摩擦约束塑性力学变分不等原理的半反推法

带摩擦约束的弹塑性接触问题,由于摩擦约束条件是一种判别性的条件,它的变分问题的逆问题的研究比较困难。本文对弹塑性接触力学中的变分不等问题的逆问题进行了研究,改进了半反推法并将其应用到弹塑性变分不等原理的研究中,导出了摩擦约束弹塑性增量广义变分不等原理中的能量泛函,消除了用拉氏乘子法可能产生的临界变分现象,在证明中,巧妙地处理了增量表示的接触摩擦边界条件,避免了使用非线性泛函分析和凸分析,简化了证明。     

弹性理论中的临界变分及消除方法

临界变分现象是拉氏乘子法的固有特性，钱伟长应用高阶拉氏乘子消除了临界变分现象。本文将提出一种新的方法－凑合反推法，这种方法摒充了拉氏乘子法，把拉氏乘子所在的项目一个待定函数Ｆ代替。这样构成的泛函，作者称之为试泛函。而待定函数Ｆ的识别类似于拉氏乘子的识别。通过该法可以方便地构造出各种多变量广义变分原理，并且可以消除临界变分现象。     

刚塑性广义变分不等原理及其在平面应变分析中的应用

首先利用Lagraingian乘子法,从势能角度出发构造了考虑摩擦效应这一能导致变分不等形式的广义能量泛函,把一般的有条件的变分原理化为无条件的变分原理来唯一确定,得出了各Lagrangian乘子所代表的物理意义,建立了刚塑性理论中的Coulomb摩擦约束的广义变分不等原理.而后基于退化的摩擦约束广义变分等式原理,对长矩形板镦粗进行了塑性加工工步分析.所得结果与经典上限法结果相吻合.     

线弹性力学广义变分原理的构造函数理论

本文从构造函数理论来讨论广义变分原理,得到了构造函数与广义变分原理之间的对应关系,以及单值构造函数的一般解.     

关于拉氏乘子的识别

本文讨论了在消元识别和换元识别过程中拉氏乘子的不唯一性问题，拉氏乘子表达方式的不唯一性反映了消元或换元方式的多样性，由于不同的消元或换元方式不改变问题，拉氏乘子的解仍是唯一的，因而由不同的拉氏乘子表达式可以得到一族具有价驻值条件的无约束驻值问题，本文用换元识别法，推导了Ｈｅｌｌｉｎｇｅｒ－Ｒｅｉｓｓｎｅｒ变分原理和胡一鹫变分原理及其更多样的形式。     

广义变分不等原理在圆柱体非均匀镦粗中的应用

基于摩擦约束广义变分不等原理,对圆柱体非均匀镦粗成形工艺进行详细的塑性加工工步分析,实践证明了广义变分不等原理理论的正确性和实际操作的可行性以及优越性。     

等价变分原理泛函之间的联系

本文证明了等价变分原理的泛函,实质上都只相差某种加权残差项,这也就表明了,在已知的泛函后附加加权残差项,是建立等价变分原理最简便的方法.     

耦合热弹性问题的分区变分原理及其广义变分原理

本文在文献[1]和[2]的基础上建立并论证了耦合热弹性问题的分区变分原理及其分区广义变分原理     

率型的广义变分原理

本文提出一个适合于率型本构方程的边值-初值问题的广义变分原理,此原理是在现时构形上以卷积形式给出,可用于建立大变形分析中适时的Lagrange描述的增量分析公式。     

薄板弯曲理论中一个新的广义变分原理及其交替max,min性质

本文在薄板弯曲的经典理论中提出一个以挠度、横向剪应变、曲率、弯矩、横向剪力、边界上的未知反力为自变函数的新的变分原理,在对自变函数预加不严厉的约束后,新泛函具有一连串交替max,min的性质。     

Generalized Baker–Ericksen Inequalities

It is well known that isotropic, nonlinearly elastic materials satisfy the Baker–Ericksen inequalities as a consequence of the strong ellipticity or rank 1 convexity. Here we present a generalization to a non-isotropic elastic material which posseses a preferred element in the symmetry group.     

粘性流体力学的变分原理及其广义变分原理

本文通过引入Laplace变换,应用变积运算方法,建立了不可压缩粘性流体力学的变分原理及其广义变分原理.     

广义变分原理的结构形状优化伴随法灵敏度分析

提出了一种利用伴随变量进行结构形状优化灵敏度分析的新方法. 基于广义变分原理,考虑了形状优化中位移边界条件的变化对结构响应的影响. 新方法弥补了Arora等人所提出的形状优化灵敏度分析变分原理中的缺陷,为采用伴随法进行灵敏度分析提供了新的框架.     

Modeling of Steady Flows in a Channel by Navier–Stokes Variational Inequalities

A mathematical model of a steady viscous incompressible fluid flow in a channel with exit conditions different from the Dirichlet conditions is considered. A variational inequality is derived for the formulated subdifferential boundary-value problem, and the structure of the set of its solutions is studied. For two-ption on the low Reynolds number is proved. In the three-dimensional case, a class of constraints on the tangential component of velocity at the exit, which guarantees solvability of the variational inequality, is found.     

带孔隙损伤的弹性固体的广义变分原理

应用弹性微结构理论,建立了具广义力场带孔隙损伤线弹性固体的基本模型．应用变积方法,同时分别建立了带孔隙损伤弹性固体四类和两类变量的广义变分原理,这些变分原理对应着带孔隙损伤弹性固体微分方程和初值边值条件．应用弹性微结构理论,建立了带孔隙损伤的弹性Timoshenko 梁的基本方程,得到带孔隙损伤的弹性Timoshenko 梁两类变量的广义变分原理．这些广义变分原理为近似求解带孔隙损伤的弹性问题提供了有效途径．     

Generalized Hessian and External Approximations in Variational Problems of Second Order

We introduce a suitable notion of generalized Hessian and show that it can be used to construct approximations by means of piecewise linear functions to the solutions of variational problems of second order. An important guideline of our argument is taken from the theory of the Γ-convergence. The convergence of the method is proved for integral functionals whose integrand is convex in the Hessian and satisfies standard growth conditions.     

Viscosity Solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles of Non-Local Type

In this paper, we study systems of nonlinear second-order variational inequalities with interconnected bilateral obstacles with non-local terms. They are of min–max and max–min types and related to a multiple modes zero-sum switching game in the jump-diffusion model. Using systems of penalized reflected backward SDEs with jumps and unilateral interconnected obstacles, and their associated deterministic functions, we construct for each system a continuous viscosity solution which is unique in the class of functions with polynomial growth.     

用广义变分原理求解多节点双模量静不定桁架

在外载荷作用下的多节点双模量静不定桁架平衡问题,是任意有限多个自变量的多元函数在任意有限多个约束条件下的极值问题,采用广义变分原理可以方便求解多节点双模量静不定桁架内力．通过求解多节点双模量静不定桁架内力的几个算例,阐述广义变分原理在计算多节点双模量静不定桁架内力中的应用．研究结果表明：采用广义变分原理求解多节点双模量静不定桁架内力的通用性较强,所求的结果是精确解析解．采用广义变分原理求解多节点双模量静不定桁架内力的方法不但克服了常规方法需利用几何关系建立协调方程的缺陷,且具有力学概念清晰直观、计算过程简便、便于工程设计人员在实际中掌握和计算等优点．     

Stochastic Variational Inequalities and Applications to the Total Variation Flow Perturbed by Linear Multiplicative Noise

In this work, we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation ${{\rm d}X(t) = {\rm div} \left[\frac{\nabla X(t)}{|\nabla X(t)|}\right]{\rm d}t + X(t){\rm d}W(t) {\rm in} (0, \infty) \times \mathcal{O},}$ where ${\mathcal{O}}$ is a bounded and open domain in ${\mathbb{R}^N, N \geqq 1}$ and W(t) is a Wiener process of the form ${W(t) = \sum^{\infty}_{k = 1}\mu_{k}e_{k}\beta_{k}(t), e_{k} \in C^{2}(\overline{\mathcal{O}}) \cap H^{1}_{0}(\mathcal{O}),}$ and ${\beta_{k}, k \in \mathbb{N}}$ are independent Brownian motions. This is a stochastic diffusion equation with a highly singular diffusivity term. One main result established here is that for all initial conditions in ${L^2(\mathcal{O})}$ , it is well posed in a class of continuous solutions to the corresponding stochastic variational inequality. Thus, one obtains a stochastic version of the (minimal) total variation flow. The new approach developed here also allows us to prove the finite time extinction of solutions in dimensions ${1\leqq N \leqq3}$ , which is another main result of this work.