Inverse obstacle scattering for elastic waves

Consider the scattering of a time-harmonic plane wave by a rigid obstacle which is embedded in an open space filled with a homogeneous and isotropic elastic medium. An exact transparent boundary condition is introduced to reduce the scattering problem into a boundary value problem in a bounded domain. Given the incident field, the direct problem is to determine the displacement of the wave field from the known obstacle; the inverse problem is to determine the obstacle's surface from the measurement of the displacement on an artificial boundary enclosing the obstacle. In this paper, we consider both the direct and inverse problems. The direct problem is shown to have a unique weak solution by examining its variational formulation. The domain derivative is derived for the displacement with respect to the variation of the surface. A continuation method with respect to the frequency is developed for the inverse problem. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.