变分问题

In order to reasonably depict four basic problems with friction, one Coulomb friction new form in first Kirchhoff stress is proposed to deal with finite deformation problems, other Coulomb friction form in incremental mode to elastoplastic flow theory; Hilbert function spaces concerning elastoplastical problems with friction are established, so it makes all operations and calculations in the treatise standardized within the scope of reasonably topologic structure; In view of functional extremum, the equivalence between generalized variational inequalities principles in elastoplasticity with friction and corresponding basic problems are testified by inducing Lagrangian multipliers, so it provides a rationally theoretical basis for numerical methods in elastoplasticity with friction; From the viewpoint of variational inequality, the theory of generalized variational inequalities in elasticity and elastoplasticity with frictional constraint is studied, and the uniqueness and existence of the solution of FEM is proofed under the proposed conditions of stress compatibility, and them FEM approximation and a discrete solution are discussed; Based on the principles of generalized variational inequalities in elastoplasticity with friction, direct generalized variational inequalities methods is pretended, which is a natural generalization and development of direct variational methods; Using generalized variational inequalities methods, some examples in metal forming including plane deformation, upset and extrusion are analyzed and the results prove that all the theories and methods in the paper are right, feasible, accurate and advanced.

主要内容有：为了合理地描述金属塑性成形中摩擦约束弹性、弹塑性基本问题，提出和研究了有限变形下以Kirchhoff第一应力表示的Coulomb摩擦定律形式和弹塑性流动理论下以增量形式表示的Coulomb摩擦定律表示形式；系统建立了摩擦约束弹塑性问题的Hilbert函数空间，使本文规范在一个具有合理的代数拓扑结构内进行一切操作和运算；利用Lagrange乘子，从泛函极值的角度系统地阐述和论证了一系列摩擦约束弹性、弹塑性广义变分不等原理与相应的实际问题之间的等价性，它为处理摩擦约束的弹塑性力学数值方法提供了合理的理论基础；从变分不等式的角度出发，阐述了对应于摩擦约束弹性、弹塑性问题的广义变分不等式理论，首次提出了在应力相容性条件下，它的有限元解具有存在唯一性，进而讨论了其有限元近似及离散解法；基于摩擦约束弹塑性广义变分不等式原理，首次提出了直接广义变分不等式方法，这一方法是直接变分法的合理推广和发展；利用直接广义变分不等式方法对金属压力加工中的平面变形问题、镦粗、挤压等塑性成形问题进行了分析计算，验证了该理论和数值算法的正确性、实用性、精确性和优越性。

In chapter 3 constructs firstlya new differential merit function by a perturbation structure of VIP and study theproperties of this merit function.Based on this merit function,a continuation-typeNewton method is proposed,which use the technique of inexact linear search forassuring its global convergence,and improves essentially the method of Taji,Fukushima and Ibaraki which can only solve the strong monotone VIP,and has locallyquadratic convergent rate under some condition.

第三章利用变分不等式问题的一种扰动结构构造了新的可微效用函数，并研究了效用函数的性质；在此基础上，给出了一类求解一般单调变分不等式问题的连续型Newton方法CN，方法采用了不精确的线性搜索技术以确保整体收敛性，从本质上改进了Taji、Fukushima和Ibaraki方法只能求解强单调变分不等式问题的局限性，同时又保持了局部二次收敛率。

This paper studies a completely generalized nonlinear quasi-variation-like inclusions problem in the setting of locally convex topological vector space s and proves the existence its solutions.

似变分不等式和广义拟似变分包含问题是研究非凸最优化问题、非凸和不可微最优化问题的重要工具。在局部凸的拓扑向量空间中研究了完全广义非线性拟似变分包含问题，并证明了其解的存在性

According to the existence results of general equilibrium problems and vector equilibrium problems have been studied more and more. Inspired and motivated by these research results, this paper is devoted to study systematically a class of equilibrium problems, which is unify and extension of a large number of known equilibrium problems and variational inequalities problems. The research is carried on from three aspects.Firstly, in finitely continuous topological spaces, we introduce four new types of the system of generalized vector quasi-equilibrium problems, and we derive some existence results of a solution for the system of generalized vector quasi-equilibrium problems via the maximal element theorems in product finitely continuous topological spaces.Secondly, in complete metric spaces, we provide the Ekeland variational principle to equilibrium problems with set-valued maps. And via the Ekeland variational principle, existence results for vector equilibrium problem with set-valued maps and the system of vector equilibrium problem with set-valued maps.

针对一般的均衡问题和向量均衡问题解的存在性，已有许多研究成果，受这些成果的启发，本文主要从理论上较为系统地研究了一类均衡问题，它统一和推广了许多已有的均衡问题和变分不等式问题，研究分有三个方面；首先，在有限连续拓扑空间中，我们提出了四类广义向量拟均衡系，并借助于有限连续拓扑空间中的极大元定理讨论了这四类均衡系问题的解的存在性问题，然后，在完备度量空间中，我们给出了关于集值均衡问题的Ekeland变分原理，并利用Ekeland变分原理分别讨论了集值向量均衡问题和集值向量均衡系问题的解的存在性。

The essence idea of restoration based on Total Variation is that translate the problem into minimum of energy functional with TV regularization, then the Partial Differential Equations are got from the principle of variation, finally after discretization the optimal resolution with the numerical methods is obtained.

基于整体变分的图像恢复概括的说就是把图像恢复问题形式化为带有整体变分正则化项的图像能量泛函极值问题，采用变分原理将其中的最小化问题转为所要求解的偏微分方程，即Euler-Lagrange方程，利用数字图像本身的离散特征使用适当的数值方法来求它的最优解，最后达到逼近原始图像的目的。

The research is carried on from four aspects. One is, based on answering the above open problem on a finite dimensional Euclidean space by means of partially ordered theory, to research the existence of solutions, global error bounds of proximal solutions and sensitivity of parametric unique solutions and present a class of variable-parameter three-step iterative algorithms for generalized set-valued variational inclusion problems by using - resolvent operator of set-valued mapping.Two is to consider the convexity, closedness and boundedness of the solution set of general set-valued variational inclusion problems and the sensitivity of the parametric solution set by means of graphical convergence theory. Three is to discuss directly the existence of solutions by using analytical methods for set-valued mixed quasi-variational-like inequalities and suggest a class of direct variable-parameter three-stepiterative algorithms for solving generalized set-valued variational inclusions.

研究分有三个方面：一是借助于偏序理论在有限维欧氏空间中解决了上述公开问题，在此基础上利用集值映射的η-预解算子，研究了广义集值变分包含问题解的存在性、逼近解的全局误差界、参数唯一解的灵敏性，并提出了一类变参数三步迭代算法；二是借助于图收敛理论研究了一般集值变分包含问题解集的凸性、闭性和有界性以及参数解集的灵敏性；三是用分析的方法直接讨论了集值混合拟类变分不等式问题解的存在性并提出了一类求解广义集值变分包含问题的直接变参数三步迭代算法。

In this paper, the space-time finite element theory and its application technologies are studied and discussed. Furthermore the space-time finite element method is used to analyze the transient characteristics of electromagnetism equipments. The work is important to develop the numerical method for the transient electromagnetic field. The detailed work is as follow.In the basis theory of finite element method used in the transient engineering problem, the functional theory and the finite element theory are applied to studied the existence condition of a generalized variation principle, which is the basis of the space-time finite element method.

本文在完善时空有限元法的基础理论和应用技术并将该方法应用于解决电磁装置的瞬态过程分析等相关方面进行了深入的研究与探讨，该工作对于发展瞬变电磁场问题的数值计算方法具有重要的意义，为此进行了如下几方面的工作：在工程瞬变问题的有限元方法的基础理论方面，本文应用泛函理论和有限元理论，系统讨论了一般工程瞬变问题的变分定理的存在条件及对应的变分定理。

With its theoretical basis as well as its wide prospects of application, this objective selecting method is probably a way towards the finding of the optimal weight factors in the variational problem.

这种最优选取方法，既有坚实的理论基础，又有广泛的应用前景，开辟了一条通向寻找变分问题最优权重因子的途径。

In the eight ieth, variational inequa1 ity hasits important development phase and there were suchvari ati ona1 inequa1 it ies as vector vari at ionalinequal ity, general variati ona1 inequal ity quasi-variationa1 inequal ity, 1 ike--variat iona1 inequal ity andgenera1 ized vari at iona1 inequa1 ity.

八十年代为变分不等式问题的很重要的发展阶段，出现了一系列的变分不等式，如向量变分不等式、一般变分不等式、拟变分不等式、类变分不等式及广义变分不等式。

According to the dynamics analysis of the cotton sliver coiling process,by considering the minimum friction of cotton sliver caused by screwy pipes as objective function the design of screwy pipes of can coiler is explained based on the theory of the variational direct-method.

对棉条圈条过程定量地进行了动力学分析，以棉条所受曲线斜管摩擦力最小为目标函数，基于变分问题直接法原理设计出圈条盘螺旋形曲线斜管。这种螺旋形曲线斜管充分利用圈条盘的转动对棉条产生的离心力甩摆作用，使棉条所受曲线斜管摩擦力较小，从而使出条更顺畅；在一定程度上降低了圈条盘螺旋形曲线斜管堵眼频率与条筒乱条几率，使其更适应现代并条机生产高速化的要求

But we don't care about Battlegrounds.

但我们并不在乎沙场中的显露。

Ah! don't mention it, the butcher's shop is a horror.

啊！不用提了。提到肉，真是糟透了。