ANEW MESHLESS METHOD TO SOLVEBOUNDARY-VALUE PROBLEMSC. Armando Dua.rtel

.J. T. Odc1l2

TICAM - T('xas Instit.ute for Comput.ational and Applied Mathema.t.icsTh<' Ullivf'rsity of Tpxas at A Ilt;t;illTaylor Ha.1I:2.400Aust.in. Texas. 7871:2, U.S.A.

ABSTRACT

This paper presents a new fa.mily of mesh less methods for the solution of boundary-valueproblems. In the h-p cloud method, the solution space is composed of radial basis functions asso-ciated with a set of nodes a.rbitrarily placed in th.~ domain. The paper describes the constructionof the h-p cloud functions using a signed partition of unity and how h, p or h-p refinements canbe implemented without a mesh. The h-p cloud functions and the Galerkin met.hod are used tosolve a. two dimensional boundary-value problem. Some properties of the h-p cloud functions arcalso discussed.

1 INTRODUCTION

In most large-scale numerical simulations of physical phenomena, a large percentage ofthe overall computat.ional effort is expended on technical details connected with meshing. Thesedetails include, in particular, grid generation, mesh adaptation to domain geometry, element orcell connectivity, grid motion, and separation of mesh cells to model fracture, fragmentation, freesurfaces, et.c. Moreover, ill most computer-aided design work, the generation of an appropriatemesh constitutes, by far, the costliest portion of the computer-aided analysis of products andprocesses. These are among the reasons that interest in so-called meshless methods has grownrapidly in recent times. In these methods, t.here may be no fixed connectivities among the nodes,unlike the finite element or finite difference methods. This feature has significant implications inmodeling some physical phenomena that arc characterized by a.cont.inuous cha.nge in the geometryof the domain under analysis.

Duarte (Duart.e, 1995) has prepared a comprehensive review of t.he mesh less met.hods

1 Research Assistant, TICAM. e-mail: armando@ticam.utexas.edu2 Professor, Director, TICAM. e-mail: oden@ticam.utexas.edu

found in the literat.ure. The connection between the mesh less met.hods and t.he correspondingunderlying approximation technique were also investigated. The conclusion of t.hat st.udy is t.hat,from the point. of view of a.ccuracy and efficiency, in spit.e of t.he variety of the meshless methodsfound in the literature, all the reviewed met.hods have serious limitations t.hat, ill most. situat.ions,can negate some of t.heir adva.nt.ages over more reliable methods such as h-p finite elenlf'nt met.hods.

This paper presents a new family of rneshless methods for the solution of boundary-valueproblems. The h-p cloud method is applicable to arbitrary domains and employs only a scatteredset of nodes to build approximate solut.ions t.o boundary-value problems. The method uses radialbasiR functions of varying size of supports and with polynomial reproducing properties of a.rbitraryorder. The.first. numerical experiment.s with this technique show very promising result.s.

The paper is organized as follows: following this introduction, we discuss the constructionof the h-p cloud space using a. signed part.ition of unity. A family of functions F~'} is defined andsome of its propert.ies a.re investigat.ed. Some illustrative examples of these functions ill t.wodimensions a.re also present.ed. Section 3 discusses the implement.ation of 11., p and h.-p refinementsill t.he h-p cloud context.. The solution of a model problem is t.he subject of Section 4. Finally, inSection 5, we present. conclusions, discuss the limit.ations of t.he met.hod and directions for furt.herresearch.

2 CONSTRUCTION OF THE H-P CLOUD SPACE

In this section, we describe t.he construction of the h-p cloud functions a.nd discllss some oft.heir properties. One key idea used is t.hat.of a signed partition of unity. This class of functions canbe used to construct. linearly independent functions that have many properties in common wit.hthe global ba..,is functions used in the finite element. method like loco.' compactness and polynomia.lreproducing properties. But, unlike the finite element basis functions, the funct.ions used in theh-p cloud met.hod can be as smooth as desired, even COO(D) functions. And. most remarkably,there is no need to partition the domain into smaller subdomains, e.g. finite elements, to const.ruct.t.he h-p cloud functions. All that is needed is an arbitrarily placed set of nodes.

2.1 THE SIGNED PARTITION OF UNITY

Let n be an open bounded domain in R'\ n = 1,2 or 3 and QN denote an arbitrarilychosen set of N points :cO' E n denoted by nodes

We associate with t.he set. QN a finite open covering of n in the following way: let Wa, 0' = 1, ... , N,denote a set. of balls centered at. :cO' and with radii hQ chosen in such a way that TN := {wa }~=lconstitut.es an open covering of n

- Nn CUa=lwa

A cla.'is of functions SN := {<.pO' }~=1 is called a signed pa1'lilion of unity subordinate to theopen covering TN if it. possesses t.he following properties:

1) <.pc< E C~(WCt), 1 ~ n ~ N

)N2 L:a=,<.po(:z~)=l, V;r~En

Figure 1: Example of an open covering ill 2D.

The signed partit.ion of unit.y used in the h-p cloud met.hod has also the following propert.y:givcn any set IP = {PI, P2, 00" Pm}, Pj : IRIl ---+ JR, of m linearly independent functions containingt.he unit.y funct.ion, the sigurd part.it.ion of unit)' can be constructed such t.hat

N

Pj(x) = L Pj(xa)lpo(x) , x E 00=1

(1 )

The following approach is used in the h-p cloud method to build the signed part.it.ion ofunity SN:

LeL W", : HC ---+ IR denote a weight.ing ftlnclion that. helongs to the space Cgo(wo) with thefollowing properties:

Vy En

where the functions ',Fa belong to the space Cgo(Bh.J and BhQ is a ball of radius ha centered at.the origin

Bha = {x E JRn : IIxIIJR" < ha:}Next we introduce a family of functionals defined over continuous functions defined on n by

N

(J,g)y := I:Wa(y)f(xa)g(xa),0=1

(2)

Assumption 1 Given a set of m functions P = {Ph P2, 00" Pm}, Pi : n ---+ 1Il, Pi E CO(O) fo!,i = 1, ... , 1n, the wei,qhling functions W", defined above and t.he functions Pi are such that V x E 0i hel'c holds

til

I:ak{Pk, Pdx - 0 fad = 1, ... , m if and only if ak - 0 fork = 1, ... , m.k=1

•Neccssary and in some cases sufficient condit.ions for t.he sat.isfaction of assumption 1 have

been showlI in (Duarte amI Oden, ]995).We are now in a posit.ion t.o define t.he signed partition of unity used in the h-p cloud

method. The function !.pry associated t.o the ball Wry is defined by

(3)

where

p,. p. E JPI' J

• P( x) := {PI ( X), P2( X), ... , Pm (X)} T, 3 Pi s.f. Pi ( X) = 1,

• Bo(x) := Wo(x)P(xo)

It ca,n be shown that the definition of <per given in (:3) satisfies the definit.ion of a signedpartition of unit.y and also has t.he property given by (1). The proofs can be found in (Duart.e andOdell. 199.5).

2.2 THE FAM1LIES :F~P

The most. important step in t.he h-]J cloud method is t.he construction of the family offunctions :F~11 using the signed partition of unity SN defined in t.he previolls section. This class offunctions can be constructed at. a very low cost and has the import.ant property t.hat for a properchoice of the base family D)(x) we can insure that Pp C span{:F~P} whcre Pp denotes the spaceof polynomials of dcgree less or equal to p. In this section, we describe the construction of :F~Pa.nd sta.te some theorems concerning fundamental properties of these functions.

Let £p denote the set of ten?or product Legendre polynomia.ls Li.j,k in IR?,

o S i,j, k S p

Ot.her sets of complete polynomials can be lIsed as well; e.g., t.he smallest. set of complete poly-nomials IIp. In the following S~ := {<P~}~=l will denote a signed partition of unity t.hat is £k -reducible for the set QN; that. is, given any element Liik E £k t.he following holds V x E f!:

N

Lijk(X) = L Lijk(Xo)<p~(x)0'=1

The family of functions :F~P is defined by

{ {<p:(x)} u {<p~Ljjl(X)}: 1 S Q S N; 0 S i,j,l S p,

i or j or I > k; p 2: k }

(4 )

(5)

The idea behind the definition in (5) is to add, hierarchically, appropriate elements t.o theset S~ such that the resulting set can reproduce, as linear combinat.ions, polynomia.1s of degreep 2: A:. Because of property 2) of a signed partition of unity, t.hose element.s are precisely theproduct of the functions <p~ wit.h t.he element.s from t.he set. £1' t.hat a.re missing from the set. £k·

For consist.ent results, regardless of t.he scale of the probkm, til(' h-p cloud funct.ionsintroduced in (5) a.re implemented using h-a,Hinc maps given by:

I

e Ewwll<'l"<'

is a spherc of ra.dius one and

is the support. of the function <Pa. .The h-p cloud function c.poLiidx) is implemented by

c.paLiil(X) := c.p...(x) (liil 0 F~l(X))

where L;i,(e) is a. tensor product Legendre polynomial defined Oil [-I, l]n.The following t.heorems are proved in (Dua.rtc and Oden, 1995).

Theorem 1 Let Pi, i= 1, ... ,111. and WO" (\' = 1, ... , N be Ihe basis fu.nrlions and the weightin,qflmctions used to collsl1'llclthr signed pa7'filion ofunitySfv. Suppose Ihat Pi, i = 1, ... ,rn E CI(n)and Wer, 0' = 1, ... , lV E Cq(n). Then the h-p cloud fune/ions defin.ed in (5) belong t.o the spaceCmiu(l,q)( n).

2.3 PLOT OF THE lI-P CLOUD FUNCTIONS

In this section, some e1emcnt.s from the family :F~') are plotted for the two dimensionalcase. In general, there is no closed form expression for t.hcse functions. The set of functions lPused t.o build the signed part.it.ion of unity st wa.<;composed of polynomials of degree less 01'

equal to I.: including the unity function PI = 1. The weighting functions Wcr were implementedusing bi-splincs (dcBoor, 1978). These functions arc piecewise polynomials and ca.n be built withany degree of regularity. For example, quart.ic hi-splines are C3(n) functions. In t.he plots showedbelow quartic bi-splines were used. An uniform node arrangement with five nodes in ea.ch directionwas used to build the partition of unity. The domain n was the square [-1,1] x [-1,1].

Figure 2(a) shows the function c.p~=Ofrom the family J=t=o,p associated to a node at theorigin. Figures 2(b) and 2( c) show the functions yc.p~=o and xyc.pj;° from the families .rt=0.P~l and:Fk=o p>2 t' IN '- respec .lve y.

3 THE H, P AND H-P VERSIONS

Oue rema.rkable feature of the h-p cloud method, besides that. it does not need a mesh tobuild the spa.ce of approximat.ing functions, is that. the implementation of h, p or h-p refinementis much easier tha.n in the finite element. method (FEM). The h version of the FEM ca.n beimplemented in several ways. One of the most successful approaches is based on the use ofconst.rained nodes (Demkowicz et a.I., 1989). This technique guarantees that the h refinement atsome region of t.he domain will not propagate throughout the entire domain only to guarantee thecont.inuity of the solut.ion (Dernkowiczet aI., 1989). In the h-7J cloud method the usc of constrainednodes is completely unnecessary. The implementat.ion of t.he II. refinement. is achieved simply byinsert.ing nodes in the regiolls of int.('rest.. There is no l1C'cd to add ext.ra. lIodes or t.o cOllst.raint.some of them only to make t.he solution cont.inuous. Figure :3(a) shows some of t.he balls used tobuild a.n ODell coveril1!!' for all L shaned domain. TIl<' hlack dot.s renresenf. the llod(~s a.nd t.he !!rav

(a) 2-D function cpj..=o from the family F~;=o,r. (b) 2-D function ycpj..=o from the family F~=O'J'~l.

(c) 2-D function xycp~=o from the family F~O.r?2.

Figure 2: Exampies of h-p cloud basis functions.

level of the shaded areas indicates the polynomial order associa.ted to each ball (in this exampleall balls have the same polynomial order). Figure 3(b) shows the h refinement of the previousdiscretization. New nodes were arbitrarily added and the polynomial order associated to each ballwas kept. fixed.

The p version of the method can have more than one variant. One, for example, can fix thesize ha of the balls and increase the pa.rameter p keeping k fixed. Anot.her possibility would be toincrease simultaneously k and p. Nonetheless, mathematical analysis and numerica.l experimentsperformed by (Duarte and Oden, 1995) ha.ve shown that the first variant. is preferable (see (Duarteand Oden, 1995) for details). Figure 3(c) shows the non uniform p enrichment of the balls shownin Figure 3( a). The different. gray levels indicate that each ball can have a di fferent. polynomialorder associated to it rega.rdless of t.he polynomia.l order associa.t.ed to lIcighboring balls.

In t.he h-p version of the method the number of nodes and the polynomial order associatedto each ball are simultancously increa,<;ed.

(a) Some of the balls used to build an open coveringfor an L shaped domain.

(b) h refinement of the discretization shown in theFigure (a).

(c) Non uniform p enrichment of the balls shown inFigure (a).

Figure :3: The h and p versions of the cloud method.

4 H-P CLOUD SOLUTION OF A BOUNDARY-VALUE PROB-LEM

In this section, we use the techniques described in Section 2 to construct appropriatefinite diincnsional subspaces of functions used in the Galerkin method. The resulting approach isdenoted by the h-p cloud method. We focus on the solution of a simple two dimensional bounda.ry-value problem, namely, the a.nalysis of a bar with equilateral triangular cross section subjected totorsion. The stress distribution 011 the cross section of the ba.r can be computed from the PrandtI'storsion function ¢J(x,y) (Boresi and Lynn, 1974) a.nd is giveIl by

T:rz = ¢J,y (6)

The other stress components a.re ident.ically zero.The Prandtl's t.orsion function can be found solving t.he following boundary-valuE' problem:

y

where G is the shcar modulus and 0is the angle of twist per unit. length oft.he bar. The domain D is illust.rated illFigure 4.

Find ¢>(.r,y) such that

2GO- 0

in non an

x

Figure 4: Domain, boundary conditionsand discretizat.ion used.

The solution of t.his problem is given by (Boresi and Lynn, 1974)

Gf) ( r.l '2(/) ( r.l 20) ( a. )<P =:-- :1: - v:3y - -;- J' + v 3.'1 - -;- ;1: + -20 .3 .J 3

(7)

The values GO = ]/2 and a = 12 were used in the calcula.t.iolls.Although t.his is a. simple boundary-value problem, it can be used t.o illustrate numerically

thc results of Theorem 1. The set. IP = {I}, that is J.~ = 0, and weighting functions Wa built from C3

splines are llsed to constfllct the signed partition of unity as described in Section 2.1. Therefore,from Theorem 1, the h-p cloud solution belongs to the space C3(D) and, consequently, all theapproximate st.resses are continuous throughout the entire domain.

Figure 4 shows the portion of the domain discrctized and the node arrangement used-one node at each corner of t.he domain. The boundary conditions applied at each port.ion of theboundary is also indicated in Figure 4. The Dirichlet boundary conditions were imposed usingLagrange multipliers.

The problem was solved using the family of cloud functions :F~=O,Jl=2 defined in previoussect.iolls. The h-p cloud solution is showed iJl Figure 5. The total number of degrees of freedomwas 31 (24 for the stiffness matrix and 7 for the Lagrange multipliers). The Loo error of the h-pcloud solution is 4.43 x 10-2.

Figure 5 shows the point.wise error Txz ..:..T::, where T:: denotes the stress componentcomput.ed using (6) and the h-p cloud solution. It can be observed that T~l:is continuous over theentire domain (it is indeed a C2 function). It should be mentioned that max IT:z:z - T;:I = 0.104and max ITyZ - T;:I = 0.120 which are regarded as very good results for such a coarse arrangementof nodes.

5 CONCLUSIONS AND DISCUSSION

A new approa.ch to solve boundary-value problems is discussed. The h-p cloud methodhas the following features:

The domain does not need to bc>part.itioned into smaller subdornajns to build the approx-imat.ing functions. Nonetheless, t.he domain may nced to be partit.ioned somehow or covered bya cell structure (Belyt.schko et a\., 1994) t.o perforlll numerical evaluatioJl of fuuct.iona.ls. In thepresent implementation of the h-p cloud method, we use a mesl) of quadrilateral cells tha.t exa.ctly

fit.s the domain t.o perform the numerical int.egration. The only requirements on the mesh is tha.tthe cells do not overlap and t.hat their unioll exactly matches t.he domain under analysis. Indeed,numerical integration for mesh less methods is an a.rea of a.ctive research .

V3

~

4.992034.656284.320S24 3.984763.649

3.313262.917492.641732.30698

•~ 1.97022"'I 1.63446

1.29870.9629470.6271890.291432

Figure 4: h-p cloud soillt.ion.

F· r::. p' . hp19ure a. omtwlse error Txz - Txz .

The meshless character of the h-p cloud method makes it very attractive to solve, forexample, large deformation problems, crack propa.gation problems and problems where fragmen-tation occurs.

The fluxes computed from the h-p cloud solution may be very smooth functions andtherefore the post-processing of these quantities and other derivatives may not be required. Thehigh regularity of the h-p cloud functions also make t.hem attractive candidates to solve thin plateand shell problems.

In t.he h-p cloud method, the nodes can be placed quite arbit.rarily in the domain. Thispropert.y Ca.1ldecrease substantially the cost of solving numerically many industrial problems. Alsothere a.re no fixed conllect.ivit.ies among t.he nodes and therdore problems like mesh entanglementin t.he analysis of large deformat.ion problems do not. exist.

To our knowledge, the h-p cloud Inethod is the only mesh less method where t.he support.of t.he shape functions Ca.1I be made of any size and al lhf :w.m.etime the shape functions call be

constructed in such it. way that. polynomials of a.ny degree can be represent.ed as linear combinationof t.hese functions. That. is, we have an h-p mct.hod in t.he same spirit as in t.he finite element.method. Both h-p cloud and finit.e element method are very similar wit.h regard to this point ofview but the practical implementation of h-p adaptivit.y in the h-p cloud method is significantlysimpler than in h-p finite element. methods.

Much addit.ional work remains to be done before the h-p cloud method can be used tosolve large scale industrial problems. The many advantages of the met.hod also come with somcchallenging problems. The required use of Lagrange multipliers to impose Dirichlet: boundaryconditions increases the solution cost of the resulting system of equations since the matrix will nomore be positive definite as in the finite element method. Another major difficulty is' the numerica.lintegration of the h-p cloud functions since thcse functions a.re not polynomials. Nonetheless thesedifficulties do no seem to be unsurmontablc and t.hc potent.ial benefits of t.he method justify furtherrcsearch.

ACKNOWLEDGMENT: The 8UPP01't of the CNPq of Brazil and the NSF of the USA undergrant lNT .9402416 iB gratefully achwwledged. Author C. Armando Duarte was supported bya CNPq Graduate Fellowship and he and .J. T. Oden through support of a project. at TICAM8pol/.80/'cd by t.he A rmy Research Olficr undcr contract DA A L03-.92-G-025,').

REFERENCES

Bclytschko, 1'., Lu, Y. Y., and Cu, L. (1.994). Element-Fcc galel'kin methods. Int.ernationa.1.Journal for Numerical Met.hods in Engineering, 37:229-2.56.

B01'esi, A. P. and Lynn, P. P. (1974). Elasticity in Engencering mechanics. P1'ellt.ice-Hall, NewJersew.

de 80m", C. (1978). A Practical Guide lo Splines. Sp1'inge1"- \le-dag, New York.

Demkowicz, L., Oden, J. T., Rachowicz, W., and IIm'dy, O. (1989). 1'owa1'd a universal h-p adap-tive finite element strategy, part 1. constrained approximation and data sl1"ucture. ComputerMethods in Applied Mechanics and Engineering, 77:79-112.

Duarte, C. A. M. (1995). A 1'eview of some mesh/ess methods to solve partial differential equations.Technical Report 95-06, TICAM, The University of Texas at Austin.

Dttarte, C. A. M. and Oden, J. T. (1995). Hp douds-a mesh/ess method to solve boundary-valueproblems. Technical Report 95-05, TICAAf, The University of Texas at Austin.