Finite Element Simulation and Error Estimation of Polymer Melt Flow

Transcription

1 ISSN Rvista Matéria, v. 9, n. 4, pp , 004 ttp:// Finit Elmnt Simulation Error Estimation of Polymr Mlt Flow ABSTRACT M. Vaz Jr., E.L. Gartnr Dpartmnt of Mcanical Enginring, Cntr for Tcnological Scincs Stat Univrsity of Santa Catarina Campus Univrsitário, Joinvill-SC Brazil -mail: Accuracy analysis is fundamntal to ascrtain t rliability of a numrical solution of any industrial opration. T analysis is oftn complx du to diffrnt rror sourcs. In tis work, two stratgis to assss t discrtization rrors is discussd: (i) Ricardson xtrapolation (ii) a postriori rror stimation basd on projcting/smooting tcniqus. T formr utiliss tr nstd mss wras t lattr is basd on t post-procssing of t problm solution ovr only on ms. T modl accounts for t full intraction btwn t trmal ffcts causd by viscous ating t momntum diffusion ffcts dictatd by a sar rat tmpratur-dpndnt viscosity. Kywords: Finit lmnts, rror stimation, polymr mlt flow. 1 INTRODUCTION Polymr tcnology mould dsign av xprincd grat dvlopmnt in rcnt yars wit t advnt of nw blnds, forming procsss computational tools. Softwar packags aiming at simulation of polymr procssing, suc as xtrusion injction moulding, av bn stadily dvlopd, most of wic abl to l applications ranging from common ousold objcts to complx arospac componnts. Du to t grat varity of forming oprations, allid to t complxity of suc problms, most commrcial programs attmpt to combin practical rological modls wit approximat computational modlling tcniqus. Suc approac as amprd mor comprnsiv studis on t polymr baviour assssmnt of t intraction btwn t problm paramtrs. On t otr, in acadmia, as arly as t 50's 60's (s Croct t al. [4] rfrncs trin), pionring computational tcniqus, tortical approacs to solid mcanics involving inlastic dformations (.g. viscolastic modls usd in polymr xtrusion), t gnralisd Nwtonian modl (fluid flow dominatd by t sar viscosity frquntly usd in injction moulding) av bn proposd, using, owvr, simplifid dscription of t matrial proprtis. In rcnt yars, advancmnts in computational powr modlling stratgis av favourd numrical studis basd on mor ralistic matrial modls. T prsnt work prsnts t numrical modlling of polymr mlt flow in closd cannls using t Finit Elmnt mtod discusss stratgis for rror stimation basd on Ricardson xtrapolation projction/smooting tcniqus. GOVERNING EQUATIONS AND POLYMER RHEOLOGY Sinc t pionring work of Hibr Sn [8], t Hl-Saw modl as bn constantly adoptd to simulat two-dimnsional injction moulding. Tis modl approximats 3D polymr mlt flows btwn two flat plats assuming tat t gap ticknss is muc smallr tn t cannl or cavity caractristic lngt. Tis approac prsnts t grat advantag of allowing t drivation of a flow govrning quation basd on a D prssur fild ovr t mid-surfac ( constant prssur at t crosssction), wos boundary conditions consist in imposing t inlt flow rat or prssur at t gat zro prssur at t flow front. T succss of tis stratgy can b masurd by t numbr of works availabl in t litratur by its us in commrcial packags as MOLDFLOW [9] C-MOLD [3]. T drawback of t original Hl-Saw modl rsids in t valuation of t so-calld fluidity (intgral masur of t invrs of t viscosity ovr t cannl cross-sction), wic is calculatd basd on t powr law dscription of t viscosity combind wit an in-plan computation of t sar strain rat. A mor ralistic formulation would rquir a viscosity dscription consistnt wit t rological modl a point-wis valuation of t sar strain rat tmpratur ovr t gap ticknss. Furtrmor, t Hl-Saw approximation is valid only for tin cannls cavitis, wic imposs rstrictions on simulations of a Autor Rsponsávl: M. Vaz Jr.

2 JR VAZ, M., GAERTNER, E.L., Rvista Matéria, v. 9, n. 4, pp , 004. wid rang of plastic products componnts. On of t objctivs of t prsnt work is to assss t mlt flow baviour in tick cannls cavitis wn t sar strain rat tmpratur varis ovr t cross-sction. Undrsting of suc dynamics will lp futur dvlopmnts of mor comprnsiv computational modls for injction moulding. Numrical modlling of polymr mlt flow in tick cannls moulds rquirs a fully coupld solution of mcanical trmal problms. T mcanical problm consists of t simultanous solution of t constitutiv law, τ η ( γ&, T ) d η( γ&, T ) γ& (1) quilibrium laws, div [ η ( & γ, T ) w] p z () t trmal problm rquirs solution of t nrgy consrvation law, [ k T ] div + τ γ& 0 (3) wr τ is t sar componnts of t strss tnsor, d is t dformation rat, γ & is t sar strain rat tnsor, η is t apparnt viscosity, T is tmpratur, p/ z is t prssur gradint, k is t trmal & 1/ conductivity, γ ( 1/ γ & : γ& ) η & is t quivalnt sar strain rat, τ γ is t quivalnt sar strss..1 Polymr Rology Equations (1)-(3) ar complmntd by dfining t rological baviour of t polymr mlt troug its sar viscosity. Attmpts to modl baviour of diffrnt matrials gav ris to a larg spctrum of quations, amongst wic t laws of Carrau (.g. Bao []) Cross (.g. Vroyn Duprt [14]) ar t most widly rfrrd. In t prsnt work, t rlationsip btwn viscosity, sar rat tmpratur for Polyactal POM-M90-44 is xprssd wit good accuracy using t Cross constitutiv quation [7], η ( γ&,t ) 1+ η ( T) 0 1 n( T ) [ λ( T) γ& ], (4) wr η 0 is t Nwtonian viscosity, n is t powr-law indx λ is a matrial paramtr, wic in turn, ar approximatd troug Arrnius law. η 0 (T) a 1 a xp T, b λ(t) b1 xp T c n( T) c1 xp T, (5) in wic a 1, a, b 1, b, c 1 c ar matrial constants. 3 FINITE ELEMENT METHOD T Finit Elmnt Mtod - FEM is a numrical tcniqu for obtaining approximat solutions basd itr on variational principls or wigtd rsiduals. T FEM solutions basd on Galrkin wigtd rsiduals av bn givn prfrnc ovr variational mtods du to t difficulty to xprss gnral at transfr fluid dynamic problms in a variational form. T Galrkin mtod assums tat t xact solution can b intrpolatd from n C 0 -continuous sap (or trial) functions N i, so tat t unknown paramtrs, i, can b obtaind by assuming tat t wigtd rsidual of t govrning quations corrsponding boundary conditions is minimum [16]. Application to a gnral quation, div[γ ]+ S 0, rprsnting () (3) yilds 454

3 JR VAZ, M., GAERTNER, E.L., Rvista Matéria, v. 9, n. 4, pp , 004. δ ( div [ Γ ] + S ) d + δ ( Γ n q ) da 0 (6) wr n is t outward normal vctor to surfac q dnots prscribd at transfr sar strss for t nrgy momntum quations rspctivly. T discrt quations ar dtrmind by applying t divrgnc torm to t scond trm of Equation (6) by assuming tat bot t problm solution,, wigt functions, δ δ, ar intrpolatd using t sam trial functions (Galrkin approximation), so tat. K 0 T 0 T ft + KW W fw 0 0, (7) in wic T W ar t nodal tmpraturs vlocitis rspctivly, K T K W ar t so-calld stiffnss matrics, K T N T k 0 0 N d k f T f W ar t forc vctors. K W ( ) T η T, & γ 0 N N d 0 η( T, & γ ), (8) f W p N d z f T N τ γ& d, (9) for t tmpratur vlocity problms rspctivly. As discussd in t prvious sctions, Equation (7) is coupld troug t viscosity viscous ating. In tis work, a staggrd approac is adoptd, in wic t tmpratur vlocity quations ar solvd in succssion until a pr-dfind rsidual is racd. Tis procss is stabl, owvr, gratr convrgnc rats can b acivd using t Nwton-Rapson itrativ procdur. 4 ERROR ASSESSMENT Error computation is on of t most important issus in numrical simulation of any industrial opration. Accuracy convrgnc analyss ar fundamntal to giv t packag usr a ncssary confidnc in t rsults. T most rlvant contribution of tis work is t comparativ assssmnt of two rror stimation stratgis. Evry numrical solution is unavoidably subjct to rrors, wic can b gnrally classifid in tr typs: modlling rrors, discrtization rrors itration rrors. Modlling rrors ar du to t diffrnc btwn t actual pysical baviour t matmatical modl. Discrtization rrors consist of t diffrnc btwn t xact solution of t govrning quations t corrsponding solution of t algbraic systm. Itration rrors ar dfind as t diffrnc btwn t itrativ xact solutions of t algbraic quation systm [5]. Tis work is particularly concrnd wit t assssmnt of t discrtization rrors. Suc rrors ar primarily associatd wit bot t ms (structurd, unstructurd, uniform, tc.) t discrtization mtod (Finit Elmnts, Finit Volums, Boundary Elmnts, Spctral mtods, tc.). Tis work utiliss two stratgis to valuat t discrtization rrors: Ricardson xtrapolation [1] a postriori rror stimation basd on smooting tcniqus [17]. Sctions prsnt a brif summary t radr is rfrrd to Obrkampf Trucano [10] Zinkiwicz Zu [17] for furtr considrations. 4.1 Ricardson xtrapolation On of t first attmpts to incras t convrgnc rat of a numrical solution was proposd by Ricardson [1], wo ascrtaind tat a igr ordr solution could b acivd assuming tat t xact solution is dscribd as. 455

4 JR VAZ, M., GAERTNER, E.L., Rvista Matéria, v. 9, n. 4, pp , 004. xact p+ ( ) p 1 ( x) ( x) + ε ( x) ( x) + α + O (10) in wic xact is t xact solution of t govrning quations at a givn point x, is t discrt solution, ε is t discrtization rror, is t ms siz, α is a constant p is t rror ordr. A saf application of t Ricardson xtrapolation rquirs tr assumptions: (i) t xact solution must b smoot noug tat t Taylor sris xpansion for t rror is justifid; (ii) t formal convrgnc ordr, p, is known; (iii) t ms spacing is sufficintly small suc tat t lading-ordr rror trm dominats t total discrtization rror, i.., t convrgnc is monotonic in t asymptotic rang [10]. From Equation (10), it is possibl to sow tat t discrtization rror, ε, can b stimatd using two diffrnt nstd mss, 1 (assuming tat / 1 > 1), as. ε xact 1 r 1 p 1 (11) wr r / 1 is t rfinmnt ratio. T xact ordr of t discrtization rror is ardly known a priori, spcially in non-linar problms. Trfor, it is always rcommndd to us an stimat,, wic can b asily dtrmind by assuming asymptotic convrgnc ( vrifid) by applying quation (11) to tr nstd mss, 1, 3, using a constant rfinmnt ratio, so tat * p p * log 3 1 log () r 3 r 1. (1) T global masur can b dfind by t rror norm as ε nod i 1 nod ( ε ) i i 1 1 p r 1 i, (13) wr nod rprsnts t total numbr of nods corrsponds to nodal vlocitis, W, or nodal tmpraturs, T. Altoug t tcniqu dscribd prviously can l only uniform mss, xtnsions av also bn dvlopd for non-uniform grids [6,13]. Dspit its simplicity widsprad us in simulations of Nwtonian fluid flow, t Ricardson xtrapolation sows som sortcomings: t xtrapolatd solution is not consrvativ in t sns of t consrvation laws t accuracy of t xtrapolation dos not apply to ig drivativs of t solution [13]. Furtrmor, t prsnc of singularitis or nonlinaritis may compromis t monotonic rat of convrgnc, as furtr discussd in sssion A postriori smooting tcniqu Error stimation as bcom an important issu in computational mcanics sinc t lat svntis wn Babuška Rinboldt [1] proposd a postriori rror stimats for t Finit Elmnt mtod using norms of Sobolv spacs. T topic gaind momntum wn Zinkiwicz Zu [17] introducd rror stimats basd on post-procssing tcniqus of t Finit Elmnt solutions, wic could asily b usd in association wit ms rfinmnt stratgis. Du to its rlativ simplicity, t original Zinkiwicz Zu s scm as bn xtndd to a wid spctrum of applications, ranging from linar solid mcanics to t igly non-linar mtal forming problms, suc as forging, blanking macining. T stratgy as also bn mployd succssfully by Wu t al. [15] for solving t incomprssibl (Nwtonian) Navir-Stoks fluid flow around a cylindr using an rror stimat basd on t Enrgy norm, E T [( τ τ ) ( γ& - γ& ) ( p p )( tr[ ε &] tr[ ε& ])] d, (13) wr τ, p, γ & ε&, ar t dviatoric strsss, prssur, sar normal strain rats of t xact & solution, τ, p, γ ε& ar tos of t Finit Elmnt solution, tr[ ] dnots t trac of a tnsor. T autors mpasis tat, for a continuous prssur fild, t diffrncs btwn p p ar vanisingly 456

5 JR VAZ, M., GAERTNER, E.L., Rvista Matéria, v. 9, n. 4, pp , 004. small t last trm of (13) may b disrgardd. Tis tcniqu can also b usd in conjunction wit otr rror norms [16]. In t prsnt work, bot t nrgy norm t L norm associatd wit t quivalnt sar strss quivalnt sar strain rat, E [( τ τ )(& γ - & γ )] d L [( τ τ )( τ τ )] d, (14) av bn usd. T nrgy norm, as dscribd in (14)a, as a spcial pysical significanc wn compard to t nrgy consrvation law (3). Tis rror masur is intrinsically associatd wit t viscous ating, wic in turn, can captur t ffcts of t vlocity tmpratur filds troug t sar strain rat viscosity. T stard post-procssing of t Finit Elmnt solution provids t sar strain rat tnsor at Gauss Points [16], so tat & 1 γ γ& : γ& τ η ( γ&, T )γ&. (15) In addition, t ky to tis mtod is t stimation of a igr ordr approximation to t xact solution at t intgration points basd on t availabl Finit Elmnt solution. T most usd tcniqu + + τ compriss two stps: (i) obtaining nodal valus, & γ, from t post-procssd Finit Elmnt solution, * * τ γ& τ, (ii) followd by an intrpolation to stimat γ at t intgration points. T formr uss itr xtrapolation, supr-convrgnt patc rcovry, local or global wigtd rsiduals (or L projction), wras t lattr can b asily prformd using t Finit Elmnt sap functions [16,17]. In tis work t Gauss point variabls ar xtrapolatd to nods using polynomials, so tat t stimatd "xact" solution is: * T + T + { τ } { N }{ τ } { N }[ C ] { τ } {& T } { }{& + γ } { N T + γ * N }[ C ] {& γ }, (16) in wic C + is t Gauss-Point-to-nod intrpolation matrix. Trfor, t stimatd global rrors for nrgy L norms ar valuatd as: E nl nl j E j 1 j 1 j * * [( τ τ )(& γ & γ )] d j L nl nl j L j 1 j 1 j * * [( τ τ )( τ τ )] d j, (17) wr j rprsnts t intgral ovr an lmnt j nl t total numbr of lmnts. T prvious scm as t advantag of computing t rror stimat using only on ms. Dspit its xtnsiv us in almost vry fild of computational mcanics, tis tcniqu as t disadvantag of stimating rrors basd on scondary masurs (strsss strain rats), i.., t convrgnc rat of t actual unknowns (nodal vlocitis nodal tmpraturs) ar not dirctly accountd for. Bao [] prsnts an altrnativ approac to rror stimation basd on Sobolv spacs for gnralizd Nwtonian fluids using Carrau viscosity law. It is worty to not tat rsarc on rror stimation for polymr moltn flow is still in its infancy furtr invstigation to assss accuracy applications to complx flows ar ndd. 5 NUMERICAL EXAMPLES T most common approac to mould filling usd in commrcial simulation packags [3,9] is basd on t Hl-Saw approximation. Introducd by Hibr Sn [8], tis modl is drivd from avragd masurs of t quivalnt sar strain rat viscosity ovr t cannl cross-sction trby rstricting its application to tin cannls cavitis. T prsnt work prsnts a brif assssmnt of polymr mlt flow in tick cannls rror stimation using fully coupld govrning quations. T xampl discusss t 457

6 JR VAZ, M., GAERTNER, E.L., Rvista Matéria, v. 9, n. 4, pp , 004. Finit Elmnt solution for a 5 x 5 mm cannl using four-noddd quadrilatral lmnts four intgration points. T matrial ms data ar prsntd in Tabl 1. Tabl 1 - Matrial constants for Polyactal POM-M90-44 otr simulation paramtrs. Paramtr Symbol Valu cross-sction H x W 5 x 5 mm Cannl lngt L 50 mm wall tmpratur Tw 493 K ( 0 C ) Flow rat Q 3,69 cm 3 /s Prssur drop Δp 11,0 MPa Trmal conductivity k 0,31 W/m.K Spcific mass ρ 1143,9 Kg/m 3 Viscosity paramtrs a1 0,0603 a 5003,01 b1 1,645E-6 b 3901,0 c1 1,3574 c 653,73 Ricardson mss 1 < < x 9 5 x 5 3 x 3 17 x 17 9 x 9 5 x 5 33 x x 17 9 x 9 65 x x x x x x Equivalnt sar strain rat polymr viscosity T formulation usd in tis work is abl to provid a point-wis distribution of t sar strain rat & polymr viscosity insid t cannl. Figur 1(a) sows a significant variation of γ ovr t cannl cross-sction, wit maximum valus at t cannl walls zro at its cntr cornrs. It is important to mntion tat t quivalnt sar strain rat is associatd wit t scond invariant, J, of t sar strain rat & 1/ 1/ tnsor, γ (1/ γ& : γ& ) (1/ J ), trfor, indpndnt of t co-ordinat systm. T polymr viscosity, as illustratd in Figur 1(b), is dirctly affctd by t tmpratur sar rat according to Equation (4). Low sar rats tmpraturs yild igr viscositis. A bttr prcption of tis baviour is prsntd in Figur, wic sows nod valus of γ &, T η along t Y-Y' symmtry lin. T simulation igligts t opposing ffcts of t sar rat tmpratur on t polymr flow, i.., at cntr t tmpratur is maximum t sar rat is zro, at t sid walls t tmpratur is minimum t sar rat is maximum. T combination of bot ffcts causs t viscosity to yild minimum valus at approximatly 1/4 of t distanc btwn t walls t cntr (y 0,65 mm). Ts rsults sows sarp contrast to t Hl-Saw assumptions strongly suggsts furtr invstigation on injction moulding of tick-walld componnts using a fully coupld matrial modl. 458

7 JR VAZ, M., GAERTNER, E.L., Rvista Matéria, v. 9, n. 4, pp , 004. Figur 1: Equivalnt sar strain rat viscosity distribution. Figur : Equivalnt sar strain rat, tmpratur viscosity along t Y-Y' symmtry lin. 5. Error assssmnt A suitabl quantification of t solution accuracy is crucial to stablis confidnc in t numrical modl. T instinctiv assumption tat by rfining t ms on obtains mor accurat rsults is not always tru. Som aspcts on tis issu wr addrssd by Pinto [11], wo dmonstratd tat t prssur fild providd by a wll-known commrcial packag was not consistnt wit subsqunt ms rfinmnt. Trfor, rror assssmnt is a fundamntal stp during t dvlopmnt of a numrical modl. Tis sction addrsss t discrtization rrors stimatd using Ricardson xtrapolation t Enrgy L rror norms dscribd in sctions rspctivly. 459

8 JR VAZ, M., GAERTNER, E.L., Rvista Matéria, v. 9, n. 4, pp , 004. Figur 3: Local rror distribution using Ricardson xtrapolation. Figur 3 illustrats local rror distributions basd on Ricardson xtrapolation using mss 1 65 x 65, 33 x x 17 nods (r ). It is intrsting to not tat rrors follow quit a distinctiv pattrn for vlocitis tmpraturs. T formr sows larg rrors nar locations of minimum viscositis (s Figur 1b) wras t lattr prsnts larg rrors for ig tmpraturs at t cannl cntr. Dtails of Ricardson xtrapolation can b obsrvd in Figur 4, wic prsnts t rror stimats for vlocitis along t Y-Y symmtry lin. T simulations sow tat ms rfinmnt sits t maximum rrors from t cntr towards t cannl walls, nar rgions of low viscositis. Figur 4: Ricardson rror along t Y-Y symmtry lin. It as bn found tat t coupld caractr of t problm t ig matrial nonlinarity can indr t monotonic rat of convrgnc rquird by Ricardson stimats. Tis difficulty was also obsrvd by Obrkampf Trucano [10] in t contxt of Nwtonian flows using Finit Diffrncs. T analyss sow points of ig convrgnc rats (p 5) clos to points of virtual divrgnc (p -1) nar t cornrs, wic suggsts gratr car on t application of Ricardson xtrapolation to tis class of problms. 460

9 JR VAZ, M., GAERTNER, E.L., Rvista Matéria, v. 9, n. 4, pp , 004. Howvr, dspit t indrancs, tis stimat bcoms attractiv du its capacity valuat rrors dirctly from primal variabls. Estimats using rror norms ar illustratd in Figur 5 for a 65 x 65 ms. It can b obsrvd tat bot norms provid vry similar rror pattrns, i.., larg rrors at t cntr nar low viscosity rgions. Tis baviour may b crditd to t combind ffct, in spcial t minimum valus, of t quivalnt sar strain rat (Figur 1a) viscosity (Figur 1b). T similarity of bot rror norms ar igligtd in Figur 6 wic sows t distributions along t Y-Y symmtry lin. T largr gradints providd by t Enrgy norm rcommnds its us in combination wit automatic r-msing procdurs. Figur 5: Local rror distributions for Enrgy L norms. Figur 6: Error norms along t Y-Y symmtry lin. Analysis of global convrgnc is important to stablis bot t robustnss of t computational modl t asymptotic rat of convrgnc. Figur 7 prsnts t global convrgnc curvs for bot Ricardson xtrapolation (Equation 13) Enrgy L norms (Equation 17). T litratur indicats tat, in t absnc of singularitis, for linar analysis of solid matrials [16] Nwtonian flows [15], t global discrtization rror using t classical FEM norms is proportional to q, wr is t lmnt siz q is 461

10 JR VAZ, M., GAERTNER, E.L., Rvista Matéria, v. 9, n. 4, pp , 004. t polynomial ordr of t approximation. Tis xampl uss linar lmnts (q 1) simulations sow tat t convrgnc rat xibitd by t Enrgy L norms follows t sam rul, so tat E L,011 8,011 0,996 1,0003 (18) Dspit t sortcomings of Ricardson stimat prviously discussd, t corrsponding global rrors for vlocity tmpratur, ε ε W T, , ,038 0,7944 [ mm / s] [ K] (19) Sow a convrgnc rat similar to Enrgy L norms. Furtrmor, t global analysis of Ricardson stimat suggsts tat t computational modl is indd a scond-ordr scm according to * p,3 Equation (1), wit an avrag convrgnc ordr p * W T, 5 for t vlocity tmpratur q quations rspctivly. It is wort noting tat is associatd wit t asymptotic rat of convrgnc, dfind as a global masur computd ovr t problm domain, wras is rlatd to t ordr of t scm, calculatd locally at t nods. T formr stimats global discrtization rrors t lattr approacs local approximation rrors. p Figur 7: Global rat of convrgnc. 6 CONCLUDING REMARKS Tis work addrsss aspcts of numrical formulation rror stimation stratgis for analysing polymr mlt flow in tick cannls cavitis. T govrning quations ar solvd using t Finit Elmnt mtod in association wit linar quadrilatral lmnts. In contrast to t classical Hl-Saw approximation, t fully coupld modl usd in t simulations is abl to captur point-wis variations of t sar strain rat viscosity ovr t cannl cross-sction. Dspit incrasing us of commrcial packags to simulat injction moulding procsss, t litratur is xtrmly poor on rror analysis of tis class of problms. Accuracy convrgnc analyss ar prformd using rror stimats basd on Ricardson xtrapolation Enrgy L norms. T formr as t advantag of valuating rrors dirctly form t primal variabls (vlocity tmpratur), owvr t rquirmnt of monotonic convrgnc can compromis analyss in igly nonlinar coupld problms. Enrgy L norms av bn widly mployd in computational mcanics av constitutd t natural coic to prform rror assssmnt wn using Finit Elmnts. Error prdictions using rror norms Ricardson xtrapolation wr found vry similar in t prsnt analysis. T maximum local rrors stimatd by Enrgy L norms occur nar rgions of low viscositis at t 46

11 JR VAZ, M., GAERTNER, E.L., Rvista Matéria, v. 9, n. 4, pp , 004. cannl cntr. T stimats basd on Ricardson xtrapolation prsnts diffrnt pattrns for vlocitis tmpraturs. T formr indicats largr rrors nar rgions of low viscositis t lattr at t cannl cntr. T global rror analysis sows tat t Finit Elmnt modl usd in t simulations approacs a scond-ordr scm (as indicatd by t Ricardson xtrapolation) prsnts an asymptotic rat of convrgnc (as vincd by t rror norms). T issus discussd in tis papr ar of paramount importanc strongly suggst furtr invstigation on rror convrgnc analysis for complx polymr flows during t mould filling stag. 7 REFERENCES [1] BABUŠKA, I., RHEINBOLDT, W.C., A Postriori Error Estimats for t Finit Elmnt Mtod, Intrnational Journal for Numrical Mtods in Enginring, v.1, p , [] BAO, W., An Economical Finit Elmnt Approximation of Gnralizd Nwtonian Flows, Computr Mtods in Applid Mcanics Enginring, v.191, p , 00. [3] C-MOLD. Dsign Guid. A Rsourc for Plastic Enginrs. Itaca: Advancd CAE Tcnology Inc., [4] CROCHET, M.J., DAVIES, A.R., WALTERS,K., Numrical Simulation of Non-Nwtonian Flow. Amstrdam: Elsvir, [5] FERZIGER, J.H., PERIĆ, M., Computational Mtods for Fluid Dynamics. nd dition. Hidlbrg: Springr-Vrlag, [6] FERZIGER, J.H., PERIĆ, M., Furtr Discussion of Numrical Errors in CFD. Intrnational Journal for Numrical Mtods in Enginring, v.3, , [7] HERRMANN, M.H., Simulação d Procssos d Injção d Polímros m Canais Circulars pags. Dissrtation (Mstrado m Ciência Engnaria d Matriais) - Stat Univrsity of Santa Catarina, Joinvill, 001. [8] HIEBER, C.A., SHEN, S.F., A Finit-Elmnt / Finit-Diffrnc Simulation of t Injction-Moulding Filling Procss. Journal of Non-Nwtonian Fluid Mcanics, v.7, p.1-3, [9] MOLDFLOW. Matrial Tsting Ovrviw. Wayl: Moldflow Corporation, [10] OBERKAMPF, W.L., TRUCANO, T.G., Vrification Validation in Computational Fluid Dynamics, Progrss in Arospac Scincs, v.38, p.09-7, 00. [11] PINTO, M.A.G.A., Anális dos Rsultados d Simulação d Injção m Aplicativo Comrcial, ntr Modlos Matmáticos Basados m Casca Média Casca Extrna pags. Dissrtation (Mstrado m Ciência Engnaria d Matriais) - Stat Univrsity of Santa Catarina, Joinvill, 001. [1] RICHARDSON, L.F., T Approximat Aritmtical Solution by Finit Diffrncs of Pysical Problms Involving Diffrntial Equations wit an Application to t Strsss in a Masonry Dam, Transactions of t Royal Socity of London, Sris A, v.10, p , [13] ROACHE, P.J., Quantification of Uncrtainty in Computation Fluid Dynamics, Annual Rviw of Fluid Mcanics, v.9, p , [14] VERHOYEN, O., DUPRET, F., A Simplifid Mtd for Introducing Viscosity Law in t Numrical Simulation of Hl-Saw Flow, Journal of Non-Nwtonian Fluid Mcanics, v.74, p.5-46, [15] WU, J., ZHU, J.Z., SZMELTER, J., ZIENKIEWICZ, O.C., Error Estimation Adaptivity in Navir- Stoks Incomprssibl Flows, Computational Mcanics, v.6, p.59-70,

12 JR VAZ, M., GAERTNER, E.L., Rvista Matéria, v. 9, n. 4, pp , 004. [16] ZIENKIEWICZ, O.C., TAYLOR, R.L., T Finit Elmnt Mtod. 5t d. London: Buttrwort- Hinmann, 000. [17] ZIENKIEWICZ, O.C., ZHU, J.Z., A Simpl Error Estimator Adaptiv Procdur for Practical Enginring Analysis. Intrnational Journal for Numrical Mtods in Enginring, v.4, p ,