三维边界元法中高阶单元上的几乎奇异积分

张耀明; 李小超; Vladimir Sladek; Jan Sladek

doi:10.6052/0459-1879-13-057

三维边界元法中高阶单元上的几乎奇异积分

1.
山东理工大学理学院应用数学所, 淄博 255049
2. 大连理工大学工业装备结构分析国家重点实验室, 大连 116024
3. Institute of Construction and Architecture, Slovak Academy of Sciences, Bratislava 84503, Slovakia

基金项目: 山东省自然科学基金资助项目(ZR2010AZ003)和工业装备结构分析国家重点实验室开放基金(GZ1307)资助项目.

详细信息

通讯作者:
张耀明

中图分类号: O342
计量
- 文章访问数: 01081
- HTML全文浏览量: 092
- PDF下载量: 0666
出版历程
- 收稿日期: 2013-03-04
- 修回日期: 2013-05-09
- 刊出日期: 2013-11-17

A GENERAL ALGORITHM FOR CALCULATING NEARLY SINGULAR INTEGRALS OVER HIGH-ORDER ELEMENTS IN 3D BEM

1.
Institute of Applied Mathematics, School of Mathematics, Shandong University of Technology, Zibo 255049, China
2. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China
3. Institute of Construction and Architecture, Slovak Academy of Sciences, Bratislava 84503, Slovakia

Funds: The project was supported by the National Natural Science Foundation of Shandong Province of China(ZR2010AZ003) and the Opening Fund of the State Key Laboratory of Structural Analysis for Industrial Equipment(GZ1307).

摘要

摘要: 三维边界元分析中，高阶几何单元上的几乎奇异积分计算是一个重要而且困难的问题，该文对此进行了研究。使用8节点四边形和6节点三角形曲面单元来描述几何边界；构造了新的距离函数；拓展原有的指数函数非线性变换到三维边界元法中，利用拓展的变换来消除被积函数的几乎奇异性。数值算例表明，该算法稳定，效率高，即使计算点到实际边界的距离很小，依然可获得令人满意的数值解。
- 三维弹性问题 /
- 边界元法 /
- 几乎奇异积分 /
- 高阶几何单元 /
- 变换法
Abstract: This work presents a general methodology to compute nearly singular integrals arising in 3D BEM using the eight-node second-order quadrilateral and six-node second-order triangular surface elements. The proposed method constructs a new distance function. The exponential transformation, which was proposed by present authors and is accurate and easy to implement according to extensive applications of 2D BEM, will be extended to 3D BEM to remove the near singularities of integrands. Several numerical examples are given to verify the high efficiency and the stability of the proposed scheme.
- BEM /
- nearly singular integrals /
- high-order geometry elements /
- transformation /
- 3D elasticity problem

HTML全文

参考文献(29)

Tanaka M, Sladek V, Sladek J. Regularization techniques applied to boundary element methods. Applied Mechanics Reviews, 1994, 47: 457-499

Chen JT, Hong HK. Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series. Applied Mechanics Reviews, 1999, 52(1): 17-33

Liu YJ. On the simple solution and non-singular nature of the BIE/BEM-a review and some new results. Engng Anal Bound Elem, 2000, 24: 286-292

Gao XW. An effective method for numerical evaluation of general 2D and 3D high order singular boundary integrals. Comput Methods Appl Mech Engrg, 2010, 199: 2856-2864

Zhang YM, Sladek V, Sladek J, et al. A new boundary integral equation formulation for plane orthotropic elastic media. Applied Mathematical Modelling, 2012, 36: 4862-4875

Sladek V, Sladek J, Tanaka M. Regularization of hypersingular and nearly singular integrals in the potential theory and elasticity. Int J Numer Meth. Engng, 1993, 36(10): 1609-1628

Chen HB, Lu P, Schnack E. Regularized algorithms for the calculation of values on and near boundaries in 2D elastic BEM. Engng Anal Bound Elem, 2001, 25(10): 851-876

Gao XW, Yang K, Wang J. An adaptive element subdivision technique for evaluation of various 2D singular boundary integrals. Engng Anal Bound Elem, 2008, 32: 692-696

Zhang XS, Zhang XX. Exact integrations of two-dimensional high-order discontinuous boundary element of elastostatics problems. Engng Anal Bound Elem, 2003, 28(7): 725-732

Niu ZR, Wendland WL, Wang XX, et al. A semi-analytical algorithm for the evaluation of the nearly singular integrals in three-dimensional boundary element methods. Comput Methods Appl Mech Engrg, 2005, 194: 1057-1074

Mukerjee S, Chati MK, Shi XL. Evaluation of nearly singular integrals in boundary element contour and node methods for three-dimensional linear elasticity. Int J Solids Struct, 2000, 37(51): 7633-7654

Liu YJ. Analysis of shell-like structures by the boundary element method based on 3-D elasticity, formulation and verification. Int J Numer Meth Engng, 1998, 41(3): 541-558 3.0.CO;2-K">

Milroy J, Hinduja S, Davey K. The elastostatic three-dimensional boundary element method analytical integration for linear isoparametric triangular elements, Appl Math Model, 1997, 21: 763-782

Nintcheu Fata S. Semi-analytic treatment of nearly-singular Galerkin surface integrals. Applied Numerical Mathematics, 2010, 60: 974-993

Zhang YM, Gu Y, Chen JT. Stress analysis for multilayered coating systems using semi-analytical BEM with geometric non-linearities. Computational Mechanics, 2011, 47: 493-504

张耀明, 孙翠莲, 谷岩.边界积分方程中近奇异积分计算的一种变量替换法, 力学学报, 2008, 40(2): 207-214 (Zhang Yaoming, Sun Cuilian, Gu Yan. The evaluation of nearly singular integrals in boundary integral equations with variable transformation. Chinese Journal of Theoretical and Applied Mechanics, 2008, 40(2): 207-214 (in Chinese))

Telles JCF. A self-adaptive coordinate transformation for efficient numerical evaluation of general boundary element integral. Int J Numer Meth Engng, 1987, 24: 959-973

Johnston PR. Application of Sigmoidal transformations to weakly singular and near singular boundary element integrals. Int J Numer Meth Engng, 1999, 45(10): 1333-1348 3.0.CO;2-Q">

Huang Q, Cruse TA. Some notes on singular integral techniques in boundary element analysis. Int J Numer Meth Engng, 1993, 36(15): 2643-2659

Ma H, Kamiya N. Distance transformation for the numerical evaluation of near singular boundary integrals with various kernels in boundary element method. Engng Anal Bound Elem , 2002, 26(4): 329-339

Qin XY, Zhang JM, Xie GZ, et al. A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element. Journal of Computational and Applied Mathematics, 2011, 235: 4174-4186

Peter R. Johnston1, David Elliott. A sinh transformation for evaluating nearly singular boundary element integrals. Int J Numer Meth Engng, 2005, 62: 564-578

Letizia S. On the computation of nearly singular integrals in 3D BEM collocation. Int J Numer Meth Engng, 2008, 74: 1733-1770

Johnston BM, Johnston PR, Elliott D. A new method for the numerical evaluation of nearly singular integrals on triangular elements in the 3D boundary element method. Journal of Computational and Applied Mathematics, 2013, 245:148-161.

Zhang YM, Gu Y, Chen JT. Boundary layer effect in BEM with high order geometry elements using transformation. Computer Modeling in Engineering and Sciences, 2009, 45(3):227-247.

Zhang YM, Gu Y, Chen JT. Boundary element analysis of the thermal behaviour in thin-coated cutting tools. Engng. Anal. Bound. Elem., 2010; 34(9):775-784.

Zhang YM, Qu WZ, Chen JT. BEM analysis of thin structures for thermoelastic problems. Engng Anal Bound Elem, 2013, 37: 441-452

Gao XW, Davies TG. Adaptive algorithm in elasto-plastic boundary element analysis. Journal of the Chinese Institute of Engineers, 2000, 23(3):349-356

Chati MK, Mukherjee S, Mukherjee YX. The boundary node method for three-dimensional linear elasticity . Int J Numer Meth Engng, 1999, 46: 1163-1184 3.0.CO;2-Y">