Comparison of $\mathcal{H}$-matrix- and FMM-based 3D-ACA for a time-domain boundary element method
For practitioners using time-domain BEM, this paper provides a practical comparison of two algebraic compression methods, enabling simulations previously infeasible.
The paper compares H-matrix-ACA and FMM-based 3D-ACA for compressing BEM matrices in time-domain boundary element method solving the wave equation, showing that both techniques make time-domain BEM feasible for sound scattering by an electric machine for the first time, with comparisons in storage and computing time.
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.