无单元法研究现状及展望

无单元法是众多无网格方法中较有代表性的一种,形式简单、明确,计算精度高。因其具有仅需离散的结点信息、解答具有高次连续性、能较好地反映应力高梯度分布并便于跟踪裂纹的扩展过程等优点,无单元法自问世以来获得了广泛的重视,已成为计算力学领域的一个研究热点。文中着重分析了无单元法研究中的热点问题及解决方法,介绍了该方法目前的一些应用范围,并指出其可能的发展方向。     

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.     

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.     

Meshless approach to shape optimization of linear thermoelastic solids

This paper presents a formulation for shape optimization in thermoelasticity using a meshless method, namely the element‐free Galerkin method. Two examples are treated in detail and comparisons with previously published finite element analysis results demonstrate the excellent opportunities the EFG offers for solving these types of problems. Smoother stresses, no remeshing, and better accuracy than finite element solutions, permit answers to shape optimization problems in thermoelasticity that are practically unattainable with the classical FEM without remeshing. For the thermal fin example, the EFG finds finger shapes that are missed by the FEM analysis, and the objective value is greatly improved compared to the FEM solution. A study of the influence of the number of design parameters is performed and it is observed that the EFG can give better results with a smaller number of design parameters than is possible with traditional methods. Copyright © 2001 John Wiley & Sons, Ltd.     

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)     

Analysis of thin plates by the element-free Galerkin method

A meshless approach to the analysis of arbitrary Kirchhoff plates by the Element-Free Galerkin (EFG) method is presented. The method is based on moving least squares approximant. The method is meshless, which means that the discretization is independent of the geometric subdivision into finite elements. The satisfaction of the C ¹ continuity requirements are easily met by EFG since it requires only C ¹ weights; therefore, it is not necessary to resort to Mindlin-Reissner theory or to devices such as discrete Kirchhoff theory. The requirements of consistency are met by the use of a quadratic polynomial basis. A subdivision similar to finite elements is used to provide a background mesh for numerical integration. The essential boundary conditions are enforced by Lagrange multipliers. It is shown, that high accuracy can be achieved for arbitrary grid geometries, for clamped and simply-supported edge conditions, and for regular and irregular grids. Numerical studies are presented which show that the optimal support is about 3.9 node spacings, and that high-order quadrature is required.Dedicated to J. C. Simo     

Topological aspects of meshless methods and nodal ordering for meshless discretizations

The meshless element‐free Galerkin method (EFGM) is considered and compared to the finite‐element method (FEM). In particular, topological aspects of meshless methods as the nodal connectivity and invertibility of matrices are studied and compared to those of the FE method. We define four associated graphs for meshless discretizations of EFGM and investigate their connectivity. The ways that the associated graphs for coupled FE‐EFG models might be defined are recommended. The associated graphs are used for nodal ordering of meshless models in order to reduce the bandwidth, profile, maximum frontwidth, and root‐mean‐square wavefront of the corresponding matrices. Finally, the associated graphs are numerically compared. Copyright © 2001 John Wiley & Sons, Ltd.     

Limit analysis of plates using the EFG method and second‐order cone programming

The meshless element‐free Galerkin (EFG) method is extended to allow computation of the limit load of plates. A kinematic formulation that involves approximating the displacement field using the moving least‐squares technique is developed. Only one displacement variable is required for each EFG node, ensuring that the total number of variables in the resulting optimization problem is kept to a minimum, with far fewer variables being required compared with finite element formulations using compatible elements. A stabilized conforming nodal integration scheme is extended to plastic plate bending problems. The evaluation of integrals at nodal points using curvature smoothing stabilization both keeps the size of the optimization problem small and also results in stable and accurate solutions. Difficulties imposing essential boundary conditions are overcome by enforcing displacements at the nodes directly. The formulation can be expressed as the problem of minimizing a sum of Euclidean norms subject to a set of equality constraints. This non‐smooth minimization problem can be transformed into a form suitable for solution using second‐order cone programming. The procedure is applied to several benchmark beam and plate problems and is found in practice to generate good upper‐bound solutions for benchmark problems. Copyright © 2009 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     

Element-Free Galerkin modelling of composite damage

Delamination and matrix cracking are routine damage mechanisms, observed by post-mortem analysis of laminated structures containing geometrical features such as notches or bolts. Current finite element tools cannot explicitly model an intralaminar matrix microcrack, except if the location of the damage is specified a priori. In this work, a meshless technique, the Element-Free Galerkin (EFG) method, is utilized for the first time to simulate delamination (interlaminar) and intralaminar matrix microcracking in composite laminates.     

Application of reproducing kernel particle method and element-free Galerkin method on the simulation of the membrane of capacitive micromachined microphone in viscothermal air

Simulation of the microphone membrane determines whether highest yield and sensitivity is attainable when it comes to the field of microelectromechanical system (MEMS) capacitive microphone design. Consequently, it is significantly critical to predict and understand the behavior of the membrane in the air. The reproducing kernel particle method and element-free Galerkin, RKPM and EFG respectively, are introduced to differentiate from the traditional finite element method (FEM) since RKPM and EFG models are meshless to greatly improve the problems of FEM with large size aspect ratio. The result from a numerical axisymmetry model of 1 mm radius and 10 μm thickness membrane with fixed boundary condition upon 1 mm thickness viscothermal air is identical to that from the theoretical model. Finally, a MEMS axisymmetry model of a 180 μm radius and 10 μm thickness membrane upon 10 μm thickness viscothermal air is simulated in this paper.     

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.     

Enrichment and coupling of the finite element and meshless methods

A mixed hierarchical approximation based on finite elements and meshless methods is presented. Two cases are considered. The first one couples regions where finite elements or meshless methods are used to interpolate: continuity and consistency is preserved. The second one enriches a finite element mesh with particles. Thus, there is no need to remesh in adaptive refinement processes. In both cases the same formulation is used, convergence is studied and examples are shown. Copyright © 2000 John Wiley & Sons, Ltd.     

RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D analysis of multilayered composite and FGM plates

A meshless collocation (MC) and an element-free Galerkin (EFG) method, using the differential reproducing kernel (DRK) interpolation, are developed for the quasi-three-dimensional (3D) analysis of simply supported, multilayered composite and functionally graded material (FGM) plates. The strong and weak formulations of this 3D static problem are derived on the basis of the Reissner mixed variational theorem (RMVT) where the strong formulation consists of the Euler–Lagrange equations of the problem and its associated boundary conditions, and the weak formulation represents a weighted-residual integral in which the differentiation is equally distributed among the primary field variables and their variations. The early proposed DRK interpolation is used to construct the primary field variables where the Kronecker delta properties are satisfied, and the essential boundary conditions can be readily applied, exactly like the implementation in the finite element method. The system equations of both the RMVT-based MC and EFG methods are obtained using these strong and weak formulations, respectively, in combination with the DRK interpolation. In the illustrative examples, it is shown that the solutions obtained from these methods are in excellent agreement with the available 3D solutions, and their convergence rates are rapid.     

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。     

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.     

Improved element-free Galerkin method for two-dimensional potential problems

Potential difficulties arise in connection with various physical and engineering problems in which the functions satisfy a given partial differential equation and particular boundary conditions. These problems are independent of time and involve only space coordinates, as in Poisson's equation or the Laplace equation with Dirichlet, Neuman, or mixed conditions. When the problems are too complex, they usually cannot be solved with analytical solutions. The element-free Galerkin (EFG) method is a meshless method for solving partial differential equations on which the trial and test functions employed in the discretization process result from moving least-squares (MLS) interpolants. In this paper, by using the weighted orthogonal basis function to construct the MLS interpolants, we derive the formulae of an improved EFG (IEFG) method for two-dimensional potential problems. There are fewer coefficients in the improved MLS (IMLS) approximation than in the MLS approximation, and in the IEFG method fewer nodes are selected in the entire domain than in the conventional EFG method. Hence, the IEFG method should result in a higher computing speed.     

An algorithm to automatically convert distorted finite elements into meshless particles during dynamic deformation

This paper presents an explicit 2D Lagrangian algorithm to automatically convert distorted elements into meshless particles during dynamic deformation. It also provides the contact and sliding algorithms to link the particles to the finite elements. For this approach the initial grid is composed entirely of finite elements and the response is computed with finite elements until portions of the grid become highly strained. When finite elements on the boundaries reach a user-specified plastic strain they are converted to particles and linked to the adjacent finite element grid. This approach allows for the use of accurate and efficient finite elements in the lower distortion regions, and for the use of meshless particles in the higher distortion regions. Example computations are presented to demonstrate the accuracy and utility of this approach.