A numerical comparison of two different approximations of the error in a meshless method

Meshless methods still require considerable improvement before they equal the prominence of finite elements in computer science and engineering. In the Element Free Galerkin (EFG) method, it is obviously important that the error of approximation should be estimated, as it is in the Finite Element Method (FEM).In this paper we compare two different procedures to approximate the a posteriori error for the EFG method, both procedures are recovery based errors. The performance of the two different approximations of the error is illustrated by analysing different examples for 2-D potential and elasticity problems with known analytical solutions, using regular and irregular clouds of points. For irregular clouds of points, it is recommended to use smooth transition of nodes, thus creating areas of decreasing nodal densities.     

流动问题无网格Galerkin方法的稳定化方案研究

直接运用无网格Galerkin方法求解对流占优的非线性对流扩散方程及纯对流方程,会出现数值伪振荡现象。本文基于无网格Galerkin方法,构造了MFSUPG(Meshfree Streamline Upwind Petrov-Galerkin),MF-GLS(Meshfree Galerkin Least-Square),MFSGS(Meshfree Sub-Grid Scale)及MFLS(Meshfree Least-Square)四种稳定化方案。数值实验表明:四种稳定化方案中,MFLS的通用性最强。耦合MFLS的无网格Galerkin方法能很好地求解对流占优的非线性对流扩散方程及纯对流方程,具有计算精度高、稳定性好、前后处理方便、算法实施简单的优点,并能捕捉解的大梯度变化。     

一种FEM－EFGM耦合技术及其应用

利用面力耦合(traction matching)技术，导出了增量形式的一般固体力学问题的FEM—EFGM耦合求解格式。在相应的域内分别建立FE和EFG的离散方程，利用交界面上的连续条件，可方便地建立耦合求解方程，简明有效，易于编程计算。数值算例给出令人满意的结果。     

NEW TREATMENT OF ESSENTIAL BOUNDARY CONDITIONS IN EFG METHOD BY COUPLING WITH RPIM

One of major difficulties in the implementation of meshfree methods using the moving least square (MLS) approximation, such as element-free Galerkin method (EFG), is the imposition of essential boundary conditions as the approximations do not pass through the nodal parameter values. Another class of meshfree methods based on the radial basis point interpolation can satisfy the essential boundary conditions exactly since its approximation function passes through each node in an influence domain and thus its shape functions possess the properties of delta function. In this paper, a coupled element-free Galerkin(EFG)-radial point interpolation method (RPIM) is proposed to enhance their advantages and avoid their disadvantages. Discretized equations of equilibrium are obtained in the RPIM region and the EFG region, respectively. Then a collocation approach is introduced to couple the RPIM and the EFG method. This method satisfies the linear consistency exactly and can maintain the stiffness matrix symmetric. Numerical tests show that this method gives reasonably accurate results consistent with the theory.     

质点积分无单元伽辽金法及其在金属挤压过程中的应用

无单元伽辽金法需要在背景网格上积分,计算量大.节点积分无单元伽辽金法把对求解域的积分转化为对节点的求和,效率高,但因零能模态不受控制而会产生不稳定现象,需要采取一定的稳定化方案.本文采用应力点思想,通过Newtor-Cotes法计算积分,建立了质点积分无单元伽辽金法,并通过小变形弹性静力学问题说明了该方法具有良好的稳定性,且计算效率远高于无单元伽辽金法.最后本文将质点积分无单元伽辽金法成功地应用于三维金属挤压成型过程的数值模拟,显示了该方法在分析此类问题时的优势和潜力.     

A simple and efficient error analysis for multi‐step solution of the Navier–Stokes equations

A simple error analysis is used within the context of segregated finite element solution scheme to solve incompressible fluid flow. An error indicator is defined based on the difference between a numerical solution on an original mesh and an approximated solution on a related mesh. This error indicator is based on satisfying the steady‐state momentum equations. The advantages of this error indicator are, simplicity of implementation (post‐processing step), ability to show regions of high and/or low error, and as the indicator approaches zero the solution approaches convergence. Two examples are chosen for solution; first, the lid‐driven cavity problem, followed by the solution of flow over a backward facing step. The solutions are compared to previously published data for validation purposes. It is shown that this rather simple error estimate, when used as a re‐meshing guide, can be very effective in obtaining accurate numerical solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

无单元伽辽金法的并行计算

对无单元伽辽金法的并行计算进行了详细研究,并将其应用于弹性动力学问题。使用并行桶搜索算法进行节点搜索,使用并行几何搜索算法进行样点搜索,讨论了移动最小二乘MLS(Moving Least Squares)形函数及其导数的并行计算和方程组的并行求解,并利用多层图形划分实现负载平衡。最后给出了并行无单元伽辽金法应用于弹性动力学的计算流程和实例。计算结果表明无单元伽辽金法具有很高的并行性和很好的并行效率,对其进行并行计算具有非常重要的意义。     

波前法在无网格伽辽金法中的应用

有别于有限元法,无网格法采用基于点的近似,可彻底或部分地去除网格（只保留积分所需的背景网格）,在保证计算精度同时降低计算难度。无网格伽辽金法（Element Free Galerkin method, EFG）是一种基于移 动最小二乘近似（Moving Least-Squares, MLS）的全局弱式无网格法,广泛应用于计算力学等领域,该方法的一个缺点是：计算过程中产生的系数矩阵含有的非零元数量比有限元法多,即使处理中等规模模型时,也要求计算机有很大的存储空间,并且计算时间长。波前法在有限元法中已有很成熟的应用,但至今没有应用于无网格方法。本文介绍了波前法在无网格伽辽金法中的应用方法,编写了相应的计算程序,并以弹性力学为例做了验算。     

自适应一致性高阶无单元伽辽金法

近来提出的一致性高阶无单元伽辽金法通过导数修正技术大幅度减少了所需积分点数目,并能够精确地通过线性和二次分片试验,显著改善标准无单元伽辽金法的计算效率、精度和收敛性.本文在此基础之上,充分利用无单元法易于在局部区域添加节点的优势,发展了一致性高阶无单元伽辽金法的h型自适应分析方法.根据应变能密度梯度该方法自适应地确定需节点加密的区域,基于背景积分网格的局部多层细化要求生成新的计算节点,同时考虑了节点分布由密到疏渐进过渡的情形.采用相邻两次计算的应变能的相对误差作为自适应过程的停止准则,将所发展自适应无网格法应用于由几何外形、边界外载和体力等因素造成的应力集中问题的计算分析.数值结果表明,所发展方法能够自适应地对高应力梯度区域进行节点加密,自动给出合理的计算节点分布.与已有的标准无网格法的自适应分析相比,所发展方法在计算效率、精度和应力场光滑性等方面均展现出显著优势.与采用节点均匀分布的一致性高阶无单元伽辽金法相比,它大幅度地减少了计算节点数目,有效提高了一致性高阶无单元伽辽金法在分析应力集中等存在局部高梯度问题时的计算效率和求解精度.     

基于无单元法的平面框架地震反应分析

无单元法在动力计算领域中的应用目前多以梁、板等简单构件的自振特性分析为主,对建筑结构的动力计算,尤其是地震反应分析尚未涉及.文中推导了无单元法的离散动力方程,以轴力杆、欧拉梁模型为基础,将简单构件的无单元动力计算推进到实际结构的地震反应分析,并给出了数值实现的基本流程.最后以一个十一层钢筋混凝土高层结构为算例,实现了多层多跨平面框架的无单元地震反应分析,达到了预期的效果,验证了无单元法应用于平面框架地震反应分析的可行性和有效性.     

极限下限分析的正交基无单元Galerkin法

基于极限分析的下限定理，建立了用正交基无单元Galerkin法进行理想弹塑性结构极 限分析的整套求解算法．下限分析所需的虚拟弹性应力场可由正交基无单元Galerkin法直接 得到，所需的自平衡应力场由一组带有待定系数的自平衡应力场基矢量的线性组合进行模 拟．这些自平衡应力场基矢量可由弹塑性增量分析中的平衡迭代得到．通过对自平衡应力场 子空间的不断修正，整个问题的求解将化为一系列非线性数学规划子问题，并通过复合形法 进行求解．算例表明该方法有效地克服了维数障碍问题，使计算效率得到了充分的提高，是 切实可行的．     

A proper orthogonal decomposition variational multiscale meshless interpolating element-free Galerkin method for incompressible magnetohydrodynamics flow

In the recent decade, the meshless methods have been handled for solving most of PDEs due to easiness of the meshless methods. One of the popular meshless methods is the element-free Galerkin (EFG) method that was first proposed for solving some problems in the solid mechanics. The test and trial functions of the EFG are based on the special basis. Recently, some modifications have been developed to improve the EFG method. One of these improvements is the variational multiscale EFG procedure. In the current article, the shape functions of interpolation moving least squares approximation have been applied to the variational multiscale EFG technique for solving the incompressible magnetohydrodynamics flow. In order to reduce the elapsed CPU time of simulation, we employ a reduced-order model based on the proper orthogonal decomposition technique. The current combination can be referred to as the reduced-order variational multiscale EFG technique. To illustrate the reduction in CPU time used as well as the efficiency of the proposed method, we applied it for the two-dimensional cases.     

Global interpolating meshless shape function based on generalized moving least-square for structural dynamic analysis

A global interpolating meshless shape function based on the generalized moving least-square (GMLS) is formulated by the transformation technique. Both the shape function and its derivatives meet the Kronecker delta function property. With the interpolating GMLS (IGMLS) shape function, an improved element-free Galerkin (EFG) method is proposed for the structural dynamic analysis. Compared with the conventional EFG method, the obvious advantage of the proposed method is that the essential boundary conditions including both displacements and derivatives can be imposed by the straightforward way. Meanwhile, it can greatly improve the ill-condition feature of the standard GMLS approximation, and provide good accuracy at low cost. The dynamic analyses of the Euler beam and Kirchhoff plate are performed to demonstrate the feasibility and effectiveness of the improved method. The comparison between the numerical results of the conventional method and the improved method shows that the proposed method has better stability, higher accuracy, and less time consumption.     

A total linearization method for solving viscous free boundary flow problems by the finite element method

In this paper a total linearization method is derived for solving steady viscous free boundary flow problems (including capillary effects) by the finite element method. It is shown that the influence of the geometrical unknown in the totally linearized weak formulation can be expressed in terms of boundary integrals. This means that the implementation of the method is simple. Numerical experiments show that the iterative method gives accurate results and converges very fast.     

Crack parameter estimation in structures using finite element modeling

This paper addresses the problem of linear crack quantification, crack depth estimation and localization, in structures. An optimization technique based on a finite element model for cracked structural elements is employed in the estimation of crack parameters for beam, truss and two-dimensional frame structures. The modal data for the cracked structures are obtained by solving the corresponding eigenvalue problem. The error in the modal data is simulated by an additive noise that follows the normal distribution. The simulated reduced modal data is expanded using the eigenvector projection method. Numerical examples showed that this technique gives good results for cracks with high depth ratio. The accuracy of the estimated crack parameters depends on (1) the number of modes used, (2) the error level in the cracked structure modal data and (3) the number of measured degrees of freedom in the case of reduced modal data.     

环氧树脂涂层焦散线法及用于测定应力强度因子

本文提出了一种新的实验方法——涂层焦散线法,可用于难加工的非透明材料。作为应用,对铝试件的应力强度因子进行了测定并与有限元法的计算结果作了对比。在数据处理中采用多点法构造目标函数,并引入具有选代过程的统计试验法对其进行求解,从而提高了实验精度。     

h-自适应边界元方法的插值残差计算及误差估计

本文提出了一种用于估计ｈ－自适应边界元过程解误差的新方法,这种方法基于ｈ－自适应边界元过程生成的离散网络,通过计算近似解的插值残差,以此作为误差估计的数据。此外,这种误差分析方法易于程序化,可以很方便地接入现有ｈ－自适应边界元计算机程序（简单细分或分层细分）,而对原程序不作大的改动。通过对二个经典的弹性静力学问题的分析表明：本文的方法能较好地估计边界元解的误差,并使ｈ－自适应边界程序的分析更加有效     

An h-adaptive modified element-free Galerkin method

In this work an h-adaptive Modified Element-Free Galerkin (MEFG) method is investigated. The proposed error estimator is based on a recovery by equilibrium of nodal patches where a recovered stress field is obtained by a moving least square approximation. The procedure generates a smooth recovered stress field that is not only more accurate then the approximate solution but also free of spurious oscillations, normally seen in EFG methods at regions with high gradient stresses or discontinuities.The MEFG method combines conventional EFG with extended partition of unity finite element (EPUFE) methods in order to create global shape functions that allow a direct imposition of the essential boundary conditions.The re-meshing of the integration mesh is based on the homogeneous error distribution criterion and upon a given prescribed admissible error. Some examples are presented, considering a plane stress assumption, which shows the performance of the proposed methodology.     

一类非线性发展方程的全离散有限体积元方法及其分析

给出了求解一类非线性发展方程的全离散一次有限体积元格式,并给出了L2和H1误差估计。最后通过数值例子说明了该方法的有效性。     

A combined approach of EFGM and precise algorithm in time domain solving viscoelasticity problems

In this paper, element free Galerkin (EFG) method is combined with a precise algorithm in the time domain for solving viscoelasticity problems. By expanding variables at discretized time intervals, an initial boundary value problem is converted into a series of recurrent boundary value problems which can be conveniently solved by EFG/EFG-FE with a self-adaptive computing process. There is no requirement of iteration for the solution of non-linear cases. Satisfactory numerical results are obtained for both static and dynamical viscoelasticity problems.