Journal of Guangxi Normal University(Natural Science Edition) ›› 2012, Vol. 30 ›› Issue (1): 15-21.

Previous Articles     Next Articles

A Two-Level Method for Locking-free Finite Element Discretizationin Linear Elasticity

ZHANG Hong-mei1, XIAO Ying-xiong2, OUYANG Yuan3   

  1. 1.School of Science,Hunan University of Technology,Zhuzhou Hunan 412007,China;
    2.College of Civil Engineering and Mechanics,Xiangtan University,XiangtanHunan 411105,China;
    3.Hunan Valin Lianyuan Iron and Steel Company Limited,Loudi Hunan 417009,China
  • Received:2012-01-09 Online:2012-01-20 Published:2018-12-03

Abstract: Higher-order conforming finite elements can effectively overcome the poisson-Locking in linear elasticity,which is call and Locking-free finite elements.But when compared with the linear element,it often requiresmore computer storage and has a higher computational complexity.For the Locking-free(quartic) finite element discretization in linear elasticity,a general two-level method is proposed by analyzing the relationship between the quadraticfinite element space and the quartic finite element space and by taking advantage of the special nature of the finite element's basi functions,such as compactly supported.First,the quadratic element is chosen as the coarse level space.Second,by combining the selective reduced integration and some efficient smoothers,then,obtain the two-level method is obtained in which the element is chosen as the coarse level space for the Locking-free finite element discretization with better robustness and high efficiency.The numerical results show the efficiency of the resulting method.

Key words: elasticity problem, higher-order finite element, locking phenomenon, two-level method, selective reduced integration

CLC Number: 

  • O343
[1] BABUSKA I,SURI M.On locking and robustness in the finite element method[J].SIAM Journal on Numerical Analysis,1992,29:1261-1293.
[2] BABUSKA I,SURI M.Locking effects in the finite element approximation of elasticity problems[J].Numerische Ma-thematik,1992,62(1):439-463.
[3] ARNOLD D N.Discretization by finite elements of a model parameterdependent problem[J].Numerische Mathematik,1981,37(3):405-421.
[4] ARNOLD D N,FALK R S.A new mixed formulation for elasticity[J].Numerische Mathematik,1988,53(1):13-30.
[5] MORLEY M.A family of mixed finite elements for linear elasticity[J].Numerische Mathematik,1989,55(6):633-666.
[6] BRENNER S C,SUNG L Y.Linear finite element methods for planar linear elasticity[J].Mathematics of Computation,1992,59(200):321-338.
[7] WANG Lie-heng,HE Qi.A locking-free scheme of noncomforming rectagular finite element for the planar elasticity[J].Journal of Computational Mathematics,2004,22(5):641-650.
[8] LEE C O,LEE J W,SHEEN D W.A locking free nonconforming finite element method for planar linear elasticity[J].Advances in Computational Mathematics,2003,19(1):277-291.
[9] SCOTT L R,VOGELIUS M.Norm estimates for a maximal right inverse ofthe divergence operator in spaces of piecewise polynomials[J].RAIRO Math Modeling Numer Anal,1985,19(1):111-143.
[10] SCOTT L R,VOGELIUS M.Conforming finite element methods for imcompreeeible and nearly incompressible continua[M]//ENGQUIST B E,OSHER S,SOMERVILLE R C J.Large Scale Computations in Fluid Mechanics,Part 1:Lectures in AppliedMathematics Vol 22.Providence,RI:AMS,1985:221-244.
[11] ZIENKIEWICZ O C,TAYLOR R L,TOO J M.Reduced integration techniquein general analysis of plates and Shells[J].International Journal for Numerical Method in Engineering,1971,3(2):275-290.
[12] BRANDT A.Algebraic multigrid theory:the symmetric case[J].Applied Mathematics of Computation,1986,19(14):23-56.
[13] RUGE J W,STUBEN K.Algebraic multigrid,in Multigrid Methods[M].Philadelphia,PA:Society for Industrial and Applied Mathematics,1987.
[14] RUGE J,McCORMICK S,MANTEUFFEL T,et al.AMG for higher-order discretizations of second-order elliptic PDES[C/OL]//Abstracts of the Eleventh Copper Mountain Conference on Multigrid Methods.Copper Mountain,CO.2003[2011-12-10].http://www.mgnet.org/mgnet/Conferences/CopperMtn03/Talks/ruge.pdf.
[15] HEYS J J,MANTEUFFEL T A,MCORMICK S F,et al.Algebraic multigrid (AMG) for higher-order finite elements[J].Journal of Computational Physics,2005,204:520-532.
[16] XIAO Ying-xiong,SHU Shi,ZHAO Tu-yan.A geometric-based algebraic multigrid for higher-order finite element equations in two dimensional linear elasticity[J].Numerical Linear Algebra with Applications,2009,16(7):535-559.
[17] SCHO¨BERL J.Multigrid methods for a parameter dependent problem in primal variables[J].Numerische Mathematik,1999,84(23):97-119.
[18] XIAO Ying-xiong,SHU Shi,ZHANG Hong-mei,et al.An algebraic multigrid method for nearly incompressible elasticity problem in two dimensions[J].Advances in Applied Mathematics and Mechanics,2009,1(1):69-88.
[19] 欧阳媛.一种求解二维弹性可压和几乎不可压问题的代数多层网格法[D].湘潭:湘潭大学数学与计算机科学学院,2008.
[20] WANG Lie-heng,HE Qi.On Locking finite element schemes for the pure displacement boundary value problem in the planar elasticity[J].MathematicaNumerica Sinica,2002,24(2):243-256.
[21] 陈翠玲,李明,梁家梅,等.Wolfe线搜索下一类新的共轭梯度法及其收敛性[J].广西师范大学学报:自然科学版,2010,28(3):24-28.
No related articles found!
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed   
No Suggested Reading articles found!