线法二阶常微分方程组有限元分析的结点精度修正及其超收敛计算  

NODAL ACCURACY IMPROVEMENT AND SUPER-CONVERGENT COMPUTATION IN FEM ANALYSIS OF FEMOL SECOND ORDER ODES

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》2022年第S01期9-14,34,共7页Engineering Mechanics

基　　金：国家自然科学基金项目(51878383,51378293)。

摘　　要：采用m次单元对线法二阶常微分方程组(ODEs)进行有限元(FEM)求解,其单元内部位移为m+1阶收敛,而端结点位移收敛阶可达2 m阶。单元能量投影(EEP)超收敛计算恢复的单元内部位移精度一般为(m+2,2m)阶,此收敛阶既受益于也受限于有限元端结点位移的精度。该文提出了一种修正EEP法(M-EEP),利用EEP超收敛解,先对端结点位移进行修正,再用其恢复单元内部位移。广泛的数值试验表明:对端结点位移修正后的收敛阶可达2m+2阶,再次修复的单元内部位移始终可达m+2阶收敛,摆脱了2 m阶收敛精度的限制。对于线性元,修正后结点位移的精度翻倍,单元内部M-EEP位移亦摆脱了原FEM解2阶收敛精度的限制,升到3阶收敛,基本达到二次元的收敛精度,效果显著。Elements with degree m is used in finite element method(FEM)to solve the second order ordinary differential equations(ODEs)derived from the FEM of lines(FEMOL).The interior displacement of elements generally has a convergence order of m+1,while the nodal displacements can achieve a convergence order of 2m.The super-convergence computation using the element energy projection(EEP)method usually has a convergence order of min(m+2,2m),which benefits from the nodal displacements of a higher convergence order but also limits its accuracy by the nodal displacements of elements with lower degrees.In this paper,a modified EEP(M-EEP)method is proposed.With the EEP solution,the nodal displacement accuracy is improved first,and then the interior displacement of elements is recovered,which leads to a modified EEP solution.Numerical experiments show that improved nodal displacements can achieve a convergence order of 2m+2,and the interior displacements of elements always have a convergence order of m+2 without the constraint of order 2m.For linear elements,the interior displacement of M-EEP solution does not have the limitation of second-order convergence from the traditional FEM solution and can achieve the remarkable third-order convergence,equivalent to the convergence order of quadratic elements.

关 键 词：有限元法 二阶常微分方程组 超收敛 单元能量投影(EEP) 修正的EEP法(M-EEP) 有限元线法(FEMOL) 

分 类 号：O241.81[理学—计算数学]