力学与实践 ›› 2020, Vol. 42 ›› Issue (6): 689-694.DOI: 10.6052/1000-0879-20-106

• 应用研究 • 上一篇    下一篇

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   

  1. 清华大学土木工程系,北京 100084
  • 收稿日期:2020-03-18 出版日期:2020-12-20 发布日期:2020-12-20
  • 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
  • 基金资助:
    1) 国家自然科学基金资助项目(51878383,51378293)。

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

YUAN Si2), YUAN Quan   

  1. Department of Civil Engineering, Tsinghua University, Beijing 100084, China
  • Received:2020-03-18 Online:2020-12-20 Published:2020-12-20

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。

关键词: 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.

Key words: matrix displacement method, finite element method, fixed-end solutions, a prior error estimate, mesh adaptivity

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