再谈从矩阵位移法看有限元位移精度的损失与恢复<sup>1)</sup>

袁驷; 袁全

doi:10.6052/1000-0879-20-106

再谈从矩阵位移法看有限元位移精度的损失与恢复¹⁾

袁驷^2), ,,
袁全

清华大学土木工程系,北京 100084

基金项目: 1) 国家自然科学基金资助项目(51878383，51378293)。

详细信息

通讯作者:
2)袁驷,教授,主要研究方向为结构工程。E-mail: yuans@tsinghua.edu.cn

中图分类号: O302
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出版历程
- 收稿日期: 2020-03-17
- 发布日期: 2020-12-19

REVISITING THE LOSS AND RECOVERY OF DISPLACEMENT ACCURACY IN FEM AS SEEN FROM MATRIX DISPLACEMENT METHOD ¹⁾

YUAN Si^2), ,,
YUAN Quan

Department of Civil Engineering, Tsinghua University, Beijing 100084, China

摘要

摘要: 本文是文献[1]的续篇。文献[1]以一维有限元为例,揭示了其误差主要来自于各个单元的"固端解"。其后,基于这一思想的超收敛计算的单元能量投影(element energy projection,EEP)法得以创立和发展,并有效地用于自适应有限元求解。近期的反思发现,前文的思想精华还有发扬空间：既然单元"固端解''是有限元误差的主要来源,就可以用EEP公式简便地事先求出来,从而可以不经有限元计算而一举得到所需的网格划分。本文简要介绍这一最新方法的思路和机理,并给出初步的数值结果。
- 矩阵位移法 /
- 一维有限元法 /
- 固端解 /
- 先验定量估计 /
- 自适应网格划分
Abstract: This paper is a revisit of Ref.[1], where it is shown that the errors from one-dimensional finite element (FE) results mostly come from the element fixed-end solutions. Based on this concept, the element energy projection (EEP) method for the super-convergence calculation is developed. Moreover, when the EEP technique is applied to the adaptive FE analysis to estimate and control the errors in FE solutions, the solutions satisfying the user pre-set error tolerances in the maximum norm can be obtained. Recent introspection leads to a realization that the essence in Ref.[1] has not been fully exploited: since the element fixed-end solutions are the major source of errors, then it is possible to calculate the errors a priori by using the EEP method, immediately generating a desirable mesh without the need for the FE analysis in advance. This paper gives a brief introduction to this novel idea and some initial numerical results are given to show the validity and effectiveness of the proposed technique.
- matrix displacement method /
- finite element method /
- fixed-end solutions /
- a prior error estimate /
- mesh adaptivity

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参考文献(20)

1 袁驷. 从矩阵位移法看有限元应力精度的损失与恢复. 力学与实践, 1998, 20(4): 1-6
Yuan Si.The loss and recovery of stress accuracy in FEM as seen from matrix displacement method. Mechanics in Engineering, 1998, 20(4): 1-6 (in Chinese)

2 龙驭球, 包世华, 袁驷. 结构力学教程(II). 北京: 高等教育出版社, 2019
Long Yuqiu, Bao Shihua, Yuan Si. Structural Mechanics (II). Beijing: Higher Education Press, 2019 (in Chinese)

3 Strang G, Fix G.An Analysis of the Finite Element Method. New Jersey: Prentice-Hall, 1973

4 Douglas J, Dupont T.Galerkin approximations for the two point boundary problems using continuous piecewise polynomial spaces. Numerische Mathematik, 1974, 22(2): 99-109

5 袁驷, 王枚. 一维有限元后处理超收敛解答计算的EEP法. 工程力学, 2004, 21(2): 1-9
Yuan Si, Wang Mei.An element-energy-projection method for post-computation of super-convergent solutions in one-dimensional FEM. Engineering Mechanics, 2004, 21(2): 1-9 (in Chinese)

6 袁驷, 林永静. 二阶非自伴两点边值问题Galerkin有限元后处理超收敛解答计算的EEP法. 计算力学学报, 2007, 24(2): 142-147
Yuan Si, Lin Yongjing.An EEP method for post-computation of super-convergent solutions in one-dimensional Galerkin FEM for second order non-self-adjoint Boundary-Value Problem. Chinese Journal of Computational Mechanics, 2007, 24(2): 143-147 (in Chinese)

7 袁驷, 王枚, 和雪峰. 一维C¹有限元超收敛解答计算的EEP法. 工程力学, 2006, 23(2): 1-9
Yuan Si, Wang Mei, He Xuefeng.Computation of super-convergent solutions in one-dimensional C¹ FEM by EEP method. Engineering Mechanics, 2006, 23(2): 1-9 (in Chinese)

8 袁驷, 肖嘉, 叶康生. 线法二阶常微分方程组有限元分析的EEP超收敛计算. 工程力学, 2009, 26(11): 1-9
Yuan Si, Xiao Jia, Ye Kangsheng.EEP super-convergent computation in FEM analysis of FEMOL second order ODEs. Engineering Mechanics, 2009, 26(11): 1-9 (in Chinese)

9 袁驷, 王枚, 王旭. 二维有限元线法超收敛解答计算的EEP法. 工程力学, 2007, 24(1): 1-10
Yuan Si, Wang Mei, Wang Xu.An element energy projection method for super-convergent solutions in two-dimensional finite element method of lines. Engineering Mechanics, 2007, 24(1): 1-10 (in Chinese)

10 Yuan S, Wu Y, Xu JJ, et al.A super-convergence strategy for two-dimensional fem based on element energy projection technique. Journal of Nanoelectronics and Optoelectronics, 2017, 12(11): 1284-1294

11 Yuan S, Wu Y, Xing QY.Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy projection technique. Applied Mathematics and Mechanics, 2018, 39(7): 1031-1044

12 Zhao QH, Zhou SZ, Zhu QD.Mathematical analysis of EEP method for one-dimensional finite element post processing. Applied Mathematics and Mechanics, 2007, 28(4): 441-445

13 袁驷, 邢沁妍, 叶康生. 一维C¹有限元EEP超收敛位移计算简约格式的误差估计. 工程力学, 2015, 32(9): 16-19
Yuan Si, Xing Qinyan, Ye Kangsheng.An error estimate of EEP super-convergent displacement of simplified form in one-dimensional C¹ FEM. Engineering Mechanics, 2015, 32(9): 16-19 (in Chinese)

14 Yuan S, He XF.Self-adaptive strategy for one-dimensional finite element method based on element energy projection method. Applied Mathematics and Mechanics, 2006, 27(11): 1461-1474

15 Yuan S, Xing QY, Wang X, et al.Self-adaptive strategy for one-dimensional finite element method based on EEP method with optimal super-convergence order. Applied Mathematics and Mechanics, 2008, 29(5): 591-602

6 Yuan S, Ye KS, Wang YL, et al.Adaptive finite element method for eigensolutions of regular second and fourth order Sturm-Liouville problems via the element energy projection technique. Engineering Computations, 2017, 34(8): 2862-2876

17 Yuan S, Du Y, Xing QY, et al.Self-adaptive one-dimensional nonlinear finite element method based on element energy projection method. Applied Mathematics and Mechanics, 2014, 35(10): 1223-1232

18 Jiang KF, Yuan S, Xing QY.An adaptive nonlinear finite element analysis of minimal surface problem based on element energy projection technique. Engineering Computations, 2020 (in press)

19 Yuan S, Dong YY, Xing QY, et al.Adaptive finite element method of lines with local mesh refinement in maximum norm based on element energy projection method. International Journal of Computational Methods, 2020, 17(4): 209-222

20 Dong YY, Yuan S, Xing QY.Adaptive finite element analysis with local mesh refinement based on a posteriori error estimate of element energy projection technique. Engineering Computations, 2019, 36(6): 2010-2033