基于单元的子区间摄动有限元方法研究

针对区间相关性导致区间扩张的问题,探讨了区间数之间的相关性并给出了降低区间扩张的子区间摄动方法。文中给出了基于单元的子区间摄动有限元计算公式和子区间划分数目的近似计算公式,同时文中讨论了区间有限元计算精度问题,给出了可提高计算效率的一些措施。对桁架结构和平面应力问题梁结构算例分析结果表明文中方法可以达到一定的计算精度,并且是合理可行的。     

FUZZY ARITHMETIC AND SOLVING OF THE STATIC GOVERNING EQUATIONS OF FUZZY FINITE ELEMENT METHOD

The key component of finite element analysis of structures with fuzzy parameters, which is associated with handling of some fuzzy information and arithmetic relation of fuzzy variables , was the solving of the governing equations of fuzzy finite element method. Based on a given interval representation of fuzzy numbers, some arithmetic rules of fuzzy numbers and fuzzy variables were developed in terms of the properties of interval arithmetic. According to the rules and by the theory of interval finite element method, procedures for solving the static governing equations of fuzzy finite element method of structures were presented. By the proposed procedure, the possibility distributions of responses of fuzzy structures can be generated in terms of the membership functions of the input fuzzy numbers. It is shown by a numerical example that the computational burden of the presented procedures is low and easy to implement. The effectiveness and usefulness of the presented procedures are also illustrated.     

基于区间分析的工程结构不确定性研究现状与展望

随机分析方法、模糊分析方法是已经广泛使用的工程结构不确定性分析方法, 近年来区间分析方法逐渐为人们所熟知并成为是一种新的工程结构不确定性分析方法,它主要用来研究具有区间特性的工程结构. 区间分析方法在统计信息不足以描述不确定参数的概率分布或隶属函数、工程单位仅提供不确定参数的区间范围而想获得结构响应的区间范围时就发挥了其优点. 综述了区间分析方法及其在工程结构不确定性分析中的应用状况, 将基于区间分析的工程结构不确定性问题研究归结为以下4个方面: 不确定性结构系统的区间有限元分析; 基于区间的非概率可靠性分析; 工程结构区间反演分析; 基于区间参数的结构优化设计. 分析评价了国内外在这几个方面的研究成果及其最新进展, 同时指出目前研究中存在的问题和研究的方向.      

Study on spline wavelet finite-element method in multi-scale analysis for foundation

A new finite element method （FEM） of B-spline wavelet on the interval （BSWI） is proposed. Through analyzing the scaling functions of BSWI in one dimension, the basic formula for 2D FEM of BSWI is deduced. The 2D FEM of 7 nodes and 10 nodes are constructed based on the basic formula. Using these proposed elements, the multiscale numerical model for foundation subjected to harmonic periodic load, the foundation model excited by external and internal dynamic load are studied. The results show the pro- posed finite elements have higher precision than the tradi- tional elements with 4 nodes. The proposed finite elements can describe the propagation of stress waves well whenever the foundation model excited by extemal or intemal dynamic load. The proposed finite elements can be also used to con- nect the multi-scale elements. And the proposed finite elements also have high precision to make multi-scale analysis for structure.     

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.     

区间可靠性分析方法及在地下隧道结构计算中的应用

基于工程结构不确定性的区间分析方法,本文将区间分析方法与可靠性分析方法相结合,探讨了一种可以获得区问可靠指标的可靠性分析方法.依据结构失效准则确定的功能函数在一定区间内变化,进而得出了可靠指标的变化区间,在得到区间可靠指标的同时也得到了一种反映结构稳健性的稳健可靠指标.结合区间有限元的优化计算方法,对某地下隧道结构进行了区间可靠性分析,所得区间可靠性指标合乎规律.     

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。     

Coupling of high order multiplication perturbation method and reduction method for variable coefficient singular perturbation problems

Based on the precise integration method （PIM）, a coupling technique of the high order multiplication perturbation method （HOMPM） and the reduction method is proposed to solve variable coefficient singularly perturbed two-point boundary value prob lems （TPBVPs） with one boundary layer. First, the inhomogeneous ordinary differential equations （ODEs） are transformed into the homogeneous ODEs by variable coefficient dimensional expansion. Then, the whole interval is divided evenly, and the transfer ma trix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efficient.     

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。     

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.     

On the use of characteristic‐based split meshfree method for solving flow problems

This study presents characteristic‐based split (CBS) algorithm in the meshfree context. This algorithm is the extension of general CBS method which was initially introduced in finite element framework. In this work, the general equations of flow have been represented in the meshfree context. A new finite element and MFree code is developed for solving flow problems. This computational code is capable of solving both time‐dependent and steady‐state flow problems. Numerical simulation of some known benchmark flow problems has been studied. Computational results of MFree method have been compared to those of finite element method. The results obtained have been verified by known numerical, analytical and experimental data in the literature. A number of shape functions are used for field variable interpolation. The performance of each interpolation method is discussed. It is concluded that the MFree method is more accurate than FEM if the same numbers of nodes are used for each solver. Meshfree CBS algorithm is completely stable even at high Reynolds numbers. Copyright © 2007 John Wiley & Sons, Ltd.     

基于单元的静力区间有限元法

在许多工程问题中 ,结构参数和荷载具有某种程度的误差或不确定性。若不将它们定量化或模型化加以考虑 ,就不能作出合理的分析和设计。考虑到有限元法在科学界和工程界的广泛应用 ,本文以连续梁结构为例 ,建立了基于单元的静力区间有限元法。为了说明本方法的有效性 ,本文给出了一个数值例子 ,并把所得结果与文献 [1 2 ]进行了比较。     

自然单元法研究进展

自然单元法是一种基于Voronoi图和Delaunay三角化几何结构,以自然邻点插值为试函数的一种新型数值方法.其既具有无网格方法和经典有限元方法的优点,又克服了两者的一些缺陷,是一种发展前景广阔的求解微分方程的数值方法.自然单元法的形函数满足插值性质,可以像有限元法一样直接施加本质边界条件,不存在基于移动最小二乘拟合的无网格方法不能直接施加本质边界条件的难题.由于自然单元法是无网格方法,可以方便处理有限元方法较难处理的一些问题,例如移动边界和大变形等问题.自然单元法与其他数值方法的最根本区别于其插值格式的不同.将自然邻点插值用于Galerkin过程,就得到基于Voronoi结构的自然单元Galerkin法.自然邻点插值有自然邻点Sibson插值和Laplace插值(非Sibson插值)两种.Laplace插值比Sibson插值在计算上要简单的多,并且不论对凸的或非凸的区域都能精确施加本质边界条件.以Laplace插值为试函数的自然单元法在数值实施上比以Sibson插值为试函数的自然单元法简单.本文对基于Voronoi结构的自然邻点插值和自然单元法的基本思想作了介绍,综述了国内外关于自然单元法的研究成果,总结了自然单元法的优点和尚需解决的问题.      

Flexural wave bandgap properties of phononic crystal beams with interval parameters

Uncertainties are unavoidable in practical engineering, and phononic crystals are no exception. In this paper, the uncertainties are treated as the interval parameters,and an interval phononic crystal beam model is established. A perturbation-based interval finite element method(P-IFEM) and an affine-based interval finite element method(A-IFEM) are proposed to study the dynamic response of this interval phononic crystal beam, based on which an interval vibration transmission analysis can be easi...     

Self-adaptive one-dimensional nonlinear finite element method based on element energy projection method

The element energy projection （EEP） method for computation of super- convergent resulting in a one-dimensional finite element method （FEM） is successfully used to self-adaptive FEM analysis of various linear problems, based on which this paper presents a substantial extension of the whole set of technology to nonlinear problems. The main idea behind the technology transfer from linear analysis to nonlinear analysis is to use Newton＇s method to linearize nonlinear problems into a series of linear problems so that the EEP formulation and the corresponding adaptive strategy can be directly used without the need for specific super-convergence formulation for nonlinear FEM. As a re- sult, a unified and general self-adaptive algorithm for nonlinear FEM analysis is formed. The proposed algorithm is found to be able to produce satisfactory finite element results with accuracy satisfying the user-preset error tolerances by maximum norm anywhere on the mesh. Taking the nonlinear ordinary differential equation （ODE） of second-order as the model problem, this paper describes the related fundamental idea, the imple- mentation strategy, and the computational algorithm. Representative numerical exam- ples are given to show the efficiency, stability, versatility, and reliability of the proposed approach.     

The finite element method based on interpolating with wavelet basis function

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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.     

Velocity–pressure integrated versus penalty finite element methods for high-Reynolds-number flows

Velocity–pressure integrated and consistent penalty finite element computations of high-Reynolds-number laminar flows are presented. In both methods the pressure has been interpolated using linear shape functions for a triangular element which is contained inside the biquadratic flow element. It has been shown previously that the pressure interpolation method, when used in conjunction with the velocity-pressure integrated method, yields accurate computational results for high-Reynolds-number flows. It is shown in this paper that use of the same pressure interpolation method in the consistent penalty finite element method yields computational results which are comparable to those of the velocity–pressure integrated method for both the velocity and the pressure fields. Accuracy of the two finite element methods has been demonstrated by comparing the computational results with available experimental data and/or fine grid finite difference computational results. Advantages and disadvantages of the two finite element methods are discussed on the basis of accuracy and convergence nature. Example problems considered include a lid-driven cavity flow of Reynolds number 10 000, a laminar backward-facing step flow and a laminar flow through a nest of cylinders.     

一种分析区间参数结构可靠性的减基快速计算方法

将减基法与蒙特卡洛模拟结合,提出了一种快速计算区间不确定结构可靠性的方法。该方法分为离线和在线计算两个阶段,离线阶段利用减基法建立减基空间,进而形成减缩模型;而在线阶段则将减缩模型融入蒙特卡洛方法,进而快速求解区间失效概率及相应的区间参数失效域。该方法在减基空间而非有限元空间中分析区间结构的可靠性,减少了求解时间,提高了计算效率,通过算例验证了该方法的可行性和有效性。     

多变量样条有限元法

本文提出了一种基于胡海昌－鹫津久一郎三类变量广义变分原理［１，２］的多变量样条有限元法，文中应用乘积型二元三次Ｂ样条插值函数来构造板壳的广义变量场函数，由三类变量广义变分原理来建立多变理样有限元模型。在计算种种场变量时，既不必求导，又无需用应力应变关系式，可直接算得其结果，因而对各种场变量均有足够的精度，文中，还解算了振动与稳定特征值问题，均得到了精度较好的数值结果。