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

Fabien Caubet (1) , Marc Dambrine (1) , Djalil Kateb (2) , Chahnaz Zakia Timimoun (3)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 LMAC - Laboratoire de Mathématiques Appliquées de Compiègne
3 Département de Mathématiques [Oran]

The aim of our work is to reconstruct an inclusion immersed in a fluid flowing in a larger bounded domain via a boundary measurement. Here the fluid motion is assumed to be governed by the Stokes equations. We study the inverse problem 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.

Keywords: geometric inverse problem; order two shape sensitivity; shape calculus; stationary Stokes problem; Kohn-Vogelius functional

Domains: Optimization and Control [math.OC]

Fabien Caubet, Marc Dambrine, Djalil Kateb, Chahnaz Zakia Timimoun. A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid. Inverse Problems and Imaging , 2013, 7 (1), pp.123--157. ⟨10.3934/ipi.2013.7.123⟩. ⟨hal-00678036⟩