INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng (2012)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4427

The natural radial element method

J. Belinha1,*,, L. M. J. S. Dinis2 and R. M. Natal Jorge2

1IDMEC, Institute of Mechanical Engineering, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal2Faculty of Engineering of the University of Porto, FEUP, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

SUMMARY

In this work an innovative numerical approach is proposed, which combines the simplicity of low-order finiteelements connectivity with the geometric flexibility of meshless methods. The natural neighbour concept isapplied to enforce the nodal connectivity. Resorting to the Delaunay triangulation a background integrationmesh is constructed, completely dependent on the nodal mesh. The nodal connectivity is imposed throughnodal sets with reduce size, reducing significantly the test function construction cost. The interpolationsfunctions, constructed using Euclidian norms, are easily obtained. To prove the good behaviour of theproposed interpolation function several data-fitting examples and first-order partial differential equationsare solved. The proposed numerical method is also extended to the elastostatic analysis, where classic solidmechanics benchmark examples are solved. Copyright 2012 John Wiley & Sons, Ltd.

Received 18 February 2012; Revised 18 July 2012; Accepted 11 September 2012

KEY WORDS: meshless method; finite element method; natural neighbour; elastostatic analysis

1. INTRODUCTION

The finite element methods [1] have been successfully applied to solve a variety of problems inengineering. However, the FEMs dependence on the mesh leads to some limitations. Because it isused in FEM, a mesh-based interpolation, the element distortion and/or the solid domain complexgeometries can lead to high errors and loss of accuracy. Therefore, in recent years several meshlessmethods [25] were developed with the objective of eliminating part of the FEM shortcomings. Inmeshless methods the field functions are approximated within a flexible influence domain ratheran element; therefore, the nodes discretizing the problem domain can be randomly distributed. Inmeshless methods the influence domain, which defines the field function applicability space, mayand must overlap each other, in opposition to the no-overlap rule between elements in the FEM.Meshless methods present various advantages over the FEM [6] and some can be listed:

(i) Present a higher accuracy for structures with complex geometries and use high-ordercontinuous weight functions in a compact support to construct the test functions.

(ii) Handle more easily large deformation problems and moving discontinuities (crackpropagation).

(iii) Permit to add more nodes where refinement is required to achieve a more accurate solution.Meshless methods that use the weak form solution can be divided in two categories: the ones that

use approximation functions and the others that use interpolation functions. One of the first devel-oped approximant meshless methods is the smooth particle hydrodynamics (SPH) [7], initially usedto modulate astrophysical phenomena, the SPH is based on kernel estimation [8]. Initially proposed

*Correspondence to: J. Belinha, IDMEC, Institute of Mechanical Engineering, Rua Dr. Roberto Frias, 4200-465 Porto,Portugal.

E-mail: jorge.belinha@fe.up.pt

Copyright 2012 John Wiley & Sons, Ltd.

J. BELINHA, L. M. J. S. DINIS AND R. M. NATAL JORGE

for surface fitting [9], the diffuse element method [10] was the first to use the moving least-squaresapproximants in the construction of the approximation function. With the inclusion of the omittedapproximation function derivatives and the use of a more accurate integration procedure, the elementfree Galerkin method was developed [11]. At the same time, the reproducing kernel particle method[12] was developed through the introduction of a correction function for the kernel approximationon the SPH. In parallel, the meshless local PetrovGalerkin method [13] was presented.

To solve the lack of the delta Kronecker property of approximation meshless methods, severalnew meshless methods using interpolation functions were developed in the last few years. One ofthe firsts efficient interpolation meshless method was the natural element method [14, 15], whichuses the Sibson interpolation functions and the Vorono diagram to impose the nodal connectivity.Another interpolator meshless method, the Point Interpolation Method [16] ensures a polynomialinterpolation with Kronecker delta function property, based only on a group of arbitrarily distributedpoints. Later, a radial basis function was added to the basis of the interpolation functions allow-ing the development of the Radial Point Interpolation Method (RPIM) [17]. More recently, usingthe advantages of the natural neighbours on the imposition of nodal connectivity and the radialpoint interpolator technique, the natural neighbour radial point interpolation method (NNRPIM)was developed [18, 19].

In this work a new meshless method is proposed, the Natural Radial Element Method (NREM).The main advantage of this numerical approach is the combination between the geometric flexibilityof a meshless method and the connectivity simplicity of a low-order finite element. The nodalconnectivity on the NREM is obtained using the Vorono diagram [20]. First the Vorono cells areestablished and then, using the Delaunay triangulation [21], the integration mesh is constructed.With the NREM the integration mesh is completely dependent on the nodal mesh. Using theDelaunay triangulation, small size influence-domains are determined, each one with only n D d C1nodes, being d the problem domain dimension, Rd . The NREM interpolation functions,used in the Galerkin weak form, are constructed using a Euclidean norm basis function (ENBF).The interpolation function construction process is very similar with the radial point interpolators[17, 18]; however, the ENBF does not require any shape parameter as the radial basis functionsused in [17, 18]. In this work the NREM is presented in Section 2. The nodal connectivity andthe integration mesh determination are extensively described, as well as the NREM interpolationfunction construction. Also, the most relevant interpolation function properties are demonstrated.In Section 3 the proposed interpolation functions aptitude for data-fitting problems is presented andthe partial differential equations are solved. In Section 4 linear elastostatic benchmark examples aresolved considering the two-dimensional analysis and the three-dimensional analysis. The work endswith the conclusions and remarks in Section 5.

2. NATURAL RADIAL ELEMENT METHOD

In this section the enforcement of the nodal connectivity used in the proposed meshless methodis explained and the construction of the background integration mesh is presented. Then, adetailed description of the construction and the properties of the proposed interpolation functionsare presented.

2.1. Natural element construction

First, the natural neighbour concept was introduced by Sibson [22] for data fitting and fieldsmoothing. Consider the nodal set N D n1, n2, : : : , nN discretizing the space domain Rd inX D x1, x2, : : : , xN 2 . The Vorono diagram of N is the partition of the function space dis-cretized by X in subregions Vi , closed and convex. Each subregion Vi is associated with the nodeni in a way that any point in the interior of Vi is closer to ni than any other node nj 2 N ^ j i .The set of Vorono cells V defined the Vorono diagram, V D V1, V2, : : : , VN . The Voronocell is defined by Vi WD

xI 2 Rd W kxI xik >>>>>>=>>>>>>>>>;

D

8>>>>>>>>=>>>>>>>>>;

. (6)

The constant basis reduces the size of the equation system to nC 1 linear independent equations,because it only conserves the constant part of the linear polynomial basis. The absence of polynomialbasis simplifies the size of the equation system to n, G a D f . Notice that G matrix is symmetric,because the L2-norm is directional independent and also Gij D 0, 8i D j . Solving the Equation (4),

a

b

DG PT

P 0

1 f

0

D M1

f

0

. (7)

Substituting in Equation (1), f .xI / D g.xi xI / p.xI /a bT, the solution found inEquation (7), it is possible to obtain,

f .xI / Dg.xi xI / p.xI /

M1

f

0

. (8)

Thus, the interpolation function '.xI / can be obtained with,

.xI / Dg.xi xI / p.xI /

M1, (9)

with .xI / D '1.xI / '2.xI / '3.xI / '.nCm/.xI /. To obtain the interpolation function'.xI / only the n first components of .xI / are relevant; the others do not possess applicable

Copyright 2012 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2012)DOI: 10.1002/nme

J. BELINHA, L. M. J. S. DINIS AND R. M. NATAL JORGE

(a) (b) (c) (d)Figure 3. Components of the constructed interpolation function for an interest point xI obtained (a) with n1,

(b) with n2 and (c) with n3. (d) Node ni complete interpolation function.

physical meaning. Therefore, '.xI / D '1.xI / '2.xI / 'n.xI /. In Figures 3(a), (b) and (c),for the two-dimensional space, it is possible to observe for an interest point xI , the components ofthe interpolation function '.xI / D '1.xI / '2.xI / '3.xI / using the nodal set

n1 n2 n3

.

The partial derivative of the interpolation function to a variable is easily obtained,

',.xI / D

g

,.xi xI / p,.xI /M1. (10)

The partial derivatives of the Euclidean norm square given naturally by the L1-norm,

g,.xi xI / D 2.i I /. (11)

The obtained interpolation function is a C 2-function. If, in Equation (2), the exponent p 2Rn2ZC 1 W p > 0 then the interpolation function is in fact a C1-function,

g,.xi xI / D p.i I /g.xi xI /p2. (12)

However, as presented in Section 4.3, results have shown that only the p D 2 leads to stable andaccurate results.

2.5. Interpolation functions propertiesThe obtained interpolation function, Equation (9), possesses several interesting properties,

(i) Local compact support(ii) Simple derivatives

(iii) Delta Kronecker property(iv) Partition of unity(v) Reproducibly

The first mentioned property is the local compact support. The interpolation of each interest pointis determined within a localized influence-domain. Notice that the size of the influence-domainin this numerical method is extremely reduced; the interpolation function of each interest point isobtained using only n D d C 1 nodes, d being the dimension of the function space T in which isdiscretized. This property permits to obtain a sparse and banded system matrix.

As was shown in Equations (10) and (11) the interpolation function partial derivatives are simpleto obtain and to compute.

The most important property displayed by the obtained interpolation function is certainly the deltaKronecker property, which can be easily demonstrated. For simplicity sake consider g W R2 7! Rand the absence of a polynomial base. Because of the chosen dimension of the function space Tin which is discretized, only n D d C 1 D 3 nodes contribute to the interpolation function. Thegeometric matrix G is then defined using Equation (5),

G D24g11 g12 g13g21 g22 g23

g31 g32 g33

35D

24 0 g12 g13g12 0 g23

g13 g23 0

35 (13)

Copyright 2012 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2012)DOI: 10.1002/nme

THE NATURAL RADIAL ELEMENT METHOD

with gij being defined with Equation (2). The geometric matrix G is always invertible, as long asgij 0 W i j , that is, there are no coincident nodes inside the influence-domain.

G1 D 12g12g13g23

264

g223 g13g23 g12g23g13g23 g213 g12g13g12g23 g12g13 g212

375 . (14)

If an interest point xI is considered coincident with one of the nodes inside the influence-domain,lets say node 1, then with Equation (9) it is possible to obtain,

'.xI / D g.xi xI /G1 D0 g12 g13

G1 D 1 0 0 (15)

proving the delta Kronecker property. In Figures 3(a), (b) and (c) a schematic plot of all possibleinterpolation function components obtained in a function space T 2R2 is presented. In Figure 3(d)it is shown all the possible interpolation functions obtained for node ni , considering all theinfluence-domains containing node ni .

To determine if the geometric matrix G is well-conditioned, the condition number of G must bedetermined. The condition number is obtained with Cond.G / D G G1, with the matrixnorm defined by GD max

16j6n

PniD1 jGij j. The nodal spatial disposition minimizing Cond.G / is

an equidistant nodal distribution, for which, considering again g W R2 7! R and Equation (13), thegeometric matrix G would be defined with g12 D g13 D g23. However, in the proposed meshlessmethod, nodes can be arbitrarily distributed, therefore Cond.G / is maximized when two nodes areextremely close to each other when compared with a third node. In this case the geometric matrixG would be defined with g13 D g23 >> g12. Therefore,

GD max* jg12j C jg13j

jg12j C jg23jjg13j C jg23j

+D 2jg13j (16)

G1D 12g12g13g23

max

* g223 C jg13g23j C jg12g23jjg13g23j C

g213 C jg12g13jjg12g23j C jg12g13j C

g212 +

D 1g12

C 12g13

(17)

the geometric matrix G condition number is defined as, Cond.G / D 1 C 2g13=g12. Notice thatCond.G / is proportional to the relation between the Euclidean norms determined for the nodes ofthe influence-domain, Equation (2). The lowest condition number is achieved when all nodes areequidistant, Cond.G / D 3, which indicates a well-conditioned matrix. The previous demonstrationis valid for 8T 2Rd .

Regarding the two last mentioned interpolation function properties, the obtained interpolationfunction possess the partition of unity property,

PniD1 'i .xI / D 1, and the reproducing property,Pn

iD1 'i .xI /xi D xI . These last two properties are facilitated by the delta Kronecker property andare always true if at least the linear polynomial basis is included in the formulation [27].

3. DATA FITTING AND SOLVING PARTIAL DIFFERENTIAL EQUATION

In this section the proposed interpolation functions aptitude for solving data-fitting problems isstudied and partial differential equations are solved. Concerning the use of a polynomial basis, onlythe constant basis is used, because the absence of basis leads to worse results and the use of a linearpolynomial basis does not add accuracy to the result. It was verified that the results obtained with thelinear polynomial basis match exactly the results obtained with the use of the constant basis. As wasexplained in the previous section, it is preferable to use the constant basis because the interpolationequation system becomes significantly smaller.

Copyright 2012 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2012)DOI: 10.1002/nme

J. BELINHA, L. M. J. S. DINIS AND R. M. NATAL JORGE

3.1. Data-fittingThe ability of the proposed interpolation functions within the curve, surface and volume fittingproblems is evaluated through functions fk W Rd 7! R, with d D

1 2 3

. Three functions

are studied: a cubic polynomial function, f1.x/, a sinusoidal function, f2.x/, and an exponentialfunction, f3.x/, Equation (18).8