无单元Galerkin方法中周期边界条件的处理

本文研究了无单元Galerkin方法中周期边界条件的处理技术,将Lagrange乘子法用于周期边界条件的处理.数值计算结果表明,该方法具有较高的计算精度.另外,它与无单元Galerkin方法中本质边界条件处理的Lagrange乘子法具有统一性,对于周期、本质混合型边界条件的处理尤为方便.     

弹性地基板计算的无单元法

本文用天单元法研究了弹性半空间地基板的弯曲问题，由滑动最小二乘法和变分原理导出了地基极的无单元法刚度矩阵，编制相应的无单元法计算程序，并给出算例。结果表明方法可行，且具有更加广泛的工程应用前景。     

小波基无单元法及其工程应用

论述数值计算中新的小波基无单元方法,即用小波基函数取代传统无单元方法中的幂级数基之后,使无单元法具有了小波变换的局域化和多分辨率等优良特性,并能有效地克服有限单元法的网格敏感性和单元之间应力不连续现象,从而不但拓展和丰富了无单元法的理论内容,也为其工程应用开辟了新的途径。     

小波基无单元法及其与有限元法的比较

分析了以小波基取代传统无单元法中多项式基的原因,阐述了小波基无单元法的基本原理,并将其与有限元法在基本思路、形函数、位移边界条件、计算精度和适用范围等方面的区别进行了比较研究,最后通过算例验证了B-样条小波基在无单元法中应用的可行性以及小波基无单元法相对于有限元法的优势。     

压电结构动力分析的无网格自然单元法

自然单元法是一种新兴的无网格数值计算方法,在本质边界条件的施加上较采用移动最小二乘法的无网格法具有明显的优势。将无网格自然单元法与精细积分法相结合,提出了压电结构动力响应分析的一条新途径。在空间域上采用自然单元法离散,并运用加权余量法推导了压电结构动力分析的离散控制方程。然后,采用精细积分法在时间域上进行求解。最后给出了数值算例,并验证了所提方法的有效性和正确性。     

无单元法追踪裂纹扩展

无单元法以滑动最小二乘法为基础，可以求解复杂的边值问题，无单元法只需在计算域及边界上布置节点，而不需单元信息，尤其适合于裂纹扩展问题，本文用无单元法实现了对复杂受力条件下的复合型裂纹的开裂追踪，并在算例中得到了非常理想的结果。     

基于双变量无单元法的欧拉梁动力特性计算与分析

以广义移动最小二乘法为理论基础,将同时考虑挠度和转角双变量的无单元法运用于欧拉梁的动力特性计算与分析。以罚函数法引入位移边界,建立欧拉梁无单元法质量矩阵和刚度矩阵的计算方法。运用双变量无单元法计算了四种不同边界条件欧拉梁的自振圆频率和振型,通过与理论解、有限元解、单变量无单元解的比较,表明该法较单变量无单元法具有更高的插值精度,在各种复杂边界条件下均能获得准确的计算结果。特别是在高阶振型中,计算精度明显优于有限元解。最后,通过试算法对多项式基的阶次进行了讨论,给定了在动力计算中的合理取值。     

弹塑性力学问题的无单元伽辽金法

用无单元伽辽金法(EFGM)求解了弹塑性平面问题。EFGM采用移动最小二乘函数近似试函数,并用罚函数法施加本质(位移)边界条件,这是一种与单元划分无关的无网格方法。文中采用了Newton-Raphson增量迭代法进行计算。算例表明:EFGM在求解弹塑性问题时仍具有稳定性好,收敛快的优点。     

两种减少无单元法计算中数值误差的方法

基于无单元法的发展历史和基本理论,讨论了在无单元法计算中起重要作用的A矩阵的几种取值情况,及其对滑动最小二乘法模拟精度的影响,并修正了滑动最小二乘法计算过程中容易产生数值误差的地方。确定了影响A矩阵的几种极端的布点形式,说明了形函数的值与计算点坐标无关的而只与插值点与计算点的相对坐标相关的性质,并给出了数学理论上或数值上的证明。这对无单元法模拟函数滑动最小二乘法的模拟精度有重要的理论价值和实践意义。     

基于代理模型的指数型功能梯度板固有频率优化

针对指数型功能梯度板的固有频率优化问题,提出一种基于代理模型的优化方法。采用一种基于自然邻近插值的无网格数值方法——自然单元法,建立功能梯度板的一阶剪切变形理论的自由振动分析格式。选取样本点,用自然单元法计算相应固有频率,然后基于此建立梯度指数-频率代理模型,采用Nelder-Mead单纯形法优化梯度指数使板的固有频率达到预设值。通过算例验证算法的计算效率和精度。结果表明,基于分段三次Hermite插值构造的代理模型具有较高的精度;基于代理模型的优化结果避免了多次调用自然单元法计算频率,提高了计算效率。      

无网格Galerkin法的理论进展及其应用研究

无网格Galerkin(Element-free Galerkin,EFG)法是无网格方法中应用比较广泛的一种,在介绍其基本特点和原理的基础上,对其移动最小二乘近似过程中涉及到的基函数、权函数的选择、影响域半径的确定等方面取得的新进展进行了介绍.并针对本征边界条件的满足,离散和积分方案的实施,自适应分析及误差分析的应用等一系列相关问题的研究现状及取得的成果进行了详细阐述.同时以受均布载荷的悬臂梁为例,编制了EFG平面弹性程序,验证了EFG法的可行性.最后针对EFG法存在的不足,提出了几个研究方向.     

A natural neighbour-based moving least-squares approach for the element-free Galerkin method

The element-free Galerkin method (EFG) and the natural element method (NEM) are two well known and widely used meshless methods. Whereas the EFG method can represent moving boundaries like cracks only by modifying the weighting functions the NEM requires an adaptation of the nodal set-up. But on the other hand the NEM is computationally more efficient than EFG. In this paper a new concept for the automatic adjustment of nodal influence domains in the EFG method is presented in order to obtain an efficiency similar to the NEM. This concept is based on the definition of natural neighbours for each meshless node which can be determined from a Voronoi diagram of the nodal set-up. In this approach adapted nodal influence domains are obtained by interpolating the distances to the natural neighbours depending on the direction. In the paper we show that this concept leads, especially for problems with grading node density, to a reduced number of influencing nodes at the interpolation points and consequently a significant reduction of the numerical effort. Copyright © 2006 John Wiley & Sons, Ltd.     

Heat transfer analysis of composite slabs using meshless element Free Galerkin method

This paper deals with three dimensional heat transfer analysis of composite slabs using meshless element free Galerkin method. The element free Galerkin method (EFG) method utilizes moving least square (MLS) approximants to approximate the unknown function of temperature Tx). These approximants are constructed by using a weight function, a basis function and a set of coefficients that depends on position. Penalty and Lagrange multiplier techniques have been used to enforce the essential boundary conditions. MATLAB codes have been developed to obtain the EFG results. Two new basis functions namely trigonometric and polynomial have been proposed. A comparison has been made among the results obtained using existing (linear) and proposed (trigonometric and polynomial) basis functions for three dimensional heat transfer in composite slabs. The effect of penalty parameter on EFG results has also been discussed. The results obtained by EFG method are compared with those obtained by finite element method     

A posteriori error approximation in EFG method

Recently, considerable effort has been devoted to the development of the so‐called meshless methods. Meshless methods still require considerable improvement before they equal the prominence of finite elements in computer science and engineering. One of the paths in the evolution of meshless methods has been the development of the element free Galerkin (EFG) method. In the EFG method, it is obviously important that the ‘a posteriori error’ should be approximated. An ‘a posteriori error’ approximation based on the moving least‐squares method is proposed, using the solution, computed from the EFG method. The error approximation procedure proposed in this paper is simple to construct and requires, at most, nearest neighbour information from the EFG solution. The formulation is based on employing different moving least‐squares approximations. Different selection strategies of the moving least‐squares approximations have been used and compared, to obtain optimum values of the parameters involved in the approximation of the error. The performance of the developed approximation of the error is illustrated by analysing different examples for two‐dimensional (2D) potential and elasticity problems, using regular and irregular clouds of points. The implemented procedure of error approximation allows the global energy norm error to be estimated and also provides a good evaluation of local errors. Copyright © 2003 John Wiley & Sons, Ltd.     

A proposal to deal with contact and friction by blending meshfree and finite element methods in forming processes

In this work a computational model based on a meshless method, the Element Free Galerkin Method (EFG), is applied in the simulation of forging processes. Contact and friction are handled by blending finite elements with the EFG in order to overcome the difficulty of meshless methods in dealing with essential boundary conditions. Special interface finite elements are established between the tools and the workpiece, allowing to impose directly those conditions. An application example in forging is analyzed and the numerical solution of the proposed model is compared with experimental results as well as with a traditional finite element solution.     

Sensitivity analysis and shape optimization based on FE–EFG coupled method

By use of 4-node isoparametric quadrangle interface element between finite element (FE) and meshless regions, a collocation approach is introduced to couple firstly FE and element-free Galerkin (EFG) method in this paper. By taking derivative of discreteness equilibrium equation at interface element with respect to design variable, a numerical method for discreteness-based shape design sensitivity analysis in interface element is obtained. The design sensitivity analysis (DSA) of coupled FE–EFG method is achieved by employing the DSA of nodal displacement at the interface element. The numerical method presented is testified by examples. It can be observed excellent agreement between the numerical results and the analytical solution. Finally the shape optimization of fillet is achieved by using coupled FE–EFG method. The result obtained show that imposing of the essential boundary condition is easy to implement, the computational time is reduced and the distortion of mesh is avoided.     

Computational algorithms and applications of element-free Galerkin methods for nonlocal damage models

In using complex material models, especially the strain-gradient-dependent damage models, the convergence of the finite element computation increasingly becomes a problem. Due to large strains in damaging elements the computation may often result in non-convergence. For the higher-order gradient plasticity the special element formulation would often be necessary, which causes additional difficulties in implementation and computations. In recent years, meshless methods have been developed as an alternative to the finite element method (FEM) and can overcome some known shortcomings of FEM. In the present paper an algorithm of element-free Galerkin (EFG) methods for strain-gradient based nonlocal damage models has been developed and used to simulate ductile material damage. The method provides a reliable and robust results for material failure with large damage zones. The strain gradient-dependent terms can be evaluated from the direct differentiation. The investigation confirms that the nonlocal damage model with element-free Galerkin method is suitable for computing the damage problems and predicting the size effects. With the help of the meshless method, material failure in specimens as well as the size effects are predicted accurately.     

数值方法进展:从连续介质到离散粒子模型

数值方法经历了由连续介质到离散粒子模型的进展过程。无网格粒子方法正是离散粒子模型发展的产物,它在纳米时代显示出具大的发展潜能。介绍了无网格粒子方法的背景、原理及其与其他数值方法的区别,探讨了无网格法的基函数、权函数、影响半径、本质边界条件、积分与离散方案等热点问题,列举了这种数值方法的应用现状。最后,介绍了自然单元法、多尺度计算概念、中值定理与局部边界积分方程等,并对无网格粒子方法在未来的发展进行展望。     

Application of meshless EFG method in fluid flow problems

This paper deals with the solution of two-dimensional fluid flow problems using the meshless element-free Galerkin method. The unknown function of velocity u(x) is approximated by moving least square approximants u^h(x). These approximants are constructed by using a weight function, a monomial basis function and a set of non-constant coefficients. The variational method is used for the development of discrete equations. The Lagrange multiplier technique has been used to enforce the essential boundary conditions. A new exponential weight function has been proposed. The results are obtained for a two-dimensional model problem using different EFG weight functions and compared with the results of finite element and exact methods. The results obtained using proposed weight functions (exponential) are more promising as compared to those obtained using existing weight functions (quartic spline and Gaussian)