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.     

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.     

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.     

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.     

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.      

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.     

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...     

Variational Principles for Eigenvalues of Self-adjoint Operator Functions

Variational principles for eigenvalues of certain functions whose values are possibly unbounded self-adjoint operators T() are proved. A generalised Rayleigh functional is used that assigns to a vector x a zero of the function T()x, x), where it is assumed that there exists at most one zero. Since there need not exist a zero for all x, an index shift may occur. Using this variational principle, eigenvalues of linear and quadratic polynomials and eigenvalues of block operator matrices in a gap of the essential spectrum are characterised. Moreover, applications are given to an elliptic eigenvalue problem with degenerate weight, Dirac operators, strings in a medium with a viscous friction, and a Sturm-Liouville problem that is rational in the eigenvalue parameter.     

The Extended Legendre Transform and Related Variational Principles

Mathematical Notes - Variational principles for functionals on the space C(X) of continuous functions that can be written as a representation of a functional in the form of the Legendre transform...     

Variational Principles in Nonlinear Analysis and Their Generalization

Mathematical Notes -     

On Vector Form of Differential Variational Principles of Mechanics

We introduce variation of a vector δx which can be interpreted either as a virtual displacement of a system, or as variation of the velocity of a system, or as variation of the acceleration of a system. This vector is used to obtain a unified form of differential variational principles of mechanics from the scalar representative equations of motion. Conversely, this notation implies the original equations of motion, which enables one to consider the obtained scalar products as principles of mechanics. Using the same logical scheme, one constructs a differential principle on the basis of the vector equation of constrained motion of a mechanical system. In this form of notation, it is proposed to conserve the zero scalar products of reactions of ideal constraints and the vector δx. This enables one to obtain also the equations involving generalized constrained forces from this form of notation.     

Dual Variational Principles and Pad?-type Approximants

Families of approximants for <?, f> are derived from dualvariational functionals associated with the linear equation(1+yL)?=f. Expansions in both ascending and descending powersof y are considered, and the approximants are identified eitheras one- or two-point Pad? approximants, or as approximants ofa closely related type. Compact formulae are obtained for theapproximants, and their duality and bounding properties areexhibited. Attention is paid to the special situations occurringwhen the non-negative self-adjoint operator L (i) has a zeroeigenvalue, (ii) is bounded.     

Geometric Aspects of Higher Order Variational Principles on Submanifolds

The geometry of jets of submanifolds is studied, with special interest in the relationship with the calculus of variations. A new intrinsic geometric formulation of the variational problem on jets of submanifolds is given. Working examples are provided.      

Genericity of Well-Posedness, Perturbations and Smooth Variational Principles

We show that three important topics in nonlinear analysis and optimization are intimately related: the theory of perturbations, the notion of well-posedness and variational principles in the sense of Ekeland, Borwein–Preiss and Deville–Godefroy–Zizler. The concept of genericity and the new notion of flexible perturbation play a key role in these connections. This notion enables one to consider topologies on spaces of functions which have been introduced recently. A link between the Asplund and Ekeland–Lebourg methods and the Palais–Smale condition, another important topic in nonlinear analysis, is pointed out.     

Nonsmooth Characterizations of Asplund Spaces and Smooth Variational Principles

We show that the Asplund property of Banach spaces is not only sufficient but also a necessary condition for the fulfillment of some basic results in nonsmooth analysis involving Fréchet-like normals and subdifferentials as well as their sequential limits. In this way we obtain new characterizations of Asplund spaces within the framework of nonsmooth analysis. Then we study several versions of smooth variational principles in Asplund spaces, provide necessary and sufficient conditions for the validity of such principles, and establish their relationships with certain subdifferential properties of lower semicontinuous functions.     

Variational Principles for Maximization Problems with Lower-semicontinuous Goal Functions

Set-Valued and Variational Analysis - Let X be a completely regular topological space and f a real-valued bounded from above lower semicontinuous function in it. Let C(X) be the space of all...     

大变形对称弹性理论的广义变分原理

本文以陈至达提出的变形几何非线性理论￣［１］为基础，应用Ｌａｇｒａｎｇｅ乘子法，对大变形对称弹性力学问题进行了研究，给出了相应的位能广义变分原理、余能广义变分原理，以及动力学问题的广义变分原理；同时，文中还证明了位能广义变分原理和余能广义变分原理的等价性。