结构安定分析的Galerkin边界元方法

基于Melan静力安定定理,利用Galerkin边界元方法建立了多组交变载荷作用下结构安定分析的下限计算格式.在给定载荷域的载荷角点所对应载荷作用下,采用Galerkin边界元法计算相应的虚拟弹性应力场,并且利用结构在Galerkin边界元弹塑性增量计算中同一增量步中不同迭代步之间的应力差作为自平衡应力场的基矢量,通过这些基矢量的线性组合构造了自平衡应力场,大大降低了所形成的数学规划问题的未知变量数.并通过复合形法对非线性规划问题直接进行求解,得到了结构在交变载荷作用下的下限安定乘子.计算结果表明,所采用的方法具有较高的精度和计算效率.     

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．     

轴对称结构极限下限分析的自然单元法

作为一种基于自然邻近插值的新型无网格法,自然单元法克服了大多数无网格法难以施加本质边界条件的困难．将自然单元法与减缩基技术相结合,建立了一种轴对称结构极限下限分析的数值格式和求解算法．通过不断修正自平衡应力场,轴对称结构极限下限分析可转化为一系列的非线性数学规划子问题,并由复合形法求解．在每个非线性规划子问题中,自平衡应力场表示为一组带有待定系数的自平衡应力场基矢量的线性组合,并且这些自平衡应力场基矢量可由弹塑性增量分析的平衡迭代结果得到．算例结果表明,本文所提的轴对称结构极限下限分析方法行之有效．     

弹塑性结构安定下限分析的无网格局部

将基于Voronoi结构的无网格局部Petrov-Galerkin法与减缩基技术相结合,建立了一种安定下限分析的新方法．为了克服移动最小二乘近似难以准确施加本质边界条件的缺点,采用了自然邻近插值构造试函数.通过引入基准载荷域上载荷角点的概念,消除了安定下限分析中由时间参数所引起的求解困难．利用减缩基技术,将安定分析问题化为一系列未知变量较少的非线性规划子问题．在每个非线性规划子问题中,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,而这些自平衡应力场基矢量可应用弹塑性增量分析中的平衡迭代结果得到．算例结果证明了提出的分析方法的有效性．      

弹塑性结构安定下限分析的无网格局部Petrov-Galerkin法

将基于Voronoi结构的无网格局部Petrov-Galerkin法与减缩基技术相结合,建立了一种安定下限分析的新方法.为了克服移动最小二乘近似难以准确施加本质边界条件的缺点,采用了自然邻近插值构造试函数.通过引入基准载荷域上载荷角点的概念,消除了安定下限分析中由时间参数所引起的求解困难.利用减缩基技术,将安定分析问题化为一系列未知变量较少的非线性规划子问题.在每个非线性规划子问题中,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,而这些自平衡应力场基矢量可应用弹塑性增鼍分析中的平衡迭代结果得到.算例结果汪明了提出的分析方法的有效性.     

非协调元在复合材料安定下限分析中的应用

采用应力函数法,结合均匀化理论和应变法,在细观层次上研究了复合材料的极限和安定分析,获取复合材料代表性体积元在载荷加载历史未知下的容许承载域。利用8节点非协调等参元离散结构,获取弹性应力场和自平衡残余应力场,建立复合材料在细观层次上安定下限的优化格式。在满足计算精度的同时,大大降低了优化规模。以周期性纤维增强金属基复合材料为例,验证了该单元在安定下限分析中的有效性和可靠性。     

线性强化材料弹塑性分析的自然单元法

自然单元法(NEM)是一种求解偏微分方程的无网格数值方法,其形函数兼具无网格法的特点和传统有限元法的优点.本文基于塑性增量理论,将自然单元法应用于弹塑性问题的分析计算中.为实现近似函数在非凸边界上的线性变化,采用约束的自然单元法(C-NEM)进行形函数计算.给出了增量切线刚度法求解非线性控制方程的相关公式,并对加载状态的确定和过渡状态下比例因子的计算方法等问题进行了深入的研究.编制了Von-Mises屈服准则下线性强化材料模型的二维弹塑性分析计算程序.算例分析表明,用自然单元法分析弹塑性力学问题是可行的,具有前处理过程简单、可以方便地准确施加本质边界条件等优点.     

循环接触下安定状态问题的研究

基于线性随动强化理论,运用算子分离技术,研究将弹塑性问题转换为弹性问题和残余问题的分析方法,且针对循环载荷接触安定状态,建立了计算机分析程序,该研究能够分析计算弹塑性接触载荷在安定状态下的应力、残余累积应变及残余应力,分析计算了不同载荷的安定状态,并探讨其残余应力场的分析方法。     

无单元法求解正交各向异性板自由振动问题

无单元法是一种新出现的数值方法。本文对无单元法的数学基础—滑动最小二乘法进行了详细的研究,推导了无单元法的形函数,并对一些关键问题,如权函数的选取,正交基函数,边界条件,数值实现方法等得出了研究结论。用无单元法研究了正交各向异性板的自由振动问题,由动力学变分原理和滑动最小二乘法导出了正交各向异性板的无单元法质量矩阵和刚度矩阵,编制了相应的计算程序,通过计算实例验证了该方法的有效性。     

基于区间参数摄动法的三维弹塑性区间有限元探讨

首次探讨了弹塑性问题的区间分析,推导了三维弹塑性问题的区间参数摄动法有限元计算公式,研制了相应的弹塑性区间有限元程序,通过算例分析得到结论:(1)采用增量法对弹塑性问题进行区间分析时,增量位移离差随着增量位移均值的增大而增大.(2)单元高斯点应力没有达到屈服极限之前,高斯点的应力离差很小;当单元高斯点应力达到屈服极限之后,高斯点应力离差逐渐增大.(3)是否考虑塑性矩阵的应力分量为弹塑性参数的隐函数,对应力高差计算的影响较小.     

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.     

Thermal solution of cylindrical composite systems using meshless method

This paper presents the thermal solution of cylindrical composite systems using meshless element free Galerkin (EFG) method. The EFG method utilizes the moving least square approximants, which are constructed by using a weight function, a basis function and a set of non-constant coefficients to approximate the unknown function of temperature. Dirichlet (essential) boundary conditions have been enforced using Lagrange multiplier and penalty methods. Existing rational weight function has been modified and used in the present analysis. MATLAB codes have been developed to obtain the numerical solution. The EFG results have been obtained using cubicspline, quarticspline, Gaussian, quadratic, hyperbolic, exponential, rational and cosine weight functions for a model problem. The results obtained using different EFG weight functions are also compared with those obtained by finite element method. The effect of scaling and penalty parameters has also been studied in detail.     

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.     

Analysis of piezoelectric ceramic multilayer actuators based on an electro-mechanical coupled meshless method

This paper presents an efficient meshless method for analyzing cracked piezoelectric structures subjected to mechanical and electrical loading. In this method, an element free Galerkin (EFG) formulation, an enriched basic function and some special shape functions that contain discontinuous derivatives are employed. Based on the moving least squares (MLS) interpolation approach, the EFG method is one of the promising methods for dealing with problems involving progressive crack growth. Since the method is meshless and no element connectivity data are needed, the burdensome remeshing procedure required in the conventional finite element method (FEM) is avoided. The numerical results show that the proposed method can yield an accurate near-tip stress field in an infinite piezoelectric plate containing an interior hole. In another example studying a ceramic multilayer actuator, the proposed model was found to be accurate in the simulation of stress and electric field concentrations arround the abrupt end of an internal electrode. The project supported by the National Natural Science Foundation of China (10025209, 10132010, and 90208002), and the Research Grants Council of the Hong Kong Special Administrative Region, China (HKU 7203/03E). The English text was polished by Yunming Chen.     

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.     

Numerical simulations of viscous flows using a meshless method

This paper uses the element‐free Galerkin (EFG) method to simulate 2D, viscous, incompressible flows. The control equations are discretized with the standard Galerkin method in space and a fractional step finite element scheme in time. Regular background cells are used for the quadrature. Several classical fluid mechanics problems were analyzed including flow in a pipe, flow past a step and flow in a driven cavity. The flow field computed with the EFG method compared well with those calculated using the finite element method (FEM) and finite difference method. The simulations show that although EFG is more expensive computationally than FEM, it is capable of dealing with cases where the nodes are poorly distributed or even overlap with each other; hence, it may be used to resolve remeshing problems in direct numerical simulations. Flows around a cylinder for different Reynolds numbers are also simulated to study the flow patterns for various conditions and the drag and lift forces exerted by the fluid on the cylinder. These forces are calculated by integrating the pressure and shear forces over the cylinder surface. The results show how the drag and lift forces oscillate for high Reynolds numbers. The calculated Strouhal number agrees well with previous results. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

局部正交无网格伽辽金法的研究及其在含多裂纹多孔结构中的应用

通过对无网格法中正交基函数的研究,提出局部正交无网格伽辽金法,局部正交基函数保持原正交基函数的性质,但其导数具有了通式,简洁明了,易于编程实现,计算效率高,并将其应用到求解含多裂纹多孔均匀拉伸板的应力强度因子中,计算结果与用正交无网格伽辽金法和有限元法得到的结果进行比较,证明了局部正交无网格伽辽金法的可行性和正确性。     

弹塑性体脆断相场模型的一致性无单元伽辽金法

采用无单元伽辽金法（EFG）对弹塑性体脆性断裂的相场模型进行了数值实现。利用无单元法便于构建高阶近似函数的优势，位移和相场均采用二阶移动最小二乘（MLS）近似。刚度阵的数值积分采用更为高效的二阶一致三点积分格式QC3（Quadratically Consistent 3-point integration scheme）。本构算法采用Newton-Raphson迭代和弹塑性一致性切线模量。数值结果表明了本文方法模拟弹塑性体脆性断裂的有效性。