An enhancement of the fast time-domain boundary element method for the three-dimensional wave equation
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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 <cit.>. 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+δ N_ t), where N_ s and N_ t denote the number of boundary elements and the number of time steps, respectively, and δ 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.
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