工程力学 ›› 2019, Vol. 36 ›› Issue (12): 7-14.doi: 10.6052/j.issn.1000-4750.2019.01.0005

• 基本方法 • 上一篇    下一篇

二阶非线性常微分方程边值问题有限元p型超收敛计算

叶康生, 邱廷柱   

  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:2019-02-26 修回日期:2019-05-15 出版日期:2019-12-25 发布日期:2019-05-31
  • 通讯作者: 叶康生(1972-),男,江苏人,副教授,博士,主要从事结构工程研究(E-mail:yeks@tsinghua.edu.cn). E-mail:yeks@tsinghua.edu.cn
  • 作者简介:邱廷柱(1993-),男,云南人,硕士生,主要从事结构工程研究(E-mail:1091138280@qq.com).
  • 基金资助:
    清华大学自主科研计划项目(2011THZ03)

A p-TYPE SUPERCONVERGENT RECOVERY METHOD FOR FE ANALYSIS ON BOUNDARY VALUE PROBLEMS OF SECOND-ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

YE Kang-sheng, QIU Ting-zhu   

  1. Department of Civil Engineering, Tsinghua University, Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Beijing 100084, China
  • Received:2019-02-26 Revised:2019-05-15 Online:2019-12-25 Published:2019-05-31

摘要: 该文提出二阶非线性常微分方程边值问题有限元求解的p型超收敛算法。该法基于有限元解答中结点解的超收敛特性,以单元端部的有限元解作为单元边界条件,通过泰勒展开技术在单个单元上建立了单元解近似满足的线性常微分方程边值问题,对该局部线性边值问题采用单个高次元进行有限元求解获得该单元上的超收敛解,对每个单元实施上述过程可获得全域的超收敛解。该法为后处理法,且后处理计算仅在单个单元上进行,通过很少量的计算即能显著提高解答的精度和收敛阶。数值结果显示,该法高效、可靠,是一个颇具潜力的方法。

关键词: 非线性, 常微分方程, 边值问题, 有限元, p型超收敛

Abstract: It presents a p-type superconvergent recovery method for the finite element analysis on two-point boundary value problems (BVPs) of second-order nonlinear ordinary differential equations. Based on the superconvergence property of nodal values, a linear two-point BVP which approximately governs the solutions on each element is set up by setting the elements' end values in FE solutions as boundary conditions and linearizing the governing differential equations via Taylor expansion technique. This local linear BVP is solved by using a higher order element from which the solution on each element is recovered. This method is a post-processing approach and the recovery computation is carried out on each element separately. It can improve the accuracy and convergence rate of the solutions significantly with a small computation. Numerical examples demonstrate that this method is efficient, reliable and potential.

Key words: nonlinearity, ordinary differential equation, boundary value problem, finite element method, p-type superconvergent recovery

中图分类号: 

  • TU311
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