边界元法中计算几乎奇异积分的一种无奇异算法

A Non-singular Algorithm for the Evaluation of the Nearly Singular Integrals in Boundary Element Methods

边界元法中存在几乎奇异积分的计算困难。引起边界单元上几乎奇异积分的因素是源点到其邻近单元的最小距离δ。本文拓展文 [1 ]的思想 ,进一步采用分部积分将δ移出奇异积分式中积分核之外 ,转换后的积分核是δ的正则函数。所以几乎强奇异和超奇异积分被化为无奇异的规则积分与解析积分的和 ,可由通常的Gauss数值积分解出。文中应用此正则化技术求解了弹性力学平面问题的近边界点位移和应力。

The nearly singular integral occurs in the boundary element formulation when a source point is close to the integration element (as compared to its size) but not on this element. This paper extends the strategy of Ref. and presents a non singular algorithm to deal with the difficult problem of the evaluation of the nearly strong singular and hypersingular integrals. The singular factor, which is expressed by the least distance from the source point to its neighboring element, is moved out from the kernel functions in the singular integrals by means of an integration by parts. The resulting kernels are regular functions of the least distance. Therefore, the nearly singular integrals are transformed into the sum of analytical integrals and non singular integrals, for which the Gaussian integration is sufficient to provide very accurate results. The regularization technique is used to analyze two dimensional elasticity problems. The numerical results demonstrate the accuracy and effectiveness of the algorithm.