纯位移线弹性方程Locking-Free非协调三棱柱单元的构造分析

数学物理学报 ›› 2019, Vol. 39 ›› Issue (2): 329-338.

纯位移线弹性方程Locking-Free非协调三棱柱单元的构造分析

孙艳萍^1,*(),陈绍春²

¹ 河南工程学院理学院郑州 451191
² 郑州大学数学与统计学院郑州 450001

收稿日期:2017-10-12 出版日期:2019-04-26 发布日期:2019-05-05
通讯作者: 孙艳萍 E-mail:yanpingsun2007@126.com
基金资助:
国家自然科学基金(11701522);河南省教育厅科学技术研究重点项目(17A110017)

A Nonconforming Locking-Free Triangular Prism Element Analysis for Linear Elasticity Problem

Yanping Sun^1,*(),Shaochun Chen²

¹ College of Science, Henan Institute of Engineering, Zhenzhou 451191
² School of Mathematics and Statistics, Zhengzhou University, Zhenzhou 450001

Received:2017-10-12 Online:2019-04-26 Published:2019-05-05
Contact: Yanping Sun E-mail:yanpingsun2007@126.com
Supported by:
the NSFC(11701522);the Science and Technology Research Program of Education Department of Henan Province(17A110017)

摘要/Abstract

摘要：

主要构造了三维空间中线弹性问题纯位移变分形式下无闭锁三棱柱单元.此单元是具有18个自由度的非协调元.单元的形函数满足位移的散度属于零次多项式空间，通过分析得到有限元解和真解误差的能量模具有一阶收敛性，L²模具有二阶收敛性.

关键词: 平面弹性问题, 非协调单元, 三棱柱单元, Strong引理, Lamé常数

Abstract:

This paper discuss the linear elasticity problem and constructs a nonconforming triangular prism element with 18 degrees of freedom. The shape functions of this element satisfy that the divergence of displacement is zeroth polynomial. We can deduce that the energy norm has the first order convergence rate and the L² norm has the second order convergence rate.

Key words: Planar elasticity problem, Nonconforming element, Triangular prism element, Strong lemma, Lamé constant

中图分类号:

O242.21

孙艳萍,陈绍春. 纯位移线弹性方程Locking-Free非协调三棱柱单元的构造分析[J]. 数学物理学报, 2019, 39(2): 329-338.

Yanping Sun,Shaochun Chen. A Nonconforming Locking-Free Triangular Prism Element Analysis for Linear Elasticity Problem[J]. Acta mathematica scientia,Series A, 2019, 39(2): 329-338.

参考文献

1	Brenner S C , Li Y S . Linear finite element methods for planar linear elasticity. Mathematics of Computation, 1992, 59: 321- 330
2	Falk R S . Nonconforming finite element methods for the equations of linear elasticity. Mathematics of Computation, 1991, 57: 529- 550
3	Chen S C , Zheng Y J , Mao S P . New second order nonconforming triangular element for planar elasticity problems. Acta Mathematica Scientia, 2011, 31B (3): 815- 825
4	Babu?ka I , Suri M . Locking effects in the finite element approximation of elasticity problems. Numer Math, 1992, 62 (1): 439- 464
5	Brenner S C , Scott L R . The Mathematical Theory of Finite Element Methods. New York: Springer-Verlag, 2002
6	Kouhia R , Stenberg R . A linear nonconforming finite element method nearly incompressible elasticity and stokes flow. Comput Methods Appl Mech Engi, 1995, 124 (3): 195- 212
7	Zhang Z M . Analysis of some quadrilateral nonconforming elements for incompressible elasticity. SIAM J Numer Anal, 1997, 34 (2): 640- 663
8	明平兵.非协调元vs Locking问题[D].杭州:浙江大学, 1999
8	Ming P B. Nonconforming Finite Elements vs Locking Problem[D]. Hangzhou: Zhejiang University, 1999
9	Lee C O , Lee J , Sheen D . A locking-free nonconforming finite element method for planar linear elasticity. Adv Comput Math, 2003, 19: 277- 291
10	Cheng X L , Hong H C , Zou J . Quadrilateral finite elements for planar linear elasticity problem with large Lamé constant. Journal of Computational Mathematics, 1998, 16: 357- 366
11	王烈衡, 齐禾. 关于纯位移边界条件的平面弹性问题Locking-Free有限元格式. 计算数学, 2002, 24 (2): 243- 256
11	Wang L H , Qi H . On locking-free finite element schemes for the pure displacement boundary value problem in the planar elasticity. Mathematica Numerica Sinica, 2002, 24 (2): 243- 256
12	Wang L H , Qi H . A locking-free scheme of nonconforming rectangular finite element for the planar elasticity. Journal of Computational Mathematics, 2004, 22 (5): 641- 650
13	Qi H , Wang L H , Zheng W Y . On locking-free finite element schemes for three-dimensional elasticity. Journal of Computational Mathematics, 2005, 23 (1): 101- 112
14	Shi D Y , Mao S P , Chen S C . A locking-free anisotropic nonconforming finite element for planar linear elasticity problem. Acta Mathematica Scientia, 2007, 27B (1): 193- 202
15	Shi D Y , Wang C X . A locking-free anisotropic nonconforming rectangular finite element approximation for the planar elasticity problem. Appl Math J Chin Univ, 2008, 23 (1): 9- 18
16	陈绍春, 任国彪. 一个二阶收敛的平面弹性问题Locking-Free三角形元. 工程数学学报, 2009, 26 (3): 509- 518
16	Chen S C , Ren G B . A locking-free triangular element with second-order convergence for the planar elasticity problem. Chinese Journal of Engineering Mathematics, 2009, 26 (3): 509- 518
17	陈绍春, 肖留超. 平面弹性的一个新的Locking-Free非协调有限元. 应用数学, 2007, 20 (4): 739- 747
17	Chen S C , Xiao L C . A new locking-free nonconforming finite element for the planar elasticity. Mathematica Applicata, 2007, 20 (4): 739- 747
18	Sun H X , Xiao L C . Locking-free DSP element for linear elasticity problem. Chinese Journal of Engineering Mathematics, 2011, 28 (1): 118- 128

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