On the Resolution of an Inverse Problem by Shape Optimization Techniques

and Applied Analysis 3 For u ∈ H1(Ω), we recall that the shape derivative is defined by u󸀠 = ?̇? − ∇u ⋅ V (15)


Introduction
The problem of detecting an inclusion  immersed in a fluid flowing in a greater bounded domain Ω has been researched by many authors.In [1], Alvarez et al. investigated this problem to find the location and the shape of  using the measurement of the velocity of the fluid and the Cauchy forces on the boundary Ω.Badra et al. [2] studied the same problem using the least-squares functional and Caubet et al. in [3] solved the problem using the Kohn-Vogelius functional with Dirichlet boundary conditions.
In this work we assume that the fluid is governed by Stationary Stokes equations with homogeneous Neumann boundary condition on the interior boundary and nonhomogeneous Dirichlet boundary condition on the exterior boundary.We solve our inverse problem by minimizing the Kohn-Vogelius cost functional.Then we characterize the gradient of this functional.
The paper is organized as follows: in the first part of the paper, we introduce the notations and the overdetermined problem that we consider.In the second part we state the main results of this work and we compute the first order derivative of the cost functional.
In order to do so, we need to fix some notation and definitions.For a bounded Lipschitz open subset Ω ⊂ R  ( = 2 or 3) with a smooth boundary Ω, n represents the external unit normal to Ω, and for a smooth enough function , we note, respectively,    and  2  , the normal derivative and the second normal derivative of .Recall that    fl ∇ .The tangential differential operators which will be noted by the subscript Γ are defined on Ω as follows: where ⊗ denotes the tensor product.For more details on tangential differential operators, we refer to [4,Section 5.4.3].
Finally, for a nonempty open subset  of Ω, we recall that

The Problem Setting
Let Ω be a bounded, connected and Lipschitz open subset of R  ( = 2 or 3).Given  > 0, consider O  as the set of admissible geometries such that For  ∈ O  , we consider the overdetermined Stokes boundary values problem: where f ∈ H 1/2 (Ω) such that f ̸ = 0 and the compatibility condition is fulfilled; that is, and g ∈ [ 1/2 00 ]  () is an admissible boundary measurement.Here [ 1/2 00 ]  () stands for the classical dual space of  1/2 00 ().The constant  > 0 represents the kinematic viscosity of the fluid, the vectorial function u represents the velocity of the fluid, and the scalar function  represents the pressure.
Note that we assume that there exists an admissible geometry  * ∈ O  such that (5) has a solution.So that, the geometric inverse problem under consideration reads Find  ∈ O  and (u, ) which satisfies the overdetermined system (5) .

(7)
Our purpose here is to solve the inverse problem of reconstructing  using shape optimization techniques.The reader will find all the ingredients for shape differentiation in the papers of Jacques Simon ( [5,6]) and the books of Henrot and Pierre [4] and of Sokolowski and Zolesio [7].
To recover the shape of the inclusion , we adopt the usual approach by minimizing a shape functional.We follow the classical technique of optimization; that is, we evaluate an explicit formula of the gradient of the shape functional which can be used in numerical experiments.Many choices of shape functionals are possible.For instance in [2], Badra et al. investigate the problem of the detection of an obstacle in a fluid by boundary measurement, using the least-squares cost functional.
In this paper, following previous works by Caubet et al. in [3], we will solve the inverse problem by using the Kohn-Vogelius cost functional where (u  ,   ) ∈ H 1 (Ω\ ) × L 2 (Ω\ ) is the unique solution of the Stokes problem with mixed boundary conditions given by and is the unique solution of the following Stokes problem with Neumann boundary conditions: For the results of existence, uniqueness, and regularity of the solutions of the Stokes problem with Neumann boundary conditions, one can refer to [2].Also the existence result for the mixed boundary conditions is well known.For the sake of clarity, we will recall that result in Appendix.
In order to determine the shape of , we try to minimize the Kohn-Vogelius cost functional   (): Indeed, if  * solves (11) with   ( * ) = 0, then this domain  * is a solution of the inverse problem (7).Conversely, if  * . is solution of the inverse problem (7), then   ( * ) = 0 and (11) holds.
The Needed Functional Tools.The velocity method is used to define the shape derivatives.For this purpose, we introduce the following space of admissible deformations: Then consider for any V ∈  the following application: with  > 0 being a fixed and small number.Let us notice that, for  small enough, () is a diffeomorphism of R  and   (0) = V vanishes on Ω.Now for  ∈ [0, ), we define where V is a perturbation direction.
Abstract and Applied Analysis 3 For  ∈ H 1 (Ω), we recall that the shape derivative is defined by where For more details on the differentiation with respect to the domain, see [4][5][6][7].

Identifiability Result
This section is devoted to new identifiability result for the mixed case.
Hence the solution of problem ( 7) is unique since, for a fixed , the same measure  yields the same geometry  in O  .

Shape Derivatives of the States
The following proposition states that the solutions (u  ,   ) and (u  ,   ) are differentiable with respect to the domain.Moreover, we obtain a characterization of the shape derivatives of these solutions.This result is based on [2, Proposition 2.5].
Proposition 2 (first-order shape derivatives of the states).Let V ∈  be an admissible deformation.The solutions (u  ,   ) and (u  ,   ) are differentiable with respect to the domain and the shape derivatives (u   , and We aim to compute the gradient of the Kohn-Vogelius functional.

Shape Derivative of the Kohn-Vogelius Cost Functional
We consider for  ∈ O  , the Kohn-Vogelius cost functional To simplify the expressions, we use the following notations: where (u  ,   ) solves ( 9) and (u  ,   ) solves (10).
Proposition 3 (first-order shape derivative of the functional).
For V ∈ , the Kohn-Vogelius cost functional   is differentiable at  in the direction V with where (, ) is defined by ( 22).
Proof.From Hadamard's formula (see [4, Theorem 5.2.2]), we have Apply now Green Formula for ∫ Ω\ ∇ : ∇u Since div  = 0 in Ω \  with −  u   +    n = 0 on Ω and from (20) thus we get From ( 27)-(30), we get Hence the first-order shape derivative of the functional is To recover the shape of the inclusion , we adopt the usual approach by minimizing a shape functional.We follow the classical technique of optimization; that is, we evaluate an explicit formula of the gradient of the shape functional which can be used in numerical experiments.The gradient is computed component by component using its characterization (see Proposition 3,formula (23).The optimization method used for the numerical simulations is the classical gradient algorithm which is the descent method: For a given initial shape  0 , we can compute the following iteration by the algorithm  +1 =   −   ∇  (  ) where   is a satisfying step length, until obtaining the stopped criterion.For more details see [3].

Conclusion
We solved our inverse problem using shape optimization methods to detect an inclusion immersed in a fluid.We use here the functional Kohn-Vogelius; we compute the first shape derivative of this functional which can be used in numerical experiments.