Interval Arithmetic and Static Interval Finite Element Method

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线性区间有限元静力控制方程的组合解法

区间有限元的静力控制方程常被归结为区间方程组来求解。但实际上两者并不等价。本文根据不确定结构有限元分析的力学背景，直接从问题的基本参量的不确定性出发，将基本区间参量的边界组合与求解区间方程组的有关解法相结合，提出了线性区间有限元静力控制方程的两种组合解法－参量边界全组合法和组合迭代法。可以以较小的计算量获得或逼近位移和应力区间的准确界限。且不受基本参量变化范围的限制。算例分析表明文中方法是实用和可行的。     

A method of solving the fuzzy finite element equations in monosource fuzzy numbers

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基于单源模糊数的模糊随机动态有限元方程的解法

本文提出一种模糊随机动态有限元方程的解法,指出利用单源模糊数和它的运算法则,可以把一个不含阻尼项的模糊随机动态有限元平衡方程转化为两类不同集合下的方程组,一种是模糊数方程,另一种是普通的动态有限元平衡方程。前者可用模糊数运算法则求解。通常这类方程的表达式非常简单,故很容易求解,后者可利用现有的求解随机动态随机有限元平衡方程的方法计算,这时求解该方程的计算量几乎等同于求解相应的普通随机动态有限元平衡方程的计算量。最后的算例表明,本文提出的方法与通常所用的γ截集法计算结果基本相同,而且所用的计算量远远小于用γ截集法所用的计算量。     

非确定结构系统区间分析的泛灰求解方法

工程中的不确定性问题可以用区间分析、概率理论或模糊理论来求解。采用泛灰区间分析法来处理结构静力分析和设计中的不确定性问题。将结构系统中的不确定性参数用区间数来表示，用有限元法建立系统的控制方程。该控制方程是线性区间方程组。然后，在概述泛灰数的概念及其运算规则的基础上，介绍了泛灰数与区间数的转化，利用泛灰数的可扩展性对区间进行分析，研究了泛灰线性方程求解，然后将它应用于结构静力分析和设计中的不确定性问题，泛灰数不仅具有区间分析的功能，而且能解决区间分析所不能解决的问题。文中给出了两个算例，列出了本文算法与其他算法的结果比较。     

Fuzzy variational principle and its applications

Linear and non-linear peaky fuzzy numbers and their arithmetic operations are constructed for the analysis of engineering structures with fuzzy characteristic quantities. Fuzziness of the corresponding quantities is consistently incorporated into the functional of the total potential energy. A set of deterministic recursive equations is obtained as the alternative expressions of the fuzzy variational principle by means of the second-order perturbation technique. The fuzzy Ritz method and the fuzzy finite element method are presented as the applications of the fuzzy variational principle. Accordingly, the roundabout procedures frequently used in the formulations of the fuzzy finite element method are avoided. A benchmark problem of a bending beam with fuzzy Young's modulus under fuzzy external loading is solved by the developed fuzzy numerical methods. Numerical examples show that results determined by these two fuzzy methods are both little conservative, and are in good agreement with those obtained by the analytical method. Moreover, the fuzzy Ritz method or the fuzzy finite element method can provide more valuable information than the conventional deterministic methods.     

模糊桁架结构有限元分析的随机因子法

研究模糊桁架结构在模糊力作用下的有限元分析方法。考虑桁架结构材料物理参数、几何尺寸和外荷载同时为模糊变量，利用信息熵将模糊变量转变为随机变量。基于随机因子法，利用代数综合法推导出结构位移和应力响应的均值、方差的计算表达式。通过算例，分析了结构物理参数、几何尺寸和外荷载的模糊性对结构位移和应力响应的影响，并验证了本文模型和方法的合理性与可行性。本方法的优点是能够反映结构某一参数的模糊型对结构响应的影响。     

Effective numerical methods for elasto-plastic contact problems with friction

Several effective numerical methods for solving the elasto-plastic contact problems with friction are presented. First, a direct substitution method is employed to impose the contact constraint conditions on condensed finite element equations, thus resulting in a reduction by half in the dimension of final governing equations. Second, an algorithm composed of contact condition probes and elasto-plastic iterations is utilized to solve the governing equation, which distinguishes two kinds of nonlinearities, and makes the solution unique. In addition, Positive-Negative Sequence Modification Method is used to condense the finite element equations of each substructure and an analytical integration is introduced to determine the elasto-plastic status after each time step or each iteration, hence the computational efficiency is enhanced to a great extent. Finally, several test and practical examples are presented showing the validity and versatility of these methods and algorithms. The Project Supported by National Natural Science Foundation of China.     

随机杆系结构几何非线性分析的递推求解方法

建立了随机静力作用下考虑几何非线性的随机杆系结构的随机非线性平衡方程. 将和 位移耦合的随机割线弹性模量以及随机响应量表示为非正交多项式展开式，运用传统的摄动方法获 得了关于非正交多项式展式的待定系数的确定性的递推方程. 在求解了待定系数后，利用非 正交多项式展开式和正交多项式展开式的关系矩阵，可以很方便地得到未知响应量的二阶统计矩. 两杆结构和平面桁架拱的算例结果表明，当随机量涨落较大时，递推随机有限元方法比基于 二阶泰勒展开的摄动随机有限元方法更逼近蒙特卡洛模拟结果，显示了该方法对几何非线性 随机问题求解的有效性.     

Perturbational solutions for fuzzy-stochastic finite element equilibrium equations (FSFEEE)

I.IntroductionInthefieldofengineeringstructureanalysis,suchparametersasmodulusofmaterial-Poissonratio,geometricalsizesofstructure,boundaryconditionsandexternalloadingsprobablyhavefuzzinessor/andrandomness.Thestochasticfiniteelementmethod(SFEM)1'1disposeso…     

Equivalent method for accurate solution to linear interval equations

Based on linear interval equations, an accurate interval finite element method for solving structural static problems with uncertain parameters in terms of optimization is discussed.On the premise of ensuring the consistency of solution sets, the original interval equations are equivalently transformed into some deterministic inequations.On this basis, calculating the structural displacement response with interval parameters is predigested to a number of deterministic linear optimization problems.The results are proved to be accurate to the interval governing equations.Finally, a numerical example is given to demonstrate the feasibility and efficiency of the proposed method.     

DYNAMIC CHARACTERISTIC ANALYSIS OF FUZZY- STOCHASTIC TRUSS STRUCTURES BASED ON FUZZY FACTOR METHOD AND RANDOM FACTOR METHOD

A new fuzzy stochastic finite element method based on the fuzzy factor method and random factor method is given and the analysis of structural dynamic characteristic for fuzzy stochastic truss structures is presented. Considering the fuzzy randomness of the structural physical parameters and geometric dimensions simultaneously, the structural stiffness and mass matrices are constructed based on the fuzzy factor method and random factor method; from the Rayleigh's quotient of structural vibration, the structural fuzzy random dynamic characteristic is obtained by means of the interval arithmetic; the fuzzy numeric characteristics of dynamic characteristic are then derived by using the random variable's moment function method and algebra synthesis method. Two examples are used to illustrate the validity and rationality of the method given. The advantage of this method is that the effect of the fuzzy randomness of one of the structural parameters on the fuzzy randomness of the dynamic characteristic can be reflected expediently and objectively.     

An iterative parallel algorithm of finite element method

In this paper, a parallel algorithm with iterative form for solving finite element equation is presented. Based on the iterative solution of linear algebra equations, the parallel computational steps are introduced in this method. Also by using the weighted residual method and choosing the appropriate weighting functions, the finite element basic form of parallel algorithm is deduced. The program of this algorithm has been realized on the ELXSI-6400 parallel computer of Xi'an Jiaotong University. The computational results show the operational speed will be raised and the CPU time will be cut down effectively. So this method is one kind of effective parallel algorithm for solving the finite element equations of large-scale structures.     

基于Hamilton体系的弹性力学问题的比例边界有限元方法

比例边界有限元方法是求解偏微分方程的一种半解析半数值解法。对于弹性力学问题,可采用基于力学相似性、基于比例坐标相似变换的加权余量法和虚功原理得到以位移为未知量的系统控制方程,属于Lagrange体系。但在求解时,又引入了表面力为未知量,控制方程属于Hamilton体系。因而,本文提出在比例边界有限元离散方法的基础上,利...     

哈密顿体系在断裂力学Dugdale模型中的应用

利用平面扇形域哈密顿体系的方程，通过分离变量法及共轭辛本征函数向量展开法，以解析的方法推导出基于Dugdale模型的平面裂纹弹塑性解析元列式。将该解析元与有限元相结合，构成半解析的有限元法，可求解任意几何形状和荷载平板裂纹的Dugdale模型问题。数值计算结果表明本文方法对该类问题的求解是十分有效的，并有较高的精度。     

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.     

Modes computation and analysis for long multi-span bundle conductors of power transmission lines with galloping control devices

A new type of element which is suitable for solving the modes of the galloping long multi-span bundle conductor structures is presented. The element is composed of all sub-conductor segments between two spacers. Based on the linearized governing differential equations of the conductors, the mass matrix and stiffness matrix of the element in consideration of the constrained relations imposed on the conductors by spacers are derived. The dynamic characteristics of the galloping control devices can be directly added to the element. The modes for an actual power line structure are computed by using the element formula and FEM procedures, where seven cases of different galloping control device allocations are considered. Compared with the measured data, the method is shown to be reliable and effective. Analysis and discussions of the computational results are given. Some hints that are helpful to further investigation of galloping are also obtained.     

A NEW SOLUTION TO STRUCTURAL FUZZY FINITE ELEMENT EQUILIBRIUM EQUATIONS (SFFEEE)

I.IntroductionInthefieldofthefuzzyfiniteelementmethod(FFEM),tostudynow,tosolvetheequilibriumequationsisofmomentoussignificance.EspeciallyfortheapplicationoftheFFEMtoengineeringproblems,thekeyworktodoistoexplorehigh-speedandhigh-effeciencysolutionstoequili…     

自由单元法及其在结构分析中的应用

通过吸收有限元与无网格法的优点,提出了一种新的数值方法------自由单元法.此方法在离散方面,采用有限元法中的等参单元,表征几何形状和进行物理量的插值;在算法方面,采用单元配点技术,逐点产生系统方程.主要特点是,在每个配置点只需要一个和周围自由选择的节点而形成的一个独立的等参单元,因而不需要考虑物理量在单元之间的相互连接关系与导数连续性问题. 本文介绍强形式与弱形式两种自由单元法,前者直接由控制方程和边界条件直接产生系统方程,后者通过在自由单元上建立控制方程的加权余量式产生弱形式积分式,并通过像传统有限元法中的积分过程建立系统方程组.本文提出的方法是一种单元配点法,对于域内点为了获得较高的导数精度,需要采用至少具有一个内部点的等参单元,为此除了可使用各阶次的拉格朗日四边形单元外, 还 给出了七节点三角形等参单元,用于模拟较为复杂的几何形状问题.      

Efficient stochastic finite element methods for flow in heterogeneous porous media. Part 2: Random lognormal permeability

Efficient and robust iterative methods are developed for solving the linear systems of equations arising from stochastic finite element methods for single phase fluid flow in porous media. Permeability is assumed to vary randomly in space according to some given correlation function. In the companion paper, herein referred to as Part 1, permeability was approximated using a truncated Karhunen-Loève expansion (KLE). The stochastic variability of permeability is modeled using lognormal random fields and the truncated KLE is projected onto a polynomial chaos basis. This results in a stochastic nonlinear problem since the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables. Symmetric block Gauss-Seidel used as a preconditioner for CG is shown to be efficient and robust for stochastic finite element method.