Inverse scattering by penetrable objects in two-dimensional elastodynamics

In this paper, a method for the shape reconstruction of a penetrable scatterer in two-dimensional linear elasticity is presented. The direct scattering problem is formulated in a dyadic form, a fact that is enforced by the dyadic nature of the free-space Greenʹs function. Approximate far-field equations with known terms the P and S parts of the fundamental dyadic solution at the radiation zone are used. The mathematical analysis presented is based on an interior transmission problem where a weak-type solution is introduced. The proposed inversion algorithm is an extension of the sampling method for the two-dimensional elastic case. The boundary of the scatterer can be found by noting that the L2-norms of the Herglotz kernels of the approximate far-field equations are not bounded as the source point of the fundamental solution approaches the boundary from inside. Numerical results for penetrable objects are presented illustrating the applicability of this method.