The problem of two-dimensional scattering of elastic waves by an elastic inclusion can be formulated in terms of a domain integral equation, in which the grad-div operator acts on a vector potential. The vector potential is the spatial convolution of a Green's function with the product of the density and the displacement over the domain of interest.

The first part of the thesis treats the numerical solution of the direct problem. Following similar work done in electromagnetics we employ a Galerkin approximation using a weak form of the integral equation and rooftop functions, as both expansion and test functions. The determination of the approximate elastic field is thus reduced to an algebraic problem. We present some numerical results for coaxially coated cylinders with constant Lamè coefficients and variable densities. The numerical results are compared with existing analytical solutions.

The second part of the thesis is concerned with the inverse problem of determining the density of an elastic inclusion from a knowledge of how the inclusion scatters known incident elastic waves. A modified gradient method which is based on the integral representation of the field is used for the solution of the inverse problem. The algorithm employed is an extension of the Kleinman-Van den Berg method to elasticity, and involves an iterative determination of both the unknown density and the shape of the inclusion. The synthetic data used in the inversion algorithm is obtained using the numerical solution of the direct problem developed in the first part of the dissertation.