Detecting an obstacle immersed in a fluid by shape optimization methods

摘要

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.