Improved Nearest-Nodes Finite Element Method with Field ...May 08, 2010 · Shape functions are constructed by the local multivariate Lagrange interpolation [6], using a set of nodes

Adv. Theor. Appl. Mech., Vol. 3, 2010, no. 6, 273 - 289

Improved Nearest-Nodes Finite Element Method

with Field-Consistent Shape Functions

Yunhua Luo

Department of Mechanical & Manufacturing EngineeringUniversity of Manitoba, Winnipeg, R3T 5V6, Canada

[email protected]

Abstract

In this paper, the recently developed nearest-nodes finite elementmethod (NN-FEM) was improved by introducing field-consistent shapefunctions. Field-consistent shape functions are constructed in two steps.First, general solutions to the homogeneous Euler-Lagrangian equationsare obtained; then, the undetermined coefficients in the general solutionsare replaced by nodal displacements from a set of nodes that are thenearest to a concerned quadrature point. Numerical results showed thatthe improved nearest-nodes finite element method has a superior con-vergence rate compared to the original NN-FEM and the conventionalfinite element method.

Keywords: Nearest-nodes finite element method, Euler-Lagrange equa-tions, Field-consistent shape functions

1 Introduction

In the finite element method, element performance is measured by convergencerate and affected by shape functions. It is well known that quadratic elementshave a superior convergence rate than linear elements. Raising the order ofshape functions is one way to improve element performance and it has beenwidely adopted in practice. Nevertheless, it is not the only way, and evennot the most efficient way, to improve element performance. Element perfor-mance is affected by not only the order of shape functions but also their qual-ity. Generally speaking, the quality of shape functions are mainly contributedfrom two factors: the location of involved element nodes and the characterof adopted polynomials. The effect of node location on element performanceis well corroborated by element distortion. Although a regular element and

274 Y. Luo

a distorted element use the same set of shape functions, the performance ofregular elements is superior than distorted ones. When the shape of an ele-ment is distorted, the nodes of that element are also in unfavorable positionsfor constructing high quality shape functions. To obtain high quality shapefunctions, all elements in the finite element mesh must be in good shapes.

In the finite element method, field functions such as displacements are im-plicitly assumed independent in constructing shape functions. But actuallythey are inter-dependent to each other. The inter-dependence of field func-tions represents an intrinsic field consistency. The effect of considering fieldconsistency on element performance has been closely investigated in [1, 2].One effective way for considering field consistency is to construct shape func-tions from the general solutions of homogeneous Euler-Lagrange equations [1].In [1, 2], by investigating straight and curved Timoshenko beam elements, itwas concluded that shear locking and membrane locking are extreme cases ofelement deficiency due to the neglect of field-consistency in shape functions.Elements with field-consistency fully considered in shape functions are not onlyfree of the above numerical deficiencies, but also have an optimal convergencerate.

In the recently developed nearest-nodes finite element method (NN-FEM)[3, 4, 5], finite elements are mainly used for numerical integration. Shapefunctions are constructed by the local multivariate Lagrange interpolation [6],using a set of nodes that are the nearest to the concerned quadrature point.NN-FEM is nearly not affected by element distortion and it provides a moreflexible way for constructing shape functions than the conventional finite el-ement method. However, there are still two issues to be addressed in themethod. First, the local multivariate Lagrange interpolation only works wellfor structured meshes. For unstructured meshes, the quality of derivativesobtained from the local multivariate Lagrange interpolation is poor. Second,field-consistency is not considered. In this paper, NN-FEM is improved byintroducing field-consistent shape functions. The layout of this paper is as fol-lows: the main idea of the nearest-nodes finite element method is described inSection 2; Construction of field-consistent shape functions is introduced in Sec-tion 3; Numerical examples are presented in Section 4, followed by concludingremarks in Section 5.

2 The nearest-nodes finite element method

In the finite element method, it is restricted that only nodes belonging toan element are used for constructing shape functions of the element. Thisrestriction is the origin of common issues in the finite element method. Elementdistortion is a major one of them. It is not necessary to have the aboverestriction. This is well confirmed by the newly emerged meshless or meshfree

Improved nearest-nodes finite element method 275

methods [7, 8, 9, 10, 11, 12, 13], where nodes for constructing shape functionsand cells for numerical integration are not related all all. A meshless methodthus has much greater flexibility in constructing shape functions. Based onthe above deduction, a different strategy is adopted in the nearest-nodes finiteelement method.

Unlike in the finite element method, where shape functions are constructedonce for a whole element by using nodes belonging to that element, in thenearest-nodes finite element method, shape functions are constructed for eachquadrature point, using a set of nodes that are the nearest to the currentlyconcerned quadrature point. Some of those nodes may not belong to theelement where the quadrature point is located. A typical scenario is illustratedin Fig.1 by using a 2-D mesh. Where, element stiffness matrix for the element

��

��

+

Nodes used in constructingshape functions at the integration point

Integration point

Figure 1: Nearest-nodes element method

with a ’+’, which is an integration or quadrature point, is being calculated.For the denoted quadrature point, a number of nearest nodes, marked as solidcircles in the figure, are selected for constructing shape functions. Amongthose nodes, not all of them belong to the shaded element. One node of theshaded element is not included, as it is not near enough to the quadraturepoint. The number of selected nodes is determined by the desired order ofshape functions.

For quadrature points in the same element, different nodes may be usedin constructing their shape functions. Therefore, the procedure of assemblingglobal stiffness matrix is slightly different from the conventional finite elementmethod. In the conventional finite element method, the assembly is actuallydone in two steps. First, contributions from all quadrature points of an elementare put into an element stiffness matrix; and then the element stiffness matrixis assembled into the global stiffness matrix. In the nearest-nodes finite elementmethod, contributions from numerical quadrature points are directly put intothe global stiffness matrix.

276 Y. Luo

3 Construction of Field-Consistent Shape Func-

tions

Field-consistent shape functions are constructed in two steps. First, generalsolutions of the homogeneous Euler-Lagrange equations are obtained; Then,undetermined coefficients in the general solutions are replaced by nodal dis-placements from a set of nearest nodes selected in the way described in Sec-tion 2.

For a linear static plane stress problem, the homogeneous equilibrium equa-tions, i. e. the body forces are not considered, are

⎧⎪⎪⎨⎪⎪⎩

∂σx∂x

+∂τxy∂y

= 0

∂τxy∂x

+∂σy∂y

= 0

(1)

where σx, σy and τxy are the three non-trivial stresses.The stresses are related to strains by

⎧⎨⎩

σx = D1εx +D2εyσy = D2εx +D1εyτxy = D3γxy

(2)

where

D1 =E

1 − ν2, D2 =

νE

1 − ν2, D3 =

E

2(1 + ν)

The linear strains are defined by displacements, u and v, as

εx =∂u

∂x, εy =

∂v

∂y, γxy =

∂u

∂y+∂v

∂x(3)

First, the strain-displacement relations in Eq. (3) are plugged into Eq. (2),and then the stress-strain relations in Eq. (2) are substituted into Eqs. (1).The above operations would yield the following homogeneous Euler-Lagrangeequations ⎧⎪⎪⎨

⎪⎪⎩D1

∂2u

∂x2+D3

∂2u

∂y2+ (D2 +D3)

∂2v

∂x∂y= 0

D1∂2v

∂y2+D3

∂2v

∂x2+ (D2 +D3)

∂2u

∂x∂y= 0

(4)

From the above Euler-Lagrange equations, it can be observed that the two dis-placements, u and v, are inter-dependent on each other, as the two equationsare coupled. In the conventional finite element method, this inter-dependenceand the intrinsic field consistency are completely ignored. The displacements


are interpolated using the same set of shape functions. This may cause numer-ical deficiencies such as shear locking or membrane locking as demonstratedin [1]. To consider field consistency, a set of solutions in polynomial form areassumed using two sets of coefficients, { a0 a1 · · ·a5 } and { b0 b1 · · · b5 },i. e., {

u = a0 + a1 x+ a2 y + a3 x2 + a4 x y + a5 y

2

v = b0 + b1 x+ b2 y + b3 x2 + b4 x y + b5 y

2 (5)

The assumed polynomials should be complete or at least symmetrical. It mustbe pointed out that, as the assumed polynomials are not the really solutions ofthe Euler-Lagrange equations, the field consistency is only partially fulfilled.As the equations in Eq. (4) are second order partial differential equations, linearpolynomials would automatically satisfy them. Higher order polynomials arepossible, but the solution process would be more complex [14]. Substitutingthe assumed polynomials in Eq. (5) into Eq. (4) leads to the following equations{

2D1a3 + 2D3a5 + (D2 +D3)b4 = 02D3b3 + 2D1b5 + (D2 +D3)a4 = 0

(6)

There are six unknown coefficients, a3, a4, a5, b3, b4, and b5, in the above twoequations. It is impossible to determine all of them. However, it is meaningfulto express two of them by the rest to reduce the number of unknown coeffi-cients. To keep the symmetry of the polynomials in Eq. (5), we solve a4 andb4 from Eqs. (6), i. e. {

a4 = −λ3b3 − λ1b5b4 = −λ1a3 − λ3a5

(7)

where the two new non-dimensional material parameters are defined as

λ1 =2D1

D2 +D3, λ3 =

2D3

D2 +D3(8)

With Eq. (7), the polynomials in Eq. (5) are transformed as{u = a0 + a1 x+ a2 y + a3 x

2 − b3λ3x y + a5y2 − b5λ1x y

v = b0 + b1 x+ b2 y − a3λ1x y + b3 x2 − a5λ3x y + b5y

2 (9)

Or put in matrix form, we have

u = p(x, y)aT (10)

where

u = [ u v ]T

p(x, y) =

[1 0 x 0 y 0 x2 −λ3xy y2 −λ1xy0 1 0 x 0 y −λ1xy x2 −λ3xy y2

]

a =[a0 b0 a1 b1 a2 b2 a3 b3 a5 b5

]T(11)

278 Y. Luo

In Eq. (10), there are ten unknown coefficients. They need be expressed bynodal displacements. For a plane problem, each node has two displacements, uand v. Therefore five nodes are needed to replace the unknown coefficients bynodal displacements. Five nodes that are the nearest to a concerned quadra-ture point are selected, and the polynomials in Eq. (10) are enforced at thefive selected nodes, i. e.

Pa = u (12)

where

P =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 x1 0 y1 0 x21 −λ3x1y1 y2

1 −λ1x1y1

0 1 0 x1 0 y1 −λ1x1y1 x21 −λ3x1y1 y2

1

1 0 x2 0 y2 0 x22 −λ3x2y2 y2

2 −λ1x2y2

0 1 0 x2 0 y2 −λ1x2y2 x22 −λ3x2y2 y2

2

· · · · · · · · · · · · · · · · · · · · ·1 0 x5 0 y5 0 x2

5 −λ3x5y5 y25 −λ1x5y5

0 1 0 x5 0 y5 −λ1x5y5 x25 −λ3x5y5 y2

5

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

u = [ u1 v1 u2 v2 u3 v3 u4 v4 u5 v5 ]T

(13)In the above expressions, x1, y1, · · · , x5, y5 and u1, v1, · · · , u5, v5

are, respectively, the coordinates and displacements of the five selected nodes.Assume that the inverse of matrix P exists, then, the unknown coefficients

are expressed by nodal displacements as

a = P−1u (14)

The displacements in Eq. (10) now have the following expression

u = p(x, y)P−1u = N(x, y)u (15)

where N(x, y) contains the shape functions

N(x, y) = p(x, y)P−1 (16)

and N(x, y) is a 2 × 10 matrix

N(x, y) = N =

[ϕu1 ψu1 · · · ϕu5 ψu5ϕv1 ψv1 · · · ϕv5 ψv5

](17)

What is different from the conventional finite element method is that thetwo displacements are now approximated by two inter-dependent polynomials.This can be more clearly seen by expanding Eq. (15),

{u = ϕu1u1 + ψu1v1 + · · · + ϕu5u5 + ψu5v5

v = ϕv1u1 + ψv1v1 + · · · + ϕv5u5 + ψv5v5(18)


The obtained shape functions are called field-consistent shape functions. Asthe ten unknown coefficients in Eq. (9) are solved from exactly ten equations,field-consistent shape functions thus have the Kronecker property, which makesthe application of essential boundary conditions more convenient, i. e.,

ϕui (xj) =

{1 (i = j)0 (i �= j)

, ψui (xj) = 0

ψvi (xj) =

{1 (i = j)0 (i �= j)

, ϕvi (xj) = 0

(19)

The derivatives of the displacements are calculated from Eq. (15),

u,α =∂u

∂α=∂N (x, y)

∂αu = p,α(x, y)P

−1u, (α = x, y) (20)

where p,x and p,y are, respectively, the derivatives of p in Eq. (11) with respectto x and y

p,x =

[0 0 1 0 0 0 2x −λ3y 0 −λ1y0 0 0 1 0 0 −λ1y 2x −λ3y 0

]

p,y =

[0 0 0 0 1 0 0 −λ3x 2y −λ1x0 0 0 0 0 1 −λ1x 0 −λ3x 2y

] (21)

The principle of minimum potential energy or the principle of virtual work isused to establish the finite element equations,

Ku = F (22)

The element stiffness matrix and element nodal vector are, respectively,

Ke =

∫Ωe

BTDBdΩ, P e =

∫Ωe

NTpdΩ +

∫Γe

NTqdΓ (23)

In Eq. (23), D is the material matrix; p and q are, respectively, the bodyforce vector and the surface traction vector. The B-matrix has the fallowingexpression

B =

⎡⎢⎣

· · · ∂ϕui

∂x

∂ψui

∂x· · ·

· · · ∂ϕvi

∂y

∂ψvi

∂y· · ·

· · · (∂ϕu

i

∂y+

∂ϕvi

∂x) (

∂ψui

∂y+

∂ψvi

∂x) · · ·

⎤⎥⎦ (i = 1, 2, · · · , n) (24)

where, n is the number of nodes involved in constructing the shape functions.The application of essential boundary conditions can be done exactly as in theconventional finite element method.

280 Y. Luo

To guarantee that matrix P in Eqs. (12) and (13) is non-singular, in fillingP matrix with items from the nearest nodes, the so-called non-singularitycriterion is applied. The criterion is based on the following theorem.

Theorem 1 (Rank in terms of determinants)[15]An m× n matrix A = [ajk] has rank r (r ≥ 1) if and only if A has an r × rsubmatrix with nonzero determinant, whereas the determinant of every squaresubmatrix with r + 1 or more rows that A has (or does not have!) is zero. Inparticular, if A is square of n× n, it has rank n if and only if det(A) �= 0.

Based on Theorem 1, in constructing matrix P , the determinant of themaximum master sub-matrix should be checked when a new node is filled in.Each node contributes two rows to matrix P . Suppose that node i has justbe filled in, the maximum master submatrix, denoted as M i

P , is a (2i× 2i)square matrix consisting of the first 2i rows and the first 2i columns of matrixP . To guarantee that matrix P is not singular, the following actions will betaken,

if | det(M iP )|

{ ≥ δ, (2i-1)-th and 2i-th rows are kept.< δ, (2i-1)-th and 2i-th rows are discarded.

(25)

where δ is a small positive real number, δ = 10−10 ∼ 10−20.

4 Numerical Investigations

The performance of the improved nearest-nodes finite element method wasinvestigated by numerical examples and compared with the original NN-FEMand the conventional finite element method. First, the properties of field-consistent shape functions were studied. Obviously, they are related to thelocations of the selected nodes. Based on Eq. (17) or (18), five nodes areneeded for constructing the second-order field-consistent shape functions. Tworepresentative scenarios shown in Fig. 2 were thus studied. In scenario (a),the five nodes are distributed regularly and symmetrically; in (b), the centralnode is intentionally dislocated and very close to a vertex node to investigate itseffect on shape functions. As the four vertex nodes are symmetrical, their shapefunctions should be similar to each other. Therefore, only shape functions atone vertex node and at the interior node are displayed, see Figs. 3 and 4.


1

4 3

2

5

1 2

4 3

5

(a) Regular distribution (b) Dislocated distribution

Figure 2: Five nodes for constructing second-order inter-dependent shape func-tions

The Kronecker property of field-consistent shape functions described inEq. (19) is verified by numerical results. By comparing the shape functionsshown in Figs. 3 and 4, it can be seen that shape functions at vertices arenot affected by the dislocated node. On the other hand, the dislocated nodedoes have effect on interior nodal shape functions, the magnitude of the shapefunction values is greatly increased. Please note that in both cases, ψu5 and φv5are actually zero.

−1

0

1

−1

0

1−0.5

0

0.5

1

φu1

−1

0

1

−1

0

1−0.2

−0.1

0

0.1

0.2

ψu1

−1

0

1

−1

0

1−0.2

−0.1

0

0.1

0.2

φv1

−1

0

1

−1

0

1−0.5

0

0.5

1

ψv1

−1

0

1

−1

0

1−1

0

1

2

φu5

−1

0

1

−1

0

1−4

−2

0

2

4

x 10−16

ψu5

−1

0

1

−1

0

1−4

−2

0

2

4

x 10−16

φv5

−1

0

1

−1

0

1−1

0

1

2

φv5

Figure 3: Field-consistent shape functions constructed from regularly dis-tributed nodes

282 Y. Luo

−1

0

1

−1

0

1−0.5

0

0.5

1

φu1

−1

0

1

−1

0

1−0.2

−0.1

0

0.1

0.2

ψu1

−1

0

1

−1

0

1−0.2

−0.1

0

0.1

0.2

φv1

−1

0

1

−1

0

1−0.5

0

0.5

1

ψv1

−1

0

1

−1

0

1−5

0

5

10

φu5

−1

0

1

−1

0

1−10

−5

0

5

x 10−15

ψu5

−1

0

1

−1

0

1−10

−5

0

5

x 10−15

φv5

−1

0

1

−1

0

1−5

0

5

10

φv5

Figure 4: Field-consistent shape functions constructed from severely dislocatednodes

Convergence of the improved nearest-nodes element method was studiedusing the cantilever beam shown in Fig. 5. The beam has the following pa-rameters: length L = 10.0, height 2b = 2.0, thickness t = 1.0, Young’s modulusE = 1000.0, Poisson’s ration ν = 0.0. A shear force of unit magnitude is ap-plied at the right end. The initial and one uniformly refined mesh are displayedin Fig. 6.

2b

L

Figure 5: A cantilever beam under shear force


(a) initial mesh (b) uniformly refined mesh

Figure 6: Initial and refined mesh

0 200 400 600 800 1000 1200 14000.984

0.986

0.988

0.99

0.992

0.994

0.996

0.998

1

1.002

Number of nodes

Nor

mal

ized

dis

pl. a

t loa

ded

end

cent

er

NN−FEM (field−consistent shape functions)NN−FEM (local multivariate Lagrange interpolation)FEM (Quadratic Triangle Elements)

Figure 7: h-Convergence

(a) Deformed configuration (b) Distribution of equivalent stress

Figure 8: Deformed configuration and distribution of equivalent stress

Convergence rate is measured by the variation of vertical displacement at

284 Y. Luo

the cross-section center of the right end of the beam with the total numberof nodes. The results are normalized using theoretical beam solutions. Theobtained results are plotted in Fig. 7. For comparison, results produced bythe original NN-FEM [3] and by ANSYS using element PLANE82 are alsodisplayed. PLANE82 is a quadratic triangle element having six nodes. Adeformed configuration and the distribution of equivalent stress are displayedin Fig. 8. As can be observed from the results, nearest-nodes finite elementmethod with field-consistent shape functions has a superior convergence rate.This confirms that whether or not considering field-consistency does have effecton convergence rate.

The second numerical example is a plate with a small hole, as shown inFig. 9. The hole has a unit radius and a unit thickness. A pair of uniform

σx

y

θr

σ

Figure 9: Plate with a hole

0 200 400 600 800 1000 1200 1400 1600 180043

44

45

46

47

48

49

50

Number of nodes

von

Mis

es s

tres

s at

hol

e ap

ex

NN−FEM (Second−order Polynomials)FEM (Quadratic Triangle Elements)

Figure 10: Variation of equivalent stress with number of nodes


5

10

15

20

25

30

35

40

45

50

(a) Deformed configuration (b) Distribution of equivalent stress

Figure 11: Deformed configuration and stress distribution

loads are applied on the opposite edges of the plate. The dimensions of theplate are much larger than the radius of the hole, so that stress distributionfar from the hole can be considered not affected by the existence of the hole.The material of the plate has a Young’s modulus E = 1000.0 and a Poisson’sratio ν = 0.3. The variation of equivalent stress at the hole apex with the totalnumber of element nodes is displayed in Fig. 10. For the reason mentionedin the introduction, the results produced by the original NN-FEM using aunstructured mesh are poor and thus not displayed in the figure. A deformedconfiguration and a distribution of equivalent stress are plotted in Fig. 11.Once again, it can be observed that the improved nearest-nodes finite elementmethod has a superior convergence rate than the conventional finite elementmethod.

The third example is related to mesh distortion in simulating manufactur-ing process such as pressing and forging. A round bar shown in Fig. 12(a)is to be pressed into a flat thick plate. Lagrangian formulation was adoptedto handle large deformation. The initial workpiece has a cross-section radiusR = 5. For simplicity, the material is assumed hyper-elastic and described byYoung’s modulus E = 1000.0 and Poisson’s ratio ν = 0.35. The initial meshis shown in Fig. 12(b). The pressing process was simulated in a number ofsteps. Several evolved configurations of the mesh during pressing are shown inFig. 13. Due to large material deformation, the quality of the finite elementmesh is gradually deteriorated. Upon Step 8, the mesh is already severelydistorted, see Fig. 13(d). A zoomed in portion of the mesh is shown in Fig. 14.

286 Y. Luo

(a) Workpiece and pressing set-up (b) Initial mesh

Figure 12: Pressing set-up

(a) Step 2 (b) Step 4

(c) Step 6 (d) Step 8

Figure 13: Evolution of finite element mesh during pressing process


Figure 14: Zoom-in on the distorted mesh in Fig. 13(d)

0 0.5 1 1.5 2 2.5 3 3.50

1

2

3

4

5

6

7

8x 10

4

Die vertical displacement

Pre

ssin

g fo

rce

Nearest−Nodes Element MethodFinite Element Method (quadratic triangle)

Figure 15: Pressing force vs die vertical displacement

It can be observed that aspect ratio of most elements is abnormally large.For the conventional isoparametric finite element method, the Jacobian ma-trix is close to be singular. The accuracy of obtained finite element solutionsis greatly reduced. Upon Step 11, for the extreme distortion of some elements,the global stiffness matrix is singular and the simulation process with the con-ventional finite element method could not be continued. While the improvednearest-nodes finite element method was nearly not affected by mesh distor-tion. The obtained pressing force vs die vertical displacement is displayed inFig. 15.

5 Concluding Remarks

The recently developed nearest-nodes finite element method is improved byintroducing field-consistent shape functions. In the method, finite elementsare mainly used for numerical integration. Shape functions are constructedin two steps. First, general solutions of the homogeneous Euler-Lagrangian

288 Y. Luo

equations are obtained; then, undetermined coefficients in the general solu-tions are replaced by nodal displacements from a set of nodes that are nearestto a concerned quadrature point. Numerical results show that the improvednearest-nodes finite element method has a superior convergence rate. Theimproved nearest-nodes finite element method is able to deal with very largedeformation without re-meshing. The only requirement on a finite elementmesh is that elements in the mesh do not overlap or penetrate to each otherto avoid difficulty in numerical integration. The above favorable features at-tribute to the way of selecting nearest nodes for constructing shape functionsand the consideration of field consistency. As field-consistent shape functionshave the Kronecker property, essential boundary conditions can be treated inexactly the same way as in the conventional finite element method.

References

[1] Y. Luo. Explanation and elimination of shear locking and membranelocking with field consistence approach. Computer Methods in AppliedMechanics and Engineering, 162:249–269, 1998.

[2] Y. Luo. On Shear Locking in Finite Elements — a field consistence solu-tion. VDM-Verlag, Germany, 2008.

[3] Y. Luo. A nearest-nodes finite element method with local multivariatelagrange interpolation. Finite Elements in Analysis and Design, 44:797–803, 2008.

[4] Y. Luo. Dealing with extremely large deformation by nearest-nodes femwith algorithm for updating element connectivity. International Journalof Solids and Structures, 45:5074–5087, 2008.

[5] Y. Luo. 3D Nearest-Nodes Finite Element Method for solid continuumanalysis. Advances in Theoretical and Applied Mechanics, 1:131–139,2008.

[6] Y. Luo. A local multivariate lagrange interpolation method for construct-ing shape functions. Communications in Numerical Methods in Engineer-ing, (Published Online: 22 Jul 2008, DOI: 10.1002/cnm.1149), 2008.

[7] R.A. Gingold and J.J. Monaghan. Smoothed particle hydrodynamics:Theory and application to non-spherical stars. Mon. Not. R. Astr. Soc.,181:375–389, 1977.

[8] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl. Mesh-less methods: An overview and recent developments. Comput. MethodsAppl. Mech. Engrg., 139:3–47, 1996.


[9] J.T. Oden, C.A. Duarte, and O.C. Zienkiewicz. A new cloud based hpfinite element method. Comput. Methods Appl. Mech. Engrg., 153:117–126, 1998.

[10] W.K. Liu, S. Jun, and Y.F. Zhang. Reproducing kernel particle methods.Int. J. Numer. Meth. Engng., 20:1081–1106, 1995.

[11] T.J. Liszka, C.A.M. Duarte, and W.W. Tworzydlo. hp-meshless cloudmethod. Comput. Methods Appl. Mech. Engrg., 139:263–288, 1996.

[12] I. Babuska, U. Banerjee, and J. Osborn. Meshless and generalized finiteelement methods: A survey of major results. In M. Griebel and M. A.Schweitzer, editors, Meshfree Methods for Partial Differential Equations.Springer, 2002.

[13] S.R. Idelsohn and E. Onate. To mesh or not to mesh. that is the question... Comput. Methods Appl. Mech. Engrg., 195:4681–4696, 2006.

[14] Y. Luo and A. Eriksson. Extension of field consistence approach intodeveloping plane stress elements. Computer Methods in Applied Mechanicsand Engineering, 173:111–134, 1999.

[15] E. Kreyszig. Advanced Engineering Mathematics. Wiley, 8 edition, 1998.

Received: January, 2010

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Improved Nearest-Nodes Finite Element Method with Field ...May 08, 2010 · Shape functions are constructed by the local multivariate Lagrange interpolation [6], using a set of nodes