二维位势边界元法高阶单元几乎奇异积分半解析算法  

A novel semi-analytic algorithm of nearly singular integrals in high order boundary element analysis of 2D potential

作　　者：胡宗军[1] 牛忠荣[1] 程长征[1] 周焕林[1] 

机构地区：[1]合肥工业大学土木与水利工程学院,合肥230009

出　　处：《计算力学学报》2014年第6期763-768,798,共7页Chinese Journal of Computational Mechanics

基　　金：国家自然科学基金(11272111;11072073)资助项目

摘　　要：准确计算几乎奇异积分是边界元法难题之一。目前,对于一般的高阶单元的几乎奇异积分尚缺乏通用高效的计算方法。本文在单元局部坐标系中表征了二维高阶单元的几何特征,提出了源点相对高阶单元的接近度概念。针对二维位势边界元法的3节点二次等参单元,构造出与单元积分核具有相同几乎奇异性的近似奇异核函数。从二维位势几乎奇异积分单元积分核中扣除近似奇异核函数,把几乎奇异积分项转换为规则积分和奇异积分两部分之和,规则积分部分用常规Gauss数值积分计算,奇异积分部分由导出的解析公式计算,从而建立了二维位势问题高阶单元几乎强奇异和超奇异积分的半解析算法。算例结果表明了本文半解析算法的有效性和计算精度。The evaluation of the nearly singular integrals on general high order element is presently one of the difficulties in boundary element method （BEM）.In this paper,the geometric feature of the high order elements in two dimensional （2D）BEM is expressed by the local coordinate,in which the relative dis-tance from a source point to high order element （named approach degree）is defined.Then,the approxi-mate kernel functions of the nearly singular integrals are constructed for 3-noded quadratic isoparametric element in 2D potential BEM,which have the same singularity as the kernel functions of the element in-tegrals.By subtracting the approximate kernel functions from the singular kernels on the nearly singular integral elements,the original nearly singular integral is divided into two parts,one is regular integral and the other is the nearly singular integral.The former can be calculated accurately by Gauss numerical quadrature.The later can be calculated by the analytic formulation established in the paper.Thus the no-vel semi-analytic algorithm is proposed to evaluate the nearly strongly singular and hyper-singular inte-grals on the high order elements in 2D potential BEM.Finally,two examples are given to demonstrate the present algorithm is efficient and accurate for treating the nearly singular integrals on the high order elements.

关 键 词：位势 边界元法 高阶单元 几乎奇异积分 半解析算法 

分 类 号：O343.1[理学—固体力学]