Lower Bound Shakedown Analysis by Using the EFG Method and Nonlinear Programming

Lower bound shakedown analysis by using the element free Galerkin method and non-linear programming

The element-free Galerkin (EFG) method is one of the widely used meshfree methods for solving partial differential equations. In the EFG method, shape functions are derived from a moving least-squares (MLS) approximation, which involves the inversion of a small matrix for every point of interest. To avoid the calculation of matrix inversion in the formulation of the shape functions, an improved MLS approximation is presented, where an orthogonal function system with a weight function is used. However, it can also lead to ill-conditioned or even singular system of equations. In this paper, aspects of the IMLS approximation are analyzed in detail. The reason why singularity problem occurs is studied. A novel technique based on matrix triangular process is proposed to solve this problem. It is shown that the EFG method with present technique is very effective in constructing shape functions. Numerical examples are illustrated to show the efficiency and accuracy of the proposed method. Although our study relies on monomial basis functions, it is more general than existing methods and can be extended to any basis functions.

A new formulation of the Element Free Galerkin (EFG) method is developed for the modelling of incompressible materials. Beginning with a mixed variational principle, a selective reduced integration procedure is developed by implementing nodal quadrature. Numerical examples are provided which compare the performance of the proposed technique to the standard EFG approximation. These studies illustrate the capability of the new formulation to eliminate volumetric locking. For the standard method, however, the degree of volumetric locking is shown to be a function of the local support sizes (domains of influence) of the EFG shape functions. Copyright © 1999 John Wiley & Sons, Ltd.

This study aims to interpret the non-linear steady-state heat conduction for temperature-dependent thermal conductivity ($k$ ($T$)) using Element-Free Galerkin (EFG) method. In this present study, a one-dimensional heat conduction problem with uniform heat generation was explicated. Moving Least Squares (MLS) approximants were applied to estimate the unknown function of temperature $T$ ($x$) with $T^h$ ($x$) using linear basis and weight functions. The variational method has been used to develop discrete equations. Essential boundary conditions are enforced by using the Penalty method. The results have been obtained for the one-dimensional model using essential MATLAB codes. The results obtained by the EFG method are compared with the analytical and finite-element method results. The results are also studied by increasing the number of nodes to study the convergence which indicated that EFG has good convergence behavior. The results have also been obtained for different values of the scaling parameter ($α_s$) and any values of αs between 1.8 and 2.0 were found suitable for providing better results in the EFG method.

Transient heat transfer analysis of orthotropic materials considering phase change process based on element-free Galerkin method

In this paper, the adaptive nodal generation procedure based on the estimated local and global error in the element-free Galerkin (EFG) method is proposed. To investigate the possibility of h-type adaptivity of EFG method, a simple nodal refinement scheme is used. By adding new node along the background cell that is used in numerical integration, both of the local and global errors can be controlled adaptively. These errors are estimated by calculating the difference between the values of the projected stresses and original EFG stresses. The ultimate goal of this study is to develop the reliable nodal generator based on the local and global errors that is estimated posteriori. To evaluate the performance of proposed adaptive procedure, the convergence behavior is investigated for several examples.

In this paper, we present a procedure to estimate the error in elliptic equations using the element‐free Galerkin (EFG) method, whose evaluation is computationally simple and can be readily implemented in existing EFG codes. The estimation of the error works very well in all numerical examples for 2‐D potential problems that are presented here, for regular and irregular clouds of points. Moreover, it was demonstrated that this method is very simple in terms of economy and gives a good performance.The results show that the error in EFG approximation may be estimated via the error estimator described in this paper. The quality of the estimation of the error is demonstrated by numerical examples. The implemented procedure of error approximation 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 redesign of cloud of points. Copyright © 2001 John Wiley & Sons, Ltd.

Meshfree analyses of cable-reinforced membrane structures by ALE–EFG method

Stability of piezoelectric FGM rectangular plates subjected to non-uniformly distributed load, heat and voltage

This thesis is devoted to the numerical analysis of mesh-free methods and, in particular, to the study of the possible advantages of the EFG (Element Free Galerkin) mesh-free method against the well-known FE (Finite Element) method. More precisely, the EFG method and the FE method behavior are compared in two particular interesting problems: (1) analysis of volumetric locking in mechanical problems and (2) accurate resolution of transient convection dominated problems. In both cases the good properties and possibilities of mesh-free methods become apparent. <br/>However, in several situations the FE method is still more competitive: for instance, the computation of the FE shape functions and its integrals are less costly, and essential boundary conditions can be easily imposed. Thus, in order to take advantage of the good properties of both methods, a mixed interpolation combining FE and EFG is proposed. This formulation can be applied in two useful situations: (i) enrichment of finite elements with EFG, and (ii) coupling of FE and EFG. An a priori error estimate for the first one is presented and proved. Several examples show the applicability of the mixed interpolation in adaptive computations. Aquesta tesi està dedicada a l'anàlisi numèrica dels mètodes sense malla i, en particular, a l'estudi dels possibles avantatges del mètode EFG (Element Free Galerkin) davant del ben conegut MEF (Mètode dels Elements Finits). Concretament, es comparen el mètode EFG i el MEF en dos problemes concrets d'interès: (1) l'anàlisi del bloqueig volumètric en problemes mecànics i (2) la resolució precisa de problemes transitoris amb convecció dominant. Les bones propietats i possibilitats dels mètodes sense malla es fan evidents en tots dos casos.<br/>Tot i així, en varis aspectes el MEF resulta més competitiu: per exemple, el càlcul de les funcions de forma i de les seves integrals es menys costós, i les condicions de contorn essencials es poden imposar fàcilment. Amb l'objectiu d'aprofitar les bones qualitats dels dos mètodes, es proposa una interpolació mixta combinant elements finits y EFG, aplicable en dues situacions: (i) enriquiment d'elements finits amb EFG i (ii) acoblament d'elements finits i EFG. Per al primer cas, es presenta i demostra una cota a priori de l'error. L'aplicabilitat d'aquesta interpolació mixta en processos adaptatius es mostra amb varis exemples. Esta tesis está dedicada al análisis numérico de los métodos sin malla y, en particular, al estudio de las posibles ventajas del método EFG (Element Free Galerkin) frente al bien conocido MEF (Método de los Elementos Finitos). Concretamente, se comparan el método EFG y el MEF en dos problemas concretos de interés: (1) el análisis del bloqueo volumétrico en problemas mecánicos y (2) la resolución precisa de problemas transitorios con convección dominante. Las buenas propiedades y posibilidades de los métodos sin malla se hacen evidentes en ambos casos.<br/>Sin embargo, en varios aspectos el MEF resulta más competitivo: por ejemplo, el cálculo de las funciones de forma y sus integrales es menos costoso, y las condiciones de contorno esenciales se pueden imponer fácilmente. Con el objetivo de aprovechar las buenas cualidades de ambos métodos, se propone una interpolación mixta combinando elementos finitos y EFG, aplicable en dos situaciones: (i) enriquecimiento de elementos finitos con EFG, y (ii) acoplamiento de elementos finitos y EFG. Para el primer caso, se presenta y demuestra una cota a priori del error. La aplicabilidad de esta interpolación mixta en procesos adaptativos se muestra con varios ejemplos.

The mathematical model of thermomechanical coupling multi-objective topology optimization for anisotropic structures was established using the element-free Galerkin (EFG) method, and the weighted function of compliance and heat dissipation was defined as the objective function. The proposed model and procedure were verified by the topological results based on the finite element method. The multi-objective EFG optimal topological structures have clearer boundary profiles even without using the sensitivity filtering technique. The effects of the weight coefficient, thermal conductivity factor, Poisson's ratio factor, off-angle and volume fraction on the multi-objective EFG optimal topological structure and multi-objective function were evaluated in detail, and reasonable ranges of the above parameters were recommended to improve the heat dissipation and mechanical performance. The multi-objective optimal results of the anisotropic structure were 3D printed and compared with the isotropic material, and their temperature, displacement and stress were improved, which reflects the advantages of orthotropic structures.

The interpolating element-free Galerkin method for three-dimensional elastoplasticity problems

Effect of an efficient numerical integration technique on the element-free Galerkin method

A computational methodology for nucleation of phase transformations in a class of grade 2, non-linearly elastic materials is presented. Nucleation is treated as an energy extremum problem. The material is assumed to be governed by a non-linear, non-local elastic constitutive relation represented by a Landau-Ginzburg potential. The extremum problem is solved using the element-free Galerkin (EFG) method and a perturbed Lagrangian technique. The EFG method is used because of its ability to handle continuity of displacement gradients required in the weak form. Applications to homogeneous nucleation in two dimensions are presented which illustrate the accuracy of the method and its suitability for problems of this type.

The element free Galerkin’s (EFG) technique is proposed for solving 2D and 3D elastic wave equation. In this paper, we compare the waveforms calculated by three methodologies; Finite Difference Method with 4th order accuracy in space (FDM4), traditional FEM and EFG. Traditional FEM and EFG are tested on two individual interpolating functions (IFUNC); 1st order (plane) IFUNC and 2nd order (curved) IFUNC. From the comparison to the exact waveforms of Lamb’s problem, the proposed algorism gives accurate waveforms. By two-dimensional implementation, FEM4 demonstrates more accurate result than traditional FEM with plane IFUNC and EFG with plane IFUNC demonstrates more accurate one than FEM4. However, by employing curved IFUNC, traditional FEM and EFG demonstrate almost same accuracy. Moreover, we expand 2D program to 3D one. On the case of 1st order (plane) interpolating function, EFG improves numerical accuracy of traditional FEM without the increase of computational memory.