THE SUPERCLOSENESS OF THE FINITE ELEMENT METHOD FOR A SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEM ON A BAKHVALOV-TYPE MESH IN 2D

For singularly perturbed convection-diffusion problems,supercloseness analysis of the finite element method is still open on Bakhvalov-type meshes,especially in the case of 2D.The difficulties arise from the width of the mesh in the layer adjacent to the transition point,resulting in a suboptimal estimate for convergence.Existing analysis techniques cannot handle these difficulties well.To fill this gap,here a novel interpolation is designed delicately for the smooth part of the solution,bringing about the optimal supercloseness result of almost order 2 under an energy norm for the finite element method.Our theoretical result is uniform in the singular perturbation parameterεand is supported by the numerical experiments.

Finite Element Analysis for Singularly Perturbed Advection-Diffusion Robin Boundary Values Problem

A singularly perturbed advection-diffusion two-point Robin boundary value problem whose solution has a single boundary layer is considered. Based on the piecewise linear polynomial approximation, the finite element method is applied to the problem. Estimation of the error between solution and the finite element approximation is given in energy norm on shishkin-type mesh.

PERTURBATIONAL SOLUTIONS FOR FUZZY-STOCHASTIC FINITE ELEMENT EQUILIBRIUM EQUATIONS (FSFEEE)

In this paper, the random interval equilibrium equations (RIEE) is obtained by lambda-level cutting the fuzzy-stochastic finite element equilibrium equations (FSFEEE). Based on the relations between the variables of equilibrium equations, solving RIEE is transformed into solving two kinds of general random equilibrium equations (GREE). Then the recursive equations of evaluating the random interval displacement is derived from the small-parameter perturbation theory. The computational formulae of statistical characteristic of the fuzzy random displacements, the fuzzy random strains and the fuzzy random stresses are also deduced in detail.

FINITE ELEMENT DISPLACEMENT PERTURBATION METHOD FOR GEOMETRIC NONLINEAR BEHAVIORS OF SHELLS OF REVOLUTION OVERALL BEDING IN A MERIDIONAL PLANE AND APPLICATION TO BELLOW (Ⅱ)

The finite_element_displacement_perturbation method (FEDPM)for the geometric nonlinear behaviors of shells of revolution subjected to pure bending moments or lateral forces in one of their meridional planes (Ⅰ) was employed to calculate the stress distributions and the stiffness of the bellows. Firstly, by applying the first_order perturbation solution (the linear solution)of the FEDPM to the bellows, the obtained results were compared with those of the general solution and the initial parameter integration solution proposed by the present authors earlier, as well as of the experiments and the FEA by others.It is shown that the FEDPM is with good precision and reliability, and as it was pointed out in (Ⅰ) the abrupt changes of the meridian curvature of bellows would not affect the use of the usual straight element. Then the nonlinear behaviors of the bellows were discussed. As expected, the nonlinear effects mainly come from the bellows ring plate,and the wider the ring plate is, the stronger the nonlinear effects are. Contrarily, the vanishing of the ring plate, like the C_shaped bellows, the nonlinear effects almost vanish. In addition, when the pure bending moments act on the bellows, each convolution has the same stress distributions calculated by the linear solution and other linear theories, but by the present nonlinear solution they vary with respect to the convolutions of the bellows. Yet for most bellows, the linear solutions are valid in practice.

FINITE ELEMENT DISPLACEMENT PERTURBATION METHOD FOR GEOMETRIC NONLINEAR BEHAVIORS OF SHELLS OF REVOLUTION OVERALL BENDING IN A MERIDIONAL PLANE AND APPLICATION TO BELLOWS (Ⅰ)

In order to analyze bellows effectively and practically, the finite_element_displacement_perturbation method (FEDPM) is proposed for the geometric nonlinear behaviors of shells of revolution subjected to pure bending moments or lateral forces in one of their meridional planes. The formulations are mainly based upon the idea of perturba_ tion that the nodal displacement vector and the nodal force vector of each finite element are expanded by taking root_mean_square value of circumferential strains of the shells as a perturbation parameter. The load steps and the iteration times are not as arbitrary and unpredictable as in usual nonlinear analysis. Instead, there are certain relations between the load steps and the displacement increments, and no need of iteration for each load step. Besides, in the formulations, the shell is idealized into a series of conical frusta for the convenience of practice, Sander's nonlinear geometric equations of moderate small rotation are used, and the shell made of more than one material ply is also considered.

Finite volume element method for analysis of unsteady reaction-diffusion problems

A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the high-gradient boundary layers.

Review: Recent Developments in the Non-Probabilistic Finite Element Analysis

Generally, the finite element analysis of a structure is completed under deterministic inputs.However,uncertainties corresponding to geometrical dimensions,material properties, boundary conditions cannot be neglected in engineering applications. The probabilistic methods are the most popular techniques to handle these uncertain parameters but subjective results could be obtained if insufficient information is unavailable. Non-probabilistic methods can be alternatively employed,which has led to the procedures for nonprobabilistic finite element analysis. Each non-probabilistic finite element analysis method consists of two individual parts,including the core algorithm and pre-processing procedure. In this context,three types of algorithms and two typical pre-processing procedures as well as their effectiveness are described in detail,based on which novel hybrid algorithms can be conceived for the specific problems and the future work in this research field can be fostered.

Uniform Convergence Analysis for Singularly Perturbed Elliptic Problems with Parabolic Layers

In this paper, using Lin＇s integral identity technique, we prove the optimal uniform convergence θ（Nx^-2ln^2Nx＋Ny^-2ln^2Ny） in the L^2-norm for singularly perturbed problems with parabolic layers. The error estimate is achieved by bilinear finite elements on a Shishkin type mesh. Here Nx and Ny are the number of elements in the x- and y-directions, respectively. Numerical results are provided supporting our theoretical analysis.

Analysis of a Streamline-Diffusion Finite Element Method on Bakhvalov-Shishkin Mesh for Singularly Perturbed Problem

In this paper,a bilinear Streamline-Diffusion finite element method on Bakhvalov-Shishkin mesh for singularly perturbed convection–diffusion problem is analyzed.The method is shown to be convergent uniformly in the perturbation parameterǫprovided only that ∈≤N^(−1).An O(N^(−2)(lnN)^(1/2))convergent rate in a discrete streamline-diffusion norm is established under certain regularity assump-tions.Finally,through numerical experiments,we verified the theoretical results.

ERROR ESTIMATES OF THE FINITE ELEMENT METHOD WITH WEIGHTED BASIS FUNCTIONS FOR A SINGULARLY PERTURBED CONVECTION-DIFFUSION EQUATION

In this paper, we establish a convergence theory for a finite element method with weighted basis functions for solving singularly perturbed convection-diffusion equations. The stability of this finite element method is proved and an upper bound O（h｜lnε｜3/2） for errors in the approximate solutions in the energy norm is obtained on the triangular Bakhvalov-type mesh. Numerical results are presented to verify the stability and the convergent rate of this finite element method.

A Finite Element Method for Singularly Perturbed Reaction-diffusion Problems

Abstract A finite element method is proposed for the singularly perturbed reaction-diffusion problem. An optimal error bound is derived, independent of the perturbation parameter.

ON THE hp FINITE ELEMENT METHOD FOR THE ONE DIMENSIONAL SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEMS

Presents information on singularly peturbed two-point boundary value problem of convection-diffusion type. Analysis of the problem; Details of an hp version finite element method on a strongly graded piecewise uniform mesh of Shiskin type; Convergence of the method with respect to the singular perturbation parameter.

基于优化分层网格的多尺度有限元求解二维奇异摄动的计算格式与效率分析

针对奇异摄动问题的二维对流扩散方程,应用多尺度有限元法在优化的分层网格上探究高效计算方案。多尺度有限元法仅需在粗网格求解子问题,详细给出了多尺度之间的数据映射关系,将相应的微观信息代入宏观尺度,用于求解降低规模的矩阵方程以节约计算资源。基于摄动系数迭代,形成自适应分层网格,能够有效地逼近奇异摄动的边界层。通过数学分析与数值实验,对比计算消耗和运行时间,验证了多尺度有限元法随着分层网格的加密,可以获得稳定、高阶、高效的一致收敛结果,凸显新方法的计算效率与应用优势。

几何参数对加筋圆柱壳轴压屈曲折减因子的影响

加筋圆柱壳广泛应用于航天运载器结构中。受制造工艺影响,实际产品总是存在无法避免的几何缺陷,导致实际航天运载器结构的轴压承载力远低于理论解,需要引入一个0-1之间的折减因子乘上理论解来进行修正,令理论解在保证安全的同时尽可能接近实验值。当结构的几何参数在设计迭代过程中调整时,折减因子的值也可能会随之改变。厘清折减因子随结构几何参数发生改变时的变化规律,有助于便捷、准确地得到合适的折减因子值,从而保证结构安全、降低重量、缩短设计周期。介绍了基于边界扰动载荷法的实现过程,针对主要分析参数进行了收敛性考察,明确了分析参数的选取原则。基于该方法研究了结构的典型参数发生变化时折减因子相应的变化规律。结果表明,折减因子随蒙皮厚度的增加而线性增加,随纵筋截面面积的增加而线性减小,即蒙皮厚度越大、纵筋越是低而薄,则折减因子值越大、结构的缺陷敏感性也越低,而壳体高度的变化则对折减因子没有显著影响。

A ROBUST FINITE ELEMENT METHOD FOR A 3-D ELLIPTIC SINGULAR PERTURBATION PROBLEM

This paper proposes a robust finite element method for a three-dimensional fourth-order elliptic singular perturbation problem. The method uses the three-dimensional Morley element and replaces the finite element functions in the part of bilinear form corresponding to the second-order differential operator by a suitable approximation. To give such an approximation, a convergent nonconforming element for the second-order problem is constructed. It is shown that the method converges uniformly in the perturbation parameter.

THEORY OF PERTURBATION FINITE ELEMENT ANALYSIS FOR SOLUTION OF NONLINEAR BUCKLING CRITICAL LOADS OF STRUCTURES

The author presents a theory, including the complete analysis and incomplete analysis,of perturbational finite element analysis for the solution of nonlinear buckling critical loadsof structures.

Structural finite element model updating using incomplete ambient vibration modal data

This paper presents an effective approach for updating finite element dynamic model from incomplete modal data identified from ambient vibration measurements.The proposed method is based on the relationship between the perturbation of structural parameters such as stiffness and mass changes and the modal data measurements of the tested structure such as measured mode shape readings.Structural updating parameters including both stiffness and mass parameters are employed to represent the differences in structural parameters between the finite element model and the associated tested structure.These updating parameters are then evaluated by an iterative solution procedure,giving optimised solutions in the least squares sense without requiring an optimisation technique.In order to reduce the influence of modal measurement uncertainty,the truncated singular value decomposition regularization method incorporating the quasi-optimality criterion is employed to produce reliable solutions for the structural updating parameters.Finally,the numerical investigations of a space frame structure and the practical applications to the Canton Tower benchmark problem demonstrate that the proposed method can correctly update the given finite element model using the incomplete modal data identified from the recorded ambient vibration measurements.

反对称铺设复合材料层合板非线性后屈曲分析

基于层合板壳理论,考虑反对称铺设层合板的拉弯耦合效应和后屈曲过程中的非线性几何变形,推导了由应力函数和挠度表示的复合材料层合板的后屈曲控制方程。引入无量纲参数对控制方程和边界条件进行无量纲化,以消除材料参数及几何尺寸对分析结果的影响。采用摄动法将无量纲的非线性控制方程及边界条件展开成一系列非齐次线性摄动方程组,分析各阶摄动方程的通解与特解的构造,并逐次求解,建立了反对称铺设复合材料层合板受单向均布压力作用的临界屈曲荷载及后屈曲平衡路径的理论解。进而运用ABAQUS软件对复合材料层合板在面内压缩载荷作用下的屈曲和后屈曲进行有限元分析,结果表明理论解与ABAQUS结果十分接近,验证了理论解的正确性。在此基础上进一步讨论了铺设角度、铺设层数和拉弯耦合效应等对层合板后屈曲性能的影响。研究发现层合板的屈曲载荷受铺设角度与层数的影响较为显著,而拉弯耦合效应使板的屈后强度大大降低。

FINITE ELEMENT ANALYSIS OF A LOCAL EXPONENTIALLYFITTED SCHEME FOR TIME-DEPENDENTCONVECTION-DIFFUSION PROBLEMS

In [16], Stynes and O' Riordan(91) introduced a local exponentially fitted finite element (FE) scheme for a singularly perturbed two-point boundary value problem without turning-point. An E-uniform h(1/2)-order accuracy was obtain for the epsilon-weighted energy norm. And this uniform order is known as an optimal one for global exponentially fitted FE schemes (see [6, 7, 12]). In present paper, this scheme is used to a parabolic singularly perturbed problem. After some subtle analysis, a uniformly in epsilon convergent order h\ln h\(1/2) + tau is achieved (h is the space step and tau is the time step), which sharpens the results in present literature. Furthermore, it implies that the accuracy order in [16] is actuallay h\ln h\(1/2) rather than h(1/2).

A Robust Modified Weak Galerkin Finite Element Method for Reaction-Diffusion Equations

In this paper,a robust modified weak Galerkin(MWG)finite element method for reaction-diffusion equations is proposed and investigated.An advantage of this method is that it can deal with the singularly perturbed reaction-diffusion equations.Another advantage of this method is that it produces fewer degrees of freedom than the traditional WG method by eliminating the element boundaries freedom.It is worth pointing out that,in our method,the test functions space is the same as the finite element space,which is helpful for the error analysis.Optimalorder error estimates are established for the corresponding numerical approximation in various norms.Some numerical results are reported to confirm the theory.