An enhancement of the fast time-domain boundary element method for the three-dimensional wave equation

TL;DR

This paper improves the stability and speed of the 3D time-domain boundary element method for wave equations by introducing Burton--Miller boundary integral equations and higher-order B-spline temporal bases, enabling efficient and stable simulations.

Contribution

It develops a stabilized and accelerated TDBEM using BMBIE and higher-order temporal bases, generalizing the fast multipole method for improved computational complexity.

Findings

BMBIE is essential for solving the homogeneous Dirichlet problem.

Higher-order bases (d≥2) are unstable for Dirichlet but stable for Neumann problems.

The method achieves the theoretical complexity of O(N_s^{1+δ} N_t).

Abstract

Our objective is to stabilise and accelerate the time-domain boundary element method (TDBEM) for the three-dimensional wave equation. To overcome the potential time instability, we considered using the Burton--Miller-type boundary integral equation (BMBIE) instead of the ordinary boundary integral equation (OBIE), which consists of the single- and double-layer potentials. In addition, we introduced a smooth temporal basis, i.e. the B-spline temporal basis of order $d$, whereas $d=1$ was used together with the OBIE in a previous study [Takahashi 2014]. Corresponding to these new techniques, we generalised the interpolation-based fast multipole method that was developed in \cite{takahashi2014}. In particular, we constructed the multipole-to-local formula (M2L) so that even for $d\ge 2$ we can maintain the computational complexity of the entire algorithm, i.e. $O(N_{\rm s}^{1+\delta} N_{\rm t})$, where $N_{\rm s}$ and $N_{\rm t}$ denote the number of boundary elements and the number of time steps, respectively, and $\delta$ is theoretically estimated as $1/3$ or $1/2$. The numerical examples indicated that the BMBIE is indispensable for solving the homogeneous Dirichlet problem, but the order $d$ cannot exceed 1 owing to the doubtful cancellation of significant digits when calculating the corresponding layer potentials. In regard to the homogeneous Neumann problem, the previous TDBEM based on the OBIE with $d=1$ can be unstable, whereas it was found that the BMBIE with $d=2$ can be stable and accurate. The present study will enhance the usefulness of the TDBEM for 3D scalar wave problems.