Composite Structures 93 (2011) 2031–2041

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

A review of meshless methods for laminated and functionally graded platesand shells

K.M. Liew a,⇑, Xin Zhao a, Antonio J.M. Ferreira b

a Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kongb Faculdade de Engenharia da Universidade do Porto, Departamento de Engenharia Mecanica, Rua Dr Roberto Frias, 4200-465 Porto, Portugal

a r t i c l e i n f o

Article history:Available online 25 February 2011

Keywords:Meshless methodsLaminated plates and shellsFunctionally graded plates and shells

0263-8223/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compstruct.2011.02.018

⇑ Corresponding author.E-mail address: kmliew@cityu.edu.hk (K.M. Liew).URL: (K.M. Liew).

a b s t r a c t

This review focuses mainly on the developments of element-free or meshless methods and their applica-tions in the analysis of composite structures. This review is organized as follows: a brief introduction toshear deformation plate and shell theories for composite structures, covering the first-order and higher-order theories, is given in Section 2. A review of meshless methods is provided in Section 3, with mainemphasis on the element-free Galerkin method and reproducing kernel particle method. The applicationsof meshless methods in the analysis of composite structures are discussed in Section 4, including staticand dynamic analysis, free vibration, buckling, and non-linear analysis. Finally, the problems and difficul-ties in meshless methods and possible future research directions are addressed in Section 5.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction astrophysical area to a variety of engineering problems, such as

The finite element method (FEM) has dominated engineeringcomputations since its invention and its application has expandedto a variety of engineering fields. In FEM, the continuum is dividedinto a finite number of elements, the behavior of each element isspecified by a finite number of parameters and the solution ofthe complete system as an assembly of its elements can be ob-tained according to the procedures as those applicable to the stan-dard discrete problems [1,2]. This discretization procedure is themajor advantage of FEM. However, there still exist some draw-backs due to its mesh-based interpolation. For instance, it suffersheavily from mesh distortion in large deformation and intensiveremeshing requirements in dealing with the structures with com-plex geometries and discontinuities.

The interpolation in meshless methods is entirely based on a setof scattered nodes instead of meshes, this unique feature enablesthem to eliminate at least part of the difficulties existing in FEM,such as mesh distortion and remeshing. One of the earliest mesh-less methods is smooth particle hydrodynamics (SPH) [3,4], whichinitially was used for modeling astrophysical phenomena. It is con-sidered an interpolation method based on kernel estimation, andthe accuracy of this interpolation method was analyzed byMonaghan [5,6]. Since then, numerous researchers have furtherdeveloped the SPH and substantial improvements [7–9] have beenmade. The applications of SPH have been extended from the initial

ll rights reserved.

high velocity impact computation [10,11], simulation of surfacetension [12], and metal forming [13].

Nayroles et al. [14] presented a meshless method called diffuseelement method (DEM) using moving least square approximations.By including the derivatives of interpolants omitted in the DEMand employing a more accurate integration method, Belytschkoet al. [15,16] proposed the element-free Galerkin method (EFG)and applied it to the analyses of thin plates [17], thin shells [18],and dynamic fracture [19,20].

In a different way, Liu et al. [21–23] developed the reproducingkernel particle method (RKPM) by introducing a correction func-tion for the kernel approximation in SPH to meet the reproducingconditions. Similarly, Liu et al. [24,25] also proposed the movingleast-square reproducing kernel methods (MLSRK) using movingleast-square interpolation function. Some applications of RKPM in-volve structure dynamics [26], large deformation analysis[22,27,28], metal forming [29,30], beams and plates [31], and thinshells [32].

In addition to EFG and RKPM, many other meshless methodshave been developed. Two meshless methods that are distinct fromEFG and RKPM are the hp-cloud method [33–35] and partition ofunity method [36–38], both of which are based on the partitionof unity concept. Contrary to the EFG method and RKPM that de-pend on global weak forms, the meshless local Petrov–Galerkinmethod (MLPG) [39–42] is based on local weak forms.

Some major advantages of meshless methods can be summa-rized as below:

(1) The methods can provide more accurate approximations forstructures with complex geometries than FEM.

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(2) The shape functions are constructed in terms of higher-ordercontinuous weight functions and possess compact support.

(3) They have great advantages in handling problems with largedeformations, moving discontinuities such as crack propaga-tion in dynamic fracture and phase transformation in thedevelopment of advanced materials.

(4) Nodes can be easily added (h-adaptivity) on portions whererefinement is needed for a higher solution accuracy.

(5) They can be coupled with FEM or boundary element meth-ods (BEM) to avoid their inherent drawbacks.

Several review articles [16,23,43,44], which cover the classifica-tion, application and computer implementation aspects of mesh-less methods, have been reported in literature. The aim of thissurvey is to provide a general overview of the developments andapplications of meshless methods originating from EFG and RKPMfor the analysis of composite structures.

The topics related to the developments of meshless methods in-clude the construction of approximations, shape functions, com-pleteness, convergence, enforcement of boundary conditions, andintegration methods. The applications involve static analysis, freevibration, buckling and postbuckling, non-linear analysis, and tran-sient dynamics. This review is organized as follows: Section 2 givesa brief introduction of shear deformation plate and shell theoriesfor composite structures. Section 3 provides a general overviewof the element-free Galerkin method and reproducing kernel parti-cle method. Section 4 describes the applications of meshless meth-ods in the analysis of structures. The last section discusses theproblems and difficulties inherent in meshless methods and possi-ble future research directions.

2. Shear deformation theories

Composite laminates are composed of stacking layers of differ-ent materials and fiber orientation, and can be treated as plate orshell elements. Therefore, the two commonly used theories forplates and shells, the classical bending theory and shear deforma-tion theories, have been employed for the analysis of compositelaminates. When a plate (shell) has a ratio of thickness to represen-tative dimensions is 1/20 or less, it is considered to be thin and theclassical bending theory will be adopted. When the ratio is greaterthan 1/20, the shear deformation theories should be used. In clas-sical (Kirchhoff) plate or shell theories, it is assumed that the nor-mals to the midplane before deformation remain straight andnormal to the plane after deformation. This assumption (Kirchhoff)is equivalent to the disregard of the effect of transverse shearforces. Another assumption is that the transverse normal stressesare neglected. In shear deformation theories, the Kirchoff hypoth-esis is relaxed, the transverse normals do not remain perpendicularto the midplane after deformation. This indicates that the effects oftransverse shear deformation are included in the theories. In thefirst-order shear deformation theory, the transverse shear strainis assumed to be constant through the thickness direction. In high-er-order theories, the transverse shear strain is expressed as afunction of the thickness coordinate. In this section, the commonlyused shear deformation theories are reviewed, and a brief surveyfor the other two three-dimensional theories, including theReddy’s layer-wise theory and zig-zag theory, are also provided.

2.1. The first-order and higher-order shear deformation theories

Reissner [45,46] proposed the first-order shear deformationplate and shell theories based on kinematics analysis to take intoaccount the transverse shear deformation, and Mindlin [47] pre-sented a first-order shear deformation plate theory that includesrotary inertia terms for free vibration analysis of plates. Because

the first-order shear deformation theories based on Reissner–Mindlin kinematics violates the zero shear stress condition onthe top and bottom surfaces of the shell or plate, a shear correctionfactor is required to compensate for the error due to a constantshear strain assumption through the thickness. For isotropic struc-tures, the shear correction factor is taken as 5/6, which is com-puted such that the strain energy due to the constant transverseshear stresses equals the strain energy due to the true transversestresses predicted by the three-dimensional elasticity theory. Fora composite structure, the shear correction factor not only dependson the material and geometric properties but also the loading andboundary conditions. The Reissner–Mindlin theory has been ap-plied to the analysis of a variety of structures. Whitney [48] exam-ined the shear correction factors for orthotropic laminates understatic loads, and investigated the effects of shear deformation inthe bending of laminated plates [49]. Whitney and Pagano [50]studied the shear deformation in heterogeneous anisotropic plates,and Reissner [51] presented a consistent treatment of transverseshear deformations in laminated anisotropic plates. Additionalapplications include the wave propagation in heterogeneous plates[52], the vibration of antisymmetrical angle-ply laminated plates[53], vibration of shear deformable curved anisotropic compositeplates [54], vibration of Mindlin plate [55], moderately thick lam-inated shells [56], and the stability of laminated plates [57].

The classical Reissner–Mindlin theory is able to provide a suffi-ciently accurate description of global response for thin to moderatelythick laminated structures. However, to obtain a better prediction ofshear deformation and transverse normal strains in laminated platesand shells, higher-order theories are required. Murthy [58] presentedan improved transverse shear deformation theory for laminatedanisotropic plates under bending. A displacement approach like thatin Reissner–Mindlin theory was used but the displacement field waschosen so that the transverse shear stress vanishes on the plate sur-faces. Thus, this improved theory eliminates the requirement forusing a shear correction factor in computing the shear stresses. Lev-inson [59] reported a similar theory for the static and dynamic anal-ysis of isotropic plates but using different displacement fieldexpressions. Later, Reddy [60] pointed out that the equilibrium equa-tions derived by Murthy [58] and Levinson [59] are variationallyinconsistent, and developed a simple higher-order theory for lami-nated composite plates, which leads to a consistent derivation ofthe displacement field and associated equilibrium equations. Thistheory is based on the assumptions that the membrane displace-ments are cubic functions of the thickness coordinate and the trans-verse displacement is constant through the thickness. The majoradvantage of this theory is that the assumed displacement fields re-sult in a parabolic shear strain distribution through the thickness,thus agreeing more closely with linear elasticity and removing theneed for using a shear correction factor. Reddy and Liu [61] alsodeveloped a similar higher-order shear deformation theory for lam-inated elastic shells. The theories [60,61] have been successfully ap-plied to the analyses of static, vibration and buckling of laminatedplates and shells [62–69]. The assessment of different shear deforma-tion theories and more applications can refer to Noor and Burton [70],Reddy [71–73], Kapania [74], and Reddy and Arciniega [75].

The aforementioned first-order and high-order shear deforma-tion theories are considered as equivalent single-layer theories.These theories are able to provide a sufficiently accurate descrip-tion of global response of thin to moderately thick laminates,including gross deflections, critical buckling loads, and fundamen-tal vibration frequencies. Among the theories, the first-order sheardeformation theory is the most popular one because it providesboth adequate accuracy and efficiency. Compared to the first-ordershear deformation theory, the higher-order shear deformationtheories offer a slight improvement in accuracy but at the expenseof an increase in computation effort.

K.M. Liew et al. / Composite Structures 93 (2011) 2031–2041 2033

2.2. The layer-wise theory of Reddy and the zig-zag theory

Although the aforementioned higher-order shear deformationtheories are capable of providing accurate descriptions of the glo-bal response of composite structures, but they often fail to provideaccurate three-dimensional stresses and strains at the ply levelnear geometric and material discontinuities. Thus the three-dimensional elasticity theory or a layerwise laminate theory thatcontains full three-dimensional kinematics and constitutive rela-tions is required. Several layerwise models were reported in liter-ature. Whitney [49] used a layerwise quadratic variation of thetransverse stresses to improve the gross response of laminatedplates. This model yields good results for deflections, natural fre-quencies, and buckling loads, but the equations of equilibriumare not consistent in an energy sense due to the adoption of equa-tions of the classical lamination theory. Reddy [76] proposed alayer-wise theory to evaluate the detailed local stress analysis. Thisrefined theory ensures an accurate description of the three-dimen-sional displacement field by expanding it as a linear combinationof thickness coordinate and unknown functions of position of eachlayer. As special cases, the classical, Reissner–Mindlin and othershear deformation theories can be deduced from this theory [77].Di Sciuva [78,79] presented an improved shear deformation theorynamed the zig-zag theory, in which the three-dimensional problemis reduced to the two-dimensional one by assuming a displace-ment field that allows piecewise linear variation of the membranedisplacement, and constant value of the transverse displacementthrough the thickness. The assumed displacement field fulfillsthe static and geometric continuity conditions between the contig-uous layers, and accounts for the distortion of the deformednormal.

The layer-wise theory has been used for the analysis of piezo-electrically actuated beams [80], modeling of delamination in com-posite laminates [81], postbuckling of laminated circularcylindrical shells [77], and composite shaft rotodynamic [82]. Theapplications of the zig-zag theory include the bending of laminatedcomposite shells [83], buckling analysis for delaminated compos-ites [84], coupled thermo-electric-mechanical smart compositeplates [85], and thick beams [86].

3. EFG and RKPM

The element-free Galerkin method is based on moving leastsquare approximations (MLS), which were initially presented byLancaster and Salkauskas [87] and were first employed by Nayroleset al. [14] for the development of the diffuse element method(DEM). By modifying and refining the DEM method, Belytschkoet al. [15] proposed the element-free Galerkin method. Recognizingthe lack of completeness of the SPH method, Liu et al. [21] derivedthe reproducing kernel particle method by adding a correct func-tion for the SPH kernel. The RKPM is intrinsically a kernel-basedmethod. It has been demonstrated that the kernel approximationthat meets consistent condition is identical to the related MLSapproximation [16,88]. Therefore, the reproducing kernel particleinterpolant can be considered another version of MLS interpolant.In this section, the constructions of shape functions in the EFGmethod and RKPM are briefly introduced, and the issues relatedto approximations, convergence, integration techniques, andenforcement of essential boundary conditions are addresses.

3.1. Meshless approximations

3.1.1. MLS approximationsIn MLS approximations, the approximation of a function u(x)

can be expressed as

uhðx; �xÞ ¼Xm

i¼1

pið�xÞaiðxÞ ð1Þ

where pið�xÞ are monomial basis functions of order m, and ai(x) arenon-constant coefficients that are functions of spatial coordinatesx. The coefficients ai(x) are computed by performing a minimizationof the difference between local approximation and the nodal values(on the weighted least-square L2 norm)

J ¼X

I

wðx� xIÞ½uhðx; xIÞ � uðxIÞ�2 ð2Þ

where w(x � xI) is called weight function. With the solution of ai(x),the final MLS approximation can be obtained as

uhðxÞ ¼Xn

I¼1

UIðxÞuI ð3Þ

where UI(x) is the shape function, and n is the number of localnodes. The detailed construction procedure has been given in refer-ences [15,16,89].

3.1.2. Reproducing kernel particle methodLiu et al. [21,26] found that the SPH kernel function fails to meet

the completeness requirement, and made a modification to thekernel function by introducing a correction function

uhðxÞ ¼Xn

I¼1

Cðx; x� xIÞwðx� xIÞuðxIÞDxI ð4Þ

where C(x; x � xI) is the correction function, which is expressed by alinear combination of polynomial basis functions

Cðx; x� xIÞ ¼ Hðx� xIÞbðxÞ ð5Þ

in which H(x � xI) is a vector of basis function and b(x) is a vector ofcoefficients, which are functions of x. The unknown coefficients b(x)can be solved by imposing the reproducing conditions, and the finalapproximation is written as

uhðxÞ ¼Xn

I¼1

WIðxÞuI ð6Þ

where WI(x) is the shape function. It has been demonstrated that,when DxI = 1 is chosen, the MLS is equivalent to the RKPM if thesame weight functions and basis functions are selected.

The shape functions constructed according to MLS or RKPMhave compact support, or domain of influence, which is identicalto the support of the corresponding weight function. The supportcan be of arbitrary shapes, such as circular, rectangular, andsquare. The two popular types of supports include the circularand rectangular. The commonly used weight functions encompassthe exponential, the cubic spline, and the quartic spline. The con-tinuity and smoothness of shape functions inherit from weightfunctions, the properties of which have been addressed in refer-ences [15,16,89–91]. The polynomial basis is the favorite amongbasis functions, it is also possible to use singular functions andother special functions as basis functions as long as they aremonomial. For instance, a trigonometric basis was used by Bely-tschko et al. [92] and Fleming [93] for the approximation of cracktip fields, and was also employed by Rao and Rahman [94] forfracture analysis of cracks. Some similar trigonometric bases wereused in Fourier analysis of RKPM [95], and computational acous-tics [96,97].

The shape functions in meshless methods possess local charac-teristics due to their compact support. It is noteworthy that, inmost cases, the parameter l is not the true nodal value (uI – uh(xI))because the shape functions lack the delta function property.

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3.2. Discontinuities in approximation

It is necessary to introduce discontinuities into meshlessapproximations for problems that involve multiple connected do-mains and various discontinuities in the solutions or their deriva-tives. The incorporation of discontinuities into approximations canbe achieved by modifying weight functions, basis functions, andapproximations. There are three ways to modify weight functionsfor the inclusion of discontinuities in meshless approximations.The first method is called the visibility method reported byBelytashko et al. [15]. In this approach, the boundaries of the bodyand any interior lines of discontinuity are considered opaque.Although this method causes undesired interior discontinuitiesand difficulties along nonconvex boundaries, it has been demon-strated that the resulting approximations from the visibility meth-od can lead to convergent solutions despite the drawbacks [90].The second method is the diffuse method [98], in which continuityis retained by wrapping the support partially around a nonconvexboundary. The method is motivated by the way light diffractsaround a sharp corner, but the equations used in constructingthe support and the weight function bear almost no relationshipto the equations of diffraction. This method is quite effective indealing with cases with smoothing boundaries and sharp cornerssuch as cracks, but it is difficult to extend it to three dimensions.The third method is the transparency method [98], in which conti-nuity is introduced by adding varying transparency to cracks neartheir tips. This method requires that nodes not be placed immedi-ately adjacent to boundaries to avoid a sharp gradient in theweight function across the line ahead of the crack.

Fleming et al. [93] augmented the basis by including functionssimilar to the near-tip asymptotic fields [99] to incorporate discon-tinuities into approximations. This method is easy to implement asit only requires a modification of the basis. However, it becomesexpensive for multiple cracks. Meanwhile, the same authors alsoproposed another approach for the introduction of discontinuities.In contrast to the above method, the trial functions are modified toinclude the first term of the near-tip asymptotic expansion for thedisplacement field [93]. The advantage of this method is that it canbe used for multiple cracks with little additional expense. Thedrawback is that it may cause poor conditioning of equations dueto the sensitivity of stress intensity factors to perturbations. Ithas been shown that both techniques offer good accuracy and effi-ciency for crack problems.

Krongauz and Belytschko [100] presented a technique for theincorporation of discontinuities in derivatives of meshless solutionsfor one- and two-dimensional problems. Special jump functions thatcontain discontinuities in derivatives are employed to enrich theapproximations. The resulting discontinuous approximation hascompact support, which leads to sparse and banded discrete equa-tions. The accuracy of this treatment of discontinuities has been ver-ified by numerical comparison with closed-form solutions. Thisapproach is also compatible with standard CAD database.

3.3. Completeness, stability, and convergence

The meshless approximations must satisfy consistency for con-vergence requirements. Since consistency is mainly used in finitedifference methods, in which a structured grid of nodes is em-ployed, it is difficult to prove the consistency for meshless methodsbecause they employ an unstructured grid of nodes. Therefore, thecompleteness, which replaces the consistency in the convergenceanalysis of finite element methods, is examined for convergenceanalysis in meshless methods. The completeness in Galerkin meth-ods plays the same role as consistency in finite difference methods.Completeness requires that the approximation or its derivativesmeet reproducing conditions.

For kernel estimation-based meshless methods, the complete-ness can be restored in kernel approximations through a correctiontransformation. Shepard functions [101] were used as correctionfunctions to meet constant completeness. By adding a correctionfunction in the SPH kernel function [21], Liu et al. [21] demon-strated that the RPKM shape functions fulfill the linear complete-ness requirement. Belytschko et al. [15,16] proved that the MLSapproximations satisfy linear completeness if the basis function in-cludes all constant and linear monomials, and the completeness oforder k can be attained when the basis is complete in the polyno-mials of order k [92].

The aforementioned completeness restoration confines to thecorrection of the approximation functions. To meet the complete-ness requirement of the derivatives of approximations, it is alsoachievable to amend the derivatives of approximations directlywithout necessarily restoring the completeness of the approxima-tion functions. Krongauz and Belytschko [102] developed a tech-nique for regaining linear completeness by constructing thederivatives of approximations from linear combinations of theShepard function derivatives. The corrected derivatives are calledpseudo-derivatives because they are not integrable. The othertechniques for correcting derivatives include the Johnson–Beisselcorrection [10], the Randles and Libersky renormalization [7],and the symmetrisation [5]. The completeness of approximationsis usually examined by performing patch test. More detailed dis-cussions on the completeness of mesh-free methods have been gi-ven by Belytschko et al. [88].

In addition to the completeness, the stability of meshless meth-ods is also important to their robustness. A variety of instabilitiesoccurs in meshless methods. The tensile instability was first iden-tified by Swegle et al. [103] in SPH. A conservative smoothingscheme [104,105] and higher-order spline kernels [106,107] wererecommended to eliminate this instability. Bessiel and Belytschko[108] found that the instabilities occurred with nodal integrationof the Galerkin form of the momentum equation when using mov-ing least square approximations. A unified stability analysis wasprovided by Belytschko et al. [109] to investigate the causes of var-ious instabilities. Two distinct instabilities, tensile instability andhigh-frequency instability, were analyzed. It was found that theformer results from the interaction of the second derivative of ker-nel, and the latter results from rank deficiency. It was revealed thatthe tensile instability occurs when Eulerian kernel is employed,and disappears if Lagrangian kernel is used; but the instabilitydue to rank deficiency, however, occurs for both Lagrangian andEulerian kernels with nodal integration. Several stabilization tech-niques were checked and their effectiveness was discussed indetail.

The convergence rate of meshless methods is established byperforming error analysis. Durate and Oden [34] presented an errorestimator for the hp-cloud approximations to study the conver-gence rate, and Liu et al. [110] reported interior error estimatesfor the investigation of convergence of the reproducing kernel par-ticle approximation using Fourier analysis for arbitrary domain.The error estimates of bounded domain and irregular particle dis-tribution were discussed by Liu et al. [24,25]. A framework for erroranalysis of RKPM is established by Han and Meng [111], they de-rived optimal order error estimates for RKPM interpolants on a reg-ular family of particle distribution, and the theoretical errorestimates were verified by the demonstration of the convergenceorders. Chung and Belytsko [112] proposed local and global errorestimates for the EFG method by comparing projected stresseswith these given directly by the EFG solution. Voth and Christon[113] investigated numerical errors associated with the RKPM dis-cretization of various wave equations. Krysl and Belytschko [90]examined the convergence rate of the EFG method for second-or-der problems in an energy norm. Estimates of the convergence rate

K.M. Liew et al. / Composite Structures 93 (2011) 2031–2041 2035

were derived for smooth and non-smooth solutions obtained fromconforming and non-conforming EFG bases.

3.4. Enforcement of essential boundary conditions

As the shape functions constructed according to MLS and RKPMdo not possess Kronecker delta property, the coefficients of theinterpolants are not equal to the nodal values, therefore, the essen-tial boundary conditions cannot be imposed directly. Several tech-niques, including Lagrange multiplier approach, penalty method,transformation method, and coupled finite element and particlemethod, have been developed to enforce the boundary conditionsin meshless methods.

The Lagrange multiplier method was employed by Belytschko etal. [15] in EFG to impose the essential boundary conditions.Mukuerjee and Mukherjee [114] presented a modified Lagrangemultiplier method by redefining the norm and using fluxes as La-grange multipliers. The Lagrange method is accurate and usefulfor two-dimensional problems with less cost, but the computa-tional cost increases significantly when problems become biggerdue to rising unknowns. The accuracy of this method, however,may suffer from poor discretization, and the stability is easily af-fected by the choice of shape functions.

The coupled finite element method was reported by Krongauzand Belytschko [115] for the enforcement of essential boundaryconditions in EFG. This technique employs a string of finite ele-ments along the essential boundaries, and the shape functionsfrom these edge finite elements are then combined with shapefunctions of the approximations so that the essential boundaryconditions can be imposed as in finite elements. This method en-sures the approximation is continuous and retains the complete-ness. This method is particularly useful when finite elements areused as a background cell for quadrature, since then a finite ele-ment mesh is already available.

Another method used in EFG for the enforcement of essentialboundary conditions is penalty method [116,117]. Compared tothe Lagrange multiplier method, this method is easily implementedand does not increase computational cost. An important consider-ation is the choice of an appropriate penalty parameter. A general re-view of the above three methods was given in reference [118].

The full transformation method was proposed by Chen et al.[22,28,29] to allow the direct imposition of the essential boundaryconditions in RKPM. This method needs to compute the transfor-mation matrix between the generalized coordinate and the nodalcoordinate for the unknown variables. This method has been usedin large deformations analysis, and it is more efficient than the La-grange multiplier method because the transformation matrix iscomputed only at initial stage and is reusable at each incrementalstep. Later, Chen and Wang [119] modified the full transformationmethod and presented the mixed transformation method and theboundary singular kernel method. In the mixed transformationmethod, the discrete nodes are separated into a boundary groupthat includes the kinematically restrained related nodes, and aninterior group that contained the remaining nodes. The transfor-mation operation is only performed on the smaller matrix formedby the nodes in the boundary group, thus the computational effortneeded is less than that in the full transformation method. In theboundary singular kernel method, singularities are introduced intothe kernel functions associated with the restrained nodes, this al-lows the direct imposition of boundary constrains. Gunther andLiu also reported a boundary transformation method [120] forthe imposition of boundary conditions in RKPM. This technique isbased on d’Alember’s principle, and was applied to the analysisof fluid–structure interaction. The other techniques used in RKPMinclude the admissible approximation method [121] and colloca-tion method [122].

3.5. Integration methods

In mesh-free methods, Gaussian quadrature is commonly em-ployed to evaluate the integrals in Galerkin weak form. A back-ground mesh or a background cell structure has to beconstructed to perform the integration [15,16,19,21]. It has beendemonstrated that the integration error could be considerable ifthe quadrature cells do not match the shape function supports[123]. Although some improvements for quadrature have beenpresented [117,123–125], in order to get more accurate results, arelatively dense quadrature cell and dense Gauss points are neededfor problems with irregular domains, this leads to the increase ofthe computational cost.

To reduce the computation effort, the nodal integration tech-niques have been developed by performing the integrals basedon nodes only instead of quardrature cells [111,124]. However,Beissel and Belytschko [108] demonstrated that the nodal inte-gration of EFG resulted in a spatial instability due to the underintegration of the weak form. Several stabilizing techniques havebeen proposed to eliminate the spatial instability. Bonet andKulasegaram [124] presented a least-square stabilization tech-nique to eliminate spurious mode in nodal integration, and Chenet al. [126] reported a stabilized conforming nodal integration ap-proach for Galerkin mesh-free methods to eradicate spatial insta-bility. In this approach, an integration constraint (IC) wasintroduced as a necessary condition for a linear exactness inmesh-free Galerkin approximation. It has been revealed that theGauss integration violates the integration constraint (IC) andleads to noticeable error for linear solutions. A strain smoothingstabilization procedure was proposed to compute the nodal strainby applying a divergence theorem. The attractive feature of thisintegration method is that the domain integration is replacedby performing a contour integration along particle cells, thusthe computation effort is significantly reduced, especially forlarge deformation problems. This approach has been successfullyapplied to the analyses of plates, shells, and large deformationproblems [127–132].

4. Applications

Numerous improvements have been made since the appearanceof EFG and RKPM [130–145], and their applications have spannedmany engineering areas [146–176]. In this section, the applicationsof EFG, RKPM, and the methods derived from them are discussed,mainly covering the static and free vibration, buckling and post-buckling, non-linear analysis, and transient dynamics of structures.

4.1. Crack problems

One of the earlier applications of meshless methods is the sim-ulation of crack growth. The EFG method with linear MLS approx-imations was used to study the two-dimensional elastostatic andelastodynamic fracture [15,16,19,93,90,116] problems, such ascrack growth from a fillet, crack propagation in concrete, andedge-cracked plate under impact loading. Krysl and Belytschko[177] conducted the modelling of arbitrary three-dimensionaldynamically propagating cracks in elastic bodies using EFG withexplicit time integration. Several examples were studied, includingthe simulation of mixed-mode growth of centre-through crack in afinite plate, mode-I surface-breaking penny-shaped crack in a cube,penny-shaped crack growing under general mixed-mode condi-tions, and torsion-tension rectangular bar with centre-throughcrack. It is found that, compared to FEM, the EFG method is moresuitable for crack problems because it does not require remeshingand avoids the need for excessive refinement near the crack front.

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Some crack problems were also dealt with by using modified EFGmethod [178–183].

The RKPM was also employed in the analysis of crack problems.Li et al. employed the RKPM to perform the simulation of dynamicshear band propagation and failure mode transition [184], andboth 2-D and 3-D shear band formations under large deformation[185]. Jun and Im [186] used a multiple-scale RKPM for the simu-lation of adiabatic shear band formation in a thermo-viscoplasticmaterial, and Hao et al. [187] carried out the modelling and simu-lation of intersonic crack growth.

Recently, Sun et al. [188] presented a meshfree simulation ofcracking and failure of structures by combining the EFG methodand a cohesive segment method, and Zhang et al. [189] investi-gated the 2D fracture problems via an improved element-freeGalerkin method. A boundary element free method, a variant ofthe EFG method, was employed for fracture analysis of 2D piezo-electric solids [190] and the interaction between collinear interfa-cial cracks [191].

4.2. Static analysis of structures

Krysl and Belytschko [17] first applied EFG to the bending anal-ysis of thin plates. The thin plate theory (or Kirchhoff plates) wasemployed and the boundary conditions were enforced by Lagrangemultipliers. Gauss integration was performed on a background cellto evaluate the stiffness matrix. The plates with different shapesand loading conditions were considered, and the effects of the reg-ular and irregular nodal distributions on the accuracy of solutionswere investigated. The same authors also conducted the analysisfor thin shells using the EFG method [18]. The geometrically exacttheory of shear flexible shells was adopted and appropriate adjust-ments were made to account for the Kirchhoff–Love hypothesis.Numerical integration was carried out on the background cells byGaussian quadrature. It was demonstrated that the method yieldsgood results for quadratic polynomial basis. The membrane lockingwhich appears in the numerical model was alleviated by enlargingthe domains of influence of the EFG nodes for the quadratic basis,and it was removed completely by using quartic polynomial basis.Noguchi et al. [192] extended EFG to the analysis of three-dimen-sional thin shell structures. The geometry of curved surface was ex-panded in a two dimensional space by using a mapping technique,and the nodes were generated on this two-dimensional mappedspace. The formulation for shells accounts for transverse shearstrains, therefore, it is applicable for both thin and relatively thickshells. The bi-cubic and quartic basis functions were adopted forthe construction of shape functions to eradicate shear and mem-brane locking. Dolbow and Belytschko [193] developed an alterna-tive implementation of the element-free Galerkin method using aselective reduced integration formulation to remove volumetriclocking in plates. The shear locking was also investigated by Huertaand Fernan dez-Mendez [194], Kanok-Nukulchai et al. [195], andAskes et al. [196].

Liew et al. [197] analyzed the laminated composite plates andbeams with piezoelectric patches using the EFG method. Thefirst-order shear deformation plate theory was employed and thefull transformations method was used to impose essential bound-ary conditions. The static shape control of piezo-laminated com-posite beams and plates was studied, and the influence ofstacking sequence on the change in shapes was examined. Numer-ical examples demonstrated that EFG produced accurate solutionsin analyzing the shape control of piezo-laminated compositebeams and plates. Peng et al. [198] carried out the bending analysisof un-stiffened and stiffened folded plates using the EFG method.The first-order shear deformation plate theory was adopted andthe folded plates were considered as assemblies of un-stiffenedor stiffened folded plates. More applications of the EFG method

encompass the analyses of laminated folded plate structures[199], stiffened corrugated plates [164,170,200,201], functionallygraded plates [169,176,202], and prestressed concrete beams[203].

The RKPM also has been used for the static analysis of platesand shells. Donning and Liu [31] carried out the static analysis ofshear deformable beams and plates via RKPM. The Mindlin–Reiss-ner theory was employed, and Cardinal splines were used to con-struct the shape functions. The shear and membrane lockingwere eliminated due to the adoption of Cardinal splines. Chenand Wang [129] developed a meshfree shell formulation in Carte-sian coordinates. Two techniques, dummy node technique andpseudo-inverse method, were used to construct constrained repro-ducing kernel shape functions. The shear and membrane lockingwere removed by using a stabilized nodal integration method.Using a kp-Ritz method derived from RKPM, Zhao et al. [204,205]discussed the thermoealstic response of functionally graded plateand shells. The first-order shear deformation theory was employedand the effects of temperature and different volume fraction expo-nents on the behavior of functionally graded plates and shells wereinvestigated. Liu and Taciroglu [206] gave a semi-analytical solu-tion for Almansi–Michell problems of piezoelectric cylinders usingthe RKPM, and Wang and Chen [127] provided a meshfreeMindlin–Reissner plate formulation using a locking-free stabilizednodal integration.

Bobaru and Mukherjee [207] presented a formulation for shapeoptimization of linear thermoelastic solids using the EFG method.They investigated the influence of the number of design parame-ters and observed that the EFG can give better results with a smal-ler number of design parameters than FEM. The shape sensitivityanalysis and shape optimization in planar elasticity were also per-formed by the same authors [208]. Zhang et al. [209] conducted theanalysis of shape optimization using RKPM and an enriched geneticalgorithm, and Kim et al. [210,211] carried out the design sensitiv-ity analysis for shell structures and three-dimensional contactproblems.

4.3. Buckling and free vibration

Liew et al. [212] studied the elastic buckling behavior of stiff-ened and un-stiffened folded plates under partial in-plane edgeloads. The formulation was based on the first-order shear deforma-tion theory and element-free Galerkin method. The stiffness andinitial stress matrices of the flat plates as determined by themesh-free Galerkin method were superposed to obtain the stiff-ness and initial stress matrices of the entire folded plate. The solu-tions show the EFG method has a good accuracy and convergencerate. Using the similar method, the authors also conducted thebuckling analysis of corrugated plates [213] and stiffened struc-tures [214]. Zhao et al. [215] investigated the mechanical and ther-mal buckling response of functionally graded plates using anelement-free kp-Ritz method. The first-order shear deformationplate theory was adopted to account for the transverse shear defor-mation and the shear locking was eliminated by using a stabilizednodal integration method. The effects of volume fraction exponenton the buckling response of functionally graded plates wereexamined.

Using the EFG method, Liu and Chen [216,217] studied thevibration response of the thin plates of complicated shape, andLiu et al. [218] investigated the free vibration of thin shells struc-tures. Zhao et al. [147,219,220] carried out the free vibration anal-ysis of laminated composite cylindrical panels using the element-free kp-Ritz method. The Love’s thin shell theory was used andthe effects of lamination schemes on the frequency characteristicsof panels were examined. Based on the Love’s thin shell theory andkp-Ritz method, A series of investigations also have been

K.M. Liew et al. / Composite Structures 93 (2011) 2031–2041 2037

conducted to describe the vibration behaviour of rotating cylindri-cal shells and panels [221], conical shells [222,223], and function-ally graded plates and shells [224]. The dynamic stability ofcylindrical shells and panels were also examined by combiningthe Bolitin method and kp-Ritz method [225,226]. A free vibrationanalysis of folded plates was provided by Peng et al. [227] using thefirst-order shear deformation theory and the EFG method, and freevibration of sandwich beams with functionally graded core wasinvestigated by Amirani et al. [228].

4.4. Non-linear analysis

Meshless methods have demonstrated great advantages andpromising potential in non-linear analysis of structures becausethey are flexible in handling discontinuities and large deformationproblems, in which severe mesh distortion usually occurs whenusing FEM. Chen et al. [22] presented a formulation for large defor-mation analysis of non-linear structures based on the RKPM, theelasto-plasticity and hyperelasticity problems were studied. Ithas been demonstrated that the RKPM shows good performanceand high solution accuracy for these kinds of large deformationproblems. The numerical results also revealed that RKPM is ableto avoid volume locking in treating incompressible materials underlarge deformation. Jun et al. [27] proposed an efficient algorithm,the explicit RKPM, for the simulation of large deformation prob-lems. In this modified RPKM, the Lagrangian description referredto the reference configuration with explicit time integration wereemployed, and the calculation of shape function and the searchingprocedure, which are the most time-consuming processes duringcomputation, are needed only once. The RKPM was also employedfor other non-linear problems, such as numerical simulation oflarge deformation of thin shell structures [32], rubbers, and shearbanding [185].

Basing on the RKPM and wavelet theory, Liu et al. [229–232]proposed a multiple-scale RKPM for large deformation problems.The essential feature of this method is that the response of a com-plex mechanical system is separated into different scales, eachscale is investigated separately, and the solution can be viewedas a sum of the low- and high-scale responses of the mechanicalsystem. For a high gradient problem, the low-scale component ofthe solution shows the fundamental mode of the total responseof the solution, while the high-scale component is dominated bythe structure of the high gradient portion of the solution. Thehigh-scale component of the solution is useful as a criterion forthe procedure of adaptivity. This multiple-scale RKPM has been ap-plied to the analyses of localized shear deformation [233], damagefracture and localization [234], contact problems [235], Solids[236] and stress concentration problems [237].

The RKPM has also been extended to metal forming analysis.Chen et al. [29] presented a meshless formulation for loading pathdependent material behavior and frictional contact conditionsbased on a Lagrangian reproducing kernel particle method. Numer-ical simulations for sheet metal stretch by a cylindrical punch,sheet metal stretch by a hemispherical punch, and ring compres-sion were performed, and the results demonstrated that the RKPMshows good accuracy and stability in dealing with this kind of largedeformation problems. Chen et al. [30] also used a collocation for-mulation in the boundary integral of the contact constraint equa-tions formulated by a penalty method to conduct metal forminganalysis. The other work on metal forming analysis using RKPM in-cludes those given by Xiong et al. [238], Yoon and Chen [239], andShangwu et al. [240].

Much effort has been devoted to the non-linear analysis of plateand shell structures using meshless methods. Liew et al. [241] pre-sented a formulation for large deformation analysis using theRKPM, and carried out a numerical simulation of thermomechani-

cal behaviors of shape memory alloys [242]. Using the element-free kp-Ritz method, Liew et al. [150,243,244] conducted the geo-metrically non-linear analysis of laminated plates based on thefirst-order shear deformation plate theory and the von Kármánstrains. The formulation was obtained by combining the RKPMand arc-length iterative algorithm, and the load–deflection re-sponses for various laminated composites were investigated. Sim-ilarly, the non-linear analyses also have been performed forcomposite cylindrical shells [245], as well as functionally gradedplates and shells [246,247].

Belinha and Dinis [248] conducted the non-linear analysis oflaminated plates using element-free Galerkin method, and Renet al. [149,249] performed the modelling and simulation ofsuperelastic behaviours of shape memory alloys. The EFG meth-od was also employed by other researcher for the non-linearanalysis of folded plate structures [250], lowerbound shakedownanalysis of structures made of elastic-perfectly plastic material[251], non-linear wave propagation in damaged hysteretic mate-rials [252], and simulation of and non-linear dynamic fracture[253].

In additional to the aforementioned meshless methods, othermesh-free methods that are based on EFG and RKPM have beendeveloped [254–259]. Additional application areas of meshlessmethods include transient analysis [260,261], probalistic mechan-ics and reliability [256], explicit dynamic analysis [262], linear andnon-linear dynamic analysis of solids [263].

5. Conclusions and discussion

Meshless methods and their applications in the analysis of engi-neering structures have been reviewed, with emphasis on the meth-ods originating from EFG and RKPM rather than other meshlessapproaches [264–273]. The aim of this survey is to provide a generaldescription of the developments and applications of meshless meth-ods, the detailed mathematical implementation is not concerned. Ithas been demonstrated that the meshless methods are able to han-dle a variety of engineering problems, and offer great advantagesover conventional numerical methods, especially in dealing withdiscontinuities and large deformation problems. However, thereare still some challenges remaining. For three-dimensional model-ling of structures, especially for thin shell structures, the computa-tional cost is still too expensive. Although the nodal integrationthat does not need background quadrature cells makes the methodcomputationally efficient, but the accuracy and stability may suffer.Therefore, efficient algorithms and stability techniques need to bedeveloped. For multidimensional problems with discontinuities, itis necessary to develop the techniques for incorporating discontinu-ous derivatives and error estimators for convergence.

An interesting area that can utilize the advantages of meshlessmethods is the multi-scale modelling of composite materials andstructures. To thoroughly understand the mechanical propertiesof composite materials, the deformation of composite materialshas to be investigated at different phases. There are two main rea-sons for employing the multiscale modeling for materials analysis:1) the existence of fundamentally multi-scale material phenom-ena; 2) the characterizations of material properties at macroscopiclevel are quite different with those at material level. Althoughmuch effort has been devoted to this area using finite elementmethods and mesh-free methods, there are still a lot of aspects thatneed to be studied.

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