The general solution for axial symmetrical bending of nonhomogeneous circular plates resting on an elastic foundation

In this paper,a new method,the exact analytic method,is presented on the basis of stepreduction method.By this method,the general solution for the bending of nonhomogenouscircular plates and circular plates with a circular hole at the center resting,on an elastfcfoundation is obtained under arbitrary axial symmetrical loads and boundary conditions.The uniform convergence of the solution is proved.This general solution can also be applieddirectly to the bending of circular plates without elastic foundation.Finally,it is onlynecessary to solve a set of binary linear algebraic equation.Numerical examples are givenat the end of this paper which indicate satisfactory results of stress resultants anddisplacements can be obtained by the present method.     

A high convergent precision exact analytic method for differential equation with variable coefficients

The exact analytic method was given by[1].It can be used for arbitrary variable coefficient differential equations and the solution obtained can have the second order convergent precision.In this paper,a new high precision algorithm is given based on[1],through a bending problem of variable cross-section beams.It can have the fourth convergent precision without increasing computation work.The present computation method is not only simple but also fast.The numerical examples are given at the end of this paper which indicate that the high convergent precision can be obtained using only a few elements.The correctness of the theory in this paper is confirmed.     

Analytical solution of basic equations set of atmospheric motion

On condition that the basic equations set of atmospheric motion possesses the best stability in the smooth function classes, the structure of solution space for local analytical solution is discussed, by which the third-class initial value problem with typicality and application is analyzed. The calculational method and concrete expressions of analytical solution about the well-posed initial value problem of the third kind are given in the analytic function classes. Near an appointed point, the relevant theoretical and computational problems about analytical solution of initial value problem are solved completely in the meaning of local solution. Moreover, for other type of problems for determining solution, the computational method and process of their stable analytical solution can be obtained in a similar way given in this paper.     

Exact analytic method for solving variable coefficient differential equation

Many engineering problems can be reduced to the solution of a variable coefficient differential equation. In this paper, the exact analytic method is suggested to solve variable coefficient differential equations under arbitrary boundary condition. By this method, the general computation format is obtained. Its convergence is proved. We can get analytic expressions which converge to exact solution and its higher order derivatives unifornuy. Four numerical examples are given, which indicate that satisfactory results can be obtained by this method.     

瞬态热传导方程的子结构精细积分方法

对线性定常结构动力系统提出的精细积分方法,在数值精度等方面表现出极大优越性,但是当矩阵尺度很大时在数值计算与存储中将产生困难,对此,本文对瞬态热传导方程,根据结构的概念,将结构分为若干个子结构,对各子结构分别进行指数矩阵运算并通过了结构间界面的物理量相联系,从而提高精细积分方法的计算效率。     

基于转换矩阵的FEM/MLPG耦合算法

首次基于有限元的转换矩阵(TMF)和无网格的转换矩阵(TMM),提出有限单元法(FEM)和无网格局部彼得罗夫-伽辽金法(MLPG)的耦合算法。编制了相应算法的三维程序,计算分析了三维柱体的拉伸和弯曲问题,并将计算结果与ABAQUS软件计算结果以及理论解进行了比较。结果表明,本文给出的耦合算法计算精度高,收敛性好,可以用以模拟裂纹扩展等问题。     

The convergent condition and united formula of step reduction method

The step reduction method was first suggested by Prof. Yeh Kai-yuan^[1]. This method has more advantages than other numerical methods. By this method, the analytic expression of solution can he obtained for solving nonuniform elastic mechanics. At the same time, its calculating time is very short and convergent speed very fast. In this paper. the convergent condition and united formula of step reduction method are given by mathematical method. It is proved that the solution of displacement and stress resultants obtained by this method can converge to exact solution uniformly. when the convergent condition is satisfied. By united formula, the analytic solution can be expressed as matrix form, and therefore the former complicated expression can be avoided. Two numerical examples are given at the end of this paper which indicate that by the theory in this paper, a right model can be obtained for step reduction method.Project Supported by Science and Technic Fund of the National Education Committee.The author would like to thank Prof. Yeh Kai-yuan for his directing.     

弹性薄板分析的条形传递函数方法

提出一种用于矩形弹性薄板变形分析的条形传递函数方法．一个矩形区域首先沿某一个方向被剖分成若干个条形子域,分割这些子域的直线称为结线,在结线上定义位移函数,它是结线坐标的一维函数,结线的两个端点称为结点．为适应复杂边界条件,在边界结线上定义若干结点,该结线的位移函数用结点位移参数插值表示．每个条形子域的变形用结线位移函数和适当的插值函数（形函数）表示．结线位移函数和结点位移参数满足的平衡微分方程及代数方程由变分原理给出     

An exact analytic method applied to nonhomogeneous ring- and stringer-stiffened cylindrical shells

In this paper, the nonlinear axial symmetric deformation problem of nonhomogeneous ring- and stringer-stiffened shells is first solved by the exact analytic method. An analytic expression of displacements and stress resultants is obtained and its convergence is proved. Displacements and stress resultants converge to exact solution uniformly. Finally, it is only necessary to solve a system of linear algebraic equations with two unknowns. Four numerical examples are given at the end of the paper which indicate that satisfactory results can be obtained by the exact analytic method.     

含3个分配点结构的弯矩分配公式法精确解

为了提高弯矩分配法的计算速度和精度,将弯矩分配传递的过程视为多个等比数列的运算过程. 在第一轮弯矩分配与传递结束后,可得出各个等比数列的首项和公比,并利用等比数列公式直接求和求得精确解.提出了含有3 个分配点结构的弯矩分配公式法,在弯矩传递中采用双向传递,并通过等比数列公式求精确解.以结构、载荷均不对称的两跨刚架为例,将手算与电算结果进行比较,验证了该方法求解的精确性和实用性.     

大型结构特征值问题的Lanczos子结构并行算法

本文利用子结构和Lanczos方法,提出了大型结构固有频率与模态的并行解法。该方法在Lanczos方法的求解过程中,仅利用子结构刚度阵和质量阵并行进行凝聚,进而求得新的迭代矢量,最终求得三对角阵对应的特征值和特征向量。该算法在西安交通大学ELXSI-6400并行计算机上程序实现,计算结果表明能有效地节省计算时间和计算机的内存,为一种有效的大型工程结构动力问题的求解方法。     

AN EXACT METHOD OF BENDING OF ELASTIC THIN PLATES WITH ARBITRARY SHAPE

ＡＮ　ＥＸＡＣＴ　ＭＥＴＨＯＤ　ＯＦ　ＢＥＮＤＩＮＧ　ＯＦ　ＥＬＡＳＴＩＣ　ＴＨＩＮ　ＰＬＡＴＥＳ　ＷＩＴＨ　ＡＲＢＩＴＲＡＲＹ　ＳＨＡＰＥＺｈｏｕＤｉｎｇ（周叮）（ＲｅｃｅｉｖｅｄＡｐｒｉｌ１０，１９９５，ＲｅｖｉｓｅｄＡｐｒｉｌ２６．１９９６．Ｃ...     

一种大型结构特征值问题的并行解法

本文提出了一种求解大型结构固有频率与模态的并行解法。该方法在子空间迭代过程中,利用子结构刚度阵和质量阵并行进行凝聚,求得下一次的迭代基矢量,直到收敛。该算法在西安交通大学ELXS1-6400并行机上程序实现,计算结果表明能大幅度节省计算时间,同时也有效地节省了内存。     

The effects of relative density of metal foams on the stresses and deformation of beam under bending

The exact analytic solution of the pure bending beam of metallic foams is given. The effects of relative density of the material on stresses and deformation are revealed with the Triantafillou and Gibson constitutive law (TG model) taken as the analysis basis. Several examples for individual foams are discussed, showing the importance of compressibility of the cellular materials. One of the objects of this study is to generalize Hill’s solution for incompressible plasticity to the case of compressible plasticity, and a kinematics parameter is brought into the analysis so that the velocity field can be determined. The English text was polished by Yunming Chen.     

Extremum problems of boundary control for steady equations of thermal convection

An inverse extremum problem of boundary control for steady equations of thermal convection is considered. The cost functional in this problem is chosen to be the root-mean-square deviation of flow velocity or vorticity from the velocity or vorticity field given in a certain part of the flow domain; the control parameter is the heat flux through a part of the boundary. A theorem on sufficient conditions on initial data providing the existence, uniqueness, and stability of the solution is given. A numerical algorithm of solving this problem, based on Newton’s method and on the finite element method of discretization of linear boundary-value problems, is proposed. Results of computational experiments on solving extremum problems, which confirm the efficiency of the method developed, are discussed.     

超越摄动:同伦分析方法基本思想及其应用

介绍一种新的、求解强非线性问题解析近似的一般方法------同伦分析方法.该方法从根本上克服了摄动理论对小参数的过分依赖, 其有效性与所研究的非线性问题是否含有小参数无关, 因此,适用范围广.此外, 不同于所有其他解析近似方法,同伦分析方法提供了一个简单的途径, 确保所得到的级数解收敛, 从而获得足够精确的解析近似.而且, 不同于所有其他解析近似方法, 同伦分析方法(HAM)提供了选取基函数之自由, 从而可以选择较好的基函数, 更有效地逼近问题的解.同伦分析方法为非线性问题的解析近似求解提供了一个全新的思路, 为非线性问题(特别是不含小参数的强非线性问题)的求解开辟了一个全新的途径.简要描述同伦分析方法的基本思想, 其在非线性力学、物理、化学、生物、金融、工程和计算数学等领域的应用举例, 以及与摄动方法、Lyapunov 人工小参数法、$\delta$展开法、Adomian 分解法、同伦摄动方法之区别和联系.      

大型结构特征值问题的混合粒度并行算法

本文提出一种求解大形结构特征值问题的粗细粒度混合并行算法：在子结构模态综合粗粒度并行算基础上,综合系统的特性值问题采用细粒度并行方式求解。细粒度并行包括子空间迭代法的子结构并行算法、雅可比分块并行计算的方法和一种Ｎｅｗｔｏｎ－Ｒａｐｈｏｎ迭代法在多处理器上任力均衡分配的有效策略。子空间迭代法的子结构并行计算的实施是利用子结构的刚度阵和质量阵而不必完全组集系统刚度阵和国求综合系统的特征值问题。利用雅     

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.     

New analytic solutions for static problems of rectangular thin plates point-supported at three corners

This paper deals with the bending of rectangular thin plates point-supported at three corners using an analytic symplectic superposition method. The problems are of fundamental importance in both civil and mechanical engineering, but there were no accurate analytic solutions reported in the literature. This is attributed to the difficulty in seeking the solutions that satisfy the governing fourth-order partial differential equation with the free boundary conditions at all the edges as well as the support conditions at the corners. In the following, the Hamiltonian system-based equation for plate bending is formulated, and two types of fundamental problems are analytically solved by the symplectic method. The analytic solutions of the plates point-supported at three corners are then obtained by superposition, where the constants are obtained by a set of linear equations. The solution procedure presented in this paper offers a rigorous way to yield analytic solutions of similar problems. Some numerical results, validated by the finite element method, are shown to provide useful benchmarks for comparison and validation of other solution methods.     

Strip method for steady heat-conduction problems

Summary An approximate solution of heat-conduction problems can be obtained by the strip method. The method consists of an application of the finite-difference approximation in one physical coordinate and an analytic solution in other coordinates. A simple illustrative example is given and the result is compared with that obtained by the exact solution. By application of this method, an approximate solution is given for the steady heat conduction through a rectangular parallel composite wall with different rates of heat generation.