An inverse problem for fluid-solid interaction

Abstract

Any acoustic plane wave incident to an elastic obstacle results in a scattered field with a corresponding far field pattern. Mathematically, the scattered field is the solution of a transmission problem coupling the reduced elastodynamic equations over the domain occupied by the obstacle with the Helmholtz equation in the exterior. The far field pattern is obtained applying an integral operator to the scattered field function restricted to a simple smooth surface surrounding the obstacle. The subject of our paper is the inverse problem, where the shape of the elastic body represented by a parametrization of its boundary is to be reconstructed from a finite number of measured far field patterns. We define a family of objective functionals depending on a non-negative regularization parameter such that, for regularization parameter zero, the shape of the sought elastic obstacle is a minimizer of the functional. For any positive regularization parameter, there exists a regularized solution minimizing the functional. Moreover, for the regularization parameter tending to zero, these regularized solutions converge to the solution of the inverse problem provided the latter is uniquely determined by the given far field patterns. The whole approach is based on the variational form of the partial differential operators involved. Hence, numerical approximations can be found applying finite element discretization. Note that, though the transmission problem in its weak formulation may have non-unique solutions for domains with so-called Jones frequencies, the scattered field and its far field pattern is unique and depend continuously on the shape of the obstacle.