a meshfree-enriched finite element method for compressible and near-incompressible elasticity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2012; 90:882–914Published online 28 March 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.3349

A meshfree-enriched finite element method for compressible andnear-incompressible elasticity

C. T. Wu1,*,†, W. Hu1 and J. S. Chen2

1Livermore Software Technology Corporation, Livermore, CA 94550, U.S.A.2Department of Civil and Environmental Engineering, University of California, Los Angeles, CA 90095, U.S.A.

SUMMARY

In this paper, a two-dimensional displacement-based meshfree-enriched FEM (ME-FEM) is presented for thelinear analysis of compressible and near-incompressible planar elasticity. The ME-FEM element is estab-lished by injecting a first-order convex meshfree approximation into a low-order finite element with anadditional node. The convex meshfree approximation is constructed using the generalized meshfree approxi-mation method and it possesses the Kronecker-delta property on the element boundaries. The gradient matrixof ME-FEM element satisfies the integration constraint for nodal integration and the resultant ME-FEM for-mulation is shown to pass the constant stress test for the compressible media. The ME-FEM interpolation isan element-wise meshfree interpolation and is proven to be discrete divergence-free in the incompressiblelimit. To prevent possible pressure oscillation in the near-incompressible problems, an area-weighted strainsmoothing scheme incorporated with the divergence-free ME-FEM interpolation is introduced to providethe smoothing on strains and pressure. With this smoothed strain field, the discrete equations are derivedbased on a modified Hu–Washizu variational principle. Several numerical examples are presented to demon-strate the effectiveness of the proposed method for the compressible and near-incompressible problems.Copyright © 2012 John Wiley & Sons, Ltd.

Received 17 June 2011; Revised 21 September 2011; Accepted 1 October 2011

KEY WORDS: meshfree; finite element; convex; volumetric locking; near-incompressible

1. INTRODUCTION

Recent developments in meshfree methods add a dimension to computational mechanics [1–5].Those methods do not rely on the traditional mesh-based approach to define approximation func-tions. In comparison with the conventional finite element methods, the characteristics of arbitrarysmoothness and conformity of the approximation, p-version-like intrinsic basis for higher conver-gence rate make the meshfree methods effective alternative numerical techniques for industrialapplications [6, 7]. Galerkin meshfree methods using moving least-squares (MLS) [8, 9] or repro-ducing kernel (RK) [4,10] approximations have been successfully applied to the solid and structuralanalyses in the past decade. However, most Galerkin meshfree methods consume higher CPU timeand demand more memory storage than the finite element methods, which greatly limits their appli-cations in solving large-scale industrial problems. One of the main computational costs in theGalerkin meshfree method is the extra effort to impose constraints and essential boundary con-ditions whenever the Kronecker-delta property does not hold in the approximation. Other costsinclude a high-order quadrature rule for the integration of weak form and a large bandwidth in thestiffness matrices. Furthermore, volumetric locking in meshfree method exists when a low-ordermeshfree approximation and small nodal supports are utilized. Relieving locking by the use of a

*Correspondence to: C. T. Wu, Livermore Software Technology Corporation (LSTC), 7374 Las Positas Road, Livermore,CA 94551, U.S.A.

†E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.

MESHFREE-ENRICHED FINITE ELEMENT METHOD 883

higher-order meshfree approximation implies larger nodal support sizes and therefore additionalcomputational loads.

Several coupling procedures for meshfree methods and finite element methods were proposed[11–13] to improve the computational efficiency and to simplify the essential boundary conditiontreatments in the meshfree methods. On the other hand, enriched finite element methods such as theextended finite element method [14] and the generalized finite element method [15] were developedbased on the partition of unity method [16] to generate desirable approximation functions for cap-turing specific physical behavior such as strong discontinuity. The enrichment function is extrinsicto the finite element basis function and introduces additional DOFs even though the meshes areindependent of the location of discontinuities. The unique feature of those enriched finite elementmethods is that the enrichment functions can in principle be arbitrary and are certainly not limitedto polynomials. A different way of constructing an effective meshfree approximation for solid andstructural analyses is to incorporate the convex approximants (non-negative and exactly reproducingaffine functions) [17,18]. The meshfree convex approximation guarantees the unique solution insidea convex hull with a minimum distributed data set and poses a weak Kronecker-delta property at theboundaries, and therefore avoids the special treatments on the essential boundaries. Wu et al. [19]provided a unified approach that can generate specific convex approximations and reproduce sev-eral existing meshfree approximations, which are referred to as the generalized meshfree (GMF)approximation. Park et al. [20] embarked on a detailed eigenanalysis for meshfree convex approx-imation and reported that a large critical time step can be used in the explicit dynamic analysis.Their dispersion analysis results also reveal that meshfree convex approximation exhibits smallerlagging phase and amplitude errors than conventional MLS approximation in the full-discretizationof the wave equation. The applications of meshfree convex approximation to the analysis of near-incompressible solids and Kirchhoff–Love shells were also presented recently [17, 18]. Meshfreemethods using convex approximations have been developed in conjunction with a high-order Gaussquadrature rule for the numerical integration. However, their efficiency improvements have notbeen investigated.

Attempts [21,22] have been made to introduce stabilization of nodal integration in meshfree meth-ods to improve the computational efficiency. Most methods were proposed in the content of strainsmoothing techniques [22, 23] using the meshfree nonconvex approximations. Chen et al. [22, 24]presented a stabilized conforming nodal integration (SCNI) method together with ReproducingKernel Particle Method (RKPM) approximation for the linear and nonlinear analyses of solids.Wang and Chen [25] extended the method to the analysis of Mindlin plate bending problems. SCNImethod was also employed in the point interpolation method [26] and radial point interpolationmethod [27] for solids and shells. A modified SCNI method [28] was developed to further suppressthe nonzero energy modes that exist in SCNI for problems having large surface to volume ratio inthe geometry. Alternatively, the strain smoothing technique in SCNI method was applied to the FEMto provide a softening effect for improving the solution accuracy in the forms of various node-based,element-based, and edge-based smoothed finite element formulations [29, 30].

It is well-known that the low-order displacement-based FEM exhibits a volumetric locking prob-lem in the near-incompressible analyses. Many approaches have been developed to overcome thisdifficulty. Among some popular methods are the methods of mixed formulation [31] and enhancedassumed strain formulation [32]. The u/p mixed formulation requires a stable inf–sup pair ofspaces for the displacement and pressure, and usually a high-order finite element is recommended.Low-order pairs are usually supplemented by stabilization or post-processing procedures [33] thatremove spurious pressure modes to counteract the lack of inf–sup stability. The u/p mixed formula-tion for Stokes or linear elasticity problems using the MINI triangular element of Arnold et al. [34]has a close relationship with the stabilized method using equal-order interpolation. The MINI ele-ment is based on an equal-order continuous piecewise linear interpolations for velocity and pressurewith velocity field enriched by bubble functions. A static condensation of bubble functions yieldsa matrix formulation, which can be made identical to the matrix obtained from stabilized methodusing equal-order interpolation [35]. On the other hand, the basic idea of enhanced strain formula-tion consists in enriching the space of discrete strains by means of suitable local modes or bubblefunctions. However, it was soon discovered that classical enhanced strain formulation suffered from

Copyright © 2012 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2012; 90:882–914DOI: 10.1002/nme

884 C. T. WU, W. HU AND J. S. CHEN

undesirable nonphysical instabilities, especially when applied to strong compression tests [36]. Inaddition, classical enhanced strain formulation experiences a decline in accuracy when the meshis distorted [37].

Displacement-based Galerkin meshfree method using a low-order approximation experiences vol-umetric locking phenomena similar to that of the standard low-order finite elements. The cause ofvolumetric locking is the inability to approximate the incompressible field, that is, the inability toconstruct a divergence-free (r � u D 0) approximation space. Huerta and Fernandez-Mendez [38]carried out a modal analysis to examine the volumetric locking in the element-free Galerkin (EFG)method. The influence of nodal supports and approximation order on the locking behavior was alsodiscussed. A pseudodivergence-free interpolation for EFG method was proposed by Vidal et al. [39]to impose the divergence-free constraint a priori in a displacement-based Galerkin meshfree formu-lation. Another Galerkin meshfree formulation was presented by Chen et al. [40] for the rubber-likeincompressible materials, where a pressure projection approach was introduced and is applicableto general large deformation. Various other approaches have also been developed to alleviate theincompressible locking [41, 42] in the framework of B-bar and mixed formulations. In the imple-mentation of mixed formulation in Galerkin meshfree, numerical inf–sup conditions indicate thatthe mixed displacement/pressure method possess the property of uniform convergence in the incom-pressible limit. Despite those successes, the development of a robust and efficient Galerkin meshfreemethod for a wide range of near-incompressible or incompressible problems is not trivial.

The purpose of this paper is to introduce a meshfree-enriched FEM (ME-FEM) that circumventsthe volumetric locking problem and lowers the high computational costs of the meshfree method.In this paper, a meshfree-enriched finite element with divergence-free property is formulated byembedding a convex meshfree approximation constructed using the GMF approximation methodinto a low-order finite element with an additional node. The element-wise meshfree interpolationpreserves the Kronecker-delta property at the boundary, and the resultant elements are naturallyconforming. This property permits direct treatment of the essential boundary conditions. The gradi-ent matrix of ME-FEM element satisfies the integration constraint [22] and the resultant ME-FEMformulation is shown to pass the linear patch test in the compressible analysis. An area-weightedstrain smoothing (AW-SS) scheme incorporated with the divergence-free ME-FEM interpolationis introduced for the smoothing in strains and pressure to suppress possible pressure oscillation inthe near-incompressible problem. An assumed strain method based on the modified Hu–Washizuvariational principle is considered to introduce the smoothed strain into the Galerkin formulation.

This paper is organized as follows: Section 2 discusses the volumetric locking problems in thestandard finite element and meshfree methods through the eigenvalue analysis. Section 3 presentsthe basic equations in the GMF approximation and the meshfree enrichment in the low-order finiteelement. Section 4 introduces a locking-free ME-FEM formulation and its divergence-free interpo-lation. In addition, an AW-SS scheme to suppress pressure oscillation is presented, and the discreteequations are derived based on a modified variational principle. In Section 5, several numericalexamples are presented to examine the effectiveness of the proposed method. Final remarks aredrawn in Section 6.

2. VOLUMETRIC LOCKING IN DISPLACEMENT-BASED FINITE ELEMENT ANDMESHFREE METHOD

Consider a linear elastic isotropic material with domain � in R2 and boundary � D @�. Let u bethe displacement vector and the strain tensor � is expressed as

� .u/D1

2

�ruC .ru/T

�. (1)

For a given body force f, the equilibrium equation in � is

r � � C f D 0, (2)



where � is the Cauchy stress tensor given by

� D 2��C � .t r�/ I . (3)

Here, I is the identity tensor, and � and � are Lamé constants.For simplicity, the displacement is assumed to satisfy the homogeneous Dirichlet boundary condi-

tion uD 0 on �D and the traction t is distributed over the Neumann boundary �N with � D �D[�N

and �D \ �N D 0. The variational form of this problem is to find u 2 V such that

A .u, v/D l.v/ 8v 2 V (4)

where the space V DH 10 .�/ consists of functions in Sobolev space H 1 .�/, which vanish on the

Dirichlet boundary and is defined by

V .�/D®v W v 2H 1, vD 0 on �D

¯. (5)

The bilinear form A .�, �/ and linear functional l../ in Equation (4) are defined by

A W V � V !R A .u, v/D 2�Z�

� .u/ W � .v/ d�C �

Z�

.r � u/ .r � v/d� (6)

l W V !R l .v/D

Z�

f � vd�C

Z�N

t � vd� (7)

The bilinear form A .�, �/ in the above equation is symmetric, V -elliptic, and continuous on V . ByKorn’s inequality and Lax–Milgram theorem, there exists a unique solution to the problem. [43]

The standard Galerkin method is then formulated on a finite dimensional subspace V h � V

employing the variational formulation of Equation (4) to find uh 2 V h such that

A�uh, vh

�D l.vh/ 8vh 2 V h. (8)

The second term on the right-hand side of Equation (6) resembles the penalty term similar to theclassical penalty method in the Stokes problems. For the energy in Equation (6) to remain finite as�!1 .or v! 0.5/, the following constraint must be enforced

r � uD 0 for u 2 V . (9)

Similarly, in the finite dimensional space, we have

r � uh D 0 for uh 2 V h. (10)

The solution of Equation (8) using a low-order approximation for displacement field in gen-eral is not capable of retaining the optimal approximation when the incompressible constraint inEquation (10) is enforced. This leads to the volumetric locking in the near-incompressible problem.In other words, volumetric locking occurs when the approximation space V h is not rich enough forthe approximation to satisfy the divergence-free condition r � uh D 0.

An eigenvalue analysis [12, 38, 44] can be employed to further reveal the locking phenomena inthe low-order displacement-based finite element and meshfree methods. The eigenvalue analysis ofone rectangular Q1 bilinear finite element with 2�2 integration is analyzed first, and two nonphysicallocking modes are obtained and shown in Figures 1(a) and (b). The eigenvalues grow unbounded asv approaches 0.5, which is not expected physically. Accordingly, eight nonphysical locking modesare obtained from the eigenvalue analysis using four rectangular Q1 elements. Similar to the resultsin the four Q1 finite element model, the same number of nonphysical locking modes are generatedby element-free Galerkin method as displayed in Figure 2 using the first-order approximation withnormalized nodal support to be 1.5. The increase of nodal support and reduction of Gauss quadratureorder does not completely eliminate the volumetric locking in EFG method. Note that although thenumber of nonphysical modes remains unchanged, the energy associated with the locking modescan be decreased by increasing the normalized nodal support [38].



(a) (b)

Figure 1. Two nonphysical modes for the Q1 element. (a) nonphysical locking mode 1 and (b) nonphysicallocking mode 2.

Figure 2. Eight nonphysical modes for 3� 3 discretization using EFG.

3. MESHFREE-ENRICHED FINITE ELEMENT INTERPOLATION

3.1. Generalized meshfree approximation

In this section, a first-order GMF convex approximation is introduced to approximate the displace-ment field for ME-FEM. The fundamental idea of the GMF approximation [19] is the introductionof an enriched basis function in the Shepard function [45] to achieve linear consistency. The choiceof the basis function determines whether the GMF approximation has convexity property. Assumea convex hull conv.ƒ/ of a node set ƒD ¹xi , i D 1, � � �nº �R2 defined by [46]

conv.ƒ/D

´nXiD1

˛ixi

ˇ̌̌ˇ̌ xi 2ƒ, ˛i 2R ,˛i > 0,

nXiD1

˛i D 1, i D 1, 2, � � �

μ. (11)

The GMF method is to construct convex approximations of a given function u in the form

uh .x/DnXiD1

‰i .x/ui (12)

with the generating function ‰i W conv .ƒ/! R satisfying the following polynomial reproductionproperty

nXiD1

‰i .x/xi D x 8x 2 conv .ƒ/ . (13)



The proof of the weak Kronecker-delta property at the boundaries in convex approximation canbe found in [19].

The first-order GMF approximation in multiple dimensions is expressed as

‰i .x,�r/D i

D

�a.xIXi /�i .Xi ,�r/PnjD1 �a.xIXj /�j .Xj ,�r/

(14)

subjected to

Rr.x,�r/DnXiD1

‰iXi D 0 (linearity constraints), (15)

where

i D �a.xIXi /�i .Xi ,�r/, (16)

DXn

iD1 i D

Xn

iD1�a.xIXi /�i .Xi ,�r/, (17)

Xi D x� xi , (18)

�a.xIXi /: the weight function of node i with support size supp .'a .xIX i //D ai ,�i .Xi ,�r/: the basis function of the GMF approximation,x: the coordinate of interior point (fixed point),xi : the coordinate of node i ,n: the number of nodes within the support size a.x/ at fixed x,�r.x/.r D 1, 2, � � � ,m/: constraint parameters, which have to be decided,m: the number of constraints (mD 1 in 1D, mD 2 in 2D, and mD 3 in 3D),

In the GMF approximation, the property of the partition of unity is automatically satisfied by thenormalization in Equation (14). The completion of the GMF approximation is achieved by finding� to satisfy Equation (15). To determine � at any fixed x in Equation (14), a root-finding algorithmis required for the nonlinear basis functions. Usually, the Newton–Raphson method is consideredfor the equation solving of the objective function in Equation (15). The partial derivative of theobjective function with respect to � is

JDnXiD1

��a�i ,�r

�˝Xi

��Rr ˝

nXjD1

��a�j ,�r

�, (19)

where J is an r � r Jacobian matrix and ˝ indicates the dyadic product of vectors. Once theconverged � is obtained, the basis functions are computed and the spatial derivative of the GMFapproximation can be obtained and given by

r‰i D‰i ,xC‰i ,�r�r ,x, (20)

where

‰i ,x D i ,x

�‰i

nXjD1

j ,x

(21)

i ,x D �a,x�i C �a�i ,x (22)

‰i ,�r D�a�i ,�r

�‰i

nXjD1

��a�j ,�r

�(23)



�r ,x D�J�1Rr ,x (24)

Rr ,x D

nXiD1

‰i ,xXi CnXiD1

‰iXi ,x (25)

The convexity of the GMF approximation is determined by the selection of a positive basis func-tion in Equation (14). Table I gives the typical basis functions used to generate the convex andnonconvex GMF approximations. A combination of those basis functions or other monotonicallyincreasing or decreasing functions can also be considered as GMF basis functions. Note that theGMF(exp) approximation in Table I is known as the ME approximation with Shannon entropy [47]and the GMF(MLS) approximation is the MLS or the RK approximation. The GMF(Renyi) approx-imation is the ME approximation with Renyi entropy. [48] In addition, the GMF approximation canbe readily extended to higher-order approximation, which can be used to improve the accuracy ofmeshfree methods. [19]

In this paper, a convex GMF approximation constructed using an inverse tangent basis functionis employed in the displacement approximation for the ME-FEM and is denoted by GMF(atan) asshown in Table I. A cubic spline function with a rectangular support is chosen to be the weightfunction in Equation (16). The corresponding derivatives of the convex GMF approximation can befound in [19].

3.2. Enriched finite element interpolation by GMF approximation

Let Mh D [eQe be the regular triangulation of the domain � into convex quadrilateral elements.Each element Qe contains four corner nodes Ni , 1 6 i 6 4 and certain enriched nodes Ki ,1 6 i 6 nb. The quantity nb is the number of enriched nodes in element Qe. The global elementQe is also the image of a reference square NQe Œ�1, 1� � Œ�1, 1� through an isoparametric mappingrepresented in a form as

x D F e .�/ , (26)

where x D Œx,y�T , � D Œ� , �T , and F e is a mapping function. In this study, we present a five-nodemeshfree enriched finite element with the fifth node located at the center of the element as shownin Figure 3.

The shape functions of the reference element are constructed by the meshfree convex approxi-mation as described in the previous subsection. In other words, we construct a meshfree approx-imation locally in each reference element. Because each reference element edge only involves

Table I. Examples of basis functions in the GMF approximations.

Convexity Basis function Abbreviation Note

e�x GMF(exp) Maximum-entropy (ME)approximation

(Shannon entropy)1C tan h.�x/ GMF(tanh) New approximation

Convex approximation 1C 2� tan�1.�x/ GMF(atan) New approximation�

1C 1�˛˛ �x

� 1˛�1

GMF(Renyi) ME approximation

(Renyi basis function) (Renyi entropy).0.5 < ˛ < 1.0/

MLS approximation(˛ D 2)

1C �x GMF(MLS) MLS approximationNonconvex approximation 1C �x3 GMF(x3) New approximation

e�x�1C �x3

GMF(exp � x3) New approximation



Figure 3. Isoparametric mapping in the five-node meshfree-enriched finite element.

two reference element nodes, the element-wise meshfree shape functions reduce to the standardbilinear finite element shape functions along the element edge, which satisfy a Kronecker-deltaproperty at the element vertices and exhibit the following convexity properties

‰i��jD ıij 16 i , j 6 4

‰5 .�/ > 0 8� 2Qe

‰5 .�/D 0 8� 2 @Qe

. (27)

Assuming that each node is assigned a weight function with the same rectangular supportsupp .�a .�I�i // D ai D a in the reference square NQe, the tensor product weight function inEquation (16) can be defined by

'a .�I�i / W NQe ! Œ0, 1� 'a .�I�i /D '

�� i

a

�'�� i

a

�(28)

On the basis of the above assumption, there exist certain symmetry properties in this five-nodeelement shape functions and their derivatives as summarized in Appendix A.1.

Giving the five-node ME-FEM shape functions, the isoparametric mapping function F e .�/ inEquation (26) can be defined as follows:

F e W NQe!Qe, x D F e .�/D�F 1e ,F 2e

D

5XiD1

xi‰i .� , / ,5XiD1

yi‰i .� , /

!for all � 2 NQe.

(29)With the above notations, we define the following approximation space for the displacement field

V h .�/D

²uh W uh 2

H 10

�2,uh

ˇ̌̌Qe

D Nuh ı F�1e , Nuh 2P1�NQe

for all Qe 2Mh

³, (30)

where P1�NQe

D span ¹‰i , i D 1, � � � , 5º denotes the space containing a set of basis functions that

reproduce the complete first-order polynomial in NQe .To this end, the fifth node plays a role similar to the bubble function in the incompatible ele-

ment [49] or enhanced strain elements [50]. Unlike the unknown coefficients of the classical bubblefunction, the fifth node carries the physical displacements and should be considered in computingthe element body force vectors as well. In comparison to the employment of element-level bubblefunctions in the low-order finite element mixed formulation, the meshfree-enriched shape functionsresult in an element formulation that can be easily fitted into conventional displacement-based finiteelement code for a general analysis in solids. Because the meshfree-enriched element formationinvolves no static condensation and the resultant discrete linear system is positive-definite, it can besolved by standard direct solver. Compared with the bubble functions in the incompatible elementand enhanced strain element, the proposed meshfree-enriched shape functions for displacements



yield a conforming approximation. According to Equation (29), the isoparametric mapping at thefifth node reads

.x5,y5/D F e.0, 0/D

5XiD1

xi‰i .0, 0/,5XiD1

yi‰i .0, 0/

!. (31)

Because each node in the reference element is assigned a weight function with the same nodalsupport size, we have ‰1.0, 0/ D ‰2.0, 0/ D ‰3.0, 0/ D ‰4.0, 0/ owing to the symmetry of thenodal positions. Consequently, the global co-coordinates of the fifth node is determined by

.x5,y5/D

0BBB@

4PiD1

‰i .0, 0/xi

.1�‰5.0, 0//,

4PiD1

‰i .0, 0/yi

.1�‰5.0, 0//

1CCCAD

0BBB@

4PiD1

‰i .0, 0/xi

4PiD1

‰i .0, 0/

,

4PiD1

‰i .0, 0/yi

4PiD1

‰i .0, 0/

1CCCAD

0BBB@

4PiD1

xi

4,

4PiD1

yi

4

1CCCA

(32)which is the centroid of the quadrilateral.

Figure 4 is a plot of GMF(atan) approximation in a five-node ME-FEM element that shows theKronecker-delta property at the boundary. The shape functions of the ME-FEM element reduce tothe standard bilinear finite element shape functions along the element edge that exhibit the convexityapproximation property. Note that the derivatives of the shape functions in the five-node ME-FEMelement do not degenerate to the finite element derivatives along the element edge because of thenonzero boundary value of derivatives of the fifth node. Because the shape functions of the ME-FEM element reduce to the standard bilinear finite element shape functions on element edge, theyare considered as C 0 continuous finite element bases.

The Jacobian matrix J � is defined to be the gradient of the mapping function F e by

Jij D@F ie .�/

@�j, i , j D 1, 2 (33)

or in matrix form

J � DGFGTx , (34)

where matrices GF and Gx are given by

GF D r .� ,�/ Œ‰i .� , /�e D

"@‰1@�

: : : @‰5@�

@‰1@�

: : : @‰5@�

#(35)

00.01

0.020.03

0.040.05 0

0.010.02

0.030.04

0

0.2

0.4

0.6

0.8

1

YX

Sha

pe fu

nctio

n of

cor

ner

node

00.01

0.020.03

0.040.05 0

0.010.02

0.030.04

0

0.2

0.4

0.6

0.8

1

YX

Sha

pe fu

nctio

n of

mid

dle

node

(a) shape function of corner node (b) shape function of central node

Figure 4. GMF(atan) convex approximation in a five-node meshfree-enriched finite element.



Gx D

�x1 � � � x5y1 � � � y5

�(36)

r .� ,�/ D

"@@�@@�

#. (37)

The analytical proof of the invertibility of mapping function F e (or nonsingularity of theJacobian) is not an easy task because of the rational nature of the meshfree shape functions andshould be addressed by the inverse function theorem. In engineering applications, it is more impor-tant to examine the invertibility at the quadrature points in the element [46]. The examination of thenonsingularity of the Jacobian in ME-FEM element is given in Appendix A.2.

We proceed to show that the isoparametric mapping in the five-node ME-FEM quadrilat-eral element preserves the global element area Ae numerically, where Ae D

RQe

d� DR 1�1

R 1�1 det .J /d�d D

4PJD1

det�J �J

. In other words, the sum of det�J �

using a 2 � 2 Gaus-

sian integration preserves the area of global element Qe despite that the ME-FEM element shapefunctions and their derivatives are rational functions, and the nodal support sizes are adjustable.

ProofFrom Equation (34), the area of global elementQe is obtained using the conventional 2�2Gaussianintegration given by

Ae D

4XJD1

det�J �JD

4XJD1

det�GF

��gJ

GTx

�

D

4XJD1

5XID1

@‰I��gJ

@�

xI �

5XID1

@‰I��gJ

@

yI

�

5XID1

@‰I��gJ

@�

yI �

5XID1

@‰I��gJ

@

xI

!.

(38)

The first-order consistency condition gives

GF��gi

GT� D I for i D 1, 2, 3, 4 (39)

G� D

��1 1 �1 �1 0

�1 �1 1 1 0

�. (40)

Using the symmetry properties in Equation (A.1) together with the first-order consistency conditionsleads to the following explicit expression of Ae

AeD.�x1Cx2Cx3 � x4/ � .�y1� y2Cy3C y4/� .�y1C y2C y3 � y4/ � .�x1� x2C x3C x4/

2(41)

Here, Ae is the exact area of global element Qe and is independent of co-coordinates of the fifthnode and nodal support size a. �

Next, we show that the five-node ME-FEM formulation preserves the linear exactness in theGalerkin approximation of a Dirichlet boundary value problem, that is,Z

�

BId�D 0 orZ�

bId�D 0 8 interior node I 2�[ � , (42)



which is known as the integration constraints [22] in meshfree Galerkin method. In Equation (42),BI is the gradient matrix, which is given by

BI .�/D

264

@‰I .�/@x

0

0 @‰I .�/@y

@‰I .�/@y

@‰I .�/@x

375 (43)

and bI .�/D

"@‰I .�/@x

@‰I .�/@y

#is a vector of the shape function derivatives of node I.

ProofFirst consider the case that interior node is the center node K1 of the global element Qe. In eachelement, we haveZQe

b5d�D

Z 1

�1

Z 1

�1

b5 det.J �/d�d

D

Z 1

�1

Z 1

�1

J�1� �Nb5 det.J �/d�d

D

Z 1

�1

Z 1

�1

J �� Nb5d�d

D

2666664

4XjD1

5XiD1

yi@‰i

��gj

@

!@‰5

��gj

@�

�

5XiD1

yi@‰i

��gj

@�

!@‰5

��gj

@

!

4XjD1

5XiD1

xi@‰i

��gj

@

!@‰5

��gj

@�

�

5XiD1

xi@‰i

��gj

@�

!@‰5

��gj

@

!3777775

D

26666664

5XiD1

0@yi 4X

jD1

@‰i

��gj

@

@‰5��gj

@�

�@‰i

��gj

@�

@‰5��gj

@

!1A

5XiD1

0@xi 4X

jD1

@‰i

��gj

@

@‰5��gj

@�

�@‰i

��gj

@�

@‰5��gj

@

!1A

37777775D

2666664

5XiD1

yigi5

5XiD1

xigi5

3777775 ,

(44)

where

NbI D r .� ,�/‰I D

2664@‰I .� , /

@�

@‰I .� , /

@

3775 (45)

J �� .�/D det.J �/J�1� .�/ (46)

giI D

4XjD1

@‰i

��gj

@

@‰I��gj

@�

�@‰i

��gj

@�

@‰I��gj

@

!. (47)

Using the symmetry properties in Equation (A.1) of Appendix A.1, a simple but tedious calculationleads to Z

Qe

b5d�D 0. (48)



Let us next consider an interior node I DN e13 connected by several neighboring elements where

the superscript e1 denotes the element number and subscript 3 is the associated element node numberas shown in Figure 5. Similar to the above derivation, we have

ZQe

be13 d�D

Z 1

�1

Z 1

�1

be13 det�J e1�

�d�d

D

26666664

4XjD1

5XiD1

ye1i@‰i

��gj

@

!@‰3

��gj

@�

�

5XiD1

ye1i@‰i

��gj

@�

!@‰3

��gj

@

!

4XjD1

5XiD1

xe1i@‰i

��gj

@

!@‰3

��gj

@�

�

5XiD1

xe1i@‰i

��gj

@�

!@‰3

��gj

@

!

37777775

D

"ye12 g

e123C y

e14 g

e143C y

e15 g

e153

xe12 ge123C x

e14 g

e143C x

e15 g

e153

#

.

(49)The result in Equation (49) is independent of the coordinates of node N e1

3 and its diagonal nodeN e11 in element e1. For the adjacent elements e1 and e2, we have

ZQe1

be13 d�C

ZQe2

be24 d�D

"ye12 g

e123C y

e14 g

e143C y

e15 g

e153

xe12 ge123C x

e14 g

e143C x

e15 g

e153

#C

"ye21 g

e214 C y

e23 g

e234 C y

e25 g

e254

xe21 ge214 C x

e23 g

e234 C x

e25 g

e254

#(50)

Using the fact xe12 D xe21 , ye12 D y

e21 and ge123 D g

e214, Equation (50) can be rewritten as

ZQe1CQe2

be1I C be2I d�D

"ye14 g

e143C y

e15 g

e153

xe14 ge143C x

e15 g

e153

#C

"ye23 g

e234 C y

e25 g

e254

xe23 ge234 C x

e25 g

e254

#(51)

Note that I D N e13 D N e2

4 from Figure 5. Subsequently, for multiple elements nl we caneasily verify �

Z�

bId�D

nlXnD1

Z 1

�1

Z 1

�1

bnI det�J n�

�d�dD 0 (52)

Figure 5. Interior node I and its neighboring elements.



4. LOCKING-FREE MESHFREE-ENRICHED FINITE ELEMENT FORMULATION

4.1. Divergence-free meshfree-enriched finite element interpolation

Recall the near-incompressible problem in Equation (8) where the error between the exact solu-tion u and the unique solution uh obtained from a standard displacement-based Galerkin method isestimated and given in the energy error norm by��u� uh��2

E6 C0 inf

vh2V h

��u� vh��2E

6 C0 ku�˘ huk2E

D C0A .u�˘ hu,u�˘ hu/

D C0

�2� ju�˘ huj

21,�C � kr � .u�˘ hu/k

20,�

�6 2�C0 ku�˘ huk

21,�C �C0 kr � .u�˘ hu/k

20,�

(53)

The energy norm in Equation (53) is defined by

kukE D .A .u,u//12 (54)

˘ h WH1 .�/! V h .�/ is an interpolation operator associated with the isoparametric mapping F e

in Equation (29) and defined by

˘ huD N̆ h .u ıF e/ ıF�1e , (55)

where N̆ h W H1�NQe

! P1

�NQe

is a projection operator that preserves polynomials of degree

6 1; that is,

N̆hp D p, 8 p 2P1

�NQe

(56)

and P1�NQe

is the space of polynomials of degree 6 1 on NQe. k � km and j � jm are Sobolev norm

and seminorm of order m, respectively, as defined in a standard way. The constant C0 in Equation(53) does not depend on the element size h and the functions considered. Note that we have usedthe well-known Céa’s lemma for the first inequality in Equation (53) to estimate the error. Let usfurther assume the regularity on the solution u 2 H 2 .�/ and use the approximation property [46]for the first-order approximation to yield the global approximation error��u� uh��2

E6 C0

�2� ku�˘ huk

21,�C � kr � .u�˘ hu/k

20,�

�6 C01�h2 juj22,�CC10�h

2 jr � uj21,�

6 C11h2�juj22,�C � jr � uj

21,�

� , (57)

where C01, C10, and C11 are positive constants independent of � and element size h.From Equation (57) it is clear that solution uh converges to the exact solution u as mesh is

refined in the compressible case. However, the convergence of uh may not be achieved by a low-order approximation as � ! 1. For the energy error norm to be bounded in the incompressiblelimit, the divergence-free condition has to be enforced strongly using a high-order approximationor be enforced weakly by a mixed/reduced integration/modified strategies guided by the inf–supcondition.

In this study, instead of enforcing the divergence-free condition weakly, we enforce thedivergence-free condition strongly but point-wise at the quadrature points. That is,

r � uh��gi

ˇ̌̌Qe

! 0 as �!18Qe 2Mh, i D 1, 2, 3, 4, (58)

where �gi are Gauss points.



Recall that the approximation of the divergence r � uh evaluated at the Gauss points can beexpressed by

r � uh��gi

D tr

�ruh

��gi

�D tr

�J�1� �r .� ,�/u

h��gi

�D tr

��GFG

Tx

��1�r .�,�/u

h��gi

�.

(59)The term r .� ,�/u

h��gi

in Equation (59) can be rewritten as

r .� ,�/uh��gi

D

24 @‰1.�gi/

@��

@‰5.�gi/@�

@‰1.�gi/@�

� � �@‰5.�gi/

@�

35264u1 v1...

...u5 v5

375DGF

��gi

ue, i D 1, 2, 3, 4,

(60)where ue is a collection of nodal displacement for the element Qe given by

uTe D

�u1 � � � u5v1 � � � v5

�(61)

To show that the divergence-free condition can be achieved point-wise using the proposed five-node ME-FEM approximation presented in Section 3.2, we revisit the eigenvalue analysis describedin Section 2.1. For convenience, let us consider a rectangular bilinear four-node finite elementQe D Œ�1, 1��Œ�1, 1� in which the global co-coordinates coincide with the reference co-coordinates.In other words, the Jacobian matrix J � in Equation (34) becomes J � DGFG

Tx D I and the expres-

sion of divergence r � uh can be degenerated to a simple form. Figure 6 illustrates one of the twovolumetric locking modes in a rectangular bilinear four-node finite element where the displacementfields are given in the global co-coordinates by

Œu, v�D Œc1xy, 0� (62)

In the above equation, c1 is a coefficient describing the deformation of the locking mode, whichis a function of nodal displacements. The divergence r � uh is not null when evaluated at all fourGauss points and interpolation does not meet the divergence-free condition of Equation (58) in theincompressible limit. On the other hand, the divergence r � uh in same deformation mode using thefive-node ME-FEM approximation evaluating at Gauss points can be expressed using Equation (59)

Figure 6. Locking mode in bilinear finite element.



by

r � uh��gi

D tr

�J�1� � r.� ,�/u

h��gi

�D tr

�I � r.� ,�/u

h��gi

�

D

5XkD1

‰k,�

��gi , gi

uk C

5XkD1

‰k,�

��gi , gi

vk for i D 1, 2, 3, 4.

(63)

According to Equation (58), the divergence-free interpolation in ME-FEM element Qe requests

r � uh��gi

D 0 for i D 1, 2, 3, 4 (64)

or in matrix form

2664‰5,�

��g1, g1

‰5,�

��g1, g1

‰5,�

��g2, g2

‰5,�

��g2, g2

‰5,�

��g3, g3

‰5,�

��g3, g3

‰5,�

��g4, g4

‰5,�

��g4, g4

3775�u5v5

�D�

2666666666664

4PkD1

�‰k,�

��g1, g1

uk C‰k,�

��g1, g1

vk

4PkD1

�‰k,�

��g2, g2

uk C‰k,�

��g2, g2

vk

4PkD1

�‰k,�

��g3, g3

uk C‰k,�

��g3, g3

vk

4PkD1

�‰k,�

��g4, g4

uk C‰k,�

��g4, g4

vk

3777777777775

.

(65)The above linear system consists of two unknowns (u5 and v5) and four constraint equations and

therefore it is underdetermined. Nevertheless, from the symmetry properties in Appendix A.1, weobserve that the above four constraint equations are not totally independent. Using Equation (A.1),we can reduce four constraint equations to two equations and determine the two unknowns by

�u5v5

�D

"c1 .�‰1,�.�g1,�g1/�‰3,�.�g1,�g1/C‰2,�.�g1,�g1/C‰4,�.�g1,�g1//

‰5,�.�g1,�g1/0

#(66)

In essence, the two additional DOFs associated with the fifth node in ME-FEM element allowan exact production of two nonlocking modes that are missing in the standard bilinear quadrilateralelement. As a result, the overall ME-FEM element behavior is volumetric locking-free. Note thatthe displacement field of the five-node ME-FEM element is divergence-free point-wise at the Gausspoints. The same analysis can be applied for the second nonphysical locking mode in Figure 1(b)to achieve a divergence-free condition. Because the shape function derivative ‰5,�

��g1, g1

in

Equation (66) is nonzero (indeed is nonzero everywhere in the element expect at the node itself)using the convex approximation, the displacement of the fifth node can be uniquely determined.Figures 7(a) and (b) present an eigenvalue analysis result using the ME-FEM. A bounded engien-value near the incompressible limit indicates the ME-FEM element is volumetric locking-free.The eigenanalysis also assures that the ME-FEM element contains no spurious zero energy mode.Because the volumetric energy term in Equation (53) is bounded in the incompressible limit, theenergy norm error of the ME-FEM solution is expected to be close to the theoretical asymptotic rateof O(h). Accordingly, we have the pressure computed by

ph D �r � uh, (67)

which remains finite in the incompressible limit, and the following L2-norm error estimate holds��r � �u� uh��0,�6 c2h jr � uj1,� . (68)



2 4 6 82

4

6

8

10

-log10

(0.5-nu)

log 10

(eig

)

2 4 6 82

4

6

8

10

-log10

(0.5-nu)

log

10(e

ig)

(a) (b)

Figure 7. Two locking-free modes in one ME-FEM element. (a) locking-free mode 1 and (b) locking-freemode 2.

The discrete divergence-free condition can also be obtained in a general quadrilateral five-nodeME-FEM element. The analytical solution of the fifth node displacements that satisfies the dis-crete divergence-free condition in a general quadrilateral five-node ME-FEM element is given inAppendix A.2.

Although the ME-FEM element does not contain any spurious zero energy mode, the ME-FEM formulation using conventional Gaussian quadrature rule experiences pressure oscillationin the near-incompressible analysis under certain conditions. The cause of pressure oscillation inME-FEM formulation using Gauss quadrature rule is different from the one in the discontinuous-pressure/displacement mixed FEM, which exhibits a rank-deficiency in the assembled pressureequations [51]. The pressure oscillation in ME-FEM may be due to the nature of the displacement-based Galerkin method with low-order approximation. In fact, the displacement derivatives inME-FEM are discontinuous across element boundaries and a strain smoothing procedure can beemployed to remove the possible oscillation, for example, the pressure smoothing techniques [51].However, additional computational effort is required and robust performance in the nonlinear analy-sis is not guaranteed. To suppress the possible pressure oscillation, an AW-SS scheme incorporatedwith the divergence-free ME-FEM interpolation is proposed.

4.2. Area-weighted strain smoothing scheme

The strain smoothing in SCNI method [22] was originally proposed to provide a strain smoothingstabilization for nodal integrated weak form and to preserve the linear exactness in the Galerkinmeshfree method. In this study, an AW-SS scheme is introduced to the five-node ME-FEM elementto suppress the pressure oscillation in the incompressible limit. The strain smoothing is definedas follows

Q�h D Qruh D1

Am

Z�m

ruhd�, (69)

where Qrdenotes the smoothed version of the gradient operatorr , andAm is the area of the smootheddomain �m. Each element contains four smoothing domains and each smoothing domain containstwo Gauss points for strain evaluation as illustrated in Figure 8(a). Accordingly, the smoothed strainin the first smoothing domain shown in Figure 8(a) is expressed by

Q�h D1

2Am

��g1, g1

det .J 1/C �

��g2, g2

det .J 2/

(70)

Am Ddet .J 1/C det .J 2/

2D

det�GFG

Tx

�1C det

�GFGx

T�2

2(71)



Figure 8. AW-SS scheme.

This leads to the smoothed divergence-free condition for the smoothed strain Q�h in one element

Qr � uh D t r�Q�h�D t r

�1

Am

Z�m

ruhd�

�

D1

Am

Z�m

t r�ruh

�d�

D1

Am

Z�m

r � uhd�

Dr � uh

��g1

det .J 1/Cr � uh

��g2

det .J 2/

det .J 1/C det .J 2/

D 0

(72)

For the adjacent elements shown in Figure 8(b), we have the smoothing area Am calculated fromthe sum of four Jacobian determinants in two elements given by

Am Ddet.J e1

2 /C det.J e13 /C det.J e2

1 /C det.J e24 /

2

Ddet

�GFG

Tx

�e12C det

�GFG

Tx

�e13C det

�GFG

Tx

�e21C det

�GFG

Tx

�e24

2,

(73)

where superscripts e1 and e2 are adjacent elements that share the same element edge m and twoconnecting nodes Is1 and Is2. Similar to the derivation in Equation (72), we can also show that thesmoothed divergence Qr � uhin the adjacent elements satisfies the divergence-free condition in theincompressible limit. With Equation (73), the weighted average of the strain-displacement matrix inthe smoothed domain �m is given by

QBI D

24 QbIx 0

0 QbIyQbIy QbIx

35 (74)

QbIx D

�bIx

e12

det�J e12

C .bIx/

e13 det

�J e13

C .bIx/

e21 det

�J e21

C .bIx/

e24 det

�J e24

2Am

(75)

QbIy D

�bIy

e12

det�J e12

C�bIy

e13

det�J e13

C�bIy

e21

det�J e21

C�byx

e24

det�J e24

2Am

, (76)



where the components in QbIx and QbIy are expressed by

².bIx/

e1i

.bIx/e1i

³D J e1�1

i

8<:

@‰e1I .�gi ,�gi/

@�@‰e1I .�gi ,�gi/

@�

9=; , i D 2 and 3 (77)

².bIx/

e2i

.bIx/e2i

³D J e2�1

i

8<:

@‰e2I .�gi ,�gi/

@�@‰e2I .�gi ,�gi/

@�

9=; , i D 1 and 4. (78)

It is easy to verify that for multiple elements nl as shown in Figure 5, the smoothed gradientmatrix of inter node I satisfies the integration constraint, that is,

Z�

QbId�D

nlXnD1

Z 1

�1

Z 1

�1

Qbn

I det.J nxi /d�dDnlXnD1

Z 1

�1

Z 1

�1

bnI det.J n� /d�dD 0. (79)

Therefore, the integration in the AW-SS scheme preserves the linear exactness and the resultingME-FEM formulation passes the patch test in the compressible case.

Remark. The AW-SS scheme introduces more neighboring nodes in the calculation of strains andtherefore a larger bandwidth in the final stiffness matrix. On the other hand, the integration in theAW-SS scheme also implies a reduction of integration points, which can reduce the computationalcosts in the assembly of the stiffness matrix. For example, there is a reduction of the total number ofintegration points in an n�n mesh from 4�n�n integration points using a 2�2 Gauss quadraturerule to 2�n� .nC 1/ integration points using the integration in the proposed AW-SS scheme. Thisallows the total number of integration points to be scaled down by a factor of two in a large-scalemodel in two dimensions.

4.3. Modified Hu–Washizu formulation

To introduce the strain smoothing formulation into the Galerkin approximation, the Hu–Washizufunctional [52] is considered and expressed as

UHW .u, Q�, Q� /DZ�

1

2Q�TC Q�d�C

Z�

Q� T .ru� Q�/d��Wext, (80)

where the displacements u, smoothed strains Q� and stresses Q� are independently varied. The termWext designates the external work as defined in Section 2. Because the smoothed strains are definedlocally on each smoothing domain �m and no continuity conditions are applied at the boundariesof �m, the approximation space of smoothed strain Q�h is defined as

Eh .�/D°Q�h W Q�h 2L2 .�/ , Q�h j�M is symmetric and contains piecewise constants

for all �m 2Mh º .(81)

If the approximation space of stresses S h .�/is chosen to be the subspace of Eh .�/, such as

Q� h D C Q�h D � h (82)

the stress fields Q� h can be eliminated from the functional leading to the following weak form ofHellinger–Reissner functional [53]

ıUHR,

�uh, Q�h

�D

Z�

ı Q�hT

C Q"hd�� ı

Z�

Q�hT

C�ruh � Q�h

�� ıWext. (83)



Because Eh contains piecewise constants in�, the second term on the right-hand side of Equation(80) can be further expressed using Equation (82) to yield

ı

Z�

Q�hT

C�ruh- Q�h

�d�D

nmXmD1

ı

Z�M

Q�hT

C�ruh- Q�h

�d�

D

nmXmD1

Q�hTC ı

�Z˝M

�ruh- Q�h

�d�

�

D

nmXmD1

Q�hTC ı

�Z�M

ruhd�� Q�hAm

�

D

nmXmD1

Q�hTC ı

�Z�M

ruhd��

Z�M

ruhd�

�D 0

. (84)

The index nm in Equation (84) denotes the total number of smoothing domain �m in �

.�D[m�m/. Equation (84) also implies an orthogonal condition between the stress field Q� andthe difference of the displacement gradient ruh and smoothed strain field Q�h, and the variationalformulation reduces to the following assumed strain variation principle of Simo and Hughes [54]:

ıUHW,mod

�uh, Q�h

�D

Z�

ı Q�hT

C Q�hd�� ıWext (85)

Here, the displacements in each element are approximated by the enriched ME-FEM interpolationand are expressed by

uh D

5XID1

‰IuI . (86)

The strains are approximated using the strain smoothing scheme described in Section 4.2 and aregiven by

Q�h D

NPXID1

QBIuI , (87)

where QB is the smoothed gradient matrix defined in Equation (74) and 5 6 NP 6 8 denotes thenumber of nodes involved in each smoothing domain �m. By introducing the displacement andstrain approximations into Equation (85), the following discrete governing equation is obtained:

QKd D f ext (88)

QK IJ D

nmXlD1

NPXID1

QBT

I CQBJAl (89)

f extI D

Z�

‰TI f d�C

Z�N

‰TI td� , (90)

where the standard 2 � 2 Gauss quadrature rule based on finite element mesh is performed inthe evaluation of Equation (90). Because ME-FEM element interpolation preserves Kronecker-delta property on the element boundary, the essential boundary conditions can be treated in thestandard way.

When the same eigenvalue analysis for a single element is reanalyzed using ME-FEM with strainsmoothing, almost identical locking-free modes as those shown in Figure 7 are obtained. For multi-ple elements the results are presented in Figure 9. Compared with the EFG results in Figure 2, eighteigenvalues of ME-FEM elements with strain smoothing are bounded in the near-incompressiblelimit and the corresponding deformation modes are nonlocking. The ability of the proposed methodto suppress pressure oscillations is demonstrated in the numerical examples in the next section.



Figure 9. Eight locking-free modes in four ME-FEM elements by the integration in AW-SS.

5. NUMERICAL EXAMPLES

In this section, six benchmark examples are analyzed to study the performance of the ME-FEMfor the compressible and near-incompressible cases. Domain integration in ME-FEM is performedusing both Gauss integration (ME-FEM-GI) and the integration in the AW-SS scheme (ME-FEM-AW) for all examples. Standard meshfree method with a 2 � 2 Gaussian integration (MM-GI) andQ1 bilinear FEM with selective reduced integration (Q1-SR) are also included for comparison. Theweight function is chosen to be the cubic B-spline kernel function with normalized support size of1.5 for ME-FEM and MM-GI. On the other hand, the result in ME-FEM is insensitive to the normal-ized support (ranged from 1.01 to 10.01 or even larger). Therefore, we simply use the normalizedsupport size the same as that in MM-GI for all numerical examples. Unless otherwise specified,the following material constants are used for all benchmark examples: Young’s modulus E=1000,Poisson’s ratio v=0.3 for compressible material and v=0.4999999 for near-incompressible material.

5.1. Constant strain patch test and element distortion test in compressible case

A constant strain patch test problem shown in Figure 10 is considered. By prescribing the valuesof the nodal coordinates to the displacements at the external nodes, the displacement fields at theinterior nodes are solved. To pass the test, the results must exactly represent the correct constantstrain within machine precision. Both ME-FEM-GI and ME-FEM-AW pass this test, which hasbeen proved theoretically in Section 3. The standard meshfree method (MM-GI) cannot pass thetest that has been reported and studied in the literature.

In the element distortion test, we design two problems with different mesh profiles as shown inFigure 11. In single element test shown in Figure 11(a), one of the exterior nodes is given by adisplacement u along the x-direction while the other exterior nodes are prescribed by appropriateconstraints to prevent rigid-body motion. The determinant of deformation gradient at four integra-tion points is examined as the element deforms, which provides the information about how severe

FEM nodes

Meshfree-enriched nodes

Figure 10. Constant stress patch test.



(a) (b)

x

y

u

2

x

yu

2

Figure 11. Element distortion tests: (a) single element test and (b) multiple elements test.

Table II. Element distortion ratio in single andmultiple elements.

Q1-SR ME-FEM-GI ME-FEM-AW

Model (a) 63% 78% 88%Model (b) 88% 94% 100%

(a)

(b)

x

y

L =10

D=2 P=-1

Figure 12. Cantilever beam. (a) geometry and boundary conditions and (b) discretization.

an element is distorted. We measure an element distortion ratio defined by c=u/2 when deformationreaches its limit and the deformation gradient is no longer computable. The results are presented inTable II.

Similarly, distortion in multiple elements shown in Figure 11(b) is studied and the results arereported in Table II. Both ME-FEM-GI and ME-FEM-AW improve element distortion ratio over theQ1-SR in single and multiple elements tests. The results suggest that ME-FEM can better handleelement distortion with the enrichment of the fifth node.

5.2. Cantilever beam

In this example, a cantilever beam problem as shown in Figure 12(a) is analyzed to study the con-vergence of ME-EFM in compressible and near-incompressible cases. Analytical displacement fieldis prescribed along x=0, and parabolic vertical traction P is applied along x=10. The analyticaldisplacement and stress solutions are given as [55]



ux D�Py

6 QEI

�.6L� 3x/ xC .2C Qv/

�y2 �

D2

4

��

uy DP

6 QEI

�3vy2 .L� x/C .4C 5v/

D2x

4C .3L� x/ x2

�

xx D�P .L� x/ y

I, yy D 0, xy D

P

2I

�D2

4� y2

�,

(91)

where QE DE=�1� v2

, Qv D v= .1� v/ under plane strain assumption and I DD3=12 is the sec-

ond moment of area of the beam. The computational domain is discretized uniformly with enrichednodes located in the middle of elements as shown in Figure 12(b) for the coarsest mesh.

The results of L2-norm and energy-norm errors in compressible case are shown in Figures 13(a)and (b), respectively. All three methods (MM-GI, ME-FEM-GI, and ME-FEM-AW) present similaroptimal rate of convergence in L2-norm error where MM-GI and ME-FEM-GI shows slightly bet-ter accuracy than ME-FEM-AW. On the other hand, ME-FEM-AW is more accurate and displays asuperconvergent rate in the energy error norm.

Figures 14(a) and (b) depict two stress components of ME-FEM-AW results in the near-incompressible case where the numerical results agree with the analytical solution very well. Thestresses shown in Figures 14(a) and (b) are computed at the integration points along the cross-sectionx=4.875 on a 40�8 uniform mesh.

Superior performance of ME-FEM over MM-GI is presented in the L2-norm and energy-norm errors of the near-incompressible case. The MM-GI suffers from a volumetric locking

(a) L2 error norm (b) energy error norm

Figure 13. Convergence of error norms in compressible case.

(a) xx component (b) xy component

Figure 14. Stress results along the cross-section x=4.875 in near-incompressible case.



(a) L2 error norm (b) energy error norm

Figure 15. Convergence of error norms in near-compressible case.

(a) geometry and boundary conditions (b) discretization

hΓ

P

Figure 16. Inflation problem.

and displays a poor convergence rate close to O(1) in L2-norm error as shown in Figure 15(a).Similar to the compressible case, a superconvergent rate in the energy error norm is observedin ME-FEM-AW as shown in Figure 15(b) when the material is close to the near-incompressiblelimit.

5.3. Inflation of infinitely long near-incompressible tube

The solution for an infinitely long tube subjected to an inner pressure P is given as [55]

r DP r21r22 � r

21

�1�

r22r2

�

� DP r21r22 � r

21

�1C

r22r2

�.

(92)

The exact pressure solution of this problem is a constant pressure field. Only the upper quadrantof the problem is modeled as shown in Figure 16(a) with dimension of inner radius r1=1.0 andan outer radius r2=2.0. A coarse and relatively uniform discretization as shown in Figure 16(b) isused for the mesh refinement in the convergence study. Only the near-incompressible case is stud-ied in this example. Because MM-GI experiences serious volumetric locking in this problem, weonly compare the solutions obtained from ME-FEM-GI and ME-FEM-AW in the energy error andpressure error norms. The pressure error norm is defined by

ehp D��p � ph��

0D

sZ�

�p � ph

2d� (93)

A convergence study is next presented using four levels of mesh refinements. Figures 17(a) and(b) show the energy and pressure error norms, respectively, computed from the ME-FEM-GI and



Figure 17. Convergence of error norms for inflation problem in near-incompressible case.

Table III. The convergence of ME-FEM-AW energy error norm in a fixed mesh.

v=0.3 0.499 0.49999 0.4999999 0.499999999

Energy error norm 1.152085475 2.023429298 2.035368298 2.035488728 2.035493517

(a) geometry and boundary conditions (b) discretization

48

16

44

f=6.25

Figure 18. Cook’s membrane problem.

ME-FEM-AW solutions. Because the exact solution is smooth, the rate of convergence in energyerror and pressure error norms using ME-FEM-GI is very close to the theoretical asymptotic rateof O(h). Better rates of convergence in energy and pressure error norms are seen in ME-FEM-AWsolution.

To show that the energy error norm is bounded using ME-FEM-AW as Poisson’s ratio approaches0.5, we monitor the energy error norm as Poisson ratio v changes from 0.3 to 0.499999999 with a16�16 fixed mesh. The results are presented in Table III. In Table III, the energy error norm con-verged to a finite value as Poisson’s ratio approaches the incompressible limit. The results indicatethat pressure computed from Equation (67) remains finite and converged as �!1.

5.4. Cook’s membrane problem in near-incompressible case

Cook’s membrane model is a classical benchmark problem used to examine the numerical perfor-mance under combined shear and bending deformation. The geometry and boundary condition areshown in Figure 18(a). The left side boundary is fully constrained, and the right side boundary issubjected to a uniformly distributed vertical traction. A typical discretization for ME-FEM withenriched nodes is plotted in Figure 18(b). Because the analytical solution is not available in thisexample, we include the solution of Q1 bilinear FEM with selective reduced integration (Q1-SR)for comparison.



Figure 19. Tip deflection of Cook’s membrane problem in near-incompressible case.


(a) 8 × 8 mesh

(b) 16 × 16 mesh

Figure 20. Pressure distribution of Cook’s membrane in near-incompressible case.

Figure 19 compares the variation of the vertical displacement of the upper right corner as mesh isrefined and all three methods generate locking-free and convergent solutions. The use of the AW-SSscheme in ME-FEM is clearly more accurate than the other two methods.

Figures 20(a) and (b) show the comparison of pressure distribution in two different discretizations8�8 and 16�16. In Q1-SR, the element pressure is presented at the element center. In ME-FEM-GI and ME-FEM-AW, the pressure is reported by averaging the pressure from element integrationpoints. As we can see from Figure 20, both Q1-SR and ME-FEM-GI produce pressure oscillationbetween elements and the pressure oscillation is not improved in a refined mesh. By contrast, theME-FEM-AW displays a smooth pressure distribution and is free of pressure oscillation.



Figure 21. Geometry, boundary condition and a typical discretization in punch problem.

16 × 4 mesh 32 × 8 mesh

Q1-SR

ME-FEM-GI

ME-FEM-AW

Q1-IE

Figure 22. Pressure distribution of punch problem in near-incompressible case.

5.5. Punch problem

The pressure oscillation is also studied in the Prandtl’s punch model. The model consists of ablock of near-incompressible material punched by a rigid, frictionless, and flat plate. The modelgeometry and the discretization are shown in Figure 21. Two uniform meshes 16�4 and 32�8are utilized to investigate the pressure oscillation in model refinement. In this example, we con-sider an incompatible element using second-order bubble functions [56] for additional comparison.This incompatible element (Q1-IE) is a standard bilinear displacement-based quadrilateral elementenriched by second-order bubble functions.

The resulting element-wise pressure distributions in the deformed configurations are shown inFigure 22. Although the Q1-SR, Q1-IE, and ME-FEM-GI do not exhibit locking behavior, visiblecheckerboard patterns in the pressure field are observed. The mesh refinement in these three methodsdoes not reduce the magnitude of checkerboard modes. The use of the AW-SS scheme in ME-FEM,on the other hand, shows a marked improvement in the pressure field. Those smooth pressure dis-tributions illustrate a checkerboard-free solution in ME-FEM-AW, as compared with the Q1-SR,Q1-IE, and ME-FEM-GI solutions. Table IV reports the convergence of the punch force as mesh isrefined. Consistent convergence and very similar results are obtained in four methods.

5.6. Driven cavity problem in near-incompressible case

We consider a unit square subjected to a unit horizontal displacement along the upper boundary asshown in Figure 23(a). This boundary condition results in corner singularities for the solution for



Table IV. Convergence of forces on the flat rigid punch.

Elements Q1-SR Q1-IE ME-FEM-GI ME-FEM-AW

8� 2 2492.4817 2184.8205 2309.2415 2327.625316� 4 1982.1401 1856.0541 1915.1849 1884.451124� 6 1865.3470 1791.0968 1826.3456 1799.678432� 8 1814.3202 1762.3510 1786.9213 1765.6699

(a) geometry and boundary condition (b) uniform mesh (c) non-uniform mesh

u0

Figure 23. Driven cavity problem.


(a)

(b)

Figure 24. Pressure distribution of driven cavity problem: (a) 16 � 16 uniform mesh (b) 16 � 16nonuniform mesh.

which the exact solution is not known. The aim here is to demonstrate numerically that the smoothpressure solution can be achieved by the AW-SS scheme in ME-FEM. Both uniform and nonuniformdiscretizations in a 16�16 mesh shown in Figures 23(b) and (c) are considered.

Pressure distributions of the problem obtained by the three methods Q1-SR, ME-FEM-GI, andME-FEM-AW are plotted on the deformed configuration and shown in Figure 24. Spurious pressuremodes are obtained using Q1-SR and ME-FEM-GI as shown in Figure 24(a) when uniform meshis adopted. The checkerboard patterns are less pronounced but still visible in a nonuniform mesh as



Figure 25. Pressure distribution at x D 0.5.

displayed in Figure 24(b). In Figure 25, the pressure distributions along x = 0.5 obtained by Q1-SRand ME-FEM-GI with 32�32 mesh does not even reproduce the sinusoidal-like pressure distribu-tion. In contrast, ME-FEM-AW with 16�16 mesh generates much better pressure distributions thanQ1-SR and ME-FEM-GI.

6. CONCLUSIONS

We have presented a two-dimensional displacement-based ME-FEM for the analysis of compress-ible and near-incompressible planar elasticity. The approximation in ME-FEM is a bilinear Q1element enriched by an additional node, which is constructed using a first-order convex meshfreeapproximation. This approach preserves the Kronecker-delta property at the element boundary andenhances the convergence of the bilinear Q1 element. In particular, we have shown that the proposedME-FEM formulation satisfies the integration constraint and preserves the element area when a stan-dard two-by-two Gaussian quadrature rule is performed. Additionally, we have also proved that theproposed ME-FEM interpolation is discrete divergence-free in the incompressible limit. The eigen-value analysis has confirmed that the proposed method does not contain nonphysical locking modeand spurious zero energy mode.

To suppress possible pressure oscillation in the near-incompressible problems, an AW-SS schemeis developed and incorporated with the ME-FEM interpolation for the strain and pressure smoothing.The AW-SS scheme introduces a smoothed strain field and is formulated under the assumed strainframework. The introduction of the smoothed strain field does not affect its discrete divergence-freeproperty while meeting the integration constraint. As a result, the proposed method provides thefollowing numerical properties:

1. The proposed meshfree-enriched element interpolation is conforming. Because the element-wise meshfree interpolation is constructed using a first-order convex meshfree approximation,it preserves the Kronecker-delta property at the element boundary and the resultant elementis naturally conforming. This property admits a direct treatment of the essential boundaryconditions, and hence reduces the computational cost and complexity.

2. The proposed method contains no spurious zero energy mode and passes the constant straintest in the compressible case. In addition, the meshfree enrichment enhances the element per-formance in Q1 bilinear element and the element performance because of element distortionas shown in the first numerical example.

3. The proposed method is immune to the volumetric locking. The inclusion of the enriched noderemoves two volumetric locking modes in Q1 bilinear element and the resulting interpolationis divergence-free point-wise and thus assures the locking-free property of the method.

4. No numerical control parameter is involved in the strain smoothing for a nonoscillating pres-sure field in the incompressible limit. The proposed AW-SS scheme in ME-FEM appears



to produce a smooth and checkerboard-free solution that has been demonstrated in severalnumerical examples.

5. The proposed ME-FEM with the AW-SS scheme is computationally inexpensive. Althoughthe enrichment of the fifth node in each ME-FEM element increases the total DOFs and hencea larger bandwidth in the global stiffness matrix, the integration in the AW-SS scheme scalesdown the total number of integration points by a factor of two, making it attractive for solvinglarge-scale problems.

The extension of this method to triangular element and nonlinear problems has been discussedin separate papers [44, 57]. The three-dimensional formulation of the ME-FEM was also presentedin [57] for tetrahedral element. Its application to hexahedral and high-order elements will also beconsidered in the future.

APPENDIX A.1

The symmetry properties of the shape function derivatives in the five-node ME-FEM element areexpressed by

‰1,�

��g1, g1

D�‰2,�

��g2, g2

D�‰3,�

��g3, g3

D‰4,�

��g4, g4

‰1,�

��g1, g1

D‰2,�

��g2, g2

D�‰3,�

��g3, g3

D�‰4,�

��g4, g4

‰2,�

��g1, g1

D�‰1,�

��g2, g2

D�‰4,�

��g3, g3

D‰3,�

��g4, g4

‰4,�

��g1, g1

D‰3,�

��g2, g2

D�‰2,�

��g3, g3

D�‰1,�

��g4, g4

‰2,�

��g1, g1

D‰1,�

��g2, g2

D�‰4,�

��g3, g3

D�‰3,�

��g4, g4

‰4,�

��g1, g1

D�‰3,�

��g2, g2

D�‰2,�

��g3, g3

D‰1,�

��g4, g4

‰3,�

��g1, g1

D�‰4,�

��g2, g2

D�‰1,�

��g3, g3

D‰2,�

��g4, g4

‰3,�

��g1, g1

D‰4,�

��g2, g2

D�‰1,�

��g3, g3

D�‰2,�

��g4, g4

‰5,�

��g1, g1

D�‰5,�

��g2, g2

D�‰5,�

��g3, g3

D‰5,�

��g4, g4

‰5,�

��g1, g1

D‰5,�

��g2, g2

D�‰5,�

��g3, g3

D�‰5,�

��g4, g4

‰1,�

��g1, g1

D‰1,�

��g1, g1

‰2,�

��g1, g1

D‰4,�

��g1, g1

‰5,�

��g1, g1

D‰5,�

��g1, g1

(A.1)

where �gi D��gi , gi

i D 1, 4 are Gauss points.

APPENDIX A.2

Consider a five-node ME-FEM quadrilateral element with arbitrary shape in global coordinate, asshown in Figure A.1(a), where node 1 is totally fixed and node 2 is constrained in y direction to elim-inate the rigid-body translation and rotation. The meshfree-enriched node five is the centroid of thequadrilateral. The nodal coordinates and displacement vectors are .xi ,yi / and .ui , vi /, respectively,where i D 1, 2, : : : , 5. According to our geometry and constraint setting, we have

x1D y1 D y2 D 0, x5 D.x1C x2C x3C x4/

4, y5 D

.y1C y2C y3C y4/

4

u1D v1 D v2 D 0

. (A.2)



(a) (b)

1

43

2

5

xy

1

4 3

2

5-1 1

1

-1

Figure A.1. ME-FEM quadrilateral element with arbitrary shape in (a) global coordinate and (b) referencecoordinate.

We are going to show that for any prescribed nodal displacement field Œu3, v3,u4, v4�T , there

exists a set of nodal displacement solution Œu2,u5, v5�T such that the divergence free condition

r � uD 0 is satisfied at all Gauss integration points��gj , gj

, where j D 1, 2, 3, 4, that is

5XiD1

‰i ,x��gj , gj

ui C

5XiD1

‰i ,y��gj , gj

vi D 0, for j D 1, 2, 3, 4. (A.3)

Substituting Equation (A.2) into (A.3) leads to an alternative form of equations as follows:

‰5,x��gj , gj

u5C‰5,y

��gj , gj

v5 D�

4XiD2

‰i ,x��gj , gj

ui C

4XiD3

‰i ,y��gj , gj

vi (A.4)

In the reference domain, as shown in Figure A.1(b), the derivatives of ME-FEM shape function,denoted by‰i ,�

��gj , gj

,‰i ,�

��gj , gj

for nodal index i D 1, 2, : : : , 5 and integration indexj D

1, 2, 3, 4, process the symmetry property shown in Appendix A.1 and satisfy the linear consistencycondition. Therefore, the set of‰i ,�

��gj , gj

,‰i ,�

��gj , gj

can be completely expressed by only

two parameters a1 and a2, where a1 D ‰1,�

��g1, g1

and a2 D ‰2,�

��g1, g1

. On the basis of

the isoparametric mapping in Equation (29), the set of ‰i ,x��gj , gj

,‰i ,y

��gj , gj

is able to

be expressed by a1,a2 with the global coordinates .xi ,yi /. Mathematical derivation provides thefollowing two results:

(1) "‰5,x

��g1, g1

‰5,y

��g1, g1

#D c1

"‰5,x

��g3, g3

‰5,y

��g3, g3

#

"‰5,x

��g2, g2

‰5,y

��g2, g2

#D c2

"‰5,x

��g4, g4

‰5,y

��g4, g4

# , (A.5)

where c1 and c2 are two constants.(2) If u2 satisfies the following condition:

u3y4C v4x3C u2y3 � u4y3C v3x2 � v3x4 D 0

) u2 D�u3y4C v4x3 � u4y3C v3x2 � v3x4

y3,

(A.6)

equations in (A.4) can be reduced to two equations using (A.5):"‰5,x

��g1, g1

‰5,y

��g1, g1

‰5,x

��g2, g2

‰5,y

��g2, g2

# �

u5v5

�D

�b1b2

�, (A.7)



where

‰5,x��g1, g1

D

�4 .a1C a2/ y4

�a2x3y4 � a1x4y3C a1x2y3 � a2x2y3C a1x3y4C a2x4y3C2a2x2y4 � 2a1x2y4 � x4y3C x2y3 � x2y4C x3y4 � a2x3y4

‰5,y��g1, g1

D

�4 .a1C a2/ .�x4C x2/

�a1x4y3C a1x2y3 � a2x2y3C a1x3y4C a2x4y3C2a2x2y4 � 2a1x2y4 � x4y3C x2y3 � x2y4C x3y4

‰5,x��g2, g2

D

�4 .a1C a2/ y3

a2x3y4� a2x2y3C a1x4y3� a1x3y4�a2x4y3C a1x2y3C x4y3� x3y4

‰5,y��g2, g2

D

4 .a1C a2/ x3

a2x3y4� a2x2y3C a1x4y3� a1x3y4�a2x4y3C a1x2y3C x4y3� x3y4

b1 D

a1u2y3 � a2u2y3C 3a2u2y4 � a1u2y4C 2a1y4u3 � a1u4y3Ca2u4y3C a2u4y4C a1u4y4 � 2a1v3x4C 2a1v3x2C 3a2v4x2Ca1v4x3� a2v4x3� a2v4x4C u2y3� u2y4� v4x2�a1v4x2� a1v4x4C u3y4� u4y3C v3x2� v3x4C v4x3

�a2x3y4 � a1x4y3C a1x2y3� a2x2y3C a1x3y4C a2x4y3C2a2x2y4� 2a1x2y4� x4y3C x2y3� x2y4C x3y4

b2 D

2a1u2y3C a2u3y3� a1u3y4C a2u3y4� u3y4C a1u3y3C2a1u4y3C u4y3C a1v3x2� a2v3x3C a1v3x4� a2v3x4Cv3x4� a1v3x3� a2v3x2� 2a1v4x3� v4x3

a2x3y4� a2x2y3C a1x4y3� a1x3y4� a2x4y3C a1x2y3C x4y3� x3y4

Therefore, the solution Œu2,u5, v5�T of Equations (A.6) and (A.7) is able to have discrete diver-

gence free at all integration points for arbitrary element shape and any prescribed nodal displacementfieldŒu3, v3,u4, v4�

T .On the way leading to the results above, the determinant of Jacobian at all Gauss points

��gj , gj

,

where j D 1, 2, 3, 4, is also calculated to check its singularity, is listed below:

det�J��g1, g1

D1

2.A0 �A1/ .1C a1 � a2/

det�J��g2, g2

D1

2.A0 � 2A2/ .1C a1 � a2/CA2

det�J��g3, g3

D1

2.A0 �A1/ .a2 � a1/C

1

2A1

det�J��g4, g4

D1

2.A0 � 2A2/ .a2 � a1/C

1

2A2

(A.8)

where J��gj , gj

is a Jacobian matrix at Gauss point

��gj , gj

; in global coordinate, A0 is total

element area, A1 is the area of the triangle made by nodes 1–2–4, A2 is the area of triangle made bynode 1–2–3. Considering the following conditions:

1. Element is convex) A0,A1,A2 > 02. The GMF approximation in the reference domain is convex) a1 D‰1,�

��g1, g1

< 0, a2 D

‰2,�

��g1, g1

> 0, and numerical tests show 0 < a2 � a1 < 1 for ME-FEM with the

normalized support size varying between 1.001 and 10.00,

it is trivial to prove that det�J��gj , gj

> 0 for j D 1, 2, 3, 4. Although the proof is based

on Gauss points, it can be extended to other evaluation points inside the element and show thatdet .J .� , // > 0 is actually true everywhere.



ACKNOWLEDGEMENTS

The authors gratefully acknowledge the support of Dr. John O. Hallquist of LSTC and Dr. M. Koishi ofYokohama Rubber Co, Ltd, Japan to this research.

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a meshfree-enriched finite element method for compressible and near-incompressible elasticity

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