随机度量空间上的Ekeland变分原理、Petal定理和Drop定理

Ekeland变分原理从一出现就成为非线性分析最强有力的工具之一,它在凸分析、控制论、大范围分析、不动点理论等领域有着及其广泛的应用。人们企图在每一个得到的空间上建立Ekeland变分原理。在文[1]中建立了E空问上的Ekeland变分原理。文[3]中建立了一类Menger空间上的Ekeland变分原理。由于随机变量空间即RM空间是PM空间的重要子类,而E空间又是RM空间的真子类。本文首先建立了RM空间上的Ekeland变分原理,并推出Ekeland变分原理与RM空间上的Caristi不动点定理等价。     

变形体动力学的变分原理及其应用

在变形体动力学的变分原理及其应用方面的研究工作从１９８５年开始，至今已进行了十多年，本文主要从三个方面对此进行综述。１．原理的研究，提出一条新途径，其基本思想是古典阴阳互补和现代对偶互补，它的特点是完全不用拉格朗日乘子法，但能简单而系统地从一般到特殊建立各种变分原理，通过这条新途径，作者建立了某些变形动力学的一系列基本原理，其中很多是新的结果。２．原理的应用研究，提出了分别基于简化Ｇｕｒｔｉｎ型分     

一种杂交/混合四边形分层壳元

本文从修正的Hellinger-Reissner变分原理出发,基于杂交/混合有限元及分层有限元法,建立了一种杂交/混合四边形分层壳元,用于分析层合壳结构的非线性稳定问题;对典型层合柱壳结构进行了几何非线性分析,验证了该方法的正确、有效性。     

基于几何非线性不协调元变分原理的圆柱壳非线性初始稳定性分析

本文根据几何非线性不协调元增量变分原理,按严格的壳体方程,建立了高精度的圆柱壳几何非线性20参数矩形精化不协调元RCSR4,并用于圆柱壳非线性初始稳定性分析。计算结果表明,该方法收敛性良好。     

薄板哈密顿含参变分原理

将薄板哈密顿变分原理及其泛函),,,(xxxHVMwyP推广为含两个可选参数1h和2h的薄板哈密顿含参变分原理及其含参泛函),,,(21xxxHVMwyPhh。其推导过程为:首先将薄板Hellinger-Reissner变分原理及其泛函}){,(MwHRP推广为含可选参数1h的薄板Hellinger-Reissner含参变分原理及其含参泛函}){,(1MwHRhP。然后采用消元法(消去变量yM和xyM)和换元乘子法(增加变量xy和xV)由含参泛函}){,(1MwHRhP导出含两个可选参数的薄板哈密顿含参泛函),,,(21xxxHVMwyPhh。含参变分原理是多种变分原理的组合形式,并使多种变分原理之间得到沟通和融合。通过对参数1h和2h的合理选取和赋值,可以得到含参泛函的多种退化形式,为建立多种有限元模型创造条件。     

压电材料变分原理逆问题的研究——动力学中的逆问题

研究了压电材料耦合动态场中Hamilton 型和Gurtin 型变分原理的逆问题。采用变积方法, 建立了各级变分原理和广义变分原理, 为建立横观各向同性压电材料的动力学有限元分析模型提供了依据。      

考虑阻尼力的Hellinger-Reissner变分原理及其应用

建立了圆柱坐标系下包含粘滞阻尼力的修正后的Hellinger—Reissner变分原理，推导了对应的状态向量方程。考虑阻尼力后，结构的特征方程应有复数根，因而通常用于求解多项式方程实数根的二分法不再适用。为了解决这个问题，本文结合精细积分法和米勒法，为叠层壳的阻尼自由振动提出了新的数值方法，同时，通过数值实例分析了简支边界条件下开口叠层壳的复频响应问题。目前修正后的Hellinger—Reissner变分原理将有利于复杂边界条件下阻尼叠层壳动力学问题的半解析法的推导。     

压电材料变分原理逆问题的研究

研究了压电材料变分原理的逆问题, 采用文献[ 8 ]提出的变积方法, 系统地建立了压电材料的变分原理及其广义变分原理, 除得到文献中已有的结果外, 还得到了一些新的变分原理, 为建立压电材料的有限元分析模型提供了依据。     

弹性动力学显含初始条件的广义变分原理

本文给出了显含初始条件并含有两个任意参数的弹性动力学广义变分原理,参数的不同取值以及附加不同的约束条件,可以得到多种显含初始条件的变分原理.     

一种建立分区变分原理的新方法

提出了一种建立弹性理论分区变分原理的新方法。放松了分区交界面上位移、应力连续的条件，证明了弹性理论分区求解体系的微分形式与积分形式的等价关系。本文以微分形式为前提，利用这种等价关系,在统一的构架下,导出了分区广义虚功方程和弹性理论分区变分原理。变分原理是积分形式的一种表现形式。讨论了积分形式的物理含义，提出了广义虚函数的概念。广义虚函数具有任意性、虚拟性。     

Frictional contact finite elements based on mixed variational principles

This paper presents an original approach to the numerical modelling of unilateral contact by the finite element method. The main point is the development of mixed contact finite elements in which the displacement field and the contact stress field (pressure and friction shear) are discretized independently. The theory, based on variational principles, is first presented in the framework of infinitesimal deformations and, subsequently, is extended to large inelastic strains.     

Basis of finite element methods for solid continua

Finite element methods can be formulated from the variational principles in solid mechanics by relaxing the continuity requirements along the interelement boundaries. The combination of different variational principles and different boundary continuity conditions yields numerous types of approximate methods. This paper reviews and reinterprets the existing finite element methods and indicates other alternative schemes. Plate bending problems are used to compare the relative merits of the various methods.     

On the formulation of closest‐point projection algorithms in elastoplasticity—part I: The variational structure

We present in this paper the characterization of the variational structure behind the discrete equations defining the closest‐point projection approximation in elastoplasticity. Rate‐independent and viscoplastic formulations are considered in the infinitesimal and the finite deformation range, the later in the context of isotropic finite‐strain multiplicative plasticity. Primal variational principles in terms of the stresses and stress‐like hardening variables are presented first, followed by the formulation of dual principles incorporating explicitly the plastic multiplier. Augmented Lagrangian extensions are also presented allowing a complete regularization of the problem in the constrained rate‐independent limit. The variational structure identified in this paper leads to the proper framework for the development of new improved numerical algorithms for the integration of the local constitutive equations of plasticity as it is undertaken in Part II of this work. Copyright © 2001 John Wiley & Sons, Ltd.     

A study of conservation laws of dynamical systems by means of the differential variational principles of Jourdain and Gauss

Summary In this report we consider the possibility of using the differential variational principles of Jourdain and Gauss as a starting point for the study of conservation laws of holonomic conservative and nonconservative dynamical systems with a finite number of degrees of freedom. We demonstrate that this approach has the same status as the method based on the D'Alembert's differential variational principle developed in a previous paper.This research was supported by U. S. NSF Grant JPP 524     

On an alleged philosophers' stone concerning variational principles with subsidiary conditions

In general, the Lagrange multiplier method (LMM) is used to incorporate subsidiary conditions into variational principles. The Lagrange multipliers represent an additional field of independent variables. The attempt to satisfy subsidiary conditions without employing additional independent unknowns has led to the development of simplified variational principles (SVP). They are characterized by expressing Lagrange multipliers in terms of original field variables by means of the Euler-Lagrange equations for the multipliers, providing their physical interpretation. In the first part of the theoretical investigation, systems with infinitely many degrees-of-freedom are studied. It is shown that the Euler-Lagrange equations of a LMM based on a modification of the principle of minimum of potential energy (PMIPE) do not hold unconditionally, that is, for arbitrary subsidiary conditions, for the corresponding SVP. The second part of the theoretical investigation is concerned with systems with a finite number of degrees-of-freedom. The finite element method (FEM) is employed to discuss and compare the characteristics of the LMM and of the corresponding SVP. Contrary to the former, the latter are found to be problem-dependent. Several shortcomings of the SVP are listed, including the possibility of obtaining an infinite sequence of singular coefficient matrices in the process of a systematic mesh refinement. Consequently, convergence of finite element solutions to the true solution in the limit of finite element representations is not guaranteed. The theoretical findings are corroborated by the results of a detailed numerical study.     

Hellinger–Reissner principles for plate and shell finite elements

The Hellinger–Reissner generalized variational principle is used to deduce further principles directed especially towards formulation of high integrity finite elements for plates and curved shells in polynomial parametric representation. Assumed stresses in these principles are derived from displacements and supplemented with stresses derived from stress functions.     

Complementary and dual energy finite element principles in magnetostatics

A method of providing bounded solutions to a wide range of magnetostatic field problems is outlined. The method extends complementary and dual energy variational principles to encompass the T-Ω formulation of electromagnetic field problems and shows how this leads to efficient finite element implementation of the technique. Examples are given that show clearly the bounded nature of the procedure, and indicate how it may be used to reduce the computational requirements necessary for a specific accuracy of solution.     

Variational principles and finite element analysis for polarization gradient theory

The equations of classical polarization gradient theory are studied using variational methods and finite element analysis. Variational principles are derived and specialized to represent the cubic centro-symmetric crystal structure. An isoparametric nine node axisymmetric finite element is developed and used to demostrate the application of the theory. An analysis of the effects of a point charge in a semi-infinite isotropic halfspace including surface tension effects is computed.