流体力学变分原理及有限元法研究的进展

本文对流体力学变分原理的发展,特别是近二十来年连同育限元的发展与现状,作一简要的综合评述,并展望今后的发展方向,提供若干参考意见。     

基于位移型Gurtin变分原理计算动力响应的逐步积分法

本文利用位移型Ｇｕｒｔｉｎ变分原理,在时间域上采用三次Ｈｅｒｍｉｔｅ插值函数进行离散,给出了一种计算结构动力响应的逐步积分方法。通过稳定性分析研究了该方法的稳定区情况表明,当１．６４≤θ≤２．０８时,该方法的数值计算精度很高,但是条件稳定积分格式。当θ≥４．１时,该方法是无条件稳定的积分格式,精度较高。     

由Gurtin变分原理求解一维动力响应的半解析法

基于Gurtin变分原理本文提出了求解动力响应问题的半解析法.该方法在空间域内进行有限元离散,在时间域上取级数。实例表明,该方法是求解动力响应问题的一种非常有效的方法。本文引入了一种节点位移函数,使所导出的位移模式可以求解各种初始条件和载荷的一维动力响应问题。     

Gurtin变分原理及其应用的时间有限元法

动力学Ｇｕｒｔｉｎ变分原理完整地表征了动力学的全部特征。由于在Ｇｕｒｔｉｎ变分原理的泛函中含有双重卷积，给时间域的离散带来很大困难。本文通过在Ｌａｐｌａｃｅ空间构造泛函，获得了几个具有单重卷积的Ｇｕｒｔｉｎ原理。由于卷积降阶，所给出的泛函更加便于应用。本文还通过在时间域采用适当的插值多项式逼近广义节点坐标，进一步讨论了时间有限元法实施的基本原理和步骤。     

随机有限元动力分析方法的研究进展

对随机结构动力分析方法,包括随机模拟法、摄动随机有限元法、动态随机有限元法和正交展开法等进行了评述,介绍了该领域内的研究进展,并讨论了研究中存在的问题以及发展方向．      

弯曲梁随机刚度有限元法

弯曲梁的随机材料参数和随机几何参数可结合为一个随机参数-弯曲刚度。以位移的均值和协方差函数的变分原理为基础,根据基本方程推导出具有随机刚度的静定弯曲梁的有限元法。这种方法不再采用摄动技术,可直接建立位移的均值和协方差函数。     

基于信息熵的模糊随机结构有限元法

本文利用信息熵的概念,将不确定性（模糊的、随机的）变量统一为随机变量,将模糊随机结构形式上视为随机结构进行处理,从而提出了不确定性结构有限元分析的一种新方法。当不确定结构转换的等效随机变量处于小扰动情况下,利用摄动法得到有限元递归方程组,解之可以得到响应量的均值和方差。     

非饱和多孔介质有限元分析的基本控制方程与变分原理

本文在对问题研究现状进行阐述的基础上较系统地给出了骨架可变形非饱和多孔介质的全耦合分析模型，模型中考虑了孔隙气体，水（油）流动对介质力学性能的影响，多孔介质的饱和度，渗透系数与毛吸压力的关系，由实验给出，所导出的控制方程以固体骨架的位移与孔隙流体压力为基本未知量，由于问题的非自共轭特征，文中构造了非饱和介质动力问题的参数变分形式，并在此基础上给出有限元离散方程。     

The random variational principle and finite element method

In this paper, we introduced the random materials, geometrical shapes, force and displacement boundary condition directly into the functional variational formulations and developed a unified random variational principle and finite element method with the small parameter perturbation method. Numerical examples showed that the methods have the advantages of the simple and convenient program implementation, and are effective for the random mechanics problems.     

粘弹性随机分析的虚功原理及随机有限元

利用增量法处理粘弹性本构关系中的遗传积分，将粘弹性材料的随机性、结构几何形状的随机性、外载荷的随机性引入虚功方程，应用摄动方法，研究了粘弹性随机分析的虚功原理和粘弹性随机有限元。研究发现，尽管粘弹性本构关系具有时间相依性，其随机摄动格式并不存在“长期项”的影响，算例表明，应用该方法进行粘弹性结构的随机模拟，计算效率较高、精度较高。     

PARAMETRIC VARIATIONAL PRINCIPLE BASED ELASTIC-PLASTIC ANALYSIS OF HETEROGENEOUS MATERIALS WITH VORONOI FINITE ELEMENT METHOD

The Voronoi cell finite element method (VCFEM) is adopted to overcome the limitations of the classic displacement based finite element method in the numerical simulation of heterogeneous materials. The parametric variational principle and quadratic programming method are developed for elastic-plastic Voronoi finite element analysis of two-dimensional problems. Finite element formulations are derived and a standard quadratic programming model is deduced from the elastic-plastic equations. Influence of microscopic heterogeneities on the overall mechanical response of heterogeneous materials is studied in detail. The overall properties of heterogeneous materials depend mostly on the size, shape and distribution of the material phases of the microstructure. Numerical examples are presented to demonstrate the validity and effectiveness of the method developed.     

On the local average of random field in stochastic finite element analysis

The distinguishing feature of stochastic finite element analysis is that it involves the discretization of the parameter space of random fields of material properties, the geometry of structure and / or the loads. It is shown in earlier investigations that a reasonable procedure of discretization is to take the local averages of the random fields on each element. In the present paper the formulae for the covariance of the local averages of a homogeneous random vector field on rectangular elements are generalized by relaxing the condition. For an inhomogeneous random field and /or non-rectangular elements, a procedure of using Gaussian quadrature to evaluate the means and covariances of the local averages is proposed. Thus, the stochastic finite element method (SFEM) based on the local averages of random fields is adapted to a structure with irregular shape and / or inhomogeneous random fields. The effects of the mesh geometry, the ratio of element size to the correlation scale as well as the number of Gaussian quadrature points on the convergence of SFEM are discussed. It is found that even better results could be obtained by utilizing appropriate Gaussian quadrature instead of exact local average.Project supported by National Natural Science Foundation of China.     

The random variational principle in finite deformation of elasticity and finite element method

ＴＨＥＲＡＮＤＯＭＶＡＲＩＡＴＩＯＮＡＬＰＲＩＮＣＩＰＬＥＩＮＦＩＮＩＴＥＤＥＦＯＲＭＡＴＩＯＮＯＦＥＬＡＳＴＩＣＩＴＹＡＮＤＦＩＮＩＴＥＥＬＥＭＥＮＴＭＥＴＨＯＤＧａｏＨａｎｇ－ｓｈａｎ（高行山）（ＮｏｒｔｈｗｅｓｔｅｎＰｏｌｙｔｅｃｈｎｉｃａｌＵ...     

随机结构有限元分析的递推求解方法

提出了一种新的谱随机有限元分析方法——递推求解方法。该方法将随机结构的随机响应表示成非正交多项式展式,建立了和摄动法类似的一系列确定的递推方程,并通过确定性有限元方法对这些递推方程进行静力问题求解。算例表明,当随机量出现较大涨落时,计算结果相对于传统摄动法有不小的改进。     

Stochastic finite element analysis of local average random fields

This paper discussed the concepts underlying a new approach to representation and analysis of homogeneous random Fields. An orthogonalized isoparametric local average model of random fields is proposed, and the optimum selection of random variables is determined based on the theory of nonlinear programming. The finite element method is used to develop a methodology for random field analysis. Numerical examples showing the accuracy and efficiency of the method are studied.     

Stochastic finite element method based on local averages of random fields

The stochastic finite element method (SFEM) based on the local, averages of random fields, which was proposed in [5], is now generalized to analyze the structures with several correlated random parameters. The covariance matrix of the local averages of a random vector field is derived. The SFEM based on the local averages of random vector fields is formulated. The numerical examples show that the generalized SFEM preserves the advantages of the original one, i. e., rapid convergence, good accuracy and insensitivity to the correlation structures of random parameters.Project supported by National Natural Science Foundation of China.     

The optimal control variational principle and finite elements analysis for viscoplastic dynamics

This paper presents the optimal control variational principle for Perzyna model which is one of the main constitutive relation of viscoplasticity in dynamics. And it could also be transformed to solve the parametric quadratic programming problem. The FEM form of this problem and its implementation have also been discussed in the paper.     

多自由度非线性随机参数振动系统响应分析的概率摄动有限元法

提出了一般概率摄动有限元法，并用以解决了具有向量值和矩阵值函数的多自由度非线性随机结构系统承受随机激励的响应分析问题，应用Kronecker代数，矩阵微分理论，向量值和矩阵值函数的二阶矩技术，矩阵摄动理论和概率统计方法系统地扩展了国际上通用的随机有限元法，随机变量和系统导数很方便地排列到二维矩阵中，得到了优美的数学表达式。