三维近不可压缩弹性问题的罚函数有限元分析

Penalty Function Finite Element Analysis for Nearly Incompressible Elasticity Problems in Three Dimensions

摘要: 对三维近不可压缩弹性问题，利用常规有限元进行求解时会出现体积闭锁现象，需要采用某些特殊的方法。罚函数协调有限元法具有程序实现简单、罚数易于确定以及不改变泛函驻值性质等特点，是克服体积闭锁现象的一种有效方法。本文，针对混合边界条件的三维近不可压缩问题，详细推导了罚函数有限元法的计算格式，分析该方法实施成功的条件，并通过数值实验验证了该方法对解决体积闭锁现象的有效性和鲁棒性。在三维有限元分析中，剖分网格的质量将对计算精度和求解效率产生很大影响，实际计算时若能采用各向同性网格，则对问题的分析将具有更好的收敛性。

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.