A stabilized finite element method for convection–diffusion problems

A stabilized finite element method (FEM) is presented for solving the convection–diffusion equation. We enrich the linear finite element space with local functions chosen according to the guidelines of the residual‐free bubble (RFB) FEM. In our approach, the bubble part of the solution (the microscales) is approximated via an adequate choice of discontinuous bubbles allowing static condensation. This leads to a streamline‐diffusion FEM with an explicit formula for the stability parameter τ_K that incorporates the flow direction, has the capability to deal with problems where there is substantial variation of the Péclet number, and gives the same limit as the RFB method. The method produces the same a priori error estimates that are typically obtained with streamline‐upwind Petrov/Galerkin and RFB. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Dynamic Stability Analysis of Shells Using a Finite Element Based Reduction Method

A finite element formulation of a reduction method for dynamic stability analysis of imperfection-sensitive shell structures is presented. The reduction method makes use of a perturbation approach, initially developed for static buckling and later extended to dynamic buckling analysis. The single mode dynamic buckling analysis and its extension to parametric excitation analysis are described. The approach is available within a general purpose finite element code. Characteristic results for the parametric excitation analysis of a composite cylindrical shell are shown. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Free vibration and buckling analysis of plates using B-spline wavelet on the interval Mindlin element

A finite element method (FEM) of B-spline wavelet on the interval (BSWI) is used in this paper to solve the free vibration and buckling problems of plates based on Reissner–Mindlin theory. By aid of the high accuracy of B-spline functions approximation for structural analysis, the proposed method could obtain a fast convergence and a satisfying numerical accuracy with fewer degrees of freedoms (DOF). The numerical examples demonstrate that the present BSWI method achieves the high accuracy compared to the exact solution and others existing approaches in the literatures. The BSWI finite element has potential to be used as a numerical method in analysis and design.     

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.     

B-spline based finite element method in one-dimensional discontinuous elastic wave propagation

The B-spline variant of the finite element method (FEM) is tested in one-dimensional discontinuous elastic wave propagation. The B-spline based FEM (called Isogeometric analysis IGA) uses spline functions as testing and shape functions in the Galerkin continuous content. Here, the accuracy of stress distribution and spurious oscillations of the B-spline based FEM are studied in numerical modeling of one-dimensional propagation of stress discontinuities in a bar, where the analytical solution is known. For time integration, the Newmark method, implicit form of the generalized-α method, the central difference method and the predictor/multi-corrector method are tested and compared. The use of lumped and consistent mass matrices in explicit time integration is discussed. Due to accuracy, the consistent mass matrix is preferred in explicit time integration in IGA.     

Band structure computation of two-dimensional and three-dimensional phononic crystals using a finite element-least square point interpolation method

In the present study, a finite element-least square point interpolation method (FE-LSPIM) is proposed for calculating the band structures of in-plane elastic waves in two-dimensional (2D) and three-dimensional (3D) phononic crystals (PCs). This method utilizes new shape functions by combining mesh-free shape functions and finite element shape functions to exploit the specific advantages of the mesh-free method and finite element method (FEM). As a result, FE-LSPIM inherits the completeness properties of the mesh-free method and the compatibility properties of FEM, and thus the solutions obtained tend to be more accurate. Indeed, according to our previous research, the present method obtains excellent accuracy, especially in the high-frequency domain. The proposed FE-LSPIM was combined with Bloch's theory and applied to compute the band gaps (BGs) for 2D PCs and 3D PCs in the present study, where several PCs were investigated to verify the high accuracy when computing the BGs. Numerical analysis showed that the proposed method can predict the BGs more precisely compared with the FEM and modified FEM.     

Machine learning aided static structural reliability analysis for functionally graded frame structures

A novel machine learning aided structural reliability analysis for functionally graded frame structures against static loading is proposed. The uncertain system parameters, which include the material properties, dimensions of structural members, applied loads, as well as the degree of gradation of the functionally graded material (FGM), can be incorporated within a unified structural reliability analysis framework. A 3D finite element method (FEM) for static analysis of bar-type engineering structures involving FGM is presented. By extending the traditional support vector regression (SVR) method, a new kernel-based machine learning technique, namely the extended support vector regression (X-SVR), is proposed for modelling the underpinned relationship between the structural behaviours and the uncertain system inputs. The proposed structural reliability analysis inherits the advantages of the traditional sampling method (i.e., Monte-Carlo Simulation) on providing the information regarding the statistical characteristics (i.e., mean, standard deviations, probability density functions and cumulative distribution functions etc.) of any concerned structural outputs, but with significantly reduced computational efforts. Five numerical examples are investigated to illustrate the accuracy, applicability, and computational efficiency of the proposed computational scheme.     

AN ERROR ANALYSIS METHOD SPP-BEAM AND A CONSTRUCTION GUIDELINE OF NONCONFORMING FINITE ELEMENTS FOR FOURTH ORDER ELLIPTIC PROBLEMS

Under two hypotheses of nonconforming finite elements of fourth order elliptic problems,we present a side-patchwise projection based error analysis method(SPP-BEAM for short).Such a method is able to avoid both the regularity condition of exact solutions in the classical error analysis method and the complicated bubble function technique in the recent medius error analysis method.In addition,it is universal enough to admit generalizations.Then,we propose a sufficient condition for these hypotheses by imposing a set of in some sense necessary degrees of freedom of the shape function spaces.As an application,we use the theory to design a P3 second order triangular H2 non-conforming element by enriching two P4 bubble functions and,another P4 second order triangular H2 nonconforming finite element,and a P3 second order tetrahedral H2 non-conforming element by enriching eight P4 bubble functions,adding some more degrees of freedom.     

Stability and accuracy of adapted finite element methods for singularly perturbed problems

The stability and accuracy of a standard finite element method (FEM) and a new streamline diffusion finite element method (SDFEM) are studied in this paper for a one dimensional singularly perturbed connvection-diffusion problem discretized on arbitrary grids. Both schemes are proven to produce stable and accurate approximations provided that the underlying grid is properly adapted to capture the singularity (often in the form of boundary layers) of the solution. Surprisingly the accuracy of the standard FEM is shown to depend crucially on the uniformity of the grid away from the singularity. In other words, the accuracy of the adapted approximation is very sensitive to the perturbation of grid points in the region where the solution is smooth but, in contrast, it is robust with respect to perturbation of properly adapted grid inside the boundary layer. Motivated by this discovery, a new SDFEM is developed based on a special choice of the stabilization bubble function. The new method is shown to have an optimal maximum norm stability and approximation property in the sense that where u _N is the SDFEM approximation in linear finite element space V ^N of the exact solution u. Finally several optimal convergence results for the standard FEM and the new SDFEM are obtained and an open question about the optimal choice of the monitor function for the moving grid method is answered. This work was supported in part by NSF DMS-0209497 and NSF DMS-0215392 and the Changjiang Professorship through Peking University.     

The Boundary Finite Element Method for the Efficient Computation of Orders and Modes of Three-Dimensional Stress Singularities

In this contribution the B oundary F inite E lement M ethod (BFEM) is employed for the computation of the orders of stress singularities for several three-dimensional stress concentration problems in linear elastic fracture mechanics. The BFEM combines the advantages of both the FEM and the BEM: while only a discretization on a structural boundary is required, the actual surface mesh consists of standard displacement based finite elements. In contrast to the BEM, no fundamental solution is required. The BFEM is an ef.cient analysis tool which leads to highly accurate results with significantly lesser computational effort when compared to e.g. standard FEM procedures. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

ON THE STABILITY OF THE RESIDUAL-FREE BUBBLES FOR THE NAVIER-STOKES EQUATIONS

This paper considers the Calerkin finite element method for the incompressible Navier-Stokes equations in two dimensions, where the finite-dimensional spaces employed consist of piecewise polynomials enriched with residual-free bubble （RFB） functions. The stability features of the residual-free bubble functions for the linearized Navier-Stokes equations are analyzed in this work. It is shown that the enrichment of the velocity space by bubble functions stabilizes the numerical method for any value of the viscosity parameter for triangular elements and for values of the viscosity parameter in the vanishing limit case for quadrilateral elements.     

A time finite element method for structural dynamics

A time (Galerkin) finite element method (time FEM) for structural dynamics is proposed in this paper. The key lies in a variational formulation that is well-posed and equivalent to the conventional strong form of governing equations of structural dynamics. Based on the variational formulation, a time finite element formulation is naturally established and its convergence property is easily derived through an a priori error analysis. Technical details on practical implementation of the time FEM are presented. Numerical examples are studied to verify the proposed time FEM.     

Dynamic responses of Euler–Bernoulli beam subjected to moving vehicles using isogeometric approach

Dynamic analysis of beam structures subjected to moving vehicles using an isogeometric Euler–Bernoulli formulation is presented in this paper. The method utilizes B-Splines or Non-Uniform Rational–Splines (NURBS) as the basis functions for both geometric and analysis implementation. The rotation-free technique has been incorporated into the formulation by using only one deflection variable with excluding the rotational degrees of freedom adopted for each control point. Then, it enables to use a few degrees of freedom (Dofs) to achieve a highly accurate solution. The validations of the proposed method included a complicated moving vehicle and rough pavement effects are compared to the precisely analytical results. Compared with most existing methods of finite element method (FEM) and readily analytical solutions, the present technique indicated the effectiveness of present isogeometric method and its well accurate prediction for suitable simulating the interaction model of the bridge structures and complicated vehicles.     

The construction of plane elastomechanics and Mindlin plate elements of B-spline wavelet on the interval

Based on two-dimensional tensor product B-spline wavelet on the interval (BSWI), a class of C0 type plate elements is constructed to solve plane elastomechanics and moderately thick plate problems. Instead of traditional polynomial interpolation, the scaling functions of two-dimensional tensor product BSWI are employed to form the shape functions and construct BSWI elements. Unlike the process of direct wavelets adding in the previous work, the elemental displacement field represented by the coefficients of wavelets expansions is transformed into edges and internal modes via the constructed transformation matrix in this paper. The method combines the versatility of the conventional finite element method (FEM) with the accuracy of B-spline functions approximation and various basis functions for structural analysis. Some numerical examples are studied to demonstrate the proposed method and the numerical results presented are in good agreement with the closed-form or traditional FEM solutions.     

Analysis of the cell boundary element methods for convection dominated convection-diffusion equations

We introduce two kinds of the cell boundary element (CBE) methods for convection dominated convection-diffusion equations: one is the CBE method with the exact bubble function and the other with inexact bubble functions. The main focus of this paper is on inexact bubble CBE methods. For inexact bubble CBE methods we introduce a family of numerical methods depending on two parameters, one for control of interior layers and the other for outflow boundary layers. Stability and convergence analysis are provided and numerical tests for inexact bubble CBEs with various choices of parameters are presented.     

区间运算和静力区间有限元

用均值和离差两参数表征区间变量的不确定性,根据区间运算规则,论证了区间变量的运算特性.将区间分析和有限元方法相结合,提出了非概率不确定结构的一种区间有限元分析方法.将区间有限元静力控制方程中n自由度不确定位移场特征参数的求解归结为求解一2n阶线性方程组.实例分析表明文中方法是有效和可行的.     

区间有限元方法及其在抗滑稳定性分析中的应用

通过区间值函数和实值函数的关系探讨了区间相关性导致的区间扩张的问题,给出了保证区间计算获得足够精度的计算方法;提出了基于单元的子区间摄动有限元计算方法,并给出了提高计算效率的一些方法和获得较好计算精度时的子区间数目的近似计算公式．结合工程实例,基于单元的子区间有限元方法和抗滑稳定性分析方法给出了稳定性的区间范围,为更合理地估计和评价结构的抗滑稳定性提供一定的依据．     

Static and dynamic analysis of two-layer Timoshenko composite beams by weak-form quadrature element method

To improve the efficiency in predicting the dynamic mode and static response of the two-layer partial interaction composite beams, this paper utilizes the differential quadrature technique to approximate derivatives of the primary unknowns with adaptive order of precision, rather than the low and constant order of interpolation used in the conventional finite element method (FEM). A degree-of-freedom-adaptive weak-form quadrature element (WQE) for dynamic analysis is formulated and implemented based on the principle of virtual work. For the purpose of comparison, a parabolic displacement-based finite element is also provided, thus (1) the predicted deflections and natural frequencies of the composite beams are verified; (2) the smoothness of the internal forces and stresses generated by WQE method and FEM are compared, and (3) the convergent rates of higher order free vibration modes are also examined. Numerical results show that the efficiency of the proposed WQE method has, on the one hand, significantly triumphed over that of FEM on analyses including static response, natural frequencies and higher order free vibration modes, on the other hand, the smoothness of results, including internal forces and stresses, is greatly refined.     

稳定化有限元方法中逆估计常数的确定

0．弓I言稳定化有限元方法［2］［4］［12］［15］ro［18］在固体和流体力学的数值计算中构造有效的格式发挥着很大的作用．从理论分析的角度看，该方法（Galerkin一局部最小二乘方法）是完备的、确定的，但是在实际计算中稳定化参数。E（0，CI）的如何选取直接影响到逼近解的质量．数值实验【9川叫'川'到表明。取得太小会造成。伪l。压力模式rI．因此，对of的估计是一个值得注意的重要问题．文［8］虽然估训一了一些逆常数，但其未能给出确定逆估计常数的一般公式，而且技巧性太强，过于依赖区域剖分的性质，使得逆常数的计算复杂化，不…     

二阶椭圆问题基于泡函数的简化的稳定化二阶混合有限元格式

本文研究二阶椭圆方程基于泡函数的稳定化的二阶混合有限元格式,通过消去泡函数导出一种自由度很少的简化的稳定化的二阶混合有限元格式, 误差分析表明消去泡函数的简化格式与带有泡函数的格式具有相同的精度而可以节省6N_p个自由度(其中N_p三角形剖分中的顶点数目).