Multiple multipole expansions for elastic scattering

Abstract

This paper presents a new approach to solve scattering of elastic waves in two dimensions. Traditionally, wave fields are expanded into an orthogonal set of basis functions. Unfortunately, these expansions converge rather slowly for complex geometries. The new approach enhances convergence by summing multiple expansions with different centers of expansions. This allows irregularities of the boundary to be resolved locally from the neighboring center of expansion. Mathematically, the wave fields are expanded into a set of nonorthogonal basis functions. The incident wave field and the fields induced by the scatterers are matched by evaluating the boundary conditions at discrete matching points along the domain boundaries. Due to the nonorthogonal expansions, more matching points are used than actually needed, resulting in an overdetermined system which is solved in the least-squares sense. Since there are free parameters, such as location and number of expansion centers, as well as kind and orders of expansion functions used, numerical experiments are performed to measure the performance of different discretizations. An empirical set of rules governing the choice of these parameters is found from these experiments. The resulting algorithm is a very general tool to solve relatively large and complex two-dimensional scattering problems.