A wavelet-based stochastic finite element method of thin plate bending

A wavelet-based stochastic finite element method is presented for the bending analysis of thin plates. The wavelet scaling functions of spline wavelets are selected to construct the displacement interpolation functions of a rectangular thin plate element and the displacement shape functions are expressed by the spline wavelets. A new wavelet-based finite element formulation of thin plate bending is developed by using the virtual work principle. A wavelet-based stochastic finite element method that combines the proposed wavelet-based finite element method with Monte Carlo method is further formulated. With the aid of the wavelet-based stochastic finite element method, the present paper can deal with the problem of thin plate response variability resulting from the spatial variability of the material properties when it is subjected to static loads of uncertain nature. Numerical examples of thin plate bending have demonstrated that the proposed wavelet-based stochastic finite element method can achieve a high numerical accuracy and converges fast.     

基于参数变分原理的非均质材料弹塑性有限元分析的Voronoi单元法

在非均质材料的有限元数值模拟中,采用了Voronoi单元(VCFEM)以克服经典位移元的局限性．基于参数变分原理和二次规划法进行了Voronoi单元的二维弹塑性分析A·D2推导了有限元列式并形成最终的二次规划求解模型．研究了非均质材料微观夹杂对整体力学性能的影响．数值算例证明了该方法的正确和可行性．     

对流-扩散问题的Galerkin部分迎风有限元方法

时(其中h表示典型的网格尺寸),将会出现数值解的伪振荡.为了克服这种数值不稳定性,人们提出了多种解决途径,例如采用迎风型的差分格式.Zienkiewicz等人首先提出用Petrov-Galerkin有限元法求解对流-扩散问题.他们通过分别选择解空间和检验函数空间,克服了数值不稳定性.但这类方法由于解空间和检验函数空间的基函数比较     

The discrete maximum principle for linear simplicial finite element approximations of a reaction-diffusion problem

This paper provides a sufficient condition for the discrete maximum principle for a fully discrete linear simplicial finite element discretization of a reaction-diffusion problem to hold. It explicitly bounds the dihedral angles and heights of simplices in the finite element partition in terms of the magnitude of the reaction coefficient and the spatial dimension. As a result, it can be computed how small the acute simplices should be for the discrete maximum principle to be valid. Numerical experiments suggest that the bound, which considerably improves a similar bound in [P.G. Ciarlet, P.-A. Raviart, Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Eng. 2 (1973) 17-31.], is in fact sharp.     

基面力单元法在空间几何非线性问题中的应用

基于基面力的概念,并结合Euler角的位移描述方法,提出了适用于几何非线性计算的空间6结点余能基面力单元.使用MATLAB语言编程并对典型梁、板结构进行弹性大变形数值模拟.由计算结果可以看出,基于余能原理的基面力元法(BFEM)在计算构件的空间大变形时有较好的计算精度,对比传统有限元计算方法具有网格尺寸影响小和抗畸变能力强的特点,有良好的计算性能.     

THE UPWIND FINITE ELEMENT SCHEME AND MAXIMUM PRINCIPLE FOR NONLINEAR CONVECTION-DIFFUSION PROBLEM

In this paper, a kind of partial upwind finite element scheme is studied for twodimensional nonlinear convection-diffusion problem. Nonlinear convection term approximated by partial upwind finite element method considered over a mesh dual to the triangular grid, whereas the nonlinear diffusion term approximated by Galerkin method. A linearized partial upwind finite element scheme and a higher order accuracy scheme are constructed respectively. It is shown that the numerical solutions of these schemes preserve discrete maximum principle. The convergence and error estimate are also given for both schemes under some assumptions. The numerical results show that these partial upwind finite element scheme are feasible and accurate.     

弹性厚板的分区广义变分原理

本文提出弹性厚板分区广义变分原理,其要点如下:1.各分区可任意定为势能区或余能区.分区势能、分区余能、分区混合变分原理是它的三种特殊形式.2.每个分区中独立变分变量的个数可任意规定.每个分区可定为单类变量区、二类变量区或三类变量区.3.每个交界线上的位移和力的连接条件可以放宽.这个原理为非协调元的厚板有限元法提供理论基础.各种厚板有限元模型可看作这个原理的特殊应用.特别是弹性厚板分区混合变分原理的提出为分区混合有限元法应用于厚板问题打下了基础.     

更一般的杂交广义变分原理及有限元模型

本文根据钱伟长教授提出的更一般的广义变分原理,给出了适用于有限元法中的更广义杂交变分原理,并由此建立了新的广义杂交模理. 进一步以变厚度薄板弯曲单元为例,对基于各种不同的广义变分原理建立的各种杂交元做了比较.     

Richard courant and the finite element method: A further look

The finite element method has in recent years become one of the most popular and effective numerical procedures for boundary value and eigenvalue problems. In the early 1960s it was found that the method was an independent rediscovery of a simpler idea proposed in 1943 by Richard Courant. The importance of Courant's 1943 paper has been emphasized by G. Strang in 1973 and by others. This note seeks to describe briefly Courant's finite element work which led to his publication of 1943. It is shown that Courant used the finite element ideas as early as 1922 in a proof employing Dirichlet's principle.     

Hybrid stress finite element formulation based on additional internal displacements – application of microscopic geometric perturbation and extension to linear dynamics

Based on the variational principle for hybrid stress finite element method, the assumed stress distribution can be determined through a variational process which includes a microscopic geometric perturbation. An interpretation of such geometric perturbation is discussed. For the creation of, what is called, hyper-bioelement and for the purpose of maintaining invariance, tensor notations of the physical quantities are represented by the use of covariant and contravariant basis vectors of the natural coordinate system. This process is illustrated by a simple 4-node plane element. Furthermore, based on the legitimate variational principle for linear dynamics and the presently proposed method to choose the assumed stress distribution, the finite element formulation different from the ordinary formula is advocated.     

Stability analysis of frame structures by bubble-function method

In the stability analysis of frame structures, the results by conventional finite element method (FEM) in which one member is taken as one element are sometimes unavailable. This paper took a new basic function system with bubble functions as the shape function of a bar element to develop a bubble function finite element method (BFEM), in which the bending and the geometric stiffness matrices were derived from the principle of virtual work. Bubble functions are finite element modes that are located entirely within a single element and are zero on boundaries of the element, but are nonzero at the other points. BFEM is as concise as conventional bar FEM but has better accuracy, and is adaptable to the buckling analysis of all kinds of frame structures. The use of bubble functions significantly improves the convergence of finite element analysis, and efficiently reduces the computation cost for the buckling analysis of frame structures. Numerical results show that using bubble functions in finite element for the stability analysis of structures is very efficient, especially for high-rise and large-scale frame structures.     

瞬变温度场问题的有限元解法和最大模原理

在二维区域Ω(边界为?Ω)上,描述变温过程的抛物型方程及其初边值条件为式中,T为温度,α为导温系数,Q(x,y,t)为相应于热源的已知函数,T_0(x,y)为初始温度,g(x,y,t)为已知的边界温度. 用有限元法求解瞬变温度场问题,通常从变分原理出发,假设温度对时间的导数与温度一样为分片线性函数.它导出的热容量矩阵是非对角型的,我们称之为算法I.采用     

Convergence of Three-Step Taylor Galerkin Finite Element Scheme Based Monotone Schwarz Iterative Method for Singularly Perturbed Differential-Difference Equation

Monotone Schwarz iterative methods for parabolic partial differential equations are well known for their advantage of eliminating the search for an initial solution. In this article, we propose a monotone Schwarz iterative method for singularly perturbed parabolic retarded differential-difference equations based on a three-step Taylor Galerkin finite element scheme. The stability and ε-uniform convergence of the three-step Taylor Galerkin finite element method have been discussed. Further, by using maximum principle and induction hypothesis, the convergence of the proposed monotone Schwarz iterative method has been established.     

Numerical analysis of incompressible wormhole propagation with mass-preserving characteristic mixed finite element procedure

In this article, we construct a new combined characteristic mixed finite element procedure to simulate the incompressible wormhole propagation. In this procedure, we use the classical mixed finite element method to solve the pressure equation and a modified mass-preserving characteristic finite element method for the solute transport equation, and solve the porosity function straightly by the given concentration. This combined method not only keeps mass balance globally but also preserves maximum principle for the porosity. We considered the corresponding convergence and derive the optimal L²-norm error estimate. Finally, we present some numerical examples to confirm theoretical analysis.

     

A note on the determination of influence lines and surfaces using finite elements

This paper presents a simple yet general procedure to determine influence lines and surfaces for frames, beams, trusses, and plates. The method is based on the application of the Muller-Breslau principle and finite elements. It can be easily implemented with most commercial finite element codes.     

应力协调迭代法在高速冲击动力有限元中的应用

在EPIC[2][3],NONSAP[4]等弹塑性撞击动力有限元程序中,有一个共同的弱点是都采取了静力有限元方法,把位移函数用线性插值表示.单元之间应力是非协调的.因此应用虚功原理的基础不合理.为了克服以上困难,本文引入一个新的方法,即协调应力迭代法.实例表明,这种方法在冲击动力有限元计算中是稳定和精确的,同时具有减小单元刚度的作用.     

广义变分原理在有限元半分析法中的应用

本文是在文献[1]中所指出的广义变分原理在合理处理有限元法的边限条件的应用价值还没有受到足够的重视这一思想启发下,应用广义变分原理,选用样条函数与正弦(或余弦)函数乘积型的级数形式再加上多项式,作为板壳的逼近函数,以薄板弯曲问题为例较好地解决了有限元半分析法中出现的耦联问题.由于其未知数个数比有限元法、有限条法均少很多,而精度更高,故为用微机解决一类工程问题,提供了一个有效的方法.     

Combined higher order finite volume and finite element scheme for double porosity and non-linear adsorption of transport problem in porous media

In this work, a dual porosity model of reactive solute transport in porous media is presented. This model consists of a nonlinear-degenerate advection-diffusion equation including equilibrium adsorption to the reaction combined with a first-order equation for the non-equilibrium adsorption interaction processes. The numerical scheme for solving this model involves a combined high order finite volume and finite element scheme for approximation of the advection-diffusion part and relaxation-regularized algorithm for nonlinearity-degeneracy. The combined finite volume-finite element scheme is based on a new formulation developed by Eymard et al. (2010) [10]. This formulation treats the advection and diffusion separately. The advection is approximated by a second-order local maximum principle preserving cell-vertex finite volume scheme that has been recently proposed whereas the diffusion is approximated by a finite element method. The result is a conservative, accurate and very flexible algorithm which allows the use of different mesh types such as unstructured meshes and is able to solve difficult problems. Robustness and accuracy of the method have been evaluated, particularly error analysis and the rate of convergence, by comparing the analytical and numerical solutions for first and second order upwind approaches. We also illustrate the performance of the discretization scheme through a variety of practical numerical examples. The discrete maximum principle has been proved.     

一种新型的高次多结点薄壁杆件单元

应用Hermite插值多项式以及最小势能原理导出了一种新型的高次多结点薄壁杆件单元,这种单元的特点是精度高,并可应用于桥梁、高层建筑巨型结构的有限元分析中.     

连续体力学中有限变形与转动的计算增量法

目前在非线性弹塑性力学计算中常用的经典非线性大变形理论由于内在的数学缺点,当变形量与转动很大时.往往误差达到不许可的程度.本文采用作者的有限变形力学理论表述了增量法.在作者与尚勇、谢和平联合研究的另二篇论文中,详细叙述这个新方法在工程中的应用,结果证明从微小变形过渡到大变形,计算结果总是可以满意地符合实验.