弹性力学广义混合变分原理及有限元广义混合法

近些年弹性力学中出现一种新型的变分原理,称为广义混合变分原理.特点是其泛函中包含某些可以任意选择的附加函数,称为分裂因子.新原理将弹性理论中现有的各主要变分原理都统一在一个框架中,并揭示出它们之间更深一层的相互关系.在应用方面,它提供了一个新的数学手段以建立有限元分析中的新模式.这些新模式已经显示出它们的优点:适当调节分裂因子,它们给出更好的数值解答,特别是,可用它们来处理有限元方法中棘手的病态问题.本文综述了线性及非线性弹性理论中的这种新型变分原理并就其在有限元中的应用作了讨论. 

GENERALIZED MIXED VARIATIONAL PRINCIPLE AND FEM IN ELASTICITY

A new type of variational principle in elasticity, called the generalized mixed variational principle, has been developed in recent years. Its distinguished feature is that its functional contains some arbitrary additional function, called the splitting factor. This new principle unifies all the main existing variational principles in elasticity into one frame and brings to light a more profound relationship among them. In application, it offers us a fresh mathematical means to establish new models for finite clement analysis. These models have shown their advantages, by regulating the splitting factor, they give better numerical solutions, and especially, can be used to cope with the thorny ill-conditioned problems in FEM.This paper gives a summary of the new variational principles in both linear and nonlinear theories of elasticity and presents some discussion on their applications to FEM.