刚体动力学的拟变分原理及其应用

为了适应航天事业发展的需要,极有必要开展多柔体系统的理论分析.作为应用非保守系统的拟变分原理进行多柔体动力学的理论分析的组成部分,研究了刚体动力学的拟变分原理及其应用:建立了刚体动力学的拟变分原理,推导出刚体动力学的拟变分原理的拟驻值条件; 建立了刚体动力学的广义拟变分原理,说明了应用广义拟变分原理求得问题的解析解和数值解的途径; 最后,借助算例说明了应用变分方法来研究刚体动力学问题的优越性.      

弹性理论中广义变分原理的研究及其在有限元计算中的应用

本文的目的在于说明怎样系统地建立各种广义变分原理,怎样合理地使用各种广义变分原理来改进有限元计算的成效。为了易于说明问题,本文只局限于弹性理论的各种广义变分原理,但其推广并不困难。本文指出,广义变分原理的泛函,可以系统地采用拉格朗日乘子法,把一般有条件的变分原理化为无条件的变分原理来唯一地决定的。拉格朗日乘子所代表的物理量,可以通过变分求极值或驻值的过程求得,从而消除了在建立广义变分原理的泛函时,人们经常陷入的象猜谜一样的困境。本文也指出:我们同样可以用拉格朗日乘子法把一般有多个条件的变分原理,化为条件个数较少的变分原理。我们称变分条件减少了的变分原理为各级不完全的广义变分原理。凡是把全部变分条件都消除了的变分原理,称为完全的广义变分原理,或简称广义变分原理;实际上是完全无条件的变分原理。本文建立了弹性小位移变形理论中的各级不完全的广义位能原理,和各级不完全的广义余能原理,包括从最小位能原理和最小余能原理分别导出的最完全的广义变分原理;并且证明了这两个弹性力学广义变分原理的泛函是等同的。在这些广义变分原理中,包括了Hellinger-Reissner(1950),胡海昌-鹫津久一郎(1955)的广义变分原理。本文也建立了弹性大位移变形理论中的位能原理和余能原理,并建立了有关位能余能的各级不完全的广义变分原理,包括以大位移变形的最小位能和最小余能原理分别导出的弹性力学广义变分原理,并且也证明了在大位移变形情况下,这两个弹性力学的广义变分原理也是等同的。本文除了列举广义变分原理在有限元法上的众所周知的应用外,还补充了三个比较重要的应用范围。     

带孔隙损伤的弹性固体的广义变分原理

应用弹性微结构理论,建立了具广义力场带孔隙损伤线弹性固体的基本模型．应用变积方法,同时分别建立了带孔隙损伤弹性固体四类和两类变量的广义变分原理,这些变分原理对应着带孔隙损伤弹性固体微分方程和初值边值条件．应用弹性微结构理论,建立了带孔隙损伤的弹性Timoshenko 梁的基本方程,得到带孔隙损伤的弹性Timoshenko 梁两类变量的广义变分原理．这些广义变分原理为近似求解带孔隙损伤的弹性问题提供了有效途径．     

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.     

摩擦约束广义变分不等原理的临界变分现象及分析

对于具有摩擦约束的弹塑性接触问题,由于边界接触面上的摩擦力由不等式表示,导致得到包含摩擦约束的广义变分原理为广义变分不等原理.广义变分不等原理通过将摩擦力纳入问题的能量泛涵,可避免考虑摩擦力变化的具体过程,便于数值方法如有限元等在弹性接触问题上的应用.但是,通过对广义变分不等原理的研究,发现在弹性力学广义变分不等原理中,势能型和余能型广义变分不等原理,均存在临界变分现象,即变分时拉格朗日乘子为零,变分失败;或者得到的能量泛函变分后得不到问题的欧拉方程.在对弹性力学广义变分不等原理临界变分现象进行分析后,提出了避免发生临界变分现象的方法.实际应用证明了方法的有效性.通过避免临界变分现象的发生,可以保证拉格朗日乘子方法的有效使用.     

包含终值条件为驻值条件的哈氏型弹性动力学变分原理

对于弹性动力学的终值问题,给出了包含终值条件为驻值条件的Hamilton型弹性动力学变分原理,其中有位能作用量变分原理、余能作用量变分原理,广义位能作用量变分原理及广义余能作用量变分原理。通过一个简例说明了所给原理的应用。     

分析力学初值问题的一种变分原理形式

明确了分析力学初值问题的控制方程,按照广义力和广义位移之间的对应关系,将 各控制方程卷乘上相应的虚量,代数相加,进而在 原空间中建立了分析力学初值问题的一种变分原理形式,即建立了分析力学初值问题的卷积 型变分原理和卷积型广义变分原理. 推导了分析力学初值问题卷积型变分原理和卷积型广义 变分原理的驻值条件. 在建立分析力学初值问题的一种变分原理形式的同时, 将变积方法推广为卷变积方法.     

弹性扁壳的广义变分原理及扁壳理论的某些问题

本文导出了一个以应力函数及挠度为变量函数的弹性扁壳的广义变分原理。在这个变分原理中,扁壳全部基本方程都是Euler方程,全部边界条件都是自然边界条件。 应用这个变分原理,我們討論了以下問題: 1.用应力函数及挠度表示几何边界条件的問題; 2.多連通扁壳的位移单位条件問題。 文内还导出了大挠度情形的广义变分原理。     

粘性流体动力学有限元变分原理

本文把建立有限元变分原理的一种新方法“N>2直接方法”从固体力学推广到流体力学,并用该方法把粘性流体动力学的广义功率消耗原理和广义变分原理发展成为有限元变分原理。还在论证中发现,相邻有限元交界面上的应力协调条件会自然地满足而无需引进任何拉民乘子。本文还建立了混合杂交非协调元的变分原理和广义变分原理,它解除了全部协调性约束条件和其它的边界性约束条件,但是并不增加待定的拉氏乘子,因此使有限元计算得到简化。本文结果可以作为粘性流体动力学有限元计算的基础定理。     

基于Herglotz型微分变分原理研究相空间中非保守系统的守恒律

为了进一步探究非保守系统的Herglotz变分问题与其守恒律之间的关系,该文提出并研究基于Herglotz型微分变分原理构建相空间中非保守力学系统的守恒律.首先,基于相空间中非保守系统的Herglotz变分问题,建立该系统的Herglotz型微分变分原理;其次,利用广义变分与经典等时变分之间的关系,给出微分变分原理不变性条件的变换,并建立非保守系统的守恒定理,得到了该系统基于Herglotz变分问题的守恒量及其存在条件;再次,导出守恒定理的逆定理,由相空间中非保守系统的已知守恒量可找到无限小变换的空间和时间的生成元.文末举例说明结果的应用.     

Periodic solutions of rigid body–viscous flow interaction

This paper describes the work on extending the finite element method to cover interactions between a viscous flow and a moving body. The problem configuration of interest is that of an arbitrarily shaped body undergoing a simple harmonic motion in an otherwise undisturbed incompressible fluid. The finite element modelling is based on a primitive variables representation of the Navier-Stokes equations using curved isoparametric elements. The non-linear boundary conditions on the moving body are obtained using Taylor series expansion to approximate the velocities at the fixed finite element grid points. The method of averaging is used to analyse the resulting periodic motion of the fluid. The stability of the periodic solutions is studied by introducing small perturbations and applying Floquet theory. Numerical results are obtained for several example body shapes and compared with published experimental results. Good agreement is obtained for the basic non-linear phenomenon of steady streaming.     

矩形空腔内Stokes流的状态空间有限元法

基于Hellinger-Reissner二类变分原理,从平面Stokes流问题的平衡方程、连续性要求和边界条件出发,得到相应的Hamilton函数,建立Hamilton正则方程后,采用分离变量法对场变量进行离散求解:在x方向采用有限元插值,在y方向采用状态空间法给出控制坐标方向的解析解。计算过程中的指数矩阵均采用精细积分法求解,使得本文算法具有高效率、高精度、对步长不敏感的优点。通过对侧边自由液面边界条件的单板驱动矩形空腔Stokes流问题的求解,得到与文献相同的结果,从而验证了本文方法的有效性。本文旨在将弹性力学状态空间有限元法的思想引入到低雷诺数流体力学中,为Hamilton体系下研究复杂边界Stokes流问题提供新的途径。     

The application of generalized variational principle in finite element-semianalytical method

The method developed in this paper is inspired by the viewpoint in ref. [1] that sufficient attention has not been paid to the value of the generalized variational principle in dealing with the boundary conditions in the finite element method. This method applies the generalized variational principle and chooses the series constituted by spline function multiplied by sinusoidal function and added by polynomial as the approximate deflection of plates and shells. By taking the deflection problem of thin plate, it shows that this method can solve the coupling problem in the finite element-semianalytical method. Compared with the finite element method and finite stripe method, this method has much fewer unknown variables and higher precision. Hence, it proposes an effective way to solve this kind of engineering problems by minicomputer.     

三维电磁场辛有限元列式及电磁共振腔的计算

针对三维共振腔的电磁场分析,利用Maxwell方程的对偶方程体系形式,从其相应的对偶变量变分原理出发,导出了三维电磁场辛有限单元的详细列式。为了有限元列式的保辛,变分原理被积函数可导向对于对偶变量为对称的形式。变分原理的边界积分项对于相邻单元相互抵消。由于采用了对偶变量的插值函数,使得电磁场单元构造可以在层面上进行,从而避免了所谓的连续性问题。无物理意义的零本征解可采用奇异值分解加以排除。文末分别对矩形及圆柱形的共振腔做了数值计算并与解析解和棱边元计算结果进行对比,算例表明了列式及算法的有效性。     

Subregion generalized variational principles for elastic thick plates

Tn this paper, the subregion generalized variational principle for elastic thick plates is proposed. Its main points may be stated as follows:1. Each subregion may be assigned arbitrarily as a potential region or complementary region. The subregion variational principles of potential energy, complementary energy and mixed energy represent three special forms of this principle.2. The number of independent variational variables in each sub-region may be assigned arbitrarily. Any one of the subregions may be assigned as a one-variable-region, two-variable-region or three-variable-region.3. The conjunction conditions of displacements and stresses on each interline of neighbouring subregions may be relaxed. On the basis of this principle the finite element analysis of non-conforming elements for thick plates can be formulated.Different finite element models for thick plates can be obtained by different applications of this principle. In particular,the subregion mixed variational principle for thick plates may be applied to formulating the subregion mixed finite element method for thick plates.     

电磁共振腔辛有限元法

将电磁场的基本方程导向了对偶方程形式。给出了推导电磁场有限元所需相应的对偶变量变分原理。为了有限元列式的保辛，交分原理被积函数可导向对于对偶变量为对称的形式。交分原理的边界积分项对于相邻单元互相抵消。对偶变量有限元推导可避免所谓的C1连续性问题。采用对偶变量离散分析了共振腔本征值问题，离散后再消去一类变量可导出普通的广义本征值问题而求解。算例表明了对偶变量有限元分析的有效性。     

Variable penalty method for finite element analysis of incompressible flow

A new scheme is applied for increasing the accuracy of the penalty finite element method for incompressible flow by systematically varying from element to element the sign and magnitude of the penalty parameter λ, which enters through ?.v + p/λ = 0, an approximation to the incompressibility constraint. Not only is the error in this approximation reduced beyond that achievable with a constant λ, but also digital truncation error is lowered when it is aggravated by large variations in element size, a critical problem when the discretization must resolve thin boundary layers. The magnitude of the penalty parameter can be chosen smaller than when λ is constant, which also reduces digital truncation error; hence a shorter word-length computer is more likely to succeed. Error estimates of the method are reviewed. Boundary conditions which circumvent the hazards of aphysical pressure modes are catalogued for the finite element basis set chosen here. In order to compare performance, the variable penalty method is pitted against the conventional penalty method with constant λ in several Stokes flow case studies.     

Development of new triangular elements for free surface flows

New finite elements have been developed to simulate steady and unsteady two-dimensional free surface flows. The depth-averaged velocity components with the free surface elevation have been used as independent variables in the model. The differences between the various elements presented lie in the choice of velocity approximation. The Newton–Raphson method has been used to solve the non-linear system of equations. Emphasis is put on bench-mark examples to assess the accuracy and efficiency of the elements. A simple stable new element tested herein shows promising advantages for industrial finite element codes.