工程力学 ›› 2019, Vol. 36 ›› Issue (7): 8-17.doi: 10.6052/j.issn.1000-4750.2018.06.0351

• 基本方法 • 上一篇    下一篇

自适应有限元线法在二维无穷域问题中的应用

董义义, 邢沁妍, 方楠, 袁驷   

  1. 清华大学土木工程系, 土木工程安全与耐久教育部重点实验室, 北京 100084
  • 收稿日期:2018-06-26 修回日期:2018-09-25 出版日期:2019-07-06 发布日期:2019-07-06
  • 通讯作者: 邢沁妍(1981-),女,辽宁人,讲师,博士,主要从事结构工程研究(E-mail:xingqy@tsinghua.edu.cn). E-mail:xingqy@tsinghua.edu.cn
  • 作者简介:董义义(1991-),男,安徽人,博士生,主要从事结构工程研究(E-mail:dongyy17@mails.tsinghua.edu.cn);方楠(1981-),男,黑龙江人,博士,主要从事结构工程研究(E-mail:fangnan99@mails.tsinghua.edu.cn);袁驷(1953-),男,北京人,教授,博士,主要从事结构工程研究(E-mail:yuans@tsinghua.edu.cn).
  • 基金资助:
    国家自然科学基金项目（51508305，51378293，51078199）

Application of adaptive finite element method of lines in 2D unbounded domain problems

DONG Yi-yi, XING Qin-yan, FANG Nan, YUAN Si   

  1. Department of Civil Engineering, Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Tsinghua University, Beijing 100084, China
  • Received:2018-06-26 Revised:2018-09-25 Online:2019-07-06 Published:2019-07-06

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摘要： 无穷域问题广泛存在于实际工程中，半解析、半离散的数值计算方法—有限元线法（Finite ElementMethod of Lines，简称FEMOL）对其具有较好的适应性。在已有的映射型FEMOL无穷单元理论的基础上，基于单元能量投影（Element Energy Projection，简称EEP）法的自适应FEMOL被应用于二维无穷域问题的求解。用户只需输入稀疏的初始网格和误差限，算法即自动生成优化的FEMOL网格，该网格上常规单元和无穷单元的FEMOL解均按最大模度量满足给定误差限。文中首先介绍二维FEMOL的原理策略、无穷单元的构建，然后概述基于EEP法的自适应FEMOL算法，并讨论其对无穷域问题的适用性，之后对圆柱绕流的Poisson方程问题、带孔无穷大板单向拉伸的弹性力学平面问题、受圆形均布荷载半空间体的三维轴对称问题进行了自适应分析，最终不仅给出了满足误差限的函数（位移）解，也给出了具有优良性态的导数（应力）解，从而为无穷域问题的求解提供了一种高效可靠的新途径。

关键词: 无穷域问题, 自适应, 有限元线法, 无穷单元, 单元能量投影

Abstract: Unbounded domain problems are frequently encountered in engineering. As a semi-analytical and semi-discretized numerical method, Finite Element Method of Lines (FEMOL) has shown good performance on this type of problems. Based on the proposed theory of infinite elements with mapping technique, the adaptive FEMOL with Element Energy Projection (EEP) super-convergent method is applied to the solution of 2D unbounded domain problems, in which users are only required to pre-specify an error tolerance and a rough initial mesh, and then an adaptive FEMOL mesh is automatically produced by the algorithm, on which the accuracy of FEMOL solution with both regular elements and infinite elements satisfies the specified error tolerance in maximum norm. An introduction of the theory of FEMOL and the infinite elements are given firstly, and then the strategy of adaptive FEMOL based on EEP method is presented. The feasibility of applying the adaptive FEMOL to unbounded domain problems is analyzed. Then three unbounded domain problems are adaptively solved, including the Poisson equation of flow around a circular cylinder, the plane problem of uniaxial tension of infinite plate with a circle hole in elasticity, and the semi-infinite half space body under uniformly distributed circular load. Finally the displacements (function solutions) satisfying the error tolerance can be obtained and the stresses (derivative solutions) with superior accuracy can be calculated. Therefore the adaptive FEMOL can be taken as a new approach for solution of unbounded domain problems.

Key words: unbounded domain problems, self-adaptivity, finite element method of lines (FEMOL), infinite elements, element energy projection (EEP)

中图分类号: 

  • O241.82
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