Deriving generalized variational principles in general mechanics by using Lagrangian multiplier method

By using the involutory transformations, the classical variational principle—Hamiltonian principle— of two kinds of variables in general mechanics is advanced and by using undetermined Lagrangian multiplier method, the generalized variational principles and generalized variational principles with subsidiary conditions are established. The stationary conditions of various kinds of variational principles are derived and the relational problems discussed. Project supported by the National Natural Science Foundation of China (Grant No. 19872022) and the Doctoral Education Foundation of China (Grant No. 97021710).     

On the minimal solution for some variational inequalities

Applying an asymptotic method, the existence of the minimal solution to some variational elliptic inequalities defined on bounded or unbounded domains is established. The minimal solution is obtained as limit of solutions to some classical variational inequalities defined on domains becoming unbounded when some parameter tends to infinity. The considered quasilinear operators are only monotone (not strictly) and noncoercive. Some related comparison principles are also investigated.     

功的互等定理和线弹性变分原理

本文从功的互等定理系统地导出了线弹性变分原理并且给出了边界条件变化的混合变分原理.     

弹性正交索网结构动力学的各类非传统Hamilton型变分原理

根据古典阴阳互补和现代对偶互补的基本思想,通过罗恩早已提出的一条简单而统一的新途径,系统地建立了正交索网结构几何非线性弹性动力学的各类非传统Hamilton型变分原理．这种新的非传统Hamilton型变分原理能反映这种动力学初值-边值问题的全部特征．文中首先给出正交索网结构几何非线性动力学的广义虚功原理的表式,然后从该式出发,不仅能得到正交索网结构几何非线性动力学的虚功原理,而且通过所给出的一系列广义Legendre变换,还能系统地成对导出正交索网结构几何非线性弹性动力学的5类变量、4类变量、3类变量和2类变量非传统Hamilton型变分原理的互补泛函、以及相空间非传统Hamilton型变分原理的泛函与1类变量非传统Hamilton型变分原理势能形式的泛函．同时,通过这条新途径还能清楚地阐明这些原理的内在联系．     

Unconventional Hamilton-type variational principles for nonlinear coupled thermoelastodynamics

According to the basic idea of classical yin-yang complementarity and modern dual-complementarity, in a simple and unified new way proposed by Luo, the unconventional Hamilton-type variational principles for geometrically nonlinear coupled thermoelastodynamics can be established systematically. The new unconventional Hamilton-type variational principle can fully characterize the initial-boundaty-value problem of this dynamics. In this paper, an important integral relation is given, which can be considered as the expression of the generalized principle of virtual work for geometrically nonlinear coupled thermodynamics. Based on this relation, it is possible not only to obtain the principle of virtual work in geometrically nonlinear coupled thermodynamics, but also to derive systematically the complementary functionals for eight-field, six-field, four-field and two-field unconventional Hamilton-type variational principles by the generalized Legendre transformations given in this paper. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly.     

用变分方法求解大变形对称弹性力学问题

文中以经典力学的数学理论和陈氏定理为基础,用变分的方法求解大变形对称弹性力学问题,得出了以瞬时位形为基准的位能广义变分原理和余能广义变分原理,以及两个变分原理的等价性;此外,还给出了以瞬时位形为基准的动力学问题的广义变分原理.     

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.     

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.     

基于改进集的集值Ekeland变分原理

Ekeland变分原理在最优化理论及应用研究中具有十分重要的作用.利用非线性标量化函数及相应的非凸分离定理建立了基于改进集的集值Ekeland变分原理.新的Ekeland变分原理包含了一些经典的Ekeland变分原理作为其特例.     

Reissner夹层板动力学的非传统Hamilton型变分原理

根据古典阴阳互补和现代对偶互补的基本思想,通过早已提出的一条简单而统一的新途径,系统地建立了Reissner夹层板动力学的各类非传统Hamilton型变分原理．这种新的非传统Hamilton型变分原理能反映这种动力学初值-边值问题的全部特征．文中首先给出一个Reissner夹层板广义虚功原理的表式．然后从该式出发,不仅能得到Reissner夹层板动力学的虚功原理,而且通过所给出的一系列广义Legendre变换, 还能系统地成对导出五类变量、 二类变量和一类变量非传统Hamilton型变分原理的互补泛函．同时,通过这条新途径还能清楚地阐明这些原理的内在联系．     

功率型变分原理和功能型拟变分原理及其应用

自从钱伟长建立了功率型变分原理以来,功率型变分原理和功能型变分原理在理论方面和应用方面有什么区别和联系,成为学术界关注的课题.应用变积方法,根据Jourdain原理和d’Alembert原理,建立了不可压缩黏性流体力学的功率型变分原理和功能型拟变分原理,推导了不可压缩黏性流体力学的功率型变分原理的驻值条件和功能型拟变分原理的拟驻值条件.研究了不可压缩黏性流体力学的功率型变分原理在有限元素法中的应用.研究表明,功率型变分原理与Jourdain原理相吻合,功能型变分原理与d’Alembert原理相吻合.功率型变分原理直接在状态空间中研究问题,不仅在建立变分原理的过程中可以省略在时域空间中的一些变换,而且给动力学问题有限元素法的数值建模带来方便.     

On extended versions of Dancs-Hegedüs-Medvegyev’s fixed-point theorem

In this article, we establish some fixed-point (known also as critical point, invariant point) theorems in quasi-metric spaces. Our results unify and further extend in some regards the fixed-point theorem proposed by Dancs, S.; Hegedüs, M.; Medvegyev, P. (A general ordering and fixed-point principle in complete metric space. Acta Sci. Math. 1983;46:381–388), the results given by Khanh, P.Q., Quy D.N. (A generalized distance and enhanced Ekeland?s variational principle for vector functions. Nonlinear Anal. 2010;73:2245–2259), the preorder principles established by Qiu, J.H. (A pre-order principle and set-valued Ekeland variational principle. J. Math. Anal. Appl. 2014;419:904–937) and the results obtained by Bao, T.Q., Mordukhovich, B.S., Soubeyran, A. (Fixed points and variational principles with applications to capability theory of wellbeing via variational rationality. Set-Valued Var. Anal. 2015;23:375–398). In addition, we provide examples to illustrate that the improvements of our results are significant.     

Reduced Variational Formulations in Free Boundary Continuum Mechanics

We present the material, spatial, and convective representations for elasticity and fluids with a free boundary from the Lagrangian reduction point of view, using the material and spatial symmetries of these systems. The associated constrained variational principles are formulated and the resulting equations of motion are deduced. In addition, we introduce general free boundary continua that contain both elasticity and free boundary hydrodynamics, extend for them various classical notions, and present the constrained variational principles and the equations of motion in the three representations.     

Variational principles for Ginzburg–Landau equation by He’s semi-inverse method

Via the semi-inverse method of establishing variational principles proposed by He, a generalized variational principle is established for Ginzburg–Landau equation. The present theory provides a quite straightforward tool to the search for various variational principles for physical problems. This paper aims at providing a more complete theoretical basis for applications using finite element and other direct variational methods.     

非局部微极线性弹性介质理论中的各种互易定理和变分原理

在本文的第一部份中.我们扩展了经典的卷积和I.Hlavácěk给出的“卷积数积”的概念,提出了“卷积向量”和“卷积向量点积”的概念.从而可使我们把具有算子系数的方程的初值问题和初值——边值问题推广到具有算子系数的方程组的相应问题中去.在本文的第二部份中,以卷积向量和卷积向量点积的概念为基础导出了非均匀的各向异性固体的非局部微极线性弹性动力学的两种基本型式的互易定理.在本文的第三部份中,利用一和二中卷积向量和卷积向量点积的概念和结论及由钱伟长提出的Lagrange乘子法给出了非局部微极线性弹性动力学的四种主要型式的广义变分原理.它们是与经典弹性理论中的胡海昌-鹫津久一郎型的、Hellinger-Reissner型的和Gurtin型的以及局部微极弹性理论和非局部弹性理论中的Hlavácěk型的和Iesan型的广义变分原理相应的各种变分原理.最后还指出了这里提出的后两种主要型式的广义变分原理是等价的.     

Minimax principles for elliptic mixed hemivariational–variational inequalities

In this paper, minimax principles are explored for elliptic mixed hemivariational–variational inequalities. Under certain conditions, a saddle-point formulation is shown to be equivalent to a mixed hemivariational–variational inequality. While the minimax principle is of independent interest, it is employed in this paper to provide an elementary proof of the solution existence of the mixed hemivariational–variational inequality. Theoretical results are illustrated in the applications of two contact problems.     

Homogenization of Cahn–Hilliard-type equations with unbounded potentials

In this note we recall two approaches to evolutionary Γ-convergence for classical gradient systems: evolutionary variational inequalities versus energy-dissipation principles. Following [1] we apply both approaches to homogenize Cahn–Hilliard-type equations with rapidly oscillating coefficients and quite general potential. Here the focus is on the Γ-convergence of the energy functional, which we prove without assuming any growth conditions for the convex part of the potential. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The method of integrodifferential relations for analysing the natural oscillations of membranes

An approach to solving of problems of the free oscillations of membranes, based on the method of integrodifferential relations, is proposed. The variational properties of the integrodifferential problems are investigated and their relation to classical variational principles is demonstrated. Integral and local criteria for the quality of the approximate solutions are proposed. A numerical-analytical algorithm is developed for finding the characteristic frequencies and modes of oscillations. Problems involving the free transverse motions of circular and elliptic membranes are considered as an example.     

On potentiality conditions for partial differential-difference equations

We obtain necessary and sufficient conditions for the existence of variational principles with respect to a classical bilinear form for boundary value problems for partial differential-difference equations.     

离散型固体力学及其间断型变分原理

本文对即将形成的一门新的学科——离散型固体力学的基本假设,微分形式的方程,及其间断型变分原理进行了论述.离散型固体力学是离散介质力学的一个分支,是近期以来力学的一个发展方向.它是基于固体体系具有离散性、可分性,待解函数在定义城内具有各种不同的间断性,以及定义域的边界可动性的基础上,为解决种种情况下的固体的应力,位移、应变所形成的力学系统.当待解函数在整个定义域内为充分光滑的函数类和略去边界可动性的影响时,则离散型固体力学就退化为连续介质力学范畴的古典固体力学.它所属的变分原理,在相应的情况下,也就退化为古典与非占典变分原理.