Penalty Function Finite Element Analysis for Nearly Incompressible Elasticity Problems in Three Dimensions
Abstract: The locking phenomenon will appear when the commonly used finite elements are applied to the solution of nearly incompressible problems in three dimensions. It is necessary to use some special methods. The penalty function conforming finite element method is an effective method to overcome this locking phenomenon since it is simple for the realization of the resulting program and easy to determine the penalty number and it also does not change the functional stationary value properties. In this paper, the computing format of penalty function finite element method is carefully derived, the conditions for success of the resulting method is analyzed and the effective-ness and robustness of this method are finally verified by some numerical experiments for nearly incompressible elasticity problems. The quality of the mesh used in three-dimensional finite ele-ment analysis has a great effect on the accuracy and computational efficiency. If the isotropic grids can be used in the practical calculations, the method will have better convergence.
文章引用: 肖映雄 , 周 磊 (2014) 三维近不可压缩弹性问题的罚函数有限元分析。 力学研究， 3， 43-54. doi: 10.12677/IJM.2014.34005
 Cheung, Y.K. and Chen, W. (1989) Hybrid element method for incompressible and nearly incompressible materials. International Journal of Solids and Structures, 25, 483-495.
 Morley, M. (1989) A mixed family of elements for linear elasticity. Numerische Mathematik, 55, 633-666.
 Chama, A. and Reddy, B.D. (2013) New stable mixed finite element approximations for problems in linear elasticity. Computer Methods in Applied Mechanics and Engineering, 256, 211-223.
 Brenner, S.C. and Scott, L.R. (1998) The mathematical theory of finite element methods. Springer-Verlag.
 Wang, L.H. and Qi, H. (2004) A locking-free scheme of nonconforming rectangular finite element for the planar elasticity. Journal of Computational Mathematics, 22, 641-650.
 Falk, R.S. (1991) Nonconforming finite element methods for the equations of linear elasticity. Mathematics of Computation, 57, 529-556.
 Brenner, S.C. (1994) A nonconforming mixed multigrid method for the pure traction problem in planar linear elasticity. Mathematics of Computation, 63, 435-460.
 Scott, L.R. and Vogelius, M. (1985) Conforming finite element methods for incompressible and nearly incompressible continua. In: Large Scale Computations in Fluid Mechanics. Lectures in Applied Mathematics, Vol. 22, AMS, Providence, 221-244.
 Stenberg, R. and Suri, M. (1996) Mixed h-p finite element methods for problems in elasticity and Stokes flow. Numerische Mathematik, 72, 367-390.
 Oden, J.T. and Carey, G.F. (1981) Finite elements (Vol: V). Prentice-Hall, Upper Saddle River.
 Hughes, T.J.R. (1977) Equivalence of finite elements for nearly incompressible elasticity. Journal of Applied Mechanics, 44, 181-183.
 Malkus, D.S. and Hughes, T.J.R. (1978) Mixed finite element methods-reduced and selective integration techniques: A unification of concepts. Computer Methods in Applied Mechanics and Engineering, 15, 63-81.
 Qi, H., Wang, L.-H. and Zheng, W.-Y. (2005) On locking-free finite element schemes for three-dimensional elasticity. Journal of Computational Mathematics, 23, 101-112.
 Du, Q. and Wang, D.S. (2003) Tetrahedral mesh generation and optimization based on Centroidal Voronoi Tessellations. International Journal for Numerical Methods in Engineering, 56, 1355-1373.
 王勖成 (2006) 有限单元法. 清华大学出版社, 北京.