多极子去奇异边界元法在三维水波问题中的应用

Application of the fast multipole desingularized BEM to the 3D water wave problems

在众多水波问题,尤其是非线性水波问题求解中,巨大计算量和存储量一直是人们试图克服和解决的重点与难点。本文通过对简单Green函数基本解1/r做多极子展开,并应用到去奇异边界元法中求解三维水波问题。通过无限区域中水流对圆球绕射算例的数值计算,验证了多极子去奇异边界元法能给出高精度的满意结果,与传统边界元方法相比避免了边界积分中奇异性的单独处理并将计算量和存储量均降到了O(NIgN)和O(N)数量级,同时对完全非线性水波问题的时域模拟也得出了令人满意的结果,验证了该方法在处理大计算量问题中所具有的优势。

For solving various water wave problems, especially nonlinear ones, people have been attempting to overcome the difficulties caused by the huge computational cost and storage in the computer. By the fast multipole expansion of simple Green function 1/r, this paper applies the fast multipole method （FMM） to the desingularized BEM to resolve the water wave problems in three-dimensions. Satisfactory calculation results are obtained by the present method, on the flow diffraction from a stationary sphere in an unbounded domain, in which the singularity is avoided and the computation cost and storage of the computer are reduced to O（N） order compared with the traditional BEM. Meanwhile perfect results are also got in the calculation of fully nonlinear water wave problems in time domain. It proves that the FMM desingularized BEM is superior to the traditional one in dealing with large-scale problems.