Fast multipole method applied to 3-D frequency domain elastodynamics

This article is concerned with the formulation and implementation of a fast multipole-accelerated BEM for 3-D elastodynamics in the frequency domain, based on the so-called diagonal form for the expansion of the elastodynamic fundamental solution, a multilevel strategy. As usual with the FM-BEM, the linear system of BEM equations is solved by GMRES, and the matrix is never explicitly formed. The truncation parameter in the multipole expansion is adjusted to the level, a feature known from recently published studies for the Maxwell equations. A preconditioning strategy based on the concept of sparse approximate inverse (SPAI) is presented and implemented. The proposed formulation is assessed on numerical examples involving O (10⁵) BEM unknowns, which show in particular that, as expected, the proposed FM-BEM is much faster than the BEM traditional, and that the GMRES iteration count is significantly reduced when the SPAI preconditioner is used.