工程力学 ›› 2019, Vol. 36 ›› Issue (2): 17-25.doi: 10.6052/j.issn.1000-4750.2018.10.0439

• 基本方法 • 上一篇    下一篇

基于EEP超收敛解的自适应有限元法特性分析

孙浩涵, 袁驷   

  1. 清华大学土木工程系, 北京 100084
  • 收稿日期:2018-08-03 修回日期:2018-10-16 出版日期:2019-02-22 发布日期:2019-02-22
  • 通讯作者: 袁驷(1953-),男,北京人,教授,博士,中国土木工程学会副理事长、中国力学学会副理事长,从事结构工程研究(E-mail:yuans@tsinghua.edu.cn). E-mail:yuans@tsinghua.edu.cn
  • 作者简介:孙浩涵(1994-),男,河南人,博士生,从事结构工程研究(E-mail:shh16@mails.tsinghua.edu.cn).
  • 基金资助:
    国家自然科学基金项目(51378293,51078199)

PERFORMANCE OF THE ADAPTIVE FINITE ELEMENT METHOD BASED ON THE ELEMENT-ENERGY-PROJECTION TECHNIQUE

SUN Hao-han, YUAN Si   

  1. Department of Civil Engineering, Tsinghua University, Beijing 100084, China
  • Received:2018-08-03 Revised:2018-10-16 Online:2019-02-22 Published:2019-02-22

摘要: 基于EEP (单元能量投影)超收敛计算的自适应有限元法,已对一系列问题取得成功,但其自适应特性尚缺乏相关研究。该文以二阶常微分方程为模型问题,同时考察基于EEP和SPR (超收敛分片恢复)超收敛解的自适应分析方法,与有限元最优网格进行了比较分析,进而提出反映自适应有限元收敛特性的估计式,并给出了自适应收敛率β的定义。该文给出的数值试验表明:采用m次单元,对于解答光滑的问题,SPR法与EEP法均可有效用于自适应求解,其位移可按最大模获得m+1的自适应收敛率;对于奇异因子为α(<1)的奇异问题,SPR法失效,而基于EEP法的自适应求解,其位移按最大模可获得m+α的自适应收敛率,远高于α的常规有限元收敛率。

关键词: 有限元法, 超收敛, 自适应求解, EEP法, SPR法, 最优网格

Abstract: The adaptive finite element method (AFEM) based on the element energy projection (EEP) technique has succeeded in solving a wide range of problems, while few studies have been done on its adaptive characteristics. Taking the second-order ordinary differential equation as the model problem, the adaptive methods based on both the EEP and super-convergent patch recovery (SPR) solutions were studied. The meshes generated by the two adaptive methods were compared with the optimum mesh. Furthermore, an estimate formula reflecting the characteristics of AFEM was proposed with an adaptive convergence rate β. Numerical experiments show that, with the element of degree m, both SPR and EEP can be well applied for the smooth problem and the convergence rate achieved m+1. For the singular problem with a singularity factor α(<1), SPR failed while EEP-based adaptive procedure gained the convergence rate of m+α, a much higher rate than that of the conventional finite element method α.

Key words: FEM, super-convergence, adaptive procedure, EEP, SPR, optimum mesh

中图分类号: 

  • TU311.4
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