An Inversion Algorithm in Two-Dimensional Elasticity

In this paper scattering problems for the rigid body and the cavity in two-dimensional linear elasticity are considered. In each case the corresponding far-field scattering amplitudes are presented and the Herglotz condition and Herglotz wavefunctions are introduced. A pair of integral equations are constructed in the far-field region. The properties of the Herglotz functions are used to derive solvability conditions and to built approximate far-field equations. A method for solving inverse scattering problems is proposed, and the support of the scattering obstacle is found by noting the unboundedness of the L2-norm of the Herglotz densities as an interior point approaches the boundary of the scattering object from inside the scatterer. Illustration of the unboundedness property on the boundary is carried out for rigid circular cylinders and cavities. Numerical results for rigid bodies are also given, showing the applicability of this method.