meshfree-enriched simplex elements with strain smoothing for the finite element analysis of...

Comput. Methods Appl. Mech. Engrg. 200 (2011) 2991–3010

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Comput. Methods Appl. Mech. Engrg.

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Meshfree-enriched simplex elements with strain smoothing for the finiteelement analysis of compressible and nearly incompressible solids

C.T. Wu ⇑, W. HuLivermore Software Technology Corporation, Livermore, CA 94550, USA

a r t i c l e i n f o

Article history:Received 2 March 2011Received in revised form 23 June 2011Accepted 25 June 2011Available online 8 July 2011

Keywords:MeshfreeTriangular elementFinite elementConvexVolumetric locking

0045-7825/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.cma.2011.06.013

⇑ Corresponding author. Address: Livermore Softw(LSTC), 7374 Las Positas Road, Livermore, CA 94551,fax: +1 925 449 2507.

E-mail address: [email protected] (C.T. Wu).

a b s t r a c t

This paper presents a meshfree-enriched finite element formulation for triangular and tetrahedral ele-ments in the analysis of two and three-dimensional compressible and nearly incompressible solids.The new formulation is first established in two-dimensional case by introducing a meshfree approxima-tion into a linear triangular finite element with an enriched node. The interpolation functions of the four-noded triangular element are constructed by the meshfree convex approximations and are completed toa polynomial of degree one. The reference mapping using the constructed interpolation functions isshown to be invertible everywhere in the element and the global element area is proven to be conservedunder a standard three-point integration rule. The triangular element formulation is extendable to thetetrahedral element in three-dimensional case. To provide a locking-free analysis for the nearly incom-pressible materials, an area-weighted strain smoothing is developed in conjunction with the enrichedinterpolation functions to yield a discrete divergence-free property at the integration point. The resultantelement formulation with strain smoothing is shown to pass the patch test. To introduce the smoothedstrain into Galerkin formulation, a modified Hu–Washizu variational principle is adopted to formulatethe discrete equations. Since the Kronecker-delta property in element interpolation is held along the ele-ment boundary using meshfree convex approximation, boundary conditions can be treated in a standardway. Several numerical benchmarks are provided to demonstrate the effectiveness and accuracy of theproposed method.

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1. Introduction

The simplicity and economy of low-order Lagrange elementshas made them attractive and has been widely used in the finiteelement analyses of problems in solid mechanics. However low-or-der Lagrange elements based on the standard displacement-basedformulation exhibit a poor performance and lead to volumetriclocking in the incompressible regime. Even in the compressiblecase, those elements could cause shear locking in bending-domi-nated problems when a coarse mesh is used.

Many techniques have been proposed to overcome thosenumerical difficulties in particular the volumetric locking. Amongthem are reduced/selective integration [28], u/p mixed formulation[58], enhanced strain method [45] and more recently the averagenodal pressure element [8]. The approach based on u/p mixed for-mulation has a close link with Hughes’s reduced/selective integra-tion [37]. However u/p mixed formulation requires a stable inf-suppair of spaces for the displacement and pressure, and the

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higher-order finite element is generally recommended. Since theu/p mixed method involves more unknowns, it may result in anindefinite discrete linear system. On the other hand, Simo and Rif-ai’s enhanced strain method [45] is often regarded as a generaliza-tion of Hu–Washizu three-field formulation [19]. This method canalso be considered as a special case of nonconforming method [36]in quadrilateral meshes. Classical enhanced strain method usuallysuffers from undesirable instability in compression and requires astabilization procedure [42] to remove the instability. In addition,the method experiences a decline in accuracy when the initial ele-ment is distorted. This deficiency was later improved by replacingthe quadrilateral element using the affine element [4]. Other low-order nonconforming methods [9,24,51] based on the Crouzeix–Raviart element have also been developed and successfully appliedto the near-incompressible elasticity. In comparison with the low-order Lagrange elements, these low-order nonconforming ele-ments give rise to a weaker form of continuity and attention waspaid to fulfill the discrete Korn’s inequalities [22] for the stabilityrequirement. Since the pioneering work by Bonet and Burton [8]various average nodal pressure formulations have been developed(cf. e.g., [1,29]) to overcome volumetric locking. A priori error esti-mate [30] using primal and dual meshes reveals that original aver-age nodal pressure formulation does not satisfy a uniform inf-sup

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2992 C.T. Wu, W. Hu / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2991–3010

condition [6]. In order to have a stable formulation, the linear dis-placement space needs to be enriched with bubble functions as inthe mini-element [2]. This analysis yields a consistent variationalframework [31] for a link to the stabilized nodally integrated tetra-hedral presented by Puso and Solberg [40]. Another pressure aver-aging approach [23,25] based on macroelement technique [46] alsohas been developed for linear-elastic problems leading to the uni-form convergence in the nearly incompressible case.

Recent developments in meshfree method have provided analternative for solving problems describing incompressible ornearly incompressible materials. However displacement-basedGalerkin meshfree method using a low-order approximation expe-riences volumetric locking phenomena similar to that of standardlow-order finite element method. The cause of volumetric lockingin Galerkin meshfree method is the inability of the meshfreeapproximation spaces to satisfy the incompressible constraint im-posed on the solution near the incompressible limit. Therefore, alocking-free displacement-based meshfree formulation requiresthe approximation field to be divergence-free verifying $ � uh ¼ 0.Huerta and Fernandez-Mendez [26] carried out a model analysisto examine the volumetric locking in the element-free Galerkin(EFG) method. The influence of nodal supports and approximationorder on the locking behavior was also discussed. A pseudo-diver-gence-free interpolation for EFG method was proposed by Vidal etal. [49] to diffuse the divergence-free constraint which can be im-posed a priori in a displacement-based Galerkin meshfree formula-tion. Another locking-free displacement-based Galerkin meshfreeformulation was presented by Chen et al. [13] for the rubber-likematerials. With an appropriate decomposition of the strain energydensity function and a selection of particular pressure interpola-tion function, a simple pressure projection formulation wasobtained and can be easily implemented into the displacement-based Galerkin meshfree code. Subsequently, various approacheshave also been developed to alleviate the incompressible locking[18,20] in the framework of B-bar or u/p mixed formulations.Although the implementation of mixed formulation in Galerkinmeshfree method is not straightforward, numerical inf-sup studies[18] indicate that the mixed u/p meshfreee formulation possess theproperty of uniform convergence in the incompressible limit.

Since Kronecker-delta property does not hold in the conven-tional meshfree approximations such as moving least-squares(MLS) [7] or reproducing kernel (RK) [11,35] approximations, spe-cial techniques [12,50] are needed to impose constraint and essen-tial boundary conditions in the Galerkin meshfree method.Alternatively, several convex approximations were introduced[3,47] to simplify the essential boundary condition treatment inthe meshfree methods. Meshfree convex approximations guaran-tee the unique solution inside a convex hull with a minimum dis-tributed data set and poses a weak Kronecker-delta property at theboundaries, and therefore avoid the special treatments on theessential boundaries. Recently Wu et al. [55] provided a unified ap-proach that can generate specific convex approximation as well asreproduce several existing meshfree approximations. The applica-tion of meshfree convex approximation using u/p mixed formula-tion [39] was presented recently to the analysis of compressibleand nearly incompressible elasticity.

Despite those successes, development of an efficient and stableGalerkin meshfree method for compressible and near-incompress-ible problems is not trivial. Additional emphasis has to be placedon the issue of numerical integration of the weak form in Galerkinmeshfree method [14,21]. The inaccuracy in numerical integrationof Galerkin meshfree formulation is due to the characteristics of ra-tional and overlapping properties in the meshfree shape functions.As a result, the method fails to pass the patch test, a test that hasbeen widely used in the engineering community to check the sta-bility of the formulation. An improved numerical integration

scheme that considers the overlapping domain and the intersec-tion with boundaries was developed in the method of finitespheres by De and Bathe [17]. On the other hand, an integrationconstraint (IC) was introduced by Chen et al. [14] as a necessarycondition for a linear exactness in the Galerkin meshfree approxi-mation. In order to satisfy the IC, a strain smoothing technique wasproposed and led to the stabilized conforming nodal integration(SCNI) method [14]. The strain smoothing technique of SCNI wasalso employed in the point interpolation method [33] and radialpoint interpolation method [57] for solids and shells analyses. Amodified SCNI method [41] was developed incorporating a stiff-ness-based gradient stabilization to further enhance meshfree per-formance in large deformation analysis. Alternatively, the strainsmoothing technique of SCNI method was applied to the finite ele-ment method providing a softening effect to improve the solutionaccuracy and led to various node-based, element-based and edge-based smoothed finite element formulations of Liu and co-workers[16,32,34].

The aim of this paper is to develop a low-order displacement-based finite element formulation that utilizes the meshfreeapproximation and strain smoothing technique to circumvent thevolumetric locking as well as to acquire the desired accuracy andconvergence for the compressible and near-incompressible analy-ses. We refer this formulation as the meshfree-enriched finite ele-ment method (ME-FEM). The reminder of the paper is outlined asfollows: In the next Section, we define the boundary-value prob-lem of linear elasticity and review the volumetric locking in thestandard displacement-based Galerkin method. In Section 3, wepresent a meshfree-enriched finite element interpolation for thetriangular elements. In Section 4, we introduce an area-weightedstrain smoothing to the Galerkin approximation and utilize theHu–Washizu variational principle to formulate the discrete equa-tions. In addition, locking-free modes of the proposed triangularelement formulation are studied via eigenvalue analysis and proofsare provided to verify their discrete divergence-free conditions inthe nearly incompressible limit. A discussion on the stability con-dition in general cases is also provided. Finally, the triangular ele-ment formulation is extended to tetrahedral element for the three-dimensional analysis. Several numerical examples are presented inSection 5 to illustrate the accuracy and robustness of the method.Final remarks are drawn in Section 6.

2. Preliminaries

In this section we consider the static response of an elastic bodyunder plain strain conditions. We assume the domain X � R2 be abounded polygon with boundary C = oX. Also, let u be the dis-placement and further assume that the Dirichlet boundary condi-tions are applied on CD and the Neumann boundary conditionsare prescribed on CN. For a prescribed body force f, the governingequilibrium equation and boundary conditions are written as:

� $ � rðuÞ ¼ f in X;

u ¼ g on CD

r � n ¼ t on CNCD [ CN ¼ C; CD \ CN ¼ ;ð Þ;

ð1Þ

where g is the prescribed displacement on CD, and t is theprescribed traction and n is the outward unit normal to the bound-ary CN.

The infinitesimal strain tensor e(u) is defined by

eðuÞ ¼ 12

$uþ ð$uÞT� �

: ð2Þ

In the case of linear isotropic elasticity, the Cauchy stress tensor rand strain tensor e have the following relationship:

C.T. Wu, W. Hu / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2991–3010 2993

r ¼ Ce ¼ 2leþ ktrðeÞI; ð3Þ

where C is the fourth-order elasticity tensor and I is the identitytensor. The positive constants l and k are called the Lamé constantswhich are related to the Young’s modulus E and Poisson ratio v by

l ¼ E2ð1þ vÞ ; k ¼ vE

ð1þ vÞð1� 2vÞ : ð4Þ

The variational form of this problem is to find the displacementu 2 Vg = {m 2 H1(X): m = g on CD} such that for all m 2 V:

Aðu; mÞ ¼ lðmÞ; ð5Þ

where the space V ¼ H10ðXÞ consists of functions in Sobolev

spaceH1(X) which vanish on the boundary and is defined by

VðXÞ ¼ m : m 2 H1; m ¼ 0 on CD

n o: ð6Þ

The bilinear form A(�, �) and linear functional l(�) in Eq. (5) are de-fined by

Aðu; mÞ ¼ 2lZ

XeðuÞ : eðmÞdXþ k

ZX

$ � uð Þ $ � mð ÞdX; ð7Þ

lðmÞ ¼Z

Xf � mdXþ

ZCN

t � mdC: ð8Þ

The bilinear form A(�, �) in the above equation is symmetric, V-ellip-tic and continuous. By Lax–Milgram theorem, there exists a uniquesolution u 2 Vg to the problem [15]. Moreover, let u 2 Vg\H2ðf 2 L2ðXÞ; g ¼ w1jCD

where w1 2 H2(X), and t ¼ w2jCNwhere

w2 2 H1(X)), we have the following elliptic regularity estimate [9]:

uk k2 þ k $ � uk k1 6 C1 fk k0 þ w1k k2 þ w2k k1ð Þ; ð9Þ

where k�km is Sobolev norm of order m as defined in a standard way.The constant C1 in Eq. (9) does not dependent on k and l.

For simplicity, we assume the homogeneous problemVg ¼ V ¼ H1

0ðXÞ. The standard Galerkin method is then formulatedon a finite dimensional subspace Vh � V employing the variationalformulation of Eq. (5) to find uh 2 Vh such that:

A uh; mh� �

¼ lðmhÞ 8 mh 2 Vh: ð10Þ

As k ?1 (or m ? 0.5), the following constraint must be enforced:

$ � u ¼ 0 for u 2 V : ð11Þ

Similarly in the finite dimensional space, we have the followingconstraint:

$ � uh ¼ 0 for uh 2 Vh: ð12Þ

The solution of Eq. (10) using a low-order approximation fordisplacement field in general is not capable of retaining the opti-mal approximation when the incompressible constraint in Eq.(12) is enforced. This phenomenon is called ‘‘volumetric locking’’in the literature. In other words, volumetric locking occurs whenthe approximation space Vh is not rich enough for the approxima-tion to verify the divergence-free condition $ � uh ¼ 0 [26].

3. Meshfree-enriched finite element interpolation in triangularelement

This section devotes to the development of meshfree-enrichedfinite element interpolation for the ME-FEM triangular element.Let’s assume X is a convex polygon and Mh = [e Te is a regular fam-ily of triangulations of X. Each triangular element Te contains threecorner nodes xi, 1 6 i 6 3 and one enriched meshfree node x4. Theglobal element Te is also the image of a reference triangle Te

through an isoparametric mapping represented in a vector form as:

x ¼ FeðnÞ; ð13Þ

where x = [x,y]T and n = [n,g]T. Here, the mapping function Fe mapsthe reference triangle Te to the general global triangular element Te.

Since the natural coordinate system of triangular elements doesnot form an independent set, it can not be employed to the ele-ment mapping for the ME-FEM triangular elements. On the otherhand, the direct mapping of triangular elements using isoparamet-ric approach introduces an idiosyncrasy which is not associatedwith the quadrilateral elements and needs to be modified to pre-serve the nodal sequencing in the parametric mapping for the cur-rent development. Therefore, an orientation-preserving mappingfor this four-noded meshfree-enriched triangular element is intro-duced and depicted in Fig. 1. In Fig. 1, the reference element Te isan equilateral triangle, with dark circles denoting the finite ele-ment node and open circles denoting the enriched meshfree node.The location of the enriched meshfree node in reference element Te

is given by

n4; g4ð Þ ¼X3

i¼1

ni

,3;X3

i¼1

gi

,3

!; ð14Þ

which is the centroid of the reference element. ni = (ni,gi), i = 1, 2, 3,4 are nodal co-coordinates of the reference element.

In Fig. 1, each node is assigned to a weight function with samecircular support in the reference element Te. The shape functions ofthe reference element are constructed using a meshfree convexapproximation. In this study, we employ the generalized meshfreeapproximation (GMF) method [55] to obtain the meshfree convexapproximation. The convex GMF approximation is constructedusing the inverse tangent basis function and the cubic spline win-dow function is chosen to be the weight function in GMF method. Abrief introduction of the first-order GMF formulation is given inAppendix A. The detail derivation of GMF method and the corre-sponding mathematical properties can be found in [55].

Since the shape functions constructed using the GMF methodare convex, they exhibit the following convexity properties:

WiðnjÞ ¼ dij 1 6 i; j 6 3;

W4ðnÞ > 0 8 n 2 Te;

W4ðnÞ ¼ 0 8 n 2 oTe;

Wiðn4Þ > 0 1 6 i 6 3 and W4ðn4Þ–1:

ð15Þ

Giving the four-noded ME-FEM shape functions, the mappingfunction Fe(n) in Eq. (13) can be defined as follows:

Fe : Te ! Te; x ¼ FeðnÞ ¼ F1e ; F2

e

� �¼

X4

i¼1

xiWi n;gð Þ;X4

i¼1

yiWi n;gð Þ !

8 n 2 Te; ð16Þ

with the above notations, we define the following approximationspace for the displacement field:

VhðXÞ¼ mh : mh 2H10ðXÞ; mh

��Te

¼ �mh �F�1e ; �mh 2P1ðTeÞ 8 Te 2Mh

� �;

ð17Þ

where P1ðTeÞ ¼ spanfWi; i ¼ 1; . . . ;4g denotes the space contains aset of basis functions in Te. In general, Wi, 1 6 i 6 4 are not mutuallyð�; �ÞVh -orthonormal basis functions.

The Jacobian matrix Jn is defined to be the gradient of the map-ping function Fe by

Jij ¼oFi

eðnÞonj

; i; j ¼ 1;2; ð18Þ

or in matrix form:

Jn ¼ GF GTx ; ð19Þ

Fig. 1. Isoparametric mapping in the 4-noded meshfree-enriched triangular finite element.


where matrices GF and Gx are given by

GF ¼ $ðn;gÞ Wi n;gð Þ½ �e ¼oW1on � � � oW4

on

oW1og � � � oW4

og

" #; ð20Þ

Gx ¼x1 � � � x4

y1 � � � y4

; ð21Þ

$ðn;gÞ ¼oon

oog

" #: ð22Þ

To this end, the fourth node plays a role similar to the bubblefunction in the incompatible [53] or enhanced strain elements[10] in finite element method. Unlike the incompatible element,the ME-FEM element equipped with the bubble-like meshfree-en-riched shape function is compatible. In other words, the fourthnode of the ME-FEM element carries the physical displacementsand should be considered in computing the element body forcevectors as well. According to Eq. (16), the mapping at the fourthnode reads:

ðx4; y4Þ ¼ Feð0;0Þ ¼X4

i¼1

xiWið0;0Þ;X4

i¼1

yiWið0;0Þ !

: ð23Þ

Consequently, the global co-coordinates of the fourth node is deter-mined by

x4; y4ð Þ ¼P3

i¼1Wið0;0Þxi

1�W4ð0;0Þð Þ ;P3

i¼1Wið0; 0Þyi

1�W4ð0; 0Þð Þ

!

¼P3

i¼1Wið0;0ÞxiP3i¼1Wið0;0Þ

;

P3i¼1Wið0; 0ÞyiP3

i¼1Wið0; 0Þ

!

¼P3

i¼1xi

3;

P3i¼1yi

3

!; ð24Þ

which is the centroid of the global triangle.We proceed to show that the mapping in Eq. (16) is invertible

i.e., the determinant of the Jacobian matrix in Eq. (18) is positiveeverywhere in the reference element.

Theorem 3.1. The Jacobian of the reference mapping for meshfree-enriched triangular element is invertible.

Proof 1. The zero and first-order consistency conditions in theME-FEM element give:

1 � � � 1n1 � � � n4

g1 � � � g4

24 35 W1ðnÞ...

W4ðnÞ

8><>:9>=>; ¼

1ng

8<:9=;: ð25Þ

One can rewrite Eq. (25) by using Eq. (14) to yield:

1 1 1n1 n2 n3

g1 g2 g3

24 35 W1ðnÞþW4ðnÞ3

W2ðnÞþW4ðnÞ3

W3ðnÞþW4ðnÞ3

8><>:9>=>;¼

1 1 1n1 n2 n3

g1 g2 g3

24 35 bW1ðnÞbW2ðnÞbW3ðnÞ

8><>:9>=>;¼

1n

g

8<:9=;;ð26Þ

where

bWiðnÞ ¼ WiðnÞ þW4ðnÞ

3; i ¼ 1;2;3; ð27Þ

are the modified shape functions. Since the modified shape func-tions of triangular element fulfill the zero and first-order exactness,they reproduce the standard linear finite element shape functions.

Using Eqs. (19) and (27), we can further express the Jacobianmatrix Jn by

Jn ¼ GF GTx ¼ bGF

bGTx : ð28Þ

Accordingly, the Jacobian determent jJnj is:

Jn

�� ¼ GF GTx

�� ¼ bGFbGT

x

�� ¼ CeAe ¼ constantð Þ > 0; ð29Þ

where

bGF ¼ $ðn;gÞ bWi n;gð Þh i

e¼

obW1on

obW2on

obW3on

obW1og

obW2og

obW3og

264375; ð30Þ

bGx ¼x1 x2 x3

y1 y2 y3

; ð31Þ

Ae is the element area and Ce ¼ 4=ffiffiffi3p

is the coefficient correspond-ing to the mapping adopted in Fig. 1. Since the Jacobian determentis a positive constant for all n 2 Te, the mapping Fe is invertible. h

Fig. 2 shows the shape functions contours of a four-noded ME-FEM element using GMF (atan) approximation where the Kroneck-er-delta property is satisfied at the boundary. The shape functionsof the ME-FEM element reduce to the standard linear finite ele-ment shape functions along the element edge that exhibit theconvex approximation property. Consequently, the finite dimen-sional subspace Vh is continuous along the global element edgeand thus H1-conforming.

Fig. 2. GMF (atan) convex approximation in a 4-noded meshfree-enriched finiteelement.


Remark 1. Due to the enrichment of meshfree node, the ME-FEMelement is no more a constant strain element. The approximationspace is also not a divergence-free space.

Fig. 3. Strain smoothing in ME-FEM triangle elements.

4. Strain smoothing and modified variational formulation

The strain smoothing in SCNI method [14] was originally pro-posed to provide a strain smoothing stabilization for nodal integra-tion and to preserve the linear exactness in the Galerkin meshfreemethod. The original formulation for the strain smoothing in SCNImethod is defined by Chen et al. [14]:

~eh ¼ Ph e uh� ��

¼ 1Am

ZXm

e uh� �

UmðxÞdX; ð32Þ

where Ph denotes the smoothed strain operator. e(uh) is the stan-dard strain computed from displacement by compatibility,ei;j ¼ uh

i;j þ uhj;i

� �=2. Xm is a representative domain of node m which

is constructed using Voronoi diagrams in SCNI method. Am ¼R

XmdX

is the associated area of Xm. Um(x) is the characteristic or smooth-ing function of the nodal representative domain Xm defined by

UmðxÞ ¼1; if x 2 Xm;

0; else:

�ð33Þ

In SCNI method, a divergence theorem is applied to Eq. (32) to yield:

~eh ¼ 1Am

ZXm

12

ouhi

oxjþ

ouhj

oxi

!UmðxÞdX ¼ 1

2Am

ZCm

uhi nj þ uh

j ni

� �dX;

ð34Þ

where Cm is the boundary of representative domain of node m withni and nj denoting the components of the surface normal on Cm.Since the evaluation of Eq. (34) has been transformed from areaintegration into the line integration, the computation of the

smoothed strain in SCNI only involves the evaluation of shape func-tions instead of derivatives of shape functions.

4.1. Strain smoothing in meshfree-enriched triangular elements

The proposed strain smoothing departs from the original strainsmoothing in Eq. (32). In this section we first construct a newstrain smoothing domain based on the meshfree-enriched finiteelement triangulation. Secondly, an area-weighted averaging isperformed on strains in each strain smoothing domain. This leadsto a constant strain represented in each strain smoothing domain.Based on this idea, we rewrite Eq. (32) in discrete form as:

eeh ¼ 1Am

Xj¼1

e uh xjm� ��

Ajm; ð35Þ

where xjm is the position of sampling point for area-weighted strainsmoothing and Ajm is the associated area or weight of each samplingpoint. Am is now the area of the new strain smoothing domain ob-tained by Am ¼

PjAjm.

The construction of new strain smoothing domain is describedas follows: Let xj

i;1 6 i 6 3 be the positions of three corner nodesof a finite element triangular element Tj 2Mh. xj

4 is the positionof the enriched meshfree node locating at the centroid of the trian-gular element Tj. We connect xj

4 to the three vertices of the triangleby straight lines to divide the triangle into three sub-triangles Sl,Sm

and Sn as shown in Fig. 3. Each sub-triangle Sl shares the elementedge l of the triangle and carries one sampling point for the strainsmoothing. Since each sub-triangle occupies the same area in tri-angular element Tj, the value of smoothing area Ajm in Eq. (35)for sampling point xjm is one-third of the element area Aj

e ¼ 3Ajm.There are several ways to choose the sampling point in each sub-triangle of the triangular element Tj. In this study we select thelocations of the sampling points xjm, 1 6m 6 3 to locate at the mid-dle of element edge as shown in Fig. 3. This leads to the followingrelationship of three sampling points in terms of shape functionderivatives defined in the reference co-coordinate system:X3

m¼1

oWIðnjmÞon

¼ 0; ð36Þ

andX3

m¼1

oWIðnjmÞog

¼ 0: ð37Þ


Now the new strain smoothing domain Xm is defined by

Xm ¼ [S2Sm S; ð38Þ

which is the collection of the sub-triangle Sm of adjacent elementssharing the same element edge m.

Subsequently, the smoothed strain in Eq. (35) can be expressedby

~eh ¼ 1Am

Xj¼1

e uh xjm� ��

Ajm

¼ 1Am

Xj2edge m

X4

I¼1

BjI xjm� �

uhI

� �Ajm

¼ 13Am

Xj2edge m

X4

I¼1

BjI xjm� �

uhI

� �Aj

e

¼ 13Am

XNP

I¼1

Xj2edge m

BjI xjm� �

Aje

� �uh

I ¼XNP

I¼1

eBIuhI ; ð39Þ

where NP is the number of nodes involved in each smoothing do-main. Apparently, NP is determined by the number of sub-trianglesof adjacent elements in the smoothing domain. In general NP = 6since each pair of sub-triangles of adjacent elements involves 6 totalnumber of nodes in the smoothed strain computation. If thesmoothing domain only contains one sub-triangle, NP becomes 4.This is the case when the element edge m contains a piecewise bof global boundary oX = [bmb and is not sharing with otherelements.

The smoothed strain gradient matrix eBI in Eq. (39) is defined by

eBI ¼X

j2egde m

BjI xjm� �

Aje

3Am

!; ð40Þ

where BjIðxjmÞ is the standard strain gradient matrix of node I in ele-

ment Tj evaluated at xjm and is defined by

BIðxÞ ¼

oWIðnÞox 0

0 oWIðnÞoy

oWIðnÞoy

oWIðnÞox

26643775: ð41Þ

We continue to show that the smoothed four-noded ME-FEMformulation preserves the linear exactness in the Galerkin approx-imation of Dirichlet boundary value problem. i.e.,Z

X

eBIdX ¼ 0 8 interior node I 2 X [ C; ð42Þ

which is known as the integration constraints (IC) [14] in meshfreeGalerkin method.

Theorem 4.1. The smoothed gradients of the meshfree-enrichedtriangular elements satisfy the IC and the resultant element formu-lation passes the patch test.

Proof 2. First consider the case that interior node is the enrichedmeshfree node of the element. Substituting Eq. (40) into Eq. (42)and using Eqs. (29), (36), (37) and (41) to yield the expression ofintegration constraint for the enriched meshfree node I in triangu-lar element Tj byZ

X

eBIdX ¼Z

X

Xj2egde m

BjIxjmAj

e

3Am

!dX ¼

X3

m¼1

Xj2egdem

BjI xjm� �

Aje

3

!

¼X3

m¼1

BI xjm� �

Ae

3¼Z Z

bI njm

� �dndg ¼

X3

m¼1

bI njm

� �3Ce

¼ 0; ð43Þ

where

bIðnÞ ¼

oWIðnÞon 0

0 oWIðnÞog

oWIðnÞog

oWIðnÞon

26643775; ð44Þ

Ce is the coefficient related to the isoparametric mapping and Ae isthe element area of the interior node I which is given in Eq. (29).Consequently, for any interior finite element node in X, we have:Z

X

eBIdX ¼Z

XBIdX ¼

Xnelem

j¼1

X3

m¼1

BjI xjm� �

Aje

3¼Xnelem

j¼1

X3

m¼1

bjI njm

� �3Ce

; ð45Þ

where nelem is the total number of elements containing the com-mon finite element node I in X. Using the relationships in Eq.(27), we can obtain the shape function derivatives of finite elementnodes defined in the reference co-coordinates by

oWiðnÞon

¼ o bWiðnÞon

� oW4ðnÞ3on

; 1 6 i 6 3; ð46Þ

oWiðnÞog

¼ o bWiðnÞog

� oW4ðnÞ3og

; 1 6 i 6 3: ð47Þ

Note that W4 is the meshfree-enriched finite element shape func-tion of the enriched meshfree node. bWi;1 6 i 6 3; are the standardlinear finite element shape functions of the triangular elementwhich satisfies the integration constraint byZ

X

bBIdX ¼Xnelem

j¼1

X3

m¼1

bBjI njm

� �Aj

e

3¼Xnelem

j¼1

X3

m¼1

bbjI njm

� �3Ce

¼ 0; ð48Þ

where bBI is the standard strain gradient matrix of linear triangularfinite element and bbI is the corresponding matrix defined in Eq.(44). Substituting Eqs. (46) and (47) into Eq. (44) leads to:

bI ¼ bbI �b4

3; 1 6 I 6 3: ð49Þ

The integration constraint of interior finite element node in X canbe obtained by substituting Eq. (49) into Eq. (45) and using theproperties in Eqs. (36), (37), (43) and (48) to yield:Z

X

eBIdX ¼Xnelem

j¼1

X3

m¼1

bjI njm

� �3Ce

¼Xnelem

j¼1

X3

m¼1

b̂jI njm

� �3Ce

�X3

m¼1

bj4 njm

� �9Ce

!¼ 0� 0 ¼ 0: ð50Þ

Eqs. (43) and (50) show that the integration constraints hold for theME-FEM triangular elements and the resultant ME-FEM formulationwill pass the patch test in the compressible case. Hence, the prooffollows. h

It is interesting to note that the above proof does not involve thenodal support. Since the smoothed strains are defined locally oneach smoothing domain Xm and no continuity conditions are ap-plied at the boundaries of Xm, the approximation space ofsmoothed strain ~eh can be defined by

EhðXÞ ¼ ~eh : ~eh 2 L2ðXÞ; ~ehjXMis symmetric and contains

npiecewise constants for all Xm 2 Mh

o: ð51Þ

4.2. Modified Hu–Washizu formulation

To introduce the strain smoothing formulation into the Galerkinapproximation, the Hu–Washizu functional [52] is considered andexpressed as:


UHW u; ~e; ~rð Þ ¼Z

X

12

~eT C~edXþZ

X

~rT ru� ~eð ÞdX�Wext; ð52Þ

where the displacements u, smoothed strains ~e and stresses ~r areindependently varied. The term Wext designates the external workas defined in Section 2. If the approximation space of stressesSh(X) is chosen to be the subspace of Eh(X), namely Sh � CEh, wehave:

~rh ¼ C~eh ¼ rh: ð53Þ

Accordingly, the stress fields ~rh can be eliminated from the func-tional leading to the following Hellinger–Reissner functional [19]:

UHR; uh; ~eh� �

¼Z

X

12

~ehTC~ehdX�

ZX

~ehTC $uh � ~eh� �

dX�Wext: ð54Þ

Since Eh contains piecewise constants in X, the second term on theRHS of Eq. (54) can be further expressed using Eq. (35) to yield:Z

X

~ehTC $uh � ~eh� �

dX ¼Xnm

m¼1

~ehTCZ

XM

$uh � ~eh� �

dX

¼Xnm

m¼1

~ehTCZ

XM

$uhdX� ~ehAm

�

¼Xnm

m¼1

~ehTCZ

XM

$uhdX�Z

XM

$uhdX�

¼ 0: ð55Þ

The index nm in Eq. (55) denotes the total number of smoothing do-main Xm in X(X = [mXm). Eq. (55) implies an orthogonal conditionbetween the stress field ~r and the difference of the displacementgradient $uh and smoothed strain field ~eh. This is equivalent tothe assumed strain variation principle of Simo and Hughes [44].After eliminating the stress components from Eq. (54) using Eq.(55), the following modified Hu–Washizu functional is obtaineddepending only on strain and displacement fields:

UHW ;mod uh; ~eh� �

¼Z

X

12

~ehTC~ehdX�Wext: ð56Þ

The above modified functional can be rewritten as a primal varia-tional problem by

eUmod uh� �

¼Z

X

12

Ph eT uh� ��

CPh e uh� ��

dX�Wext; ð57Þ

where the smoothed strain operator Ph is defined in Eq. (32). SinceEh is a piecewise constant space, there exist positive constants Cp1

and Cp2 such that Ph is continuous as a local L2 operator satisfying[30]:

Cp1 wk k0 6 Phwk k0 6 Cp2 wk k0 8 w 2 L2: ð58Þ

We also have the following error estimate from the approximationproperty:

kðI �PhÞwk0 6 Cahkwk1 8 w 2 H1; ð59Þ

where Ca is generic constants independent of element size h and k.Taking variation of Eq. (57) gives the following reduced problemwritten in abstract form: find uh 2 Vh such that:

Ah uh; mh� �

¼ l mh� �

8 mh 2 Vh; ð60Þ

where the modified bilinear form Ah(�, �) is defined by

Ah uh; mh� �

¼Z

XCPh e uh

� �� : Ph e mh

� �� dX: ð61Þ

Remark 2. The orthogonal condition in Eq. (55) allows us torewrite Eq. (61) as:

Ah uh; mh� �

¼Z

XCPh e uh

� �� : e mh� �

dX; ð62Þ

which gives the same variation space for displacement and there-fore the RHS of Eq. (60).

Using the Cauchy–Schwarz inequality and inequality (58), it isready to show that the modified bilinear form Ah(�,�) is boundedon Vh � Vh.

Theorem 4.2. The modified bilinear form Ah(�,�) defined in Eq. (61) isbounded on Vh � Vh, i.e., there exists a positive constant Cb such that:

jAh uh; mh� �

j 6 Cbkuhk1kmhk1 8 uh; mh 2 Vh: ð63Þ

Proof 3.

Ah uh;mh� �� 6Z

XCPh e uh

� �� : Ph e mh

� �� dX

6 cmax Cð ÞZ

XPh e uh

� �� 0 Ph e mh

� �� 0dX

6 cmax Cð ÞZ

XPh e uh

� �� 2

0dX� 1=2 Z

XPh e mh

� �� 2

0dX� 1=2

6 cmax Cð ÞC2p2

ZX

e uh� �� 2

0dX� 1=2 Z

Xe mh� �� 2

0dX� 1=2

;

ð64Þ

where cmax is the largest eigenvalue of C.Using triangle inequality it follows:Z

XeðuhÞ�� 2

0dX ¼Z

X

X2

i;j¼1

eij uh� �� 2

dX

¼X2

i;j¼1

ZX

12

ouhi

oxjþ

ouhj

oxi

!" #2

dX

6

X2

i;j¼1

ZX

12

ouhi

oxj

� 2

þouh

j

oxi

!224 35dX ¼ juhj21: ð65Þ

Substituting inequality (65) into inequality (64) to yield:

Ah uh; mh� �� 6 cmax Cð ÞC2

p2 uh��

1 mh��

1

6 Cb uh��

1 mh��

1 8 uh; mh 2 Vh�: ð66Þ

The coercivity of the modified bilinear form Ah(�, �) on Vh � Vh

can also be proved using Korn’s inequality [15] and inequality(58). For simplicity, we only prove the coercivity of the reducedproblem under pure Dirichlet boundary conditions (CD = C).

Theorem 4.3. The modified bilinear form Ah(�, �) defined in Eq. (61) iscoercive on Vh � Vh, i.e., there exists a positive constant Cc such that:

Ah mh; mh� �

P Cckmhk21 8 mh 2 Vh: ð67Þ

Proof 4.

Ah mh; mh� �

P cmin Cð ÞZ

XPh e mh

� �� : Ph e mh

� �� dX

P cmin Cð ÞCcp1

ZXe mh� �

: e mh� �

dX

Pcmin Cð ÞCcp1

Ckkmhk2

1 ¼ Cckmhk21 8 mh 2 Vh; ð68Þ

where cmin is the smallest eigenvalue of C and Ck is the constantfrom Korn’s inequality. Cc is the coercive constant which is positiveand depends on l and k. h


According to Lax–Milgram theorem, a unique solution exists tothe reduced problem in Eq. (60) due to inequalities (63) and (67)with respect to each k > 0.

Since the coercive constant Cc is function of k, the internal en-ergy may not be bounded in the near-incompressible limit ask ?1 unless the constraint equation of Eq. (12) is satisfied. Inother words, in order to have a convergent solution, the volumetricenergy needs to be finite in the nearly incompressible regime, i.e.,the volumetric term on the RHS of Eq. (61) has to satisfy:

kZ

XtrPh e uh

� �� trPh e mh

� �� dX! 0 as k!1; ð69Þ

or satisfy the following discrete condition:

trPh e uhðni

� �jTeÞ

� �! 0 as k!1 8 Te 2 Mh;

nii ¼ 1;2;3 are sampling points: ð70Þ

Eq. (70) is called the discrete divergence-free condition in this pa-per. The fulfillness of the discrete divergence-free condition in theproposed method will be further discussed in the next section.

Now we discretize the displacement-based variational formula-tion in Eq. (60) by approximating the displacements in each ele-ment using the enriched ME-FEM interpolation:

uh ¼X4

I¼1

WIuI: ð71Þ

The strains are approximated using the strain smoothing schemedescribed in Section 4.1 and are given by

Ph e uh� ��

¼XNP

I¼1

eBIuI; ð72Þ

where eBI is the smoothed gradient matrix defined in Eq. (40) and4 6 NP 6 6 is the number of nodes involved in each smoothing do-main Xm. By introducing the displacement and strain approxima-tions into Eq. (60), the following discrete governing equation isobtained:eK d ¼ f ext; ð73ÞeK IJ ¼

ZX

eBTI CeBJdX; ð74Þ

f extI ¼

ZX

WIf dXþZ

CN

WItdC: ð75Þ

Since ME-FEM element interpolation preserves Kronecker-deltaproperty at the element boundary, the essential boundary condi-tions can be treated in the standard way. From Eq. (15) it is not hardto see that the computed displacements of meshfree-enrichednodes are not the real nodal displacements. This is because theshape function of the enriched node does not possess Kronecker-delta property. An element-wise nodal transformation [11] can beeasily performed to obtain the real nodal displacements for en-riched nodes as needed.

4.3. Eigenvalue analysis and stability condition in near-incompressiblelimit

An eigenvalue analysis [26,27] using the LHS of Eq. (10) can beemployed to reveal the locking phenomena in the low-order dis-placement-based finite element and ME-FEM methods. The eigen-value analysis of one rectangular Q1 bilinear finite element isanalyzed first and two non-physical locking modes are obtainedand shown in Fig. 4(a). In Fig. 4(a), the eigenvalues grow un-bounded as v approaches to 0.5, which is not expected physically.When the rectangular Q1 bilinear element is replaced by a patchconsisting of two standard triangular P1 linear elements, twonon-physical locking modes are also observed as displayed in

Fig. 4(b). The volumetric locking in P1 linear elements is clearlydue to the inability of the constant strain interpolation to producethe divergence-free condition at the Gauss point. We further con-sider the NICE-T3 element (nodally integrated continuum element)of Krysl and Zhu [29], where the nodal strain–displacement matri-ces are constructed as averages of the strain–displacement matri-ces from the connected elements. Similarly, two volumetriclocking modes are generated from the eigenvalue analysis usingtwo NICE-T3 elements as shown in Fig. 4(c).

When the same eigenvalue analysis is reanalyzed using twoME-FEM triangular elements with strain smoothing, two locking-free modes as shown in Fig. 5 are obtained. In other words, thedivergence-free condition in Eq. (70) can be achieved point-wiseat the integration points. Note that no spurious zero energy modeis found in the analysis using ME-FEM triangular elements.

A direct proof of discrete divergence-free properties for the firstlocking-free mode is given in Appendix B. Similar proof can be ap-plied to the second locking-free mode. The locking-free result inME-FEM elements indicates that the enrichment of meshfree nodeand strain smoothing can enlarge the space of displacement diver-gence and result in a relaxation of the incompressible constraint.Nevertheless, the direct proof of discrete divergence-free proper-ties may not be available for general cases.

Remark 3. Compared to the ME-FEM triangular element, the ME-FEM quadrilateral element [56] embeds a divergence-free spaceand therefore a direct proof of discrete divergence-free propertiesin ME-FEM quadrilateral element for general cases can be providedwithout difficulty.

Since the pressure does not appear explicitly in the reducedproblem of Eq. (60), the convergence analysis for the nearly incom-pressible problem using ME-FEM triangular elements with strainsmoothing can alternatively be transformed to the study of the sta-bility condition for a saddle point problem of penalized Stokesequations. To start with, we consider the following mixed formula-tion of linear elasticity problem under pure Dirichlet boundaryconditions assumption: Find ðuh; phÞ 2 Vh � Ph such that:

A uh; mh� �

þ B div mh;ph� �

¼ l mh� �

8 mh 2 Vh;

B div uh; qh� �

� 1k

B ph; qh� �

¼ 0 8 qh 2 Ph;ð76Þ

where

A uh; mh� �

¼ 2lZ

XPh e uh

� �� : Ph e mh

� �� dX;

B p; qð Þ ¼Z

XpqdX;

div uh ¼ trPh e uh� ��

:

ð77Þ

Due to the pure Dirichlet boundary condition assumption, we havethe following additional scaling condition in the incompressiblelimit:Z

XpdX ¼

ZC

u � nds ¼ 0: ð78Þ

Therefore we have to restrict the pressure space Ph to be a subset ofL2

0ðXÞ ¼ q 2 L2ðXÞ;R

X qdX ¼ 0n o

. Apparently if we select the pres-sure space Phto contain the piecewise constants defined by

Ph ¼ qh : qh 2 L20ðXÞ; qhjXm

2 P0ðXmÞ 8 Xm 2 Mh

n o; ð79Þ

then we can apply the static condensation and eliminate the pres-sure in smoothing domain Xm to obtain an equivalent displace-ment-based formulation of the reduced problem in Eq. (60).Hence the stability condition of the reduced problem in Eq. (57)resembles the inf-sup condition in the penalty formulation [38] of

Fig. 4. Two non-physical locking modes.

Fig. 5. Two physical locking-free modes in two ME-FEM triangular elements.


Fig. 6. Reference mapping in the 5-noded meshfree-enriched tetrahedral finiteelement.

Fig. 7. Strain smoothing in ME-FEM tetrahedral elements.


the Stokes equations. In other words, the well-posedness of the re-duced problem in near-incompressible regime is still subject to astability condition between the displacement space Vh and an im-plicit pressure space Ph induced by the following equation:

ph ¼ �ktrPh e uh� ��

¼ �kdiv uhð Þ in Ph: ð80Þ

An essential step for the stability analysis of the mixed problemin Eq. (76) is to verify the following discrete inf-sup condition forbilinear function Bðdivuh; qhÞ:

infqh2Phnf0g

supmh2Vhnf0g

RX qhdiv mhð ÞdXk~e mhð Þk0kqhk0

P bk; ð81Þ

or equivalently:

infqh2Phnf0g

supmh2Vhnf0g

RX qh kdiv mhð Þ

� �dX

kk~e mhð Þk0kqhk0P bk; ð82Þ

where the inf–sup constant bk is a positive number independent of

mesh size h. Here the norm k~eðmhÞk0 ¼R

X

P2i;j¼1k~eijk2

0dX� �1=2

is de-

fined in terms of strain components in L2 space. Under this circum-

stance, a satisfaction of inf-sup condition for a pair of spaces Vh � Ph

would imply a discrete divergence-free condition in Eq. (70) for thedisplacement-based formulation. A discrete divergence-free solu-tion for the reduced problem of Eq. (60) will converge linearly tothe exact solution in the energy norm independent of k using thefirst-order meshfree convex approximation [56]. The failure to sat-

isfy inf-sup condition indicates the pair of spaces Vh � Ph is not sta-ble. As a result, numerical solutions may exhibit locking andpossible pressure oscillation and optimal convergence rates areusually not available. Using Eq. (80), it is recognized that:Z

Xqh kdiv mhð Þ� �

dX ¼ B qh; qh� �

¼ qh�� 2

0 > 0: ð83Þ

Therefore, a direct proof of inf-sup condition requires the followinginequality:

k ~eðmhÞ��

0 61bk

qh��

0: ð84Þ

The proof of the inequality (84) relies on the macroelementtechnique [46] which is difficult to be applied using the meshfreeshape functions. In addition, an analytical proof of discrete inf-sup condition for general cases involving different mesh orienta-tions and boundary conditions is also not an easy task. For the sim-plicity of exposition, we will study the rate of convergence forenergy and pressure error norms using several benchmark exam-ples. Additionally, we will conduct the numerical inf-sup test ofBathe [5] for those problems whose analytical solutions are notavailable. Since the pass of numerical inf-sup test is not a necessarycondition for the stability of the numerical method, we will alsoexamine if locking and pressure oscillation exist under variousloading and boundary conditions to determine the stability of theproposed method. Those will be demonstrated through the numer-ical examples in Section 5.

4.4. Extension to tetrahedral elements

The meshfree-enriched finite element formulation for triangu-lar element is extendable to tetrahedral element in three-dimensional case. The construction of meshfree-enriched finiteelement interpolation in tetrahedral element and its referencemapping follow the same procedure of triangular element as de-scribed in Section 3. For simplicity, we present the reference map-ping of the 5-noded ME-FEM tetrahedral element in Fig. 6. Sincethe bubble-like shape function of the enriched node is continuous

across the element boundaries and the element is compatible, theJacobian of the reference mapping for the 5-noded ME-FEM tetra-hedral element retains invertibility as proven in Theorem 3.1.

The strain smoothing in the 5-noded ME-FEM tetrahedral ele-ments utilizes the same strain-smoothing technique in the 4-noded ME-FEM triangular elements as presented in Section 4.1.The strain smoothing in the adjacent ME-FEM tetrahedral elementsis illustrated in Fig. 7. The satisfaction of linear exactness in theME-FEM tetrahedral formulation can be easily verified followingsimilar proof given in Theorem 4.1. The numerical inf-sup testand the examination of locking and pressure oscillation will alsobe conducted for the three-dimensional example given in the fol-lowing section.

5. Numerical examples

In this section, we analyze five linear benchmark examples intwo and three-dimensions to study the performance of the mesh-free-enriched simplex elements with area-weighted strainsmoothing (ME-Tri-AW) in the compressible and nearly incom-pressible cases. As comparison, we also provide the results using:(1) the 4-noded Q1 bilinear element and 8-noded Q1 tri-linear ele-ment with the selected reduced integration (Q1-SR), (2) the 3-noded triangular element with nodal integration (NICE-T3) [29].In meshfree-enriched shape function, the weight function is cho-sen to be the cubic B-spline kernel function with normalized sup-port size equal to 1.35. Unless otherwise specified, the followingmaterial constants are used for all benchmark examples: Young’smodulus E = 1000, Poisson ratio m = 0.3 for compressible materialand m = 0.4999999 for nearly incompressible material. The


following L2 and energy error norms are used for the investigationof convergence rates:

ehuL2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZX

uhx � ux

� �2 þ uhx � ux

� �2h i

dX

s;

ehuE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZX

e uhð Þ � e uð Þð Þ : C : e uhð Þ � e uð Þð Þ½ �dX

s:

ð85Þ

5.1. Cantilever beam

Consider a cantilever beam problem, as shown in Fig. 8(a), incompressible and near-incompressible cases. Analytical displace-ment field is prescribed along x = 0, and parabolic vertical traction

Fig. 8. Cantilever beam.

Fig. 9. Convergence of error norms in compressible case.

P is applied along x = 10. The analytical displacement and stresssolutions are given as [48]:

ux ¼ �Py

6eEIð6L� 3xÞxþ ð2þ ~vÞ y2 � D2

4

!" #;

uy ¼P

6eEI3vy2ðL� xÞ þ ð4þ 5vÞD

2x4þ ð3L� xÞx2

" #;

rxx ¼ �PðL� xÞy

I; ryy ¼ 0; rxy ¼

P2I

D2

4� y2

!;

ð86Þ

Fig. 10. Stress results along the cross-section x = 4.875 in nearly incompressiblecase.

Fig. 11. Deformation plot using ME-Tri-AW in nearly-incompressible case (scaledby 5 times): initial (dash lines), analytical (thick red lines), numerical (thick bluelines). (For interpretation of the references to color in this figure legend, the readeris referred to the web version of this article.)

Fig. 12. Convergence of error norms in near-incompressible case.

Fig. 13. Inflation problem.

Fig. 14. Convergence of error norms for inflation problem in nearly incompressiblecase.


where eE ¼ E=ð1� m2Þ; ~m ¼ m=ð1� mÞ under plane strain assumptionand I = D3/12 is the moment of the inertia of the beam. Thecomputational domain is discretized uniformly with enriched nodeslocated in the middle of triangular elements as shown in Fig. 8(b)for the coarsest mesh.

The results of L2-norm and energy-norm errors against the to-tal number of degrees of freedom in compressible case are shownin Fig. 9(a) and (b), respectively. All methods present similar opti-mal rate of convergence in L2-norm error. ME-Tri-AW is moreaccurate and displays a superconvergent rate in the energy errornorm.

Fig. 10(a) and (b) depict two stress components of ME-Tri-AWresults in the near-incompressible case where the numerical re-sults agree with analytical solution very well. The stresses shownin Fig. 10(a) and (b) are plotted at the sampling points along thecross-section x = 4.875 on a 40 � 8 uniform mesh.

Superior performance of ME-Tri-AW is presented in deforma-tion plot (Fig. 11) and convergence of the L2-norm and energy-norm errors (Fig. 12) for near-incompressible case. Similar to thecompressible case, a superconvergent rate in the energy errornorm is observed using ME-Tri-AW as shown in Fig. 12(b) whenthe material is close to the near-incompressible limit.

5.2. Inflation problem in nearly incompressible case

The solution for an inflation problem subjected to an inner pres-sure P is given as [48]:

rr ¼Pr2

1

r22 � r2

1

1� r22

r2

� ; rh ¼

Pr21

r22 � r2

1

1þ r22

r2

� : ð87Þ

The exact pressure solution of this problem is a constant pressurefield. Only the upper quadrant of the problem is modeled as shownin Fig. 13(a) with dimension of inner radius r1 = 1.0 and an outer ra-dius r2 = 2.0. A coarse and relatively uniform discretization of ME-Tri as shown in Fig. 13(b) is used for the mesh refinement in theconvergence study. Only the nearly incompressible case is studiedin this example. The pressure error norm is defined by


ehp ¼ p� ph

�� 0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZX

p� phð Þ2dX

s: ð88Þ

A convergence study is next presented using four levels of meshrefinements. Fig. 14(a) and (b) show the energy error and pressureerror norms respectively computed from the NICE-T3 and ME-Tri-AW solutions. Both methods achieve superconvergent rate inenergy error and pressure error norms, and ME-Tri-AW obtainsbetter accuracy than NICE-T3. Fig. 15 shows the deformation plotfor both semi-uniform and random discretization. The ME-Tri-AW results are in very good agreement with analytical solution.

In order to show that the energy error norm is bounded usingME-Tri-AW as Poisson ratio approaches to 0.5, we monitor the en-ergy error norm as Poisson ratio v changes from 0.3 to

Fig. 15. Deformation plot using ME-Tri-AW in nearly-incompressible case: initial (dash lthe references to color in this figure legend, the reader is referred to the web version of

Table 1The convergence of ME-Tri-AW energy error norm in a fixed mesh.

m = 0.3 0.499

Energy error norms 0.6809066430 1.184904133

Fig. 16. Cook’s mem

0.499999999 with a 16 � 16 fixed mesh. The results are presentedin Table 1. In Table 1, the energy error norm converged to a finitevalue as Poisson ratio is approaching to the incompressible limit.The results indicate that the energy error norm computed fromEq. (85) will remain finite and converged as k ?1.

5.3. Cook’s membrane problem in nearly incompressible case

Cook’s membrane model is a classical benchmark problem usedto examine whether the numerical method is likely to provide alocking-free and non-oscillating solution in nearly incompressiblematerial under combined shear and bending deformation. Thegeometry and boundary condition are shown in Fig. 16(a). The leftside boundary is fully constrained, and the right side boundary is

ines), analytical (thick red lines), numerical (thick blue lines). (For interpretation ofthis article.)

0.49999 0.4999999 0.49999999

1.192336610 1.192409294 1.192411627

brane problem.

Fig. 18. Pressure distribution of Cook’s membrane problem in nearly inco

Fig. 17. Tip deflection results for Cook’s membrane in nearly incompressible case.


subjected to a uniform distributed vertical traction. A typical dis-cretization for ME-Tri-AW with enriched nodes is plotted inFig. 16(b).

Fig. 17 compares the variation of the vertical displacement ofthe upper right corner as mesh is refined and all methods generatelocking-free and convergent solutions. NICE-T3 with nodal integra-tion provides soft response as compare to results from Q1-SR andME-Tri-AW. ME-Tri-AW is clearly more accurate than the othertwo methods.

Fig. 18(a) and (b) show the comparison of pressure distributionin three discretizations: 8 � 8, 16 � 16 semi-uniform meshes, andhighly non-uniform mesh. In Q1-SR, the element pressure ispresented at the element center. The pressure of NICE-T3 andME-Tri-AW is reported by averaging the pressure from elementintegration points and sampling points respectively. As we cansee from Fig. 18, Q1-SR produce pressure oscillation between

mpressible case: (a) 8 � 8 mesh, (b) 16 � 16 mesh, (c) random mesh.

Fig. 19. Geometry, boundary condition and a typical discretization in punch problem.

Fig. 20. Pressure distribution of punch problem in nearly incompressible case.

Table 2Convergence of forces on the flat rigid punch in 2D.

Elements Q1-SR NICE-T3 ME-Tri-AW

Q1 Tri

8 � 2 32 2146.5615 5402.6086 1679.860716 � 4 128 1637.0067 1449.3848 1473.610624 � 6 288 1521.8001 1318.3118 1423.430932 � 8 1152 1472.0337 1298.5309 1401.8714

Fig. 21. Numerical inf-sup test result of ME-Tri-AW for the 2D punch problem.


elements and the pressure oscillation is not improved in a refinedmesh or random mesh. Compared to Q1-SR, although NICE-T3 ob-tains less oscillated pressure results, the pressure distribution isstill not a smooth field. By contrast, the ME-Tri-AW displays asmooth pressure distribution and is free of pressure oscillation.

5.4. Punch problem

The pressure oscillation is also studied in the 2D Prandtl’spunch model. The model consists of a block of nearly incompress-ible material punched by a rigid, frictionless and flat plate with aprescribed displacement. The model geometry and the discretiza-tion of ME-Tri-AW are shown in Fig. 19. Two uniform meshes16 � 4 and 32 � 8 are utilized to investigate the pressure oscilla-tion in different mesh size.

The resulting element-wise pressure distribution is plotted onthe deformed domain and shown in Fig. 20. Although the Q1-SRdoes not exhibit volumetric locking behavior, visible checkerboardpatterns are observed in the pressure field. On the other hand, theNICE-T3 using nodal integration not only shows a strong oscillationin pressure field but also produces an unstable result in thedeformation. The mesh refinement in Q1-SR and NICE-T3 methodsdoes not improve the oscillation in the pressure field. A smoothedpressure distribution of ME-Tri-AW shown in Fig. 20 clearly dem-onstrates its superior performance over the other two methods.Table 2 reports the convergence of the punch force in threemethods as mesh is refined. Consistent convergence results are ob-tained in Q1-SR and ME-Tri-AW. Although NICE-T3 also presents aconvergent result, it suffers from volumetric locking and behavesover stiffly in the case of coarse mesh. A soft force response

Table 3Convergence of forces on the flat rigid punch in 3D.

Elements Q1-SR ME-Tri-AW

Q1 Tet

4 � 4 � 2 192 6548.8263 9318.34376 � 6 � 3 648 5614.1421 6925.04018 � 8 � 4 1536 5232.3181 6071.067910 � 10 � 5 3000 5037.9441 5642.3964

Fig. 23. Numerical inf-sup test result of ME-Tri-AW for the 3D punch problem.

Fig. 22. 3D punch problem.

Fig. 24. Driven cavity problem: (a) geometry and boundary condit


predicted by NICE-T3 in the case of using a finer mesh confirms theunstable deformation mode observed in Fig. 20.

The numerical inf-sup test is conducted with continuous meshrefinement for the stability study. The computed inf-sup parame-ters [5] using ME-Tri-AW are plotted in Fig. 21, where N is numberof finite element nodal intervals along the thickness. As we can see,ME-Tri-AW passes the numerical inf-sup test.

We also analyze a similar 3D punch problem using ME-Tri-AWand Q1-SR (8-noded Q1 tri-linear element with selected reducedintegration), where a quarter model is setup as in Fig. 22(a).Fig. 22(b) plots the deformed configuration using ME-Tri-AW with10 � 10 � 5 mesh. Since NICE elements behave poorly in this 3Dexample, we do not consider it for the comparison.

Table 3 reports the convergence of the punch force of Q1-SR andME-Tri-AW in 3D case. Similar to the results in Table 2, both meth-ods obtain consistent convergence results. Fig. 23 shows ME-Tri-AW passes the numerical inf-sup test in 3D case.

5.5. Driven cavity problem in nearly incompressible case

In this last example, we consider the problem of a unit squaresubjected to a unit horizontal displacement along the upperboundary as shown in Fig. 24(a). This boundary condition resultsin corner singularities for the solution for which the exact solutionis not known. The aim here is to demonstrate numerically thesmooth pressure solution can be achieved by the ME-Tri method.Both uniform and non-uniform discretizations in a 16 � 16 meshare considered and shown in Fig. 24(b) and (c).

Pressure distributions of the problem obtained by three meth-ods Q1-SR, NICE-T3 and ME-Tri-AW are plotted on the deformedconfiguration and shown in Fig. 25. NICE-T3 fails to deliver reason-

ion, (b) uniform discretization, (c) non-uniform discretization.

Fig. 25. Pressure distribution of driven cavity problem.

Fig. 26. Pressure distribution at x = 0.5.

Fig. 27. Numerical inf-sup test result of ME-Tri-AW for the cavity problem.


able solutions as the computed pressure is several orders of mag-nitude higher than the other two methods. Spurious pressuremodes are obtained using Q1-SR as shown in Fig. 25(a) when uni-form mesh is adopted. The checkerboard patterns are less pro-nounced but still visible in a non-uniform mesh as displayed inFig. 25(b).

In Fig. 26, the pressure distributions at x = 0.5 obtained by Q1-SR does not even reproduce the sinusoidal-like pressure character-istic of the problem. In contrast, ME-Tri-AW generates much betterpressure distributions than Q1-SR despite the use of different dis-cretizations. A smooth sinusoidal-like pressure distribution atx = 0.5 can be restored using ME-Tri-AW as shown in Fig. 26.Fig. 27 shows ME-Tri-AW passes the numerical inf-sup test in thisproblem.

6. Conclusion

We have presented a displacement-based meshfree-enriched fi-nite element method (ME-FEM) using simplex elements for thetwo and three-dimensional linear analyses of compressible andnearly-incompressible elasticity. The key step of this developmentlies in the extra freedom in the approximation of functions whichallows the enriched and smoothed strains to achieve the discretedivergence-free properties in the nearly incompressible regime.This is done by adding an enriched meshfree node into the stan-dard linear finite element using a first-order convex meshfree


approximation and performing an area-weighted strain smoothingusing a simple integration rule. The shape function of enrichedmeshfree node can be considered as a compatible bubble function.Different from the coefficients of the classical finite element bubblefunctions, the enriched mesfree nodes directly carry physical quan-tities. Since meshfree-enriched finite element approximation pre-serves the Kronecker-delta property at the element boundary, theresultant element formulation is conforming and the impositionof essential boundary conditions is straightforward. The smoothedgradient matrix obtained from the area-weighted strain smoothingusing meshfree-enriched finite element approximation is provento satisfy the integration constraint (IC). This property guaranteesthat the proposed method will pass the patch test. Another favor-able property of this method is no control parameter is involved inthe strain smoothing. Through the eigenvalue analysis, we haveverified the discrete divergence-free conditions in the incompress-ible limit and have shown that the proposed method does not con-tain non-physical locking mode and spurious zero energy mode.The proposed method is robust since it appears to produce uni-formly converged and checkerboard-free solutions as demon-strated in several numerical examples.

The extension of this method to the quadrilateral elements in-volves different meshfree enrichment in the finite element approx-imation and thus the strain smoothing. This will be discussed in aseparate paper [56]. While the results generated from the proposedmethod in the linear context is promising, much theoretical worksremain to be done in the error analysis and non-linear formulation.

Acknowledgements

The authors thank Dr. John O. Hallquist of LSTC for his supportto this research. The supports from Yokohama Rubber Co., Ltd., Ja-pan and NEC Corporation, Japan are also gratefully acknowledged.

Appendix A. Generalized meshfree (GMF) approximation

In this study, a first-order generalized meshfree (GMF) convexapproximation is introduced to approximate the displacementfield for meshfree-enriched finite element method. The implemen-tation of meshfree convex approximation implies a weak Kroneck-er-delta property at the boundaries and therefore the essentialboundary condition can be easily treated. The fundamental ideaof the GMF approximation [54,55] is the introduction of an en-riched basis function in the Shepard function [43] and its satisfac-tion of linear consistency. The choice of the enriched basis functiondetermines whether the GMF approximation has convexityproperty. Giving a convex hull conv (K) of a node setK ¼ fxi; i ¼ 1; . . . ng � R2 defined by [55]:

convðKÞ¼Xn

i¼1

aixijxi 2K;ai 2R;ai P 0;Xn

i¼1

ai¼1; i¼1;2; . . .

( );

ðA:1Þ

the GME method is to construct a convex approximation of a given(smooth) function u in the form:

uhðxÞ ¼Xn

i¼1

WiðxÞui; ðA:2Þ

with the generating function Wi: conv(K) ? R satisfying the follow-ing polynomial reproduction property:Xn

i¼1

WiðxÞxi ¼ x 8 x 2 convðKÞ: ðA:3Þ

The proof of the weak Kronecker-delta property at the boundariesin convex approximation can be found in [55].

The first-order GMF approximation in multi-dimension is ex-pressed as:

Wi x; krð Þ ¼ wi

w¼ /a x; Xið ÞCi Xi; krð ÞB Xi; kr ;Hð ÞPn

j¼1/a x; Xj� �

Cj Xj; kr� �

B Xj; kr ;H� � for fixed x;

ðA:4Þ

subjected to Rrðx; krÞ ¼Xn

i¼1

WiXi ¼ 0 ðlinearity constraintsÞ;

ðA:5Þ

where wi ¼ /aðx; XiÞCiðXi; krÞBðXi; kr ;HÞ; ðA:6Þ

w ¼Xn

i¼1

wi ¼Xn

i¼1

/aðx; XiÞCiðXi; krÞBðXi; kr ;HÞ; ðA:7Þ

Xi ¼ x� xi; ðA:8Þ

/a(x;Xi): the weight function of node i with support sizesupp(/a(x;Xi)) = ai.Ci(Xi,kr): 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.kr(x)(r = 1,2, . . . ,m): constraint parameters which have to bedecided.m: the number of constraints (m = 1 in 1D, m = 2 in 2D, andm = 3 in 3D).B(Xi,kc,H): the blending function and H is internal functions orvariables.

In the GMF approximation, the property of the partition of unityis automatically satisfied by the normalization in Eq. (A.4). Theblending function B(Xi,kc,H) is introduced to modify the basisfunction for the generation of a particular GMF approximation withspecial purposes such as satisfying the weak Kronecker-delta prop-erty for the non-convex approximation [55]. For constructing afirst-order convex approximation, the blending function B(Xi,kc,H)is unity. The GMF approximation is completed by finding kc to sat-isfy Eq. (A.5), a constraint equation which guarantees the GMFapproximation has linear consistency. To determine kc at any fixedx in Eq. (A.4), a root-finding algorithm is required for the non-lin-ear basis functions.

The convexity of the GMF approximation is determined by theselection of a positive basis function in Eq. (A.4). In addition, theGMF approximation can be readily extended to the higher-order,which can be used to improve the accuracy of meshfree methods[55]. In this paper, a convex GMF approximation constructed usingan inverse tangent basis function is employed in the displacementapproximation for the ME-FEM method and is denoted by GMF(atan). A cubic spline window function with a circular support ischosen to be the weight function in Eq. (A.6). The correspondingderivatives of the convex GMF approximation can also be foundin [55].

Appendix B. Proof of discrete divergence-free properties in ME-FEM triangular elements

In the following appendix, we will provide the direct proof ofdiscrete divergence-free properties for the first locking-free modeshown in Fig. 5. Let M = T1 [ T2 be a patch that is a unit squareand consists of two ME-FEM triangular elements as shown inFig. B.1. The locking-free mode of the patch considered in the proofis plotted in shaded area with the finite element nodal displace-ments provided by

u1x2 ¼ u2

x3 ¼ �u1x1 ¼ �u2

x1 ¼ �u2x2 ¼ �u1

x3 ¼ ci; ðB:1Þ

Fig. B.1. A patch of ME-FEM triangular elements and one of its locking-free modesin eigenvalue analysis.

Fig. B.2. Reference mapping in ME-FEM elements.


where 0 < ci < 1 is a coefficient describing the deformation of thelocking-free mode in the unit square M.

For the simplicity of the proof, let M ¼ T1 [ T2 be the referencepatch as shown in Fig. B.2. A mapping eF eðn;gÞ : M ! M is definedto map the reference patch onto the global patch such thatTi ¼ eF eðTiÞ; i ¼ 1;2. In other words, for element T1; eF eðn;gÞ ¼Feðn;gÞ, where Fe(n,g) is defined in Eq. (16). For element T2, themapping eF eðn;gÞ is just a rotated and translated Fe(n,g). UsingEq. (28), it is readily to observe that the following relationshipholds for elements T1 and T2:

J1n ðnÞ ¼ J2

nðnÞ ¼ JnðnÞ; for all n 2 M; ðB:2Þ

where the superscript 1 and 2 denote the element T1 and elementT2 respectively. One can also obtain the following symmetry prop-erties in shape function derivatives:

W1;n n11ð Þ ¼ W2;n n22ð Þ ¼ �W2;n n11ð Þ ¼ �W3;n n22ð Þ;W1;n n12ð Þ ¼ W3;n n21ð Þ ¼ �W2;n n13ð Þ ¼ �W2;n n23ð Þ;W2;n n12ð Þ ¼ W2;n n21ð Þ ¼ �W1;n n13ð Þ ¼ �W3;n n23ð Þ;W3;n n12ð Þ ¼ W1;n n21ð Þ ¼ �W3;n n13ð Þ ¼ �W1;n n23ð Þ;W4;n n11ð Þ ¼ W4;n n21ð Þ ¼ �W4;n n13ð Þ ¼ �W4;n n23ð Þ;W3;n n11ð Þ ¼ W4;n n11ð Þ ¼ W1;n n22ð Þ ¼ W4;n n22ð Þ:

ðB:3Þ

Note that the symbol njm in Eq. (B.3) denotes the mth samplingpoint in element Tj. In addition, strain smoothing is applied to thesmoothing domain that contains the common element edge asshown in the shaded area in Fig. B.2. Note that the element systemdescribed above is different from the one in macroelement [46] inwhich the piecewise linear function is adopted in the mapping andpressure is assumed to contain the piecewise constants in themacroelement.

The divergence $ � uh in Fig. B.1 evaluating at integration pointscan be expressed by

$ � uh njit� �

¼ tr $uh nji

� �� ¼ tr J�1

n � $ n;gð Þuh nji

� �� ; ðB:4Þ

where $ðn;gÞuhðnjiÞ is calculated by

$ n;gð Þuh nji

� �¼

oW1 njið Þon � � � oW4 njið Þ

on

oW1 njið Þog � � � oW4 njið Þ

og

264375

ux1 uy1

..

. ...

ux4 uy4

26643775¼GF nji

� �ue; i¼1;2;3:

ðB:5Þ

GF is defined as in Eq. (20), and ue is a collection of nodal displace-ment for the element Te given by

uTe ¼

ux1 � � � ux4

uy1 � � � uy4

: ðB:6Þ

Accordingly, the divergence-free condition of element T1 evaluatingat sampling points n11 and n12 gives:

J�111 W4;n n11ð Þþ J�1

12 W4;g n11ð Þ J�121 W4;n n11ð Þþ J�1

22 W4;g n11ð ÞJ�1

11 W4;n n12ð Þþ J�112 W4;g n12ð Þ J�1

21 W4;n n12ð Þþ J�122 W4;g n12ð Þ

" #u1

x4

u1y4

( )

¼�

J�111

P3i¼1

Wi;n n11ð Þu1xi

� þ J�1

12

P3i¼1

Wi;g n11ð Þu1xi

� þ J�1

21

P3i¼1

Wi;n n11ð Þu1yi

� þJ�1

22

P3i¼1

Wi;g n11ð Þu1yi

� J�1

11

P3i¼1

Wi;n n12ð Þu1xi

� þ J�1

12

P3i¼1

Wi;g n12ð Þu1xi

� þ J�1

21

P3i¼1

Wi;n n12ð Þu1yi

� þJ�1

22

P3i¼1

Wi;g n12ð Þu1yi

�

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;:

ðB:7Þ

Using Eq. (B.1), one can solve the above linear equation and obtainthe nodal displacements of the enriched node u1

4 ¼ u1x4;u

1y4

h i. Simi-

larly, one can solve for u24 ¼ u2

x4;u2y4

h iin element T2. Using the prop-

erties in Eq. (B.3), one can further find that:

u1x4 ¼ u2

x4 and u1y4 ¼ u2

y4: ðB:8Þ

Analogously, the smoothed divergence tr(Phe) evaluated at thesampling point n13 = n21 can be expressed by

tr Phe uh n13ð Þ� ��

¼ tr ~eh� �¼ tr

1Am

ZXm

$uhdX�

¼ 1Am

ZXm

tr $uh� �

dX

¼tr J�1

n �$ n;gð Þu1e

��n13

� det J1ð Þþ tr J�1

n �$ n;gð Þu2e

��n21

� det J2ð Þ

det J1ð Þþdet J2ð Þ: ðB:9Þ

Substituting Eqs. (B.1) and (B.8) into Eq. (B.9) and using the proper-ties in Eqs. (B.2) and (B.3) leads to:

tr Phe uh n13ð Þ� ��

¼ 0: ðB:10Þ

Eq. (B.10) implies the discrete divergence-free condition is holdingfor the smoothed strains as well.

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meshfree-enriched simplex elements with strain smoothing for the finite element analysis of...

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