18
PERGAMON Computers and Structures 77 (2000) 215-232 Computers & Structures www.e!sevier.com/loca te/col1lpst ruc Generalized finite element nlethods for three-dilnensional structural mechanics pro blen1s C.A. Duarte a "*, I. Babuska h, J.T. Oden "COMCO, 111e.. 7800 Shoal Creek Bird. .s·l/ill' 290E, Amlin, TX 78757, USA bTICAAI. The Universitr or Texas ill AI/Slill, SJJe 310. AUSlill. TX 787/2. USA Received 26 February 1999: accepted 23 August 1999 Abstract The present paper summarizes the generalized finite element method formulation and demonstrates some of its advantages over traditional IInite element methods to solve complex. three-dimensional (:'ID) structural mechanics problems. The structure of the stilrness matrix in the GFEM is compared to the corresponding FEM matrix. The perfonnance of the GFEM and FEM in the solution of a 3D elasticity problem is abo compared. The construction of p-orlhotropic approximations on tetrahedral meshes and the usc of a-priori knowledge about the solution of elasticity equations in three-dimensions are also presented. :'C) 2000 Elsevicr Science Ltd. All rights reserved. t. Introduction The analysis of complex three-dimensional (3D) structural components has become a common task in recent years in several manufacturing industries. How- ever. thc analysis of this class of problcms using tra- ditional finite element methods still poses several difficulties. It is a common practice to use automatic tetrahedral mesh generators to discrctize complex 3D structural components. This type of mesh generators can handle very complex geometries with a minimum of human intervention (as compared to. e.g .. the man- ual generation of a mesh of hexahedral elements). The main drawbacks of any automatic mesh gcnerators are: I. The need of an excessive numher of elements in order to keep the aspect ratios of the finite elements * Corresponding author. Tel.: + 1-512-467-0618: fax: + 1- 512-467-1382. E-mail address: [email protected] (CA. Duarte). within reasonable bounds. This is specially true when the component has transition zones from bulky to slender parts 2. Polynomial approximations. as used in traditional finite element methods, require the use of a large number of elements in order to capture stress con- centrations and singularities at corners and edges of the domain. Meshes that are insufficiently refined at these regions are often used in order to keep the number of degrees of freedom (dof) below a reason- able level or simply because the available mesh gen- erators can not create an appropriate mesh where needed without creating an exccssive numbcr of el- ements evcrywhere. This practice may lead to results of poor quality even at points in the domain distant from the singularities. duc to numerical pollution. 3. Another drawback of automatic mesh generators, especially when tetrahedral elements are Llsed. is that they preclude the use of p anisotropic approxi- mations, that is, approximations that have dilTerent polynomial orders associated with each direction. Problems where boundary layers occur, such as in 0045-7949/00;$ - see front matter ,<.) 2000 Elsevier Science Ltd. All rights reserved. PH: S0045-7949(99)0021 1-4

Generalized finite element nlethods for three …users.ices.utexas.edu/~oden/Dr._Oden_Reprints/1999-015.generalized...Generalized finite element nlethods for three-dilnensional structural

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PERGAMON Computers and Structures 77 (2000) 215-232

Computers& Structures

www.e!sevier.com/loca te/col1lpst ruc

Generalized finite element nlethods for three-dilnensionalstructural mechanics pro blen1sC.A. Duartea"*, I. Babuska h, J.T. Oden

"COMCO, 111e.. 7800 Shoal Creek Bird. .s·l/ill' 290E, Amlin, TX 78757, USAbTICAAI. The Universitr or Texas ill AI/Slill, SJJe 310. AUSlill. TX 787/2. USA

Received 26 February 1999: accepted 23 August 1999

Abstract

The present paper summarizes the generalized finite element method formulation and demonstrates some of itsadvantages over traditional IInite element methods to solve complex. three-dimensional (:'ID) structural mechanicsproblems. The structure of the stilrness matrix in the GFEM is compared to the corresponding FEM matrix. Theperfonnance of the GFEM and FEM in the solution of a 3D elasticity problem is abo compared. The constructionof p-orlhotropic approximations on tetrahedral meshes and the usc of a-priori knowledge about the solution ofelasticity equations in three-dimensions are also presented. :'C) 2000 Elsevicr Science Ltd. All rights reserved.

t. Introduction

The analysis of complex three-dimensional (3D)structural components has become a common task inrecent years in several manufacturing industries. How-ever. thc analysis of this class of problcms using tra-ditional finite element methods still poses severaldifficulties. It is a common practice to use automatictetrahedral mesh generators to discrctize complex 3Dstructural components. This type of mesh generatorscan handle very complex geometries with a minimumof human intervention (as compared to. e.g .. the man-ual generation of a mesh of hexahedral elements). Themain drawbacks of any automatic mesh gcneratorsare:

I. The need of an excessive numher of elements inorder to keep the aspect ratios of the finite elements

* Corresponding author. Tel.: + 1-512-467-0618: fax: + 1-512-467-1382.

E-mail address: [email protected] (CA. Duarte).

within reasonable bounds. This is specially truewhen the component has transition zones frombulky to slender parts

2. Polynomial approximations. as used in traditionalfinite element methods, require the use of a largenumber of elements in order to capture stress con-centrations and singularities at corners and edges ofthe domain. Meshes that are insufficiently refined atthese regions are often used in order to keep thenumber of degrees of freedom (dof) below a reason-able level or simply because the available mesh gen-erators can not create an appropriate mesh whereneeded without creating an exccssive numbcr of el-ements evcrywhere. This practice may lead to resultsof poor quality even at points in the domain distantfrom the singularities. duc to numerical pollution.

3. Another drawback of automatic mesh generators,especially when tetrahedral elements are Llsed. isthat they preclude the use of p anisotropic approxi-mations, that is, approximations that have dilTerentpolynomial orders associated with each direction.Problems where boundary layers occur, such as in

0045-7949/00;$ - see front matter ,<.) 2000 Elsevier Science Ltd. All rights reserved.PH: S0045-7949(99)0021 1-4

216 C.A. Dl/aNc eI al. : CampI/leI's alld SII'llC1l1H'S 77 (2000 J 215-232

the analysis of orthotropic materials or when one ofthe dimensions of the structural part is much smal-ler than the others, are examples in which p-ortho-tropic approximations may lead to t:onsiderablesavings in the number of dof needed to achieve<Kccptable accuracy.

The generalized finite element method (GFEM) wasproposed independently by Babuska and colleagues[2..\.21 J (under the names 'special finite elementmcthods'. 'gencralized linite element method' and'tinite element partition of unity method') and byDuarte and Oden [9-12,25] (under the names 'hpclouds' and 'c1oud-based lip tinite element method').Several of the so-called meshless methods proposed inret:ent years can also be viewed as special cases of thegeneralized tinite element method. Recent surveys onmesh less methods can be found in [6,8]. The key fea-ture of these methods is the use of a partition of unity(PU), which is a set of functions whose values sum tothe unity at e.lch point x in a domain Q. The analysisof the performance and computational cost of lipclouds. element free Galerkin [7J, dilruse element [22]and reproducing kernel particle [I8J methods can befound in [12]. It was found in that study that the inte-gration of the stiffness matrix in these methods can beconsiderably more expensive than in traditional liptinite element methods, depending on the choice of thepartition of unity, the Moving Least Squares PU [17],being among the most expensive.

The present paper summarizes the ideas behind theGFEM formulation and demonstrates through numeri-cal examples some of the advantages of GFEM overtraditional FEM for solving complex, 3D structuralmechanics problems. In Section 5, the structure of thestillness matrix in the GFEM is compared to the corre-sponding FEM matrix. In Section 6, we comparc the

performance of the GFEM and FEM in the solutionof a representative 3D clasticity problem. The use of p-orthotropic approximations on tetrahedral mcshes andthe use of a-priori knowledge about the solution of thcelasticity equations in threc-dimensions is also demon-strated in that section. Finally, in Section 7. majorconclusions of the study arc given.

2. Formulation of gcncnllized finite clement methods

In this section. \lie review the basic ideas behind theconstruction of generalized finite element approxi-mations in a one-dimensional (ID) setting using a 10linear finite element partition of unity. In Section 2.1.the GFE formulation in an /I-dimensional setling ispresented. In Sections 5 and 6, we use generalized tet-rahedral finite elements to discretize 3D problems. Fora more detailed discussion on the theoretical aspects ofG FE approximations, see Refs. [3.10,2 1,25J and thereferences therein.

Let 11 be a function detined on a domain Q C IR.Suppose that we build an open covering

vDc UflJ,

"=1

of Q consisting of N supports w, (often called douds)with centers at .\' ,,'Y. = 1, ... ,IV .

Let 11, be a local approximation of II that belongs toa local space I.,(w,) defined on the support Wa. It ispresumed that each space i'.,(w,), 'Y. = 1. ... ,N, can bechosen such that there exists a II, E I.,(w,) that can ap-proximate well 111,." in some sense. Fig. I illustrates thedefinitions given above. In this case, the supports OJ,

are opcn intervals with centers .\'".

..,

~ .\.'\\ ,UN

... ...... --"

,

Ua

,

l"

• IXu2

(JJu

• IXN• I •

(ON

Fig. I. Lo\:al approximations defined on lhe supports (I)".

CA. Duarte d 01. iCOlIIl'uters and S1I'l/clures 77 (2000) 215-232 217

The local approximations II~. 0: = I, ... ,N. have to

be somehow combined together to give a global ap-proximation IIhp of 11. This global approximation hasto be built sllch that the difrerence between IIhp and 11,

in a given norm, be bounded by the local errors 11-

11~. In partition of unity methods, this is accomplishedlIsing functions defined on the supports (!)~,

1'J. = I .... ,N, and having the following property

qJ, E C;;(w,), .1';::0, I <:;;o:sN

Lrp,(x) = I "Ix E Q,

The functions (P~are called a partitioll or IInity slibordi-111lteto the opell corering .'7 .... Examples of partitionsor unity are Lagrangian tinite elements, the various'reproducing' functions gencrated hy moving leastsquares methods and Shepard functions [11. I7].

In the case of finite element partitions of unity, thesupports (clouds) (J)x are simply the union of thc finiteelements sharing a vertex node 'Xx (see, for exampleRefs. [21,25]), In this case, the implementation of themethod is essentially the same as in standard finite el-emcnt codes, the main difrerence being the detinitionof the shape functions as cxplaincd below. This choiceof partition of unity avoids the problem of integrationassociated w'ith the use of moving least squaresmethods or Shepard partitions of unity used in severalmesh less methods. Here, the integrations are per-formed with the aid of the so called master elements,as in classical finite elemcnts. Thereforc. the GFEMcan use existing infrastructure and algorithms devcl-oped for the classical finitc element method.

Fig. 2 shows a one-dimensional finite element discre-\\'la\ion. The partition of unity functions qJx arc theusual global finitc element shape functions, the classi-cal 'hat-functions', associatcd with node X,. The sup-port (!)x is thus the union of the elements T'_I and T~.

Consider now the element Tx with nodes X ~ ami.\>+t as depicted in Fig. 2. Suppose that the followingshape fUIll;tions are used on this element

That is, the elemcnt T" has a total of six shape func-tions (three at each node) built from the product ofthe standard Lagrangian finite element shape functions(a partition of unity), and the local approximationsu~, 11,+1 that. by assumption, can approximate well thefunction u over the tinite element Tx. Of coursc, we canfurther generalize this idea by increasing the numberof functions tI, and IIx+l, resulting in a space Sx of stilllarger dimension. This approach is discussed in thenext section.

Thanks to the partition of unity property of thefinite elemcnt shape functions, we can easily show thatlinear combinations of thc shape functions detinet!above can reproduce the local approximations IIx, U,+I.

that is.

qJ,U~ + rp'Hl Ux = 1I~(qJ, + (P~+I)= II, (no sum on 1'J.)

In other words,

Ux• U,+I E span{S,I.

Thc basic idea in partitIOn of unity methods, and inparticular. in the GFE method, is to use a partition ofunity to paste together local approximation spaces.The shape functions are built such that they can repro-duce, through linear combinations. the local approxi-mations defincd on each cloud w,. The approximationproperties of such functions arc discusscd in the nextsection.

2.1. Generalizedfinite element shape jilllctions, the./illlii/y of junctiolls :?Ft.

In this section, we define generalized I1nite elemcntshape functions in an II-dimensional setting using thesame ideas outlined in the previolls section.

Let the functions qJ" 0: = \' ... .fll. denote a finite el-ement partition of unity subordinate to the open co v-

Fig. 2. One-dimensional finite element partition of unity.

218 C.A. Duarte et al.': COlllplllen' and Sll'lIclures 77 (2000) 215-232

ering .!Is = (OJ,I:=1 of a domain Q C ~". II = I. 2, 3.Here, N is the number of vertcx nodes in the finite el-ement mesh. The cloud (J), is the union of the finite el-ements sharing the vertex node Xx.

Let X,(w.) = span{L,., },E.f'(X) denote local spacesdefined on OJ., rx= I, ... ,N. where .F(:x). rx= I. ... ,N,are index sets and Li' denotes local approximationfunctions analogous with the functions 11, mentionedin the previous section. Possible choices for these func-tions are discussed below.

Suppose that the finite element shape functions, f/J"are linear functions and that

9'1'-1 (w.) c X.(wx):x = 1, .... N.

where 9'p-J denotes the space of polynomials of degreeless or equal to p - I. The generalized finite elementshape functions of degree p are defined by [21,25]

(I)

Note that there is considerable freedom in the choiceof the local spaces X,. The most obvious choice for abasis of x. is polynomial functions which can approxi-mate well smooth functions. In this case, the GFEM isessentially identical to the classical FEM. The im-plementation of hp adaptivity is, however, greatly sim-plified by the PU framework. Since each basisfunctions. (Li.JiE.I(.). a = I .... ,N, can have a differentpolynomial order for each a, we can have differentpolynomial orders associated with each vertex node ofthe finite element mesh [25]. The approximations canalso be non-isotropic (i.e., different polynomial ordersin different directions), regardless of the choice of finiteelement partition of unity (hexahedral, tetrahedral,etc.). Examples of p-orthotropic approximations builton a tetrahedral mesh are given in Section 6. The con-cept of edge and middle nodes, which is used in con-ventional p FEMs. is not needed in the framework ofGFEM. The implementation of h adaptivity is alsosimplified in PU methods since it needs to be doneonly on the partition of unity (linear finite elements inthis case). Therefore. the implemenLation of h adaptiv-ity for high-order approximations is the same as forlinear approximations (there is no need. for example.of using high-order constraints as done in traditionallip finite clement methods).

There are many situations in which the solution of aboundary value problem is not a smooth function. Inthese situations. the use of polynomials to build theapproximation space, as in the FEM. may be far fromoptimal and may lead to poor approximations of thesolution u unless carefully designed meshes are used.In the GFEM, we can use any a-priori knowledgeabout the solution to make better choices for the localspaces I.,. For example, in Section 6. we solve a

boundary value problem in which the solution pos-sesses point or lines of singularities at some parts ofthe domain. Then we can use the local spaces X, tobuild generalized finite element shape functions thatrepresent these singularities much more effectively thanpolynomial functions. The construction of these custo-mized shape functions is discussed in details in Section4.

Detailed convergence analysis of the generalizedfinite element method can he found in [11. I2,20,21].Several a-priori error est imates along with numericalexperiments demonstrating their accuracy can also befound in those works. Below. we summarize the maina-priori error estimate for the generalized finite elementmethod.

Suppose that the partitions of Q into finite elementswhich form the partition of unity satisfy the usualregularity assumptions in two and three dimensions.and. in one dimension, the ratio between the lengths ofneighboring elements is bounded. We denote

h = max 11,'=t.....N

and

x"P = span!</>n, a = I. .... tV, iEI(a)

where the GFE shape functions ¢~ are defined in Eq.(I). In addition, suppose the unity function, I, belongsto the set of local approximation functions, i.e.,

1 EI.,(W,) :x= I. .... N,

and that there exists a quantity £ depending of a, h. p,and 11, such that

1Il/-II,IIEIDrIw,):-::;((:x.I1.P,II) :x= 1, .... N

Then, it can be proved that [I 1,21] 311hl' E X"P suchthat

where the constant C is independent of II, h. p.

3. Quadratie GFE shapc fUllclions for tctrahedralelements

In this section, quadratic GFE shape functions fortetrahedral elements are defined and analyzed. Theissue of linear dependence of GFE shape function and

CA. Duarle e1 al.,I lOll/plllers and Slrt/clures 77 (201111)215-232 219

how to solve the resulting positive semi-definite systemof equations are also discussed.

Quadratic GFE shape functions for tctrahedral el-emcnts are built from thc product of trilincar tetrahe-dral shape functions and linear monomials as follows

titian of unity functions 1fJ" ;t = I, 2, 3.4. However, aswe demonstratc below. this is not sullicient to avoidlinear dependcncies.

Theorem 1. LeI S, ([/ld 5r he liS defined ([hol'e. Then

:;(= I. .... N (2)(i)

I ' "Ispan(S,) = span I . .\",Y, z. x~, xy, .c, ,I", .rz. z- = .'!I'2

where cp" (1. = I..... N, are standard trilinear Lagran-gian tetrahedral shape functions (e.g .. Refs. [4,5.32]),x, =(x,. ,1',. z,) are thc coordinates of the node ex.11, isthe diameter of the largest finite clement sharing thenode :;( and N is the number of nodes in the mesh.This transformation is used to minimize rouncl-offerrors. For details see, for example. Refs. [10, I I].

In the notation of Section 2.1 wc havc

(ii)

span{S, 1= span{ 1..\", y. z. x2, xy ..c. /.r::.,,,") .~l,'" 137.' !

~- \" \"j.' \ .•• I" \'- \. l'-~' .- \. '-1' \',1." = ,)7>• ,. .. ." ", '. ". ...• ." 3

S, = 1fJ, x I!..\". y, z I ;t = I. 2. 3.4

Proof. In order to simplify the notation, we considerhere the case in which the hasis functions. L;" of X,are given by {I. x. y. z}. Therefore. in this case,

Since 1fJ"

0: = I, 2, 3, 4, are trilinear tctrahedral shapefunctions. there exist constants (1;. a,,', li". :;(= I. 2, 3.4, such that Yx E r.

'/ = span(L;,) = span { I. x -/ x" y ~ Y', z ~ z, }A'J. let 'J. :x

;t= I, ... ,N

Consider now the case of a tetrahedral element r with.for example, nodes XI, X2. Xl, X4. Then the GFE quad-ratic shape functions for this element are given by

{

X - x, y -v, z - z, }S, = 1fJ, X I, -/ . -/ ' -, ;t = I, 2, 3. 4

1, I, 1,

Each quadratic tetrahedral element has therefore six-teen shape functions (instead of ten as in classicalFE\'Is). It should be noted that all shape functions areassociated with the vertices of the element as in a lin-ear element and there is no need for the conccpt of anedge. face or interior node, as in classical high ordcrtri-dimensional finite elements. Also. the support(cloud) of the higher order shape functions (i.e., thedomain on which the function is non-zero) is identicalto the support of the trilinear shape functions {cPt, 1fJ2'1fJ3' 1fJ4}' This fact. as we demonstra te in Section 5, hasimportant implications on the structure of the stiffnessmatrix.

Generalized tetrahedral shape functions that canreproduce quadratic polynomials can also be built asfollows. Suppose that the following basis is used forthe local spaces X" a = I, 2. 3. 4

4

I>p,(x) = I,=1

4

Ll{;CP,(X) = x1=1

4"I'L)'i lp,(X) = .1',=1

4

L(~lfJl(X) = ::1=1

Therefore

! I. x, .1', z Ic span{S,}

(3)

(4)

(5)

(6)

0: = l. 2, 3,4From the equations above \\le have

where ~ = (x - .\",)/11" 'I = LV - .1',)/11"

, = (z - z,)/h,.Then the shape functions for an element r with nodesXI. X2, X3, X4, are given by

4 4

LlT~'(IfJ,x) = x La~lfJ, = X~~~I ::r:=l

Note that we excluded from the basis of Z, the el-ements ~. '/, C since they can be reproduced by the par-

S- {I ;;'" ~.. 2 .. Y'lf = 1fJ, X • ., ", 1;1/, .. ~, 'I . '/~'~- , ex = l. 2, 3. 4

4 4

La/(ip,X) = .\'La/cp, = xy::x==l 1=1

and similarly for the terms X2', .1'2, ye. ::2. Therefore

(7)

220 C.A, DuaNe el al. !eOlllpuler.\· and Sll"l/('ll//,{'S 77 (2000) 215-231

(S)

Let us now show that S, is a linear dependent set andthat the rank dellciency 01" 5jr is cqu;tl to 6. Lsing thepartition of unity property (3) of the functions (P>:

4 4

LUP,X) = x LIp~ = X,=1 ~=t

f'=Tf

where

,i,-.jT;.j= ~

If 1\;, i

ThcnUsing Eq. (4) and the above

Ku=f' ( 10)4 4

LUP,\) - L([~Ip, = 0,=t ~=t

and similarly for the terms y and ::. Therefore, thedimension of a basis of span(S,} is less or equal to 13.But

4 4

""'(["('~ 1') = I' ""'axm = I'X~,:x't'Y.. .. ~).'t') •

~=I ~=t

Using Eq. (7) and the above

4 4

Lai\((p~:q - La,;UP"y) = (J,=1 ,=1

and similarly for the terms -'.r and ::.1'. Therefore, thedimension of a basis of span{Srl is Jess or equal to 10.But from Eq. (8) the dimension of this sct is greater orequal to 10. This proves (i). The proof of (ii) followsthe same steps.

The linear dependencies that appear above are aconsequence of the fact that hoth, the partition ofunity and thc basis of the local spaces /.,. are poly-nomial functions. Babuska and colleagues [15.29] haveproposed two approaches to solve the system ofequations

(9)

where K is a positive semi-ddinitc stilrness matrix builtfrom generalized finite element shape functions. Thcfirst approach consists simply of using a direct solverfor symmetric indelinite systems like Rcfs. [13,28].Similar implementations arc prcscntly uscd in severalcommercial FEM solvers. The second approach con-sists of the following itera tive algori tl1m [15.llJ.29]:

Let

K=TKT

The above tmnsformation leads to a stiffness matrixwhose diagonal entries are equal to I. This scaledmatrix is then perturbed as follows

K. =K+d, (>0, 1,-.j=(),,;

The matrix K, is positive delinite and hence non-singu-lar. The solution u to thc system (10) is then computedusing the following sequence:

1'0 = f' - Kuo

Let eo = 1I - Uo thcn

K.eo ~ Ke() = KII- Kuo = 1'(1

therefore

Compute

i-I

rj =C() - LKeij=U

i-I

IIi = Uo + LCij=O

for i~ I until

I ci~ei IlIiKui

is sutliciently small. Numerical experiments performedby Strouhoulis et al. [15,2lJ] show that in practice, with( = Io-tu, a single iteration is suHicient when usingdouble precision accuracy. Each iteration involves amatrix-vector multiplication. <I forward- and back-sub-stitutions and lower order operations. Therefore. the

C.A. Duarfc ('f (II. !CVIII{Jlllers (Jlld S//'lI{'lures 77 (2000) 2/5-232 221

- ; (I Icos (I' (1) - ')) 0 I'} . '} -.

,;~"I[ ]lI~/I)(,..O)= 2G I\-Q)I)(i.)1I+I) codjtJo

where (I', O. () are the cilyndrical coordinates relatives\0 the system shown in Fig. 3. lI~(r. 0). 11'1(1'. 0) and11;;(", 0) are Cartesian components of u in the ~-, 11- and(-direclions. respectively.

Assuming that the houndary is traction-free and. b d (' If' (I) (1) (2) I" Ineglectmg 0 y lorces, t le unctIons lI~i ' II,!; , II!;) • 1I"i

are given by [30,31]

(II)

[ IIN)(r. 0) I= L.

DC

.4;1) 1I~)(r, 0)1=1 0

111,(1-. OJ I

u(r. O. n = 11,/(1',0)u~(r. 0)

4. Customized shape functions fur an edge in 3D

ii = Tu

There are many classes of problems for which Ihcstructure of the underlying partial differential equationcan be exploited. aden and Duarte [24.26] havedemonstrated how knowledge of the solution of theelasticity equations ncar a corner in 20 space can beused in a partition of unity method to efficiently modelthe singularities that occur in this class of problems.This type of singularity is resolved very poorly bypolynomial functions such as are used in traditionalfinite element methods. unless a very refined mesh isused. In this section, the formulation propllscd byaden and Duarte [24,26] is extended to thc case ofedges in 3D problems. Numerical examples are pre-sented in Section 6.

Consider an straight edge in 3D space as depicted inFig. 3. In the figure. 2n -:x is the opening angle. As-sociated with the edge, there is a Cartesian local coor-dinate systcm (~. II, 0 and a cylindrical coordinatesystem (I', O.n with origins at (Ox' 0,.. 0,).

The displacement field u(r, O. ,,) in the neighbor-hood of the edge (for points far from its vertices) canbe written as [30.31]

computational cost of each iteration is negligihle whencompared to the t~lctorization of the stitrness matrix.The solution to thc original system (9) is then given by

<2I(.. r;:" 1[' 1"1(' ,(2) )]. ,(2)II~ I, 0) = 2G ). - Q; I'j + I Sill Aj (}

- ;.Ylsin (i? - 2)0 I

yt

z~x

/1)

lI\~'(r, 0) = 12~1[1\ + QYt1j!) + I) ]sin i,;IIO

·(11 . ('11' ) I+ I'j SIl1~'j - 2 (I

'CI

1Iln(r (I) = -~ I[,.+ QI21(A.12)+ I)]cos ;l2l()'II ' , 2G \ } I . '"J

+ iY1cos(;';1l - 2)0 Iwhere the eigenvalues i.;I). ;.;2i are found by solving

Fig. 3. Coordinate systems associated with an edge in 3Dspace.

222 C.A. Dliaue el ul.: eOlll(llllerS and Sl/'l/cllifCS 77 (20()(j) 2/5-232

In the case of a crack. where rx= 2n. the eigenvalues. ·1 I, '12), . ~

ale /'j = 11.; = /.j

J.j=i+ I .2 /22

Ii]) =v

otJl,.1:-SI"1 ;(3)() . I2G' "I / = . 3, 5, ...

lO,1,.12G cos ;)"0 i= 2.4.6 ....

For the edges in the mechanical part analyzed Section6, where ':J. = 4.525 264 2.

i.\11 = 0.564 34') 91)3 512569

i/J = O.9X5 954 537 70H 805

The material constant" and G are

where

Ii!. i = 1. 2. ..,x

Before using the above functions to huild customizedGFE shape functions. they have first to be transformedto the physical coordinates (x. y. z) as follows:

Define

COS(i'J(II- l)rx/2 . sin(;jLl- 1)'x/2Q'I) = - = -AI"

I cos(J.t+l)rx/2 J sin(A;ll+I}X/2

where E is thc Young's modulus and v is the Poisson'sratio. This assumes a statc of plane strain which is agood approximation for thc strcss stale in the neigh-borhood of a straight edge in three-dimensions. Forpoints close to [he vertex of the edge [he stress state ismore complex (see for example Rei's. [14.23]).

II' 1" .The parameters Qj and Q/ are gIven by

T-t (J: v ( .) "')I : '>, 'I. ~)~ r. t. (,

-(')(" V) (.1'1 T-l): . V)".J .;;.1/, ~ = "~i 0 I (<,.". ~

'(.')(l' V) I') 1,-1 (l' V)II/Ii .". 11,~ = "'li 0 I .".", ~

.(3' ( ): Y) (s) 1'- I( V " )ulj <" 11 • ." = l/U C I .;;. 'I· <,

( 12)

(13)

s=I.2 j=I ..... M

Je +112

arctan(% )

(I~I

E(i = 2(1 + v)

I': =:'1 - 4v

The coordinate system (~. 'I. 0 is shown in Fig. 3.Next dellne.' ('121 I) I") . (,12) I) /1• sm "i - ':J. - CDS I',J' - rx ~Q(-) = _. = _/1(21

I sin( At + I )x/2 I cos ( A;2) + I )rx/2

where

. .1')AI.'.I - /"j - I

J -- 11.') S = I ')'''.J + I .~

-I.') ..• - - '(.'J T-1 . , _":.i(.\.J.-)-l/~j C 2 (.\.),~)

'(.'1 . -(.0) 1,-1 )l/'Ij (\. y. ::) = "/Ii 0 2 (x. y. Z

.(3 , . , _ _ .(3) -I ._IIOJ (.\. L ~) - 1I'li 0 T2 (X, .I'. -)

s = I, 2 ) = I, .... M

(14)

In the casc of a crack.

Qlt,={-I )=3.5,7. ..J _/11\),- 1 "). } / - . _, 4. 6, ..

Q(2) _ { -I i = I. 2. 4. 6I - -tf' j = 3. 5. 7.....

For the edges in the mechanical part analyzed in Scc-lion 6.

Ql}I=-O.032551169

The functions liS I are given by Rcf. 130] (assuming th<l1the boundary is traction-free :Ind neglecting bodyforces)

(15)

where R:;-' E R' x ~]. with rows given by thc basc vec-tors of thc coordinate system (~. 1/. 0 written withrespect of the base vectors of the coordinate system (x.y, '::), and 0 = (0\..0,. OJ are the coordinates of anarbitrary point along the edge (cC Fig. 3). In theabove. ii~/(x. y . .::).ll~:/(X, y. z), S = I. 2 and 11~~I(x.y. z)are the components of the displacement vectors in thedirections e, II and (. respectively, written in terms ofthe physical coordinates x, y and z. To get the com-ponents of u in the directions x, y and z the followingtransformation needs to be applied:

CA. DlIartl' <'I 01. i Complltcrs and Structures 77 (2000) 215 -232 223

Fig. 4. Tctrahedral me,h u,ed to investigatc thc structure of the ,tilrness matrix in OFE/vb.

.\'= \,2 j= \, .... M

where R2 = (R;-t?The construction of customized GFE sh~lpe func-

tions using singular functions as defined above, followsthe same approach as in the case polynomial typeshape functions. The singular functions are multipliedby the partition of unity functions 1fJ, associated withnodes ncar an edge. In the computations of Section 6,the following singular functions are used in the con-struction of customized GFE shape functions

the stiffness matrix in the GFEM compares to that ofthe corresponding FEM mntrix. More specifically,given a mesh. we analyze how the size and sparsity ofthe two matrices compare. In the case of linear ap-proximations. the two matrices are of course identical.However, for higher degree elements the corrcspondingmatrices can be quite dilrerent. To illustrate, considerthe mcsh of tetrahedral clements shown in Fig. 4. Thestructures of the stiffness matrices corresponding toten-node quadratic tetrahedral tlnite clements andfour-node quadratic generalized finite clement, asdefined in Eq. (2). are compared in this section.

The tetrahedral mesh shown in Fig. 4 is composed

( 16)

The customized GFE shape functions are then built as

Here, rx in the index of a finitc element vertex node onor near an edge in 3D.

which arc given by

5. The structure of the stiffness matrix ill GFEMs

In this section, we investigate how the structure ofFig. 5. Non-zero enlries of the stiffnes, matrix for the meshshown in Fig. 4 with linear elel1lents.

224 CA. Dllarle el a/.! COli/pilfers alld Slrllc/llres 77 (201l0j 2/5-2n

(b)

Fig. 6. Non-zcro cntries of the stilflh':ss matrix for the mesh shown in Fig. 4: (a) matrix slruclure for ten-node quadratic tetrahedralclements; (h) l1latrix strueture tor gencralizcd quadratic tetrahcdral elemenls as defined in Eq. (2).

of 30 elements :lIld 24 vertex nodes. Fig. 5 shows thestructure of thc stilrness matrix t"or this mesh with lin-ear tetrahedral ~lcments. The m:Llrix has 72 rows and819 non-zero cntries (represented by the black part ofthe picture). Fig. 6(a) shows the structure of Ihe still-ness matrix when ten-node finite elements are used.The matrix has 297 rows and 7.R57 non-zero entries. Ifgeneralized quadratic (inite eleml:nt approximations areused. a stiflness matrix is ohtaincd with the struetur~shown in Fig. 6(b). The matrix structure for IheGFEM is sccn to he simpler than that of the FEM.The GFE matrix has 288 rows and 12,672 non-zeroentries. These data are shown on Table I. The tablcalso contains thc number of floating point operationsrcquired for thl: numerical factorization of the matriccsand the computed strain energy when the model islixed at the left end and a uniform pressure 'T =(0.0. - 1.0.0.0) is applied at the right tip. The material

Table IFE and GFE results for thc c'llllikver hcam shown in Fig. 4"

is assumed to be linearly clastic with Young's modulusE = 1.000,000 PA and Poisson's ratio I' = 0.3. Thefollowing observations can he made:

I. In the case of classical finite elements, the non-zeropattern of the matrix changes substantially when theapproximation is enriched from p = 1 to P = 2.This is due to the fact that nev,,' nodes along theedg~s of the tetrahedral clements must he created.As disl:ussed in Section 2, the p enrichment of theapproximation in thc GFEM does not require thecreation of new nodes and therefore the structure ofthe matrix is unchanged with p enrichment. as isobserved by comparing Figs. 5 and 6(b). It is alsoohservcd that there ar~ scveral adjacent columns inthc matrix shown in Fig. 6(h) with exactly the samenumber of non-zero entries. Several modern linearsolvcrs can take advantage of this type of structureto improve the performance of the factorization

Method

Numher of equal ionsNUl1lhcr of nOll-zeros lhsNum. Float. PI. Op.Strain cnergy

FErvl/GFEM.1' = I

728191.595 Ie + 0459. 102 124 91

FEM. f' = 2

2'>77.8573.587 7e + OS224.LJ9803

GFE\1.1' = 2

28812.6727,S514e+05244.978 20

"The beam is lixcd at the left and a uniform pressurc T = (0.0, - 1.0. 0.0) is applied at the right tip.

CA. Duane ('I al. i Complllers alld Sll"lICll/res 77 (2()()O) 215-232 225

...

process. One example is the Boeing sparse linear sol-vcr [28] which is used in our implementations.

2. The number of equations. for the same mesh andthe same degree of approximation, is smaller in theGFEM than in the FEM (288 versus 297 equations).This seems to be somewhat contradictory with thefact that a quadratic generalized tetrahedral elementhas 16 shape functions instead of 10 as in the classi-cal FEM. However. in the GFEM all the dol' are as-sociated with vertex nodes which are shared byseveral clements (especially in a tetrahedral mesh)and therefore the assembled system of equationswill in gcneral be smaller than in the classical FEMwhere some nodes are shared by less elemcnts, Thcratio bctwecn the number of elemcnts and the num-ber of nodes for the mesh shown in Fig. 4 is 30;24= 1.25 which is quite low. [n more realistic meshes,this ratio can be much larger, and consequently. thedifference bctween the dimcnsions of the stiffnessmatrix in the FEM and GFEM is much more pro-nounced. One illustrative example is given in thenext secLion,

3. The number of non-zeros in the stiffncss matrix forthe G FEM is substantially larger than in the corre-sponding matrix for the FEM (12,672 versus 7.857).This is a consequence of the fact that the support ofthe higher order shape funcLions in the G FEM islarger than the one in the dassical FEM. The sup-port of a shape functions (in the FEM or GFEM) isequal to the union of all elements sharing the node

UniformPressureX Direction

associated with the shape function. In the GFEM,we have only vertex nodes which are, in general.shared by more elements than edge. face and bubblcnodes used in high order finite elements. In the nexlsection, we demonstrate that the structure of theGFEM matrices, combined with thc fact that forthe same mesh and order of approximation theirdimensions are smaller in the GFEM than in theFEM. more than compensatc for their having morenon-zeros hel()}'e factorization. Our numerical exper-iments show that t he factorization time is, for theclass of problems and meshes analyzed, much smal-ler for the GFEM.

6. Analysis of a structural component using GFEmethods

In this section. we analyze the structural part shownin Fig, 7 using traditional FE and GFE methods. Themesh used has 15,527 tetrahedral elements and 3,849vertex nodal points. The mesh and problcm size arerepresentative of those used in. c.g .. automotive andaerospace industry. The large ratio of the number ofelements and number of nodes is typil:al for this dassof problems.

The material is assumed to be linearly elastic withYoung's modulus E = 100,000 Pi\, and Poisson's ratio

Fig. 7. Boundary conditions and mesh for structural parI.

226 CA. Dillin£' er 1111CVII/jllllers (/Ila SII'IU'IIl1"l'S 77 (2000) 215-232

Tahle 2finite elements results using four- and tell-llode tetrahedral el-Cl1lents

This value is uscd to compute the error in the energynorm luI' all FE and GFE n1lldels analyzed in this sec-lion, In Tablcs 2. 3. 5 and 6. 'Build, K, F(CPU. sfdenotes the CPU time, in seconds. fur the numericalintegration of stilTness matrix and load vector. 'Num-ber of Non-zeros K" denotes the number of non-zeroentries in the stilrness matrix, ·Num. Fact. K(CPU, sfdenotes the CPU time. in seconds. for the numericalI:lctorization of the slifrness matrix, 'Num. Float. PI.

I' = n.33. The boundary conditions are those rep-resented in Fig. 7. The component is fixed at both sup-pOrlS and thcre is a uniformly distributed load p = 1.0in the negative x-direction applied at its upper part(sec Fig. 7). The aspect ratio of the elements in themesh are quitc good. However. as wc demonstratebelow, if classical FEMs are used, this mesh can notproperly model the singularities prescnt in the model.

Linear and quadratic finite clements results 'lreshown in Table 2. The value of thc cxact strain encrgywas estimated using an lip adapted finite element meshwith 99.990 dol' and the a-posteriori error estimationcapabilities of I'hlex Solids [271. In our computations,we adopt the following value for thc exact straincnergy

( 18)~= I.... ,.11/

where. as in Eq. (2). CP•• ~ = L .... N, are standard tri-linear Lagrangian tetrahedral shape functions. x. = (x •.J, . .::.l are the coordinates of the node (1,. 11, is the di-ameter of the largest llnite element sharing the node IX

and N is the numher of nodes in the mesh. Theseshape functions are quadratic in the x-direction andlinear in the other directions since they arc built fromthe product of a polynomial (linear) partition of unityand the polynomial x - x,. We use the notation p =I +px, I +1\, I +p: to denote the polynomial order ofa GFE approximation. the 'Is' indicating the linearorder of the functions delining the partition of unityand P\'o P.r' /I: denote the degrees of the polynomials

Op: denotes the number of floating points operationsrequired for the factorization of the stilTness matrix'M Flops' is the computational rate attained in mega-flops and ·Conel. Number Estim.' is an estimate of thecondition number for the stiffness matrx. All the com-putations were performed on an Hewlett-Packardworkstation model 735/125 running HP-UX 10.20.

The lincar FE discretization leads to 11,547 dofwhich can be factorized in 36.21 s, using a sparse lin-ear solver [28]. Howevcr, this discretization yields asolution with a relative error of 36.7'10 in the energynorm which is unacceptable for practical purposes.The quadratic discretization has an error of 12.95%.hut it leads to a substantial increase in the number ofdof (76.797) and in the solution time (2.737.8 s, whichis 75.6 times the factorization time of the linear discre-tization) Moreover. this particular discretization cannot adequately capture thc stress singularities that existin the model and p-enrichment will lead only to asmall algebraic convergence rate.

In the GFE analysis of Ihis problem. thc linear tetra-hedral !inite element discretization is used as the par-tition of unity. In the first GFE discretization, theshape functions are as follows

(17)

2

2.729.7176.7972.965.0772737.83 (45.6 min)7.859c + 1028.712.699 130.129483

271.3011.547218.84436.219.422e'" 0826.022.375 330.367041

U(u) = 2.74515

p

Build K. F(CPlJ. s)Number of cquationsNumbcr of non-zeros KNUI1l. Fact. K (CPL:. s)Num. Float. PI. Op.MPiopsStrain energyIklldllllilE

Tahle 3Gcn~r;t1izedJinite dcments rcsults using p-ortholropic approximations built on a tetrahedral mesh

p",p, .1': I + 0, I + O. I + 0 I + J, 1 + 0, I + 0 I + I. I + I. 1+

Build K. F(CPU, s)Numher of elJuationsNumber of non-zeros K;'\Jum. Faet. K(CPL.s)Num. Float 1'1. 01'.MFlopsStrain energy"'''It/luIiFCondo Number !::slim.

57.9211.547218.84434.209.422e + 0827.552,375 350.3670296.5270e+04

141.9723.ll94~63.R29276.837.511Ie+0927.102.550 130.2(,6 537

366.1046.1883.432,2222062.13 (34.4 min)5.986e+ 1029.032.674910.159969

CA. DW1f1(' el a/. / COll/pllTers aJld STmclllre.\· 77 (200()! 2/5-232 227

in Eq. (16). These functions are built from the productof the trilincar tetrahedr.d shape functions and the firstterms of the modc I. II and III asymptotic expansionsof the elasticity solution in the neighborhood of ancdge. At the vertcx of those edgcs the elasticity sol-ution is marc complex than along the edges (see fore.g .. ReI's. [14.23]). This fcaturc is ignored in the calcu-lations to be descrihed: the same set of shape functionsare used at all nodes Illcated along the Edges 1 and 2.The enrichmenl of these nodes is justilied by the fal:lthat a large fraction thc error is ncar those elements.Fig. 8 shows the error indicators computed for theIlnite element discretization with p = I. The error indi-cators are computed using the element residual method(see, for example, Ref. [I] and the references therein)implemented in PhlexSolids [27]. Fig. 9 shows the errorindicators on the bllundary faces of the elements witherror indicators largcr than 50% of the largest elementerror indicator. It is ohserved that only a small frac-lion of the elements carries at least 50% of the totalerror and that the elements on Edges I and 2 areamong them.

The distribution of the error in the domain can alsohe analyzed by l:onsidcring the following sets of liniteelements:

used in the construction of the GFE shape functions inthe x, y, z directions. rcspcctively. Therefore, if onlythe partition of unity is used as a basis. we have p = 1+ O. I + O. I + O. For the shape functions defined inEq. (18) we have p = 1 + \, I + 0, I + O. This dis-cretization is used to illustratc that in the GFE it ispossible to build p-orthotropic approximations regard-less of the underlying finite element mesh used. Thischoice is also motivated by the fact that thc gcometry.material properties, and boundary conditions of theproblem do not change substantially in the z-direction.

In the second GFE discretization. the shape func-tions defined in Eq. (2) are used. In our notation. thisdiscretization is of degree p = I + I. I + I, J + I.i.e .. it is a quadratic approximation in all directions(see Theorem I). The GFE discretizations have dol'only at the vertices of the tctrahedral elements (fourdof for each component of the solution in the case ofthe quadratic discretization). That is, there are no dofalong the edges or in the interior of the element. The p= I + I, I + O. I + 0 discretization has 23,094 dol'(exactly twice the linear discretization) and a relativeerror in the energy norm of 26.6'Yo. The quadraticG FE discretization has 46.18H dof which is only 60'%or the number of dof in the finite element discretiza-tion. The total CPU time for Ihe factorization of theresulting system or equations is 2,062.1 s. whieh isabout 25°;;, smaller than in the case of quadratic finiteelements. The relative error in the energy norm for thisdiscretization is 16.0% which is ahout 235% largerthan in the case of quadratic finite elements.

The GFE results are summarized in Table 3. Thequadratic GFEM discretization leads to a smaller sys-tem of equations than comparable FEM. However. theGFEM matrix has about 16% more non-zero entriesthan the FEM counterpart hejorr! faCTorization. None-theless, being a smaller matrix with a more favorablesparse structure, the GFEM matrix can be factorizedusing about 24% less floating point operations.

Set 0Sct I

Set 2

Set 3

Sct 4

Set 5

Set 6

all elements in the mesh,elements ull1neded to one of the two verticesof Edge I.elements connedecl to the Edge I but not con-nected to one of its vertices,elements connected La one of the two verticesof Edge 2,elements conncctcd to thc Edge 2 but not con-nected to one of its vertices.e1cments connected to one of the rims of theroles where displacement boundary conditionis applied.elements not in Sets I. 2. 3. 4 or 5.

The following quanttties wcre computed using theclassical finite clement discretization with p = I and p= 2. The sum of lhe error.

6./. Modeling of singularities using specialfimctiol/S

Polynomial approximations. as used in the FE andGFE discretizations described above, can not ellicientlyapproximate the solution in the neighborhood of geo-metric edges, such as Edges I and 2 shown in Fig. 7.However, asymptotic expansions of the elasticity sol-ution in the neighborhood of such edges are wellknown and the GPE framework allows a straightfor-ward inclusion of these asymptotic lields in the GFEapproximation spaces, as dcscrihcd in Section 4. Thisapproach is demonstrated in the solution of the pro-blem represented in Fig. 7.

The same GFE discretizations described previouslyare used, but the nodes located along the Edges I and2 are enriched with the GFE shape functions defined

(

. 1/2

e'x =: LIle;lI~) Set ~ = 0, .... 6iE.lZ

the average error.

(L lledl}) 1/2

e" =' iE,r. card ,P Set,~ = o..... 6

and the normalized average error.

(19)

(20)

228 C'.II. Dllarll' 1'1 al. :' COlllpllll'rS and Slrllcillres 77 (20(}O) 2/5-232

Fig. 8. Error indicalOrs computed for the finite elel1lent discretization with I' = I.

0.045

0.040

0.035

0.030

0.025

0.020

0.015

0.010

0.005

0.000

Table 4Average error indicalors for several selS oj' elements when classicalllnile elements with I' = I and 2 are used"

Sel (a) 0 I 2 3

e" (p= I) 1.567 1.5nc-01 4.518c-Ol 1.298e-01e' (p = I) 1.258c-02 3.475c-02 4541e-02 2.41 k-02e' (p = I) 1.0 2.763 3.611 1.917c' (I' = 2) 2.716c-01 3.507c-02 9.788e-02 2.723c-02c' (p =2) 2.180e-03 7.652e-03 9.837e-03 5.057e-03~' (I' = 2) 1.0 3.511 4.513 2.320

., The quantities e', e' and e' arc defined in Eqs. (19). (20) and (21). respectively.

4

4.42ge-0 14.204c-023.3439248c-028.778e-034.027

5

3.298c-012.045c-021.6266.872c-024.2fi2e-031.955

6

1.380e+ 001.126e-020.8962.212e-OI1.806c-030.829

C ..-I. Dllal'll' el al.,' Complilers alll/ SII"l/('/lIr".\' 77 (2()(){)) 2/5-232 229

0.044

0.042

0,040

0.038

0036

0.034

0.032

0.030

0.028

0.026

0.024

0.022

0.020

Fig. 9. Error indicators on the boundary races of thc elemcnts with error indicators larger than 50% of the largest elcmcnt crror in-dicator.

Table 5Generalized linite elements results using p-orthotropic approximations and singular shape i'ullctiollS at nodes along Edges 1 and 2

px. 1',., Po

Build K. F(CPU,s)Number or equationsNUl1lber or nOll-zeros KNUIl1. Fact. K(CPU,s)NUI1l. Floul. PI. Op.MFlopsStrain cncrgyIJeIlE/liuliECondo Number Estim.

I ~ O. I + ~ I + 0

201.3911.943244,36841.229,476c+OH22.992.435 7770.3357077.3942e + 04

I + I. I + O. I + 0

42H.3923.490904.635316.237.355e + 0923.262.586 0490.240746

1 + 1. I + 1, I +

936.3246.5843.503,5922.156.57 (35.9min)6.260e+ 1029.032.7110880.111 399

230 CA. f)1/1I1"1e' el III.': Com!'l/le,'s and S1I"lICII//"e'.I' 77 (2000j 215-232

Table 6Generalized finite c1emelHs results using singular functions atnodes along Edges I and 2 and at nodes connected to theseedges by a finite clement and located at the boundary of thepar!

0.323 301 2640.334 580 5030.335447 0630.335 573 2100.335 535 1420.335 619 7720.335 669 9540.335 707 5780.335 728 2820.335 743 8830.335 757 904

2.45R 221 2122.437 R51 04R2.436 257 1602.436 024 7902.431i 094 9212.435 93R 9982.435 846 5232.435 777 1802.435 739 0192.435 710 2612.435684414

Strain energyNum. Int. Pts.

4II2445216343512729100013311728

Table 7Computed strain energy and discretization error for variouschoices of quadrature. The rules with 4.11.24 and 45 pointsare Keast [16] quadraturc rulc). (thc others are tensor productGaussian quadrature rule. The discretization used is GFEwith I' = I + O. I + 0, I + 0 and singular functions atnodcs along Edges I and 2

The key feature of the Generalized Finite ElementI'vlethod is the use of a partition of unity to buildthe approximation spaces. This partition of unity

puted strain energy and relative discretization errorlIel£/IIIIIiE for the GFE discretization p = I + 0, I +O. I + 0 enriched with singular functions along EdgesI and 2 as a function of the number of integrationpoints used in the elements with nodes carrying singu-lar functions. The rules with 4, II. 24 and 45 pointsare Keast [16] quadraturc rules and the others are ten-sor product Gaussian quadrature. The discretizationerror is computed using thc reference value (I7) for theexact strain energy. Based on these computations, andthe fact that we use discretizations with an error ofmore than 5%. we employ tensor product Gaussianquadrature with 1000 points on the elements withnodes carrying singular functions. A more computa-tionally emcient approach is. of course. to use adaptiveintegration on those elements and set the tolerance ofthe numerical integration according to the required ac-curacy of the approximate solution.

Another important issue is the eOcct or the enrich-ment with singular functions on the condition numberof the stilTness matrix. Estimates of the condition num-ber are shown in Tables 5 and 3. It can be observedthat the enrichment with singular functions of thenodes along Edges I and 2 has no detrimental efrecton the conditioning of the stiffness matrix. The cnrich-ment with singular functions does not lead to lineardependences as in the case of polynomial enrichment.This last case is discussed in Section 3.

7. Conclusions

(2\ )

I + I, I + I. I + I

1.276.3647,44R3,646.4762.072.94 (34.5 min)6.043e+ 1029.152.71256390.10R 959 R

Sct ,:1. = O....• 6

Build K. F(CPU. s)Number of equationsN Ulll bel' of non-zeros KNum. Fact. K(CPU, s)Num. Float. Pt. Or.r."IFlopsStrain energyIIdlE/III/IiE

where ei is the error on the elemenl i, ,j« denotes anindex set for the Set 'Y. and II· II[; denotes tbe energynorm, The results are shown on Table 4.

It is observed that the avcrage error i? for the el-emcnts along Edges I and 2 (Sets I, 2. 3. 4) is con-siderably larger than the average crror on the wholedomain (Set 0). In addition. [hose elements representonly a small fraction of the total number of elementsin the mesh. which justifies our choice or using custo-mized singular shape functions only along those edges.

The results for tbe enriched GFE discretizations areshown in Table 5, It is observed that [be addition ofthe singular functions along the Edges I and 2increases the total number of dol' by only 396. This isonly I % more dof in the case of the GFE discretiza-tion with p = I -1- I, I + I, I + I which increasesthe solution time by only 4.5°/.,. The eOect of thisenriched shapc functions on the discretization error.hO\vevcr. is quite noticeablc. They lead to a decrease ofabout 30% in the discretization crror for GFE with p= I + I, I + I, I T I. We also investigate tbe effectof additionally enriching thc nodes connected to EdgesI and 2 hya finite element ([lit/located at the boundaryof the part. The results lar tbis discretization areshown in Table 6. It is observed that the effect of thisadditional enricbment is not substantial.

The enrichment of the G FE discretization withsingular functions leads to the issue of numerical inte-gration of thesc functions, In this work, our main goalis to investigate the benefits of adding these functionsin terms of controling the discretization error. A sufll-ciently high quadrature rule was used on tbe elementswith nodes carrying singular functions. The order ofthe quadrature rule was chosen so that the numericalintegration errors were small enough to not arrect thecomputed discretization errors. Table 7 shows the com-

CA. DU{[rle el 0/: C;OI1lPUl<TS([ut! Sln/(lIIres 77 '/2000j 215-232 231

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[II] Duarte CAM. Oden .IT. III' clouds·- an hp l1leshlcssmethod. Numerical Methods for Partial DifferentialEquations 1996:12:673..705.

[12] [)uarle CA. The hI' cloud method. PhD dissertation.Universily of Texas at Austin, Austin. TX. USA.December 1996.

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[15J Bahusk:i I. Strouboulis T, Copps K, Uanganl SK,Upadhyay CS A-posteriori error estllnation for linite el-ement and g.;neralized tinite clement method. Availablefrom htl p:.:,"yoyodyne.tamu.edu!researehierrorigfem_fran-ce.pdf

[16J Keast P. Moderate-degree tetrahedral quadrature for-

...

framework has several powerful properties such as1he ability to produce seamless /rp finite elemen1 ap-proximations with nonuniform hand p. the abilityto develop customized approximations for specificapplications, the capability to build p-orthotropicapproximations on, e.g., 3D tetrahedral meshes, etc,Several of the so-called meshless methods proposedin recent years also make use, explicitly or im-plicitly, of a partition of unity to build the approxi-mation spaces [7,11.18,22J. The fundamentaldifference between these methods and the GFEM isin the choice of the partition of unity. In theGFEM, the partition of unity is provided by con-ventional finite element methods. In this case. theimplementation of the method is essentially thesame as in standard finite element codes. the maindifferencc being the definition of the shape func-tions. This choice of parti1ion of unity avoids theproblem of numerical integration associated with theuse of moving least squares or Shepard partitionsof unity common to scveral mesh less methods. Inaddition, the use of a finite element partition ofunity allows easy implementation of essential bound-ary conditions, which may not be a straightforwardproposition in other techniques using moving leastsquares partition of unity.

In this paper, we investigate several importantpractical aspects of the gcneralized finite elementmethod in a 3D setting. The structure of the stiff-ness matrix in the GFEM is compared with thefinite element counterpart when the same mesh andpolynomial degree of the approximation is used, Weshow that the structure of the matrix in the GFEMdoes not change with p-enrichment. in contmst withthe classical FEM case. Modern linear solvers cantake adv<mtage of the type of structure in the GFEmatrices to improve the performance of the factoriz-ation process. It is also demonstrated that for thesame mesh and degree of approximation, theGFEM leads to a smaller system of equations thanin the classical FEM. This difference in the numberof equations is specially pronounced in mesheswhere the ratio between the number of elementsand nodes in the mesh is high, as in the case oftetrahedral meshes of complex structures. This istranslated into faster solution times for the GFEM.

The GFEM framework allows for straightforwardconstruction of special solution spaces using a-prioriknowledge of properties of the solution of the pro-blem. In this paper. this is demonstrated for the cascof elasticity equations in three dimensions. This pro-cedure does not require the modilication of the finiteelement mesh. In general, these properties allows us toobtain the solution of 3D problems with higher accu-racy and less computational effort than the classicalfinite element method.

Acknowledgements

The work of J.T.ported by ArmyDAAH04-96-1-0062.

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