The method of finite spheresS. De, K. J. Bathe

Abstract The objective of this paper is to present some ofour recent developments in meshless methods. In partic-ular, a technique is given ± the method of ®nite spheres ±that is truly meshless in nature in the sense that the nodesare placed and the numerical integration is performedwithout a mesh. The method can be viewed as a specialcase of the general formulation known as the meshlesslocal Petrov±Galerkin (MLPG) procedure. Some of thenovel features of the method of ®nite spheres are the nu-merical integration scheme and the way in which theDirichlet boundary conditions are incorporated. A newway of modeling doubly-connected domains is also pre-sented. Various example problems are solved to demon-strate the method.

List of symbolsA Interior of a set in Rn; n 1; 2; 3:oA Boundary of set AA = A [ oA Closure of set Asuppv = fx 2 A; vx 6 0g Support of a function vX Open bounded domain in Rn; n 1; 2; 3:C = oX Boundary of X (assumed to be Lipschitz

continuous)n Outward unit normal de®ned on CBxI ; rI = fx 2 A; kxÿ xIk0 < rIg Open sphere of radius

rI centered at xI in n-dimensional Euclideanspace (n 1; 2 or 3)

SxI ; rI = fx 2 A; kxÿ xIk0 rIg Surface of then-dimensional sphere of radius rI centeredat xI .

h a global measure of the support radii(determined by the values of rI)

hImx = The mth shape function at node ICmA Space of functions with continuous derivatives

up to order m on A

Cm0 A = fvjv 2 CmA; supp(v) is a compact subset

of AgQmA Space of polynomials of degree m de®ned

on AVh;p The two-parameter global approximation

space (depending on parameters h and p).vh;p 2 Vh;p

a; A bilinear form from Vh;p Vh;p # R

Vh;pI The two-parameter local approximation space.

vh;pI 2 V

h;pI

1IntroductionIn spite of the great success of the ®nite element methodand the closely related ®nite volume method as effectivenumerical tools for the solution of boundary value prob-lems on complex domains, there has been a growing in-terest in the so-called ``meshless'' methods over the pastdecade. A number of methods have been proposed so farincluding the smooth particle hydrodynamics (SPH)method (Monaghan, 1988), the diffuse element method(DEM) (Nayroles et al., 1992), the element free Galerkin(EFG) method (Belytschko et al., 1994), the reproducingkernel particle method (RKPM) (Liu et al., 1993), themoving least-squares reproducing kernel method(MLSRK) (Li and Liu, 1996; Liu et al., 1997) the partitionof unity ®nite element method (PUFEM) (Melenk, 1992;Melenk and Babuska, 1996; Babuska and Melenk, 1997;Melenk and Babuska, 1997), the hp-clouds method (Duarteand Oden, 1996a; Duarte and Oden 1996b), the repro-ducing kernel hierarchical partition of unity method (Liand Liu, 1999a, b), the ®nite point method (OnÄate et al.,1996), the local boundary integral equation (LBIE) method(Zhu et al., 1998b) and the meshless local Petrov-Galerkin(MLPG) method (Atluri et al., 1999a, b; Atluri and Zhu1998a, b).

The primary reason for the interest in meshless meth-ods is that in the ®nite element/®nite volume method amesh is required. The automatic generation of goodquality meshes presents signi®cant dif®culties in theanalysis of engineering systems (especially in three di-mensions) and these dif®culties are circumvented when nomesh is needed.

The established computational methods which arebased on the weighted residual technique have three keyingredients:Interpolation: An expansion of the unknown ®eld vari-able/s in terms of trial basis/shape functions and unknownparameters,

Computational Mechanics 25 (2000) 329±345 Ó Springer-Verlag 2000

329

Received 26 October 1999

S. De, K. J. Bathe (&)Department of Mechanical Engineering,Massachusetts Institute of Technology,77 Massachusetts Avenue,Cambridge, MA 02139, USA

The authors would like to thank Dr. M.A. Srinivasan of theResearch Laboratory of Electronics, M.I.T. for the support ofthis work, and Professors S.N. Atluri, I. Babuska, T. Belytschko,W.K. Liu and J.T. Oden for their comments on an earlier versionof this paper.

Integration: The determination of the governing algebraicequations by setting the residual error orthogonal to a setof test functions which may or may not coincide with thetrial functions, andSolution of the algebraic equations: The solution of thegoverning equations for the unknown parameters.

If the ®rst two steps can be performed without a mesh,then what results is a ``truly meshless'' method. Many ofthe early ``meshless'' techniques such as the DEM, EFGM,hp-clouds method etc. are not truly meshless since even ifthe interpolation is independent of a background mesh,the integration is not.

In the ®nite element/®nite volume methods the inter-polation functions are polynomials (or mapped polyno-mials) and the numerical integration is performed mostef®ciently using Gauss-Legendre product rules on n-dimensional cubes or tetrahedra (in the mapped isopara-metric space). Failure to perform the integration accu-rately results in loss of accuracy and possibly stability ofthe solution scheme (Bathe, 1996). In most mesh-freetechniques, however, complicated non-polynomial inter-polation functions are used which render the integrationof the weak form rather dif®cult (Dolbow and Belytschko,1999).

Another dif®culty lies in the accurate imposition of theDirichlet boundary conditions. The Kronecker deltaproperty of the ®nite element/®nite volume shape func-tions allows the incorporation of these conditions veryef®ciently (Bathe, 1996). But the interpolation functionsused in most meshless schemes do not satisfy this condi-tion at the nodes and hence the imposition of Dirichletboundary conditions becomes complicated. Popular withresearchers are techniques involving Lagrange multipliers(Belytschko, 1994), penalty formulations (Atluri and Zhu,1998a, b), use of ®nite elements along Dirichlet boundaries(Krongauz and Belytschko, 1996), modi®ed variationalprinciples (Lu et al., 1994), corrected collocation tech-niques (Wagner and Liu, 1999) and transformationmethods (Atluri et al., 1999a).

A survey of meshless techniques may be found inDuarte (1995) and Belytschko et al. (1996). The majormeshless methods described in these two review works,namely the RKPM, SPH method, DEM and EFG methodare based on three classes of interpolation functions:wavelets, moving least squares functions (used in the DEMand the EFG method), and the partition of unity (PU) orhp-clouds functions (used in the PUFEM and hp-cloudsmethods). All these methods are really ``pseudo meshless''since they use a background mesh for the numerical in-tegration (and sometimes even for imposing the Dirichletboundary conditions).

The ®nite point method (OnÄate et al., 1996) is a trulymeshless scheme. The method uses a weighted leastsquares (WLS) interpolation and point collocation, thusbypassing integration. However, methods based on pointcollocation are notorious for the sensitivity of the solutionon the choice of ``proper'' collocation points.

As a truly meshless technique, the meshless local Pet-rov±Galerkin (MLPG) method (Atluri and Zhu, 1998a)seems to be the most promising. The technique is based ona weak form computed over a local sub-domain, which can

be any simple geometry like a sphere, cube or an ellipsoidfor ease of integration. The trial and test function spacescan be different or may be the same. Any class of functionswith compact support satisfying certain approximationproperties (like the MLS functions or PU functions) can beused as trial and test functions (Atluri et al., 1999a). Thismethod has been successfully applied to a wide range ofproblems (Atluri et al., 1999a, b; Atluri and Zhu, 1998a, b)and is of very general nature. A method using a similarapproach but boundary integral techniques is the localboundary integral equation (LBIE) method (Zhu et al.,1998b).

However, although considerable efforts have been madein the development of meshless methods, the currentlyavailable techniques are still computationally much lessef®cient than the well established ®nite element/®nite vol-ume procedures. The primary reason is that complicated(non-polynomial) shape functions are employed and therequired numerical integration is very dif®cult to performef®ciently. Hence some researchers (refer to Oden et al.,1998; Duarte et al., 1999) have reverted back todeveloping ®nite element techniques incorporating certainaspects of the meshless methods. In this paper, however, weconcentrate attention on methods which are truly meshlessand endeavor to make them as ef®cient as possible.

The objective of this paper is to present some devel-opments using the MLPG approach with the aim ofreaching a powerful solution technique. In the method of®nite spheres, the MLPG concept is used with a speci®cchoice of geometric sub-domains, test and trial functionspaces, numerical integration technique, and a procedurefor imposing the essential boundary conditions.

The method of ®nite spheres uses compactly sup-ported PU functions to form a globally conforming ap-proximation space and a Bubnov±Galerkin formulationas the weighted residual scheme. Figure 1 shows aschematic of the method as a natural generalization ofthe ®nite element method. In the classical ®nite element/®nite volume method the support of a shape functioncorresponding to a node is usually a polytope in n-di-mensions whereas the method of ®nite spheres usesn-spheres as supports. Integration is performed on then-dimensional spheres or spherical shells using special-ized cubature rules.

A brief outline of the paper is as follows. In Sect. 2 wediscuss in detail our justi®cation for the use of PU basisfunctions based on Shepard partitions of unity and sum-marize key results relating to consistency and a-priori erroranalysis. In Sect. 3 we derive the weak form for a symmetricsecond-order differential operator. In Sect. 4 we discuss theimposition of Neumann and Dirichlet boundary conditions.Even though we primarily concentrate on the n-dimen-sional sphere as our integration domain, we realize that todeal with doubly-connected domains ef®ciently, the ideas ofsupport and integration domain have to be decoupled. Toaddress the solution of such problems we present, in Sect. 5,our developments using n-dimensional spherical shells.In Sect. 6 we deal with issues related to numerical integra-tion on the n-dimensional spheres and spherical shells.Finally, in Sect. 7, several numerical solutions are providedto demonstrate the method of ®nite spheres.

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2The interpolation schemeThe ®rst step in the Bubnov±Galerkin procedure is toconstruct ®nite dimensional subspaces Vi of a Sobolevspace, in which the weak solution is assumed to exist. In alarge variety of practical applications, namely, problemsassociated with second-order partial differential operators,the Sobolev space under consideration is H1X, the ®rst-order Hilbert space. We summarize the theory in thatsetting, extension to higher-order spaces can be achieved.We are interested in approximation spaces which satisfythe following minimal requirements:

Consistency or polynomial reproducing property Theconsistency condition is related to the degree of the gov-erning partial differential equation. For example, whensolving an elasticity problem using a displacement-basedformulation, the approximating functions should not onlybe able to reproduce constant functions (the so-called``rigid body'' modes) but also linear functions (the ``con-stant strain'' states), i.e., we look for at least ®rst-orderconsistency. Thus, we should be able to reproduce poly-nomials to a certain order to satisfy the consistencyrequirement.

Local approximability This is a more general requirementthan just consistency and is related to the reproducingproperties of the trial functions. If we know the nature ofthe solution in certain subdomains of X, we should be ableto incorporate speci®c functions in the global approxi-mation space in order to enrich this space to closely rep-resent the solution. There are certain situations wheresingularities arise naturally in the solution of the govern-ing differential equations and polynomials perform poorlyin resolving such singularities. The idea is to use availableanalytical solutions to improve numerical predictions.

Continuity The approximation functions should satisfycertain minimal continuity conditions.

Localization by compact support One of the major rea-sons of the success of the ®nite element procedure is theconcept of approximations using piece-wise polynomials

on compactly supported domains (elements). The mainadvantages of using compactly supported functions, i.e.functions that are nonzero only on small subsets of X, arethat (1) they allow the localization of the approximation(and hence steep gradients can be handled by using morefunctions locally), (2) they result in banded system ma-trices since only a few of the supports overlap at any givenpoint of the domain and (3) they allow a natural means ofcontrolling the rate of convergence of numerical schemesthrough h, p or hp-type re®nements.

There is no unique way of constructing the approxi-mation spaces. The current interest in the so-calledmeshless methods has been primarily spurred by theability to construct non-polynomial approximation spaceswith compact support without the need for a backgroundmesh. As we mentioned already, examples of suchapproximation functions are compactly supported waveletfunctions, the MLS (moving least squares) functions(Nayroles et al., 1992; Belytschko et al., 1994), the PUfunctions of Babuska and Melenk (Melenk and Babuska,1996) and the hp-cloud functions of Duarte and Oden(Duarte and Oden, 1996a, b). In the wavelet-based meth-ods, compactly supported functions with desirable prop-erties are developed using FIR (®nite impulse response)®lter-banks. The dif®culties of using wavelets as basisfunctions are that they are designed to have desirableorthogonality properties in L2X but not in higher-orderHilbert spaces, the computation of inner products through``connection coef®cients'' is very cumbersome and theapplication of wavelets to arbitrary domains is still beingresearched.

The moving least squares technique of generatingcompactly supported functions having desirable repro-ducing properties is quite appealing, but has some draw-backs, most important of which is the need to invert amm matrix (when m basis functions are used to gen-erate the MLS shape functions). Besides increasing thecomputational cost for any m > 1, this requirement meansthat at each evaluation point at least m weight functionsshould be nonzero for the matrix to be invertible.

Of the existing techniques for the generation of com-pactly supported basis functions, the methods developedusing the partition of unity (PU) paradigm appear to be

Fig. 1a, b. Discretization of a domain W in R2 by the ®nite element method (a) and the method of ®nite spheres (b). In (a) the domain isdiscretized by quadrilateral elements with a node at each vertex point. The ®nite element shape function hI is shown at node I. In (b)the domain is discretized using a set of nodes only. Corresponding to each node I, there is a sphere (i.e. a disk in R2), centered at the node,which is the support of a set of shape functions corresponding to that node. One such shape function, hI0, is shown in the ®gure

331

the most general. They possess the desirable properties wehave listed above and it is possible to generate low-costpartitions of unity (without the need to invert matrices)(Duarte and Oden 1996a). Moreover, the ®nite element/®nite volume basis functions and the MLS basis functionsmay be looked upon as specialized applications of thepartition of unity construction technique.

We have adopted the PU paradigm to generate the shapefunctions for the method of ®nite spheres. We brie¯ysummarize the construction of these basis functions andstate important results in the following two sections. Fordetails regarding the theory of the PU method as well as forproofs of the theorems see Melenk (1992), Melenk andBabuska (1996), Duarte and Oden (1996a).

2.1The partition of unity functionsThe ®rst step in the PU method is of course the generationof the partition of unity functions.

De®nition 2.1 Let X 2 Rn n 1; 2; 3 be an openbounded domain. Let a family of open sets fXI ; I 1; 2; . . . ;N;N Nh; h > 0g form a covering for X, i.e.,X SN

I1 XI . Then there exists a system of functionsf/IgN

I1 of Cs0Rn; s 0 such that

1.PN

I1 /Ix 1 8 x 2 X.2. supp/I XI .

Then this system of functions f/IgNI1 is de®ned as a

partition of unity subordinate to the open cover fXIg(Yosida, 1978).

The sets fXIg are also called ``clouds'' or ``patches''(Melenk and Babuska, 1996; Duarte and Oden, 1996a).In the method of ®nite spheres (MFS) we choose XI BxI ; rI, which geometrically represents a sphere. Weassociate a ``node'' with the center, xI I 1; 2; . . . ;N,of each sphere. By CI we denote the surface of the sphere,SxI ; rI. We use a particular family of (nonpolynomial) PUfunctions, called the Shepard partitions of unity (Shepard,1968) having zeroth-order consistency (i.e., these functionscan only reproduce constant functions exactly). Let WIxdenote a weighting function with the following properties:

1. WIx 2 Cs0Rn; s 0:

2. suppWI XI .

The Shepard partition of unity functions are de®ned as

u0I x

WIPNJ1 WJ

: 1

Important considerations in the choice of the weightfunctions are the continuity class to which they belong andhow easily they can be differentiated and integrated. Wefollow Duarte's work (Duarte and Oden, 1996a) andchoose quartic spline weight functions (which belong toC1

0X):

WIs 1ÿ 6s2 8s3 ÿ 3s4 0 s < 10 s 1

where s kxÿ xIk0=rI :

2.2The approximation spaces V h;p

Having chosen the Shepard functions which satisfy the PUrequirement, we are in a position to develop approxima-tion spaces which are subspaces of H1X. LetV

h;pI H1X \ XI be a two parameter family of function

spaces, the parameters being h (the size of the sphere) andp (the polynomial order) (Melenk and Babuska, 1996;Duarte and Oden, 1996a), such that

Qp spanVh;pI 8 I

then we de®ne the two-parameter global approximationspace Vh;p as

Vh;p XN

I1

u0I V

h;pI H1X :

Since Vh;pI spanm2Ipmx, where I is an index set and

pmx is a member of the local basis, any functionvh;p

I 2 Vh;pI can be expressed as vh;p

I x P

m2I aImpmx,for aIm 2 R. If we multiply each pmx by u0

I x, the re-sulting function has the same support as u0

I x. The globalapproximation space is constructed using such products.Hence, any function vh;p 2 Vh;p can now be written as

vh;px XN

I1

Xm2I

hImxaIm 2

where

hImx u0I xpmx

and hIm is a basis/shape function associated with the mthdegree of freedom of node I. We now state, without proof,some important properties of the discretization scheme(for proofs see Melenk and Babuska, 1996; Duarte andOden, 1996a). We use the symbol C to denote a genericpositive constant which may take different values at suc-cessive occurrences (including in the same equation).

Theorem 2.1 (Reproducing property). If any functionpmx is included in the local bases, it is possible to exactlyreproduce it.

Corollary 2.1 (Consistency). If Qm spanVh;pI 8 I, then

Qm spanVh;p.

Theorem 2.2 (Continuity). Let WI ; I 1; 2; . . . ;N 2 Cs

0XI and let pmx 2 ClX for s; l 0; then theshape functions hImx satisfy hImx 2 C

mins;l0 XI \ X.

Theorem 2.3 (Approximation error estimate). Let u bethe function to be approximated, and let the given PUfunctions f/Ixg satisfy

k/IkL1Rn C ;

kr/IkL1Rn C

rI:

Assume that the local approximation spaces Vh;pI have

the following properties: On each patch XI \ X; u can beapproximated by a function vh;p

I 2 Vh;pI such that

332

kuÿ vh;pI kL2XI\X 1I; h; p; u ;

kruÿ vh;pI kL2XI\X 2I; h; p; u :

then there is a function vh;p 2 Vh;p satisfying

kuÿ vh;pkL2X CXN

I1

1I; h; p; u2 !1=2

kruÿ vh;pkL2X XN

I1

C

rI

2

1I; h; p; u2

C 2I; h; p; u 2!1=2

:

Theorem 2.4 (Convergence rate of the h-version). Letu 2 HkX; k 2. Let V

h;pI have the following approxima-

tion properties:

1I; h; u Crl1I ku kHkXI\X

2I; h; u CrlI ku kHkXI\X

for some appropriate l > 0. If uh;p is the numerical solu-tion, then

kuÿ uh;pkL2X Chl1kukHkX;

kruÿ uh;pkL2X Chlku kHkX :

Theorem 2.1 states that if we include a-priori knowledgeof the solution in local subdomains, then this knowledgewill enhance the approximation capability because thefunctions representing this knowledge can be reproduced.Corollary 2.1 assures that it is possible to obtain any orderof consistency, at least theoretically. It turns out that forour choice of the PU functions, the functions hIm are lin-early independent, i.e. Vh;p spanhImx; as long as thelocal bases are linearly independent. Theorem 2.2 tells uswhat order of continuity is obtained by the global ap-proximation. Theorem 2.3 is of very general nature andprovides an interpolation error estimate if the local ap-proximation behavior is known. Theorem 2.4 is an appli-cation of the previous theorem to obtain a bound on thesolution error. Speci®cally if a polynomial basis of degreep is used as the local approximation space, and k p 1,then l p and an Ohp1 convergence in the solutionvariable is predicted. Duarte and Oden have pointed outthat the use of Shepard functions to generate the partitionsof unity is probably the least expensive for a given level ofaccuracy (Duarte and Oden, 1996a).

3The weak form for a n-dimensional sphereIn this section, we develop the weak form and discretizedequations of the governing differential equation by inte-grating over each n-dimensional sphere centered around anode. We focus on a second-order partial differentialequation in a single variable. Extension to multiple vari-ables and higher-order differential operators can be di-rectly achieved.

Consider the operator equation

Au f in X 3where A: DA H2X# L2X is a second-ordersymmetric positive de®nite differential operator withdomain of de®nition DA and f 2 L2X is the forcingfunction, with

A ÿXd

i;j1

ooxi

aijx ooxj cx

where d is the dimensionality of the problem, aijx andcx are bounded measurable coef®cients. Assume thatNeumann boundary conditions are prescribed over theboundary CfXd

i;j1

aijx ou

oxjni f s on Cf 4

where ni is the component of the outward unit normal onthe boundary along the ith direction (see Fig. 2), andDirichlet boundary conditions are provided on the boun-dary Cu

u us on Cu 5where C Cu [ Cf and Cu \ Cf 0. In the Bubnov±Galerkin procedure, we ®nd the approximation uh;p 2 Vh;p

to the true solution u by making the residual Auh;p ÿ f orthogonal to the basis functions fhImg. Hence, corre-sponding to node I, we generate the following set ofequations:

Auh;p ÿ f ; hIm 0; m 2 I :

Using Eq. (2) and Green's Theorem, we obtain the mthequation corresponding to the Ith node asXN

J1

Xn2I

KImJnaJn fIm fIm 6

Fig. 2. General three-dimensional body discretized using a setof nodes. Associated with each node I is a sphere WI. Spheres thatlie completely inside the domain are called ``interior spheres''while those which intersect the boundary of the domain, C, arecalled ``boundary spheres''. Dirichlet boundary conditions areprescribed over a portion Cu of the boundary while Neumannboundary conditions are prescribed over Cf ; C Cu [ Cf ;Cu \ Cf 0

333

where

KImJn ahIm; hJn Z

XI\XcxhImhJn dX

Xd

i;j1

ZXI\X

aijx ohIm

oxi

ohJn

oxjdX ;

fIm Z

XI\Xf hIm dX ;

fIm Xd

i;j1

ZCI

hImniaijx ouh;p

oxjdC :

In the formulation we distinguish between ``interiorspheres'' and ``boundary spheres''. An interior sphere haszero intercept with the boundary, i.e., XI \ X XI (seeFig. 3a). A boundary sphere has a nonzero intercept withthe boundary (see Fig. 3b, c). For an interior sphere,therefore, f Im 0 due to compact support and Eq. (6)reduces to:XN

J1

Xn2I

KImJnaJn fIm :

4Imposition of boundary conditionsIn this section we discuss how the boundary conditions,given by Eqs. (4) and (5), can be incorporated ef®ciently.

4.1Natural boundary conditionsIn the ®nite element/®nite volume method, due to theKronecker delta property of the shape functions, only thenodes on the boundary are subjected to the appliedboundary conditions. But in the MFS, the basis functions,de®ned on the spheres, do not satisfy the Kronecker deltacondition and hence, any sphere, with nonzero interceptwith the boundary contributes to the boundary integral.Let CfI be the intercept of the sphere I with the boundaryCf , see ®gure 3b, then Cf [I2Nf

CfI , where Nf is theindex set of nodes considered. For such a sphere, Eq. (6)applies with

f Im Z

CfI

hImf s dC :

4.2Essential boundary conditionsThe imposition of the essential boundary conditions ismore dif®cult in the absence of the Kronecker deltaproperty than the imposition of the natural boundaryconditions. In the following we present a technique for theincorporation of the essential boundary conditions andshow that a speci®c arrangement of nodes on the boun-dary may emulate Kronecker-delta-like properties.

Referring to Fig. 3c we note that any node with nonzerointercept of its sphere with the boundary Cu contributes tothe boundary integral in Eq. (6). Let CuI

be the intercept ofthe sphere I with the boundary Cu, then Cu [I2Nu

CuI,

where Nu is the index set of nodes considered. Makinguse of the chain rule of differentiation, we may now writef Im as

f Im XN

J1

Xn2I

KUImJnaJn ÿ fUIm ;

where

KUImJn Xd

i;j1

ZCuI

ooxj

aijxhImhJnni

ÿ dC ;

fUIm Xd

i;j1

ZCuI

us ooxj

aijxnihIm

ÿ dC :

We note that KUImJn is a symmetric stiffness term(KUImJn KUJnIm) and fUIm is a (known) forcing term.Hence, Eq. (6) becomes

XN

J1

Xn2IKImJn ÿ KUImJnaJn fIm ÿ fUIm : 7

This procedure for imposing the Dirichlet boundary con-ditions is quite general but may be somewhat dif®cult toimplement. Namely, if the nodes are distributed on andnear the boundary at random and the boundary is a

Fig. 3a±c. Figure showing ``interior spheres'' (a) and ``boundary spheres'' (b) & (c). Integration is performed on WI in (a) and onXI \ X in (b) & (c)

334

complex (d-1) dimensional surface, then the computationof the intercepts of the spheres with the boundary surfacemay become computationally intensive.

To circumvent this dif®culty, we propose the specialdistribution of the boundary spheres shown in Fig. 4. Inthis construction, we assume that the nodes are placed onthe boundary such that the distance between two succes-sive nodes is the radius of the spheres and that there areno nodes whose spheres intercept the boundary other thanthose that are on the boundary. This nodal arrangementovercomes the problem of ®nding the intercept of theboundary spheres with complex boundaries. The ar-rangement also gives rise to a Kronecker delta-like prop-erty. Then at any such boundary node I, the basisfunctions hIm are such that

hI0xI u0I xI

and

hImxI 0; m 6 0 :

From Eq. (1), by construction,

u0I xI WIPN

J1 WJ

1 :

Hence, the basis function hI0 at node I enjoys the Krone-cker delta property

hI0 1 at node I0 at all other nodes

nwhereas the higher-order basis functions exhibit theproperty

hIm 0 at node I0 at all other nodes

nfor m 6 0 :

Hence

vh;px xI aI0 :

Thus the speci®ed value of the ®eld variable u at node I onthe Dirichlet boundary is taken up by the coef®cient of hI0.The implications are that for speci®ed homogeneous(zero) Dirichlet conditions, we simply remove, from thestiffness matrix, all the rows and columns correspondingto the Shepard functions associated with the nodes that areon the Dirichlet boundary and solve the resulting set ofreduced Eq. (7). If inhomogeneous Dirichlet conditions

are speci®ed, we also remove the rows and columns cor-responding to the Shepard functions associated with thenodes on the Dirichlet boundary but need to bring theeffect of the nonzero prescribed displacements to the righthand side of the governing equations. Hence, Eq. (7) be-comes (m 6 0 with I 2NuXN

J1

Xif J2Nu

n2I

n60

KImJn ÿ KUImJnaJn fIm ÿ fUIm: ÿ fUIm

where

fUIm X

J2Nu

KImJ0 ÿ KUImJ0usxJ

and xJ is the coordinate of node J. Of course fUIm 0when zero Dirichlet conditions are prescribed.

5Doubly-connected domains:the n-dimensional spherical shellSo far we have concentrated on nodes whose integrationdomains are singly-connected and therefore coincide withthe support. There are certain situations, however, for ex-ample, a hole in a plate, or a spherical cavity inside a three-dimensional continuum, when it would be effective to beable to directly model doubly-connected domains. Theerror introduced in modeling the boundaries of thesecavities by placing nodes along their periphery is theneliminated and hence less nodes are required to model suchgeometries. Also, the known behavior of the solution of thegoverning equations can be included in the local bases ofthese nodes and thus higher convergence rates can beattained. To be able to model doubly-connected domains,we decouple the regions of support and integration.

Assume that there is a spherical cavity of radius ri andcenter xI inside the domain X (see Fig. 5a). We place anode, I, at the center of the cavity and associate with ita weight function WI such that suppWI BxI ; rI, butwe choose the integration domain for this node as

XI BxI ; ronBxI ; rifor some ri < ro rI . We see that Eq. (6) applies with theintegral in f Im written as the sum of two integrals (ap-plying contour integration as shown in Fig. 5a)

f Im Xd

i;j1

ICIo

aijxhImniouh;p

oxjdC

Xd

i;j1

ICIi

aijxhImniouh;p

oxjdC ; 8

where CIo SxI ; ro and CIi SxI ; ri. We consider twocases:

Case (1) ro rI : The ®rst integral in Eq. (8) is zero due tothe property of compact support and we have

f Im Xd

i;j1

ICIi

aijxhImniouh;p

oxjdC :

Fig. 4a, b. Nodal arrangement for easy incorporation of Dirichletboundary conditions on (a) convex and (b) concave boundaries

335

Usually we have some boundary data prescribed on theinside surface of the cavity which can be incorporatedusing the techniques described in the previous section.

Case (2) ro < rI : In this case we have to use Eq. (8) in itsfull form.

6Numerical integrationNumerical integration is an important ingredient of ameshless method. In the ®nite element method the func-tions to be integrated are usually polynomials (or mappedpolynomials) whereas in the MFS, the PU shape functionsare rational (nonpolynomial) functions. In the ®nite ele-ment method, the elements can be mapped to n-dimen-sional cubes and hence Gauss-Legendre product rules areprobably most appropriate (Stroud, 1971). Since in theMFS, the integration domains are spheres or sphericalshells for interior domains and general sectors for boun-dary spheres, a separate class of integration rules is re-quired. In Appendix A we state, without proof, Gaussianproduct rules of cubature on two-dimensional sectors andannuli. Even though cubature rules exist for circular an-nuli as well as spheres and hyperspheres (Peirce, 1957a, b;Lether, 1971; Stroud, 1971; Stoyanova, 1997), rules forannular sectors seem not to have been published. Note thatthe integration rules we are using are different from thosein Atluri et al. (1999a).

Interior disk For an interior disk (see Fig. 6a, b), we useCorollary A.1 and are able to integrate with any givenprecision. The integration points are on equally spacedradii and the integration weights are independent of an-gular position (Gauss-Chebyshev rule in the h-direction).For the two-dimensional solutions considered in Sect. 7, arather large number of integration points are used per disk(6 integration points along each of 24 radial directions).

Boundary disk We categorize the boundary sectors intotwo major groups depending on the angle /0 that the radiijoining the center of the disk to the two intercepts of thedisk on C make interior to the domain:

Type I sector: /0 p (see Fig. 7) In Appendix A we statethe rule that allows us to perform numerical cubature onthis sector to any desired order of accuracy. But roots ofhigher-order orthogonal polynomials for every boundary

Fig. 6a, b. An interior disk is shown in (a). In (b) we display theintegration points (schematically) corresponding to an accuracyof order k=7. Note that the integration points are on equallyspaced radii and the integration weights are independent ofangular position

Fig. 5a, b. A domain W with aspherical cavity of radius ri is shown.Node I is placed at the center of thecavity. The weight function, WI, atnode I has a support radius of rI. Theintegration domain associated withthe node I is a spherical shell of innerradius ri and outer radius ro

336

sector need be evaluated. To circumvent this expense, wepropose an ``engineering solution'' and use Gauss-Legendre quadrature in the h-direction.

Type II sector : /0 > p (see Fig. 8) This type of boundarysector is more expensive to handle. We decompose a TypeII sector into a sector for which the rules of the Type Isector can be used and a triangle as shown in Fig. 8b. Forthe triangle we use a product rule based on Gauss±Legendre quadrature.

7Numerical examplesIn this section we present numerical examples in one andtwo dimensions demonstrating the above formulation. Asimple problem involving a bar with distributed loading issolved in one-dimension, followed by a one-dimensionalhigh Peclet number ¯ow problem. In two-dimensions we®rst solve a Poisson problem with mixed boundary con-ditions and then two linear elasticity problems. Corre-sponding to each distinct type of equation solved by theMFS, a patch test was performed and the method passedthe patch test in each case.

7.1The MFS in R1: a bar with distributed loading

7.1.1FormulationIn R1 the ``spheres'' reduce to line-segments (as shown inFig. 9a). We solve the following problem of a bar of unitlength, subjected to a distributed loading:

d2uxdx2

f x 0 in X 0; 1u us at x 0

du

dx f s at x 1

The parameters us and f s and the function f x are chosenso that the analytical solution u is given by the followingexpression:

ux 1

2xÿ x3

3

2x 1 :

The weight function de®ned in Sect. 2.1 is used to generatethe Shepard partition of unity. At each node I, the fol-lowing shape functions were used;

fu0I x;u0

I xxÿ xI=rIgto attain linear consistency. Figure 9a shows a plot of theseshape functions for a typical node within the domain. Thediscretized equation corresponding to the Ith node andmth degree of freedom is given by Eq. (6) where the in-tegrals are

KImJn Z x2

x1

dhIm

dx

dhJn

dxdx

fIm Z x2

x1

f xhIm dx

fIm

0 for an ``interior sphere''

f shImx1 for a sphere on theNeumann boundaryPN

J1

Pn2I

KUImJnaJnÿ fUIm for a sphere on theDirichlet boundary

8>>>>><>>>>>:where

KUImJn ÿ d

dxhImhJn

x0

fUIm ÿus dhIm

dx

x0

it being understood that x1 max0; xI ÿ rI andx2 min1; xI rI.

7.1.2Numerical resultsBoth regular and arbitrary distributions of nodes havebeen used to solve the problem. Figure 9b shows the result

Fig. 8a, b. Type II boundary sector (a) and the distribution ofintegration points (b)

Fig. 7a, b. Type I boundary sector (a) and the distribution ofintegration points (b)

337

when six nodes are distributed regularly. Figure 9c showsthe solution when the same problem is solved using ®venodes with arbitrary distances between them.

7.2The MFS in R1: a high Peclet number flow problem

7.2.1FormulationWe consider the following problem of steady state heatconduction in one dimension with prescribed velocity v asdiscussed in Bathe (1996). The temperature is prescribedat two points, x 0 and x L and we intend to computethe temperature in 0; L. The governing differentialequation for the temperature, h, is

d2hdx2 Pe

L

dhdx

in 0; L

with the boundary conditions;

h hL at x 0

h hR at x L

where the Peclet number is de®ned as Pe vL=a (a isthe thermal diffusivity of the ¯uid). The exact solution tothis problem is given by

hÿ hL

hR ÿ hL

exp PeL x

ÿ 1

expPe ÿ 1:

With increase in the Peclet number, the solution curveshows a strong boundary layer at x L. The solutionusing a simple Galerkin ®nite element scheme leads tosevere numerical dif®culties and a variety of upwind-typeprocedures have been proposed to solve the problem (seeBathe, 1996). In this work we apply the MFS and simplyuse as our local approximation spaces

Vh;pI spanf1; expPe x=Lg :

7.2.2Numerical resultsFigure 10 shows the numerical solutions obtained usingthe MFS, with quite arbitrarily spaced nodes, plotted onthe analytical solution curves for Pe 1; 10; 20 and 50.Due to the solution space chosen we expect a very accurateresponse with no ``wiggles'', and this is the case, seeFig. 10. It seems that the method of ®nite spheres hassigni®cant potential for the development of ¯uid me-chanics solution schemes to solve two- and three-dimen-sional ¯ow problems.

7.3The MFS in R2: the Poisson equationon the bi-unit square

7.3.1FormulationWe consider a problem in a single ®eld variable de®ned onR2. We seek a function ux; y satisfying the Poissonequation

u(x)

Ω3

h30h20 h40

32 4

h31

AnalyticalMFS

3.5

3.0

2.5

2.0

1.5

1.00 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

3.5

3.0

2.5

2.0

1.5

1.0

u(x)

x x

Analytical

MFS

a

b c

Fig. 9a±c. A bar of unit length with dis-tributed loading. In (a) a part of the bar isshown with 3 nodes. The Shepard func-tions hI0 (I= 2,3,4) are plotted at eachnode. At node 3, a higher order shapefunction h31= ((x-x3)/r3)h30 is alsoplotted. The sphere at each node I (WI)reduces to a line segment in one-dimen-sion. In (b) and (c) the displacement ®eldu(x) is plotted as a function of the dis-tance along the bar corresponding to theboundary conditions and loading givenin section 7.1.1. The numerical result in(b) corresponds to a regular distributionof 6 nodes on the bar while that in (c)corresponds to an arbitrary distributionof 5 nodes

338

o2u

ox2 o2u

oy2 f x; y 0 in X ÿ1; 1 ÿ1; 1

subject to the boundary conditions (see Fig. 11)

u usx; y on Ca

ou

oy 0 on Cb

ou

ox ÿf sx; y on Cc

ou

oy 0 on Cd :

The functions f x; y; f sx; y and usx; y are chosen suchthat the analytical solution ux; y is

ux; y 7x x7 cospy :The discretized equation corresponding to the mth degreeof freedom of the Ith node is given by Eq. (6) where theintegrals are

KImJn Z

XI\X

ohIm

ox

ohJn

ox ohIm

oy

ohJn

oy

dX

fIm Z

XI\Xf x; yhIm dX

f Im

0 for an ``interiorsphere''

0 for sphereson Cb and CdR

CcIf shIm dX for a sphere on CcPN

J1

Pn2IKUImJnaJnÿ fUIm for a sphere on Ca

8>>>>>>>><>>>>>>>>:

where

KUImJn Z

CaI

ooxhImhJndC

fUIm Z

CaI

us ohIm

oxdC

and Ca [I2NaCaI and Cc [I2NcCcI ;Na and Nc beingthe index set of nodes whose spheres have nonzero in-tercepts with the boundaries Ca and Cc respectively.

7.3.2Numerical resultsFigure 11 shows the discretization of the domain using aregular arrangement of 36 nodes. In Fig. 12 we present theMFS shape functions at an interior node. The nonpoly-nomial nature of the shape functions is quite evident. InFig. 13a we present the MFS solution ux superposed onthe analytical one along two lines y 0 and y 1. InFig. 13b the solution uy (as computed by the MFS) isshown as a function of the spatial coordinate y along theboundary Ca (i.e. x 1 together with the analyticalsolution. Note that the essential boundary conditionprescribed along this boundary is exactly satis®ed at thenodes but only approximately (in a weak sense) in-between the nodes.

7.4Linear elasticity problems in R2

7.4.1FormulationIn this section we derive the weak form and the discretizedequations for a linear elastic continuum in 2D. The system

Fig. 11. A regular arrangement of 36 nodes is shown on thedomain on which the Poisson problem is de®ned. Some of theinterior disks as well as some type I boundary disks are alsoplotted. Some selected integration points are shown for one of theinterior disks as crosshatched circles Ä (the actual number ofintegration points used in this example was 144 per interior disk)

Fig. 10. Results of the high Peclet number ¯ow problem. Thenormalized temperature distribution is plotted against normal-ized distance along the ¯ow direction for four different Pecletnumbers (Pe = 1, 10, 20 and 50). Continuous lines correspond tothe analytical solution. The solution obtained using the method of®nite spheres is plotted corresponding to arbitrary distributionsof nodes along the domain

339

of governing differential equations and the boundaryconditions can be written as:

Equilibrium equations:

oT s fB 0 in X 9

Strain±displacement relationships:

ou in X 10Linear elastic constitutive relationship:

s C in X 11Boundary conditions:

Ns f s on Cf 12u us on Cu 13In the Eqs. (9) to (13), u, and s are the displacement,stress and strain vectors, C is the elasticity matrix, f s is theprescribed traction vector on the Neumann boundaryCf ; us is the vector of prescribed displacements on theDirichlet boundary Cu (note that the domain boundaryC Cf [ Cu; fB is the body force vector (including inertiaterms), o is a linear gradient operator and N is the matrixof direction cosine components of a unit normal to thedomain boundary (positive outwards). In R2 these vectorsand matrices are written as:

u ux; y vx; yT 14 exx eyy cxyT 15s sxx syy sxyT 16f s f s

xx; y f sy x; yT

us usx; y vsx; yT

o o=ox 0

0 o=oyo=oy o=ox

24 35 17

N nx 0 ny

0 ny nx

18

C c11 c12 0c12 c11 00 0 c33

24 35where

c11 E

1ÿ m2; c12 Em

1ÿ m2and c33 E

21 mfor plane stress conditions

c11 E1ÿ m1 m1ÿ 2m ; c12 Em

1 m1ÿ 2m and

c33 E

21 m for plane strain conditions

E and m being the Young's modulus and Poisson's ratio ofthe material, respectively.

We have the following approximation for the displace-ment ®eld

ux; y 'XN

J1

Xn2I

HJnx; yaJn Hx; yU ; 19

where

U a10 a11 a12 aJn Tis the vector of nodal unknowns (not nodal displacementsunless the Kronecker delta property is satis®ed by theshape functions), and

aJn u Jn v Jn

Fig. 12a±c. Three shape functions (hI0, hI1 and hI2) at an interiornode are shown. hI0 is the Shepard function at the node, while hI1

= ((x-xI)/rI) hI0 and hI2=((y-yI)/rI) hI0

340

is the vector of nodal unknowns at node J correspondingto the nth degree of freedom (uJn and v Jn are the nodalvariables for the x and y direction displacements at node Jcorresponding to the nth degree of freedom). The nodalshape function matrix corresponding to the nth degree offreedom is

HJnx; y hJnx; y 00 hJnx; y

: 20

Hence, the discretized versions of Eqs. (10) and (11) are

x; y 'XN

J1

Xn2I

BJnx; yaJn Bx; yU 21

and

sx; y 'XN

J1

Xn2I

CBJnx; yaJn CBx; yU ; 22

where the strain±displacement matrix Bx; y in Eq. (21) ispartitioned as

Bx; y B10x; y B11x; y BJnx; y where

BJnx; y oHJnx; y ohJn=ox 0

0 ohJn=oyohJn=oy ohJn=ox

24 35 :

23At each node I, we may write the Bubnov±Galerkin weakform corresponding to the mth degree of freedom as (fromEq. (9))Z

XI\XHImoT

s dXZ

XI\XHImfB dX 0 : 24

In 2D these are two equations corresponding to the twocoordinate directions. Applying Green's theorem andEqs. (22) and (23) we obtain the discretized system ofalgebraic equationsXN

J1

Xn2I

KImJnaJn f Im f Im 25

which is the vector form of Eq. (6). In Eq. (25), the variousmatrices and vectors are as follows;

KImJn Z

XI\XBT

ImCBJn dX 26

f Im Z

XI\XHImfB dX 27

f Im Z

CI

HImNs dC 28

It is only this last Eq. (28) that is different for the differenttypes of spheres (disks) that we encounter.

If I is a node associated with an ``internal sphere'', then

f Im 0

from compact support.If I is a node with an annular integration domain (see

Sect. 5), with internal boundary CIi and external boundaryCIo, then

f Im I

CIo

HImNs dCI

CIi

HImNs dC : 29If I is a node on the Neumann boundary, then fromEq. (12),

f Im Z

CfI

HImf s dC 30

where Cf [I2NfCfI

, Nf being the index set of nodeswith spheres having nonzero intercepts on Cf .

If I is a node on the Dirichlet boundary, then

f Im XN

J1

Xn2I

KUImJnaJn ÿ fUIm 31

where

KUImJn Z

CuI

HImNCBJn dCZ

CuI

BTImCNTHJn dC

32and

fUIm Z

CuI

BTImCNTus dC 33

where Cu [I2NuCuI

, Nu being the index set of nodeswith spheres having nonzero intercepts on Cu. Note thatthe stiffness matrix KU is symmetric.

Analytical

MFS

y=1 on Γb

y x

u(x)

y=0

-10 0

-8

0

1

10

-2

-1

-4

-6

8

6

4

2 u(y)

Analytical

MFS between nodes

10

-10

0

-1 0 1

-8

-6

-4

-2

8

6

4

2

MFS at nodes

a b

Fig. 13a, b. Results of the Poissonproblem. In (a) the ®eld variable u(x) isplotted as a function of x for two valuesof y (= 0, 1). In (b) the Dirichletboundary Ca is considered. The MFSsolution is plotted as a function of ywith the analytical solution 341

It is interesting to note that this formulation gives riseto the same system matrices as obtained when the applieddisplacements are introduced into a variational formula-tion of the problem via Lagrange multipliers and theLagrange multipliers are replaced by their ``physicalsigni®cances'' as described by Washizu (1975). This for-mulation has also been adopted by Lu et al. (1994) in their

implementation of essential boundary conditions in thecontext of the element free Galerkin methods.

Another point to note is that we may incorporate theDirichlet conditions by the special arrangement of nodeson the boundary as discussed in Sect. 4.2.

Patch tests were performed on a bi-unit square (both inplane stress and in plane strain conditions) using theabove formulation and they were passed with as few asfour nodes placed at the corners.

7.4.2Numerical results

Thick walled pressure vessel in plane stress We considera thick-walled pressure vessel of external radius Ro 10and internal radius Ri 5, subjected to uniform internalpressure pi and external pressure po. The material prop-erties of the cylinder wall are chosen as E 100 andm 0:3. As shown in Fig. 14, one quadrant of the cylinderis discretized. The nodes for all the disks are superimposedat the origin. All nodes are associated with annular inte-gration domains, and the integration annuli have the sameinner radius (Ri) but different outer radii. To be able toincorporate the Neumann boundary conditions at Ro,some annuli are chosen to have outer radii greater than Ro

(see Fig. 14). Figures 15 and 16 present results of twonumerical experiments. In the ®rst one (Fig. 15) po pi 1:0: In the second experiment (Fig. 16) pi 10 andpo 0. The computed radial displacement ®eld is plottedagainst the analytical solution in Fig. 16a. Figure 16bshows the radial (rrr) and hoop (rhh) stresses normalizedwith the internal pressure (pi). In this solution 12 nodeswere used.

Fig. 14. A quadrant of the thick-walled pressure vessel (in planestress). All the nodes are placed at the origin of the coordinatesystem. The integration domain corresponding to each node is anannular sector of inner radius Ri

Fig. 15a±d. Results of the pressurevessel problem for pi=po=1.0. Theradial displacement (ur) , the nor-malized radial stress (srr), hoopstress (shh) and shear stress (srh)are plotted along a radius of thecylinder in (a), (b), (c) and (d)respectively. Continuous linesindicate analytical solution whereasthe MFS solution is plotted withasterisks

342

A cantilever plate in plane strain with uniformlydistributed loading In this problem we consider a canti-lever plate in plane strain conditions as shown in Fig. 17a.The material properties of the plate are chosen as E 100and m 0:3. A uniformly distributed load of magnitudew 1:0 per unit length is applied as shown. Figure 17bshows the convergence in strain energy when a h-typeuniform re®nement is performed. In this analysis the localbasis of each node has been chosen as a complete poly-nomial of second degree. The strain energy of the limit

solution of the system is obtained by solving the sameproblem using a 50 50 mesh of 9-noded ®nite elements.An order of convergence of 4.078 is observed whichcompares very well with the theoretical value of 4. Thismeasured rate of convergence is much better than the oneobtained in the classical ®nite element analysis when un-graded meshes are used, and might indicate a robustnessof the MFS.

8Concluding remarksThe objective of this paper was to present some of ourlatest developments in meshless methods. We reviewedseveral existing meshless techniques and concluded thatnone of the methods is effective for general applications.However, we found the meshless local Petrov±Galerkin(MLPG) method to be conceptually the most attractive. Weare therefore using this concept in our research to obtainan effective meshless solution procedure.

In this paper we propose a specialized form of theMLPG approach ± the method of ®nite spheres (MFS) ±which appears to have many valuable attributes. Themethod is truly meshless and the boundary conditions areincorporated relatively easily, in particular when a specialarrangement of nodes on the boundary is used. Moreover,a new technique for modeling doubly-connected domainsis presented in this paper using the concept of n-dimen-sional spherical shells as integration domains. The solu-tions of various example problems are given todemonstrate the MFS.

An important feature of the method is the numericalintegration scheme used. We employ Gaussian productrules for integration on two-dimensional annuli andannular sectors. A generalization would be used in threedimensions. The number of integration points employedis, however, rather large compared to those used in theconventional ®nite element method. This is attributed tothe fact that in the conventional ®nite element method thefunctions to be integrated are polynomials (or mappedpolynomials) but in the MFS complicated rational (non-polynomial) functions have to be integrated.

Considerable improvements in the choice of functionsand the numerical integration schemes must be achievedbefore the method would become, in ef®ciency, compet-itive with the existing ®nite element procedures. Such

5 6 7 8 9 100.65

0.7

0.75

0.8

0.85

0.9

0.95

1

r4 6 8 10

-1

-0 .5

0

0.5

1

1.5

2

r

ur

ip

i

rr

p

a b

τ

τ

θθ

Fig. 16. (a) Radial displacement ®eld ur

and (b) normalized radial (srr andhoop (shh) stresses in the cylinder wallcorresponding to pi=10 and p0=0. Con-tinuous lines indicate analytical solutionwhereas the MFS solution is indicated byasterisks (), triangles D, or squares (h)

Fig. 17. (a) Cantilever plate (L=2.0) in plane strain. Uniformlydistributed load of magnitude w=1.0 per unit length is applied.E=100, v=0.3. In (b) the convergence of the strain energy (Eh)with decrease in radius of support (h) is shown. The strain energyof the limit solution, E, is obtained by solving the same problemusing a 50 ´ 50 mesh of 9-noded ®nite elements

343

ef®ciency improvements should be possible. In addition,of course, tests and mathematical analyses of the methodshould be performed. While these research tasks stillrequire signi®cant effort, we do believe that the MFS hasan excellent potential for applications in solid and ¯uidmechanics.

Appendix ACubature on annular sectors in R2

In this appendix we state without proof a product cubaturerule for the integralZZ

Xf x; ydx dy:

Xi

Xj

Dijf xi; yj 34

with an accuracy of k. The region under considerationis the annular sector with inner radius Ri and outerradius Ro and angular span of ÿho; ho (see Fig. A1).The dot over the equality signi®es that the relationshipis a strict equality if the function f x; y is a polynomialof order at most k in x and y, otherwise it is anapproximation.

Cubature Rule If it is required that the rule in Eq. 34 haveaccuracy k 4m 3, m 0; 1; 2; . . . ; in x r cos h andy r sin h, and if it is required to have a minimumnumber of evaluation points which are taken at theintersection of concentric arcs (radius rj) with rays ema-nating from the origin (angle hi), then it is both necessaryand suf®cient for the existence of a unique set of weightsDij 2 R that the following two conditions be satis®ed:

1. Let za be the k 1=2 zeros of the polynomial ppzof degree p k 1=2 which is orthogonal to all poly-nomials of inferior degree on ÿ1; 1 relative to the weight

wz 1 zÿ12

3 cos ho

1ÿ cos hoÿ z

ÿ12

:

Let angular positions ua be de®ned by the followingrelationship

cos ua 1 cos ho

2

ÿ 1ÿ cos ho

2

za

a 1; 2; . . . ; k 1=2 :

Then the k 1 angular positions hi of the evaluationpoints are given by

hi ui i 1; 2; . . . ; k 1=2ÿuiÿk1=2 i k 3=2; . . . ; k 1

2. The radial positions of the evaluation points rj are the

positive square roots of the m 1 zeros of Pm1r2, theLegendre polynomial in r2 of degree m 1, orthogonalizedon R2

i ;R2o.

The (unique) weights Dij are equal to AiBj; where

Ai R 1ÿ1 wnlin2 dn i 1; 2; . . . ; k 1=2R 1ÿ1 wnliÿk1=2n2 dn i k 3=2; . . . ; k 1

(

and

Bj 1

2P0m1r2j Z R2

o

R2i

Pm1r2r2 ÿ r2

j

dr2 j 1; 2; . . . ;m 1 :

The functions liz being given as

liz ppzzÿ zip0pzi :

An important corollary of this rule is the case whenho p, i.e. when we have internal disks or annuli. Thecomplicated polynomial ppz in the above rule reduces tothe well known Chebyshev polynomial of the ®rst kind andthe quantities hi and Ai are known in closed form. We statethe corollary formally:

Corollary A.1 For the special case of ho p, the abovetheorem yields the following cubature ruleZZ

X

f x; ydx dy:X4m1

i1

Xm1

j1

Cjf rj cos hi; rj sin hi ;

where

1. hi ip=2m 1 ;2. Cj 1

4m1P0m1r2jR R2

o

R2i

Pm1r2r2ÿr2

j dr2 ;

3. Pm1r2 is the Legendre polynomial in r2 of degreem 1, orthogonalized on R2

i ;R2o:

4. Pm1r2j 0:

The case ho p has been previously derived by Peirce(1957a).

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