二维边界元法高阶元几乎奇异积分半解析算法

牛忠荣; 胡宗军; 葛仁余; 程长征

doi:10.6052/0459-1879-13-215

二维边界元法高阶元几乎奇异积分半解析算法

合肥工业大学土木与水利工程学院, 合肥 230009

基金项目: 国家自然科学基金资助项目(11272111, 11102056).

详细信息

通讯作者:
牛忠荣，教授，主要研究方向：计算力学。E-mail：niu-zr@hfut.edu.cn

中图分类号: O343.1
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出版历程
- 收稿日期: 2013-07-02
- 修回日期: 2013-07-31
- 刊出日期: 2013-11-17

A NEW SEMI-ANALYTIC ALGORITHM OF NEARLY SINGULAR INTEGRALS IN HIGH ORDER BOUNDARY ELEMENT ANALYSIS OF 2D ELASTICITY

School of Civil Engineering, Hefei University of Technology, Hefei 230009, China

Funds: The project was supported by the National Natural Science Foundation of China (11272111,11102056).

摘要

摘要: 针对边界元法中高阶单元中几乎奇异积分计算难题，解剖了二维边界元法高阶单元的几何特征，定义源点相对高阶单元的接近度。将高阶单元上奇异积分核函数用近似奇异函数逼近，从而分离出积分核中主导的奇异函数部分，其奇异积分核分解为规则核函数和奇异核函数两项积分之和。规则核函数用常规高斯数值积分，再对奇异核函数积分导出解析公式，从而建立了一种新的半解析法，用于高阶边界单元上几乎强奇异和超奇异积分计算。给出3个算例，采用边界元法高阶单元的半解析法计算了弹性力学薄体结构和近边界点位移/应力，并与线性边界元正则化算法结果作了比较，结果表明提出的二次元的半解析算法更加有效。特别是分析薄体结构，采用正则化算法的线性边界元分析比有限元有显著优势，而用提出的二次边界元半解析算法分析比其线性元的有效接近度又减小了4个量级。
- 弹性力学 /
- 边界元法 /
- 高阶单元 /
- 几乎奇异积分 /
- 半解析法
Abstract: The calculation of the nearly singular integrals on high order elements is difficult in boundary element method (BEM) at present. In this paper, a new semi-analytic algorithm is established to deal with the nearly strongly singular and hyper-singular integrals for high order elements in two dimensional (2D) BEM. By analyzing the geometric feature of high order elements by local coordinates, the relative distance from a source point to the element is defined. For the nearly singular integrals of the high order elements, the leading singular part of the integral kernel function is separated into the explicit formulation by a series of deduction. Then the nearly singular integrals on the high order elements close to the source point are transformed to both the non-singular part and singular part by the subtraction, where the former is computed by the numerical quadrature and the later is evaluated by the analytic algorithm. Consequently, the quadratic element with the new semi-analytic algorithm was applied to calculate the displacements and stresses very close to the boundary and thin-walled structures in the boundary element analysis of 2D elasticity. Three examples were given to demonstrate that the computed results of the quadratic element with the semi-analytic algorithm are more accurate than those of the linear element with the analytic algorithm for the nearly singular integrals. In fact, the boundary element analysis with the linear element has been greatly more advantageous compared with the finite element method in analyzing the thin bodies.
- elasticity /
- BEM /
- high order element /
- nearly singular integrals /
- semi-analytic method

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参考文献(15)

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