平面弹性问题自适应有限元方法的收敛性分析

刘春梅; 钟柳强; 舒适; 肖映雄

doi:10.3879/j.issn.1000-0887.2014.09.003

平面弹性问题自适应有限元方法的收敛性分析

doi: 10.3879/j.issn.1000-0887.2014.09.003

¹湖南科技学院数学与计算科学系计算数学研究所, 湖南永州 425199;²华南师范大学数学科学学院, 广州 510631;³湘潭大学数学与计算科学学院, 湖南湘潭 411105;⁴湘潭大学土木工程与力学学院, 湖南湘潭 411105

基金项目: 湖南省自然科学基金(14JJ3135); 国家自然科学基金(11201159);全国博士学位论文作者专项资金(201212);广东省高等学校优秀青年教师培养计划(Yq2013054); 广州市珠江科技新星项目(2013J2200063)

详细信息

作者简介:
刘春梅(1981—), 女, 山西人, 讲师, 博士(E-mail: liuchunmei0629@163.com);舒适(1962—), 男,湖南人, 教授, 博士, 博士生导师(通讯作者. E-mail: shushi@xtu.edu.cn).

中图分类号: O241.8;O242
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出版历程
- 收稿日期: 2014-01-20
- 刊出日期: 2014-09-15

Convergence of an Adaptive Finite Element Method for 2D Elasticity Problems

¹Institute of Computational Mathematics, Department of Mathematical Sciences, Hunan University of Science and Engineering, Yongzhou, Hunan 425199, P.R.China;²School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.R.China;³School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, P.R.China;⁴College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan, Hunan 411105, P.R.China

Funds: The National Natural Science Foundation of China(11201159)

摘要

摘要: 针对平面弹性问题,首先采用基于最新顶点二分法的网格加密方法,给出一种不需要标记振荡项和加密单元、不需要满足“内节点”性质的自适应有限元方法.其次,通过对各层网格上解函数和误差指示子的分析,利用相邻网格层上解函数的正交性、解函数和真解函数的能量误差的上界估计、相邻网格层上误差指示子的近似压缩性等结果,从理论上严格证明了该自适应有限元方法是收敛的.最后数值实验验证了该自适应有限元方法是收敛的和鲁棒的.
- 平面弹性问题 /
- 自适应有限元方法 /
- 收敛性
Abstract: For 2D linear elasticity problems, firstly, a standard adaptive finite element method (AFEM) was developed based on the newest vertex bisection grid refinement, which was marked only according to the error estimators without special treatment of the oscillation terms and intended conformance to the interior node properties. Secondly, through analysis of the numerical solution functions and error indicators at all the grid levels, the AFEM was strictly proved to be convergent by means of the orthogonality between the numerical solution functions at adjacent grid levels, the upper bound estimation of the energy errors between the true solution functions and the numerical solution functions, and the approximate compressibility of the error indicators between adjacent grid levels. Finally, several numerical experiments confirm that the presented AFEM is convergent and robust.
- 2D elasticity problem /
- adaptive finite element method /
- convergence

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参考文献(12)

[1]	Senturia D S, Aluru N, White J. Simulating the behavior of MEMS devices: computational methods and needs[J].Computational Science & Engineering, IEEE,1997,4(1): 30-43.
[2]	Brenner S C, Li Y S. Linear finite element methods for planar linear elasticity[J].Mathematics of Computation,1992,59(200): 321-338.
[3]	Cai Z Q, Korsawe J, Starke G. An adaptive least squares mixed finite element method for the stress displacement formulation of linear elasticity[J].Numerical Methods for Partial Differential Equations,2005,21(1): 132-148.
[4]	陈竹昌, 王建华, 王卫中. 自适应多层网格有限元求解应力集中问题[J]. 同济大学学报, 1994, 22(3): 203-208.(CHEN Zhu-chang, WANG Jian-hua, WANG Wei-zhong. Adaptive multigrid FEM for stress concentration[J].Journal of Tongji University,1994,22(3): 203-208.(in Chinese))
[5]	梁力, 林韵梅. 有限元网格修正的自适应分析及其应用[J]. 工程力学, 1995,12(2): 109-118.(LIANG Li, LIN Yun-mei. Adaptive mesh refinement of finite element method and its application [J].Engineering Mechanics,1995,12(2): 109-118.(in Chinese))
[6]	Whiler T P. Locking-free adaptive discontinuous Galerkin FEM for linear elasticity problem[J].Mathematics of Computation,2006,75(255): 1087-1102.
[7]	Lonsing M, Verfürth R. A posteriori error estimators for mixed finite element methods in linear elasticity[J].Numerische Mathematik,2004,97(4): 757-778.
[8]	刘春梅, 肖映雄, 舒适, 钟柳强. 弹性力学问题自适应有限元及其局部多重网格法[J]. 工程力学, 2012,29(9): 60-67.(LIU Chun-mei, XIAO Ying-xiong, SHU Shi, ZHONG Liu-qiang. Adaptive finite element method and local multigrid method for elasticity problems[J].Engineering Mechanics,2012,29(9): 60-67.(in Chinese))
[9]	Carstemsen C. Convergence of adaptive finite element methods in computational mechanics[J].Applied Numerical Mathematics,2009,59(9): 2119-2130.
[10]	Cascon J, Kreuzer C, Nochetto R, Siebert K. Quasi-optimal convergence rate for an adaptive finite element method[J]. SIAM Journal on Numerical Analysis,2008,46(5): 2524-2550.
[11]	Bonsch E. Local mesh refinement in 2 and 3 dimensions[J].IMPACT of Computing in Science and Engineering,1991,3(3): 181-191.
[12]	Scott L R, Zhang S. Finite element interpolation of nonsmooth functions satisfying boundary conditions[J].Mathematics of Computation,1990,54(190): 483-493.