轴对称弹塑性扭转问题的数值解

在很多自由边界问题的研究中,变分不等式是一个有力的工具,它不但可以用来研究解的存在唯一性、正则性等理论问题,而且还提供了有效的数值方法(见1-3]).对轴对称机轴的弹塑性扭转问题,4,5]用变分不等式研究了解的存在唯一性和正则性,在此基础上,6]建议用有限元法求解等价的障碍问题.该法的缺点是,事先要解一个一阶非线性偏微分方程的Cauchy问题以求出障碍函数,并且此Cauchy问题的解一般不唯一.本文的方法是直接将原来的变分不等式问题作有限元离散,再将离散问题化成鞍点问题,然后采用Uzawa型算法求解.这就避免了6]中方法的上述困难.

THE NUMERICAL SOLUTION OF AN AXISYMMETRIC ELASTIC-PLASTIC TORSION PROBLEM

For the elastic-plastic torsion problem of an axisymmetric shaft we propose inthis paper that the original problem (instead of the equivalent obstacle problem) besolved by the FEM. The existence and uniqueness of the solution for the FE discreteproblem is proved under some conditions. The corresponding saddle point problem ofthe discrete problem is established, and the convergence is proved for an algorithm ofUzawa's type for the saddle point problem.