meshless computation

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Engineering Computations,

Vol. 18 No. 1/2, 2001, pp. 170-192.

# MCB University Press, 0264-4401

Towards an efficient meshlesscomputational technique: the

method of finite spheresSuvranu De and Klaus-Ju rgen Bathe

Department of Mechanical Engineering, Massachusetts Institute ofTechnology, Cambridge, Massachusetts, USA

Keywords Finite element method, Meshless method

Abstract Computational efficiency and reliability are clearly the most important requirementsfor the success of a meshless numerical technique. While the basic ideas of meshless techniquesare simple and well understood, an effective meshless method is very difficult to develop. The

efficiency depends on the proper choice of the interpolation scheme, numerical integrationprocedures and techniques of imposing the boundary conditions. These issues in the context of themethod of finite spheres are discussed.

1. IntroductionOver the past decade several meshless techniques have been proposedincluding the smoothed particle hydrodynamics (SPH) method (Lucy, 1977), thediffuse element method (DEM) (Nayroles et al., 1992), the element free Galerkin(EFG) method (Belytschko et al., 1996), the reproducing kernel particle method(RKPM) (Liu et al., 1993), the partition of unity finite element method (PUFEM)(Melenk and Babuska, 1996), the hp-clouds method (Duarte and Oden, 1996), thefinite point method (Onate et al., 1996), the local boundary integral equation(LBIE) method (Zhu et al., 1998), the meshless local Petrov-Galerkin (MLPG)method (Alturi and Zhu, 1998) and the method of finite spheres (De and Bathe,2000).

The primary reason that there is significant interest in meshlesscomputational procedures is that the well established and successful numericaltechniques like the finite element/finite volume methods require a mesh. Theautomatic generation of a good quality mesh poses a significant problem in theanalysis of practical engineering systems. Moreover, the simulation andanalysis of certain types of problems (like dynamic crack propagation) requirean expensive remeshing operation.

Meshless techniques overcome these difficulties associated with meshing byeliminating the mesh altogether. Interpolation is performed in terms of nodalpoints ``sprinkled'' on the analysis domain using functions having compactsupport. A weighted residual technique is used to generate the discrete setof equations corresponding to the governing partial differential equations.While the basic idea behind meshless techniques is rather straightforward,the different meshless techniques found in the literature differ in theimplementation details; namely, the choice of the interpolation functions,the particular form of the weighted residual scheme employed, the procedure

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of imposing the boundary conditions and the numerical integration rulesadopted.

A review of the literature on meshless techniques reveals that the currenttrend is towards application of the new techniques to diverse problem areas in

engineering. But, it is clear that for general applications none of these methodsis computationally as efficient as the traditional finite element/finite volumetechniques. The issue of efficiency has been given due consideration onlyrecently. Breitkopfet al. (2000), for example, have proposed a way of computingthe moving least squares interpolants and their derivatives using a consistencybased approach (Breitkopfet al., 2000). However, for any meshless technique tofind eventually wide application, it must be reasonably efficient compared tothe now classical finite element/finite volume techniques and it should, ofcourse, be reliable. With these considerations in mind, we proposed a trulymeshless technique the method of finite spheres (De and Bathe, 2000).

In the method of finite spheres the discretization is performed usingfunctions that are compactly supported on general n-dimensional spheres andthe Galerkin weak form of the governing partial differential equations isintegrated using specialized numerical integration rules. The primary reasonfor the selection of the spherical support is that the relative orientation and theregion of overlap of two spheres are completely determined by the coordinatesof their centers and their radii.

In the traditional finite element/finite volume methods, the interpolationfunctions are piece-wise continuous polynomials (or mapped polynomials) andsatisfy the Kronecker delta property at the nodes. This ensures that Dirichletboundary conditions are enforced rather easily. Moreover, numerical

integration is performed most efficiently using Gauss-Legendre product ruleson integration domains that are n-dimensional cubes or tetrahedra. The Gauss-Legendre quadrature rule ensures arbitrary polynomial accuracy and thereforethe stiffness terms (for undistorted elements) are exactly integrated withminimal cost (Bathe, 1996).

In the method of finite spheres, however, the interpolation functions do notsatisfy the Kronecker delta property at the nodes. Hence, efficient imposition ofDirichlet boundary conditions is an important issue. Moreover, theinterpolation functions are rational (nonpolynomial) functions and theintegration domains are spheres, spherical shells or general sectors. Henceeffective numerical integration rules have to be developed and exact integration

can hardly be achieved. It is interesting to note that some of the earlierproposed meshless techniques use compactly supported functions forinterpolation, but require a background mesh for numerical integrationpurposes, and are therefore not truly meshless as is the method of finitespheres.

In this paper we discuss how computational efficiency may be achieved inthe method of finite spheres by proper choice of the interpolation functions,techniques of imposing the Dirichlet boundary conditions and efficientnumerical integration rules. The organization of this paper is as follows. In


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section 2 we review the interpolation scheme used in the displacement-basedmethod of finite spheres and compare it with the other popular interpolationschemes. In section 3 we address the important issue of numerical integrationand introduce several different integration rules. In section 4 we briefly reviewthe imposition of Dirichlet boundary conditions. In section 5 we estimate thecomputational cost in the method of finite spheres and compare the method inthis respect with other meshless techniques currently available and also withthe classical finite element technique.

2. Choice of interpolation schemeIn the method of finite spheres the interpolation functions are generated usinga partition of unity paradigm (Melenk and Babuska, 1996) based on theShepard partition of unity functions (Shepard, 1968). In this section wereview some of the most popular interpolation schemes and show that the

one used in the method of finite spheres offers a comparatively lessexpensive means of generating approximation spaces with a given order ofconsistency.

Let P Rd d 1Y 2 or 3 be an open bounded domain and let S be itsboundary (see Figure 1). Let a family of open spheres fBxIY rI;

I 1Y 2Y F F F YNg form a covering for , i.e. &N

I1BxIY rI, where xI and rIrefer to the center and radius of the sphereI respectively. We associate a ``node''with the geometric center xI of each sphere. By SxIY rI we denote the surfaceof sphere I. The spheres may be entirely within the domain (interior spheres)or may have nonzero intercepts with the boundary (boundary spheres) (seeFigure 1).

In order to obtain localized approximations we define a radial weightingfunction (or window function) WIx compactly supported on the spherecentered at nodeIsuch that:

Figure 1.(a) A schematic of themethod of finite spheresand (b) some shapefunctions in twodimensions


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(1) WIx P Cs0BxIY rI,s ! 0;

(2) suppWI & BxIY rI; and

(3) WIx ! 0 Vx P .

We concentrate on radial functions of the form WIx WsI, wheresI

kxxIk0rI

.

2.1 Moving least squares shape functionsThe moving least squares (MLS) method of generating smooth approximationspaces was used by Nayroles et al. (1992) in the context of the diffuse elementmethod and was later adopted by Belytschko et al. (1996) in their element freeGalerkin method. The MLS method has also been used by Atluri and Zhu(1998) in their implementation of the meshless local Petrov-Galerkin method).The MLS and closely related techniques described in this section require an

expensive matrix inversion/equation solution process at every evaluation pointand therefore result in shape functions that are computationally veryexpensive.

In the moving least squares method, an element vh of the globalapproximation space Vh (the subscript h is a measure of the size of the spheres)can be written as:

vhx NI1

hIxI Y 1

where hIx is the shape function associated with the nodal degree of freedom

I. Ifvhx P spanfpmxgn

m1 Vx P , where pmx is a polynomial or otherfunction, then the MLS shape function corresponding to nodeIis expressed as:

hIx WIxxTe

1xxI Y 2

where:

xT p1x p2x F F Fpnx Y 3

and:

ex N

I1

WIxxIxIT X 4

The derivative ofhIx with respect to a spatial variablexi (i 1Y 2 or 3) is:

dhIx

dxi WIx

dxT

dxie

1xxI

xTe1xdWIx

dxi WIx

dex

dxie

1x

xIX

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From equation (2) we observe that supphI suppWI. We make thefollowing remarks (for details, see Belytschko et al., 1996).

Remark 2.1. Reproducing property: pmx NI1 hIxpmxI for

m 1Y F F F Y n.This directly follows from equations (2) and (4). Hence to obtain linear

consistency inR2 for example, it is sufficient to choosexT 1YxYy.An interesting case arises whenx f1g and zeroth order consistency is

assured by the shape functions in equation (2) which assume the particularlysimple form:

hIx WIxN

J1 WJxX 6

These functions are also known as the Shepard partition of unity functions

(also described in section 2.2).Furthermore, if 1 P x then the shape functions form a partition of unity

subordinate to the cover fBxIY rIgNI1, i.e.:N

I1

hIx 1X 7

Remark 2.2. Continuity: let WI P Cs0BxIY rIYI 1Y 2Y F F F YN and let

pix P Cl, i 1Y F F F Y m for sY l ! 0; then the shape functions hIx in

equation (2) satisfy hIx P CminsYl0 BxIY rI .

This property may be used to choose the functional form ofW.Remark 2.3. Invertibility of ex: a necessary condition for the matrix

ex P Rnn to be invertible at every x P is:

x P

JP

BxJY rJ Y 8

where is an index set with cardfg ! n.This directly follows from equation (4), which representsex as the sum of

rank one dyadic products. In practice the matrix ex should be invertible atthe integration points and therefore at every integration station at least n

spheres (to be consistent in the treatment of the various meshless schemes, weassume that the support of the weighting function, WI, is a sphere throughoutour discussion) should have nonzero support. To obtain second orderconsistency, for example, six spheres in R2 and ten spheres in R3 should havenonzero support at every integration point. This is quite a formidablerequirement and of course it is not sufficient to ensure invertibility.

If x f1g then condition (8) is also a sufficient condition forinvertibility. For d b 1 if linear consistency is desired then an additionalcondition is required, namely, no three nodes J P should be colinear. If the


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nodal arrangement is close to this pathological condition, then the matrixexbecomes ill-conditioned resulting in large approximation errors.

The fact that the n n matrixex has to be generated and then inverted atevery integration point makes the MLS approach of generating shape functions

very expensive.Remark 2.4. The weighted least squares (WLS) interpolation scheme used in

the finite point method (On ate et al., 1996) is related to the MLS technique. Amultivalued global interpolation is obtained by requiring that:

Vx P BxIY rI vhx NJ1

hIJxJ Y 9

where hIJx and J are the shape function and degree of freedom, respectively, atnodeJ such that xJ P BxIY rI and:

hIJx WxJ xIxTe1I xI Y 10

where:

xT p1x p2x F F Fpnx Y

as before and:

eI NI1

WxJ xIxIxITX 11

Note that a discontinuous global approximation is obtained. The generalizedfinite difference scheme of Liszka and Orkisz (1980) may be considered as aspecial case of the WLS technique where the weights are so chosen that theirsupports always include the same number of nodal points.

Remark 2.5. The so-called ` kernel interpolation techniques'' like thesmoothed particle hydrodynamics (SPH) (Lucy, 1977) and the reproducingkernel particle methods (RKPM) (Liu et al., 1993) generate nodal shapefunctions that are very similar to those obtained using the MLS technique. TheSPH shape functions provide zeroth order consistency and hence if thefunctions WIx are used as the discrete kernels then the SPH shape functionsare identical to the Shepard partition of unity functions (see equation (6)). The

RKPM shape functions provide higher order consistency by applying acorrection to the SPH kernel. Of course, if the same function WIx is used asthe kernel function and the consistency requirement of order p is imposed, theresulting RKPM shape functions are identical to the MLS shape functions inequation (2) satisfyingpth order consistency (Belytschko et al., 1996).

2.2 Method of finite spheres shape functions: the partition of unity paradigmIn this section we describe the procedure we have adopted in generating lowcost shape functions in the method of finite spheres (De and Bathe, 2000) using


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the partition of unity paradigm (Melenk and Babuska, 1996). We consider herea pure displacement-based discretization scheme. The displacement/pressuremixed discretization scheme is more expensive since both displacements andpressures are discretized but is required in incompressible (or nearlyincompressible) analyses (De and Bathe, nd).

The first step is to define, at each nodeI, a basis function, 90Ix, satisfying:

(1)N

I1 90Ix I Vx P;

(2) supp90Ix & BxIY rI; and

(3) 90Ix P Cs0R

nYs ! 0X

This system of functions f90IxgNI1 is said to form a ``partition of unity''

subordinate to the open cover fBxIY rIg.The functions f90Ixg satisfy zeroth order consistency, i.e. they ensure that

rigid body modes are exactly represented. To attain higher order consistency,

at each node I, a local approximation space VhI spanmPsfpmxg is defined,where pmx is a polynomial or other function and s is an index set. Thesuperscript h is a measure of the size of the spheres.

The global approximation space Vh is generated by multiplying thepartition of unity function at each nodeIwith the functions from the local basis:

Vh NI1

90IVh

I X

Hence, any function vh P Vh can now be written as:

vhx NI1

mPs

hImxIm Y 12

where:

hImx 90Ixpmx Y 13

and hIm is a basis/shape function associated with the mth degree of freedom Im

of nodeI.It should be observed that no inversion of matrices is involved in the

generation of the shape functions and the computation of the shape functions

and their derivatives:

dhImdxi

90Ixdpmx

dxipmx

d90Ix

dxiY i 1Y 2 or 3Y 14

is rather straightforward.Remark 2.6. Reproducing property: if a function pix is included in the local

basis of all the nodes, then it is possible to exactly reproduce pix on the entiredomain.


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The proof directly follows from equations (12) and (13) and noting thatNI1 9

0Ix I Vx P .

In our work we use the Shepard partitions of unity functions (Shepard, 1968)defined by:

90Ix WIN

J1 WJX 15

Important consideration should be given to the choice of the functions WIxso that a low cost partition of unity is obtained. We have chosen a cubic splineweight function of the following form:

WsI

23

4s2I 4s3I 0 sI `

12

43

4sI 4s2I

43s3I

12

` sI 10 sI ` 1

V

X16

The choice of the functional form ofWI is guided by the following remark.Remark 2.7. Continuity: let WI P C

s0BxIY rIYI 1Y 2Y F F F YN and let

pmx P Cl for sY l ! 0; then the shape functions hImx satisfy

hImx P CminsYl0 BxIY rI .

The proof is immediate from equation (13). It therefore follows that:

(1) Displacement continuity. The displacement field is continuous so long asthe functions WI andpmx are continuous.

(2) Stress continuity. The stress fields, obtained by differentiating thedisplacement field (12), are continuous on if each of the functions WIhas zero slope at the center, xI, and on the surface, SxIY rI of the sphereon which it is defined, and provided the functions pmx and theirderivatives are sufficiently smooth.

For sufficiently smooth functions pmx, the stress fields are continuousprovided the derivatives of WI with respect to the spatial coordinates

xii P f1Y 2Y 3g:

dWsI

dxi

xi xIir2I

1

sI

dWsI

dsI

!Y 17

are continuous inBxIY rI and on SxIY rI.This derivative exists as sI 3 0 if WI has zero slope at the center of thesphere. Moreover, the derivative in equation (17) is continuous on SxIY rI, i.e.assI 3 1 ifWI has zero slope on the surface SxIY rI.

Equation (17) introduces two conditions on the first derivative of thefunction WI if a continuous stress field is to be obtained. A third conditionarises from the constraint that the function WI vanishes on SxIY rI, i.e.WsI 1 0. To satisfy these three conditions, the function WI needs to beat least a cubic insI. This justifies the choice of the cubic spline in equation (16).


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Remark 2.8. It was pointed out by Duarte and Oden that the use of Shepardfunctions to generate the partitions of unity is probably the least expensive fora given level of accuracy (Duarte and Oden, 1996). However, the more generalhp-clouds functions of Duarte and Oden use moving least squares functions

9pIx satisfyingpth order consistency to provide the partition of unity, i.e.:

Vh NI1

9pIVh

I Y 18

where:

9pIx WIxx

Te

1xxI Y 19

with x containing complete polynomials of order p and ex defined inequation (4). It should be recognized that this procedure of generating the

partition of unity functions is extremely expensive and we do not considerthese functions in our discussion in section 5.

3. Choice of numerical integration schemesIn the finite element techniques, the functions to be integrated are usuallypolynomials (or mapped polynomials), the elements do not overlap and theycan be mapped to n-dimensional cubes. Hence Gauss-Legendre product rulesare used for numerical integration with relatively low computational cost andhigh accuracy.

In the method of finite spheres, however, the shape functions are rational(non-polynomial) functions and the integration domains are spheres, spherical

shells or general sectors. Moreover, the overlaps of spheres give rise to generallens shaped regions. Hence, a separate class of integration rules is required.Exact integration of the stiffness terms is not possible since the integrationrules can, at best, guarantee polynomial accuracy. Hence the focus is onobtaining rules that use a minimal number of integration points for sufficientaccuracy.

In this section we discuss the numerical integration rules for two-dimensionalintegration domains in the method of finite spheres. First we introduce efficientnumerical integration rules on interior disks and boundary disks. We realizethat integration on the lens shaped region of overlap of two disks is animportant issue and propose and compare two different integration rules.

3.1 Integration on an interior diskIt is possible to develop integration rules having arbitrary polynomial accuracyon the unit disk (see Figure 2(a)). In this section we state such a product rulewith polynomial accuracy k. The rule has the following form:

fxYydxdy X

i

j

DijfxiYyj X 20


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The region under consideration is a disk with radius Ro (see Figure 2). Thedot over the equality signifies that the relationship is a strict equality if thefunction fxYy is a polynomial of order at most k in x and y, otherwise it is anapproximation. The following theorem is due to Peirce (Peirce, 1957).

Theorem 3.1. If it is required that the rule in equation (20) has accuracyk 4m 3Y m 0Y 1Y 2Y F F F Y in x ros and y rsin , and if it is requiredto have a minimum number of evaluation points which are taken at theintersection of concentric arcs (radius rj) with rays emanating from the origin

Figure Integration on ``interior'' disk

radius 1


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(angle i), then it is both necessary and sufficient for the existence of a uniqueset of weightsDij P Rthat:

i1 i 2%

k 1i 1Y 2Y F F F Y k Y 21

and that the r2i be the m 1 zeros ofPm1r2, the Legendre polynomial in r2 of

degree m 1, orthogonalized on 0YR2o. The (unique) weights Dij are of theformAiBj, where:

Ai 2%

k 1i 1Y 2Y F F F Y k 1 Y 22

and:

Bj 12P

H

m1r2

j

R2o0

Pm1r2

r2 r2

j

dr2 j 1Y 2Y F F F Y m 1 X 23

The integration points are on equally spaced radii and the integrationweights are independent of angular position (Gauss-Chebyshev rule in the -direction).

To demonstrate the efficiency of this rule we compare it with the Gauss-Legendre product rule on the disk:

fxYydxdy

Ro

yRo

Xy

xXyfxYy dxdy 9

Nxi1

Nyj1

DijfxiYyj Y 24

where Nx and Ny are the number of integration points chosen along the x and y-directions, respectively, and Dij W

xi

Wyj

is the product of the usual Gaussianweights Wxi and W

yj in the x and y-directions. It is not possible to guarantee

exact integration of polynomials of any degree using this rule. Fordemonstration we consider the simple problem of computing the area of a unitcircle (wherefxYy 1) in Figure 2(c).

3.2 Integration on a boundary sectorWe categorize the boundary sectors into two major groups depending on theangle 90 20 that the radii joining the center of the disk to the two interceptsof the disk on Smake interior to the domain.

Type I sector: 90 % (see Figure 3(a)). The rule that allows us to performnumerical cubature on this sector to any desired order of accuracy iscomputationally expensive since it requires the evaluation of the zeros of adifferent orthogonal polynomial for every 90 (De and Bathe, 2000). We,therefore, propose the following ``engineering solution'':


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fxYydxdy 94m1i1

m1j1

Dijfri os jY ri sin j Y 25

where Dij AiBj with Ai being the usual Gaussian weights on an interval

Figure

Integration points onboundary sect


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0Y 0 and Bj as in equation (23). The integration points ri are the positivesquare roots of the zeros of the Legendre polynomial Pm1r

2 in r2 of degreem 1 (k 4m 3 ), orthogonalized on 0YR2o as in theorem 3.1. However, theradial coordinates of the integration points j are chosen as the zeros of theLegendre polynomial P4m1 in of degree 4m 1, orthogonalized on0Y 0.

This rule guarantees arbitrary polynomial accuracy if the integrand in equation(25) is a purely radial function. Hence, as Figure 3(c) shows, the area of the sector(fxYy 1) can be exactly integrated but if the integrand is fxYy x, forexample, exact integration is not possible. Notice, however, that this rule is betterthan a pure Gauss-Legendre product rule on the sector as in equation (24).

Type II sector: 90 b % (see Figure 3(b)). This type of boundary sector is moreexpensive to handle. We decompose a type II sector into a sector for which therules of the type I sector can be used and a triangle as shown in Figure 3(b). Forthe triangle we use a product rule based on Gauss-Legendre quadrature.

3.3 Integration on the lensTo be able to integrate efficiently on the lens shaped region of intersection oftwo disks, we need specialized integration rules. In this section we propose andcompare two schemes for numerically evaluating:

IJ

fxYydxdy Y

where IJ BxIY rI BxJY rJ T 0.

Scheme 1. In this scheme (see Figure 4(a)) we use a Gauss-Legendre productrule of the form:

IJ

fxYydxdy

yoyyo

X2yxX1y

fxYy dxdy 9Nxi1

Nyj1

DijfxiYyj Y 26

where Nx and Ny are the number of integration points chosen along the x and y-directions, respectively and Dij W

xi W

yj is the product of the usual Gaussian

weights Wxi and Wyj

in thex andy-directions.Scheme 2. In this scheme (see Figure 4(b)), we map the domain IJ onto a

unit disk and compute the resulting integral using the scheme in theorem 3.1:IJ

fxYydxdy

1&0

2%0

F&Y J& d& d Y 27

whereJ is the Jacobian of the transformation.We note that none of the schemes can guarantee polynomial accuracy for the

integrand fxYy. This is because the integration limits in rule (26) are not


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simple. On the other hand, the Jacobian, in equation (27), is a complex functionto integrate. In Figure 4(c) the absolute error in integrating the area of theintersection region is shown as a function of the number of integration points.Note that neither scheme can integrate fxYy 1 exactly on the domain. In

Figure 4(d) the absolute value of the error in integrating a more complexfunction fxYy

1 x2 y2p

is shown as a function of the number ofintegration points. Such numerical experiments show that scheme 1 requiresfewer integration points for the same accuracy than scheme 2.

4. Imposition of essential boundary conditionsIn a weighted residual technique, the governing differential equation is notsatisfied point-wise in the interior of the domain. Point-wise satisfaction of theessential boundary conditions on a general Dirichlet boundary is similarly not

Figure Integration on the le


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possible. Considering the finite element technique, the shape functions satisfythe Kronecker delta property at the nodes (i.e. the shape function at any node isunity at that node and is zero at all other nodes). Furthermore, only those nodesthat lie on the Dirichlet boundary participate in the imposition of the Dirichlet

boundary conditions.Therefore, if homogeneous Dirichlet boundary conditions are prescribed,

then these conditions can be exactly satisfied along element edges on theDirichlet boundary. When a nonhomogeneous boundary condition isprescribed the finite element approximation along the element edges on theDirichlet boundary uhx jSu

hIx jSu uI (where hIx jSu is the trace of

the finite element shape function hIx on the Dirichlet boundary Su and uI isthe prescribed boundary condition at nodeI) converges to the applied boundarycondition in a weak sense. For example, if the finite elements exhibitpolynomial consistency of order p then ku uhk0 Ch

p1 (h denotes theelement size and C is a constant depending on the problem considered but isindependent ofh).

In the method of finite spheres the shape functions do not satisfy theKronecker delta property at the nodes. This is also true for the MLS (andrelated) shape functions. Moreover, nodes not lying on the Dirichlet boundarybut with nonzero intercepts of their spheres with the Dirichlet boundary arealso involved in enforcing the boundary conditions. Indeed, to retain theflexibility of ``sprinkling'' the nodes relatively arbitrarily on the domain, weshould be able to satisfy the Dirichlet conditions (in some sense) without even asingle node directly on the Dirichlet boundary (so long as the spheres cover thedomain). Therefore, rather than trying to satisfy the Dirichlet boundary

conditions point-wise at the nodes it is more important to be able to enforcethem in a weak sense along the boundary. We have therefore not considered inour work the so-called collocation techniques (Zhu and Atluri, 1998; Wagnerand Liu, nd).

Some of the other procedures of imposing Dirichlet boundary conditions thathave been employed in the context of meshless methods are techniquesinvolving Lagrange multipliers, penalty formulations, use of finite elementsalong Dirichlet boundaries and modified variational principles (Belytschko etal., 1996). The use of Lagrange multipliers results in indefinite systems ofequations and increases the number of unknowns considerably. Penaltyformulations can result in ill-conditioned matrices. The use of finite elements

along the Dirichlet boundaries destroys the meshless character of theapproximation.

A technique of much potential is the use of modified variational principles.In the method of finite spheres we have used this technique for the impositionof boundary conditions (for details, refer to De and Bathe, 2000). This techniqueenforces the Dirichlet boundary conditions in a weak sense without increasingthe number of unknowns. Furthermore, we have also shown that for a specialarrangement of nodes on the Dirichlet boundary, it is possible to exactly satisfythe Dirichlet conditions at the nodes (De and Bathe, 2000).


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5. Computational costsIn this section we estimate the computational cost in the displacement-basedmethod of finite spheres and compare this cost with the expense of a similarmeshless scheme using the moving least squares approximants as well as the

classical finite element techniques. We adopt the y-notation to imply theasymptotic upper bound to within a constant, i.e. for a given function gn, wedenote by ygn the set of functions:

ygn ffn X W constants cY no b 0 such that 0 fn cgnVn ! n0gX

For example, iffn aon a1 with ao b 0, then we may write fn P yn.A constant is represented as y1.

We assume that the major computational cost may be broken down into:

(1) cost of computation of the global stiffness matrix; and

(2) cost of solution of the resulting set of algebraic equations.

This approach neglects the computation of the loads, the application of theboundary conditions, memory traffic and other overheads associated with theexecution of a general numerical software and therefore gives only a roughestimate of the efficiency. Moreover, we do not consider the preprocessing orpostprocessing time. However, a main advantage of the meshless techniquesover the traditional finite element techniques is that preprocessing time isreduced as no mesh is required. Postprocessing in meshless techniques is alsorelatively straightforward since, for example, no additional stress smootheningis required.

We consider a general elliptic problem in d dimensions (d 1Y 2 or 3) and

assume a discretization scheme using N nodal points and a consistency oforder p. The superscripts MLS, FEM and MFS will be used in the followingdiscussion to differentiate the same variable for different methods. We assumebanded symmetric matrices with constant column height and a constant halfbandwidth mK. Each node is assumed to have an average connectivity of M,i.e. the support of each node is assumed to have nonzero overlaps with anaverage ofM 1 other nodal supports.

5.1 Cost of computation of the global stiffness matrixThe finite element method. In the finite element method, we assume that thestiffness matrix has ``dNFEM'' rows and ``dMFEM'' non zero columns per row.

Hence, the computational time for the global stiffness matrix may be assessedas

TFEMK yd2NFEMMFEMTFEMKij Y 28

where TFEMKij is the computational time for a single term of the stiffness matrix,which is assumed to be a volume integral of the sum of inner products of thederivatives of the shape functions. This integral is evaluated using numericalintegration over each finite element. Let NFEMg denote the number of Gaussian


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integration points per finite element (NFEMg p 12

for a tensor product elementinR2 using a complete polynomial of orderp). We model TFEMKij as:

TFEMKij yNFEMg T

FEMh Y 29

where TFEMh is the average computational time for a finite element shapefunction (or its derivative). We neglect the fact that multiple derivatives need betaken and different elements connected to a nodal point contribute. Therefore:

TFEMK yd2NFEMg N

FEMMFEMTFEMh X 30

Meshless method using moving least squares interpolants. We consider ameshless method which uses a moving least squares type of interpolant. Thestiffness matrix has ``dNMLS'' rows and ``dMMLS'' non zero columns per row.Hence, the computational time for the global stiffness matrix may be modeled

as:TMLSK yd

2NMLSMMLSTMLSKij Y 31

where TMLSKij is the computational time for a single term of the stiffness matrix.This integral is evaluated using numerical integration. Let NMLSg denote thenumber of integration points per sphere. We model TMLSKij as:

TMLSKij yNMLSg T

MLSh Y 32

where TMLSh is the average computational time for a MLS shape function (or itsderivative). The MLS shape functions and their derivatives are very complex

(see equations (2) and (5)) and involve the inversion of a n n matrix at eachintegration point, where n p 1p 2a2Y if a consistency of order p isrequired inR2. We may model TMLSh as:

TMLSh yn2MMLSTW Y 33

where TW is the computational cost of evaluating the weighting function inequation (16) or its derivative at a single evaluation point. This is justified sincematrix inversion and matrix-matrix multiplies are yn3 operations and

MMLS yn at least to ensure invertibility of the ex matrix in equation(4). Therefore:

TMLSK yd2n2NMLSg NMLSMMLS2TWX 34

Method of finite spheres. In the method of finite spheres with n functions in thelocal basis enforcing pth order consistency, the global stiffness matrix has` dnNMFS'' rows and ` dnMMFS'' non zero columns per row. Hence, thecomputational time for the global stiffness matrix may be modeled as:

TMFSK yd2n2NMFSMMFSTMFSKij Y 35


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where TMFSKij is the computational time for a single term of the stiffness matrix. LetNMFSg denote the number of integration points per sphere. We model T

MFSKij

as:

TMFSKij yNMFSg T

MFSh Y 36

where TMFSh is the average computational time for a MFS shape function (or itsderivative). The MFS shape functions and their derivatives are much simpler tocompute than the MLS shape functions (see equations (13) and (14)) and do notinvolve matrix inversions. The most expensive operation is the computation ofthe Shepard partition of unity functions (and their derivatives) which requiresthe evaluation of MMFS weighting functions at each integration point.Therefore we may model TMFSh as:

TMFSh yMMFSTW Y 37

where TW is the computational cost of evaluating the weighting function inequation (16) or its derivative at a single evaluation point. Therefore:

TMFSK yd2n2NMFSg N

MFSMMFS2TWX 38

Comparisons. It is interesting to observe that estimates (34) and (38) have thesame form. It may appear that the cost advantage in computing the simplerstiffness terms is lost in the number of terms that have to be computed in theMFS. This is true if the same number of nodes, connectivity and, of coursenumber of integration points, are used in both techniques. But due to thecondition mentioned in remark 2.3 regarding the invertibility of the exmatrix, MMLS yn, while essentially MMFS y1. Indeed, it has beenreported that MMLS for the MLS functions used in the element free Galerkinmethod can be as high as 50 in R2 (Krysl and Belytschko, 1995). In the methodof finite spheres, however, MMFS can be four. Furthermore, the invertibility oftheex matrix also necessitates thatNMLS ) NMFS for comparable accuracy(for example, in the method of finite spheres, the problem in Figure 5(a) may besolved with quadratic consistency using only four nodes at the four corners).Therefore:

TMLSK

TMFS

K

) 1X0X 39

Next, we compare the computation time estimates (30) and (38):

TMFSKTFEM

K

On2NMFSg N

MFSMMFS2TW

NFEMg NFEMMFEMTFEM

h

2 3X

It is true that TMFSK aTFEMK b 1X0 but the ratio is not very large since

NMFS ( NFEM for the same accuracy in solution. This is because in the


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method of finite spheres, the observed convergence rate is much better than inthe finite element method (De and Bathe, 2000).Let us consider an example of the square cantilevered plate in plane

strain shown in Figure 5(a) with uniformly distributed loading on the topsurface. The convergence in strain energy corresponding to a uniform h-typerefinement is presented in Figures 5(b) and (c) when the problem wassolved using nine-noded finite elements and the method of finite spheresrespectively. It is observed that a 8 8 regular nodal arrangement (withquadratic consistency) provides a solution (in strain energy) which is

Figure 5.Solving a problem usingnine-noded finiteelements and themethod of finite spheres


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comparable in accuracy with the solution provided by a 25 25 mesh ofnine-noded finite elements. Hence, for comparable accuracy and uniformmeshes, we require 5,202 degrees of freedom in the finite element technique and768 (64 nodes 12 degrees of freedom per node) degrees of freedom in themethod of finite spheres. We estimate:

NFEM 2Y601Y NMFS 64Y

NFEMg 9Y NMFSg 144Y

MFEM $ 25Y MMFS $ 4Y

n 6 Y

40

and obtain:

TMFSKTFEMK

$ 9 Y 41

assuming that TFEMh and TW are of the same order of magnitude. This estimateis quite close to the one obtained by comparing the actual computational times(the observed ratio is around eight).

5.2 Cost of solution including solving the set of algebraic equationsGiven the dimensions of the stiffness matrices considered in the previoussection, the solution times for the three techniques may be assessed as:

(1) The finite element method:TFEMs ydN

FEMmFEMK 2X 42

(2) Meshless method using moving least squares interpolants:

TMLSs ydNMLSmMLSK

2X 43

(3) Method of finite spheres:

TMFSs ydnNMFSmMFSK

2X 44

We are interested in the order of magnitude estimates and not any particularsolution technique.

From our discussion above we realize that for a large problem, the halfbandwidth mMFSK ( m

MLSK and N

MFS ( NMLS. Therefore it is reasonable toestimate that TMFSs ( T

MLSs .

Comparing estimates (40) and (42) we observe that NMFS ( NFEM forcomparable accuracy and mMFSK ( m

FEMK X For example, from the data in

equation (40) we see that:


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NMFS 64Y NFEM 2Y601Y

mFEMK 209Y mMFSK 119

and therefore:

TMFSsTFEMs

$ 4X8 102X 45

Estimates (41) and (45) are important in comparing the total computationalcosts of the method of finite spheres (TMFStotal ) and the finite element technique(TFEMtotal ). We assume:

TFEMtotal TFEMK T

FEMs

TMFStotal TMFSK T

MFSs X

46

We observe, by solving the problem in Figure 5(a) using a sequence of nine-noded finite elements using the commercial finite element software packageADINA, that for fine discretizations:

TFEMKTFEMs

$ 1X0X 47

From equation (46) we have:

TMFStotal

TFEM

total

TMFSK aT

FEMK

1 TFEM

s aTFEM

K

TMFSs aT

FEMs

1 TFEM

K aTFEM

s

X 48

From equation (45) it may be observed that the ratio TMFSs aTFEMs is very small

compared to the denominator (which is of the order of two). From equation (41)we observe that the ratio TMFSK aT

FEMK is of the order of ten. Therefore, it is

reasonable to estimate that the method of finite spheres is about five (or sayten) times slower than the finite element technique for elastostatic problems intwo dimensions.

6. Concluding remarksOver the past several years an increasing interest in meshless computational

techniques is observed. While current research efforts are primarily aimed atapplying these techniques to the analysis of various complex engineeringproblems, not enough emphasis is placed on computational efficiency.However, for a meshless technique to be widely applicable in industry,computational efficiency is clearly a most important factor to be considered.We have developed the method of finite spheres with this issue in mind. In thispaper we have discussed various aspects of the efficiency of the method.

We note that even though multiple shape functions are used at a node themethod of finite spheres is more efficient than the techniques based on the


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moving least squares interpolants since no matrix inversion is required atevery integration point and there are no stringent overlap criteria. In thecurrent form of implementation, when measured roughly theoretically and asseen in an example solution, the method of finite spheres is about five timesslower than the traditional finite element techniques for representativeproblems in two-dimensions since it offers comparable accuracy in solutionwith considerably fewer nodes on the domain. This is quite encouraging sincethe preprocessing time is considerably less than in the finite elementtechniques. However, future developments to obtain more efficient shapefunctions and integration rules along the lines discussed in this paper would beof much value.

References

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meshless computation

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