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 引用本文: 张耀明 孙翠莲 谷岩. 等价间接规则化边界积分方程中的拟奇异积分[J]. 力学学报, 2008, 40(2): 207-214.
Yaoming Zhang, Cuilian Sun, Yan Gu. The evaluation of nearly singular integrals in the boundary integral equations with variable transformation[J]. Chinese Journal of Theoretical and Applied Mechanics, 2008, 40(2): 207-214.
 Citation: Yaoming Zhang, Cuilian Sun, Yan Gu. The evaluation of nearly singular integrals in the boundary integral equations with variable transformation[J]. Chinese Journal of Theoretical and Applied Mechanics, 2008, 40(2): 207-214.

## The evaluation of nearly singular integrals in the boundary integral equations with variable transformation

• 摘要: 准确估计近奇异边界积分是边界元分析中一项很重要的课题，其重要性仅次于对奇异积分的处理.近年来已发展了许多方法，都取得了一定程度的成功，但这个问题至今仍未得到彻底的解决. 基于一种新的变量变换的思想和观点，提交了一种通用的积分变换法，它非常有效地改善了被积函数的震荡特性，从而消除了积分的近奇异性，在不增加计算量的情况下, 极大地改进了近奇异积分计算的精度. 数值算例表明，其算法稳定，效率高，并可达到很高的计算精度，即使区域内点非常地靠近边界，仍可取得很理想的结果.

Abstract: The numerical solution of boundary value problemsusing boundary integral equations demands the accurate computationof the integral of the kernels, which occur as the nearly singularintegrals when the collocation point is close to the element ofintegration but not on the element in boundary element method(BEM). Such integrals are difficult to compute by standardquadrature procedures, since the integrand varies very rapidlywithin the integration interval, more rapidly the closer thecollocation point is to the integration element. Practice showsthat we can even obtain the results of superconvergence for thecomputed point far enough from the boundary; however, usingstandard quadrature procedures, which neglect the pathologicalbehavior of the integrand as the computed point approaches theintegration element, will lead to a degeneracy of accuracy of thesolution, even no accuracy, which is the so-called boundarylayer effect''. To avoid the boundary layer effect'', the accuratecomputation of the nearly singular boundary integrals would bemore crucial to some of the engineering problems, such as thecrack-like and thin or shell-like structure problems.The importance of the accurate evaluation of nearlysingular integrals is considered to be next to the singularboundary integrals in BEM, and great attentions have beenattracted and many numerical techniques have been proposed for itin recent years. These developed methods can be divided on thewhole into two categories: indirct algorithms'' and directalgorithms'', which have obtained varying degree of success, butthe problem of the nearly singular integrals has not beencompletely resolved so far. In this paper, a new efficienttransformation is proposed based on a new idea of transformationwith variables. The proposed transformation can remove the nearlysingularity efficiently by smoothing out the rapid variations ofthe integrand of nearly singular integrals, and improve theaccuracy of numerical results of nearly singular integrals greatlywithout increasing the computational effort. Numerical examples ofpotential problem with their satisfactory results in both curvedand straight elements are presented, showing encouragingly thehigh efficiency and stability of the suggested approach, even whenthe internal point is very close to the boundary. The suggestedalgorithm is general and can be applied to other problems in BEM.

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