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 引用本文: 张耀明 谷岩 陈正宗. 位势边界元法中的边界层效应与薄体结构[J]. 力学学报, 2010, 42(2): 219-227.
Yaoming Zhang, Yan Gu, Jeng-Tzong Chen. Boundary layer effect and thin body structure in bem for potential problems[J]. Chinese Journal of Theoretical and Applied Mechanics, 2010, 42(2): 219-227.
 Citation: Yaoming Zhang, Yan Gu, Jeng-Tzong Chen. Boundary layer effect and thin body structure in bem for potential problems[J]. Chinese Journal of Theoretical and Applied Mechanics, 2010, 42(2): 219-227.

## Boundary layer effect and thin body structure in bem for potential problems

• 摘要: 边界层效应与薄体结构问题的数值分析是边界元法的难点之一，其实质是近奇异积分的精确计算. 现有的处理近奇异积分的多数方法，特别是精确积分法，通常考虑的是线性几何单元.然而，多数工程问题的几何区域是十分复杂的，采用高阶几何单元近似显然能更好地逼近问题的真实边界，所得结果也将更加精确. 但由于高阶几何单元下的雅可比及被积函数形式的复杂性，相应的近奇异积分的精确计算一直是一个非常困难的问题. 提出一种新的反插值思想和方法，将被积函数中的规则部分用反插值多项式近似，从而导出计算近奇异积分的精确表达式. 数值算例表明，该算法稳定，效率高，在不增加计算量的前提下，极大地改进了近奇异积分计算的精度，成功地解决了边界层效应与薄体结构问题.

Abstract: In boundary element analyses, when a considered fieldpoint is very close to an integral element, the kernels' integration wouldexist various levels of near singularity, which can not be computedaccurately with the standard Gaussian quadrature. As a result, the numericalresults of field variables and their derivatives may become lesssatisfactory or even out of true. This is so-called boundary layereffect''. Therefore, the accurate evaluation of nearly singular integralsplays an essential role to obtain highly accurate and reliable results byusing boundary element method (BEM). For most of the current numericalmethods, especially for the exact integration method, the geometry of theboundary element is often depicted by using linear shape functions whennearly singular integrals need to be calculated. However, most engineeringprocesses occur mostly in complex geometrical domains, and obviously, higherorder geometry elements are expected to be more accurate to solve suchpractical problems. Thus, efficient approaches for estimating nearlysingular integrals with high order geometry elements are necessary both intheory and application, and need to be further investigated. As is wellknown, for high order geometry elements, the forms of Jacobian andintegrands are all complex irrational functions, and thus for a long time,the exact evaluation of nearly singular integrals is a difficult problem oreven impossible implementation. In this paper, a new exact integrationmethod for element integrals with the curvilinear geometry is presented. Thepresent method can greatly improve the accuracy of numerical results ofnearly singular integrals without increasing other computational efforts.Numerical examples of potential problems with curved elements demonstratethat the present algorithm can effectively handle nearly singular integralsoccurring in boundary layer effect and thin body problems in BEM.

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