is directed inward the inclusion ω. We set ω ^ε := ω\ω ^ε _int and we denote its boundary by Γ ^ε . We also denote by n ^ε the inward unit normal vector to Γ ^ε . In the sequel, we use the lower index e for all quantities related to Ω\ω and the lower index i for all quantities related to ω _int ^ε .

We introduce the following Sobolev space H ¹ _Γ

_D

(Ω\ω) :=

v ∈ H ¹ (Ω\ω) ; v = 0 on Γ D .

Let f ∈ L ² (Ω\ω) be some exterior forces and a load g ∈ H

^−1/2

(Γ N ). We are concerned with the following transmission problem



 

 

 

 



 

 

 

 



−div A ext e(u ^ε _e ) = f in Ω\ω

−div A _int e(u ^ε _i ) = 0 in ω ^ε _int u ^ε _e = 0 on Γ _D (A _ext e(u ^ε _e )) n = g on Γ _N (A _int e(u ^ε _i )) n = (A _ext e(u ^ε _e )) n on Γ

u ^ε _i = u ^ε _e on Γ A int e(u ^ε _i )n ^ε = 0 on Γ ^ε ,

(1.2)

where A int represents the Hooke’s law associated to ω _int ^ε with Lamé coefficients µ int > 0 and λ int > 0. The solution of such a problem exists, is unique and belongs to H ¹ _Γ

_D

(Ω\ω ∪ ω _int ^ε ) thanks to the Lax-Milgram theorem and the Korn’s inequality.

We introduce the following Hilbert space V(Γ) =

ψ ∈ L ² (Γ) ; e

_Γ

(ψ) ∈ L ² (Γ) , endowed with the graph norm ||ψ|| ²

_V(Γ)

:=

||ψ|| ²

_L2

(Γ) + ||e

_Γ

(ψ)|| ²

_L2

(Γ)

^1/2 where e

_Γ

(ψ) := 1

2 Π d ∇ Γ ψ + ^t ∇ Γ ψ

Π d and Π d := I d − n ⊗ n. (1.3) We denote its dual space by V

⁰

(Γ). Then, we set

H(Ω\ω) :=

v ∈ H ¹ _Γ

_D

(Ω\ω); v

_|Γ

∈ V(Γ) . The space H(Ω\ω) endowed with the graph norm kvk

_H(Ω\ω)

:=

kvk ²

_H1

(Ω\ω) + ||v|| ²

_V(Γ)

^1/2 is a Hilbert space.

We prove below (see Proposition 2.1) that the solution of the transmission problem (1.2) can be approximated using an asymptotic analysis by the solution u ∈ H(Ω\ω) of the following problem with Generalized Impedance Boundary Conditions (GIBC) on Γ:



 



 



−div (A ext e(u)) = f in Ω\ω u = 0 on Γ _D (A _ext e(u)) n = g on Γ _N (A _ext e(u)) n − εdiv _Γ (σ

_Γ

(u)) = 0 on Γ.

(1.4)

Here, we have used the following notations:

σ

_Γ

(u) := λ int (div Γ u)Π d + 2µ int e

_Γ

(u) = A int e

_Γ

(u)

A int ξ := 2 µ int ξ + λ int Tr(ξ) Π d , (1.5) where µ int and λ int := _λ ^2λ

^int

^µ

^int

int

+2µ

int

are the modified Lamé constants of the thin layer. The well- posedness of this problem is studied in Appendix A (see Proposition A.1).

Remark 1.1. (i) The negative Wentzell-type operator ψ 7→ −div _Γ (σ

_Γ

(ψ)) is a positive symmet- ric operator bounded from V(Γ) to V

⁰

(Γ). Indeed we have

h−div Γ (σ

_Γ

(ψ)), ϕi

_V⁰

_(Γ),V(Γ) = Z

Γ

λ int (div Γ ψ)(div Γ ϕ)ds + 2µ int

Z

Γ

e

_Γ

(ψ) : e

_Γ

(ϕ)

= hψ, −div Γ (σ

_Γ

(ϕ))i

V(Γ),V⁰

(Γ) .

(ii) For any ψ ∈ L ² (Γ), let set ψ _n := ψ · n and ψ _t := ψ − ψ _n n. Then we get the following inclusion

ψ ∈ L ² (Γ) ; ψ _t ∈ H ¹ (Γ) ⊂ V(Γ) thanks to the equality e

_Γ

(ψ) = e

_Γ

(ψ _t ) + ψ _n [D ² b] .

Moreover, using the formulas given in [21, page 88], one can check that the operator ψ ∈ V(Γ) 7→ e

_Γ

(ψ) ∈ L ² (Γ) corresponds to the operator γ αβ defined in [21, Theorem 2.7.1].

Using an atlas for the boundary Γ, one can derive an inequality of Korn-type on 2-dimensional compact manifolds without boundary from [21, Theorem 2.7.1] ensuring that there exists a constant c 0 > 0 depending on Γ such that

kψ _t k ²

_H1

(Γ) + kψ _n k ² _L

2

(Γ)

^1/2

≤ c ₀ ||ψk

V(Γ)

.

In other words, the two spaces coincide, i.e.

ψ ∈ L ² (Γ) ; ψ _t ∈ H ¹ (Γ) = V(Γ).

We summarize the notations concerning the domains in Figure 1 below.

∂Ω Ω\ω

ω ^ε ω _int ^ε

Γ

Γ ^ε ε

(a)

Ω\ω

ω

Γ

N

Γ

D

(b)

Figure 1: Notations for the thin layer model (left) and the obtained GIBC model (right).

Main aims and results of the paper. The first aim of this work is to derive the GIBC on Γ given in Problem (1.4). To do this, we perform an asymptotic analysis on the transmission problem (1.2) with respect to the thickness of the layer. We then obtain the following main result of this paper which gives the rate of convergence of this analysis. This part is treated in Section 2 and uses asymptotic results given in Appendix B.

Theorem 1.2. If f ∈ H ^1/2 (Ω\ω), then there exists a constant C, depending only on the domains Ω and ω, such that

ku − u ^ε _e k

_H1

(Ω\ω) ≤ C ε ² ,

where u ∈ H(Ω\ω) is the solution of Problem (1.4) and u ^ε _e ∈ H ¹ _Γ

_D

(Ω\ω) is defined by Prob- lem (1.2).

The second main aim is to make a shape sensitivity analysis of this problem. Hence we prove the existence of shape derivatives with respect to the shape ω and characterize the shape derivative of Problem (1.4) (we refer to Section 3.2 for some reminders about the definition of the shape derivative). Then we obtain the following second main result.

Theorem 1.3. Let V ∈

θ ∈ C ^2,∞ = C ² ∩ W ^2,∞ ( R ^d ); supp(θ) ⊂ Ω _δ ; kθk _2,∞ < min ^δ ₃ , 1 and let u ∈ H(Ω\ω) the solution of Problem (1.4). The shape derivative u

⁰

of u, which belongs to L ² (Ω\ω), exists. Moreover, if u ∈ H ² (Γ), then u

⁰

∈ H(Ω\ω) is the only solution of the following boundary value problem



 



 



−div (A ext e(u

⁰

)) = 0 in Ω\ω u

⁰

= 0 on Γ D

(A ext e(u

⁰

)) n = 0 on Γ N

(A ext e(u

⁰

)) n − εdiv Γ (σ

_Γ

(u

⁰

)) = ξ(u, V n ) on Γ,

(1.6)

with

ξ(u, V n ) := V n f + div Γ

V n Π d (A ext e(u)) Π d

− εdiv Γ

V n ([D ² b] − H Π d )σ

_Γ

(u)

− εdiv Γ

V n A int Π d 1

2 [D ² b]∇ Γ u + ^t ([D ² b]∇ Γ u) Π d

+ εdiv Γ σ

_Γ

(V n ∂

n

u) +ε [D ² b] + n div Γ Π d

div Γ (V n σ

_Γ

(u)) +εdiv Γ

A int 1

2 ([∇ Γ u]n ⊗ ∇ Γ V n + ∇ Γ V n ⊗ [∇ Γ u]n) , (1.7) where V n := V · n, where H is the mean curvature of Γ and b is the signed distance.

We also provide an asymptotic analysis on the shape derivatives for the transmission problem (1.2) and compare it with the shape derivatives of Problem (1.4). This analysis concerning the shape calculus is done in Section 3.

We conclude this paper with an example of application of this work concerning the minimization of the compliance in Section 4. We compute the shape derivative of the compliance through the introduction of an adjoint problem and make some numerical simulations in the two-dimensional case in order to illustrate and validate our theoretical results. Thus, defining the compliance of the structure Ω\ω as

J (Ω\ω) :=

Z

Ω\ω

A ext e(u) : e(u),

where u ∈ H(Ω\ω) solves Problem (1.4), we obtain the following expression of the shape gradient of this cost functional.

Theorem 1.4. Let V ∈

θ ∈ C ^2,∞ = C ² ∩ W ^2,∞ ( R ^d ); supp(θ) ⊂ Ω δ ; kθk 2,∞ < min ^δ ₃ , 1 . If u ∈ H ² (Γ), then the shape derivative of the cost functional can be written as

∇J (Ω\ω) · V = Z

Γ

V n ξ

^∗

(u, p + u)ds + Z

Γ

V n (A ext e(u) : e(u)) , (1.8)

where u ∈ H(Ω\ω) is the solution of Problem (1.4) and where ξ

^∗

(u, v ) := − (Π _d (A _ext e(u)) Π _d ) : ∇ _Γ v + ε n

([D ² b] − H Π _d )σ

_Γ

(u)

: ∇ _Γ v + ([D ² b]∇ _Γ u) : σ

_Γ

(v) + (∂

_n

u)div _Γ (σ

_Γ

(v)) + σ

_Γ

(u) : (∇ _Γ ([∇ _Γ v]n)) + div _Γ (σ

_Γ

(v)[∇ _Γ u]n) o

+ f · v and p ∈ H(Ω\ω) is solution to the following adjoint problem



 



 



−div (A ext e(p)) = f in Ω\ω

p = 0 on Γ D

(A _ext e(p)) n = g on Γ _N (A _ext e(p)) n − εdiv _Γ (σ

_Γ

(p)) = 2 (A _ext e(u)n) on Γ,

(1.9)

with V n := V · n and b denoting the signed distance.

Remark 1.5. We are confident that the regularity assumption u ∈ H ² (Γ) made in the two previous theorems is useless since automatically satisfied. Unfortunately, up to our knowledge, the arguments to obtain it are not trivial and we postpone this study to a forthcoming work.

2 Derivation of the GIBC model and convergence analysis

This section is devoted to the derivation of the GIBC model (1.4) analyzed in this paper and to the proof of Theorem 1.2. To do this, we make an asymptotic analysis with respect to the thin layer of the transmission problem (1.2).

2.1 Asymptotic analysis of the transmission problem

Let N ∈ N . We want to approximate the solution u ^ε _e ∈ H ¹ _Γ

_D

(Ω\ω) of the transmission problem (1.2) by the solution v ^ε _{[N ]} of some boundary value problems of the form



 

 



 

 



−div A _ext e(v ^ε _{[N ]} ) = f in Ω\ω v ^ε _[N] = 0 on Γ _D

A ext e(v ^ε _{[N ]} )

n = g on Γ N

C

ε, A ext e(v ^ε _[N] )n, v ^ε _{[N ]}

= 0 on Γ,

(2.1)

where ||u ^ε _e − v ^ε _[N] ||

_H¹

_(Ω\ω) = O(ε ^{N +1} ) and C

ε, A _ext e(v ^ε _[N] )n, v ^ε _[N]

is a so-called Generalized Impedance Boundary Condition (GIBC). To do so, we follow the procedure described for example in [46, 37]. For any x ∈ Γ and s ≥ 0, we set u(x + sn(x)) =: u(x, s) and we use the change of variables y = x + sn(x) = x + εSn(x) with S ∈ [0 , 1]. We set u(x, s) = u(x, εS ) =: U(x, S).

Firstly we obtain the following asymptotic expansion when ε → 0 (see Appendix B for details):

div A int e(u)(x + εSn(x)) = 1 ε ²



Λ 0 ∂ _S ² + εΛ 1 ∂ S + X

n≥2

ε ⁿ Λ n



 U(x, S), (2.2) where

Λ 0 := (λ int + 2µ int )n ⊗ n + µ int I d − n ⊗ n ,

Λ 1 U := µ int HU + (λ int + µ int ) n div Γ U + ∇ Γ (U · n) , Λ 2 U := Λ 2,2 U + SΛ 2,1 ∂ S U,

with

Λ 2,1 U := −µ int Tr([D ² b] ² )U − (λ int + µ int ) n(Tr([D ² b]∇ Γ U)) + [D ² b]∇ Γ (U · n)

,

Λ _2,2 U := (λ _int + µ _int )∇ _Γ div _Γ U + µ _int ∆ _Γ U − λ _int n(Tr([D ² b]∇ _Γ U)) − µ _int ([D ² b]∇ _Γ U])n.

Moreover the traction trace operator is defined on Γ, i.e. for S = 0, by A _int e(U) := 1

ε Λ ₀ ∂ _S U + λ _int ndiv _Γ U + µ _int [∇ Γ U]n

and it admits the following expansion on the surface Γ ^ε , i.e. for S = 1, (see also Appendix B for details):

A _int e(U _i (·, 1)) = 1

ε Λ ₀ ∂ _S U _i (·, 1) + B ⁰ _t U _i (·, 1) + X

`≥1

ε ^{`</sup> B <sup>`} _t U _i (·, 1), with

B ⁰ _t U := λ int ndiv Γ U + µ int [∇ Γ U] n, B ¹ _t U := −

λ int n(Tr([D ² b]∇ Γ U)) + µ int ([D ² b]∇ Γ U)n .

Secondly we set u ^ε _e (x) := P

n≥0

ε ⁿ u ⁿ _e (x) in Ω\ω and u ^ε _i (x, s) := U ^ε _i (x, S ) = P

n≥0

ε ⁿ U ⁿ _i (x, S) in Γ × [0, 1]. Hence the transmission problem (1.2) can be rewritten as follows:



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

− P

n≥0

ε ⁿ div A _ext e(u ⁿ _e ) = f in Ω\ω

P

n≥0

ε ⁿ ∂ _S ² Λ ₀ U ⁿ _i = − P

n≥1

ε ⁿ Λ ₁ ∂ _S U ⁿ⁻¹ _i − P

n≥2

ε ⁿ Λ ₂ U ⁿ⁻² _i − · · · in Γ × (0, 1) P

n≥0

ε ⁿ u ⁿ _e = 0 on Γ D

P

n≥0

ε ⁿ (A _ext e(u ⁿ _e )) n = g on Γ _N

P

n≥1

ε ⁿ (A _ext e(u ⁿ⁻¹ _e ))n = P

n≥0

ε ⁿ ∂ _S Λ ₀ U ⁿ _i + P

n≥1

ε ⁿ B ⁰ _t U ⁿ⁻¹ _i on Γ × {0}

P

n≥0

ε ⁿ Λ 0 u ⁿ _e = P

n≥0

ε ⁿ Λ 0 U ⁿ _i on Γ × {0}

P

n≥0

ε ⁿ ∂ S Λ 0 U _i ⁿ = − P

n≥1

ε ⁿ B ⁰ _t U ⁿ⁻¹ _i − P

n≥2

ε ⁿ B ¹ _t U ⁿ⁻² _i + · · · on Γ × {1} . Then we prove the following result.

Proposition 2.1. The GIBC, defined on Γ, modeling the interior thin layer effects of the inclu- sion ω for N = 0, 1, are given by

C

ε, A ext e(v ^ε _{[N ]} )n, v ^ε _{[N ]} :=

A ext e(v ^ε _{[N ]} )n

+ C ^ε,N v ^ε _[N] , where C ^ε,0 := 0 and C ^ε,1 v := −εdiv _Γ A _int e

_Γ

(v)

.

Proof. The rank n = 0 allows us to compute U ⁰ _i only. The solution U _i ⁰ is characterized by



 

 

∂ _S ² Λ 0 U ⁰ _i = 0 in Γ × (0, 1)

∂ S Λ 0 U ⁰ _i = 0 on Γ × {1}

Λ 0 U ⁰ _i = Λ 0 u ⁰ _e on Γ × {0} . We deduce Λ ₀ U ⁰ _i (x, S) = Λ ₀ u ⁰ _e|Γ (x).

When n = 1, we obtain the two systems



 

 

∂ _S ² Λ 0 U ¹ _i = −Λ 1 ∂ S U ⁰ _i = 0 in Γ × (0, 1)

∂ _S Λ ₀ U ¹ _i = −B ⁰ _t U ⁰ _i = −B ⁰ _t u ⁰ _e|Γ on Γ × {1}

Λ 0 U ¹ _i = Λ 0 u ¹ _e on Γ × {0}

and 

 



 



−div A _ext e(u ⁰ _e ) = f in Ω\ω u ⁰ _e = 0 on Γ _D A _ext e(u ⁰ _e )

n = g on Γ _N

(A _ext e(u ⁰ _e ))n = ∂ _S Λ ₀ U ¹ _i + B ⁰ _t U ⁰ _i on Γ.

(2.3)

We obtain Λ ₀ U ¹ _i (x, S) = −SB ⁰ _t u ⁰ _e|Γ (x) + Λ ₀ u ¹ _e|Γ (x) and

(A ext e(u ⁰ _e ))n = ∂ S Λ 0 U ¹ _i ( · , 0) + B ⁰ _t U ⁰ _i ( · , 0) = 0. (2.4) In this case we approximate u ^ε _e by the function v ^ε _[0] = u ⁰ _e and then (A _ext e(u ^ε _e −v ^ε _[0] ))n = 0 = O(ε).

When n = 2, we obtain the two systems



 

 

∂ _S ² Λ 0 U ² _i = −Λ 1 ∂ S U ¹ _i − Λ 2,2 U ⁰ _i in Γ × (0, 1)

∂ S Λ 0 U ² _i = −B ⁰ _t U ¹ _i − B ¹ _t U ⁰ _i on Γ × {1}

Λ 0 U ² _i = Λ 0 u ² _e on Γ × {0}

and 

 



 



−div A ext e(u ¹ _e ) = 0 in Ω\ω u ¹ _e = 0 on Γ D

A ext e(u ¹ _e )

n = 0 on Γ N

(A ext e(u ¹ _e ))n = ∂ S Λ 0 U ² _i + B ⁰ _t U ¹ _i on Γ.

(2.5)

We compute ∂ _S ² Λ 0 U ² _i (x, S) = Λ 1 Λ

⁻¹

₀ B ⁰ _t u ⁰ _e|Γ (x) − Λ 2,2 u ⁰ _e|Γ (x) and we obtain Λ 0 U ² _i ( · , S) =

S ² 2 − S

Λ 1 Λ

⁻¹

₀ B ⁰ _t − Λ 2,2

u ⁰ _e

_|Γ

+ S(B ⁰ _t Λ

⁻¹

₀ B ⁰ _t u ⁰ _e|Γ − B ⁰ _t u ¹ _e|Γ − B ¹ _t u ⁰ _e|Γ ) + Λ 0 u ² _e|Γ and

(A _ext e(u ¹ _e ))n = ∂ _S Λ ₀ U ² _i ( · , 0) + B ⁰ _t U ¹ _i ( · , 0) = − Λ ₁ Λ

⁻¹

₀ B ⁰ _t − Λ _2,2 − B ⁰ _t Λ

⁻¹

₀ B ⁰ _t + B ¹ _t u ⁰ _i

_|Γ

.

We find

B ⁰ _t Λ

⁻¹

₀ B ⁰ _t =

λ int ndiv Γ · +µ int [∇ Γ · ]n λ int

λ _int + 2µ _int ndiv Γ · +[∇ Γ · ]n

= λ ² _int λ int + 2µ int

Hndiv _Γ + λ _int n(n · ∆ _Γ ) + λ _int nTr([D ² b]∇ Γ · ) + λ int µ int

λ _int + 2µ _int ∇ Γ div Γ − µ int ([D ² b]∇ Γ · )n, Λ 1 Λ

⁻¹

₀ B ⁰ _t = Λ 1

λ int

λ int + 2µ int

ndiv Γ · +[∇ Γ · ]n

= λ _int nHdiv _Γ + µ _int H[∇ _Γ · ]n + (λ _int + µ _int )n(n · ∆ _Γ ) + λ int (λ int + µ int )

λ _int + 2µ _int ∇ _Γ div _Γ + (λ _int + µ _int )nTr([D ² b]∇ _Γ · ), Λ ₁ Λ

⁻¹

₀ B ⁰ _t − B ⁰ _t Λ

⁻¹

₀ B ⁰ _t + B ¹ _t = λ _int nHdiv _Γ + µ _int H[∇ Γ · ]n + µ _int n(n · ∆ _Γ )

+ λ ² _int λ int + 2µ int

(∇ Γ div Γ − Hndiv Γ ) − (λ int − µ int )nTr([D ² b]∇ Γ · ).

We finally use

∇ Γ div Γ − Hndiv Γ = div Γ

(div Γ )(I d − n ⊗ n) , H[∇ _Γ · ]n + n(n · ∆ _Γ ) + nTr([D ² b]∇ _Γ · ) + ([D ² b]∇ _Γ )n = div _Γ (n ⊗ [∇ _Γ · ]n + [∇ _Γ · ]n ⊗ n) to get

(A ext e(u ¹ _e ))n = div Γ

2λ int µ int

λ int + 2µ int

(div Γ u ⁰ _e )Π d + µ int Π d

[∇ Γ u ⁰ _e ] + ^t [∇ Γ u ⁰ _e ] Π d

. (2.6)

In this case we approximate the solution u ^ε _e by the function v ^ε _[1] that satisfies Problem (2.1) with the following boundary condition on Γ

(A _ext e(v ^ε _[1] ))n − εdiv _Γ

2λ int µ int

λ _int + 2µ _int (div _Γ v ^ε _[1] )Π _d + µ _int Π _d

[∇ _Γ v ^ε _[1] ] + ^t [∇ _Γ v ^ε _[1] ] Π _d

= 0.

Setting u ^ε _[1] := u ⁰ _e + εu ¹ _e , we deduce the following estimate which permits to prove Theorem 2.3 below.

Lemma 2.2. We have

(A _ext e(u ^ε _[1] − v ^ε _[1] ))n + C ^ε,1 (u ^ε _[1] − v ^ε _[1] )

= ε ² div Γ

2λ int µ int

λ int + 2µ int

(div Γ u ¹ _e )Π d + µ int Π d

[∇ Γ u ¹ _e ] + ^t [∇ Γ u ¹ _e ] Π d

= O(ε ² ). (2.7)

2.2 Convergence analysis (Proof of Theorem 1.2)

We focus now on the remainder r ^ε _[N] := v ^ε _{[N ]} − u ^ε _e for N = 0, 1. We obtain the following result which proves Theorem 1.2.

Theorem 2.3. Let N = 0, 1. If f ∈ H ^1/2 (Ω\ω), then there exists a constant C _Ω\ω independent on ε such that

r ^ε _{[N ]}

_H1

(Ω\ω) ≤ C _Ω\ω ε ^N+1 .

Proof. We follow the procedure described in the proof of [23, Theorem 4.1]. The idea is to de- compose the remainder, for N = 0, 1, as r ^ε _{[N ]} := v ^ε _{[N ]} − u ^ε _{[N ]} + u ^ε _[N] − u ^ε _e , where u ^ε _[0] := u ⁰ _e and u ^ε _[1] := u ⁰ _e + εu ¹ _e .

Firstly we set R ^N _ε := u ^ε −

N

X

n=0

ε ⁿ u ⁿ and we denote by R ^N _ε,e and R ^N _ε,i the restriction of R ^N _ε respectively to Ω\ω and to ω _ext . It is straightforward to show that R ^N _ε = (R ^N _ε,e , R ^N _ε,i ) is a solution of the following system



 

 

 

 

 

 

 

 

 

 

−div A _ext e(R ^N _ε,e ) = 0 in Ω\ω

−div A int e(R ^N _ε,i ) = O(ε ^N

⁻¹

) in ω _int ^ε

R ^N _,e = 0 on Γ D

A ext e(R ^N _ε,e ) = 0 on Γ N

A ext e(R ^N _ε,e )n = A int e(R ^N _ε,i )n + O(ε ^N ) on Γ Λ ₀ R ^N _ε,e = Λ ₀ R ^N _ε,i on Γ A int e(R ^N _ε,i )n = O(ε ^N ) on Γ ^ε .

The remainders are considered as distributions in H

⁻¹

(ω ^ε _int ), H

^−1/2

(Γ) and H

^−1/2

(Γ ^ε ) respectively.

Taking the variational form for the transmission problem, one can prove the uniform coercivity with respect to ε of the associated bilinear form and the continuity of the bilinear and linear forms. We get, thanks to the Lax-Milgram theorem and the Korn’s inequality, that there exists a constant C, depending on N but independent on ε, such that

R ^N _ε,e

_H1

(Ω\ω) ≤ C ε ^N−1 . To improve the precision of the remainder, we write

R ^N _ε,e = R ^N _ε,e ⁺² + ε ^{N +1} u ^N+1 _e + ε ^N+2 u ^N _e ⁺² and get the estimation

u ^ε _e − u ^ε _[N]

_H1

(Ω\ω) = R ^N _ε,e

_H1

(Ω\ω) ≤ C ε ^{N +1} . (2.8)

Secondly let us focus on d _{[N ]} := u ^ε _[N] − v ^ε _{[N ]} . For N = 0, we have u ^ε _[0] = u ⁰ _e = v ^ε _[0] , i.e. d _[0] = 0.

Hence, thanks to the previous estimate (2.8), there exists a positive constant C 0 , independent of ε,

such that

r ^ε _[0]

_H1

_(Ω\ω) ≤ C ₀ ε.

For N = 1, d _[1] = u ^ε _[1] − v ^ε _[1] satisfies the following system



 



 



−div A ext e(d [1] ) = 0 in Ω\ω d [1] = 0 on Γ D

A _ext e(d _[1] )n = 0 on Γ _N (A _ext e(d _[1] )n + C ^ε,1 (d _[1] ) = ε ² div _Γ (σ

_Γ

(u ¹ _e )) on Γ.

Let us notice that the last equality comes from Lemma 2.2 and the right hand side is considered as a distribution in V

⁰

(Γ). Writing the variational form, we deduce, thanks to Proposition A.1 and Remark 1.1, that there exist two positive constants C, C

⁰

> 0 such that

d _[1]

H(Ω\ω)

≤ Cε ² ||div Γ (σ

_Γ

(u ¹ _e ))||

_V⁰

_(Γ) ≤ C

⁰

ε ² ||u ¹ _e ||

_V(Γ)

.

Moreover, thanks to the condition f ∈ H ^1/2 (Ω\ω), the local regularity in a neighborhood of ω ensures that the solution u ⁰ _e of Problem (2.3) (with the boundary condition given by (2.4)) belongs to H ^5/2 (Ω δ \ω) for some smooth open set Ω δ such that ω ⊂⊂ Ω δ ⊂⊂ Ω\ω. Then u ⁰ _{e Γ} ∈ H ² (Γ) and div Γ (σ

_Γ

(u ⁰ _e )) _Γ ∈ L ² (Γ). Using the local regularity of the solution u ¹ _e of Problem (2.5) with the Neumann boundary condition (2.6), we obtain that there exists C

⁰⁰

> 0 such that

||u ¹ _e ||

_H3/2

(Ω

δ\ω)

≤ C

⁰⁰

||div Γ (σ

_Γ

(u ⁰ _e ))||

_L²

_(Γ) .

Finally, using the continuity of the trace operator and the continuous embedding H ¹ (Γ) ⊂ V(Γ), we obtain that there exists C

⁰⁰⁰

> 0

d _[1]

_H(Ω\ω)

≤ C

⁰⁰⁰

ε ² ||div Γ (σ

_Γ

(u ⁰ _e ))||

_L²

_(Γ) .

Using the triangular inequality and inequality (2.8), we conclude that there exists a positive con- stant C 1 , independent of ε, such that

r ^ε _[1]

_H1

(Ω\ω) ≤ C 1 ε ² .

Remark 2.4. The regularity results used in the above proof can be found for example in [43, Chapter 4] or in [13, Theorems 9.25 and 9.26, and Remark 24] (notice that the mentioned results in [13] concern global regularity results but the local regularity results are then classically deduced using a cut-off function).

3 Shape sensitivity analysis

This section is devoted to the proof of the existence of the shape derivative of the state u, solution of Problem (1.4). This step is done following the general method exposed for example in [35, Chapter 5] which consists in using a change of variable and the implicit function theorem. Then we prove the characterization of the shape derivative u

⁰

claimed in Theorem 1.3. We conclude this section with an asymptotic analysis for the shape derivatives of the transmission problem (1.2) which shows that we can interchange the asymptotic and shape sensitivity analysis.

We have first to introduce some notations. Let us define Ω δ an open set with a C

^∞

boundary such that

{x ∈ Ω ; d(x, ∂Ω) > δ/2} ⊂ Ω δ ⊂ {x ∈ Ω ; d(x, ∂Ω) > δ/3} . Then, in order to make a shape sensitivity analysis, we define

U :=

θ ∈ C ^2,∞ = C ² ∩ W ^2,∞ ( R ^d ); supp(θ) ⊂ Ω _δ and kθk 2,∞ < min δ

3 , 1

(3.1)

as the space of admissible deformations. This space enables us to perturb only the object ω and

not the fixed domain Ω.

3.1 Existence of the shape derivatives for the GIBC model

In this section, we prove that the solution u ∈ H(Ω\ω) of the GIBC Problem (1.4) is differentiable with respect to the shape ω. Notice that if θ ∈ U, then (I + θ) is a diffeomorphism. For such a θ ∈ U and ω ∈ O δ , we check that Ω = (I + θ)(Ω) and we define the perturbed domain

ω θ := (I + θ)(ω) ∈ O δ .

First we need to introduce the following Piola transform and prove the following result.

Proposition 3.1. Let θ ∈ U. The adjoint Piola transform of (I + θ)

⁻¹

defined by P

^∗

(θ) : H(Ω\ω) −→ H(Ω\ω _θ )

u 7−→

(I d + ∇θ)

⁻¹

u

◦ (I + θ)

⁻¹

is an isomorphism and we denote by P

^−∗

(θ) its inverse.

Proof. The only point is to prove that P

^∗

(θ)ψ belongs to H(Ω\ω θ ) for all ψ ∈ H(Ω\ω). Firstly, for ψ ∈ H(Ω\ω), it is obvious that P

^∗

(θ)ψ belongs to H ¹ (Ω\ω θ ) and that ψ = 0 on Γ D . Sec- ondly, the restriction of (I _d + ∇θ)

⁻¹

to the boundary Γ maps

ψ ∈ H ¹ (Γ); ψ · n = 0 into H ¹ (Γ).

Moreover, we have [35, Proposition 5.4.14]

n θ := P

^∗

(θ)n

kP

^∗

(θ)nk . (3.2)

Then P

^∗

(θ) is an isomorphism from V(Γ) to V(Γ θ ) (i.e. the tangential component remains in

ψ ∈ H ¹ (Γ θ ); ψ · n θ = 0 ).

Let θ ∈ U. In what follows, ∇ Γ

_θ

and div Γ

_θ

are respectively the tangential gradient and surface divergence operators according to the boundary Γ θ . Let consider the solution u θ ∈ H(Ω\ω θ ) of the following perturbed problem



 

 

 

 

−div (A ext e(u _θ )) = f in Ω\ω θ

u _θ = 0 on Γ _D (A _ext e(u _θ )) n = g on Γ _N

A ext e

_Γ

θ

(u θ )

n θ − εdiv Γ

_θ

(σ

_Γ

θ

(u θ )) = 0 on Γ θ ,

(3.3)

where n _θ represents the exterior unit normal to Γ _θ given by (3.2), where e

_Γ

θ

(u θ ) := 1

2 Π d,θ ∇ Γ

_θ

u θ + ^t ∇ Γ

_θ

u θ

Π d,θ and Π d,θ := I d − n θ ⊗ n θ , and where

σ

_Γ

θ

(u θ ) := λ int (div Γ

_θ

u θ )Π d,θ + 2µ int e

_Γ

θ

(u θ ).

Then u θ ∈ H(Ω\ω θ ) is the solution of Z

Ω\ω

θ

A ext e(u θ ) : e(ϕ _θ ) − Z

Γ

N

g · ϕ _θ + ε Z

Γ

θ

σ

_Γ

θ

(u θ ) : ∇ Γ

_θ

ϕ _θ = Z

Ω\ω

θ

f · ϕ _θ ,

∀ϕ _θ ∈ H(Ω\ω θ ). (3.4) We will apply the general method exposed in [35, Chapter 5] and [27, Section 3.1] using the adjoint Piola transform and the implicit function theorem on

u ^θ := P

^−∗

(θ)u _θ ∈ H(Ω\ω).

Lemma 3.2. For θ ∈ U, u ^θ ∈ H(Ω\ω) satisfies for all ϕ ∈ H(Ω\ω)

λ _ext Z

Ω\ω

J

_θ

^t D ₁ (θ) : ∇(D ₁ (θ)u ^θ ) t D ₁ (θ) : ∇(D ₁ (θ)ϕ) + µ ext

Z

Ω\ω

J

θ

D 1 (θ)∇(D 1 (θ)u ^θ )

: D 1 (θ)∇(D 1 (θ)ϕ) + ^t (D 1 (θ)∇(D 1 (θ)ϕ))

− Z

Γ

_N

g · ϕ + ελ _int Z

Γ

w

_θ

^t D 2 (θ) : ∇ Γ (D 1 (θ)u ^θ ) t

D 2 (θ) : ∇ Γ (D 1 (θ)ϕ) + εµ int

Z

Γ

w

θ

D 2 (θ)∇ Γ (D 1 (θ)u ^θ )

: D 2 (θ)∇ Γ (D 1 (θ)u ^θ ) + ^t D 2 (θ)∇ Γ (D 1 (θ)u ^θ )

− εµ int

Z

Γ

w

θ

D 2 (θ)∇ Γ (D 1 (θ)u ^θ )n ^θ

· D 2 (θ)∇ Γ (D 1 (θ)ϕ)n ^θ

= Z

Ω\ω

(f ◦ (I + θ)) · D 1 (θ)ϕ J θ , where n ^θ := n θ ◦ (I + θ) and where

J

θ

:= det (I d + ∇θ) ∈ W ^1,∞ R ^d , w θ := det (I d + ∇θ)

t (I d + ∇θ)

⁻¹

n

∈ W ^1,∞ R ^d , D 1 (θ) := (I d + ∇θ)

⁻¹

∈ C ¹ R ^d , M d,d

, D 2 (θ) := (I d + ∇ Γ θ)

⁻¹

∈ C ¹ (Γ, M d,d ) .

Proof. Let ϕ ∈ H(Ω\ω). We define ϕ

θ

:= P

^∗

(θ)ϕ ∈ H(Ω\ω θ ). Thus, using ϕ

θ

as a test function in the variational formulation (3.4), we obtain

λ ext

Z

Ω\ω

θ

div (P

^∗

(θ)u ^θ )

(div (P

^∗

(θ)ϕ)) + 2µ ext

Z

Ω\ω

θ

∇(P

^∗

(θ)u ^θ ) : e(P

^∗

(θ)ϕ)

− Z

Γ

_N

g · (P

^∗

(θ)ϕ) + ελ _int Z

Γ

_θ

div _Γ

_θ

(P

^∗

(θ)u ^θ )

(div _Γ

_θ

(P

^∗

(θ)ϕ)) + 2εµ int

Z

Γ

θ

∇ Γ

_θ

(P

^∗

(θ)u ^θ ) : e

_Γ

θ

(P

^∗

(θ)ϕ) = Z

Ω\ω

θ

f · (P

^∗

(θ)ϕ).

Hence, using the change of variables x = (I + θ)(y) in the integrand and the following formulas

∇(v ◦ (I + θ)

⁻¹

) ◦ (I + θ) = D 1 (θ)∇v,

div (v ◦ (I + θ)

⁻¹

) ◦ (I + θ) = Tr[∇(v ◦ (I + θ)

⁻¹

)] ◦ (I + θ) = ^t D 1 (θ) : ∇v ,

∇ Γ

θ

(v ◦ (I + θ)

⁻¹

) ◦ (I + θ) = D 2 (θ)∇ Γ v,

div Γ

_θ

(v ◦ (I + θ)

⁻¹

) ◦ (I + θ) = Tr[∇ Γ

_θ

(v ◦ (I + θ)

⁻¹

)] ◦ (I + θ) = ^t D 2 (θ) : ∇ Γ v,

∇ _Γ

_θ

(u _θ ) : e

_Γ

θ

(ϕ _θ ) = ¹ ₂ ∇ _Γ

_θ

(u _θ ) : (∇ _Γ

_θ

(ϕ) + ^t ∇ _Γ

_θ

(ϕ)) − ¹ ₂ (∇ _Γ

_θ

(u _θ )n _θ ) · (∇ _Γ

_θ

(ϕ _θ )n _θ ), (P

^∗

(θ)ϕ)

_|ΓN

= ϕ

_|ΓN

,

we obtain the announced result.

Lemma 3.3. The function

θ ∈ U 7→ u ^θ ∈ H(Ω\ω)

is differentiable in a neighborhood of 0.

Proof. Let us consider the application F : U × H(Ω\ω) → L ² (Ω) defined for all ϕ ∈ H(Ω\ω) by hF (θ, v) , ϕi = λ _ext

Z

Ω\ω

J

_θ

^t D ₁ (θ) : ∇(D ₁ (θ)v) t D ₁ (θ) : ∇(D ₁ (θ)ϕ) + µ ext

Z

Ω\ω

J

θ

(D 1 (θ)∇(D 1 (θ)v)) : D 1 (θ)∇(D 1 (θ)ϕ) + ^t (D 1 (θ)∇(D 1 (θ)ϕ))

− Z

Γ

_N

g · ϕ + ελ _int Z

Γ

w

_θ

^t D 2 (θ) : ∇ Γ (D 1 (θ)v) t

D 2 (θ) : ∇ Γ (D 1 (θ)ϕ) + εµ int

Z

Γ

w

θ

(D 2 (θ)∇ Γ (D 1 (θ)v)) : D 2 (θ)∇ Γ (D 1 (θ)v) + ^t (D 2 (θ)∇ Γ (D 1 (θ)v))

− εµ int

Z

Γ

w

θ

D 2 (θ)∇ Γ (D 1 (θ)v)n ^θ

· D 2 (θ)∇ Γ (D 1 (θ)ϕ)n ^θ

− Z

Ω\ω

(f ◦ (I + θ)) · D 1 (θ)ϕ J θ . We have, for θ = 0, F(0, u ⁰ ) = F(0, u) = 0. Moreover, we prove analogously to what is done in [35, Proof of Theorem 5.3.2] that the application F is C ¹ . Finally, we compute for all w, w ˆ ∈ H(Ω),

D

u

F(0, u)(w) · w ˆ = Z

Ω\ω

A ext e(w) : e( ˆ w) + ε Z

Γ

σ

_Γ

(w) : ∇ Γ w. ˆ Thus, D

u

F(0, u) is an isomorphism.

Hence, the implicit function theorem applies and then there exists a C ¹ function θ ∈ U 7→

v(θ) ∈ H(Ω\ω) such that F(0, v(θ)) = 0 in a neighborhood of 0. Using the uniqueness of the solution of such a problem, we obtain that θ ∈ U 7→ u ^θ ∈ H(Ω\ω) is C ¹ .

Lemma 3.4. There exists u e θ an extension in R ^d of u θ such that θ ∈ U 7→ u e θ ∈ L ² ( R ^d ) is differentiable at 0.

Proof. For θ ∈ U, u _θ = P

^∗

(θ)u ^θ ∈ H(Ω\ω _θ ) ⊂ H ¹ (Ω\ω _θ ). According to the differentiability of θ 7→ u ^θ (see Lemma 3.3) and Stein’s extension theorem (see, e.g., [2, Theorem 5.24]), there exists u e ^θ , an extension of u ^θ , such that θ ∈ U 7→ u e ^θ ∈ H ¹ ( R ^d ) is differentiable at 0. Moreover,

θ ∈ U 7→ (I + θ)

⁻¹

− I ∈ C ^1,∞ ( R ^d ) and θ ∈ U 7→ (I _d + ∇θ)

⁻¹

∈ C ^1,∞ ( R ^d ) are differentiable at 0. Thus,

V 1 : θ ∈ U 7→

(I d + ∇θ)

⁻¹

u e ^θ , (I + θ)

⁻¹

− I

∈ H ¹ ( R ^d ) × C ^1,∞ ( R ^d ) is differentiable at 0. We apply [35, Lemma 5.3.9] to get that

V 2 : (h, µ) ∈ H ¹ ( R ^d ) × C ^1,∞ ( R ^d ) 7→ h ◦ (I + µ) ∈ L ² ( R ^d ) is C ¹ . By composition, θ ∈ U 7→ u e θ := V 2 ◦ V 1 (θ) ∈ L ² ( R ^d ) is differentiable at 0.

Using the Fréchet differentiability given by this lemma, we obtain the Gâteaux differentiability in the direction V ∈ U: there exists u e t , extension in R ^d of u t such that the function t ∈ [0, T ) 7→

u e t ∈ L ² ( R ^d ) is differentiable at 0 by composition. Hence, we prove the following result.

Lemma 3.5. The solution u of Problem (1.4) is differentiable with respect to the domain ω ∈ O δ .

We denote by u

⁰

its derivative at 0.

3.2 Shape derivatives for the GIBC problem (Proof of Theorem 1.3)

Since we proved in the previous section the differentiability with respect to the domain, we can now compute the shape derivatives of the state. For the reader’s convenience, let us recall the definition of the shape derivative in our situation (see [35] for details).

• If the mapping θ ∈ U 7→ u _θ ◦ (I + θ) ∈ L ² (Ω\ω) is Fréchet differentiable at 0, we say that θ 7→ u _θ possesses a total first variation (or derivative) at 0. In such a case, this total first derivative at 0 in the direction θ is denoted by u ^. _θ .

• If, for every D ⊂⊂ Ω\ω, the mapping θ ∈ U 7→ u θ

D

∈ L ² ( D ) is Fréchet differentiable at 0, we say that θ 7→ u θ possesses a local first variation (or derivative) at 0. In such a case, this local first derivative at 0 in the direction θ is denoted by u

⁰

_θ , is called shape derivative and is well defined in the whole domain Ω\ω:

u

⁰

_θ = d

dt (u _tθ

_D

) _t=0 in each D ⊂⊂ Ω\ω.

In the following, for V ∈ U , we denote by u

⁰

the local first variation u

⁰

_V which is referred as the shape derivative of the state. We also use the notation

V n := V · n.

Let T > 0. For a given perturbation V ∈ U and t ∈ [0, T ), we define ω t := (I + t V ) (ω)

with boundary Γ t and we consider the solution u t ∈ H(Ω\ω t ) of the following perturbed GIBC problem:



 



 



−div (A ext e(u _t )) = f in Ω\ω t

u _t = 0 on Γ _D (A _ext e(u _t )) n = g on Γ _N (A _ext e(u _t )) n _t − εdiv _Γ

_t

(σ

_Γ

t

(u _t )) = 0 on Γ _t ,

(3.5)

where n t represents the exterior unit normal to Γ t , where e

_Γt

(u t ) := 1

2 Π d,t ∇ Γ

t

u t + ^t ∇ Γ

t

u t

Π d,t and Π d,t := I d − n t ⊗ n t , and where

σ

_Γ

t

(u _t ) := λ _int (div _Γ

_t

u _t )Π _d,t + 2µ _int e

_Γ

t

(u _t ).

Proof of Theorem 1.3. For t ∈ [0, T ), we consider Problem (3.5). By differentiating with respect to the shape at t = 0, we classically obtain (see, e.g., [42, Theorem 3.3] or also [35, Chapter 5])







−div (A ext e(u

⁰

)) = 0 in Ω\ω u

⁰

= 0 on Γ _D (A _ext e(u

⁰

)) n = 0 on Γ _N .

Let us now compute the shape derivative of the GIBC (A _ext e(u _t )) n _t −div _Γ

_t

(σ

_Γ

t

(u _t )) = 0 on Γ _t . We define by τ t the transformation that maps the restriction of u t to Γ t onto the restriction of u t ◦ (I+ tV ) to Γ. Since we know that the shape derivative depends only on the normal component of the variations V (see, e.g., [35, Theorem 5.9.2]), we compute the material derivatives in the direction V = V n n. We first recall that we have [35, Corrolaire 5.2.5]

d dt

h τ _t u _t i

t=0

= u

⁰

+ V _n ∂

_n

u.

Then, we get d

dt h

(τ t A ext e(u t ))(τ t n t ) i

t=0

= (A ext e(u

⁰

))n − (A ext e(u))∇ Γ V n + V n ∂

n

(A ext e(u))n

= (A _ext e(u

⁰

))n − (A _ext e(u))∇ Γ V _n + V _n (div −div Γ )(A _ext e(u))

= (A ext e(u

⁰

))n − (A ext e(u))∇ Γ V n − V n f − V n div Γ (A ext e(u))

= (A _ext e(u

⁰

))n − V _n f − div _Γ (V _n A _ext e(u)).

Since (A _ext e(u))n = div _Γ (σ

_Γ

(u)), we have d

dt

h (τ _t A _ext e(u _t ))(τ _t n _t ) i

t=0

= (A _ext e(u

⁰

))n − V _n f − div _Γ

V _n Π _d (A _ext e(u))Π _d

−V n H(A ext e(u))n − div Γ (V n Π d (A ext e(u))n ⊗ n).

Moreover, we have (see [22, Theorem 5.4]) d

dt h

τ t ∇ Γ

t

τ _t

⁻¹

i

t=0 u = −V n [D ² b∇ Γ u] + n ⊗ ( ^t [∇ Γ u]∇ Γ V n ) and (see [22, Theorem 5.6])

d dt h

τ _t div _Γ

_t

τ _t

⁻¹

i

t=0

u = −V _n Tr[D ² b∇ _Γ u] + [∇ _Γ u]n · ∇ _Γ V _n

= −V n div Γ (D ² b − HΠ d )u + Hdiv Γ Π d u + Tr[(D ² b) ² ](u · n) +∇ Γ (u · n) · ∇ Γ V n − [D ² b]u · ∇ Γ V n .

Then, d dt

h

τ t div Γ

_t

τ _t

⁻¹

i

t=0 σ

_Γ

(u) =

− V _n

div _Γ (D ² b − HΠ _d )σ

_Γ

(u)

+ Hdiv _Γ (σ

_Γ

(u))

− ^t ([D ² b]σ

_Γ

(u))∇ Γ V _n . We have

σ

_Γ

t

(u) = λ _int (τ _t div _Γ

_t

τ _t

⁻¹

u) I − τ _t n _t ⊗ τ _t n _t + 2µ int I − τ t n t ⊗ τ t n t 1

2 [τ t ∇ Γ

_t

τ _t

⁻¹

u] + ^t [τ t ∇ Γ

_t

τ _t

⁻¹

u] I − τ t n t ⊗ τ t n t

and d

dt

h I − τ _t n _t ⊗ τ _t n _t i

t=0

= n ⊗ ∇ Γ V _n + ∇ Γ V _n ⊗ n.

Hence we obtain d

dt h

σ

_Γt

i

t=0 (u) = −V n λ int (Tr[[D ² b]∇ Γ u])Π d + 2µ int Π d

₁

2 [[D ² b]∇ Γ u] + ^t [[D ² b]∇ Γ u]

Π d

+ λ _int ([∇ Γ u]n · ∇ Γ V _n )Π _d + λ _int (div _Γ u) n ⊗ ∇ Γ V _n + ∇ Γ V _n ⊗ n

+ 2µ _{int 1}

2 ([∇ Γ u]n ⊗ ∇ Γ V _n + ∇ Γ V _n ⊗ [∇ Γ u]n) + 2µ _int

(n ⊗ (e

_Γ

(u)∇ Γ V _n ) + (e

_Γ

(u)∇ Γ V _n ) ⊗ n) and we deduce

div Γ

d dt

h σ

_Γ

t

i

t=0 (u) =

− div _Γ

V _n λ _int (Tr[[D ² b]∇ _Γ u])Π _d + 2µ _int Π _{d 1}

2 [[D ² b]∇ _Γ u] + ^t [[D ² b]∇ _Γ u]

Π _d + div _Γ

λ _int ([∇ _Γ u]n · ∇ _Γ V _n )Π _d + 2µ _int 1

2 ([∇ _Γ u]n ⊗ ∇ _Γ V _n + ∇ _Γ V _n ⊗ [∇ _Γ u]n) + (HΠ d + [D ² b] + ndiv Γ Π d ) σ

_Γ

(u)∇ Γ V n

.

We conclude using the chain and product rules that d

dt h

(τ t A ext e(u t ))(τ t n t ) i

t=0 + d dt

h

τ t div Γ

_t

τ _t

⁻¹

i

t=0 σ

_Γ

(u) + div Γ

d dt h

σ

_Γ

t

i

t=0

(u) + div Γ σ

_Γ

d

dt h

τ t u t

i

t=0

= 0.

3.3 Asymptotic analysis for the shape derivative of the transmission problem

This section focuses on the asymptotic analysis of the shape derivative of the solution to the thin layer transmission problem (1.2). First we give a characterization of the shape derivative as a solution to a new transmission problem with non-vanishing jumps. The existence of the shape derivative follows from standard procedures (see, e.g., [42, Chapter 3.4]). Hence we claim the following result which characterizes the shape derivative of the state (see, e.g., [42, Theorem 3.3]

and [38] for the jump formula).

Theorem 3.6. The thin layer transmission problem characterizing the first shape derivatives in the direction V is



 

 

 

 



 

 

 

 



−div A _ext e(u

^0ε

_e ) = 0 in Ω\ω

−div A _int e(u

^0ε

_i ) = 0 in ω _int ^ε

u

^0ε

_e = 0 on Γ _D

(A ext e(u

^0ε

_e )) n = 0 on Γ N

A ext e(u

^0ε

_e )n − A int e(u

^0ε

_i )n = V n f + div Γ V n Π d A ext e(u ^ε _e )Π d − div Γ V n Π d A int e(u ^ε _i )Π d on Γ u

^0ε

_e − u

^0ε

_i = −V n (∂ n u ^ε _e − ∂ n u ^ε _i ) on Γ A int e(u

^0ε

_i )n ^ε = div ^ε _Γ τ

⁻¹

_ε (V n ) A int e ^ε

_Γ

(u ^ε _i )

on Γ ^ε . (3.6) Then, for N = 0, 1, we want to approximate the shape derivative u

^0ε

, solution of Problem (3.6), by the solution w ^ε _{[N ]} of some boundary value problems of the form



 

 



 

 



−div A ext e(w ^ε _[N] ) = 0 in Ω\ω w ^ε _[N] = 0 on Γ D

A _ext e(w ^ε _[N] )

n = 0 on Γ _N C

ε, A _ext e(w ^ε _[N] )n, w ^ε _[N]

= h on Γ, where C

ε, A ext e(w ^ε _{[N ]} )n, w ^ε _{[N ]}

is a so-called Generalized Impedance Boundary Condition (GIBC).

The following result gives the GIBC modeling an approximation up to O(ε ^N+1 ) of u

^0ε

_e for N = 0, 1.

We recall that u ^ε _[0] := u ⁰ _e and u ^ε _[1] := u ⁰ _e + εu ¹ _e (see Section 2).

Proposition 3.7. The GIBC, defined on Γ, modeling the interior thin layer effects of the inclu- sion ω on the first shape derivative for N = 0, 1, are given by C

ε, A ext e(w ^ε _{[N ]} )n, w ^ε _{[N ]}

= h with

C

ε, A _ext e(w ^ε _{[N ]} )n, w ^ε _{[N ]} :=

A _ext e(w ^ε _{[N ]} )n

+ C ^ε,N w ^ε _{[N ]} , where C ^ε,0 := 0 and C ^ε,1 w := −εdiv Γ A int e

_Γ

(w)

, and with h := V n f + F ^ε,N ₁ where F ^ε,0 ₁ := div _Γ

V _n Π _d

A _ext e(u ^ε _[0] ) Π _d

= div _Γ V _n

A _ext e

_Γ

(u ^ε _[0] ) and

F ^ε,1 ₁ := div Γ

V n Π d

A ext e(u ^ε _[1] ) Π d

− εdiv Γ

V n ([D ² b] − H Π d )

A int e

_Γ

(u ^ε _[1] )

− εdiv Γ

V n

A int

h 1

2 Π d [[D ² b]∇ Γ u ^ε _[1] ] + ^t [[D ² b]∇ Γ u ^ε _[1] ] Π d

i

+ εdiv Γ

A int e

_Γ

V n ∂

n

u ^ε _[1] − Λ

⁻¹

₀ A ext e(u ^ε _[1] )n + ε [D ² b] + n div Γ Π d

div Γ

V n

A int e

_Γ

(u ^ε _[1] ) + εdiv Γ

A int

h 1

2 [∇ Γ u ^ε _[1] ]n ⊗ ∇ Γ V n + ∇ Γ V n ⊗ [∇ Γ u ^ε _[1] ]n i

.