含多个任意参数的广义变分原理及换元乘子法
Generalized Variational Principles with Several Arbitrary Parameters and the Variable Substitution and Multiplier Method
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摘要: 弹性力学变分原理的泛函变换可分为三种格式:Ⅰ、放松格式,Ⅱ、增广格式,Ⅲ、等价格式. 根据格式Ⅲ,提出含多个任意参数的广义变分原理及其泛函表示式,其中包括:以位移u为一类泛函变量的多参数广义变分原理;以位移u和应力σ为二类泛函变量的多参数广义变分原理;以位移u和应变ε为二类泛函变量的多参数广义变分原理;以位移u应变ε和应力σ为三类泛函变量的多参数广义变分原理.由这些原理可得出等价泛函一系列新形式,此外,通过参数的合理选择,可构造出一系列有限元模型. 本文还讨论了拉氏乘子法“失效”问题,指出“失效”现象产生的原因,提出乘子法“恢复有效”的作法——换元乘子法.Abstract: The functional transformations of variational principles in elasticity are classified as three patterns: Ⅰ relaxation pattern, Ⅱ augmented pattern and III equivalent pattern.On the basis of pattern Ⅲ, the generalized variational principles with several arbitrary parameters are formulated and their functionals are defined. They are: the generalized principle of single variable u with several parameters, the generalized principle of two variables u, σ with several parameters, the generalized principle of two variables u, ε with several parameters, and the generalized principle of three veriables u, ε, σ with several parameters. From these principles, a series of new forms of equivalent functionals can be obtained. When the values of these parameters are properly chosen, a series of finite element models can be formulated.In this paper, the question of losing effectiveness for Lagrange multiplier method is also discussed. In order to "recover" effectiveness for multiplier method, a modified method, namely, the variable substitution and multiplier method, is proposed.
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[1] 钱伟长,高阶拉氏乘子法和弹性理论中更一般的广义变分原理,应用数学和力学,4,2 (1983),137-150. [2] 陈万吉,更一般的杂交广义变分原理及有限元模型,合肥工业大学学报,4 (1983). [3] Taylor R.L.,O.C.Zienkiewicz,Complementary energy with penalty functions in finite element anaysis,Energy Methods in Finite Element Analysis,John Wiley & Sons(1979). [4] Oden J.T.,Penalty-finite element methods for constrained problems in elasticity,Proc.of Symposium on Finite Element Method(1982).
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