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.     

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.     

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.     

无网格伽辽金法求解平面偶应力问题

提出采用无网格伽辽金法(EFGM)求解偶应力问题,以避免有限元求解中因C¹连续要求可能引起的不便。推导了基于二次基和移动最小二乘技术的EFGM计算公式,通过计算受轴向均匀拉伸的带中心小孔无限大板和细长杆的偶应力问题,对所提方法进行了数值验证,与解析解相比结果令人满意,此外还讨论了节点密度和权函数对计算结果的影响。数值结果表明,所提算法可有效地求解平面偶应力问题。     

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.     

基于正交基无单元Galerkin法和非线性规划的安定分析方法

基于安定分析的下限定理,用正交基无单元Galerkin法建立了交交载荷作用下理想弹塑性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的载荷作用下,采用正交基无单元Galerkin法计算相应的虚拟弹性应力场.并且利用结构在正交基无单元Galerkin法弹塑性增量分析中平衡迭代结果计算得到自平衡应力场基矢量,然后由这些基矢量的线性组合模拟自平街应力场.安定分析问题最终被归结为一系列未知变量较少的非线性数学规划子问题,通过复合形法求解.算例表明该方法的计算结果是令人满意的,并且对初始复合形顶点和用于构造自平衡应力场基矢量的载荷增量是非常不敏感的.     

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.     

无网格Galerkin法GPU加速并行计算及其应用

针对无网格Galerkin法计算耗时的问题,采用逐节点对法来组装刚度矩阵、共轭梯度法求解基于CSR格式存储的稀疏线性方程组,提出了一种利用罚函数法施加本质边界条件的EFG法GPU加速并行算法,给出了刚度矩阵和惩罚刚度矩阵的统一格式,以及GPU加速并行算法的流程图。编写了基于CUDA构架平台的GPU程序,且在NVIDIA GeForce GTX 660显卡上通过数值算例对所提算法进行了性能测试与分析比较,探讨了影响加速比的因素。算例结果验证了所提算法的可行性,并在满足计算精度的前提下,其加速比最大可达17倍;同时线性方程组的求解对加速比起决定性影响。     

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.     

直接施加本质边界条件的FE／EFG耦合算法及其数值实现

提出一种可以直接施加本质边界条件的有限元与无网格Galerkin（FE／EFG）耦合算法。将问题域分成FE和EFG两种类型的子域,采用转换矩阵耦舍两子域的交界面;通过另一转换矩阵将无网格区域本质边界上的名义位移转换成真实位移,从而可在其上直接施加本质边界条件;采用二次转换实现两种转换矩阵之间的协调。提出全域统一采用单元...     

An error indicator for the element free Galerkin method

In this paper we present an error indicator for the Element Free Galerkin (EFG) method, whose evaluation is computationally so simple that it can be readily implemented in existing EFG codes. The error indicator works very well in all numerical examples for 2-D potential and elasticity problems that are presented here, for regular and irregular grid of nodes. Moreover, it was demonstrated that this method is very simple in terms of economy and gives a good performance. The results show the error in EFG approximation may be estimated via the error indicator described in this paper. The indicator allows the global energy norm error to be estimated and also gives a good evaluation of local errors. It can thus be combined with a full adaptive process of refinement or, more simply, provide guidance for cloud of points redesign.     

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．     

High-precision stress determination in photoelasticity

Stress separation is usually achieved by solving differential equations of equilibrium after parameter determination from isochromatics and isoclinics. The numerical error resulting from the stress determination is a main concern as it is always a function of parameters in discretization. To improve the accuracy of stress calculation,a novel meshless barycentric rational interpolation collocation method(BRICM) is proposed. The derivatives of the shear stress on the calculation path are determine...     

Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity

Gradient theories, as a regularized continuum mechanics approach, have found wide applications for modeling strain localization failure process. This paper presents a second gradient stress–strain damage elasticity theory based upon the method of virtual power. The theory considers the strain gradient and its conjugated double stresses. Instead of introducing an intrinsic material length scale into the constitutive law in an ad hoc fashion, a microstructural granular mechanics approach is applied to derive the higher-order constitutive coefficients such that the internal length scale parameter reflects the natural granularity of the underlying material microstructure. The derivations of the required damage constitutive relationships, the strong form governing equations as well as its weak form for the second gradient model are described. The recently popularized Element-Free Galerkin (EFG) method is then employed to discretize the weak form equilibrium equation for accommodating the resultant higher-order continuity requirements and further handling the mesh sensitivity problem. Numerical examples for shear band simulations show that the proposed second gradient continuum model can produce stable, accurate as well as mesh-size independent solutions without a priori assumption of the shear band path.     

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.     

Étude du critère de plasticité des matériaux poreux

The ductile failure of porous metallic materials is studied here using both Limit Analysis (LA) methods, a problem treated by Gurson with his famous kinematic approach in 1977. The present work is devoted to determining the strength of porous materials with long circular cylindrical cavities in the case of plane stress. The numerical methods developed here use the Hill–Mandel method based on the homogenization theory of heterogeneous media within the LA framework. The use of kinematic and static approaches gave an excellent estimation of the yield criterion for all the cases studied. The numerical results based on LA methods have been compared with analytical and semi-analytical yield domain expressions proposed by different authors. The results show that the Richmond model was the most accurate in terms of our predictions.     

On thermoelastostatics of composites with nonlocal properties of constituents II. Estimation of effective material and field parameters

One considers a linear thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated heterogeneities. It is assumed that the stress–strain constitutive relations of constituents are described by the nonlocal integral operators, whereas the equilibrium and compatibility equations remain unaltered as in classical local elasticity. The general integral equations connecting the stress and strain fields in the point being considered and the surrounding points are obtained. The method is based on a centering procedure of subtraction from both sides of a known initial integral equation their statistical averages obtained without any auxiliary assumptions such as, e.g., effective field hypothesis implicitly exploited in the known centering methods. In a simplified case of using of the effective field hypothesis for analyzing composites with one sort of heterogeneities, one proves that the effective moduli explicitly depend on both the strain and stress concentrator factor for one heterogeneity inside the infinite matrix and does not directly depend on the elastic properties (local or nonlocal) of heterogeneities. In such a case, the Levin’s (1967) formula in micromechanics of composites with locally elastic constituents is generalized to their nonlocal counterpart. A solution of a volume integral equation for one heterogeneity subjected to inhomogeneous remote loading inside an infinite matrix is proposed by the iteration method. The operator representation of this solution is incorporated into the new general integral equation of micromechanics without exploiting of basic hypotheses of classical micromechanics such as both the effective field hypothesis and “ellipsoidal symmetry” assumption. Quantitative estimations of results obtained by the abandonment of the effective field hypothesis are presented.     

Determination of V-notch SIFs in multi-material anisotropic wedges by digital correlation experiments

In this paper, theoretical formulations based on the Stroh’s complex function approach were used to find the displacement field and H-integral of a sharp V-notch formed from several anisotropic materials. Displacements from the image-correlation experiments are then substituted into the least-squares formulation to find V-notch stress intensity factors (SIFs) in multi-material anisotropic wedges. Validations using the H-integral indicate that the experimental SIFs evaluated from the proposed method of acceptable accuracy. The major advantage is that the proposed method only requires displacements inside the specimen, and displacements near the notch tip, specimen boundaries, or notch surfaces are not necessary.     

Experimentally Evaluating Equilibrium Stress in Uniaxial Tests

Many models use the equilibrium stress, also sometimes known as the back stress, in characterizing the response of both polymeric and non-polymeric materials. We study the characteristics of the equilibrium and show that the tangent modulus and local Poisson’s ratio at equilibrium both are rate independent for common modeling assumptions. This fact is used to propose a method based on uniaxial tension or compression to measure the equilibrium stress, and the associated point’s tangent modulus and local Poisson’s ratio. The method is based on cyclic loading and identification of similar states with vastly different loading rates. The method is used to characterize the equilibrium stress in glassy polycarbonate, and the results are studied in regard to the possible error for such a measurement. The method is faster than most other proposed methods for calculating the equilibrium stress, and provides additional measurements of parameters at equilibrium that are normally not obtained.     

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.论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.