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

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[1]. Corresponding to these new techniques, we generalised the interpolation-based fast multipole method that was developed in [1]. In particular, we constructed the multipole-to-local formula (M2L) so that even for d >= 2 we can maintain the computational complexity of the entire algorithm, i.e. O(N-s(1+delta) N-t), where N-s and N-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 (C) 2021 Elsevier B.V. All rights reserved.