TOPOLOGICAL SENSITIVITY ANALYSIS FOR THE ANISOTROPIC LAPLACE PROBLEM

This paper is concerned with the reconstruction of objects immersed in anisotropic media from boundary measurements. The aim of this paper is to propose an alternative approach based on the KohnVogelius formulation and the topological sensitivity analysis method. The idea is to formulate the reconstruction problem as a topology optimization one minimizing an energy-like function. We derive a topological asymptotic expansion for the anisotropic Laplace operator. The unknown object is reconstructed using level-set curve of the topological gradient. We make finally some numerical examples proving the efficiency and accuracy of the proposed algorithm.


Introduction
In this work we will establish a topological sensitivity analysis for the anisotropic Laplace operator. The topological sensitivity analysis consists of studying the variation of a given cost functional with respect to the presence of a small domain perturbation, such as the insertion of inclusions, cavities, cracks or source-terms.
In our paper we concentrate in a small Dirichlet geometric perturbation.
Let us briefly discuss the history of this method. Its main idea was originally introduced by Schumacher [22] in the context of compliance minimization in linear elasticity. In the same context Sokolowski and Zochowski [13], who studied the effect of an extract infinitesimal part of the material in structural mechanics.
Then in [17] Masmoudi worked out a topological sensitivity analysis framework based on a generalization of the adjoint method and on the use of a truncation technique. By using this framework the topological sensitivity is obtained for several equations [8,18,20,21]. For other works on the topological sensitivity concept, we refer to the book by Novotny and Sokolowski [19].
The general idea of the proposed method is to rephrase the inverse problem as a topology optimization problem, where the object immersed in the anisotropic media is the unknown variable. Our aim is to detect this unknown object immersed from over-determined boundary data.
Let H be an unknown object immersed inside the background domain Ω and having a smooth boundary Σ = ∂H. The geometric inverse problem that we consider here can be formulated as follows: • Giving two boundary data on Γ ; an imposed flux Φ ∈ H −1/2 (Γ) and a measured datum ϕ d ∈ H 1/2 (Γ).
• Find the unknown location of the object H inside the domain Ω such that the solution ϕ of the anisotropic Laplace equation satisfies the following over-determined boundary value problem where A is a symmetric positive definite matrix, n is the exterior unit normal vector and F ∈ L 2 (Ω) is a given source term.
In this formulation the domain Ω\H is unknown since the free boundary is unknown. This problem is ill-posed in the sense of Hadamard [10]. The majority of works dealing with this kind of problems fall into the category of shape optimization and based on the shape differentiation technics. It is proved in [3,7] that the studied inverse problems, treated as a shape optimization problems, are severely ill-posed (i.e. unstable), for both Dirichlet and Neumann conditions on the boundary . Thus they have to use some regularization methods to solve them numerically.
To solve this inverse problem, we extend the topological sensitivity analysis notion to the anisotropic case and we suggest an alternative approach based on the Kohn-Vogelius formulation [6] and the topological gradient method [1,2,5,8,9,16,17]. We combine here the advantages of the Kohn-Vogelius formulation as a self regularization technique and the topological gradient approach as an accurate and fast method.
The first step of our approach is based on the Kohn-Vogelius formulation which rephrase the considered geometrical inverse problem into a topology optimization one. It leads to define for any given permutation H two forward problems: The first one, called Neumann problem, is associated to the Neumann datum Φ The second one is associated to the measured data ϕ d , it is called the Dirichlet problem: One can observe that if H coincides with the exact obstacle H * , the misfit between the solutions of(P N ) and (P D ) vanishes: ϕ N = ϕ D . Starting from this observation, the inverse problem can be formulated as a topological optimization one. The unknown object will be characterized as the minimum of the following Kohn-Vogelius type functional [6] J Where the Kohn-Vogelius function is exactly More precisely, the identification problem can be formulated as follow: where D ad is a given set of admissible domains.
To solve the topological optimization problem (P) and detect the location of the unknown object we will derive a topological sensitivity analysis for the Kohn-Vogelius function J which gives the variation of a criterion with respect to the presence of a small Dirichlet geometric perturbation in the domain. A one-shot reconstruction algorithm is proposed. The main advantage of this algorithm is that, it provides fast and accurate results for detection.
The paper is organized as follows. In the next Section, we present the perturbed Neumann and Dirichlet problems. In Section 3 we study the topological sensitivity analysis for the function J . The obtained results are based on a preliminary estimate describing the perturbation caused by the presence of a small geometry modification of the background domain Ω. A simplified formulation of the shape function variation with respect to the creation of the hole H z,ε in Ω is derived in Section 4. The Section 5 is devoted to the Kohn-Vogelius type function variation. The proposed numerical algorithm and the detection results are described in Section 6.

The perturbed problems
In this section, we present the Neumann and Dirichlet problems in the perturbed domain. In the presence of a small geometry perturbation H z,ε inside the domain Ω, the Neumann problem consists in finding The Neumann problem in the non perturbed domain is: Similarly, the perturbed Dirichlet problem consists in finding ϕ ε In the absence of any perturbation (i.e ε = 0), the Dirichlet problem is: We introduce the considered shape functional J . Given a small geometrical perturbation H z,ε inside the initial domain Ω, the function J measures the difference between the Neumann and Dirichlet perturbed solutions. We define J as In the non perturbed domain (ε = 0), the function J is expressed as Our aim is to derive an asymptotic expansion for the function J and calculate the topological sensitivity function δJ .
The variation of the function J with respect to the presence of a small perturbation is given by In the following, we will derive a topological sensitivity analysis valid for all function J ε verifying the following hypothesis: -There exist a real number δJ ∈ R, independent of ε and a scalar function ρ : with ϕ ε is the solution of the perturbed anisotropic Laplace problem inside the perturbed domain Ω\H z,ε Here δJ is called the topological sensitivity function.

Topological asymptotic expansion
In this section, we derive a topological asymptotic expansion for the anisotropic Laplace operator. We start our analysis by establishing a variational formulation associated to the anisotropic Laplace system.
From the weak formulation of 2.1, we deduce that ϕ ε ∈ V ε is the unique solution to where the function space V ε , the bilinear form a ε and the linear form l ε are defined by Under the hypothesis 2.1, the variation of the shape function J reads Let v 0 ∈ V 0 be the solution to the associated adjoint problem Then, the shape function variation rewritten as Aiming to derive an asymptotic expansion for J , we examine in the next section the asymptotic behavior with respect to ε of the term a 0 (ϕ 0 − ϕ ε , v 0 ). The main result of this section is summarized in the following theorem.
Theorem 3.1. Let j a design function of the form j(Ω\H z,ε ) = J (ϕ ε ). If J satisfies the assumption 2.1 , then j has the following asymptotic expansion The term δJ is the variation of the considered cost function J .
In order to check the hypothesis 2.1, we derive an asymptotic expansion the variation of the bilinear form.
We have Using the Green formula, we obtain Next, we shall examine each term on the right hand side of 3.2 separately. The following lemma gives an estimate for the first term.

Proof
Using the change of variable x = z + εy one obtains Due the the smoothness of ϕ 0 and v 0 in H z,ε , we derive Then it follows Next to examine the second term of 3.2 we introduce the variable it is easily to show that χ ε satisfies the following system We can write χ ε as with E is the fundamental solution of the anisotropic Laplace operator [14]: where |A| is the determinant of A, A * be the positive-definite symmetric matrix such that A 2 * = A −1 . Then r ε is solution of A∇r ε · n = −A∇h ε · n on Γ n , We set Using the Green formulation, we obtain Assuming that R i (ε) = o (ρ(ε)) , i = 1, 2. We will give the proof for the two dimensional case in section 4.3.
Besides thanks to the fundamental solution, we obtain the main result presented in the following sections concerns the topological asymptotic expansion of an arbitrary design function j. Some cost function examples are presented in section 4.

A particular class of cost function
In this section, we present some useful examples of shape functions and we gives their variations δJ .
4.1. First example. This example is concerned with the L 2 −norm. We consider the shape function defined by where ϕ d ∈ H 1 (Ω) is a given desired (objective) function.
Proposition 4.1. The cost function J ε defined by satisfies the hypothesis 2.1 with 4.2. Second example. Here, we are dealing with the H 1 −semi-norm. We consider the shape function with ϕ d ∈ H 2 (Ω) is a given desired function.
Proposition 4.2. The cost function J ε defined by satisfies the hypothesis 2.1 with

Preliminary results.
The aim of this section is to give some technical results which will be used in section 4.3.

Proof of Lemma 4.2
Using the definition of h ε , we have Then we have h ε is solution of Note that, by Lemma 4.1, Then, This completes the proof. .

Proof of Lemma 4.3
Using the change of variables x = z + εy, we obtain The smoothness of ϕ 0 and E in H z,ε and H gives that and .
Thus the proof is complete.
Lemma 4.4. We have the following estimation

Proof of Lemma 4.4
We can write r ε solution of the system 3.4 as follows r ε = r 1 ε + r 2 ε where r 1 ε satisfies and r 2 ε is solution of (4.4)

Proof of theorem 3.1
We only need to prove that Remember that We have Changing variables x = z + εy and using the lemma 4.2 .
Likewise, using the same change of variables and due to lemma 4.4, it follows that which completes the proof.

Proof of proposition 4.1
The function J is differentiable with respect to ϕ and we have Computing the variation J (Ω\H z,ε ) − J (Ω) By the divergence formula and the system 3.3, we have .
A change of variable and the fact that ϕ 0 and ϕ d are of class C 2 in a neighborhood of the origin yield Hz,ε Finally, by theorem 3.1, we deduce .

Proof of proposition 4.2
The function J is differentiable and we have Moreover, we have As ϕ 0 and ϕ d are sufficiently regular in H z,ε , we obtain Hz,ε .
By the divergence formula, we have . Hence Finally, by theorem 3.1, we deduce .

The Kohn-Vogelius norms
The Kohn-Vogelius criterion [15] is used like a cost functional. Since the boundary conditions (ϕ d , Φ) are overspecified, one can define for any hole H two forward problems: • the "Dirichlet" problem: • the "Neumann" problem: The optimal hole H * coincides with the actual boundary H when the misfit between the solutions vanishes: ϕ D = ϕ N . Therefore, we propose an identification process based on the minimization of the following energy functional This is the so-called Kohn-Vogelius criterion [15]. Our approach concerns the derived topological optimization problem: min H⊂Ω J (Ω\H).
We will use the topological gradient method to solve this problem. It provides an asymptotic expansion of the function J with respect to a small topological perturbation of the domain Ω.

5.1.
Asymptotic expansion of the cost functional. The following Theorem describes the variation of the function J when creating a small hole H z,ε inside the domain Ω with a Dirichlet boundary condition on where ϕ ε N and ϕ ε D are the solutions to the systems ; 5.1.1. The three dimensional case. In this paragraph, we present the topological asymptotic expansion for the Anisotropic Laplace equations in the three dimensional case. In this case the fundamental solution of the anisotropic Laplace operator E is given by Theorem 5.1. Under the same hypotheses of theorem 3.1, the function J has the following asymptotic 5.1.2. The two dimensional case. In this paragraph, the result is obtained using the same technique described in the previous paragraph. The unique difference comes from the expression of the fundamental solution of the Anisotropic Laplace equations. In this case E is given by Theorem 5.2. Under the same hypotheses of theorem 3.1, the function J has the following asymptotic

Numerical Result
This section is concerned with some numerical investigations. We consider the bidimentional case and we present a fast and simple one-iteration identification algorithm. The unknown object H is identified using a level set curve of the topological gradient δJ . More precisely, the unknown object H is likely to be located at zone where the topological gradient δJ is more negative.
One-iteration algorithm: [11,12] • Solve the two problems (P 0 N ) and (P 0 D ) in initial domain Ω, • Compute the topological gradient function δJ (x), x ∈ Ω, • Determine the unknown object where c min is a negative constant chosen in such a way that the cost function J has the most negative value.
Next, we will present some numerical simulations using the proposed algorithm. In Figure 1, we test our algorithm on circular shape. In Figure 2, we consider the case of an elliptical shape. In Figure 3, we can notice that, when the shape is non-regular, the reconstruction is quite efficient. In the case of non trivial shape, yet we applied a one-iteration algorithm, we obtain an interesting reconstruction result (see Figure   4).
The obtained result can serve as a good initial estimate for an iterative optimization process based on the shape derivative.
The considered model can be viewed as a prototype of a geometric inverse problem valid in many applications. 6.0.1. Reconstruction of circular-shaped objects. In this case, we test our procedure to detect an object having circular-shaped. We reconstruct in this case the object H described by a disk centered at z = (2, 0) with different radius: r ∈ {0.2, 0.4, 0.6}. The obtained results are illustrated in Figure 1. One can easily observe in Figure 1, the unknown object is located in the region where the topological sensitivity function δJ is the most negative (red zone). The boundary of H * is approximated by an iso-value curve. Our one-iteration algorithm gives an efficient reconstruction results for the different chosen sizes. 6.0.2. Reconstruction of ellipse-shaped objects. In this example, we reconstruct an object described by an ellipse inserted in the disc D = B((2, 0), 1) and centered at (2, 0). We represent the results in Figure 2. In this case, we examine the numerical reconstruction of various ellipses having different directions and sizes.
As one can observe in Figure 2, the boundary of the object is again detected and located in the zone where the topological gradient is most negative (red lines). Also, our one-iteration algorithm gives quite effecient reconstruction results for different chosen ellipse-shaped objects. 6.0.3. Reconstruction of geometry with corners. We tried to apply our proposed algorithm to detect more complicated geometry. Our objective is to reconstruct an object with corners. More precisely, we are trying to detect a square and rectangle shape . We can see in the Figure 3 (a), that the unknown square H * is located in the zone where the topological gradient function δJ is the most negative (red zone) and also its boundary is approximated by an iso-value curve. So here, our one-iteration algorithm detects the location and the shape of the square. But in the case of a rectangle shape (see Figure 3 (b)) the boundary H * cannot be approached by any iso-value curve of the topological gradient function. We can remark in this case, that the one-iteration algorithm detects the zone containing the unknown geometry but the reconstruction result is not good.
6.0.4. Reconstruction of a non trivial-shaped objects. We apply now our proposed algorithm to detect a non trivial shapes. We can see in Figure 4 that the unknown shape H * is located in the zone where the topological gradient function δJ is the most negative (red iso-values) but we cannot approximate the boundary of H * by any iso-value curve of the topological sensitivity function δJ . We can improve these reconstruction results by suggesting an iterative algorithm.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.