Eigenfunction expansion method and its application to two-dimensional elasticity problems based on stress formulation

This paper proposes an eigenfunction expansion method to solve twodimensional （2D） elasticity problems based on stress formulation. By introducing appropriate state functions, the fundamental system of partial differential equations of the above 2D problems is rewritten as an upper triangular differential system. For the associated operator matrix, the existence and the completeness of two normed orthogonal eigenfunction systems in some space are obtained, which belong to the two block operators arising in the operator matrix. Moreover, the general solution to the above 2D problem is given by the eigenfunction expansion method.     

多层层合板圣维南问题的解析解

将哈密尔顿体系理论引入到多层层合板问题之中，建立了一套求解该问题的横向哈密尔顿算子矩阵的本征函数向量展开解法，并成功地求解出圣维南问题的解析解．进一步显示了弹性力学新求解体系的有效性及其应用潜力     

分离变量法与哈密尔顿体系

数学物理与力学中用分离变量法求解偏微分方程经常导致自共轭算子的sturmLiouville问题,在此基础上而得以展开求解。然而在应用中有大量问题并不能导致自共轭算子。本文通过最小势能变分原理,选用状态变量及其对偶变量,导向一般变分原理。利用结构力学与最优控制的模拟理论,导向哈密尔顿体系。将有限维的理论推广到相应的哈密尔顿算子矩阵及共轭辛矩阵代数的理论。拓广了经典的分离变量法,证明了全状态本征函数向量的共轭辛正交归一性质及按本征函数向量展开的理论。以条形板为例,说明了应用。     

矩形中厚板Hamilton正则方程的解析解

运用Fourier分析方法,建立了对边简支的矩形中厚板弯曲问题的完备的辛本征展开. 借助于Mathematica软件的帮助,得到了来源于矩形中厚板问题的Hamilton算子的本征函数. 接着证明了本征函数系的完备性,这为使用分离变量法求解相应问题提供了理论保证;进而运用完备性定理,得到了问题的解析解;一个数值算例验证了结果的正确性.     

Symplectic eigenfunction expansion theorem for elasticity of rectangular planes with two simply-supported opposite sides

The eigenvalue problem of the Hamiltonian operator associated with plane elasticity problems is investigated.The eigenfunctions of the operator are directly solved with mixed boundary conditions for the displacement and stress in a rectangular region.The completeness of the eigenfunctions is then proved,providing the feasibility of using separation of variables to solve the problems.A general solution is obtained with the symplectic eigenfunction expansion theorem.     

Moment Lyapunov exponent for three-dimensional system under real noise excitation

The pth moment Lyapunov exponent of a two-codimension bifurcation system excited parametrically by a real noise is investigated. By a linear stochastic transformation, the differential operator of the system is obtained. In order to evaluate the asymptotic expansion of the moment Lyapunov exponent, via a perturbation method, a ralevant eigenvalue problem is obtained. The eigenvalue problem is then solved by a Fourier cosine series expansion, and an infinite matrix is thus obtained, whose leading eigenvalue is the second-order of the asymptotic expansion of the moment Lyapunov exponent. Finally, the convergence of procedure is numerically illustrated, and the effects of the system and the noise parameters on the moment Lyapunov exponent are discussed.     

多层横观各向同性圆柱壳的三维弹性理论解

本文从三维弹性理论出发,用特征函数法研究多层横观各向同性圆柱壳的轴对称问题.把位移和应力分量的齐次解表达成特征函数展开式,并把特解部分用Fourier级数表示.以多层圆柱壳的内、外柱面作为齐次边界,同时考虑层间的连续条件,推导出问题的特征方程并用Muller法求解.文中运用传递矩阵技术处理多层问题,并用边界型最小二乘配点法处理端部边界条件.作为实例,对双层圆柱壳作了数值计算.     

Eigenfunction expansion method to the solution of simultaneous equations and its application in mechanics

The solution of a homogeneous vector-matrix differential equation, when the operator is of linear second order, has been established by the method of eigenfunction approach. The uniqueness of the solution has been established for both the cases when the roots of the characteristic equations are distinct and when they are repeated. Finally, the theory has been applied on two problems of mechanics and the results are compared with the existing literature.     

Third-order nonlinear singularly perturbed boundary value problem

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Completeness of system of root vectors of upper triangular infinite-dimensional Hamiltonian operators appearing in elasticity theory

This paper deals with a class of upper triangular infinite-dimensional Hamiltonian operators appearing in the elasticity theory.The geometric multiplicity and algebraic index of the eigenvalue are investigated.Furthermore,the algebraic multiplicity of the eigenvalue is obtained.Based on these properties,the concrete completeness formulation of the system of eigenvectors or root vectors of the Hamiltonian operator is proposed.It is shown that the completeness is determined by the system of eigenvectors of the operator entries.Finally,the applications of the results to some problems in the elasticity theory are presented.     

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.     

二维弹性力学问题的光滑无网格伽辽金法

计算效率低的问题长期阻碍着无网格伽辽金法(element-free Galerkin method, EFGM) 的深入发展. 为了提高EFGM 的计算速度, 本文提出一种求解二维弹性力学问题的光滑无网格伽辽金法. 该方法在问题域内采用滑动最小二乘法(moving least square, MLS)近似、在域边界上采用线性插值建立位移场函数; 基于广义梯度光滑算子得到两层嵌套光滑三角形背景网格上的光滑应变, 根据广义光滑伽辽金弱形式建立系统离散方程. 两层嵌套光滑三角形网格是由三角形背景网格本身以及四个等面积三角形子网格组成. 为了提高方法的精度, 由Richardson外推法确定两层光滑网格上的最优光滑应变. 几个数值算例验证了该方法的精度和计算效率. 数值结果表明, 随着光滑积分网格数目的增加, 光滑无网格伽辽金法的计算精度逐步接近EFGM 的, 但计算效率要远远高于EFGM的. 另外, 光滑无网格伽辽金法的边界条件可以像有限元那样直接施加. 从计算精度和效率综合考虑, 光滑无网格伽辽金法比EFGM具有更好的数值表现, 具有十分广阔的发展空间.      

Symplectic system based analytical solution for bending of rectangular orthotropic plates on Winkler elastic foundation

This paper analyses the bending of rectangular orthotropic plates on a Winkler elastic foundation.Appropriate definition of symplectic inner product and symplectic space formed by generalized displacements establish dual variables and dual equations in the symplectic space.The operator matrix of the equation set is proven to be a Hamilton operator matrix.Separation of variables and eigenfunction expansion creates a basis for analyzing the bending of rectangular orthotropic plates on Winkler elastic foundation and obtaining solutions for plates having any boundary condition.There is discussion of symplectic eigenvalue problems of orthotropic plates under two typical boundary conditions,with opposite sides simply supported and opposite sides clamped.Transcendental equations of eigenvalues and symplectic eigenvectors in analytical form given.Analytical solutions using two examples are presented to show the use of the new methods described in this paper.To verify the accuracy and convergence,a fully simply supported plate that is fully and simply supported under uniformly distributed load is used to compare the classical Navier method,the Levy method and the new method.Results show that the new technique has good accuracy and better convergence speed than other methods,especially in relation to internal forces.A fully clamped rectangular plate on Winkler foundation is solved to validate application of the new methods,with solutions compared to those produced by the Galerkin method.     

AN INTERVAL FINITE ELEMENT METHOD BASED ON THE NEUMANN SERIES EXPANSION 1)

实际工程问题中通常存在大量的不确定参数, 区间有限元方法是一种结合有限元数值计算工具对结构进行不确定性分析的区间方法. 区间有限元的目的是获得在含有区间不确定性参数条件下的结构响应上下边界, 其关键问题在于区间平衡方程组的求解, 而这属于一类往往很难求解的NP-hard问题. 本文归纳了一类工程实际中常见的结构不确定性问题, 即可线性分解式区间有限元问题, 并针对此提出一种基于Neumann级数的区间有限元方法. 在区间有限元分析中, 当区间不确定参数表示为一组独立区间变量线性叠加时, 若结构的刚度矩阵也可表示为这些独立区间变量的线性叠加形式, 则称此类区间有限元问题为可线性分解式区间有限元问题. 对于此类问题, 采用Neumann级数对其刚度矩阵的逆矩阵进行表示, 可获得结构响应关于区间变量的显式表达式, 从而可高效求解结构响应的上下边界. 最后通过两个算例验证了本文所提方法的有效性.     

一种基于Neumann级数的区间有限元方法

实际工程问题中通常存在大量的不确定参数, 区间有限元方法是一种结合有限元数值计算工具对结构进行不确定性分析的区间方法. 区间有限元的目的是获得在含有区间不确定性参数条件下的结构响应上下边界, 其关键问题在于区间平衡方程组的求解, 而这属于一类往往很难求解的NP-hard问题. 本文归纳了一类工程实际中常见的结构不确定性问题, 即可线性分解式区间有限元问题, 并针对此提出一种基于Neumann级数的区间有限元方法. 在区间有限元分析中, 当区间不确定参数表示为一组独立区间变量线性叠加时, 若结构的刚度矩阵也可表示为这些独立区间变量的线性叠加形式, 则称此类区间有限元问题为可线性分解式区间有限元问题. 对于此类问题, 采用Neumann级数对其刚度矩阵的逆矩阵进行表示, 可获得结构响应关于区间变量的显式表达式, 从而可高效求解结构响应的上下边界. 最后通过两个算例验证了本文所提方法的有效性.      

Plane infinite analytical element and hamiltonian system

IntroductionManyinfiniteproblemscanbefoundincivilengineering ,suchastunnelconstruction ,structurefoundation ,etc ..Forotherengineeringproblems,whenthephysicaldimensionsofanobjectaresmallandthesurroundingmediaorstructuresaremuchbiggerthantheobject,thenumericalcalculatingmodelcanbetreatedasoneinaninfinitefield .Sofar,onlyafewanalyticalsolutionsforinfinitefieldproblemscanbefound[1- 3].ManyprojectsrelatedtoinfinitefieldproblemsaresolvedbytheFEM ,whereinfiniteelementmethodsareused[4 ,5 ].Sometime…     

A Partially Built-in Plate under Uniform Load

A thin plate has the form of the infinite strip ?∞<x<∞, 0≤y≤aand has the edge y=abuilt-in. The edge y=0 has its right half 0<x<∞ built-in while the left half ?∞<x<0 is free. The whole plate is now subjected to a uniform load p ₀applied to its upper surface. What is the resulting deflection of the plate and what are the induced moment and shear resultants? We present a solution to this classical problem based on eigenfunction expansions. In the right and left halves of the strip, the deflection can be expanded as separate eigenfunction expansion series, but these are difficult to match across the line x=0 because of the singularity at (0,0) induced by the boundary conditions. We adopt the novel technique of expanding the field near the centre of the strip in its correct form as a series of Williams polar eigenfunctions, and then linking this expansion to the right and left eigenfunction expansions by using a special form of elastic reciprocity. These right and left reciprocity conditions give two infinite systems of linear equations satisfied by the polar expansion coefficients, and we prove that these equations are sufficient to determine these coefficients. Further applications of reciprocity give closed form expressions for the right and left eigenfunction expansion coefficients so that the whole solution is then determined. The method yields accurate results using small systems of linear equations. We present numerical results for the deflection of the plate and the induced moment and shear resultants.     

求解固体力学问题的强?弱耦合形式单元微分法

单元微分法是一种新型强形式有限单元法. 与弱形式算法相比, 该算法直接对控制方程进行离散, 不需要用到数值积分. 因此该算法有较简单的形式, 并且其在计算系数矩阵时具有极高的效率. 但作为一种强形式算法, 单元微分法往往需要较多网格或者更高阶单元才能达到满意的计算精度. 与此同时, 对于一些包含奇异点的模型, 如在多材料界面、间断边界条件、裂纹尖端等处, 传统单元微分法往往得不到较精确的计算结果. 为了克服这些缺点, 本文提出了将伽辽金有限元法与单元微分法相结合的强?弱耦合算法, 即整体模型采用单元微分法的同时, 在奇异点附近或某些关键部件采用有限元法. 该策略在保留单元微分法高效率与简洁形式等优点的同时, 确保了求解奇异问题的精度. 在处理大规模问题时, 针对关键部件采用有限元法, 其他部件采用单元微分法, 可以在得到较精确结果的同时, 极大提高整体计算效率. 在本文中, 给出了两个典型算例, 一个是具有切口的二维问题, 一个是复杂的三维发动机问题. 针对这两个问题, 分析了该耦合算法在求二维奇异问题和三维大规模问题时的精度与效率.      

Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes

A 2D, depth-integrated, free surface flow solver for the shallow water equations is developed and tested. The solver is implemented on unstructured triangular meshes and the solution methodology is based upon a Godunov-type second-order upwind finite volume formulation, whereby the inviscid fluxes of the system of equations are obtained using Roe's flux function. The eigensystem of the 2D shallow water equations is derived and is used for the construction of Roe's matrix on an unstructured mesh. The viscous terms of the shallow water equations are computed using a finite volume formulation which is second-order-accurate. Verification of the solution technique for the inviscid form of the governing equations as well as for the full system of equations is carried out by comparing the model output with documented published results and very good agreement is obtained. A numerical experiment is also conducted in order to evaluate the performance of the solution technique as applied to linear convection problems. The presented results show that the solution technique is robust. © 1997 John Wiley & Sons, Ltd.     

Solving the incompressible fluid flows by a high-order mesh-free approach

In this paper, we propose for the first time to extend the application field of the high-order mesh-free approach to the stationary incompressible Navier-Stokes equations. This approach is based on a high-order algorithm, which combines a Taylor series expansion, a continuation technique, and a moving least squares (MLS) method. The Taylor series expansion permits to transform the nonlinear problem into a succession of continuous linear ones with the same tangent operator. The MLS method is used to transform the succession of continuous linear problems into discrete ones. The continuation technique allows to compute step-by-step the whole solution of the discrete problems. This mesh-free approach is tested on three examples: a flow around a cylindrical obstacle, a flow in a sudden expansion, and the standard benchmark lid-driven cavity flow. A comparison of the obtained results with those computed by the Newton-Raphson method with MLS, the high-order continuation with finite element method, and those of literature is presented.