Stokes 流问题中的辛本征解方法

通过引入哈密顿体系，将二维Stokes流问题归结为哈密顿体系下的本 征值和本征解问题. 利用辛本征解空间的完备性，建立一套封闭的求解问题方法. 研究结果 表明零本征值本征解描述了基本的流动，而非零本征值本征解则显示着端部效应影响特点. 数值算例给出了辛本征值和本征解的一些规律和具体例子. 这些数值例子说明了端部非规则 流动的衰减规律. 为研究其它问题提供了一条路径.     

基于辛体系的正交各向异性地基梁解析解

基于二维弹性理论, 利用Hellinger-Reissner变分原理, 通过引入对偶变量, 推导 了双参数地基上正交各向异性梁平面应力问题的辛对偶方程组; 采用分离变量法和本征展 开方法, 将原问题归结为求解零本征值本征解和非零本征值本征解, 得到了适用于任意横纵 比的梁的解析解. 由于在求解过程中不需要事先人为地选取试函数, 而是从梁的基本方程出 发, 直接利用数学方法求出问题的解, 使得问题的求解更加合理化. 其中, 地基对梁的力学 行为的影响看作是侧边边界条件, 类似于外载, 可通过零本征解的线性展开来评价, 非零本 征值本征解对应圣维南原理覆盖的部分. 还利用哈密顿变分原理, 给出了两端固支梁的 一种新的改进边界条件. 编程计算了细梁和深梁等算例, 研究了地基上梁的变形沿着厚度方 向的变化特性, 验证了辛方法的有效性.     

辛体系下平面热黏弹性圣维南问题的解析解

借助积分变换,将辛体系引入平面热黏弹性问题,建立了基本问题的对偶方程,并将全 部圣维南问题归结为满足共轭辛正交关系的零本征值本征解问题. 同时,利用变量代换和本 征解展开等技术给出了一套求解边界条件问题的具体方法. 算例讨论了几种典型边界条件问 题,描述了热黏弹性材料的蠕变和松弛特征,体现了这种辛方法的有效性.     

空腔内粘性流问题与哈密顿体系方法

以双板驱动空腔粘性流问题为研究对象,根据其特点建立了哈密顿体系下的对偶正则方程,将问题归结为辛体系下的本征值问题.利用辛本征解空间的完备性、正交性和展开理论,形成一套封闭的求解问题方法.算例的数值结果揭示了一些空腔流动的特点.同时这种方法也为研究其他问题提供了一条思路.     

圆柱型正交各向异性弹性楔的佯谬解

圆柱型正交各向异性弹性楔体顶端受有集中力偶的经典解,当顶角满足一定关系时,其应力成为无穷大,这是个佯谬．该文在哈密顿体系下将该问题进行重新求解,即利用极坐标各向异性弹性力学哈密顿体系．在原变量和其对偶变量组成的辛几何空间求解特殊本征值的约当型本征解,从而直接给出该佯谬问题的解析解．结果再次表明经典力学中的弹性楔佯谬解对应的是哈密顿体系下辛几何的约当型解．     

辛方法在弹性圆板屈曲问题中的应用

在辛几何空间中将临界载荷和屈曲模态归结为辛本征值和本征解问题,从而形成一种辛方法.研究和讨论了轴对称屈曲和非轴对称屈曲问题,它们分别属于零本征值问题和非零本征值问题.以弹性圆板屈曲问题作为研究对象,借助于系统的能量构造出哈密顿体系,得到了该体系下的所有的本征解.数值结果给出了圆板和圆环板问题的临界载荷和屈曲模态.数值结果表明:对应低阶屈曲模态的临界载荷相对较小且屈曲模态在周向的波纹数也较少,说明在屈曲过程中低阶屈曲模态容易出现,特别是轴对称屈曲更容易发生;对应较大分支数的临界载荷,其值相对较大且屈曲模态在径向的波纹更加复杂;同时物理常数和几何参数也会直接影响临界载荷的大小.     

奇异Hamilton系统矩阵的精细积分法

常规位移有限元的结构振动方程是n个二阶常微分方程组.采用一般交分原理推导,将结构振动问题引入Hamiltoil体系,将得到2n个一阶常微分方程组.精细积分法宜于处理一阶方程,应用于线性定常结构动力问题求解,可以得到在数值上逼近精确解的结果.对于非齐次动力方程,当结构具有刚体位移时,系统矩阵将出现奇异.本文借鉴全元选大元高斯-约当法求解线性方程组的经验,提出全元选大元法求奇异矩阵零本征解的方法,该方法可以简便快速地寻求奇异矩阵零本征值对应的子空间.利用Hamiltoil体系已有研究成果及Hamilton系统的共轭辛正交归一关系,迅速将零本征值对应的子空间分离出来,通过投影排除奇异部分,然后用精细积分法求得问题的解.数值算例表明,该方法对Hamilton系统奇异问题,处理方便,计算量小,易于实现,同时保持了精细算法的优点.     

功能梯度材料平面问题的辛弹性力学解法

将辛弹性力学解法推广用于功能梯度材料平面问题的分析,考虑沿长度方向弹性模量为指数函数变化而泊松比为常数的矩形域平面弹性问题,给出了具体的求解步骤. 提出了移位Hamilton矩阵的新概念,建立起相应的辛共轭正交关系;导出了对应特殊本征值的本征解,发现材料的非均匀特性使特殊本征解的形式发生明显的变化.      

弹性圆柱壳在轴向冲击载荷和温度耦合作用下的屈曲

通过引入哈密顿体系,将临界载荷和临界温度及它们所对应的屈曲模态归结为辛体系下的广义本征值和本征解问题。根据辛本征解的正交性和完备性,给出了全部的且独立存在的屈曲模态。数值结果表明,在轴向冲击载荷和温度耦合作用下,弹性圆柱壳的屈曲呈现出复杂的模式,温度直接影响冲击临界载荷的大小。随着温度的增加,冲击临界荷载降低,最后,文中给出各种条件下的屈曲模态。     

哈密顿体系与弹性楔体问题

将哈密体系引入到级坐标下的弹性力学楔体问题,利用该体系辛空间的性质,将问题化为本征值和本征向量求解上,得到了完备的解空间,从而改变了弹性力学传统的拉格朗日体系以应力函数为特征的半逆法的讨论去解决该类问题的思路,给出了一条求解该类问题的直接法。     

Symplectic solution system for reissner plate bending

Based on the Hellinger-Reissner variatonal principle for Reissner plate bendingand introducing dual variables, Hamiltonian dual equations for Reissner plate bending werepresented. Therefore Hamiltonian solution system can also be applied to Reissner platebending problem, and the transformation from Euclidian space to symplectic space and fromLagrangian system to Hamiltonian system was realized. So in the symplectic space whichconsists of the original variables and their dual variables, the problem can be solved viaeffective mathematical physics methods such as the method of separation of variables andeigenfunction-vector expansion. All the eigensolutions and Jordan canonical formeigensolutions for zero eigenvalue of the Hamiltonian operator matrix are solved in detail, and their physical meanings are showed clearly. The adjoint symplectic orthonormal relation of the eigenfunction vectors for zero eigenvalue are formed. It is showed that the alleigensolutions for zero eigenvalue are basic solutions of the Saint-Venant problem and theyform a perfect symplectic subspace for zero eigenvalue. And the eigensolutions for nonzeroeigenvalue are covered by the Saint-Venant theorem. The symplectic solution method is notthe same as the classical semi-inverse method and breaks through the limit of the traditional semi-inverse solution. The symplectic solution method will have vast application.     

EIGENVALUE PROBLEM OF A LARGE SCALE INDEFINITE GYROSCOPIC DYNAMIC SYSTEM

Gyroscopic dynamic system can be introduced to Hamiltonian system.Based on an adjoint symplectic subspace iteration method of Hamiltonian gyroscopic system, an adjoint symplectic subspace iteration method of indefinite Hamiltonian function gy- roscopic system was proposed to solve the eigenvalue problem of indefinite Hamiltonian function gyroscopic system.The character that the eigenvalues of Hamiltonian gyroscopic system are only pure imaginary or zero was used.The eigenvalues that Hamiltonian function is negative can be separated so that the eigenvalue problem of positive definite Hamiltonian function system was presented,and an adjoint symplectic subspace iteration method of positive definite Hamiltonian function system was used to solve the separated eigenvalue problem.Therefore,the eigenvalue problem of indefinite Hamiltonian function gyroscopic system was solved,and two numerical examples were given to demonstrate that the eigensolutions converge exactly.     

Paradox solution on elastic wedge dissimilar materials

According to the Hellinger-Reissner variational principle and introducing proper transformation of variables , the problem on elastic wedge dissimilar materials can be led to Hamiltonian system, so the solution of the problem can be got by employing the separation of variables method and symplectic eigenfunction expansion under symplectic space, which consists of original variables and their dual variables . The eigenvalue - 1 is a special one of all symplectic eigenvalue for Hamiltonian system in polar coordinate . In general, the eigenvalue - 1 is a single eigenvalue, and the classical solution of an elastic wedge dissimilar materials subjected to a unit concentrated couple at the vertex is got directly by solving the eigenfunction vector for eigenvalue - 1. But the eigenvalue - 1 becomes a double eigenvalue when the vertex angles and modulus of the materials satisfy certain definite relationships and the classical solution for the stress distribution becomes infinite at this moment, that is, the para     

Analytical and numerical methods of symplectic system for Stokes flow in two-dimensional rectangular domain

In this paper, a new analytical method of symplectic system, Hamiltonian system, is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain. In the system, the fundamental problem is reduced to an eigenvalue and eigensolution problem. The solution and boundary conditions can be expanded by eigensolutions using adjoint relationships of the symplectic ortho-normalization between the eigensolutions. A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space. The results show that fundamental flows can be described by zero eigenvalue eigensolutions, and local effects by nonzero eigenvalue eigensolutions. Numerical examples give various flows in a rectangular domain and show effectiveness of the method for solving a variety of problems. Meanwhile, the method can be used in solving other problems.     

Analytical and numerical methods of symplectic system for Stokes flow in two-dimensional rectangular domain

In this paper,a new analytical method of symplectic system.Hamiltonian system,is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain.In the system,the fundamental problem is reduced to all eigenvalue and eigensolution problem.The solution and boundary conditions call be expanded by eigensolutions using ad.ioint relationships of the symplectic ortho-normalization between the eigensolutions.A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space.The results show that fundamental flows can be described by zero eigenvalue eigensolutions,and local effects by nonzero eigenvalue eigensolutions.Numerical examples give various flows in a rectangular domain and show effectivenees of the method for solving a variety of problems.Meanwhile.the method can be used in solving other problems.     

基于辛本征值方法的受约束介电弹性体中褶皱不稳定性的分析

辛弹性力学已广泛应用于弹性学中各种边值问题的精确解、计算表面波模式以及预测多层超弹性薄膜中的表面褶皱。本文展示了辛分析框架还可应用于受约束介电弹性体中的表面褶皱。机械和电位移向量是两个基本变量来描述介电弹性体中机械变形与电场紧密耦合。褶皱的临界电压可以通过引入基本变量的对偶变量来从辛本征值问题中解决。本文采用扩展的W-W（Wittrick-Williams）算法和精确的积分方法,准确而高效地解决制定的辛本征值问题的本征值。通过将褶皱电压和波数与有无表面能的褶皱基准结果进行比较,验证了辛分析的有效性。辛分析框架简洁且适用于其他不稳定问题,如分层电介质弹性体、磁弹性不稳定性以及层压复合结构的微观和宏观不稳定性。     

HAMILTONIAN SYSTEM AND THE SAINT VENANT PROBLEM IN ELASTICITY

ＨＡＭＩＬＴＯＮＩＡＮＳＹＳＴＥＭＡＮＤＴＨＥＳＡＩＮＴＶＥＮＡＮＴＰＲＯＢＬＥＭＩＮＥＬＡＳＴＩＣＩＴＹＺｈｏｎｇＷａｎｘｉｅ（钟万勰）；ＸｕＸｉｎｓｈｅｎｇ（徐新生）；ＺｈａｎｇＨｏｎｇｗｕ（张洪武）（ＲｅｃｅｉｖｅｄＪｕｎｅ５，１９９５）Ａｂｓ...     

On new symplectic elasticity approach for exact bending solutions of rectangular thin plates with two opposite sides simply supported

This paper presents a bridging research between a modeling methodology in quantum mechanics/relativity and elasticity. Using the symplectic method commonly applied in quantum mechanics and relativity, a new symplectic elasticity approach is developed for deriving exact analytical solutions to some basic problems in solid mechanics and elasticity which have long been bottlenecks in the history of elasticity. In specific, it is applied to bending of rectangular thin plates where exact solutions are hitherto unavailable. It employs the Hamiltonian principle with Legendre’s transformation. Analytical bending solutions could be obtained by eigenvalue analysis and expansion of eigenfunctions. Here, bending analysis requires the solving of an eigenvalue equation unlike in classical mechanics where eigenvalue analysis is only required in vibration and buckling problems. Furthermore, unlike the semi-inverse approaches in classical plate analysis employed by Timoshenko and others such as Navier’s solution, Levy’s solution, Rayleigh–Ritz method, etc. where a trial deflection function is pre-determined, this new symplectic plate analysis is completely rational without any guess functions and yet it renders exact solutions beyond the scope of applicability of the semi-inverse approaches. In short, the symplectic plate analysis developed in this paper presents a breakthrough in analytical mechanics in which an area previously unaccountable by Timoshenko’s plate theory and the likes has been trespassed. Here, examples for plates with selected boundary conditions are solved and the exact solutions discussed. Comparison with the classical solutions shows excellent agreement. As the derivation of this new approach is fundamental, further research can be conducted not only on other types of boundary conditions, but also for thick plates as well as vibration, buckling, wave propagation, etc.     

四边固支矩形薄板自由振动的哈密顿解析解

在哈密顿体系中利用辛几何方法求解了四边固支矩形薄板自由振动问题的解析解。首先,从基本方程出发,将问题表示成Hamilton正则方程,然后利用辛几何方法导出本征值问题,从而得到本征函数解,使之满足边界条件;再由方程组有非零解的条件,最终推导出四边固支矩形薄板的自振频率方程,得到频率的解析解。计算了不同长宽比情况下四边固支矩形薄板的频率,结果与已有文献完全一致。该解法有望推广至更多尚未得到解析解的矩形板的振动问题。