纯位移线弹性方程Locking-Free非协调三棱柱单元的构造分析
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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,L~2模具有二阶收敛性.
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 L~2 norm has the second order convergence rate.
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 L~2 norm has the second order convergence rate.
引文
[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] Babuska 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,1999Ming 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 Lame 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):243-256Wang 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):509-518Chen 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):739-747Chen 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
[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] Babuska 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,1999Ming 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 Lame 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):243-256Wang 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):509-518Chen 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):739-747Chen 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