Detecting an obstacle immersed in a fluid by shape optimization methods

Mehdi Badra (1) , Fabien Caubet (1) , Marc Dambrine (1)

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

The paper presents a theoretical study of an identification problem by shape optimization methods. The question is to detect an object immersed in a fluid. Here, the problem is modeled by the Stokes equations and treated as a nonlinear least-squares problem. We consider both the Dirichlet and Neumann boundary conditions. Firstly, we prove an identifiability result. Secondly, we prove the existence of the first order shape derivatives of the state, we characterize them and deduce the gradient of the least-squares functional. Moreover, we study the stability of this setting. We prove the existence of the second order shape derivatives and we give the expression of the shape Hessian. Finally, the compactness of the Riesz operator corresponding to this shape Hessian is shown and the ill-posedness of the identification problem follows. This explains the need of regularization to numerically solve this problem.

Mots clés: Stationary Stokes problem; Order two shape sensitivity; Geometric inverse problem; Sensitivity with respect to the domain; Shape calculus

Domaines: Optimisation et contrôle [math.OC]

Mehdi Badra, Fabien Caubet, Marc Dambrine. Detecting an obstacle immersed in a fluid by shape optimization methods. Mathematical Models and Methods in Applied Sciences, 2011, 21 (10), pp.2069--2101. ⟨hal-00583469⟩