二维有限元单元角结点位移精度修正之初探  

A PRELIMINARY STUDY ON ACCURACY IMPROVEMENT OF NODAL DISPLACEMENTS IN 2D FINITE ELEMENT METHOD

作　　者：黄泽敏 袁驷[1] HUANG Ze-min;YUAN Si(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,北京100084

出　　处：《工程力学》2021年第S01期1-6,共6页Engineering Mechanics

基　　金：国家自然科学基金项目(51878383,51378293)。

摘　　要：二维四边形有限元单元角结点位移相较于其他结点位移,有更高的收敛阶。对于足够光滑的问题,采用m次单元,其角结点位移收敛阶最高可达2m阶。该文以二维Poisson方程为例,在有限元解的基础上,利用单元能量投影(EEP)法的超收敛解计算残余荷载向量,在不改变整体刚度矩阵的基础上,仅需进行代数方程组回代,即可得到具有更高精度的单元角结点位移。数值结果表明:当采用EEP简约格式解计算残余荷载向量时,单元角结点位移收敛阶最高可提高为2m+2阶。特别地,对于线性元,精度翻倍,效益十分显著。The nodal displacements gain higher convergence rate compared with displacements elsewhere in Finite Element Method(FEM).When the problem is smooth enough,the errors of displacements at element corner nodes can gain convergence order of at most 2m using elements of degree m in 2D FEM.In this paper,taking the 2D Poisson equation as the model problem and based on an obtained FEM solution,residual nodal load vectors are derived by using super-convergent solutions calculated by Element Energy Projection(EEP)technique.Without changing global stiffness matrices,simple back-substitutions alone would yield nodal displacements of higher convergence rate.Numerical examples show that the nodal displacements can gain super-convergence order of 2m+2 at most when EEP simplified form is used to calculate the residual load vectors.In particular,for linear elements,the accuracy is doubled,and the benefit is very significant.

关 键 词：二维有限元 POISSON方程 单元角结点位移精度修正 单元能量投影 超收敛 

分 类 号：O242.21[理学—计算数学]