Direct and Inverse scattering problems for elastic waves
In this thesis, both direct and inverse elastic scattering problems are considered. For a given incident wave, the direct problem is to determine the displacement of wave field from the known structure, which could be an obstacle or a surface in this thesis; The inverse problem is to determine the structure from the measurement of displacement on an artificial boundary.
In the second chapter, we consider the scattering of an elastic plane wave by a rigid obstacle, which is immersed in a homogeneous and isotropic elastic medium in two dimensions. Based on a Dirichlet-to-Neumann (DtN) operator, an exact transparent boundary condition is introduced and the scattering problem is formulated as a boundary value problem of the elastic wave equation in a bounded domain. By developing a new duality argument, an a posteriori error estimate is derived for the discrete problem by using the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator which decays exponentially with respect to the truncation parameter. An adaptive finite element algorithm is proposed to solve the elastic obstacle scattering problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are chosen through the finite element discretization error.
In chapter 3, we extend the argument developed in chapter 2 to elastic surface grating problem, where the surface is assumed to be periodic and elastic rigid; Then, we treat the obstacle scattering in three dimensional space; The direct problem is shown to have a unique weak solution by examining its variational formulation. The domain derivative is studied and a frequency continuation method is developed for the inverse problem. Finally, in chapter 4, a rigorous mathematical model and an efficient computational method are proposed to solve the inverse elastic surface scattering problem which arises from the near-field imaging of periodic structures. The surface is assumed to be a small and smooth perturbation of an elastically rigid plane. By placing a rectangle slab of a homogeneous and isotropic elastic medium with larger mass density above the surface, more propagating wave modes can be utilized from the far-field data which contributes to the reconstruction resolution. Requiring only a single illumination, the method begins with the far-to-near field data conversion and utilized the transformed field expansion to derive an analytic solution for the direct problem, which leads to an explicit inversion formula for the inverse problem; Moreover, a nonlinear correction scheme is developed to improve the accuracy of the reconstruction; Numerical examples are presented to demonstrate the effectiveness of the proposed methods for solving the questions mentioned above.