INTRODUCTION

• Why BEM?

The BEM (Boundary Element Method) is capable of analyzing exterior problems (where the computational domain extends infinitely) without any artificial tricks, so-called absorbing boundary conditions (ABCs). This inherent feature is unique among modern numerical methods for PDEs.

• Why Time-Domain?

Time-domain approach is more intuitive than frequency-domain approach and can handle multiple-frequencies at one shot through signal processing (Fourier analysis). This enables to perform an acoustic (noise) simulation very efficiently.

• Why Now?

In spite of the above unique and promising features, the conventional time-domain BEM (TDBEM) is computationally expensive: it requires O(Ns^2*Nt) cost, where Ns and Nt denote the number of boundary elements (spatial DOF) and number of time-steps (temporal DOF). This fundamental issue has been resolved by developing fast (and approximated) algorithms nowadays. The present software aXustica relies on the interpolation-based fast multiple method, which is a time-domain version of the fast multipole method (FMM) and akin to the planewave time-domain (PWTD) algorithm. The reduced computation cost is O(Ns^d*Nt), where the index d is 3/2 when the boundary elements distribute uniformly in a 3D space and 4/3 when they exist on some surfaces. 

• More information?

The foundation of the aforementioned fast TDBEM was presented in the paper ``An interpolation-based fast multipole-accelerated boundary integral equation method for the three-dimensional wave equation" (arxiv; link) and improved in the paper ``An enhancement of the fast time-domain boundary element method for the three-dimensional wave equation'' (arxiv; link).

Examples

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2025/10/7

Toru Takahashi, Dr. Eng.
Mechanical Systems Engineering
Graduate School of Engineering
Nagoya University, Japan
toru.takahashi@mae.nagoya-u.ac.jp