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A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid

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  • The aim of our work is to reconstruct an inclusion $\omega$ immersed in a fluid flowing in a larger bounded domain $\Omega$ via a boundary measurement on $\partial\Omega$. Here the fluid motion is assumed to be governed by the Stokes equations. We study the inverse problem of reconstructing $\omega$ thanks to the tools of shape optimization by minimizing a Kohn-Vogelius type cost functional. We first characterize the gradient of this cost functional in order to make a numerical resolution. Then, in order to study the stability of this problem, we give the expression of the shape Hessian. We show the compactness of the Riesz operator corresponding to this shape Hessian at a critical point which explains why the inverse problem is ill-posed. Therefore we need some regularization methods to solve numerically this problem. We illustrate those general results by some explicit calculus of the shape Hessian in some particular geometries. In particular, we solve explicitly the Stokes equations in a concentric annulus. Finally, we present some numerical simulations using a parametric method.
    Mathematics Subject Classification: Primary: 49Q10, 34A55, 49Q12; Secondary: 76D07.

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