Preconditioned Conjugate Gradient Methods for the 3D Wilson Nonconforming FEM Discretizations

XIAO Ying-xiong; LI Zhen-you

doi:10.21656/1000-0887.370037

Volume 37 Issue 8

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XIAO Ying-xiong, LI Zhen-you. Preconditioned Conjugate Gradient Methods for the 3D Wilson Nonconforming FEM Discretizations[J]. Applied Mathematics and Mechanics, 2016, 37(8): 820-831. doi: 10.21656/1000-0887.370037

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Preconditioned Conjugate Gradient Methods for the 3D Wilson Nonconforming FEM Discretizations

doi: 10.21656/1000-0887.370037

College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan, Hunan 411105, P.R.China

Funds: The National Natural Science Foundation of China(10972191)

Received Date: 2016-01-25
Rev Recd Date: 2016-03-16
Publish Date: 2016-08-15

Abstract

Abstract

The nonconforming finite element method (FEM) is an efficient method to overcome the volume locking trouble in 3D elasticity problems. This method has the advantages of a few degrees of freedom and high accuracy. In order to improve the overall efficiency of the FEM analysis, it is necessary to design some faster solvers for the corresponding system of discretization equations. The faster solvers for the Wilson nonconforming FEM discretizations were considered. When Poisson’s ratio ν was close to 0.5, the resulting system of equations was symmetric positive definite and highly illconditioned, and the preconditioned conjugate gradient (PCG) method was one of the most efficient methods for solving such FEM equations. Moreover, in practical applications, anisotropic meshes are often obtained due to the specificity of the structure considered, which will greatly decrease the convergence rate of the PCG method. A type of PCG method based on the DAMG was presented and then applied to the solution of the Wilson FEM discretizations. This DAMG was an algebraic multi grid (AMG) method based on the distance matrix and can be used to solve the system of equations discretized on anisotropic meshes. The numerical results show that, in combination with the effective smoothing operators, the proposed PCG method has high efficiency and robustness for nearly incompressible problems.
- volume locking,
- Wilson nonconforming element,
- ill-conditioned matrix,
- algebraic multigrid method,
- preconditioner

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References(20)

References

[1]	Morley M. A family of mixed finite elements for linear elasticity[J].Numerische Mathematik,1989,55(6): 633-666.
[2]	Falk R S. Nonconforming finite element methods for the equations of linear elasticity[J].Mathematics of Computation,1991,57(196): 529-550.
[3]	QI He, WANG Lie-heng, ZHENG Wei-ying. On locking-free finite element schemes for three dimensional elasticity[J].Journal of Computational Mathematics,2005,23(1): 101-112.
[4]	Brenner S C. A nonconforming mixed multigrid method for the pure traction problem in planar linear elasticity[J].Mathematics of Computation,1994,63(208): 435-460.
[5]	Scott L R, Vogelius M. Conforming finite element methods for incompressible and nearly incompressible continua[C]//Large Scale Computations in Fluid Mechanics,Part 2, 1983: 221-244.
[6]	Hackbusch W.Multi-Grid Methods and Applications [M]. New York: Springer-Verlag, 1985.
[7]	Trottenberg U, Oosterlee C W, Schuler A.Multigrid[M]. Academic Press, 2001.
[8]	刘石, 陈德祥, 冯永新, 徐自力, 郑李坤. 等几何分析的多重网格共轭梯度法[J]. 应用数学和力学，2014,35(6): 630-639.(LIU Shi, CHEN De-xiang, FENG Yong-xin, XU Zi-li, ZHENG Li-kun. A multigrid preconditioned conjugate gradient method for isogeometric analysis[J].Applied Mathematics and Mechanics,2014,35(6): 630-639.(in Chinese)）
[9]	CHEN Zhang-xin, Oswald P. Multigrid and multilevel methods for nonconforming Q₁ element[J].Mathematics of Computation,1998,67(222): 667-693.
[10]	周叔子, 文承标. 抛物问题非协调元多重网格法[J]. 计算数学, 1994,16(4): 372-381.(ZHOU Shu-zi, WEN Cheng-biao. Nonconforming element multigrid method for parabolic equations[J].Mathematica Numerica Sinica,1994,16(4): 372-381.(in Chinese))
[11]	许学军. 旋转Q₁非协调元的V循环多重网格法[J] . 计算数学，1999,21(2): 251-256.(XU Xue-jun. V-cycle multigrid methods for Q₁ nonconforming finite elements[J].Mathematica Numerica Sinica,1999,21(2): 251-256.(in Chinese))
[12]	SHI Zhong-ci, XU Xue-jun. V-cycle multigrid methods for Wilson nonconforming element[J].Science in China(Series A): Mathematics,2000,43(7): 673-684.
[13]	Chartier T P, Falgout R D, Henson V E, Jones J E, Maneuffel T, McCormick S F, Ruge J W, Vassilevski P S. Spectral AMGe(ρAMGe)[J].SIAM Journal on Scientific Computing,2003,20(1): 1-20.
[14]	Brezina M, Cleary A J, Falgout R D, Henson V E, Jones J E, Manteuffel T A, McCormick S F, Ruge J W. Algebraic multigrid based on element interpolation (AMGe)[J].SIAM Journal on Scientific Computing,2000,22(5): 1570-1592.
[15]	Kolev T V, Vassilevski P S. AMG by element agglomeration and constrained energy minimization interpolation[J].Numerical Linear Algebra With Applications,2006,13(9): 771-788.
[16]	XIAO Ying-xiong, SHU Shi, ZHANG Ping, TAN Min. An algebraic multigrid method for isotropic linear elasticity on anisotropic meshes[J].International Journal for Numerical Methods in Biomedical Engineering,2010,26(5): 534-553.
[17]	El maliki A, Guénette R, Fortin M. An efficient hierarchical preconditioner for quadratic discretizations of finite element problems[J].Numerical Linear Algebra With Applications,2011,18(5): 789-803.
[18]	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.
[19]	Saad Y.Iterative Methods for Sparse Linear Systems[M]. 2nd ed. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2003.
[20]	Notay Y. An aggregation-based algebraic multigrid method[J].Institute of Computational Mathematics,2010,37: 123-146.