拉氏乘子法在广义变分原理和有限元素法中的应用

The Application of Lagrangian Multiplier Method in Generalized Variational Principles and Finite Element Method

在弹塑性力学的广义变分原理的研究中 ,广泛应用Lagrange乘子法 ,我国学者对Lagrange乘子是否唯一的问题进行了有益的讨论 .本文通过研究弹性力学的广义变分原理 ,论述了从一个角度看问题 ,La grange乘子是唯一的 ,从另一个角度看问题 ,Lagrange乘子又是不唯一的 ,两种观点反映了同一事物的两个不同的侧面 .通过研究有限元素法中的位移杂交的混合元模型和应力杂交的混合元模型 ,论述了有时在某个局部区域中 ,构成关于Lagrange乘子的定解微分或代数方程组 。

In the research of generalized variational principles in elasticity and plasticity,Lagrangian multiplier method has been used widely.The scholars of our nation discussed the problem of whether Lagrangian multiplier is unique or not.Because of the significance and comlexity of the problem,the discussion has been carried out off and on for more than 20 years.There are two viewpoints in this discussion.According to the results of generalized variational principles in elasticity,this paper expounds that Lagrangian multiplier is unique from one point of view and Lagrangian multiplier is not unique from another point of view.Those two viewpoints reflect the two different aspects of the same object.According to the results of finite element method,this paper expounds that Lagrangian multiplier is unique when there are closed differential equation grub or algebraic equation grub in some local area.