论非线性弹性理论的各种变分原理

本文从“全能量原理”出发推导出非线性弹性理论中各种可能的主要的变分原理的泛函,其中有几个是在我们所能见到的文献中还没有的.通过本文的推导过程,我们提出了胡海昌的专著[1]中表6.1的第11类和第6类变分原理不存在的猜想.     

带摩擦的弹性接触问题广义变分不等原理的简化证明

在弹性摩擦接触问题中 ,从变分原理出发来研究接触问题 ,可以将摩擦力纳入问题的能量泛函 .为了得到摩擦约束弹性接触问题的能量泛函 ,日前大多是用拉格朗日乘子法 ,但拉格朗日方法用在变分不等问题中 ,要利用非线性泛函分析和凸分析来证明 ,证明复杂 .本文利用向量分析的工具及巧妙的变换 ,对带摩擦约束的弹性接触问题的广义变分不等原理进行了严格的证明 ,由于只用到向量分析 ,简化了证明 .     

变分转化分析(Ⅱ)

本文介绍有关的一些泛系分析概念(诸如泛系、泛系空间、泛系逻辑空间、半序度量泛系等),把凸集理论、Banach完全性定理、Lax等价理论、Kuhn-Tucker型定理、Дубовицкий-Милютин型定理等推广于半序度量泛系,发展了不同于传统结果的一些形式,并研究了一般的算子方程的稳定性及逼近的MSP转化原则,另外,对古典极值定理、变分学、互易原理、二次泛函变分定理及单边变分原理给出一种统一的泛函框架并作了一些推广.     

高阶拉氏乘子法和弹性理论中更一般的广义变分原理

作者曾指出[1],弹性理论的最小位能原理和最小余能原理都是有约束条件限制下的变分原理采用拉格朗日乘子法,我们可以把这些约束条件乘上待定的拉氏乘子,计入有关变分原理的泛函内,从而将这些有约束条件的极值变分原理,化为无条件的驻值变分原理.如果把这些待定拉氏乘子和原来的变量都看作是独立变量而进行变分,则从有关泛函的驻值条件就可以求得这些拉氏乘子用原有物理变量表示的表达式.把这些表达式代入待定的拉氏乘子中,即可求所谓广义变分原理的驻值变分泛函.但是某些情况下,待定的拉氏乘子在变分中证明恒等于零.这是一种临界的变分状态.在这种临界状态中,我们无法用待定拉氏乘子法把变分约束条件吸收入泛函,从而解除这个约束条件.从最小余能原理出发,利用待定拉氏乘子法,企图把应力应变关系这个约束条件吸收入有关泛函时,就发生这种临界状态,用拉氏乘子法,从余能原理只能导出Hellinger-Reissner变分原理[2],[3],这个原理中只有应力和位移两类独立变量,而应力应变关系则仍是变分约束条件,人们利用这个条件,从变分求得的应力中求应变.所以Hellinger-Reissner变分原理仍是一种有条件的变分原理.     

一类具有变号位势Kirchhoff型方程解的存在性

该文研究了一类带有变号位势非线性项的Kirchhoff型方程的Neumann边值问题.利用变分方法,首先对空间进行分解,证明了该问题的能量泛函满足山路结构;然后证明了能量泛函的(PS)序列有强收敛的子列;最后通过Ekeland变分原理和山路引理,获得了该问题两个非平凡解的存在性.     

广义重调和算子及其在薄板弯曲中的应用

本文用δ－函数具体构造出广义重调和算子，建立相应的二次泛函表达式，并将其应用于弹性薄板的弯曲问题．结果表明．当自变量函数为广义函数时，变分泛函中的自变量函数自然就允许某种程度的不连续性，用Ｌａｇｒａｎｇｅ乘子法所得的修正变分原理实际上是文中给出的变分原理的特殊形式．     

非线性广义系统最优控制的最大值原理：无限维情形

§1.引言 对于无限维非线性最优控制问题,[1]—[3]在一定条件下证明了最大值原理。在有限维情形,[4]讨论了线性广义系统的二次型指标最优问题。关于有限维非线性广义系统的讨论见[5],[6]。而对于无限维非线性广义系统的最优控制问题,目前尚无讨论。本文利用Ekeland变分原理[7]—[10]和Fattorini引理,对具有一般目标泛函的无限维广义系统的最优控制问题给出了最大值原理。     

关于弹性力学广义变分原理的等价性定理的研究

利用场论中的不变性原理研究弹性力学广义变分原理的等价性定理,主要目的是研究弹性力学广义变分原理之间的关系;根据弹性力学广义变分原理的泛函在无穷小标度变换下的不变性,证明了这些泛函之间的等价性定理．如果这些泛函具有无穷小标度变换下的不变性,那么只有两类变量是独立的, 应力应变关系是这些泛函必须满足的变分约束条件．所得到的结果再一次证明了钱伟长教授关于所有的弹性力学广义变分原理都是等价的结论．     

含多个任意参数的广义变分原理及换元乘子法

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.     

带惩罚的有限元方法的收敛性

在用有限元方法解二阶椭圆型方程的边值问题时,首先将边值问题化为一个等价的变分问题,即泛函的极值问题。对于第二,第三边值问题,在相应的变分问题中边界条件被吸收到泛函的表示式中。因而在求泛函极值时不再对允许函数类附加边界条件的约束。在这种情况下边界条件就称为自然边界条件,相应的变分问题称为无约束变分问题。而对于第一边值问题即所谓狄氏问题则不然,相应的等价变分问题是带约束的变分问题。求泛函极值时的允许函数类必须满足强加的边界条件。例如我们考虑二维有界区域Ω上的方程     

Characterization of energy minimizers in micromagnetics

We present a new characterization of minimizing sequences and possible minimizers (all called the minimizing magnetizations) for a nonlocal micromagnetic-like energy (without the exchange energy). Our method is to replace the nonlocal energy functional and its relaxation with certain local integral functionals on divergence-free fields obtained by a two-step minimization of some auxiliary augmented functionals. Through this procedure, the minimization problem becomes equivalent to the minimization of a new local variational functional, called the dual variational functional, which has a unique minimizer. We then precisely characterize the minimizing magnetizations of original nonlocal functionals in terms of the unique minimizer of the dual variational functional. Finally, we give some remarks and ideas on solving the dual minimization problem.     

A method of solving nonlinear variational problems by nonlinear transformation of the objective functional. Part II

Summary The method of transformation of the objective functional is extended to solve nonlinear variational problems with non-differentiable objective functionals. The method is applied to the Bingham flow problem.     

Variational statement of deformation problems for a composite latticed plate with various types of lattices

The variational statement of various boundary value problems for tangential displacements and forces in a latticed plate with an arbitrary piecewise smooth contour is investigated. The lattice consists of several families of bars made of a homogeneous composite material with a matrix of relatively low shear stiffness. The energy method reduces the problem to the variational problem of minimizing the energy functional. The conditions on the plate contour are established under which the functional is minimal and positive definite, which ensures that the problem is well posed.     

Projection iterative methods for extended general variational inequalities

In this paper, we introduce and study a new class of variational inequalities involving three operators, which is called the extended general variational inequality. Using the projection technique, we show that the extended general variational inequalities are equivalent to the fixed point and the extended general Wiener-Hopf equations. This equivalent formulation is used to suggest and analyze a number of projection iterative methods for solving the extended general variational inequalities. We also consider the convergence of these new methods under some suitable conditions. Since the extended general variational inequalities include general variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems.     

A variational method of deriving the equations of the non-linear mechanics of liquid crystals

The non-linear equations of the dynamics of liquid crystals [1], derived previously by the Poisson brackets method, are derived from the Hamilton-Ostrogradskii variational principle. The variational problem of an unconditional extremum of the action functional in Lagrange variables is investigated. The difference between the volume densities of the kinetic and free energy of the liquid crystal is used as the Lagrangian. It is shown that the variational equations obtained are equivalent to the differential laws of conservation of momentum and the kinetic moment of the liquid crystal in Euler variables, while the Ericksen stress tensor and the molecular field are defined in terms of the derivatives of the free energy.     

Iterative algorithms for a new class of extended general nonconvex set-valued variational inequalities

Some new classes of extended general nonconvex set-valued variational inequalities and the extended general Wiener-Hopf inclusions are introduced. By the projection technique, equivalence between the extended general nonconvex set-valued variational inequalities and the fixed point problems as well as the extended general nonconvex Wiener-Hopf inclusions is proved. Then by using this equivalent formulation, we discuss the existence of solutions of the extended general nonconvex set-valued variational inequalities and construct some new perturbed finite step projection iterative algorithms with mixed errors for approximating the solutions of the extended general nonconvex set-valued variational inequalities. We also verify that the approximate solutions obtained by our algorithms converge to the solutions of the extended general nonconvex set-valued variational inequalities. The results presented in this paper extend and improve some known results from the literature.     

Lagrange Multipliers and mixed formulations for some inequality constrained variational inequalities and some nonsmooth unilateral problems

Abstract

This paper addresses a class of inequality constrained variational inequalities and nonsmooth unilateral variational problems. We present mixed formulations arising from Lagrange multipliers. First we treat in a reflexive Banach space setting the canonical case of a variational inequality that has as essential ingredients a bilinear form and a non-differentiable sublinear, hence convex functional and linear inequality constraints defined by a convex cone. We extend the famous Brezzi splitting theorem that originally covers saddle point problems with equality constraints, only, to these nonsmooth problems and obtain independent Lagrange multipliers in the subdifferential of the convex functional and in the ordering cone of the inequality constraints. For illustration of the theory we provide and investigate an example of a scalar nonsmooth boundary value problem that models frictional unilateral contact problems in linear elastostatics. Finally we discuss how this approach to mixed formulations can be further extended to variational problems with nonlinear operators and equilibrium problems, and moreover, to hemivariational inequalities.     

Extended general variational inequalities

In this work, we introduce and consider a new class of general variational inequalities involving three nonlinear operators, which is called the extended general variational inequalities. Noor [M. Aslam Noor, Projection iterative methods for extended general variational inequalities, J. Appl. Math. Comput. (2008) (in press)] has shown that the minimum of nonconvex functions can be characterized via these variational inequalities. Using a projection technique, we establish the equivalence between the extended general variational inequalities and the general nonlinear projection equation. This equivalent formulation is used to discuss the existence of a solution of the extended general variational inequalities. Several special cases are also discussed.