各向异性位势问题边界元法中几乎奇异积分的解析算法

导出了一种解析积分算法,精确计算了二维各向异性位势问题边界元法中近边界点的几乎奇异积分。对线性单元,几乎奇异积分可用解析公式直接计算。对二次单元,可将其细分为几个线性单元,采用该解析公式间接近似计算。当内点离积分单元较远时,仍然保持常规高斯数值积分模式;而当内点离其较近时,高斯积分结果失效,采用该解析积分取代高斯数值积分。数值算例证明了该算法的有效性和精确性。二次元比线性元计算结果更精确。

Analytical algorithm of the nearly singular integrals in boundary element method to anisotropic potential problems

A new analytical integral algorithm is proposed and applied to the evaluation of the nearly singular integrals in the Boundary Element Method for 2D anisotropic potential problems.The nearly singular integrals over the linear elements are accurately computed by new analytical formulas.For quadratic element,after it is subdivided into several linear elements,the nearly singular integrals can be approximately computed by the analytical formulas.When an interior point is far away from an element,the conventional Gaussian numerical quadrature scheme is kept.Otherwise if the point is very close to the element,the Gaussian numerical quadrature is replaced by the new analytical formulas due to the invalidation of the Gaussian quadrature.The numerical results of two examples demonstrate the effectiveness and accuracy of the algorithm.The results of quadratic elements are more accurate than those of linear elements.