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

Martin Schanz, Vibudha Lakshmi Keshava, Herbert de Gersem

TL;DR

This work tackles the high memory and compute demands of time-domain boundary element methods for hyperbolic wave problems by introducing a three-dimensional adaptive cross-approximation ($3$D-ACA) to compress the frequency-dependent data array produced by generalized convolution quadrature ($gCQ$). It compares two data-sparse realizations of the compressed data: an $H$-matrix framework with ACA and a fast multipole method (FMM) approach, applied to Dirichlet, Neumann, and mixed problems and validated on a unit cube and a realistic electric-machine scattering scenario. The results show substantial storage compression with both approaches, with the FMM-based variant offering better storage economy and the ACA-based variant sometimes benefiting more from faster convolution, collectively enabling feasible time-domain BEM for large-scale problems. The study highlights the potential of adaptive frequency selection and tensor-based compression to bring time-domain BEM into practical, industrial-scale electromagnetic/acoustic simulations, while pointing to avenues for further improvement via adaptive trees or optimized FMM parameters.

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.

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

TL;DR

This work tackles the high memory and compute demands of time-domain boundary element methods for hyperbolic wave problems by introducing a three-dimensional adaptive cross-approximation (D-ACA) to compress the frequency-dependent data array produced by generalized convolution quadrature (). It compares two data-sparse realizations of the compressed data: an -matrix framework with ACA and a fast multipole method (FMM) approach, applied to Dirichlet, Neumann, and mixed problems and validated on a unit cube and a realistic electric-machine scattering scenario. The results show substantial storage compression with both approaches, with the FMM-based variant offering better storage economy and the ACA-based variant sometimes benefiting more from faster convolution, collectively enabling feasible time-domain BEM for large-scale problems. The study highlights the potential of adaptive frequency selection and tensor-based compression to bring time-domain BEM into practical, industrial-scale electromagnetic/acoustic simulations, while pointing to avenues for further improvement via adaptive trees or optimized FMM parameters.

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 -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.
Paper Structure (11 sections, 33 equations, 11 figures, 1 table, 1 algorithm)

This paper contains 11 sections, 33 equations, 11 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Boundary element method: Discretisation
  4. Spatial discretisation
  5. Temporal discretisation
  6. Three-dimensional adaptive cross approximation
  7. Numerical examples
  8. Unit cube loaded by a smooth pulse
  9. Sound scattering of an electric machine
  10. Conclusions
  11. Parameters of the gCQ

Figures (11)

  • Figure 1: Unit cube: Geometry and discretisation parameters
  • Figure 2: Cube: $L_{\mathrm{max}}$-error versus refinement in space and time
  • Figure 3: Compression rates of 3D-ACA and 3D-FMM for both problems
  • Figure 4: Cube: Necessary number of frequencies for both approaches for the Dirichlet problem
  • Figure 5: Cube (level 4): Used complex frequencies. The colour code corresponds to the number of matrix blocks at which the frequency is active: Dirichlet problem, single-layer potential (SLP, top) and double-layer potential (DLP, bottom), 3D-FMM (left) and 3D-ACA (right)
  • ...and 6 more figures