正交各向异性位势问题边界元法中几乎奇异积分的解析算法

几乎奇异积分的计算困难阻碍了边界元法的工程应用。本文针对二维正交各向异性位势问题边界元法中近边界点的几乎奇异积分,采用分部积分法,导出一种直接的解析计算公式。该解析公式可以精确计算线性单元上的几乎奇异积分。对二次单元,可将其细分为几个线性元,采用该解析公式近似计算其边界积分。当内点离当前积分单元较远时,仍保持常规高斯数值积分模式;而当内点离其较近时,因常规高斯积分结果失效,则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。二次元计算结果比线性元计算结果更精确。

Analytical Algorithm of the Nearly Singular Integrals in Boundary Element Method to Orthotropic Potential Problems

The difficulty of the treatment of the nearly singular integrals hindered applications of Boundary Element Method(BEM). A set of new direct and analytical integral formulas are deduced with integration by parts to evaluate the nearly singular integrals in the BEM of 2D orthotropic potential problems, where the nearly singular integrals are computed exactly over the linear elements, computed approximately over the quadratic element subdivided into several linear elements. When an interior point is far away from a considered integral element, the conventional Gaussian numerical integral scheme is kept, otherwise, if the point is very close to this element, the Gaussian numerical integral is replaced by the new analytical integral due to the invalidation of the Gaussian integral. The numerical results of two examples demonstrate the effectiveness and accuracy of this algorithm, the results of quadratic elements are more accurate than those of linear elements.