高阶边界元奇异积分的一种通用高效计算方法

摘要

A general and efficient method for numerically evaluating singular integrals of various orders in boundary element method (BEM)is presented in this paper.The method is based on Taylor expansions of the integrands in polar coordinate system.Since the singular terms are extracted numerically rather than analytically as in existing methods,the present method is capable of treating various orders of sin-gularities in a general form,independent on the explicit expressions of integrands.Nearly singularities caused by the large aspect ratio of the physical elements and the closeness of the singular points to the element boundary are circumvented by a conformal transformation and a sigmoidal transformation, respectively.Numerical results show that our method could maintain high accuracy with few quadrature points regardless the shape of the element being distorted or not.Performance of this method is further verified by incorporating into a Nyström BEM to solve elastodynamic problems.Our method is imple-mented in C language and is freely available.%针对三维边界元法中曲面单元上的(弱、强、超)奇异积分提出了一种通用高效的计算方法。经极坐标变换，将奇异积分转化为常规积分；采用数值方法计算 Cauchy 主值积分和 Hadamard 有限项积分系数；引入保角变换和反曲变换消除因单元畸形或因积分点靠近单元边界而引起的周向积分奇异性。该方法可以统一处理(弱、强、超)奇异积分，并且只需要知道核函数的奇异阶数和少数几个点上的被积函数值，不依赖于积分和函数的具体选取；所需的积分点少，精度高，并且受单元畸形程度影响较小，稳定性好。采用该方法计算了声学和弹性力学中的典型奇异积分，并结合二阶 Nyström 方法求解了弹性力学的边界积分方程，验证了方法的高精度和高效性。本文数值积分程序可向作者索取。