An efficient solution of time domain boundary integral equations for acoustic scattering and its acceleration by Graphics Processing Units

Abstract

The present paper is aimed at developing a fast numerical solution of the time domain boundary integral equation (TDBIE) reformulated from the convective wave equation for large scale wave scattering and propagation problems. Historically, numerical solutions of boundary integral equation in the time domain have encountered two major difficulties. The first is the intrinsic numerical instability in the early time domain boundary integral equation formulations. And the second is the formidably high computational cost associated with the direct solution of the time-domain boundary integral equation. In this paper, both issues are addressed. A stable Burton-Miller type formulation is proposed for the time domain boundary integral equation in the presence of a mean flow. A justification for stability through the energy equation associated with the convective wave equation is given. A comparison of the current formulation with a previous one in literature is also offered. The boundary integral equation is solved by a time domain boundary element method (TDBEM), using high-order basis functions and unstructured surface elements. To significantly reduce the computational cost, a Time Domain Propagation and Distribution (TDPD) algorithm is proposed, making use of the delayand amplitude-compensated field with a mean flow. Implemented in multi-level interactions, the current algorithm shows a computational cost of O(N) per time step where N is the total number of unknowns on surface elements. Furthermore, GPU computing has been utilized to speedup the computation. Numerical aspects of the GPU computing for boundary element solutions are discussed. Comparison with CPU executions is also given. Numerical examples that demonstrate the capabilities of the proposed method are presented.