A low frequency elastodynamic fast multipole boundary element method in three dimensions

Abstract

This paper presents a fast multipole boundary element method (FMBEM) for the 3-D elastodynamic boundary integral equation in the ‘low frequency’ regime. New compact recursion relations for the second-order Cartesian partial derivatives of the spherical basis functions are derived for the expansion of the elastodynamic fundamental solutions. Numerical solution is achieved via a novel combination of a nested outer–inner generalized minimum residual (GMRES) solver and a sparse approximate inverse preconditioner. Additionally translation stencils are newly applied to the elastodynamic FMBEM and an implementation of the 8, 4 and 2-box stencils is presented, which is shown to reduce the number of translations per octree level by up to 60%. This combination of strategies converges 2–2.5 times faster than the standard GMRES solution of the FMBEM. Numerical examples demonstrate the algorithmic and memory complexities of the model, which are shown to be in good agreement with the theoretical predictions.