Hybrid element-free Galerkin-finite element method for electromagnetic field computations

An automatically adaptive element free method is presented to analyze the seismic response of liquefiable soils. The method is based on the element free Galerkin method (EFGM) and the fission procedure that is part of h-refinement, indicated by error estimation. In the proposed method, a posteriori error estimate procedure that depends on the energy norm of stress and the T-Belytschko (TB) stress recovery scheme is incorporated. The effective cyclic elasto-plastic constitutive model is used to describe the nonlinear behavior of the saturated soil. The governing equations are established by u-p formulation. The proposed method can effectively avoid the volumetric locking due to large deformation that usually occurs in numerical computations using the finite element method (FEM). The efficiency of the proposed method is demonstrated by evaluating the seismic response of an embankment and comparing it to results obtained through FEM. It is shown that the proposed method provides an accurate seismic analysis of saturated soil that includes the effects of liquefaction

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

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.

Shakedown theorems are exact theories of classical plasticity for the direct computation of the load-carrying capacity under varying loads. Based on the classical Melan’s theorem, a numerical solution procedure for determining the shakedown load of elasto-perfectly plastic structure is presented firstly making use of the element free Galerkin (EFG) method. The numerical implementation is very simple and convenient because it is only necessary to construct an array of nodes in the domain under consideration. The reduced-basis technique is adopted here to solve the mathematical programming iteratively in a sequence of reduced self-equilibrium stress subspaces with very low dimensions. The self-equilibrium stress field is expressed by linear combination of several self-equilibrium stress basis vectors with parameters to be determined. These self-equilibrium stress basis vectors can be generated by performing equilibrium iteration procedure during elasto-plastic incremental analysis. The Complex method is used to solve nonlinear programming and determine the maximal load amplifier. The numerical results show that it is efficient and accurate to solve shakedown analysis problems by using the EFG method and nonlinear programming.

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.

This paper presents a coupling technique for integrating the element‐free Galerkin method (EFGM) with the fractal finite element method (FFEM) for analyzing homogeneous, isotropic, and two‐dimensional linear‐elastic cracked structures subjected to mixed‐mode (modes I and II) loading conditions. FFEM is adopted for discretization of the domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprise both the EFG and the finite element (FE) shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM–FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no special enriched basis functions or no structured mesh with special FEs are necessary and no post‐processing (employing any path independent integrals) is needed to determine fracture parameters, such as stress‐intensity factors (SIFs) and T‐stress. The numerical results show that SIFs and T‐stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also, a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions. A numerical example on mixed‐mode condition is presented to simulate crack propagation. As in the proposed coupled EFGM–FFEM at each increment during the crack propagation, the FFEM mesh (around the crack tip) is shifted as it is to the new updated position of the crack tip (such that FFEM mesh center coincides with the crack tip) and few meshless nodes are sprinkled in the location where the FFEM mesh was lying previously, crack‐propagation analysis can be dramatically simplified. Copyright © 2010 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.

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.

In many engineering problems, the meshfree methods (MMs) have been dynamically projected and increasingly advanced in order to overwhelm some hitches in the predictable numerical methods. Over the past three decades in many different application area, MMs have found their way ranging from solid mechanics analysis, fluid problems, vibration analysis, heat transfer and optimization to numerical solutions of all kinds of (partial) differential equations. As every technique has shortcomings, the meshfree method also has drawbacks like higher computational cost and imposition of boundary condition which can be overruled by coupling it with the finite element method (FEM). In the past two decades, coupled MMs and FEM have appeared into a new session of computational methods with significant achievement. In addition, a noteworthy amount of growth has been made in addressing the major deficiencies that were present in the conventional methods and MMs at the premature phases. The objective of the present work is to provide a comprehensive review of various coupling techniques used for interface elements of MMs and FEM and general discussion on shape function formulation of FE and element free Galerkin method (EFGM). Key contribution of coupling techniques for coupled EFGM and FEM to structure mechanics application as primary concern.

Abstract:Seepage is an important issue in geotechnical engineering projects such as dams, levees, canals, reservoirs, etcetera, and has made researchers and specialists to pay special attention to this problem. Lots of destructions in buildings, towers, dams, tunnels, slopes, and etcetera have been reported in history which, have been caused by liquefaction, ground water movement or other related seepage problems. In this article, seepage in soil-water systems and their related problems are studied using Element Free Galerkin Method (EFGM). For time dependent problems, implied finite difference method is used. A program is also written for both stable and time dependent cases. With estimation of a function and comparing the results with the exact function, the quantity of the errors is calculated and the accuracy of the method is evaluated. Based on this method, various geotechnical engineering problems are solved and analyzed. The results showed that not only is the EFGM more exact and accurate in comparison with other conventional methods but the time and costs for network production is omitted. With variations in boundaries, the application of EFGM is easier than that of Finite Element Method (FEM). Also, the ability of solving seepage problems without any limitation is a preference of EFGM in comparison with other approximate methods.

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.

A topology optimization, based on a coupling method of finite element and meshless method, is proposed for continuum structure. A reasonable arrangement of the meshless domain can guarantee the accuracy of the meshless methods and meanwhile keep the computational efficiency of the finite element method. Besides, as the coupling method is adopted, the displacement boundary conditions are applied to finite elements nodes by standard finite element method. A dual-level density approximant is carried out to approximate and interpolate a unified and continuous density field. An unstable nodes phenomenon is observed in the interface domain, leading to nonconvergence of the equilibrium iterations. A smooth blending function and an energy convergence criterion are used to circumvent the convergent difficulty. Three benchmark problems of topology optimization are given to demonstrate the effectiveness of the proposed approach. Neither “islanding” phenomenon nor other common numerical instabilities of the SIMP method occur in this approach.

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

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

Application of NEM in seepage analysis with a free surface