A fast directional boundary element method for high frequency acoustic problems in three dimensions

摘要

A highly efficient fast boundary element method (BEM) for solving large-scaleengineering acoustic problems in a broad frequency range is developed andimplemented. The acoustic problems are modeled by the Burton-Miller boundaryintegral equation (BIE), thus the fictitious frequency issue is completelyavoided. The BIE is discretized by using the collocation method with piecewiseconstant elements. The linear systems are solved iteratively and accelerated byusing a newly developed kernel-independent wideband fast directional algorithm(FDA) for fast summation of oscillatory kernels. In addition, the computationalefficiency of the FDA is further promoted by exploiting the low-rank featuresof the translation matrices. The high accuracy and nearly linear computationalcomplexity of the present method are clearly demonstrated by typical examples.An acoustic scattering problem with dimensionless wave number $kD$ (where $k$is the wave number and $D$ is the typical length of the obstacle) up to 1000and the degrees of freedom up to 4 million is successfully solved within 4hours on a computer with one core and the memory usage is 24.7 GB.