Multiple Multipole Expansions For Elastic Scattering: An Aid To Understanding The Problems In "No-Record" Areas

Abstract

This paper presents a new approach to solving scattering of elastic waves in two dimensions. Wavefields are often 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 expansion. This allows irregularities of the boundary to be resolved locally from a nearby center of expansion. Mathematically, the wavefields are expanded into a set of non-orthogonal basis functions. The incident wavefield 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 non-orthogonal 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 the location and number of expansion centers as well as the 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 general tool to solve relatively large and complex two-dimensional scattering problems.