Complementary Variational Principles for Knudsen Flow Rates

A collisionless gas flows through the interior of a tube ofcircular cross-section, having both an entrance and an exitand which reflects molecules diffusely. Suitable complementaryvariational principles are used to obtain upper and lower boundsfor the transmission probability of the tube. The numericalresults, believed to be the most accurate to date, are comparedwith those of other authors.     

Complementary Variational Inequalities

A number of problems involving free and moving boundaries canbe solved by using the method of variational inequalities. Inthis paper, it is shown that an equivalent formulation can beobtained by using the theory of dual extremum principles developedby Noble & Sewell in 1972. This enables complementary variationalinequalities to be formulated for the problems under consideration.Some generalizations of existing results are also presented.     

Selecting Complementary Pairs of Literals

We present and rigorously analyze a heuristic that searches for a satisfying truth assignment of a given random instance of the 3-SAT problem. We prove that the heuristic asymptotically certainly succeeds in producing a satisfying truth assignment for formulas with clauses to variables ratio (density) of up to 3.52. Thus the experimentally observed threshold of the density where a typical formula's phase changes from asymptotically certainly satisfiable to asymptotically certainly contradictory is rigorously shown to be at least 3.52. The best previous lower bound in the long series of mathematically rigorous approximations by various research groups of the experimental threshold was 3.42. That was the first result where the probabilistic analysis was based on random formulas with a pre-specified, typical number of appearances for each literal. However, in that result, in order to simplify the analysis, the number of appearances of each literal was decoupled from the number of appearances of its negation. In this work, we assume not only that each literal has the typical number of occurrences, but that for each variable both numbers of occurrences of its positive and negated appearances are typical. By standard techniques, our algorithm can be easily modified to run in linear time. Thus not only the satisfiability threshold, but also the threshold (experimental again) where the complexity of searching for satisfying truth assignments jumps from polynomial to exponential is at least 3.52. This should be contrasted with the value 3.9 for the complexity threshold given by theoretical (but not mathematically rigorous) techniques of Statistical Physics.     

光测弹性理论中的耦联变分原理和广义耦联变分原理

在本文中,应用拉格朗日乘子法和高阶拉格朗日乘子法[1],我们系统地导出了光测弹性理论中的耦联势能原理,耦联余能原理和具有二类和三类变量的广义耦联势能原理和广义相联余能原理。     

粘性流体力学的变分原理和广义变分原理

本文建立了不可压缩和可压缩粘性流体力学问题的变分原理,即最大功率消耗原理和它们的广义变分原理.     

大位移非线性弹性理论的变分原理和广义变分原理

在前文中[1],作者首次提出了大位移非线性弹性力学的位能原理和余能原理,以及各种完全的和不完全的广义变分原理.但在约束条件和欧拉条件上,证明和叙述都不很明确,有时甚至把原来应该是欧拉方程的误认为是约束条件,如余能驻值原理中,应力位移关系原应是欧拉方程,但把它当作了变分约束条件.这就是说:我们把余能驻值原理约束得超过了必要的要求.还有,在所有变分原理中,应力应变关系式都是不参加变分的约束条件,亦即,他们是从已定应力导出应变或从已定应变导出应力的约束条件.这一点,在文[1](1979)中,并未明确指出.本文并将用高阶拉氏乘子法,导出更一般的广义变分原理(1983)[2].本文使用V.V.Novozhilov的有关非线性弹性力学的成果(1958)[3].     

Variational Principles of Contact Elastostatics

In this paper, the variational principles of contact elastostatics,which were proposed and proved by Fichera (1964) and Duvaut& Lions (1972) are developed in an engineering fashion withoutuse of functional analysis. The theory contains a number ofnew elements, but the elegant existence proofs of the French-Italianschool are missing. A start is made by extending the principleof virtual work to normal and frictional contact in such a mannerthat it needs no longer be known beforehand whether—inthe case of normal contact—actual contact is or is notestablished, or—in the case of frictional contact—slipdoes or does not occur. Then the principle of minimal potentialenergy is set up for a non-linear elastic body in contact witha rigid base. Uniqueness and minimality of the solution areproved under certain conditions, the Reissner principle is established,and the principle of minimal complementary energy is derived.Finally the principles are cast in what is termed surface mechanicalform, and two examples are given: the variational principlefor normal half-space contact problems, and a new principlefor time-dependent frictional half-space contact. Upon discretization,these principles provide a quadratic object function to be minimizedunder linear or quadratic inequality constraints. The positionsof the contact area and of the regions of slip and adhesionappear as by-products of the calculation.     

大应变固结理论的分区变分原理及其广义变分原理

土体材料本构特性的差异问题与大变形问题是分析岩土材料变形特性的基本问题．根据有限变形的描述方法构筑土体结构大变形固结方程,证明了大变形固结的变分原理A·D2应用分区子结构的连续条件,推导固结理论的分区变分原理．引用Lagrange乘子法构筑并证明了大变形固结问题在无约束状态下的广义分区变分原理．     

Variational Principles for Column Buckling

On the basis of a new variational principle an a priori estimateof the maximal deflection and the maximal slope of an elasticcolumn with variable cross-section is obtained. Also a methodto estimate the critical force is presented.     

Variational Principles in Curve Design

The paper considers Hermite interpolation for vector-valued functions. Corresponding to the interpolating functions f we define functionals I which contain function values of f^(r) and integrals of f^(r) where 0 ≤ r ≤ m for some integer m. The main purpose of the paper is to characterize those functions which satisfy the interpolation problem and have a minimal value of I. These characterizations contain several results of the literature including splines in tension and geometric splines.     

弹性理论中各种变分原理的分类

本文按弹性理论中各种变分原理的约束条件的不同,对所有变分原理进行分类.我们在前文中业已指出,应力应变关系这样的约束条件是不能用拉氏乘子法解除的.剩下的可能约束条件共有四种:(1)平衡方程,(2)应变位移关系,(3)边界外力已知的边界条件,和(4)边界位移已知的边界条件.弹性理论的各种变分原理中,有的只有一种约束条件,有的有两种或三种,最多只能有四种约束条件.这样一共可能有15种变分原理,但是每种变分原理既可以用应变能A表示,又可以用余能B表示.这样,我们一共应有30种形式完全不同的变分原理,我们全部列出了这三十种形式的变分原理.     

Incremental Variational Principles in Damage Mechanics

We propose a new kind of damage formulation which leads, in connection with relaxation techniques known in literature, to a well posed problem suitable for finite element calculations. The model is rate‐dependent and allows a consistent treatment of micro‐structures and localization phenomena.     

Continuous and Discrete Clebsch Variational Principles

The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group via a velocity map. This paper proves a reduction theorem which states that the canonical variables on the Lie group can be eliminated, if and only if the velocity map is a Lie algebra action, thereby producing the Euler–Poincaré (EP) equation for the vector space variables. In this case, the map from the canonical variables on the Lie group to the vector space is the standard momentum map defined using the diamond operator. We apply the Clebsch method in examples of the rotating rigid body and the incompressible Euler equations. Along the way, we explain how singular solutions of the EP equation for the diffeomorphism group (EPDiff) arise as momentum maps in the Clebsch approach. In the case of finite-dimensional Lie groups, the Clebsch variational principle is discretized to produce a variational integrator for the dynamical system. We obtain a discrete map from which the variables on the cotangent bundle of a Lie group may be eliminated to produce a discrete EP equation for elements of the vector space. We give an integrator for the rotating rigid body as an example. We also briefly discuss how to discretize infinite-dimensional Clebsch systems, so as to produce conservative numerical methods for fluid dynamics.      

Variational Principles and Mean Value Estimates

Journal of Optimization Theory and Applications - In this paper, for a continuous function defined on a closed subset of a Banach space and differentiable on the interior of the domain, the infimum...     

入水冲击问题变分原理及其它

首先建立入水前后两个衔接阶段的较为严密的场方程.再得到与之对应的各类变分原理,界限定理,第二阶段问题的边界积分方程.证明了解的存在性并提供了求解实施方案.最后以船舶兴波阻力问题的算例,论证了第二阶段问题的一种特殊应用及其正确性.从而为求取较为精确的入水冲击问题基本方程的变分有限元及边界元方法奠定了严密的理论基础.     

饱和多孔介质耦合系统的变分原理

本文采用变积方法,建立了等温准静态下饱和多孔介质的六类变量的广义变分原理.在此基础上,通过引入约束条件得到各级变分原理,其中包括五类变量,四类变量,三类变量和二类变量的变分原理.除得到文献中已有的变分原理外,本文给出了许多新的变分原理,为建立饱和多孔介质的有限元模型提供了基础.     

压电热弹性动力学的Gurtin型分区变分原理

建立有关压电热弹性动力学的各种Gurtin型分区变分原理,由此变分原理可以得到压电热弹性动力学所有方程式、关系式和边界条件,并且可以直接得到各相邻区域交界面上的连续条件。Gurtin型分区变分原理是压电热弹性动力学的重要组成部分,并能反映压电热弹性动力学初值-边值问题的全部特征。     

非牛顿流体力学变分原理

本文将钱伟长教授在文献[1]中提出的不可压缩粘性流的最大功率消耗原理进一步推广到本构方程为ε_ij=?τ/?σ'_ij的非牛顿流体流动问题,并采用识别的拉氏乘子法解除变分约束条件,导出其广义变分原理。