A General Algorithm for the Numerical Solution of Hypersingular Boundary Integral Equations
A General Algorithm for the Numerical Solution of Hypersingular Boundary Integral Equations
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DOI：
10.1115/1.2893766
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发表时间：
1992-09
期刊：
影响因子：
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通讯作者：
M. Guiggiani;G. Krishnasamy;T. Rudolphi;F. Rizzo
中科院分区：
文献类型：
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作者：
M. Guiggiani;G. Krishnasamy;T. Rudolphi;F. Rizzo
本文首先详细讨论了导出超奇异边界积分方程的极限过程。证明了具有超奇异核的边界积分方程即使在非光滑边界点上也是完全有意义的，并且所涉及的积分不需要特殊的解释。对极限过程的仔细分析对于适当的数值算法的发展也有很强的相关性。第二部分给出了边界元法（BEM）中求超奇异曲面积分的一种新的通用方法。所提出的方法可以系统地应用于任何边界元分析，无论是开放曲面还是封闭曲面，以及任何种类和阶次的曲面边界元（当然，前提是密度函数在每个配点处满足必要的规则性要求）。该算法在固有坐标的参数平面上进行运算，可以将边界元中的任何超奇异积分直接转化为一个二重积分和一个一维正则积分的和。因为所有的奇异积分都是解析的，所以可以使用标准的正交公式。首次给出了三维问题中弯曲（畸变）元上的超奇异积分的数值结果。
The limiting process that leads to the formulation of hypersingular boundary integral equations is first discussed in detail. It is shown that boundary integral equations with hypersingular kernels are perfectly meaningful even at non-smooth boundary points, and that special interpretations of the integrals involved are not necessary. Careful analysis of the limiting process has also strong relevance for the development of an appropriate numerical algorithm. In the second part, a new general method for the evaluation of hypersingular surface integrals in the boundary element method (BEM) is presented. The proposed method can be systematically applied in any BEM analysis, either with open or closed surfaces, and with curved boundary elements of any kind and order (of course, provided the density function meets necessary regularity requirements at each collocation point). The algorithm operates in the parameter plane of intrinsic coordinates and allows any hypersingular integral in the BEM to be directly transformed into a sum of a double and a one-dimensional regular integrals. Since all singular integrations are performed analytically, standard quadrature formulae can be used. For the first time, numerical results are presented for hypersingular integrals on curved (distorted) elements for three-dimensional problems.