On the use of a SIMD vector extension for the fast evaluation of Boundary Element Method coefficients

Abstract

The paper reports in detail a methodology to fully exploit the potential of a SIMD (Single Instruction Multiple Data) vector extension in the evaluation of certain type of integrals, which occur in the numerical solution of a Boundary Integral Equation (BIE) through the Boundary Element Method (BEM). Specifically, we present an algorithm for the fast evaluation of the integral coefficients appearing in the assembly of the BEM system matrices, which represents an extremely time-consuming task. The numerical scheme is tailored to the specific structure of the integrals associated to a wave propagation phenomenon, governed, in the time domain, by the D'Alembert equation. The reason of this choice resides in the critical importance achieved by this class of problems in many engineering applications. In particular, the application framework this work belongs to is the design of environmentally friendly commercial aircraft, for which the regulation and certification restrictions are, nowadays, a key constraint effecting even the conceptual phase of the design process. For the sake of generality, we used here only the basic features of the SIMD vector extension, common to all the specific architectures available on the market. Particular attention is payed to the accuracy-related issues arising from the use of the low-latency approximations of some of the operators involved. The resulting algorithm minimizes the number of operations involving operands belonging to the same register (''horizontal'' or ''intra-register'' operations). Preliminary numerical results reveal a remarkable speed-up of this highly-demanding part of the solution process, close, in most of the cases, to the theoretical peak. Standard multithreading techniques are additionally introduced to further increase the performance on multiprocessors machines.