部分应力杂交动力分析模型的混合变分原理

本文通过对具有单重卷积积分的二类变量的Ｇｕｒｔｉｎ型混合变分原理进行修正，采用“部分应力杂交”的概念，建立了线弹性动力分析 部分应力杂交变分格式，并在此基础上构造出时间有限元模型，这种模型对于分层复合板的动力分析尤其适用。     

耦合热弹性动力学的统一变分原理族

对耦合热弹性动力学问题,迄今文献中只建立了Ｇｕｒｔｉｎ型合卷积的统一变分原理,其缺点是只适用于常系数的线性问题,且因含卷积而使实际数值离散和求解复杂化．文中首次成功地建立了耦合热弹性动力学问题经典型（不含卷积）统一变分原理族,其关键是建议了动态差分变换和初终值条件的新处理法．该方法可以推广到各向异性材料以及非线性问题上去,同时在应用有限元法离散和求解上都比较简便．     

Gurtin变分原理在矩形板动力初值问题中的应用

结构动力分析是工程设计中的重要组成部分,传统动力分析方法并不能全面反映动力初值特征,而Ｇｕｒｔｉｎ变分原理则被认为是目前唯一能全面反映动力初值特征的变分原理。本文基于位移型Ｇｕｒｔｉｎ变分原理,对空间和时间同时离散,建立了一种求解板的动力初值问题的时空有限元法,并对两种边界情况板的振动问题进行了编程计算,计算结果表明时空元法精确度很高且稳定收敛。     

求解交变载荷下异形板动力响应的Gurtin半解析法

该方法以Ｇｕｒｔｉｎ变分原理为基础，在空间域作有限元离散，时间域取级数。本文研究了在交变载荷ｑ（ｘ，ｙ，ｔ）＝Ｐ（ｘ，ｙ）ｓｉｎωｔ作用下任意形状板时域级数的取法，使得各种支承条件的动力响应问题均可由本计算模式得到具有相当精度和效率的解。     

基于位移型Gurtin变分原理计算动力响应的逐步积分法

本文利用位移型Ｇｕｒｔｉｎ变分原理,在时间域上采用三次Ｈｅｒｍｉｔｅ插值函数进行离散,给出了一种计算结构动力响应的逐步积分方法。通过稳定性分析研究了该方法的稳定区情况表明,当１．６４≤θ≤２．０８时,该方法的数值计算精度很高,但是条件稳定积分格式。当θ≥４．１时,该方法是无条件稳定的积分格式,精度较高。     

关于线粘弹性动力学中各种变分原理

本文提出一条简单而统一的新途径,系统地建立了线粘弹性动力学中各种简化Gurtin型变分原理,文中首先给出一个很有用的以卷积表示的积分关系式,然后从该式出发,系统地导出成互补关系的五类变量、四类变量、三类变量、二类变量及一类变量简化Gurtin型变分原理,并清楚地阐明它们之间的内在联系,而且,还发现当前在国际上有广泛影响的力学变分原理方面的名著[1]及文[2]中,所给出的四个变分原理的泛函式均有误.本文除给出这四个正确的泛函式外,还建立了一些新的更一般广义变分原理。     

翼型跨音速流动计算和外形设计的变域变分有限元方法

在气体动力学问题研究中经常会碰到诸如激波、翼型设计等未知界面问题。未知界面的存在为该类问题的理论分析和数值求解带来了很大困难。刘高联针对未知界面问题发展了一种变域变分有限元方法，该方法将未知界面看作是一个变化区域的边界，采用变域变分将未知界面结合在变分泛函中，使其与求解流场的控制方程结合起来，从而将未知界面的求解和流场的求解完全耦合进行，因而是一种处理未知界面的独特工具，极适合于气动外形的设计求解。本文运用变域变分有限元方法对翼型跨音速流动正、反命题进行了数值研究。由于在跨音速翼型绕流中存在激波，所以为了得到压缩激波解，采用了“人工密度”办法。几个算例均得到了满意的计算结果和设计结果，证明了本文方法的有效性和优越性。     

流体饱和多孔介质的动力学Gurtin型变分原理和有限元模拟

基于多孔介质理论。在两相不可压和小变形的假设下，建立了流体饱和弹性多孔介质的动力学Gurtin型变分原理，并导出了以此变分原理为基础的有限元离散公式，由于Gurtin型变分原理是卷积型的空间积分泛函，空间的有限元离散导致一个关于时间的对称微分—积分方程组，在一般条件下。该积分—微分方程组可转化为对称的微分方程组，这组方程有别于标准Galerkin有限元的非对称离散方程组，作为数值例子，分析了流体饱和弹性多孔介质中一维纵向波的传播和反射，其结果进一步揭示了饱和多孔介质中波的传播特性。     

应用变域变分有限元法解平面叶栅反命题

反命题作为一种可变(未知)边界问题近年来得到了广泛的研究。本文给出了亚声速平面叶栅反命题计算的势函数变域变分有限元解法。变域变分通过把可变边界结合在变分泛函中，使其与求解流场的控制方程结合起来，从而使可变边界求解和流场分析可以完全耦合进行。本文针对亚声速平面叶栅的反命题，根据泛函的驻值必要条件，介绍了变域变分有限元方法的求解过程，最后给出了两个数值算例。这两个算例均采用四节点矩形单元的插值基函数，第一个算例用于检验程序的可靠性，第二个算例设计了一个给定叶面马赫数分布的叶型，并与试验结果进行对照。     

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.     

Two methods of solving dynamic response by discretization of Gurtin variational principle

Based on the Gurtin variational principle and the finite element method two recurrence formulae of solving dynamic response are derived in this paper. The first recurrence formula which is conditionally stable is obtained by adopting “dual temporal finite element” to discretize the Gurtin variational principle and then making use of convolution, while the second, an unconditionally stable one derived by discritizing the variational principle in each time step and taking approprlate integration parameter θ. The paper had been accepted by the XVIth International Congress of IUTAM, Lyngby, Denmark, August, 1984.     

卷积型GD半解析法及矩形薄板瞬态响应解

卷积型的Gurtin变分原理是目前在数学上唯一能和动力学初值问题完全等价的变分原理,它完全反映了有关初值问题的全部特征。GD法（General Differential Method）是从泰勒展开式出发,推出的一种求解偏微分方程的数值方法,本文系统地介绍了GD法的基本原理,以及权系数的推导。本文通过卷积将矩形薄板原始控...     

GURTIN VARIATIONAL PRINCIPLE AND FINITE ELEMENT SIMULATION FOR DYNAMICAL PROBLEMS OF FLUID-SATURATED POROUS MEDIA

Based on the theory of porous media,a general Gurtin variational principle for theinitial boundary value problem of dynamical response of fluid-saturated elastic porous media isdeveloped by assuming infinitesimal deformation and incompressible constituents of the solid andfluid phase.The finite element formulation based on this variational principle is also derived.Asthe functional of the variational principle is a spatial integral of the convolution formulation,thegeneral finite element discretization in space results in symmetrical differential-integral equationsin the time domain.In some situations,the differential-integral equations can be reduced to sym-metrical differential equations and,as a numerical example,it is employed to analyze the reflectionof one-dimensional longitudinal wave in a fluid-saturated porous solid.The numerical results canprovide further understanding of the wave propagation in porous media.     

A convolution-type semi-analytic DQ approach to transient response of rectangular plates

The convolution-type Gurtin variational principle is known as the only variational principle that is, from the mathematics point of view, totally equivalent to the initial value problem system. In this paper, the equation of motion of rectangular thin plates is first transformed to a new governing equation containing initial conditions by using a convolution method. A convolution-type semi-analytical DQ approach, which involves differential quadrature （DQ） approximation in the space domain and an analytical series expansion in the time domain, is proposed to obtain the transient response solution. This approach offers the same advantages as the Gurtin variational principle and, at the same time, is much simpler in calculation. Numerical results show that it is very accurate yet computationally efficient for the dynamic response of plates.     

UNCONVENTIONAL GURTIN-TYPE VARIATIONAL PRINCIPLES FOR FINITE DEFORMATION ELASTODYNAMICS

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 Gurtin-type variational prinicples for finite deformation elastodynamics can be established systemati-cally.In this paper,an important integral relation in terms of convolution is given,which canbe considered as the expression of the generalized principle of virtual work for finite deformationdynamics.Based on this relation,it is possible not only to obtain the principle of virtual work forfinite deformation dynamics,but also to derive systematically the complementary functionals forfive-field,three-field,two-field and one-field unconventional Gurtin-type variational principles bythe generalized Legendre transformations given in this paper.Furthermore,with this approach,the intrinsic relationship among various principles can be clearly explained.     

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.     

On the connections between the functionals of equivalent variational principles

It is proved in this paper that the functionals of equivalent variational principles are different essentially only in terms of weighted residues. Consequently, the simplest method for constructing equivalent variational principles is to add weighted residues to known functionals as first used by Courant [4].     

Various reciprocal theorems and variational principles in the theories of nonlocal micropolar linear elastic mediums

In the first part of our paper, we have extended the concepts of the classical convolution and the convolution scalar product given by I. Hlaváck and presented the concepts of the convolution vector and the convolution vector scalar product, which enable us to extend the initial value as well as the initial-boundary value problems for the equation with the operator coefficients to those for the system of equations with the operator coefficients.In the second part of this paper, based on the concepts of the convolution vector and the convolution vector scalar product, two fundamental types of reciprocal theorems of the non-local micropolar linear elastodynamics for inhomogeneous and anisotropic solids are derived.In the third part of this paper, based on the concepts and results in the first and second parts as well as the Lagrange multiplies method which is presented by W. Z. Chien, four main types of variational principles are given for the nonlocal micropolar linear elastodynamics for inhomogeneous and anisotropic solids. These are the counterparts of the variational principles of Hu-Washizu type, Hellinger-Reissner type and Gurtin type in classical elasticity as well as Hlaváck type and Iesan type in local micropolar and nonlocal elasticity. Finally, we have proved the equivalence of the last two main variational principles which are given in this paper.