Localisation of small obstacles in Stokes flow

Fabien Caubet (1) , Marc Dambrine (1)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

We want to detect small obstacles immersed in a fluid flowing in a larger bounded domain $\Omega$ in the three dimensional case. We assume that the fluid motion is governed by the steady-state Stokes equations. We make a measurement on a part of the exterior boundary $\partial\Omega$ and then have a Kohn-Vogelius approach to locate these obstacles. We use here the notion of topological derivative in order to determine the number of objects and their rough location. Thus, we first establish an asymptotic expansion of the solution of the Stokes equations in $\Omega$ when we add small obstacles inside. Then, we use it to find a topological asymptotic expansion of the considered Kohn-Vogelius functional, which give us a formula of its topological gradient. Finally, we make some numerical simulations, exploring the efficiency and the limits of this method.

Mots clés: Geometric inverse problem; topological sensitivity analysis; topological gradient; Stokes equations; Kohn-Vogelius functional

Domaines: Optimisation et contrôle [math.OC]

Fabien Caubet, Marc Dambrine. Localisation of small obstacles in Stokes flow. Inverse Problems, 2012, 28 (10). ⟨hal-00678037⟩