位势边界元法中的边界层效应与薄体结构  

BOUNDARY LAYER EFFECT AND THIN BODY STRUCTURE IN BEM FOR POTENTIAL PROBLEMS

作　　者：张耀明[1] 谷岩[1] 陈正宗[2] 

机构地区：[1]山东理工大学理学院应用数学所,淄博255049 [2]台湾海洋大学河海工程研究所,基隆20224

出　　处：《力学学报》2010年第2期219-227,共9页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家自然科学基金(10571110);山东省自然科学基金(2003ZX12);山东理工大学科学基金(2004KJZ08)资助项目~~

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.In boundary element analyses,when a considered field point is very close to an integral element,the kernels＇ integration would exist various levels of near singularity,which can not be computed accurately with the standard Gaussian quadrature.As a result,the numerical results of field variables and their derivatives may become less satisfactory or even out of true.This is so-called＂boundary layer effect＂.Therefore,the accurate evaluation of nearly singular integrals plays an essential role to obtain highly accurate and reliable results by using boundary element method（BEM）.For most of the current numerical methods,especially for the exact integration method,the geometry of the boundary element is often depicted by using linear shape functions when nearly singular integrals need to be calculated.However,most engineering processes occur mostly in complex geometrical domains,and obviously,higher order geometry elements are expected to be more accurate to solve such practical problems.Thus,efficient approaches for estimating nearly singular integrals with high order geometry elements are necessary both in theory and application,and need to be further investigated.As is well known,for high order geometry elements,the forms of Jacobian and integrands are all complex irrational functions,and thus for a long time,the exact evaluation of nearly singular integrals is a difficult problem or even impossible implementation.In this paper,a new exact integration method for element integrals with the curvilinear geometry is presented.The present method can greatly improve the accuracy of numerical results of nearly singular integrals without increasing other computational efforts.Numerical examples of potential problems with curved elements demonstrate that the present algorithm can effectively handle nearly singular integrals occurring in boundary layer effect and thin body problems in BEM.

关 键 词：边界元法 近奇异积分 位势问题 边界层效应 薄体结构 

分 类 号：O342[理学—固体力学]