二阶非线性常微分方程边值问题有限元p型超收敛计算  

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

作　　者：叶康生[1] 邱廷柱 YE Kang-sheng;QIU Ting-zhu(Department of Civil Engineering,Tsinghua University,Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry,Beijing 100084,China)

机构地区：[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

出　　处：《工程力学》2019年第12期7-14,共8页Engineering Mechanics

基　　金：清华大学自主科研计划项目(2011THZ03)

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。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.

关 键 词：非线性 常微分方程 边值问题 有限元 p型超收敛 

分 类 号：TU311[建筑科学—结构工程]