IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014 7009904

Computational Performances of Natural Element andFinite Element Methods

Yves Marechal, Brahim Ramdane, and Diego Pereira Botelho

G2Elab, Grenoble Electrical Engineering Laboratory, University of Grenoble Alpes, Grenoble F-38000,France, and CNRS, G2Elab, Grenoble F-38000, France

This paper compares the numerical performance of two numerical methods, the finite element method and the natural elementmethod (NEM). NEM is relatively recent and is based on functions belonging to the Voronoï cell family. Although it has been provedthat this method gives smoother and more accurate solutions than the finite elements, its computational cost is also known to behigher. In this paper, we compare computational efficiency, i.e., accuracy for a given cost, of finite elements and natural elements,for both Laplace and Sibson shape functions. We also bring into the comparison a Voronoï cell-based finite difference scheme whichproves to be very efficient. The error is calculated using dual formulations or analytical solutions.

Index Terms— Computational efficiency, finite difference methods, finite element methods, natural element methods.

I. INTRODUCTION

MESHLESS methods have been widely studied duringthe past 20 years [1]. They proved to bring high

accuracy solutions, but also have inherent difficulties to handleproperly boundary conditions, material jump, and connectionwith finite elements [2]. In the late ’90s, a new method callednatural element method (NEM) appears [3], [4], based on theVoronoï cell surrounding each node. This method overcomesall major drawbacks of previous meshless methods, whilekeeping the smooth and highly accurate solutions observedwith the latter. This method has also been successfully usedin the field of electrical engineering problems [5].

Still the computational cost of NEM is significantly higherthan the usual cost of finite elements. In this paper, wepropose to compare computational efficiency—i.e., accuracyfor a given cost, of finite elements and natural elements—ofboth [finite element method (FEM)] FEM and NEM, for whichwe take into consideration both Laplace and Sibson functions.Moreover, a finite difference implementation of Voronoï cellfunctions, natural finite difference method (NFDM), is alsoincluded in the comparison.

II. VORONOI CELL FUNCTIONS

A. Voronoï Diagram

The NEM uses the concept of Voronoï cell. Let us consider aset of nodes N = {n1, n2, n3, . . . nN } distributed in the wholedomain. The Voronoï diagram is a subdivision of the domaininto cells, where each cell Ti associated with node ni is suchthat any point in Ti is closer to node ni than to any other noden j for i �= j . These cells are the so-called Voronoï cells.

Considering for instance a 2-D space, Ti is the regionof the plane that contains the points x closest to node ni

than to any other node as shown in Fig. 1. The Delaunaytriangulation, which is the dual of the Voronoï diagram, is built

Manuscript received June 26, 2013; revised September 8, 2013; acceptedOctober 3, 2013. Date of current version February 21, 2014. Correspondingauthor: Y. Marechal (e-mail: yves.marechal@grenoble-inp.fr).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2013.2285259

Fig. 1. A Voronoï diagram (in blue) and its associated Delaunay triangulation(in red). Node ni , its Voronoï cell and its six natural neighbors are plotted.

Fig. 2. Sibson NEM shape function computation (n1–n4 are natural neighborsfor x, the filled area is associated with node n2).

by connecting the nodes whose Voronoï cells have commonboundaries; thus, leading to the notion of natural neighbors.

B. Sibson Shape Function

Based on the Voronoï diagram, a natural element shapefunction can be expressed. Among the most used in theliterature are the Sibson and Laplace functions [4].

For the Sibson functions, their determination may be per-formed by analogy with the classical FE shape functions. Forfirst-order triangles, it is well known that the shape functionsare given by a ratio of surfaces. For Sibson shape functions,the same principle is applied to the Voronoï cells as shownin Fig. 2. The shape function for node ni at a point x is thus

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7009904 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014

Fig. 3. Laplace NEM shape function computation (n1–n4 are naturalneighbors for x, the lengths l2 and h2 are associated with node n2).

given by (1) where each Si (x) represents the subarea of theVoronoï cell centered on x linked to the natural neighbor ni ,illustrated by the filled region in Fig. 2

�i (x) = Si (x)∑

jS j (x)

j iterating over the natural neighbors. (1)

It can be noticed that when moving to 3-D, Sibson functionwill involve relatively complex calculations of volumes.

C. Laplace Shape Function

In 2-D, Laplace functions are also based on the Voronoïcell centered on x, but they are expressed by a ratio of length(respectively, surface in 3-D). Fig. 3 shows the various lengthsused in the Laplace shape function expression given in (2)( j iterating over the natural neighbors nodes for x)

�i (x) = Li (x)∑

jS j (x)

with Li (x) = li (x)

hi (x). (2)

The Laplace function has explicit expression, for both shapeitself and its derivatives. This is of course a computationaladvantage when compared with Sibson’s functions. On theother hand, continuity property is slightly deteriorated sincethe Laplace function is C0 on all edges of circumscribedcircles, whereas the Sibson’s are mostly C1 except at nodelocations.

D. Shape Function Properties

The NEM shape function satisfies interpolation, partition ofunity and reproducing kernel properties [4]

�i (xi ) = δi j ,

n∑

i=1

�i (x) = 1 andn∑

i=1

�i (x)xi = x. (3)

Like FEM, the left part of (3) defines the interpolationcapability of NEM and this enables direct and easy impositionof the Dirichlet boundary conditions at the concerned nodesunlike the majority of meshless methods.

Fig. 4(b) depicts the Sibson shape function related to anode ni far from the boundary of the domain. Its supportis given by the union of the circumscribed circles to theDelaunay triangles passing through this node and any otherneighbor nodes of ni such that no other node falls within thecircle as shown in Fig. 4(a).

Fig. 4. (a) Support for NEM shape function. (b) Plot of the Sibson shapefunction.

Fig. 5. Accuracy of NEM, FEM, and NFDM according to the length betweennodes.

E. Finite Difference and Finite Element With VoronoïCell Functions

Both collocation for NFDM and Galerkin schemes for NEMcan be built using Voronoï cell functions. Finite elementapproach requires the integration of the functions, and dueto the rational aspect of these functions and their complexgeometric support, a relatively large number of Gauss pointsis generally used. In contrast, the finite difference approachdescribed in [6] is very efficient in terms computation timeand accuracy even on unstructured grids.

III. EFFICIENCY RESULTS ON A SINGULARITY

FREE TEST CASE

To illustrate the computational performance of each method,we are presenting a first numerical example based on a testproblem consisting in a square domain of dimension 1 on eachside, where the following equations are solved:

�u(x, y) = 2π2 sin(πx) sin(πy) in �

u(x, y) = 0 on ∂�. (4)

The whole domain is initially subdivided into a regularmesh of 32 triangular elements, and is enriched several timesby splitting each triangle into 4 new triangles. The analyticalsolution is given by (5) and allows the calculation of the localerror. This problem does not include any singularity and thus,theoretical convergence rates are reachable

u(x, y) = sin(πx) sin(πy). (5)

Fig. 5 shows the relative error on the variable, in L2 norm,computed on the whole domain as the integral of the local error

MARECHAL et al.: COMPUTATIONAL PERFORMANCES OF NEMs AND FEMs 7009904

Fig. 6. Error versus the number of degrees of freedom (DoF).

Fig. 7. Computational efficiency of all methods.

Fig. 8. Computation time spent for matrix construction.

with respect to the mean size of the element edge length. Onthe whole, NEM accuracy is globally between the first-order(FEM1) and the second-order FEM (FEM2) accuracies. Asexpected, both FEM1 and NEM (Laplace, Sibson of finitedifference) are following a first-order method slope (h2),whereas the FEM2 follows a second-order (h3) slope.

Next, Fig. 6 shows that even though the second-order FEMis obviously more accurate for a given size of element, allNEM methods are in fact performing better than FEM2 upto 200 degrees of freedom (DoF), which is a very interestingresult in favor of this interpolation approach.

Last, Fig. 7 plots the computational efficiency of all meth-ods, defined as the computational cost for a given accuracy.It clearly shows that for first-order approximation, NEM isoutperforming FEM1 of nearly two orders of magnitude and

TABLE I

BAND SIZE

Fig. 9. Condition number of the matrix system for FEM and NEM.

can even be compared with FEM2 efficiency. However, dueto a higher convergence rate, superiority of FEM 2 increaseswhen the number of DoF is getting higher.

To have a better understanding of where the computationtime is spent, Fig. 8 plots the time required to build thematrix system (integration and assembly). As expected, dueto the large amount of geometric calculation linked to theVoronoï cells, Laplace, and Sibson are the most expensivemethods. Finite difference Voronoï cell (NFDM) is, to thatconsideration, much more efficient since no integration and nogradient are required. It is useful to note that numerical inte-gration was carried out using a three-point Gauss quadrature.

Considering the matrix solving time, some preliminaryconsiderations must be introduced: first is the band size andsecond is the condition number. As shown in the Table I,the band size is significantly increasing in the case of NEM.This is in fact a direct consequence of its ability to smooththe solution by taking into account information far beyondfinite element edges and that property naturally contributes toincrease the integration and assembly times for NEM.

On the other side, the conditioning of the matrix is muchbetter for NEM which is shown in Fig. 9.

Hence, although the band size is higher, the time spentfor the matrix solution calculation (excluding integration andassembly) with NEM is the cheapest of all methods: whencompared with FEM1, NFDM matrix solving is 30% fasterand Laplace solving 5% faster.

IV. EFFICIENCY RESULTS ON A TEST CASE

WITH SINGULARITY

The second numerical example is based on an electrostaticproblem describing an L-shaped capacitance. The length ofthe domain is set to 100 in each direction. When comparedwith the previous case, this problem includes a singularity atthe corner, which will significantly reduce the convergencerate. Since no analytical solution is available, the error isdetermined by solving dual formulations.

Fig. 10 shows the computational efficiency of the four mainmethods. Globally, NEM still outperforms first-order FEM,

7009904 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014

Fig. 10. Computation efficiency for the L-shape test case.

Fig. 11. Model of die mold with the electromagnet. (a) Whole view. (b)Enlarged view.

Fig. 12. Magnetic flux distribution in the studied device obtained by(a) nonlinear NEM resolution and (b) nonlinear FEM resolution.

and second-order FEM behaves even better. However, dueto the singularity at the corner, accuracy is limited and allmethods tend to the same asymptotic behavior.

V. EFFICIENCY RESULTS ON A REAL PROBLEM

Last, to compare the computational performance on realcase problems, we have solved the team workshop 25 witha 2-D nonlinear NEM and FEM. It is a die mold with anelectromagnet for the orientation of the magnetic powderused for producing permanent magnet [7]. Fig. 11 shows thestudied device.

Fig. 12 illustrates the flux lines of the magnetic flux obtainedby NEM and FEM resolution. A good agreement between thetwo methods is observed.

Last, Fig. 13 plots the computational performance. Thereference solution is obtained using a very fine second-order

Fig. 13. Computation efficiency for TEAM 25 test case.

FEM solution. At each experience, the same mesh has beenused for all methods. In this particular case, the errors aremainly due to the singularities in the corners. It is wellknown that in such a situation, increasing the number ofnodes is much more efficient than increasing the order ofapproximation. Hence, for a given number of nodes, NEM ismore accurate than FEM2, and even if the intrinsic cost ofNEM is roughly three times more expensive, at the end,both NEM and FEM2 give almost the same computationalperformance.

VI. CONCLUSION

In this paper, the computational efficiency of the NEM hasbeen studied and compared with FEM. For a given accuracy,the methods based on Voronoï cell functions—NFDM andNEM—are significantly much faster than first-order FEM. Onthe TEAM 25 case, NEM and FEM second-order even givethe same performance.

This tends to clearly demonstrate the very interesting behav-ior of the Voronoï cell approach: the interpolation has a highdegree of continuity, it does not exhibit edge noise, and last, ithas a kind of isotropic interpolation behavior around a node,very similar to the one observed with meshless methods.

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