stabilized conforming nodal integration in the natural element method jeong.pdf

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2004; 60:861–890 (DOI: 10.1002/nme.972)

Stabilized conforming nodal integration in thenatural-element method

Jeong Wahn Yoo1,‡, Brian Moran2,∗,† and Jiun-Shyan Chen3,§

1Department of Civil Engineering, Northwestern University, 2145 Sheridan Rd., Evanston, IL 60208, U.S.A.2Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Rd.,

Evanston, IL 60208, U.S.A.3Department of Civil Engineering, UCLA, Los Angeles, CA 90095, U.S.A.

SUMMARY

A stabilized conforming nodal integration scheme is implemented in the natural neighbour methodin conjunction with non-Sibsonian interpolation. In this approach, both the shape functions and theintegration scheme are defined through use of first-order Voronoi diagrams. The method illustratesimproved performance and significant advantages over previous natural neighbour formulations. Themethod also shows substantial promise for problems with large deformations and for the computationof higher-order gradients. Copyright � 2004 John Wiley & Sons, Ltd.

KEY WORDS: natural-element method; nodal integration; non-Sibsonian interpolant; material incom-pressibility; higher-order gradients; large deformation

1. INTRODUCTION

Meshless methods have been developed in recent years to circumvent restrictions imposed byformal element connectivity requirements in traditional finite element methods. A broad class ofmethods show tremendous promise for problems involving large deformations as well as movinginterfaces or discontinuities, including the element-free Galerkin method [1], the reproducingkernel particle method [2], the natural-element method [3], h-p clouds [4], the partition of unityfinite element method [5], and the meshless local Petrov–Galerkin method [6], to mention justa few. An overview of developments in meshfree methods was given by Belytschko et al. [7]and Li and Liu [8].

∗Correspondence to: B. Moran, Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Rd.,Evanston, IL 60208, U.S.A.

†E-mail: [email protected]‡E-mail: [email protected]§E-mail: [email protected]

Contract/grant sponsor: National Science Foundation; contract/grant number: CMS-9732319Contract/grant sponsor: Office of Naval Research; contract/grant number: N00014-01-1-0953

Received 23 January 2002Revised 10 May 2002

Copyright � 2004 John Wiley & Sons, Ltd. Accepted 16 July 2003

862 J. W. YOO, B. MORAN AND J.-S. CHEN

In this paper, we present an approach for the solution of partial differential equations, withapplications to small and large strain elasticity, which belongs to the family of natural neighbourmethods. Sibson appears to have been the first to introduce the notion of natural neighbours andnatural neighbour interpolation [9]. Braun and Sambridge adopted the concept in the Galerkinmethod for partial differential equations and coined the name natural-element method (NEM)[3]. Sukumar et al. explored the application of NEM to solid mechanics [10] and showed thatthe method could be extended to problems requiring C1 continuity through use of Bézier splines[11]. Bueche et al. investigated the dispersive properties of NEM [12], and Reeves and Moranshowed the utility of NEM for contaminant transport problems [13]. Cueto et al. introduced amodification of NEM based on �-shapes, which rendered the Sibson shape functions conformingon non-convex boundaries [14]. Sukumar et al. showed that the non-Sibsonian interpolant,proposed by Belikov et al. [15], was precisely linear on non-convex boundaries, and theypresented a new implementation of NEM using the non-Sibsonian interpolant [16].

In the implementation of meshless methods, including natural neighbour methods, evaluationof the integrals in the weak form has been an issue of concern. The integration is typicallydone using Gauss quadrature over a background mesh. This integration scheme, however, hasbeen recognized to be far from optimal.

In this paper, we implement the stabilized nodal integration scheme, used by Chen et al.[17], in a natural element method (for which we use the acronym, nodal-NEM). The nodal-NEM is shown to satisfy the patch test to near machine precision. In the nodal-NEM, bothinterpolant and integration scheme are defined through use of the Voronoi diagrams. No specialprocedures are found necessary for the determination of neighbours, the implementation ofessential boundary and interface conditions, and the control of incompressibility constraints.

2. NON-SIBSONIAN INTERPOLANT

The construction of approximate solutions of differential equations by means of the nodal-NEMis based on the so-called non-Sibsonian natural neighbour interpolant [15, 18]. In this section,the construction and basic properties of the non-Sibsonian interpolants for a set of point sitesare illustrated.

2.1. Voronoi diagram

Let P := {p1, p2, . . . , pN } be a set of N distinct points, or sites, in Euclidean k-space, Ek

[9, 19, 20]. The Voronoi diagram, also known as Dirichlet tessellation, of P is defined as thesubdivision of Ek into N regions, one for each site in P . The Voronoi diagram of P is denotedby Vor(P ). The region V(pI ) that corresponds to a site pI is called the Voronoi cell of pI

and is defined by

V(pI ) = {p ∈ Ek: d(p, pI ) < d(p, pJ ) ∀J �= I } (1)

where d is Euclidean distance. Each V(pI ) is an intersection of finitely many open half-spaces,each being delimited by the perpendicular bisector hyperplane of a line segment pIpJ ; thus, itis open, convex, and polyhedral. V(pI ) is bounded if and only if pI lies in the interior of theconvex hull of P .

Copyright � 2004 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 60:861–890

STABILIZED CONFORMING NODAL INTEGRATION 863

1

p 6

p 2

p 3

p 4

p 5

p 7

p 9

p 8

p 10

p 11

p

Figure 1. The Voronoi diagram and Delaunay graph of a set of sites in E2.

2.2. Delaunay graph

Corresponding to the subdivision of Ek into the Voronoi polyhedra, a dual graph is obtainedby joining pairs of points, pI and pJ , whose Voronoi polyhedra have a hyperplane in common.The Delaunay graph DG(P ) of P is the dual graph whose joins are straight line segments andis a simplicial subdivision of Ek . Any Delaunay triangulation of P maximizes the minimumangle over all triangulations of P . Figure 1 shows the Voronoi diagram Vor(P ) and Delaunaygraph DG(P ) in E2.

2.3. Interpolation formula

A field u is interpolated from its values on a set of sites, uI = u(pI ), in Ek according to theformula

uh =n∑

I=1�IuI (2)

where �I are functions of the co-ordinates, and n is the number of neighbours. Define�V(pI , pJ ), which may be an empty set, by

�V(pI , pJ ) = {p ∈ Ek−1: p ∈ V(pI ) ∩ V(pJ ), J �= I } (3)

where V(pI ) is the closure of V(pI ) [15, 16, 18]. Then, the interpolation function �I atp employed in the non-Sibson method is obtained by treating p as a site in the Voronoitessellation and is written as

�I = (|�V(p, pI )|)/d(p, pI )∑J [(|�V(p, pJ )|)/d(p, pJ )] (4)

where | · | denotes the Lebesgue measure in Ek−1. The neighbours used in the interpolationare determined naturally from the construction of Dirichlet cells: these are the sites whose



p2

p3

p8p7

p6

p pd ( , )2

p

p p2( , )

Figure 2. The Voronoi cell and neighbours of a site p added in P .

Dirichlet cells have a common face with the V(p). Figure 2 illustrates a portion of newlycalculated Dirichlet tessellation, from which the interpolation function is obtained, as a site p

is added in P given in Figure 1. The five points, p2, p3, p6, p7, and p8, shown in Figure 2,are the natural neighbours of the site p.

2.4. Properties

The properties of the non-Sibson method are described in References [15, 16, 18]; thus, onlyselect ones are presented here without verification.

The non-Sibson interpolation satisfies the partition of unity; that is, a domain � is coveredby overlapping subdomains �I associated with �I that are non-zero only in �I and have theproperty, ∑

I

�I = 1 in � (5)

[5, 7]. In addition, the equality, ∑I

�I (x − xI ) = 0 in � (6)

also holds, where xI are the position vectors of pI . Equations (5) and (6) imply that theinterpolant spans the space of linear polynomials in Ek , which is a necessary condition ofconvergence for partial differential equations of order 2.

From (4), the non-Sibson interpolant has the property,

�I (pJ ) = �IJ (7)

Furthermore, it is precisely linear on the boundary of the domain. This makes for straightforwardimposition of essential or Dirichlet boundary conditions as in finite elements. Note also that this,in conjunction with regional interpolation, leads to satisfaction of material interface conditionsas in finite elements.



3. NATURAL NEIGHBOUR FORMULATION

The Galerkin method is used to produce nodal natural neighbour formulations. In this section,the procedure for linear elastostatics is outlined, briefly.

3.1. Linear elastostatics

Consider a domain �, bounded by �. The equilibrium equations are given by

∇ · � + b = 0 in � (8)

where � = �T is the Cauchy stress and b is body force per unit volume. When displacementgradients are everywhere small compared to unity, the classical elastic constitutive equationsare

� = C : � (9)

where � = ∇su is the small strain tensor. For an isotropic material, the elastic moduli C takethe form

Cijrs = ��ij�rs + �(�ir�js + �is�jr ) (10)

in which � and � are the Lamé constants. On the boundary � = �t ∪ �u,

� · n= t on �t (11)

u= u on �u (12)

where n is the unit outward normal, and t and u are prescribed tractions and displacementson �t and �u, respectively.

3.2. Galerkin method

The discrete equations are generated from the weak form (or principle of virtual work),∫�

∇s�u : C : ∇su d� =∫

��u · b d� +

∫�t

�u · t d� (13)

where u ∈ H 1 are trial functions, and �u ∈ H 10 are test functions. Here H 1(�) denotes the

Sobolev space of functions with square integrable derivatives in �, and H 10 (�) is the subspace

of H 1 with vanishing values on �u. We begin by establishing trial and test families of functionsin the form of Equation (2). By substituting the approximations, uh and �uh, into the weakform and invoking the arbitrariness of virtual nodal displacements, Equation (13) yields thestandard formula

Kd = f (14)

where the stiffness matrix K and external nodal force f are given by

KIJ =∫

�hBT

I CBJ d� (15)



in which B is the conventional strain-displacement matrix, and

fI =∫

�h�Ib d� +

∫�h

t

�I t d� (16)

Here �h is a finite-dimensional subspace of �, which is spanned by the shape functions:

�h = spanI

{�I } (17)

in which h refers to a measure of distance between the sites.

4. STABILIZED CONFORMING NODAL INTEGRATION

The integrals of terms in the discrete equations can be evaluated numerically, rather thananalytically. A conventional way to do this in meshfree methods is application of Gauss rulesto a background mesh for spatial integration. The integration, however, might be inaccuratesince the integrands are not polynomials and the supports of the integrands do not coincidewith the integration cells in general [21]. In this paper, a nodal integration scheme based onthe Dirichlet tessellation is used. This simplifies the data structure and leads to significantimprovements in accuracy over Gaussian quadrature on background cells.

4.1. Control of instabilities

Nodal integration typically exhibits spatial instability resulting from underintegration of theweak form [22]. In one stabilization procedure (originally developed for strain localizationproblems), the strain measure is replaced by a non-local approximation [17, 23]. The non-localform of strain is represented as a weighted average given by

�(s) =∫

�W(x; x − s)�(x) d� (18)

where W is a weight function that possesses the properties,

W � 0 and∫

�W d� = 1 (19)

and has compact support.

4.2. Integration formula

Suppose �L ⊂ � associated with the site pL to be

�L = {p ∈ Ek: p ∈ � ∩ V(pL)} (20)

and W to be a step function defined by

W(x; x − xL) ={

A−1L if x ∈ �L

0 if x /∈ �L

(21)



where AL is the Lebesgue measure of �L. Incorporating Equation (21) into Equation (18)gives

�ij (xL) = 1

AL

∫�L

(∇su)ij d� = 1

AL

∫�L

1

2(uinj + ujni) d� (22)

where �L are the boundaries of �L, and ni are the components of the unit outward normal.The divergence theorem has been used in the last of Equation (22). The stiffness matrix isobtained by substitution of uh and �uh into Equation (22) and the result into Equation (13).Then, Equation (15) becomes

KIJ =∫

�hBT

I CBJ d� = ∑L

BTI (xL)CBJ (xL)AL (23)

and∑

L �L = �h and the integrands are constant over each �L. Here the non-local strain-displacement matrix B in E2 is given by

BI (xL) = 1

AL

∫�L

�I n1 0

0 �I n2

�I n2 �I n1

d� (24)

Note that the nodal-NEM can be regarded as a class of assumed-strain methods [24] withnatural neighbour interpolants, resulting in subdomain collocation over �L.

4.3. Contour integrals

Equation (23) involves hyperplane contour integrals over �L. The integrals are approximatedby successive application of Gauss rules to each face of �L. Figure 3(a) shows the contourintegration over six faces of subdomain �8, and Figure 3(b) illustrates the integration overfive faces of subdomain �4; i.e. V(p4) cut by the domain boundary �. Note that Chenet al. [17] used a trapezoidal rule on the facets of the contour. As will be shown below, thisapproach can lead to difficulties when used in conjunction with non-Sibsonian interpolation.

5. REMARKS ON THE INTEGRATION SCHEME

5.1. Convergence

The assumed field for u has C0 continuity and contains a complete polynomial of degree 1 asdiscussed in Section 2. Furthermore, the preceding stabilized integration procedure, regardlessof the quadrature rule used for the contour integrals, satisfies the constraint,∫

�hBT

I d� = ∑L

BTI (xL)AL =

∫�h

t

�In d� (25)

which comes from the weak form representing a state of constant first derivatives of uh.See, e.g. Chen et al. [17] for a discussion of the integration constraint. The nodal-NEM,consequently, guarantees convergence to correct results for second-order PDEs.



Vor( )

p 6

p 9

p 8

p 3

p 2

p 7

p

P

11

4

p6

p5

p1

p3

Vor( )P

p

Γ

(a)

(b)

Figure 3. Contour integration over the �L (i.e. �8 and �4) of site pL:(a) V(pL) ⊂ �; and (b) V(pL) �⊂ �.

5.2. Caution for choice of quadrature

Instabilities arising from shortcomings in the nodal integration procedure are controlled byusing the smoothed strain. However, application of the trapezoidal rule to the contour integralsmight produce zero strain energy owing to the intrinsic properties of non-Sibson interpolation.Consider a square domain � ⊂ E2 with a set of five nodes, whose distribution and Voronoitessellation in � are shown in Figure 4. Because the interpolation functions �I vanish on thedomain boundary � for all interior sites pI /∈ � [16], the shape function �1 is zero at allsampling points of the trapezoidal rule which are indicated in Figure 4(a); and thus, B1 = 0at all nodes. Therefore, according to Equation (23), the strain energy in �

U� = 12 d

TKd (26)



2

p3

p5

p4

p1

p2

p3

p5

p4

p1

p

(a) (b)

Figure 4. Mesh of five sites, showing: (a) sampling points of the trapezoidal rule and its possiblemechanism; and (b) Gauss point locations for the case of one-point quadrature on facets of �L.

will vanish for modes dT = (dT1 0 0 0 0). The foregoing instabilities can appear in various

meshes; consequently, the trapezoidal rule should be employed in the nodal-NEM with caution.No zero-energy mode, on the other hand, is displayed by use of Gauss quadrature for the

contour integration. Figure 4(b) shows Gauss point locations for first-order quadrature on eachfacet of the contour, �L.

5.3. Material incompressibility

As Poisson’s ratio � approaches 0.5, penalty constraints arise naturally, which can easily leadto locking difficulties. The locking difficulties, however, are avoided for all practical purposesin the nodal-NEM by virtue of the integration scheme. To illustrate, write the stiffness matrixK in the form

KIJ = ∑L

BTI (xL)CGBJ (xL)AL + ∑

L

BTI (xL)CK BJ (xL)AL (27)

Here CG and CK are given by

CGijrs = G

(�ir�js + �is�jr − 2

3 �ij�rs

)and CK

ijrs = K�ij�rs (28)

where G is the shear modulus and K is the bulk modulus. Since K approaches infinityas � approaches 0.5, the incompressibility constraint is enforced at each node by the secondsummation in Equation (27). Thus optimal constraint ratios, r = 2 for two-dimensional problemsand r = 3 for three-dimensional problems, are usually achieved in the nodal-NEM. Here theconstraint ratio r is defined as the ratio of the number of active d.o.f. in a domain to thenumber of penalty constraints. Accordingly, no modification of the method is required to handleproblems with near incompressible behaviour as will be seen below.



5.4. Higher-order gradients

Higher-order gradients may be readily obtained in the nodal-NEM by recursive application ofthe non-local del operator ∇,

∇u(s) =∫

�W(x; x − s)∇u(x) d� (29)

Suppose �L and W to be defined by Equations (20) and (21). We first calculate nodal dis-placement gradients

J uijL = (∇uh

i (xL))j = 1

AL

∫�L

�I nj d�uiI (30)

We then obtain nodal displacement gradients of order 2,

(∇(J uij )

h(xL))k = 1

AL

∫�L

�I nk d�J uijI (31)

provided that Ju can be interpolated from nodal values. This process can be repeated until thegradients of desired order are obtained, though accuracy in the computed quantities is expectedto be decrease progressively with every step.

A class of theory, called strain-gradient plasticity, has gained popularity recently, in whichhigher-order gradient contributions are included. The common practice in finite element imple-mentation is to use C1 interpolation functions for the higher-order gradients. The continuityrequirements on the shape functions, however, might be relaxed in the present method, affordedby the above-mentioned procedure.

6. DISCRETIZATION FOR NON-LINEAR ANALYSIS

In this section, discrete momentum equations are developed for large deformation problems.We consider a Lagrangian formulation and emphasize the required kinematic expressions in thefollowing discussion.

6.1. Deformation and strain

Among several possible measures of finite strain, the deformation-gradient and Green straintensors are considered. In the following, we shall designate the reference configuration by asubscript 0. We begin by employing the same interpolations for the co-ordinates and displace-ments:

xh = ∑I

�I (X)xI (t) and uh = ∑I

�I (X)uI (t) (32)

where �I are given by Equation (4). The nodal deformation gradients are calculated using thenon-local vector operator ∇ and the position vector xh [25]:

FL = ∇0xh(XL, t) = ∑I

∇0�I (XL)xI (t) (33)



where, from Equation (29),

∇0�I (XL) =∫

�0

W(X;X − XL)∇�I (X) d�0 (34)

Using Equations (20) and (21), we have

FijL = (∇0�I (XL))j xiI = 1

A0L

∫�0L

�I nj d�0xiI (35)

From here onward, the discretization requires only straightforward manipulation. With FL asdefined in (33), the nodal Green strain tensors can be evaluated from the usual expression forthe strain tensor in terms of the deformation gradient:

EL = 12 (FT

L · FL − I) (36)

6.2. Equilibrium of forces

In the Lagrangian approach, we express the equilibrium of forces using the principle of virtualdisplacements, which is∫

�0

�u · �0b d�0 +∫

�0t

�u · t0 d�0 −∫

�0

�FT : P d�0 = 0 (37)

where �0 is the mass per unit reference volume, b is the body force per unit mass, and P isthe nominal stress. Substituting the approximations, uh and �uh, into the weak form as we didin linear analysis, we obtain

fext − f int = 0 (38)

in which the external and internal nodal forces are given by

fextI =

∫�h

0

�I�0b d�0 +∫

�h0t

�I t0 d�0 (39)

f intI =

∫�h

0

PTBI d�0 = ∑L

∫�h

0L

PTBI d�0 (40)

where BI = ∇�I . Employing nodal integration, ∇ replaces ∇ and Equation (40) becomes

f intI = ∑

L

PTLBILA0L (41)

where PL are the nodal nominal stresses, the evaluation of which depends on the materialmodel used, and BIL = ∇0�I (XL).

7. LINEARIZATION OF INTERNAL NODAL FORCES

In an incremental solution procedure, the equilibrium equation (38) is linearized about theconverged state at the end of the previous increment and an iterative procedure is used to



obtain convergence for the current increment. In the following, the linearization procedurefor the internal nodal forces is illustrated within the context of the nodal natural neighbourformulation.

7.1. Tangent stiffness

We begin by taking the time derivative of Equation (41) in which PL are the only time-dependent variables:

f intI = ∑

L

˙PT

LBILA0L = ∑L

(FL˙SL + ˙FLSL)BILA0L (42)

Note that we write PL = SL · FTL where SL are the nodal second Piola–Kirchhoff stresses.

Considering a hyperelastic material, in which stress is calculated from a strain energy function, we have

˙SL = �2(EL)

�EL�EL

: ˙EL = CSEL : ˙EL (43)

By substitution of Equation (43) into Equation (42), the rate of internal nodal forces has theform

f intiI = (Kmat

iIjJ + KgeoiIjJ )ujJ (44)

Here the material and geometric tangent stiffnesses are

KmatiIjJ = ∑

L

B∗LkliIC

SELklpqB∗L

pqjJ AL0 (45)

KgeoiIjJ = ∑

L

BLkI S

LklB

LlJ �ijA

L0 (46)

in which B∗L are defined by

B∗LijkI = sym

(i,j)

(BLiI F

Lkj ) (47)

8. EXAMPLE PROBLEMS

Benchmark tests on a variety of problems in elasticity are presented. A Young’s modulus of3×107 psi and a Poisson’s ratio of 0.25 are used unless stated otherwise. A second-order Gaussrule is applied to the contour integration around the nodal domain boundary. Three-point Gaussquadrature for triangles is employed for the NEM and constant-strain FEM when results arecompared with those of the nodal-NEM.

8.1. The patch test

The patch test was originally designed to determine the convergence behaviour of non-confor-ming elements [26]. In the context of meshless methods, the test represents a check on the



(a) (b)

(c)

Figure 5. Patches for a unit square domain: (a) 25 nodes; (b) 8 nodes; and (c) 70 nodes.

validity of an approximation and the integration of the weak form. Though a meshless methodthat fails the test may still provide convergence, the patch test serves as a sufficient conditionfor correct convergence of a meshless formulation provided that the method to be tested isstable.

Procedure: Figure 5 shows three patches for a unit square domain � ⊂ E2 in plane stressconditions, and comprised of 25, 8, and 70 nodes. A linear displacement field is imposed onboundary nodes of the patches; i.e. ui = xi . Internal nodes are neither loaded nor restrained.The L2(�) norm, defined by

‖u − uh‖L2(�) =(∫

�(ui − uh

i )(ui − uhi ) d�

)1/2

(48)

is used to examine the computed results.Results: Table I illustrates relative errors in the L2(�) norm of the results, where data for the

NEM are obtained from Reference [10] and the nodal-NEM results are presented for first- and



Table I. Relative errors in the L2(�) norm of the patch tests.

Patch NEM Nodal-NEM: order 1 Nodal-NEM: order 2

Figure 5(a) 7.5 × 10−4 2.7 × 10−16 2.4 × 10−16

Figure 5(b) 9.3 × 10−3 4.8 × 10−16 8.6 × 10−16

Figure 5(c) 4.4 × 10−3 1.2 × 10−15 1.2 × 10−15

x 1

x 2

P

D

L

Figure 6. Cantilever beam of unit thickness under transverse end load P .

second-order Gauss quadrature of the contour integral. The nodal-NEM results show significantimprovements in accuracy over the NEM with the former achieving near machine precision.Accordingly, the foregoing results suggest that the effects of inexact contour integration arequite minimal, unlike those of inexact background mesh integration.

8.2. Cantilever beam I

The cantilever beam problem shown in Figure 6 is solved using four different meshes andthree different methods for each mesh. Here, we seek the relation between the number of d.o.f.in its discretized form and solution error.

Closed-form solution: The exact solution of the problem is given by the theory of elasticityas [27]:

u1 = −Px2

6EI

[(6L − 3x1)x1 + (2 + �)x2

2 − 3D2

2(1 + �)

](49)

u2 = P

6EI[(3L − x1)x

21 + 3�(L − x1)x

22 ] (50)

where I is the moment of inertia of the beam cross section. The corresponding stresses are

11 = −P(L − x1)x2

I(51)

22 = 0 (52)

12 = P

2I

(D2

4− x2

2

)(53)



Figure 7. Mesh of 85 nodes for a cantilever beam.

h

Re

lativ

e er

ror

0.02 0.04 0.06 0.08 0.1

10-3

10-2

10-1

FEMNEMnodal-NEM

Figure 8. Convergence of uh with mesh refinement.

Procedure: Four uniform meshes of 85, 297, 1105, and 1701 nodes are used to examinethe discretization errors in the beam models. A mesh of 85 nodes is shown in Figure 7.Displacements and tractions calculated from Equations (49)–(53) are prescribed on �u and�t , respectively. Results are presented using plane stress and the parameter values, P =−1000 psi, L = 4 in, and D = 1 in.

Results: Figure 8 illustrates the convergence of uh with refinement, where relative errors inL2(�) norm are plotted against dimensionless length h, defined by

h = 1√N

(54)

Here N is the number of degrees of freedom in the domain. The results suggest that convergenceis quadratic for all three methods: the computed convergence rates of the FEM, NEM, andnodal-NEM are 2.02, 2.18, and 2.06, respectively.



x2 (in.)

No

rmal

str

ess

(psi

)

-0.5 -0.25 0 0.25 0.5-15000

-7500

0

7500

15000

Exactnodal-NEM

Figure 9. Exact and computed normal stresses in a cantilever beam.

x2 (in.)

Sh

ear

stre

ss (

psi

)

-0.5 -0.25 0 0.25 0.5-1750

-1250

-750

-250

250

Exactnodal-NEM

Figure 10. Exact and computed shear stresses in a cantilever beam.

Figures 9 and 10 show the comparison between exact and computed stresses along x1 = 2 in,where approximate stresses are computed by the nodal-NEM using a mesh of 1105 nodes. Thesolution for the nodal-NEM is in good agreement with the exact solution.



θ

σ0σ a0

r

Figure 11. Infinite plate with a hole under uniform tensile load 0.

8.3. Infinite plate with a circular hole

The classical problem of an infinite plate with a circular hole shown in Figure 11 is presentedto illustrate that ill-conditioning from material incompressibility is avoided in the nodal-NEM.

Closed-form solution: The exact solution of the problem is given by Szabó and Babuška[28]:

11 = 0

[1 − a2

r2

(3

2cos 2� + cos 4�

)+ 3

2

a4

r4cos 4�

](55)

22 = 0

[−a2

r2

(1

2cos 2� − cos 4�

)− 3

2

a4

r4cos 4�

](56)

12 = 0

[−a2

r2

(1

2sin 2� + sin 4�

)+ 3

2

a4

r4sin 4�

](57)

The displacement components corresponding to the stresses, without rigid body motionterms, are

u1 = 0a

8�

[r

a(� + 1) cos � + 2

a

r((1 + �) cos � + cos 3�) − 2

a3

r3cos 3�

](58)

u2 = 0a

8�

[r

a(� − 3) sin � + 2

a

r((1 − �) sin � + sin 3�) − 2

a3

r3sin 3�

](59)

where � is defined in terms of Poisson’s ratio by

� =

3 − 4� in plane strain

3 − �

1 + �in plane stress

(60)



Figure 12. Mesh of 1345 nodes for plate with a hole.

Procedure: The unbounded plate is truncated and modelled as a square with a central holeof diameter equal to one fifth of the square width. Symmetry is exploited by analysing onlyone quadrant of the plate. The mesh of 1345 nodes used is shown in Figure 12. Displacementboundary conditions for symmetric loading are prescribed on �u, the symmetry boundaries,and tractions corresponding to the stresses from Equations (55) to (57) are imposed on �t .Plane strain conditions are enforced. Numerical results are presented using parameter values,0 = 1 psi and a = 1 in.

Results: Figure 13 illustrates the behaviour of the error in the problem as Poisson’s ratioapproaches 0.5. What is plotted as the relative error is the relative L2(�) norm from Equa-tion (48). As expected, no evidence of mesh locking is observed in the nodal-NEM whereassignificant locking is evident for both FEM and NEM as the incompressibility limit of � = 0.5is approached.

Figures 14 and 15 show the comparison between exact and computed stresses along � = /2.The computed stresses are obtained by the NEM and nodal-NEM using � = 0.25. The nodal-NEM solutions for both 11 and 22 are in good agreement with the analytical solutions. Fornear incompressible material, numerical solutions are compared with the exact solutions inFigures 16 and 17, suggesting that the nodal-NEM yields acceptable results despite the nu-merical difficulties caused by round-off as � approaches 0.5. Note here that the constant-straintriangle elements are used for the FEM and they are known to lock. Also, in the NEMformulation, no modifications to avoid locking were implemented.



Poisson’s ratio

Rel

ativ

e er

ror

-0.02

0.02

0.06

0.1

0.14

FEMNEMnodal-NEM

0.25 0.49 0.4999999 0.499999999999

Figure 13. Errors in uh for various Poisson’s ratios.

r (in.)

Str

ess

(psi

)

1 2 3 4 50.8

1.4

2

2.6

3.2

ExactNEMnodal-NEM

Figure 14. Exact and computed stresses 11 of plate with a hole: � = 0.25.

8.4. Infinite plate with an inclusion

In the following example we illustrate the application of the nodal-NEM to a problem witha material discontinuity. The problem of an inclusion with constant eigenstrain in an infiniteplate shown in Figure 18 is considered.



r (in.)

Str

ess

(psi

)

1 2 3 4 50

0.15

0.3

0.45

ExactNEMnodal-NEM


r (in.)

Str

ess

(psi

)

1 2 3 4 50.8

1.4

2

2.6

3.2

ExactNEMnodal-NEM




r (in.)

Str

ess

(psi

)

1 2 3 4 50.8

1.4

2

2.6

3.2

ExactNEMnodal-NEM


-phase

=

=

tβ

uβ

tα

uα

β-phase

α

u at infinity0=

R

θ

rat r = R

Figure 18. Inclusion with uniform eigenstrain �∗� in an infinite plate.



Figure 19. Mesh of 625 nodes for plate with an inclusion.

Closed-form solution: The exact solution is obtained for u = 0 at r = ∞ and given byCordes and Moran [29]:

ur =

C1r if r �R

C1R2

rif r �R

(61)

u� = 0 (62)

where

C1 = (�� + ��)�∗�� + �� + ��

(63)

Here �∗� is a constant dilatational strain in the � phase.Procedure: The infinite plate is represented as a circle with diameter sufficiently large com-

pared to that of inclusion. Symmetry is exploited by analysing one quadrant of the model.The mesh used on quadrant is shown in Figure 19 and is comprised of 625 nodes: 92 in theinclusion, 13 in the interface, and 520 in the matrix. Displacement boundary conditions forsymmetric loading are imposed on �u, the symmetry boundaries, while �t are traction-free.Plane strain conditions are assumed. Results are obtained using parameter values, �� = 497.16,�� = 390.63, and �� = 338.35, corresponding to E� = 1000, �� = 0.28, E� = 900, and �� =0.33. The values �∗� = 0.01, R = 5, and plate radius of 200 are also used in the calculations.



r

Rad

ial d

isp

lace

men

t

0 25 50 75 1000

0.01

0.02

0.03

0.04

Exactnodal-NEM

Figure 20. Exact and computed displacements ur of plate with an inclusion.

r

Rad

ial s

trai

n

0 25 50 75 100-0.01

-0.005

0

0.005

0.01

Exactnodal-NEM

Figure 21. Exact and computed strains �rr of plate with an inclusion.



r

Ho

op s

trai

n

0 25 50 75 100-0.001

0.002

0.005

0.008

Exactnodal-NEM

Figure 22. Exact and computed strains �� of plate with an inclusion.

r

Rad

ial s

tres

s

0 25 50 75 100-6

-5

-4

-3

-2

-1

0

1

Exactnodal-NEM

Figure 23. Exact and computed stresses rr of plate with an inclusion.



r

Ho

op

str

ess

0 25 50 75 100-6

-3

0

3

6

Exactnodal-NEM

Figure 24. Exact and computed stresses �� of plate with an inclusion.

Results: The nodal-NEM and exact solutions are compared in Figures 20–24. The slightdiscrepancy at large radii observed in Figure 20 is due to modelling error, caused by thedifference between an infinite medium and its finite model. Apart from that, the nodal-NEMsolution is in good agreement with the exact solution, and no oscillations are observed in thecomputed results.

8.5. Cantilever beam II

The cantilever beam problem outlined in Section 8.2 is considered again using a uniform meshof 1701 nodes. The method for higher-order gradient evaluation, suggested in Section 5, isused to calculate second derivatives of displacement.

Results: Displacement gradients of order 2 are evaluated and compared with exact solutionsin Figures 25–27, along x2 = 0 in. The nodal-NEM and analytical solutions are seen to be ingood agreement.

Remarks: For the assumed-gradient fields defined in this method, the gradients appear to beoptimal (but not necessarily most accurate) sampled at the centers of �L. The gradients obtainedat pL of which V(pL) �⊂ �, thus, are often inaccurate because those nodes are unlikely thepoints of optimal accuracy for gradients. It is immediately evident that further inaccuracy ofthe same kind would be developed for higher-order gradient evaluation, which explains thediscrepancy of the nodal-NEM solution from the exact solution shown in Figure 26. Note thatsimilar problems exist and have been widely pursued in finite element analysis. Indeed a viablerecovery procedure for gradients at nodes would be worth investigating for the nodal-NEM,such as the recovery by equilibration of patches (REP) that has been devised in the finiteelement approximation.



x1 (in.)

u1,

12

0 1 2 3 4-0.0002

0.0002

0.0006

0.001

0.0014

0.0018

Exactnodal-NEM

Figure 25. Exact and computed u1,12 of a cantilever beam.

x1 (in.)

u2,

11

0 1 2 3 4-0.0018

-0.0014

-0.001

-0.0006

-0.0002

0.0002

Exactnodal-NEM




x1 (in.)

u2,

22

0 1 2 3 4-0.0005

-0.0003

-0.0001

0.0001

Exactnodal-NEM


8.6. Large deformation of a cantilever beam

The cantilever beam problem shown in Figure 6 is solved in order to test the nodal-NEM inlarge deformation problems.

Beam theory: From the solution to the problem of the elastica, expression for u2e, the verticaldisplacement of the end of the beam, is given in terms of an elliptic integral [30]:

u2e

L= 1 −

√4EI

PL2

∫ 2

ϑ

√1 − k2 sin2 t dt (64)

in which

k =√

1 + sin �e

2and ϑ = arcsin

1

k√

2(65)

Here Equation (64) can be solved by using corresponding values of P and �e, the angle ofrotation of the end of the beam, that are determined from√

PL2

EI=

∫ 2

ϑ

dt√1 − k2 sin2 t

(66)

Procedure: A beam of L = 20 in and D = 1 in is modelled by a regular mesh of 31 × 7nodes. The material is assumed to be an isotropic Kirchhoff material with E = 3 × 107 psi and



PL2 / EI

u2e

/L

0 2.5 5 7.5 100

0.3

0.6

0.9

ElasticaFEMnodal-NEM

Figure 28. Large deflections of a cantilever beam.

� = 0, defined by

SL = C : EL (67)

Here C is a constant elasticity tensor and takes an identical form to that given in Equation (10).The solution of the system of non-linear equations is obtained by the Newton–Raphson method.

Results: The numerically predicted response is compared with the response calculated usingbeam theory and FEM in Figure 28. Here the four-node quadrilateral elements are used forthe finite element solution. The results suggest that the kinematic descriptions of nodal-NEMadmit the accurate representation of the geometric non-linear behaviour.

9. CONCLUSION

We present a class of natural neighbour methods which shows significant advantages overthe FEM and previous NEM implementations. In particular, patch tests are satisfied to nearmachine precision, and near incompressible problems are solved without any modification ofthe integration scheme.

In the nodal-NEM, stresses and strains are defined and computed at nodes, which would besuitable for problems involving large deformations with adaptive refinement and path-dependentvariables. In addition, higher-order gradients are produced accurately by repeated applicationof the same non-local gradient operator on which the nodal integration scheme is based. The



method, consequently, shows substantial promise for problems in large deformation plasticitywith or without higher-order gradients.

ACKNOWLEDGEMENTS

The authors are grateful for the support of the National Science Foundation through Grant No.CMS-9732319 and the Office of Naval Research through Contract N00014-01-1-0953 to NorthwesternUniversity. Helpful discussions with Ted Belytschko and Natarajan Sukumar are gratefully acknowl-edged.

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