Comparison of $\mathcal{H}$-matrix- and FMM-based 3D-ACA for a time-domain boundary element method

TL;DR

This paper compares $\\mathcal{H}$-matrix and FMM-based 3D-ACA methods for efficient time-domain boundary element simulations of wave scattering, demonstrating algebraic compression enables feasible computations for complex engineering problems.

Contribution

It introduces a comparative analysis of $\\mathcal{H}$-matrix and FMM approaches using 3D-ACA for time-domain BEM, highlighting their efficiency and accuracy in practical applications.

Findings

Both methods reduce storage and computation time.

Algebraic compression makes time-domain BEM feasible for complex problems.

FMM and $\\mathcal{H}$-matrices show comparable performance.

Abstract

The homogeneous wave equation is solved by a time-domain boundary element method (BEM) using low-order shape functions for spatial, and the generalised convolution quadrature method (gCQ) by Lopez-Fernandez and Sauter for temporal discretisation. The three-dimensional array of BEM matrices according to a set of complex frequencies in Laplace domain is approximated by generalised Adaptive Cross Approximation (3D-ACA). Its rank is increased adaptively until a prescribed accuracy is reached, relying on a pure algebraic error criterion. The data slices for the selected frequency points are further processed by either the standard $\mathcal{H}$-matrices approach with ACA or by a fast multipole method (FMM). This paper compares both approaches with respect to their demands in storage and computing time. Both techniques are illustrated for calculating the sound scattered by an electric machine, for which the proposed algebraic compression techniques make time-domain BEM feasible for the first time.