Séminaire Modélisation, Optimisation, Dynamique

Detection of Several Obstacles in a Stokes Flow: A mixed approach

par Matias Godoy (Université de Chili et Université de Toulouse)

Europe/Paris
XLIM Salle X.203

XLIM Salle X.203

FST-Université de Limoges, 123, Av. Albert Thomas.
Description
We consider the inverse problem of detecting the location and shape of several objects immersed in a fluid flowing in a larger bounded domain Omega from boundary measurements. The fluid is governed by the steady-state Stokes equations. For this goal we consider a Kohn-Vogelius cost type function. This functional penalizes erroneous configurations for the considered system and even more, its minimization with respect of all possible admissible configurations is equivalent to the resolution of our inverse problem. In order to determine numerically the number and relative position of the objects, we perform a topological sensitivity analysis of the considered functional, obtaining an asymptotic expansion which leads to the expression of the so-called topological gradient of the cost function. As a complementary task, we compute the shape derivative of the cost function, which allows to improve the shape of the detected objects via our primary topological method. Then, we present some numerical simulations of this mixed approach which combines the topological and geometrical shape optimization methods. We finally discuss, briefly, the possibilities when there exists an inaccessible region of the boundary for the measurements which leads to a data completion problem. This is a joint work with Fabien CAUBET (IMT) and Carlos CONCA (UChile).