A boundary face method for potential problems in three dimensions
A boundary face method for potential problems in three dimensions
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DOI：
10.1002/nme.2633
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发表时间：
2009-10
影响因子：
2.9
通讯作者：
Jianming Zhang;X. Qin;Xu Han;Guangyao Li
中科院分区：
文献类型：
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作者：
Jianming Zhang;X. Qin;Xu Han;Guangyao Li
本文提出了一种用于拉普拉斯方程数值求解的边界节点法（BNM）的新实现方式。通过耦合边界积分方程和移动最小二乘法（MLS）近似，BNM是一种边界型无网格方法。然而，它仍然使用标准单元进行边界积分和几何近似，因此失去了无网格方法的优势。在我们的实现中，这里称为边界面法，边界积分是在边界面上进行的，这些边界面以参数形式精确表示，就像实体建模中的边界表示数据结构一样。被积函数的量，例如高斯积分点的坐标、雅可比行列式和外法线，是直接从面而不是从单元计算得到的。为了处理薄结构，对于狭长的面采用了一维MLS和拉格朗日多项式的混合变量插值方案。还开发了一种针对近奇异积分的自适应积分方案。数值算例表明，我们的实现比BNM能提供更精确的结果，并且在一些极端情况下，例如节点分布非常不规则和薄壳的情况，能保持合理的精度。版权所有©2009约翰威立父子有限公司
This work presents a new implementation of the boundary node method (BNM) for numerical solution of Laplace's equation. By coupling the boundary integral equations and the moving least‐squares (MLS) approximation, the BNM is a boundary‐type meshless method. However, it still uses the standard elements for boundary integration and approximation of the geometry, thus loses the advantages of the meshless methods. In our implementation, here called the boundary face method, the boundary integration is performed on boundary faces, which are represented in parametric form exactly as the boundary representation data structure in solid modeling. The integrand quantities, such as the coordinates of Gauss integration points, Jacobian and out normal are calculated directly from the faces rather than from elements. In order to deal with thin structures, a mixed variable interpolation scheme of 1‐D MLS and Lagrange Polynomial for long and narrow faces. An adaptive integration scheme for nearly singular integrals has been developed. Numerical examples show that our implementation can provide much more accurate results than the BNM, and keep reasonable accuracy in some extreme cases, such as very irregular distribution of nodes and thin shells. Copyright © 2009 John Wiley & Sons, Ltd.