Research Article | Open Access
On the Resolution of an Inverse Problem by Shape Optimization Techniques
In this work, we want to detect the shape and the location of an inclusion via some boundary measurement on . In practice, the body is immersed in a fluid flowing in a greater domain and governed by the Stokes equations. We study the inverse problem of reconstructing using shape optimization methods by defining the Kohn-Vogelius cost functional. We aim to study the inverse problem with Neumann and mixed boundary conditions.
The problem of detecting an inclusion immersed in a fluid flowing in a greater bounded domain has been researched by many authors. In , 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 Badra et al.  studied the same problem using the least-squares functional and Caubet et al. in  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 ( or ) with a smooth boundary , represents the external unit normal to , and for a smooth enough function , we note, respectively, and , the normal derivative and the second normal derivative of . Recall that . The tangential differential operators which will be noted by the subscript are defined on as follows:where denotes the tensor product. For more details on tangential differential operators, we refer to [4, ection 5.4.3].
Finally, for a nonempty open subset of , we recall that
2. The Problem Setting
Let be a bounded, connected and Lipschitz open subset of ( or ). Given , consider as the set of admissible geometries such that Take now as an open set with a boundary and satisfy the following assumption:
For , we consider the overdetermined Stokes boundary values problem:where such that and the compatibility condition is fulfilled; that is,and is an admissible boundary measurement. Here stands for the classical dual space of The constant represents the kinematic viscosity of the fluid, the vectorial function represents the velocity of the fluid, and the scalar function represents the pressure.
Note that we assume that there exists an admissible geometry such that (5) has a solution. So that, the geometric inverse problem under consideration reads
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  and of Sokolowski and Zolesio .
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 , 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 , we will solve the inverse problem by using the Kohn-Vogelius cost functional where is the unique solution of the Stokes problem with mixed boundary conditions given byand 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 . 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 :
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 the following application: with being a fixed and small number. Let us notice that, for small enough, is a diffeomorphism of and vanishes on . Now for , we define where is a perturbation direction.
3. Identifiability Result
This section is devoted to new identifiability result for the mixed case.
Theorem 1 (identifiability result). Let , ( or be a bounded Lipschitz domain and be a nonempty open subset of . Let and with satisfying the flux condition Let for be a solution of Assume that are such that Then .
This result is directly adapted from heorem 2.2 given in  to our problem.
Hence the solution of problem (7) is unique since, for a fixed , the same measure yields the same geometry in .
4. Shape Derivatives of the States
The following proposition states that the solutions and 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, roposition 2.5].
Proposition 2 (first-order shape derivatives of the states). Let be an admissible deformation. The solutions and are differentiable with respect to the domain and the shape derivatives and belong to . The couples and are, respectively, the only solutions of the following boundary value problems:and
We aim to compute the gradient of the Kohn-Vogelius functional.
5. Shape Derivative of the Kohn-Vogelius Cost Functional
Proposition 3 (first-order shape derivative of the functional). For , the Kohn-Vogelius cost functional is differentiable at in the direction withwhere is defined by (22).
Proof. From Hadamard’s formula (see [4, heorem 5.2.2]), we have because on As we apply Green Formula for Since in with on and
on , Apply now Green Formula for to get Since in with on
and from (20) thus we getFrom (27)-(30), we getHence 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 , we can compute the following iteration by the algorithm where is a satisfying step length, until obtaining the stopped criterion. For more details see .
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.
Result on the Stokes Problem with Mixed Conditions
Defineand denote, respectively, by and the duality product between and and the duality product between and .
Theorem A.1 (existence and uniqueness of the solution). Let be a bounded Lipschitz open set of ( and let be a Lipschitz open subset of such that is connected. Let be an open subset of the exterior boundary and . Let , , , and . Then, the problem.admits a unique solution .
Proof. According to [8, emma 3.3], consider such that , on , and on such that Then the couple satisfies From the first equation we obtain, for ,Apply now Green Formula to getSince we havethen we obtainFrom the conditions on the boundary we get From Lax-Milgram’s Theorem, there exists a unique such that, for all , one hasIn particular (A.9) is true for all Then using De Rham’s theorem (see ), there exists , up to an additive constant, such that, for all ,Using [8, emma 3.3] (or [10, héorème 3.2]), we define such that in , on , and on with . Let such that on and on and define From [8, emma 3.3] (see also [10, héorème 3.2]), we define such that , where satisfies the following equality: Then, using (A.9) and (A.10), it yields Choose the additive constant for such that Hence, we prove that there exists a unique pair such that, for all with on and on ,which complete the proof.
No data were used to support this study.
Conflicts of Interest
The author declares that she has no conflicts of interest.
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Copyright © 2019 Chahnaz Zakia Timimoun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.