P-version hybrid analytical finite element method for plate bending problems

Two sets of trial functions with different variables are constructed for the admissible space of the finite element analysis. The trial functions satisfy the equilibrium differential equation inside elements, while the deflections and rotations on the edges of the elements are approximated by the Peano hierarchical interpolation functions. Then, a generalized variational principle is applied to set up the p-version hybrid analytical finite element method for plate bending problems. The accuracy of finite element computation can be improved by increasing the order of the interpolation polynomials with fixed mesh. In the finite element formulation, to obtain the stiffness matrices and the load vectors, it is only necessary to perform quadrature over the edges of the elements. These matrices and vectors possess an embedding structure. The conformability between the elements can be controlled automatically.This work is supported by the Natural Science Foundation of China and the Aeronautical Science Foundation of China.     

弹性薄板弯曲问题的等价的直接变量边界积分方程

建立平面弹性薄板弯曲问题理论中具有直接变量的等价边界积分方程，传统的直接变量边界积分方程，它们都不是等价的，对此进行了深入的讨论。     

弹性薄板弯曲问题的双互易杂交边界点法

基于一种板的修正变分泛函,将杂交边界点法与双互易法结合,用于薄板弯曲问题的分析。该方法将问题的解分为齐次方程的通解和非齐次的特解两部分,特解采用径向基函数插值得到,而通解则使用杂交边界点法求解。在杂交边界点法用于求解通解的列式过程中,边界变量采用移动最小二乘近似,域内变量则采用基本解插值。与有限元法相比,该方法仅需要边界上离散点的信息,无论插值还是积分都不需要网格,域内点仅用来插值非齐次项,因而仍是一种纯边界类型的无网格方法。数值算例表明,本文方法能以很少的计算自由度获得与其它方法同样的计算精度,且具有前后处理简单、收敛速度快等优点,适合于求解工程中各种薄板的弯曲问题。     

The 3-D boundary element method of roller bearing by plate element analogue

The mill roller bearing is made up of an internal ring, middle rolls and an external ring, the analysis of which is a multi-bodies contact problem. In this paper, based on the three-dimensional elastic contact BEM without friction, and using the structural characteristics of roller bearings, middle rolls are described by elastic plate units of different shapes, which is placed on the internal ring. The discontinuous traction problem of the contact element can be dealt with by the traction sub-element method. Therefore the contact problem can be changed from one of multi-bodies to one of two-bodies. As an elastic deformation of the plate element, it can be calculated by an elastic contact formula and is regarded as an equivalent gap to be put into the total matrix equality. The face-to-face contact condition of displacement and traction on the element will serve as the convergence judging condition. So convergence solution can be obtained before any geometry inconsistent on the contact element may take place. Therefore, the three-dimensional distribution of load of the mill roller bearing can be achieved. It has been proved that the calculating model of this method is visualized and simple, has fast convergence solution and higher accuracy, making it an effective numerical method of designing and analyzing load characteristics of the mill roller bearing. Project supported by the National Natural Science Foundation of China (50075075).     

AN EXACT ELEMENT METHOD FOR THE BENDING OF NONHOMOGENEOUS REISSNER’S PLATE

In this paper,based on the step reduction method and exact analytic method,a new method,the exact element method for constructing finite element,is presented.Since the new method doesn’t need variational principle,it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficients.By this method,a triangle noncompatible element with15 degrees of freedom is derived to solve the bending of nonhomogenous Reissner’s plate.Because the displacement parameters at the nodal point only contain deflection and rotation angle.it is convenient to deal with arbitrary boundary conditions.In this paper,the convergence of displacement and stress resultants is proved.The element obtained by the present method can be used for thin and thick plates as well,Four numerical examples are given at the end of this paper,which indicates that we can obtain satisfactory results and have higher numerical precision.     

A new type of plate bending element

Based on a two-field generalized variational principle, a new type of arbitrary quadrilateral plate bending element, with four nodes and with the effect of transverse shear deformation taken into account, is proposed. The element is applicable to a wide range of plate thickness and an explicit expression of stiffness matrix can be obtained. Therefore, it possesses the distinguished features of general applicability, high precision and less computer time.     

On a method of comparison for plate elements in finite element engineering software programs

The purpose of this paper is to propose a method for the evaluation of plate elements in the finite element engineering software comparatively to the Kirchhoff and Reissner plate theories. The method is based on the study of transverse deflection near a crack tip. A numerical work has been conducted for the shell elements in Abaqus software with different crack lengths and various thicknesses. The deflection can be written as a function of the exponent of the distance between the crack tip and one point along the crack. This exponent is accepted to describe the behaviour of finite elements.     

A higher-order boundary element method for three-dimensional potential problems

A method of eliminating the singularities involved in boundary element methods for three-dimensional potential problems is presented and the non-singular expressions of integrals on an element on which the singular point is situated are given for linear and quadratic interpolation functions. Numerical examples are compared with analytical solutions to show that the higher-order interpolations have better precision.     

The general boundary element method and its further generalizations

In this paper, the basic ideas of the general boundary element method (BEM) proposed by Liao [in Boundary Elements XVII, Computational Mechanics Publications, Southampton, MA, 1995, pp. 67–74; Int. J. Numer. Methods Fluids, 23 , 739–751 (1996), 24 , 863–873 (1997); Comput. Mech., 20 , 397–406 (1997)] and Liao and Chwang [Int. J. Numer. Methods Fluids, 23 , 467–483 (1996)] are further generalized by introducing a non‐zero parameter . Some related mathematical theorems are proposed. This general BEM contains the traditional BEM in logic, but is valid for non‐linear problems, including those whose governing equations and boundary conditions have no linear terms. Furthermore, the general BEM can solve non‐linear differential equations by means of no iterations. This disturbs the absolutely governing place of iterative methodology of the BEM for non‐linear problems. The general BEM can greatly enlarge application areas of the BEM as a kind of numerical technique. Two non‐linear problems are used to illustrate the validity and potential of the further generalized BEM. Copyright © 1999 John Wiley & Sons, Ltd.     

Equivalent Boundary Integral Equations with indirect unknowns for thin elastic plate bending theory

Equivalent Boundary Integral Equations (EBIE) with indirect unknowns for thin elastic plate bending theory, which is equivalent to the original boundary value problem, is established rigorously by mathematical technique of non-analytic continuation and is fully proved by means of the variational principle. The previous three kinds of boundary integral equations with indirect unknowns are discussed thoroughly and it is shown that all previous results are not EBIE. Paper from SUN Huan-chun, Member of Editorial Commitee, AMM Biography: ZHANG Yao-ming (1962-)     

An exact element method for bending of nonhomogeneous thin plates

In this paper, based on the step reduction method, a new method, the exact element method for constructing finite element, is presented. Since the new method doesn't need the variational principle, it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficient. By this method, a triangle noncompatible element with 6 degrees of freedom is derived to solve the bending of nonhomogeneous plate. The convergence of displacements and stress resultants which have satisfactory numerical precision is proved. Numerical examples are given at the end of this paper, which indicate satisfactory results of stress resultants and displacements can be obtained by the present method.     

基于单元子分法的结构多尺度边界单元法

建立在基于单元子分法的一种有效自适应格式以及多区域边界元三步求解技术基础上提出了一种计算结构多尺度问题的多区域边界元法。首先,通过高斯积分误差分析公式确定边界单元在满足精度要求下所需要的高斯点数,当所需高斯点数超过规定数目时该单元就被自动划分成一定数量的子单元,从而消除结构多尺度所引起的近奇异性。在单元子分技术的基础上采用多区域边界元三步求解技术来处理材料非均质问题:第一步消除各子域的内部未知量,第二步消除各子域独自拥有的边界未知量,第三步根据位移相容性条件和面力平衡条件建立系统方程组并求解公共界面节点位移以及每个子域的其他未知量。数值算例结果表明本方法可以用较少的计算时间得到满意的结果,是处理结构多尺度问题的一种有效方法。     

Homogenization-based method for bending analysis of perforated plate

Owing to the existence of distributed holes, it is difficult to solve the bending problem of perforated plates by the conventional finite element method. A homogenization-based method for this problem is presented in this paper. As an example, the bending analysis of a circular perforated plate with distributed step-wise cylindrical holes is made. The deflection and the fundamental frequency obtained by present method are in good agreement with experimental data, this implies that the method is effective. This project is supported by the National Natural Science Foundation (19602007) and National Outstanding Youth Foundation (19525206).     

Stochastic boundary element method for reliability analysis of plate and beams composite structures

I.IntroductionTheengineeringstructuresareoftells,'1>:'rectcdtotheactionofthestochasticloadingthatvarieswiththetime,forexample,theengineeringstructuresactedonbytheearthquake,theoceanstructuresactedonbydynamicpressure,andthevehiclesofthetransportationinflue…     

一种考虑横向剪切变形的三角形板弯曲单元

本文从部分协调的三角形薄板弯曲单元出发,并假设横向剪切应变在单元内线性变化,提出了一种考虑横向剪切变形的具有15个自由度的三角平板弯曲单元。该单元应用于薄板和中等厚度板分析均有较高的精度,计算效率高,可用于工程中具有复杂形状的薄板和中等厚度板结构的分析。     

Solving the elastic bending problem for a plate with mixed boundary conditions

The bending problem for an arbitrarily outlined thin plane with mixed boundary conditions is solved. A technique based on the methods of potentials and balancing loads is proposed for constructing Green’s function for the Germain-Lagrange equation. This technique ensures high accuracy of approximate solutions, which is checked against Levi’s solution for rectangular plates __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 104–112, May 2006.     

自然单元法在Winkler地基薄板计算中的应用

自然单元法是一种基于Voronoi图及Delaunay三角形剖分图,以自然邻接点插值为试函数的一种无网格数值方法.本文以目前该方法中自然邻接点的Laplace插值形函数为基础,求出了其一阶及二阶导函数,建立了Winkler地基上薄板弯曲挠度的自然单元法求解控制方程,并编制了相应的计算程序,通过算例分析表明了本文方法的可行性和有效性.     

The boundary element method applied to elastic contact problems with friction

In the current paper the boundary integral equations (BIE) for elastic contact problems with friction are derived from the incremental virtual work principle. After introducing contact conditions of adhesion and slip into BIE all variants on boundary are made to discretize by quadratic isoparametric boundary element. In the current paper not only an auto-increment loading law is presented but also the iterative calculation laws for open, slip and adhesion condition are given. The results of numerical examples are satisfactory.     

A necessary and sufficient set of boundary integral equations with indirect unknowns for plate bending problem

A boundary integral representation of plane biharmonic function is established rigorously by the method of unanalytical continuation in the present paper. In this representation there are two boundary functions and four constants which bear a one to one correspondence to biharmonic functions. Therefore the set of boundary integral equations with indirect unknowns based on this representation is equivalent to the original differential equation formulation.     

一种处理弹性动力边界元法中奇异积分的方法

利用边界元法求解瞬态弹性动力学问题时,时域基本解函数的分段连续性和奇异性为该问题的求解带来很大的困难。为了解决时域基本解中的奇异性问题,本文依据柯西主值的定义,对经过时间解析积分之后的时域基本解进行奇异值分解,将其分成奇异和正则积分两部分;其中正则部分可通过采用常规高斯积分方法来计算,而奇异部分具有简单的形式,可以利用解析积分计算。经过上述操作之后,就可以达到直接消除时域基本解中奇异积分的目的。和传统方法相比,本文方法并不依赖静力学基本解来消除奇异性,是一种直接求解方法。最后给定两个数值算例来验证本文提出方法的正确性和可行性,结果表明使用本文算法可以解决弹性动力学边界积分方程中的奇异性问题。