Locking in the incompressible limit for the element‐free Galerkin method

Volumetric locking (locking in the incompressible limit) for linear elastic isotropic materials is studied in the context of the element‐free Galerkin method. The modal analysis developed here shows that the number of non‐physical locking modes is independent of the dilation parameter (support of the interpolation functions). Thus increasing the dilation parameter does not suppress locking. Nevertheless, an increase in the dilation parameter does reduce the energy associated with the non‐physical locking modes; thus, in part, it alleviates the locking phenomena. This is shown for linear and quadratic orders of consistency. Moreover, the biquadratic order of consistency, as in finite elements, improves the locking behaviour. Although more locking modes are present in the element‐free Galerkin method with quadratic consistency than with standard biquadratic finite elements. Finally, numerical examples are shown to validate the modal analysis. In particular, the conclusions of the modal analysis are also confirmed in an elastoplastic example. Copyright © 2001 John Wiley & Sons, Ltd.     

A new reduced integration solid‐shell element based on EAS and ANS with hourglass stabilization

In this paper, a novel reduced integration eight‐node solid‐shell finite element formulation with hourglass stabilization is proposed. The enhanced assumed strain method is adopted to eliminate the well‐known volumetric and Poisson thickness locking phenomena with only one internal variable required. In order to alleviate the transverse shear and trapezoidal locking and correct rank deficiency simultaneously, the assumed natural strain method is implemented in conjunction with the Taylor expansion of the inverse Jacobian matrix. The projection of the hourglass strain‐displacement matrix and reconstruction of its transverse shear components are further employed to avoid excessive hourglass stiffness. The proposed solid‐shell element formulation successfully passes both the membrane and bending patch tests. Several typical examples are presented to demonstrate the excellent performance and extensive applicability of the proposed element. Copyright © 2015 John Wiley & Sons, Ltd.     

On elimination of shear locking in the element‐free Galerkin method

In this study, a method for completely eliminating the presence of transverse shear locking in the application of the element‐free Galerkin method (EFGM) to shear‐deformable beams and plates is presented. The matching approximation fields concept of Donning and Liu has shown that shear locking effects may be prevented if the approximate rotation fields are constructed with the innate ability to match the approximate slope (first derivative of displacement) fields and is adopted. Implementation of the matching fields concept requires the computation of the second derivative of the shape functions. Thus, the shape functions for displacement fields, and therefore the moving least‐squares (MLS) weight function, must be at least C¹ continuous. Additionally, the MLS weight functions must be chosen such that successive derivatives of the MLS shape function have the ability to exactly reproduce the functions from which they were derived. To satisfy these requirements, the quartic spline weight function possessing C² continuity is used in this study. To our knowledge, this work is the first attempt to address the root cause of shear locking phenomenon within the framework of the element‐free Galerkin method. Several numerical examples confirm that bending analyses of thick and thin beams and plates, based on the matching approximation fields concept, do not exhibit shear locking and provide a high degree of accuracy for both displacement and stress fields. Copyright © 2001 John Wiley & Sons, Ltd.     

Adaptive crack propagation analysis with the element‐free Galerkin method

In this paper, an adaptive analysis of crack propagation based on the error estimation by the element‐free Galerkin (EFG) method is presented. The adaptivity analysis in quasi‐static crack propagation is achieved by adding and/or removing the nodes along the background integration cells, those are refined or recovered according to the estimated errors. These errors are obtained basically by calculating the difference between the values of the projected stresses and original EFG stresses. To evaluate the performance of the proposed adaptive procedure, the crack propagation behaviour is investigated for several examples. The results of these examples show the efficiency and accuracy of the proposed scheme in crack propagation analysis. Copyright © 2002 John Wiley & Sons, Ltd.     

Development of shear locking‐free shell elements using an enhanced assumed strain formulation

The degenerated approach for shell elements of Ahmad and co‐workers is revisited in this paper. To avoid transverse shear locking effects in four‐node bilinear elements, an alternative formulation based on the enhanced assumed strain (EAS) method of Simo and Rifai is proposed directed towards the transverse shear terms of the strain field. In the first part of the work the analysis of the null transverse shear strain subspace for the degenerated element and also for the selective reduced integration (SRI) and assumed natural strain (ANS) formulations is carried out. Locking effects are then justified by the inability of the null transverse shear strain subspace, implicitly defined by a given finite element, to properly reproduce the required displacement patterns. Illustrating the proposed approach, a remarkably simple single‐element test is described where ANS formulation fails to converge to the correct results, being characterized by the same performance as the degenerated shell element. The adequate enhancement of the null transverse shear strain subspace is provided by the EAS method, enforcing Kirchhoff hypothesis for low thickness values and leading to a framework for the development of shear‐locking‐free shell elements. Numerical linear elastic tests show improved results obtained with the proposed formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

One‐dimensional dispersion analysis for the element‐free Galerkin method for the Helmholtz equation

The standard finite element method (FEM) is unreliable to compute approximate solutions of the Helmholtz equation for high wave numbers due to the dispersion, unless highly refined meshes are used, leading to unacceptable resolution times. The paper presents an application of the element‐free Galerkin method (EFG) and focuses on the dispersion analysis in one dimension. It shows that, if the basis contains the solution of the homogenized Helmholtz equation, it is possible to eliminate the dispersion in a very natural way while it is not the case for the finite element methods. For the general case, it also shows that it is possible to choose the parameters of the method in order to minimize the dispersion. Finally, theoretical developments are validated by numerical experiments showing that, for the same distribution of nodes, the element‐free Galerkin method solution is much more accurate than the finite element one. Copyright © 2000 John Wiley & Sons, Ltd.     

Aspects of the use of orthogonal basis functions in the element‐free Galerkin method

The element‐free Galerkin (EFG) method is probably the most widely used meshless method at present. In the EFG method, shape functions are derived from a moving least‐squares approximation using a polynomial basis, a calculation involving the inversion of a small matrix. A new implementation of the EFG method was published soon after the original where an alternative approach using an orthogonal basis was proposed to avoid matrix inversion in the formulation of the shape functions. In this paper we revisit this topic and show that the difficulties associated with the use of a polynomial basis remain present in the orthogonal case. We also show that certain terms in the derivative expressions are omitted in the new implementation of the EFG, which can lead to errors. Finally, we propose a new approach that avoids inversion while maintaining accuracy. Copyright © 2009 John Wiley & Sons, Ltd.     

A gradient‐based adaptation procedure and its implementation in the element‐free Galerkin method

A gradient‐based adaptation procedure is proposed in this paper. The relative error in the total strain energy from two adjacent adaptation stages is used as a stop‐criterion. The refinement–coarsening process is guided by the gradient of strain energy density, based on the assumption: a larger gradient needs a richer mesh and vice versa. The procedure is then implemented in the element‐free Galerkin method for linear elasto‐static problems. Numerical examples are presented to show the performance of the proposed procedure. Copyright © 2003 John Wiley & Sons, Ltd.     

An improved crack analysis technique by element‐free Galerkin method with auxiliary supports

In this study, an improved crack analysis technique by element‐free Galerkin method (EFGM) with auxiliary supports is proposed. To efficiently model the singularity and the discontinuity of the crack, a singular basis function which varies only on the auxiliary supports is added to enrich the standard EFG approximation and the discontinuous shape function is used in the vicinity of the crack surface. The proposed technique improves the accuracy in the near tip field, by using only an initial node arrangement without any modification until the completion of an analysis. A parametric study, which can guide the analyst on the reasonable choice for the formulation and modelling parameters to be used in the technique, is performed on a relative stress norm error and stress intensity factor. In addition, some numerical examples are analysed to verify the effectiveness of the proposed technique for a crack problem. Copyright © 2003 John Wiley & Sons, Ltd.     

A reduced integration solid‐shell finite element based on the EAS and the ANS concept—Geometrically linear problems

In this paper a new reduced integration eight‐node solid‐shell finite element is presented. The enhanced assumed strain (EAS) concept based on the Hu–Washizu variational principle requires only one EAS degree‐of‐freedom to cure volumetric and Poisson thickness locking. One key point of the derivation is the Taylor expansion of the inverse Jacobian with respect to the element center, which closely approximates the element shape and allows us to implement the assumed natural strain (ANS) concept to eliminate the curvature thickness and the transverse shear locking. The second crucial point is a combined Taylor expansion of the compatible strain with respect to the center of the element and the normal through the element center leading to an efficient and locking‐free hourglass stabilization without rank deficiency. Hence, the element requires only a single integration point in the shell plane and at least two integration points in thickness direction. The formulation fulfills both the membrane and the bending patch test exactly, which has, to the authors' knowledge, not yet been achieved for reduced integration eight‐node solid‐shell elements in the literature. Owing to the three‐dimensional modeling of the structure, fully three‐dimensional material models can be implemented without additional assumptions. Copyright © 2009 John Wiley & Sons, Ltd.     

A reduced integration solid‐shell finite element based on the EAS and the ANS concept—Large deformation problems

In this paper we address the extension of a recently proposed reduced integration eight‐node solid‐shell finite element to large deformations. The element requires only one integration point within the shell plane and at least two integration points over the thickness. The possibility to choose arbitrarily many Gauss points over the shell thickness enables a realistic and efficient modeling of the non‐linear material behavior. Only one enhanced degree‐of‐freedom is needed to avoid volumetric and Poisson thickness locking. One key point of the formulation is the Taylor expansion of the inverse Jacobian matrix with respect to the element center leading to a very accurate modeling of arbitrary element shapes. The transverse shear and curvature thickness locking are cured by means of the assumed natural strain concept. Further crucial points are the Taylor expansion of the compatible cartesian strain with respect to the center of the element as well as the Taylor expansion of the second Piola–Kirchhoff stress tensor with respect to the normal through the center of the element. Copyright © 2010 John Wiley & Sons, Ltd.     

Dispersion analysis and element‐free Galerkin solutions of second‐ and fourth‐order gradient‐enhanced damage models

Gradient‐dependent damage formulations incorporate higher‐order derivatives of state variables in the constitutive equations. Different formulations have been derived for this gradient enhancement, comparison of which is difficult in a finite element context due to higher‐order continuity requirements for certain formulations. On the other hand, the higher‐order continuity requirements are met naturally by element‐free Galerkin (EFG) shape functions. Thus, the EFG method provides a suitable tool for the assessment of gradient enhanced continuum models. Dispersion analyses have been carried out to compare different gradient enhanced models with the non‐local damage model. The formulation of the additional boundary conditions is addressed. Numerical examples show the objectivity with respect to the discretization and the differences between various gradient formulations with second‐ and fourth‐order derivatives. It is shown that with the same underlying internal length scale, very different results can be obtained. Copyright © 2000 John Wiley & Sons, Ltd.     

Bounds for quantities of interest and adaptivity in the element‐free Galerkin method

A novel approach to implicit residual‐type error estimation in mesh‐free methods and an adaptive refinement strategy are presented. This allows computing upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest. The proposed approach precludes the main drawbacks of standard residual‐type estimators circumventing the need of flux‐equilibration and resulting in a simple implementation that avoids integrals on edges/sides of a domain decomposition (mesh). This is especially interesting for mesh‐free methods. The adaptive strategy proposed leads to a fast convergence of the bounds to the desired precision. Copyright © 2008 John Wiley & Sons, Ltd.     

A quadtree data structure for the coupled finite‐element/element‐free Galerkin method

In the present paper, we derive an efficient data structure for the organization of the nodes in the coupled finite element/element‐free Galerkin method. With respect to its implementation, we compare various approaches of recursive spatial discretizations that facilitate most flexible handling of the nodes. The goal of the paper is to refine the implementation issues of the data structure which is fundamental to the element‐free Galerkin method and thus to speed‐up this otherwise computationally rather expensive meshfree method. Copyright © 2001 John Wiley & Sons, Ltd.     

The dimension splitting and improved complex variable element‐free Galerkin method for 3‐dimensional transient heat conduction problems

In this paper, by combining the dimension splitting method and the improved complex variable element‐free Galerkin method, the dimension splitting and improved complex variable element‐free Galerkin (DS‐ICVEFG) method is presented for 3‐dimensional (3D) transient heat conduction problems. Using the dimension splitting method, a 3D transient heat conduction problem is translated into a series of 2‐dimensional ones, which can be solved with the improved complex variable element‐free Galerkin (ICVEFG) method. In the ICVEFG method for each 2‐dimensional problem, the improved complex variable moving least‐square approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions. Finite difference method is used in the 1‐dimensional direction, and the Galerkin weak form of 3D transient heat conduction problem is used to obtain the final discretized equations. Then, the DS‐ICVEFG method for 3D transient heat conduction problems is presented. Four numerical examples are given to show that the new method has higher computational precision and efficiency.     

Moving kriging interpolation and element‐free Galerkin method

A new formulation of the element‐free Galerkin (EFG) method is presented in this paper. EFG has been extensively popularized in the literature in recent years due to its flexibility and high convergence rate in solving boundary value problems. However, accurate imposition of essential boundary conditions in the EFG method often presents difficulties because the Kronecker delta property, which is satisfied by finite element shape functions, does not necessarily hold for the EFG shape function. The proposed new formulation of EFG eliminates this shortcoming through the moving kriging (MK) interpolation. Two major properties of the MK interpolation: the Kronecker delta property (?_I( s _J)=δ_IJ) and the consistency property (∑_Iⁿ?_I( x )=1 and ∑_Iⁿ?_I( x )x_Ii=x_i) are proved. Some preliminary numerical results are given. Copyright © 2002 John Wiley & Sons, Ltd.     

An element‐free Galerkin method for 3D crack propagation simulation under complicated stress conditions

This study develops an element‐free Galerkin method based on the moving least‐squares approximation to trace three‐dimensional crack propagation under complicated stress conditions. The crack surfaces are modelled by a collection of planar triangles that are added when cracks propagate. The visibility criterion is adopted to treat the screening effect of the cracks on the influenced domain of a Gaussian point. Cracks are assumed to propagate in the perpendicular planes at crack front points when the strain energy release rates reach the material fracture toughness. This method is unique in that it uses a nonlinear contact iterative algorithm to consider contributions of crack surface interaction to the global equilibrium equations, so that crack opening, sliding and closing under complicated stress states can be efficiently modelled. Two numerical examples of three‐dimensional quasi‐static crack propagation were modelled with satisfactory results. Copyright © 2012 John Wiley & Sons, Ltd.     

An adaptive stabilization strategy for enhanced strain methods in non‐linear elasticity

This paper proposes and analyzes an adaptive stabilization strategy for enhanced strain (ES) methods applied to quasistatic non‐linear elasticity problems. The approach is formulated for any type of enhancements or material models, and it is distinguished by the fact that the stabilization term is solution dependent. The stabilization strategy is first constructed for general linearized elasticity problems, and then extended to the non‐linear elastic regime via an incremental variational principle. A heuristic choice of the stabilization parameters is proposed, which in the numerical examples proved to provide stable approximations for a large range of deformations, different problems and material models. We also provide explicit lower bounds for the stabilization parameters that guarantee that the method will be stable. These are not advocated, since they are generally larger than the ones based on heuristics, and hence prone to deteriorate the locking‐free behavior of ES methods. Numerical examples with two different non‐linear elastic models in thin geometries and incompressible situations show that the method remains stable and locking free over a large range of deformations. Finally, the method is strongly based on earlier developments for discontinuous Galerkin methods, and hence throughout the paper we offer a perspective about the similarities between the two. Copyright © 2009 John Wiley & Sons, Ltd.     

An element‐free method for in‐plane notch problems with anisotropic materials

This study developed an element‐free Galerkin method (EFGM) to simulate notched anisotropic plates containing stress singularities at the notch tip. Two‐dimensional theoretical complex displacement functions are first deduced into the moving least‐squares interpolation. The interpolation functions and their derivatives are then determined to calculate the nodal stiffness using the Galerkin method. In the numerical validation, an interface layer of the EFGM is used to combine the mesh between the traditional finite elements and the proposed singular notch EFGM. The H‐integral determined from finite element analyses with a very fine mesh is used to validate the numerical results of the proposed method. The comparisons indicate that the proposed method obtains more accurate results for the displacement, stress, and energy fields than those determined from the standard finite element method. Copyright © 2013 John Wiley & Sons, Ltd.     

A new one‐point quadrature enhanced assumed strain (EAS) solid‐shell element with multiple integration points along thickness—part II: nonlinear applications

In this work the recently proposed Reduced Enhanced Solid‐Shell (RESS) finite element, based on the enhanced assumed strain (EAS) method and a one‐point quadrature integration scheme, is extended in order to account for large deformation elastoplastic thin‐shell problems. One of the main features of this finite element consists in its minimal number of enhancing parameters (one), sufficient to circumvent the well‐known Poisson and volumetric locking phenomena, leading to a computationally efficient performance when compared to other 3D or solid‐shell enhanced strain elements. Furthermore, the employed numerical integration accounts for an arbitrary number of integration points through the thickness direction within a single layer of elements. The EAS formulation comprises an additive split of the Green–Lagrange material strain tensor, making the inclusion of nonlinear kinematics a straightforward task. A corotational coordinate system is used to integrate the constitutive law and to ensure incremental objectivity. A physical stabilization procedure is implemented in order to correct the element's rank deficiencies. A variety of shell‐type numerical benchmarks including plasticity, large deformations and contact are carried out, and good results are obtained when compared to well‐established formulations in the literature. Copyright © 2006 John Wiley & Sons, Ltd.