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.     

A 2‐D meshless model for jointed rock structures

According to the characteristic structural features of jointed rock structures, a meshless model is proposed for the mechanics analysis of jointed rock structures based on the moving least‐squares interpolants. In this model, a jointed rock structure is regarded as a system of relatively intact rock blocks connected by joints or planes of discontinuity; these rock blocks are modelled by general shaped anisotropic blocks while these joints and planes of discontinuity are modelled by interfaces. The displacement field of each block is constructed by the moving least‐squares interpolants with an array of points distributed in the block. To deal with the discontinuities of rock structures, the displacement fields are constructed to be discontinuous between blocks. The displacement fields and their gradients are continuous in each block, hence no post processing is required for the output of strains and stresses. The finite element mesh is totally unnecessary, so the time‐consuming mesh generation is avoided. The rate of convergence can exceed that of finite elements significantly, and a high resolution of localized steep gradients can be achieved. Furthermore, the discontinuities of rock structures are also fully taken into consideration. The present method is developed for two‐dimensional linear elastic analysis of jointed rock structures, and can be extended to three‐dimensional and non‐linear analysis. Copyright © 2000 John Wiley & Sons, Ltd.     

An enriched meshless method for non‐linear fracture mechanics

This paper presents an enriched meshless method for fracture analysis of cracks in homogeneous, isotropic, non‐linear‐elastic, two‐dimensional solids, subject to mode‐I loading conditions. The method involves an element‐free Galerkin formulation and two new enriched basis functions (Types I and II) to capture the Hutchinson–Rice–Rosengren singularity field in non‐linear fracture mechanics. The Type I enriched basis function can be viewed as a generalized enriched basis function, which degenerates to the linear‐elastic basis function when the material hardening exponent is unity. The Type II enriched basis function entails further improvements of the Type I basis function by adding trigonometric functions. Four numerical examples are presented to illustrate the proposed method. The boundary layer analysis indicates that the crack‐tip field predicted by using the proposed basis functions matches with the theoretical solution very well in the whole region considered, whether for the near‐tip asymptotic field or for the far‐tip elastic field. Numerical analyses of standard fracture specimens by the proposed meshless method also yield accurate estimates of the J‐integral for the applied load intensities and material properties considered. Also, the crack‐mouth opening displacement evaluated by the proposed meshless method is in good agreement with finite element results. Furthermore, the meshless results show excellent agreement with the experimental measurements, indicating that the new basis functions are also capable of capturing elastic–plastic deformations at a stress concentration effectively. Copyright © 2003 John Wiley & Sons, Ltd.     

A moving Kriging interpolation‐based meshless method for numerical simulation of Kirchhoff plate problems

This paper mainly proposes an alternative way for numerical implementation of thin plates bending based on a new improvement of meshless method, which is combined between the standard element‐free Galerkin method and one different shape functions building technique. The moving Kriging (MK) interpolation is applied instead of the traditional moving least‐square approximation in order to overcome Kronecker's delta property where the standard method does not satisfy. Obviously, the deflection of the thin plates is approximated via the MK interpolation. To illustrate this approach, numerical analysis is examined in both regular and irregular systems. Three examples with different geometric shapes of thin plates undergoing a simply supported boundary are performed. In addition, two important parameters of the present method are also analyzed. A good agreement can be found among the proposed, analytical and finite element methods. Copyright © 2008 John Wiley & Sons, Ltd.     

A perturbation method for stochastic meshless analysis in elastostatics

A stochastic meshless method is presented for solving boundary‐value problems in linear elasticity that involves random material properties. The material property was modelled as a homogeneous random field. A meshless formulation was developed to predict stochastic structural response. Unlike the finite element method, the meshless method requires no structured mesh, since only a scattered set of nodal points is required in the domain of interest. There is no need for fixed connectivities between nodes. In conjunction with the meshless equations, classical perturbation expansions were derived to predict second‐moment characteristics of response. Numerical examples based on one‐ and two‐dimensional problems are presented to examine the accuracy and convergence of the stochastic meshless method. A good agreement is obtained between the results of the proposed method and Monte Carlo simulation. Since mesh generation of complex structures can be a far more time‐consuming and costly effort than the solution of a discrete set of equations, the meshless method provides an attractive alternative to finite element method for solving stochastic mechanics problems. Copyright © 2001 John Wiley & Sons, Ltd.     

A residual‐based finite element method for the Helmholtz equation

A new residual‐based finite element method for the scalar Helmholtz equation is developed. This method is obtained from the Galerkin approximation by appending terms that are proportional to residuals on element interiors and inter‐element boundaries. The inclusion of residuals on inter‐element boundaries distinguishes this method from the well‐known Galerkin least‐squares method and is crucial to the resulting accuracy of this method. In two dimensions and for regular bilinear quadrilateral finite elements, it is shown via a dispersion analysis that this method has minimal phase error. Numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods. It is found that even for unstructured meshes this method retains a high level of accuracy. Copyright © 2000 John Wiley & Sons, Ltd.     

A procedure for approximation of the error in the EFG method

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.     

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.     

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.     

Least‐squares collocation meshless method

A finite point method, least‐squares collocation meshless method, is proposed. Except for the collocation points which are used to construct the trial functions, a number of auxiliary points are also adopted. Unlike the direct collocation method, the equilibrium conditions are satisfied not only at the collocation points but also at the auxiliary points in a least‐squares sense. The moving least‐squares interpolant is used to construct the trial functions. The computational effort required for the present method is in the same order as that required for the direct collocation, while the present method improves the accuracy of solution significantly. The proposed method does not require any mesh so that it is a truly meshless method. Three numerical examples are studied in detail, which show that the proposed method possesses high accuracy with low computational effort. Copyright © 2001 John Wiley & Sons, Ltd.     

Dispersion analysis of the gradient weighted finite element method for acoustic problems in one,two, and three dimensions

This paper reports a detailed analysis on the numerical dispersion error in solving one-, two-, and three-dimensional acoustic problems governed by the Helmholtz equation using the gradient weighted finite element method (GW-FEM) in comparison with the standard FEM and the modified methods presented in the literatures. The discretized system equations derived based on the gradient weighted operation corresponding to the considered method are first briefed. The discrete dispersion relationships relating the exact and numerical wave numbers defined in different dimensions are then formulated, which will be further used to investigate the dispersion effect mainly caused by the approximation of field variables. The influence of nondimensional wave number and wave propagation angle on the dispersion error is detailedly studied. Comparisons are made with the classical FEM and high-performance algorithms. Results of both theoretical and numerical experiments show that the present method can effectively reduce the pollution effect in computational acoustics owning to its crucial effectiveness in handing the dispersion error in the discrete numerical model.     

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.     

A meshless singular boundary method for three‐dimensional elasticity problems

This study documents the first attempt to extend the singular boundary method, a novel meshless boundary collocation method, for the solution of 3D elasticity problems. The singular boundary method involves a coupling between the regularized BEM and the method of fundamental solutions. The main idea here is to fully inherit the dimensionality and stability advantages of the former and the meshless and integration‐free attributes of the later. This makes it particularly attractive for problems in complex geometries and three dimensions. Four benchmark 3D problems in linear elasticity are well studied to demonstrate the feasibility and accuracy of the proposed method. The advantages, disadvantages, and potential applications of the proposed method, as compared with the FEM, BEM, and method of fundamental solutions, are also examined and discussed. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability and accuracy of finite element methods for flow acoustics. II: Two‐dimensional effects

This is the second of two articles that focus on the dispersion properties of finite element models for acoustic propagation on mean flows. We consider finite element methods based on linear potential theory in which the acoustic disturbance is modelled by the convected Helmholtz equation, and also those based on a mixed Galbrun formulation in which acoustic pressure and Lagrangian displacement are used as discrete variables. The current paper focuses on the effects of numerical anisotropy which are associated with the orientation of the propagating wave to the mean flow and to the grid axes. Conditions which produce aliasing error in the Helmholtz formulation are of particular interest. The 9‐noded Lagrangian element is shown to be superior to the more commonly used 8‐noded serendipity element. In the case of the Galbrun elements, the current analysis indicates that isotropic meshes generally reduce numerical error of triangular elements and that higher order mixed quadrilaterals are generally less effective than an equivalent mesh of lower order triangles. Copyright © 2005 John Wiley & Sons, Ltd.     

A meshless method for axi‐symmetric poroelastic simulations: numerical study

A meshless model, based on the meshless local Petrov–Galerkin (MLPG) approach, is developed and implemented for the solution of axi‐symmetric poroelastic problems. The solution accuracy and the code performance are investigated on a realistic application concerning the prediction of land subsidence above a deep compacting reservoir. The analysis addresses several numerical issues, including the parametric selection of the optimal size of the local sub‐domains for the weak form and the nodal supports, the appropriate integration rule, and the linear system solver. The results show that MLPG can be more accurate than the standard finite element (FE) method on coarse discretizations, with its superiority decreasing as the nodal resolution increases. This is due to both a slower convergence rate and a progressively higher computational cost compared to FE. These drawbacks can be partially mitigated by improving the efficiency of the numerical integration and the system solver with the aid of projection techniques based on Krylov subspace methods. The outcome of the present analysis supports the development of coupled methods where a limited number of MLPG nodes are used to locally improve a FE solution. Copyright © 2006 John Wiley & Sons, Ltd.     

A point interpolation method for two‐dimensional solids

A point interpolation method (PIM) is presented for stress analysis for two‐dimensional solids. In the PIM, the problem domain is represented by properly scattered points. A technique is proposed to construct polynomial interpolants with delta function property based only on a group of arbitrarily distributed points. The PIM equations are then derived using variational principles. In the PIM, the essential boundary conditions can be implemented with ease as in the conventional finite element methods. The present PIM has been coded in FORTRAN. The validity and efficiency of the present PIM formulation are demonstrated through example problems. It is found that the present PIM is very easy to implement, and very flexible for obtained displacements and stresses of desired accuracy in solids. As the elements are not used for meshing the problem domain, the present PIM opens new avenues to develop adaptive analysis codes for stress analysis in solids and structures. Copyright © 2001 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.     

Implementation of meshless LBIE method to the 2D non‐linear SG problem

In this paper the meshless local boundary integral equation (LBIE) method for numerically solving the non‐linear two‐dimensional sine‐Gordon (SG) equation is developed. The method is based on the LBIE with moving least‐squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables. The approximation functions are constructed entirely using a set of scattered nodes, and no element or connectivity of the nodes is needed for either the interpolation or the integration purposes. A time‐stepping method is employed to deal with the time derivative and a simple predictor–corrector scheme is performed to eliminate the non‐linearity. A brief discussion is outlined for numerical integrations in the proposed algorithm. Some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped SG equation is investigated. The final numerical results confirm the ability of method to deal with the unsteady non‐linear problems in large domains. Copyright © 2009 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.