Conjunction of Displacement Fields of the Element Free Galerkin Method and Finite Element Method

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1 amkang Journal of Scence and Engneerng, Vol. 10, No 1, pp (2007) 41 Conjuncton of Dsplacement Felds of te Element Free Galerkn Metod and Fnte Element Metod Cen-Hsun Ln* and Can-Png Pan Department of Constructon Engneerng, Natonal awan Unversty of Scence and ecnology, ape, awan 106, R.O.C. Abstract A prncple weakness of te element free Galerkn metod, a metod wdely dscussed over te past decade, as been ts computaton effcency. s paper descrbes a new smple and effcent metod tat overcomes ts weakness by combnng togeter te fnte element and element free Galerkn metods. No transmsson zones are requred n ts new metod. e new metod ntroduced dffers from metods suc as Lagrange multplers or te penalty metod n tat so-called vrtual partcles are defned to approac te compatblty between two dsplacement felds. Vrtual partcles are derved from te fnte element formulaton and used as te partcles n te element free Galerkn formulaton. Key Words: Element Free Galerkn Metod, Fnte Element Metod, Compatblty, Dsplacement Felds 1. Introducton e fnte element metod s a well-developed and wdely used metod of structural analyss. However, ts metod s unsutable for use wt problems nvolvng dscontnuous geometry, as te defned mes requres updates durng analyss procedures and, as a result, demands sgnfcant uman nterventon and computng tme. Also, as te convergence rate may deterorate for problems wt g stress gradents, te defned mes must be dvded nto very small elements n tese regons. Belytscko, Lu and Gu [1] proposed te element free Galerkn metod. Wle ts metod andles dscontnuous geometry and g stress gradent problems relatvely smply and metod formulaton establses a contnuous dsplacement feld wt contnuous dervatves, ts computaton effcency s nferor to tat of te fnte element metod. Also, te assumed dsplacement feld does not fully matc boundary condtons. Convergence stablty s deterorated by a complcated formulaton *Correspondng autor. E-mal: D @mal.ntust.edu.tw tat uses a relatvely greater number of parameters n ts wegt functon. erefore, wt te excepton of certan specal cases (e.g., crack propagaton problems), ts metod s not sutable to replace te fnte element metod n most practcal applcatons. Efforts ave been devoted to develop procedures capable of combnng te fnte element metod and element free Galerkn metod nto one analytcal metod. Belytscko, Organ and Krongauz [2] suggested a transton zone n te dsplacement felds establsed by tese two metods. e transton zone dsplacement feld actually represents a combnaton of two dsplacement felds troug a ramp functon. erefore, t s very complcated n ts form and n all computatons. Also, transton zone wdt represents a new parameter; canges n wc affect computaton results. e stablty of solved results s nferor to tat obtaned usng te element free Galerkn metod. e Lagrange multpler metod or penalty metod were suggested by Hegen [3] to combne tese two dsplacement felds. Independent nodes are requred to defne bot dsplacement felds. Lagrange multplers also represent addtonal degrees of freedom. Lagrange multplers run te postve defnte caracter-

2 42 Cen-Hsun Ln and Can-Png Pan stc of te stffness matrx, and requre tat te computaton procedure must be modfed for ts cange. Rao and Raman [4] suggested an addtonal rectangular element for conjuncton. s addtonal element contans four nodes, two of wc belong to te fnte element dsplacement feld and two of wc belong to te element free Galerkn dsplacement feld. e sape functons of tese four nodes are defned by te lnear combnaton of te two. s metod s smlar to tat of Belytscko, Organ and Krongauz [2], te dfference beng tat te addtonal element replaces te transton zone. Ho, Yang, Wong and Wang [5] use combned dsplacement felds n small regons, wt compatblty obtaned troug te use of brdge scales. Huerta and Fernandez-Mendex [6] also use combned dsplacement felds. e element free Galerkn metod s undetermned coeffcents are determned usng a reproducblty condton to obtan dsplacement feld compatblty. We suggest a new approac to combne te fnte element dsplacement feld and element free Galerkn dsplacement feld n ts paper. s approac does not requre transton zones or addtonal nodes n jont lnes. e compatblty of tese two dsplacement felds can be aceved automatcally n te mes refnement procedure. A bref ntroducton to bot metods s presented n te second secton below, were topcs related to te combnaton of two dsplacement felds are dscussed. A suggested approac to conjunct dsplacement felds s explaned n te trd secton. e fourt secton uses several two-dmensonal problems to llustrate te developed metods. e fft secton s te concluson. 2. Formulaton of Dsplacement Felds e metod suggested n ts paper can be appled to any two- or tree-dmensonal structural problems. e two-dmenson plane stress problem s used as te llustraton n ts paper Element Free Galerkn Metod Detaled formulaton of te element free Galerkn metod can be found n Belytscko et al. [1,711], Zu and Atlur [12], Mukerjee and Mukerjee [13], Gavete et al. [14], Fernandez-Mendez and Huerta [15], Lee and Yoon [16], Gu and Lu [17] and Rao and Raman [18]. s paper refers only to te formulaton related to te developed metod. e assumed dsplacement feld s a polynomal functon wt varable coeffcents. e formula can be sown as: (1) e {p(x)} s called a base functon vector. e number of base functons, m, can be vared as desred. However, complete polynomals are most commonly used. e {A(X)} s a vector contanng undefned coeffcents tat vary trougout te wole problem doman. Coeffcents are determned by mnmzng te followng wegted dscrete least-squares norm: (2) e w(x X ) s called a wegt functon, and u represents partcle dsplacements. Several wegt functons were suggested by varous references. e modfed exponental functon s adopted n ts paper. e formula s as follows: (3) e sequental dstances between partcles determne te parameter c. Assumng tat c I s te (m 1), te smallest dstances of all partcles, ten c can be defned as tmes c I. e defnton sould reveal an adequate number of partcles to determne undetermned coeffcents. d s te dstance between any pont X and partcle. d m s te sze of te support for te wegt functon. e coeffcent vector can be solved by functonal varatons. e assumed dsplacement feld can ten be expressed as u ( X) { p( X)} [[ P][ W][ P] ] [[ P][ W]]{ u} ( X){ u} (4) as m u ( X) p ( X) a ( X) { p( X)} { A( X)} n 1 j1 j J w( X X )[ u ( X ) u ] j 2 2 ( d / c) ( dm / c) e e f 2 W( X X ) ( dm / c) 1 e 0 f 1 e sape functons of eac partcle can be obtaned 2 d d d d m m

3 Conjuncton of Dsplacement Felds of te Element Free Galerkn Metod and Fnte Element Metod 43 1 ( X) { p( X)} [[ P][ W][ P] ] [[ P][ W]] (5) e assumed dsplacement feld n te element free Galerkn metod s formulated for te entre problem doman. e fnte element metod formulates te dsplacement feld only for an element. e assumed dsplacement feld n eac partcle does not concde wt te partcle dsplacements. erefore, u may be called dsplacement parameters rater tan partcle dsplacements. 2.2 Fnte Element Metod e assumed dsplacement feld n eac element s defned by te multplcaton of sape functons and undetermned nodal dsplacements. e formula can be expressed as Cook, Malkus and Plesa [19] 8 j(, ) j j1 u N u (6) 2.3 Solve for Undetermned Coeffcents If te analyss problems are restrcted to lnear elastc, te mnmum total potental teorem can be appled to determne tose undetermned coeffcents n te dsplacement feld. e dervaton can follow te vrtual work prncples, wc can be appled to nonlnear problems as well. e fnal form of te soluton can be as follows: K u f (7) e stffness matrx [K] conssts of many 2 2 submatrces K IJ, IJ I J I J u u K B DB d S d s 0 s 1 f prescrbed u on d x S 0 s y s 1 f prescrbed u on d x x u y y u (8) e equvalent load matrx {f} conssts of many 2 1 submatrces f I, f td bd Su d I I t I I I u t u (9) t s te tracton force along a lne, b s te body force on a volume, and u I represents specfed dsplacement boundary condtons. 3. e Conjuncton of wo Dsplacement Felds A conjunct dsplacement feld must be developed n order to secure te advantages of bot metods. e conjuncton of two dsplacement felds can be eter a transton zone or a lne (for plane problems). A transton zone wll make te dsplacement feld muc more complcated, and te stran energy n ts zone must be accounted for n calculatons. e metod developed n ts paper uses smply a lne to connect bot felds. e nodes used by te fnte element metod are generated as usual and tose nodes n te fnte element metod are used to generate te partcles used n te element free Galerkn metod. ese partcles are called vrtual partcles n order to dstngus tem from normal partcles. e program developer s free to decde te number of vrtual partcles n a mes. s paper studes te nfluence of te number of vrtual partcles. wo vrtual partcle arrangements were tred n ts study. e frst arrangement puts all vrtual partcles n te conjuncton lne. e second tral arranges some vrtual partcles n te terrtores of bot metods. e purpose of te second arrangement s to aceve compatblty between te dsplacement dervatves. e dsplacement parameters, u v, of vrtual partcle are calculated by nodal dsplacements n te fnte element metod. e formula can be expressed as: u N, u v j j j1 (10) B s a matrx, wc relates strans and dsplacements. D s te consttutve matrx. s te doman of wole structure. s a penalty number. u s te essental boundary condton. u j are nodal dsplacements of te element, wc s adjacent to te conjunct lne. N j (, ) s te sape functon at node j. e dsplacement feld for te entre problem can be

4 44 Cen-Hsun Ln and Can-Png Pan dvded nto two parts: te frst one, u FE, s defned by te fnte element metod and te second part, u EF, s defned by te element free Galerkn metod. However, te relatve partcles nclude bot normal and vrtual partcles. e expresson of te wole dsplacement feld s: ufe u FE N 0 0 u EF u 0 EF EF v u v (11) u FE are nodal dsplacements u EF are dsplacement parameters of normal partcles u v are dsplacement parameters of vrtual partcles EF are sape functons of te element free Galerkn metod calculated for normal partcles v are sape functons of te element free Galerkn metod calculated for vrtual partcles N are sape functons of te fnte element metod. All nodes used n te FEM feld ave ndependent degrees of freedom. e assemblages of stffness matrx and load vector are te same as sown n Eqs. (8) and (9). Partcles used n te EFGM feld nclude bot normal and vrtual partcles, te latter of wc are derved from nodes and, terefore, do not ave ndependent degrees of freedom. Stffness values calculated for vrtual partcles are added to correspondng nodes. e degrees of freedom n Eq. (8) can be dvded nto four parts: e frst part s related to normal partcles I and J IJ I J I J u u K B DB d S d (12-a) e second part s related to vrtual partcle I and normal partcle J lj Il I J Il I J u u K B DB d S d (12-b) e trd part s related to normal partcle I and vrtual partcle J In I J Jn I J u Jn u K B DB d S d (12-c) e fourt part s related to vrtual partcle I and vrtual partcle J Jn ln e transformaton matrx s as follows: N Il I J Jn Il I J u Jn u K B DB d S d n1 J J n8 J J (12-d) (, ) 0 N (, ) 0 0 Nn1( J, J) 0 Nn8( J, J) Jn s relatve vrtual partcle J n element n. N n s te sape functon for node n element n. ( J, J ) are coordnates for vrtual partcle J n element n. Smlarly, te calculaton of force vector Eq. (9) can be expressed as follows e frst part s related to normal partcles I f td bd Su d I I I I I u t u e second part s related to vrtual partcle I (13-a) l Il I Il I Il I I u t u f td bd Su d (13-b) e compatblty of two dsplacement felds s aceved by te ncrease of vrtual partcles. Because te wegted dscrete least-squares norm s mnmzed durng te refnement procedure, te dsplacement feld n te element free Galerkn metod wll approac tat of all partcle dsplacements. e number of vrtual partcles used n te calculaton correlates postvely wt compatblty. Even te conjuncton of two dsplacement felds s restrcted to a lne, and te dstrbuton of vrtual partcles can go beyond te conjuncton lne. Some vrtual partcles are dstrbuted deep nto bot felds n ts study. e purpose of ts tral was to obtan a dsplacement feld tat was compatble not only n te dsplacements but also n te dervatves. e results wll be dscussed n te llustrated problems below. 4. Illustrated Problems e problems sown n ts paragrap are assumed to be sotropc lnear elastc materals. e modulus of elastcty s E = e Posson s rato s = 0.25.

5 Conjuncton of Dsplacement Felds of te Element Free Galerkn Metod and Fnte Element Metod 45 e parameters used for te element free Galerkn metod are sown below. e complete lnear polynomals are used for base functons (m = 3). e coeffcent s taken as one. In order to study te compatblty errors generated n te conjunctons, an error norm s defned to do te comparson. e formula for ts error norm s UERR F& E 2 u L u FE FE u u EF EF d F& E (14) F & E s te conjuncton lne. L s te lengt of te conjuncton lne. 4.1 Replacement of Part of an Element e problem uses a square plate wt a sde lengt equal to tree and a tckness equal to one. Unform dstrbuted loads are appled along te rgt sde wt a densty equal to one. Only one plane stress element wt egt nodes s used n te fnte element analyss. e mes s sown n Fgure 1, wt an assumpton tat results requre addtonal refnement. s sngle element s ten subdvded nto nne equal areas. e central area wll be replaced by te element free Galerkn metod. e oter egt areas wll reman as te fnte element metod. wo arrangements of partcles were tred n ts study. ere are 32 vrtual partcles arranged along four conjuncton lnes. e normal partcles are 9 and 16 for eac arrangement. Results and errors are sown n able 1. e results wt 16 normal partcles are better tan te results wt 9 normal partcles. Maxmum error for stresses s about sx percent and errors n te element free Galerkn metod area are greater tan tose n te fnte element metod areas Axal Beam Subjected to Unform Loadng e lengt of ts axal beam s egt and bot te dept and wdt of te beam secton equal one. A unform tensle stress = 1 s appled at te beam end. e fnte element mes contans 2 10 elements, wt sx replaced by te element free Galerkn metod and te oter 14 remanng n te fnte element zone. e number of normal partcles s fxed as 20. e numbers of vrtual partcles are 21, 49, and 101 n tree dfferent trals. Fgure 2 sows te arrangements. e close form soluton Fgure 1. Mes arrangement for te sngle orgnal element. able 1. Errors of dsplacement u x and stress xx for te sngle element case 9 vrtual partcles and 32 normal partcles 16 vrtual partcles and 32 normal partcles Coordnates (x, y) FEM Error of xx (%) EFGM Error of xx (%) Coordnates (x, y) FEM Error of xx (%) (1.000,1.000) (1.000,1.000) (1.000,1.125) (1.000,1.125) (1.000,1.250) (1.000,1.250) (1.000,1.375) (1.000,1.375) (1.000,1.500) (1.000,1.500) Note: Error of u x = u u exact / u exact, Error of xx = exact / exact EFGM Error of xx (%)

6 46 Cen-Hsun Ln and Can-Png Pan Fgure 2. Mes arrangements for te cantlever beam. from beam teory s: ux E XX x ( xy, ) (15-a) (15-b) 4.3. Cantlever Beam Subjected to Concentrated p Loadng e geometry of te cantlever beam s smlar to te e error norm UERR (Fgure 3) s 0.02 for 21 vrtual partcles. However, te UERR s for 49 vrtual partcles. e UERR for 101 vrtual partcles s below e trend of tese results sows tat compatblty along te conjuncton lne can be aceved by ncreasng te number of vrtual partcles. e error of analyzed stress falls below 1% as well (able 2). Fgure 3. UERR of u x on te conjuncton lne for te cantlever beam. able 2. Errors of dsplacement u x and stress xx for te cantlever beam 21 vrtual partcles 49 vrtual partcles 101 vrtual partcles Coordnates (x, y) FEM Error of xx (%) EFGM Error of xx (%) FEM Error of xx (%) (1.60,0.00) (1.60,0.50) (6.40,0.00) (6.40,0.50) (2.40,0.50) (4.00,0.50) (5.60,0.50) Note: Error of u x = u u exact / u exact, Error of xx = exact / exact EFGM Error of xx (%) FEM Error of xx (%) EFGM Error of u x (%) Error of xx (%)

7 Conjuncton of Dsplacement Felds of te Element Free Galerkn Metod and Fnte Element Metod 47 axal beam noted n Secton 4.2. A concentrated load, P = 50, s appled to te beam tp. Fgure 2 sows te problem arrangement. e close form soluton s obtaned from mosenko and Gooder [20]. e vertcal dsplacement and axal stress are coped ere: u y ( y 3 Dy D )( L x) P 2 6EI (4 5 ) D x( L x)3x 4 3 P 1 XX ( xy, ) ( Lx)( y D) I 2 (16-a) (16-b) Fgure 4. Dsplacement u v on te lne y = 0.5 for te cantlever beam. 3 BD I s te moment of nerta. B and D are beam 12 wdt and beam dept. e arrangement of te fnte element mes and element free partcles s te same as tat noted n Secton 4.2. Fgure 4 sows comparsons of analyzed results and analytcal results. e results of 49 vrtual partcles and 101 vrtual partcles matc analytcal results exactly. Only te results obtaned from 21 vrtual partcles sows dfferences. A study of UERR (Fgure 5) sows tat te error norm for 49 vrtual partcles falls below However, te error norm for 101 vrtual partcles approxmates te value of tat of te 49 vrtual partcles. Small UERR values ndcate tat compatblty between two dsplacement felds approxmates well. A comparson of stress results s sown n Fgure 6. e comparsons are also very good. All results mentoned before were obtaned by arrangng vrtual partcles n te conjuncton lnes. Dfferent arrangements were tred for ts problem and addtonal vrtual partcles were used n addton to te orgnal 101 vrtual partcles. However, te addtonal vrtual partcles fell besde te conjuncton lne, postoned eter n te fnte element terrtory or n bot terrtores. e purpose of tese trals was to assess te compatblty of te dsplacement gradents. We dd not obtan te expected mprovements n results. Our comparson of results s sown n able 3. Errors of stresses were even greater tan tose of te 101 vrtual partcles Hole n an Infnte Plate Consder an nfnte plate wt te tckness = 1. A Fgure 5. UERR of u y for te cantlever beam. Fgure 6. Error of xx ontelney=0fortecantlever beam. crcular openng exsts n te plate center. A unform stress = 1 s dstrbuted along bot left and rgt edges. We assumed a fnte area wt B = W = 10 for ts analyss. Only a quarter of te plate s modeled n ts analyss due to ts double symmetrc caracterstcs. wenty-one plane stress elements are adopted n te edge area. wenty-nne partcles are adopted to model te area close to te openng. Vrtual partcles are 17, 33, and 65 n tree dfferent arrangements. e arrangements are llustrated

8 48 Cen-Hsun Ln and Can-Png Pan able 3. Errors of xx (dsplacement gradent) for te cantlever beam Coordnates (x, y) 101 vrtual partcles on te conjuncton lne Error of xx (%) 201 vrtual partcles on two lnes ( x = 0.05) Error of xx (%) 201 vrtual partcles on two lnes ( x =0.1) Error of xx (%) (1.60,0.00) (1.60,0.25) (6.40,0.00) (6.40,0.25) Note: Error of xx = EF FE / exact n Fgure 7. e close form solutons obtaned from mosenko and Gooder [20] are: u x u y a 2 1 r cos 2 1 cos 1 r (17-a) E a 1 a cos3 cos3 3 2 r 2 r a 2 r sn 12 sn 1 r E a 1 a sn 3 sn r 2 r (17-b) Fgure 8 sows te comparsons of UERRs for dsplacements along conjuncton lnes n tree dfferent arrangements. e errors of 65 vrtual partcles were 2% for vertcal dsplacement, and 0.3% for orzontal dsplacement. Agan, te compatblty approxmated well. We also made a comparson of dsplacement for a stragt lne,x=2(fgure 9). e lengt of Y = 0~2 represents te conjuncton lne. e lengt of Y = 2~5 s n te fnte element terrtory. e results of 65 vrtual partcles approxmate tose of 33 vrtual partcles. e dfferences of analyss results and exact solutons could come from te nfnte plate assumed n te exact soluton. In order 2 4 a 3 3a XX ( xy, ) 1 cos 2 cos 4 cos r 2 2r (17-c) Fgure 8. UERR for te nfnte plate problem. Fgure 7. Mes arrangement for an nfnte plate. Fgure 9. Dsplacement u x on te lne x = 2.

9 Conjuncton of Dsplacement Felds of te Element Free Galerkn Metod and Fnte Element Metod 49 to ceck te correctness of two dsplacement felds used n ts analyss we conducted a complete fnte element analyss for comparson. Egt elements were replaced nto te element free Galerkn area. Results for te pure fnte element approxmated te converged results of te two dsplacement felds. Fgure 10 sows comparsons of stresses along te stragt lne atx=0.eresults of 33 and 65 vrtual partcles and te pure fnte element metod approxmate te exact soluton very well. Results of 17 vrtual partcles sow larger errors and unstable varatons. e potental for local refnement of ts metod s llustrated n te followng calculatons. Four elements are used to establs te orgnal mes (Fgure 11). Several suggested error norms can be used to locate areas of defcent accuracy. We pcked out te element close to te crcular openng for modfcaton and te element free dsplacement feld was replaced n ts area wt 29 normal partcles. e vrtual partcles used n te conjuncton lnes were 17 and 65, respectvely. e element free dsplacement feld was used wt te oter tree orgnal fnte elements, wt results sown n Fgure 12. e combned dsplacement felds sow very good comparsons wt te sngle fnte element feld wt 29 elements. Fgure 13 sows te comparsons of stresses along te stragt lne at X = 0.e results of 17 and 65 vrtual partcles wt 3 elements approxmate te exact soluton very well. 5. Concluson s paper develops a smple metod for combnng te fnte element dsplacement feld and element free dsplacement feld. e vrtual partcles along conjunc- Fgure 10. Stress XX ontelnex=0fortenfnte plate. Fgure 12. Dsplacement u x on te lne x = 2 for te nfnte plate (4 orgnal elements). Fgure 11. An nfnte plate solved usng te suggested procedure. Fgure 13. Stress XX ontelnex=0fortenfnte plate (4 orgnal elements).

10 50 Cen-Hsun Ln and Can-Png Pan ton lnes are te man tools employed by ts metod. e examples sown n ts paper llustrate good comparson, wt close form solutons. Replacement of te dsplacement feld can be done for a wole or partal element. e results sow tat replacng a wole element s preferable to replacng a partal element. We tred to poston te vrtual partcles on bot te conjuncton and n one or bot dsplacement felds. e purpose of tese trals was to mprove te compatblty of dsplacement dervatves. However, tese trals dd not aceve expected results. References [1] Belytscko,., Lu, Y. Y. and Gu, L., Element-Free Galerkn Metods, Internatonal Journal for Numercal Metods n Engneerng, Vol. 37, pp (1994). [2] Belytscko,., Organ, D. and Krongauz, Y., A Coupled Fnte Element-Element Free Galerkn Metod, Computatonal Mecancs, Vol. 17, pp (1995). [3] Hegen, D., Element-free Galerkn Metods n Combnaton wt Fnte Element Approaces, Computer Metods n Appled Mecancs Engneerng, Vol. 135, pp (1996). [4] Rao, B. N. and Raman, S., A Coupled Mesless- Fnte Element Metod for Fracture Analyss of Cracks, Internatonal Journal of Pressure Vessels and Ppng, Vol. 78, pp (2001). [5] Ho, S. L., Yang, S., N, G., Wong, H. C and Wang, Y., Numercal Analyss of n Skn Depts of 3-D Eddy-Current Problems Usng a Combnaton of Fnte Element and Mesless Metods, IEEE ransactons on Magnetcs, Vol. 40, pp (2004). [6] Huerta, A. and Fernandez-Mendex, S., Enrcment and Couplng of te Fnte Element and Mesless Metods, Internatonal Journal for Numercal Metods n Engneerng, Vol. 48, pp (2000). [7] Lu, Y. Y., Belytscko,. and Gu, L., A New Implementaton of te Element Free Galerkn Metod, Computer Metods n Appled Mecancs Engneerng, Vol. 113, pp (1994). [8] Belytscko,., Lu, Y. Y. and Gu, L., Crack Propagaton by Element-Free Galerkn Metods, Engneerng Fracture Mecancs, Vol. 51, pp (1995). [9] Krysl, P. and Belytscko,., Analyss of n Plates by te Element-Free Galerkn Metods, Computatonal Mecancs, Vol. 17, pp (1995). [10] Flemng, M., Cu, Y. A., Moran, B. and Belytscko,., Enrced Element-Free Galerkn Metods for Crack p Felds, Internatonal Journal for Numercal Metods n Engneerng, Vol. 40, pp (1997). [11] Krongauz, Y. and Belytscko,., Enforcement of Essental Boundary Condton n Mesless Approxmaton Usng Fnte Elements, Computer Metods n Appled Mecancs Engneerng, Vol. 131, pp (1996). [12] Zu,. and Atlur, S. N., A Modfed Collocaton Metod and a Penalty Formulaton for Enforcng te Essental Boundary Condtons n te Element Free Galerkn Metod, Computatonal Mecancs, Vol. 21, pp (1998). [13] Mukerjee, Y. X. and Mukerjee, S., On Boundary Condtons n te Element-Free Galerkn Metod, Computatonal Mecancs, Vol. 19, pp (1997). [14] Gavete, L., Bento, J. J., Falcon, S. and Ruz, A., Implementaton of Essental Boundary Condtons n a Mesless Metod, Communcatons n Numercal Metods n Engneerng, Vol. 16, pp (2000). [15] Fernandez-Mendez, S. and Huerta, A., Imposng Essental Boundary Condtons n Mes-free Metods, Computer Metods n Appled Mecancs Engneerng, Vol. 193, pp (2004). [16] Lee, S. H. and Yoon, Y. C., Numercal Predcton of Crack Propagaton by an Enanced Element-Free Galerkn Metod, Nuclear Engneerng and Desgn, Vol. 227, pp (2004). [17] Gu, Y.. and Lu, G. R., A Coupled Element Eree Galerkn/Boundary Element Metod for Stress Analyss of wo-dmenson Solds, Computer Metods n Appled Mecancs Engneerng, Vol. 190, pp (2001). [18] Rao, B. N. and Raman, S., An Enrced Mesless Metod for Non-Lnear Fracture Mecancs, Internatonal Journal for Numercal Metods n Engneerng, Vol. 59, pp (2004). [19] Cook, R. D., Malkus, D. S. and Plesa, M. E., Concepts and Applcatons of Fnte Element Analyss, rd edton, Jon Wley & Sons, New York (1989). [20] mosenko, S. P. and Gooder, J. N., eory of Elastcty, rd edton, McGraw-Hll, New York (1986). Manuscrpt Receved: Feb. 15, 2006 Accepted: Apr. 19, 2006