Fast direct integral equation-based analysis for acoustic scattering using non-uniform grid accelerated matrix compression

An approach for the fast direct solution of scattering problems via compression of boundary element method (BEM) matrices is presented. Such an approach is advantageous for large resonant problems where iterative solvers converge poorly, or if the solutions are sought for multiple directions of incidence, so that the computational cost becomes proportional to the number of desired solutions. The compression is achieved by revealing the ranks of interactions between source and observation subdomains via algebraic analysis of the source subdomains' interactions with coarse non-uniform grids (NGs) surrounding the observation subdomains and vice versa. The NG field representation, originally developed for acceleration of iterative solvers [Y. Brick and A. Boag, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 57, 262–273 (2010)], is used to facilitate the computation of interacting and non-interacting mode sets that serve as a basis for a subsequent transformation of the BEM matrix. Only a highly compressed system linking the interacting modes is solved prior to the solution's extraction for the non-interacting ones. Further compression and acceleration of a multilevel scheme is suggested based on a “layered-localized” field representation via modes sorted according to their radiation, directivity, evanescence, and focusing properties, which are also computed using the NG approach.