Zeitschrift Kunststofftechnik Journal of Plastics Technology

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1 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. Zetschft Kststofftechk Joal of Plastcs Techolog Pof. D. Tm A. Osswald, D. Iva Lóez-Gómez, Polme Egeeg Cete, Uvest of Wscos, Madso, USA D. Iva Lóez-Gómez, Oma A. Estada, Plastc ad Rbbe Isttte ICIPC, Medell, Colmba egeecht/haded : ageomme/acceted: Modelg ad Smlato of Polme Pocessg sg the Radal Fctos Method Ths ae descbes the alcato of the Radal Fctos Method to model ad smlate olme ocessg. The coled sstem of eeg ad moto eqatos fo lama two-dmesoal flows stead state of comessble o-ewtoa flds s cosdeed. The eslts delveed b RFM ae comaed wth mecal eslts of othe methods, ad wth aaltcal soltos based o smlfed mathematcal models. The cases eseted ae: o-sothemal esse flow thogh a slt fo o- ewtoa flds ad o-sothemal owe law flow the caledeg ocess. Ato/atho Pof. D. Tm A. Osswald Polme Egeeg Cete 1513 Uvest Avee Uvest of Wscos-Madso Madso, WI D. Iva Lóez-Gómez, Oma A. Estada Plastc ad Rbbe Isttte ICIPC Ca 495 S-19 Medell, Colmba E-Mal-Adesse: [email protected] Websete: Tel.: E-Mal-Adesse: [email protected] Websete: Tel.: Fa: Wsseschaftlche Abetskes de Uvestäts- Pofessoe de Kststofftechk achvete, ezesete Iteetzetschft des Wsseschaftlche Abetskeses Kststofftechk WAK achval, evewed ole Joal of the Scetfc Allace of Polme Techolog Cal Hase Velag Zetschft Kststofftechk/Joal of Plastcs Techolog 3 7

2 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. MODELIG AD SIMULATIO OF POLYMER PROCESSIG USIG THE RADIAL FUCTIOS METHOD Ivá Lóez-Gómez a b a, b, Oma Estada a, Tm Osswald b Plastc ad Rbbe Isttte ICIPC, Medell, Colomba. Polme Egeeg Cete, Uvest of Wscos - Madso, USA. Ths ae descbes the alcato of the Radal Fctos Method to model ad smlate olme ocessg. The coled sstem of eeg ad moto eqatos fo lama two-dmesoal flows stead state of comessble o-ewtoa flds s cosdeed. The eslts delveed b RFM ae comaed wth mecal eslts of othe methods, ad wth aaltcal soltos based o smlfed mathematcal models. The cases eseted ae: osothemal esse flow thogh a slt fo o-ewtoa flds ad osothemal owe law flow the caledeg ocess. 1 ITRODUCTIO Mathematcal modelg attemts to mmc the actal ocess wth eqatos b meas of mateal, eeg ad mometm balaces, alog wth a sees of assmtos that smlf the model. The moe comle the mathematcal model, the moe accatel t eesets the actal solto. Evetall, the comlet s so hgh that we mst esot to mecal smlato to model the ocess. Fo the atcla case of olme flows, the modelg eslts hghl olea atal dffeetal eqatos de to the o-ewtoa ad vscoelastc behavo of olmes. Addtoall, ma ocesses, eeg ad moto eqatos ae stogl coled, de to hgh Bkma B ad Péclet P é mbes, obtag models that ae dffclt to solve. mecal soltos fo the atal dffeetal eqatos that descbe olme flows have bee stded b meas of fo wdel kow methods: Fte Dffeeces FDM, Fte Volmes FVM, Fte Elemets FEM ad Boda Elemets BEM. Howeve, these methods have ot bee comletel satsfacto to solve olme ocessg models, eqg the eloato of othe alteatves. FEM, FDM ad FVM ae methods that eqe the geeato of meshes o fom gds. These meshes ae comosed of odes ad elemets, eqg a 6 1

3 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. comle fomlato to descbe the coectve elato betwee odes ad the heach betwee the elemets [4]. Those methods ae atclal lmted fo solvg oblems wth domas that sffe hgh defomatos, becase the meshes ad gds devate accetabl the eslts [8]. Hgh defomato domas ae commol fod olme ocessg, sch as that ecoteed tw scew etso. The Boda Elemet Method BEM s a owefl tool fo the solto of atal dffeetal eqatos scece ad egeeg. Oe of the advatages of ths method s the edcto of the sace dmesos b oe t fo homogeeos oblems. Howeve, the beefts of sg BEM dmsh fo olea o o-homogeeos oblems sch as the smlato of o- ewtoa flds, becase fomato of the doma s eqed to solve the volme tegals [19, 18] comlcatg the e-ocess. Whe dealg wth hghl o-lea behavo of olmes, the alcato of BEM s lmted. I the last 15 eas, meshless methods based o adal bass fctos have attacted atteto to solve PDEs. I the begg, the adal bass fctos wee develoed to teolate mltvaate data ad mltdmesoal oblems. I 199, Kasa [1] oosed the se of RBFs fo the solto of atal dffeetal eqatos. A smmetcal coeffcets mat s obtaed whe ths method s aled. The comtatoal mlemetato of Kasa s method s athe staghtfowad de to the smle stcte of the RBF fomlato, makg t sefl ot ol to mathematcas, bt to othe ses as well [5]. Ma- De ad Tae [17] sed the techqe to model o-ewtoa fld flow o shea thg ad vscoelastc lqds. The ma advatage of the adal fctos method RFM s that t s a techqe that does ot eed doma o boda meshes as eqed wth FEM ad BEM, o homogeeos gd ots as FDM, to solve atal dffeetal eqatos. Essetall, t s a meshless techqe based o collocato methods. The method has ove to be ve accate comaed to othe mecal techqes, eve fo a small mbe of collocato ots [1]. The se of collocato methods wth adal bass fctos fo the solto of atal dffeetal eqatos has bee stded b ma athos sch as Kasa [1, 14], Che [3], Fasshae [7, 8, 9], Fake [1], Powe [1], Schaback [3] ad Wedlad [5]. KASA S USYMMETRIC METHOD Collocato techqes ae based o the fact that a feld vaable a cotos sace ca be aomated wth lea teolato coeffcets ad bass fctos evalated o dscete ots skled wth the doma. I ths wa, f some vales of the feld ae kow, a vale at a ot of the doma ca be estmated [5]. The bass fctos of Kasa s method ae the adal bass fctos RBF [, 3, 1, 13]. The fomlato s: 6

4 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. F, = α 1 whee s the dstace betwee the odes ad, eesets the mbe of collocato ots ad s the adal bass fcto. If we al a dffeetal oeato L o eq. 1 we ca teolate the solto fo the oeated feld: { } α L{ F, } = L I ths wa, oeated felds ca be teolated allowg the solto of atal dffeetal eqatos PDEs, ad hece oblems that ae modeled b meas of PDEs.[7, 8,, 5, 6, 16]. Thee ae ma tes of adal bass fctos of global sot that ca be sed, sch as Polhamoc Sle SP, Polhamoc Th - Plate Sle TPS, Mltqadcs MQ, Wedlads, Gassa ad othes [, 3, ]. Fake [11] evalated some adal bass fctos fom the teolato accac ot of vew, fdg the best eslts sg the Polhamoc Th - Plate Sle TPS. TPS s gve b: a = l 3 3 RFM I POLYMER PROCESSIG Whe solvg the eeg ad moto eqatos that defe a olme flow, fo dffeet ma state vaables mst be smltaeosl solved fo: temeate, esse ad the two comoets of veloct, as show esectvel 4, 5, 6 ad 7. T = = T α 4 = = β 5 λ 6 ξ 7 Addtoall, the adal bass fctos ca be sed to estmate the devatves of oetes that ae a fcto of the ma vaables. I ths atcla case, the 6 3

5 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. vscost feld deeds o the temeate ad the defomato ate, theefoe, we ca wte:, = δ 8 δ δ I the above eqatos, v, T ad ae the adal bass fctos to teolate kow felds, that ma ot be the same wth the am of bette accac. Ths deeds o the dffeetal oeatos volved the goveg eqatos [6]. The coeffcets α, β, λ, ξ ad δ ae the teolato coeffcets fo temeate, esse, -comoet of veloct, -comoet of veloct ad vscost, esectvel. Fall, s the mbe of ots o the boda ad wth the doma ad s the mbe of ots wth the doma ad o the Dchlet esse bodaes. 3.1 Eeg Eqato A geeal stead-state eeg balace fo olme flows -D domas cldg heat geeated b vscos dssato ad the tasot of eeg b covecto s eseted 11 ad ts fomlato tems of Radal Bass Fctos ad teolato coeffcets eq 1. ρc T T 4 6 = k T γ T T ρc k T α = γ 1 The boda codtos mst also be wtte RFM fom. Dchlet ad ema boda codtos ae cosdeed. The Dchlet boda codto s descbed b eq.13 ad ts eesetato RFM b eq. 14. The ema boda codto s descbed b eq.15 ad ts eesetato RFM b eq. 16, whee s the t vecto omal to the boda the vct of ode. T Γ D = Ta ; 13

6 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 6 5 a T = T α 14 q T T k Γ ; = 15 q k T = α Cott ad Moto Eqatos I a smla fasho as wth the eeg eqato, we ca also aomate the cott eqato ad the eqato of moto sg adal bass fctos. The cott eqato fo a comessble fld eseted eq. 17 ca be aomated sg the RFM b e-wtg t as eq. 18. = 17 = ξ λ 18 Fo the eqato of moto, bod foces ad the tasot of mometm b covecto ae eglected, de to the low Reolds mbe of olme flows. The vscost s cosdeed as a fcto of ate of defomato ad temeate. The ad comoets of the moto eqato ae eseted eqs 19 ad ad the eesetato RBFs eqs 1 ad. = 19 = = β ξ λ ρ 1 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele.

7 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 6 6 = β λ ξ ρ Fo the oblems dscssed ths wok, Dchlet boda codtos of veloct ad esse ae cosdeed. The Dchlet veloct boda codtos ae gve b eqs. 3 ad 4 ad the ae aomated sg RBFs eqs. 5 ad 6 D a Γ ; = 3 D a Γ ; = 4 a = λ 5 a = ξ 6 The esse Dchlet boda codto, fo a fll develoed veloct ofle, ca be wtte as D a Γ ; = 7 D Γ ; = 8 D Γ ; = 9 I tems of RBF the eesetatos ae: a = β 3 = λ 31 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele.

8 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. ξ = 3 The eeg ad moto eqatos have olea tems. The eesetato b meas of RBF, as eqatos 1, 18, 1 ad, allows the leazato sch a wa that the esltg sstem of eqatos ca be solved sg a method fo the solto of lea sstems fo each ste a teatve ocess. I o case, the sstem coveges b meas of a Jacob teato o sg the ewto-rhaso method. Althogh the Jacob teato method eqes a hghe mbe of teatos to covege, the comtg tme s less tha the eqed fo the ewto-rhaso method, de to the fact that the latte seds hgh comtatoal esoces bldg the Jacoba mat. 3.3 Rheologcal Models The ewtoa, eq.33, Powe Law, eq.34 ad Caea, eq.35 models ae sed to eeset the fld vscost. = a T 33 = a m 1 T γ 34 a A = T 1 Ba c Tγ 35 The Ahes model s sed to calclate the temeate shft facto: a T = e U 1 1 R T T ef 36 The magtde of the defomato ate s calclated sg the secod vaat of the defomato ate teso: 1 1 γ = γ : γ = II

9 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. 4 APPLICATIOS I POLYMER PROCESSIG 4.1 o-sothemal esse flow betwee aallel lates of ewtoa ad o-ewtoa flds I ma flows olme ocessg, vscos dssato s sgfcat B > 1, wth a esltg temeate se that affects the flow thogh a temeate deedet vscost. To smlate ths heomeo, the coled moto ad eeg eqatos of a slt flow wee solved, cosdeg the heat geeato fo vscos dssato wth a doma of sze.15m.15m. I Fg.1 the geomet ad the locato of the coodate sstem ae llstated. Fo the smlato, two dffeet tes of ode dstbtos wee sed: a tagla dstbto ad a adom dstbto, both wth 74 odes, as dected Fg.. The adom dstbto s actall a sem-adom dstbto becase a mmal dstace betwee odes s mosed ode to avod lea deedeces betwee the eqatos of ve close odes, ad to gaatee a good dstbto ove the whole doma. Fo ths atcla case, the mmm 4 dstace betwee odes s.8 1 m. Fge 1: Geomet of the esse flow betwee aallel lates T h T L The boda codtos fo the mometm balace ae: v, = m/s 38 v,.15 = m/s 39, = = Pa 4.15, = L = 15 Pa 41 Ad fo the eeg balace: T, = T = C 4 T, = T = C 43 h 8 6 L h

10 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. [m].1.5 T, T L, = = a. Aaged Dstbto 74 odes [m] [m] b. Radom Dstbto 74 odes [m] Fge : Dstbto of odes fo esse flow betwee aallel lates The solto was obtaed sg a secod-ode TPS to teolate the esse ad temeate felds, ad a thd-ode TPS to teolate the veloct feld. To comae the eslts, a solto sg the FDM s fomlated. The FDM solto s cosdeed as the efeece, becase t ses a oe-dmesoal fomlato ad a hgh mbe of odes to aoach the solto 1 odes. The FDM solto s fomlated wth the followg eqatos: P v v = 44 L T k γ = 45 The fld has a dest ρ = 7 kg/m 3, a secfc heat C = 1 J/kg-K ad a themal codctvt k =.8 W/m-K. Fo the heolog, fo dffeet cases ae cosdeed: CASE 1: = µ. The vale of the vscost s assmed eqal to a costat, ths case = Pa-s. CASE : = f T. The vscost has a ewtoa behavo ad the temeate deedece s descbed b the Ahes eqato, eq.36, wth U = J/mol, µ = Pa-s ad T ef = 493 K. CASE 3: = f γ The vscost esets o-ewtoa behavo, whch s eeseted b meas of the Caea model, eq.35, wth: A = Pa-s, B =.77 s ad c = 1 =.661.

11 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. CASE 4: = f γ, T. The vscost deeds o temeate ad ate of defomato. The shea thg behavo s modeled b meas of the Caea eqato ad the temeate shft facto s estmated sg the Ahes model. The aametes ae the same as cases ad 3. The eslts of the veloct felds ad the temeate felds fo the dffeet cases ae eseted Fgs. 3 ad 4 The soltos of FDM ad RFM ae ve smla, dcatg that RFM gves a coect aoach to the solto of osothemal flow, obtag the bggest dffeece whe the vscost s a fcto of both temeate ad ate of defomato. The aaged ad adom dstbtos delveed good eslts, dcatg that the adom dstbto s feasble as log as a mmm dstace betwee odes s ket. [m] RFM adom RFM aaged FDM solto =ft =µ v[m/s] Fge 3: Veloct feld fo a o-sothemal esse flow betwee aallel lates fo ewtoa ad o-ewtoa flds [m] RFM adom RFM aaged FDM solto =ft =µ 1 6 =fγ =fγ =ft,γ =fγ,t Temeate [ C] Fge 4: Temeate feld fo a o-sothemal esse flow betwee aallel lates fo ewtoa ad o-ewtoa flds

12 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. 4. Caledeg ocess fo ewtoa ad shea thg olme melts I the caledeg ocess, vaables sch as the fal thckess of the sheet ad the foce eeted b the melt o the olls ae motat to calclate. Soltos sg lbcato aomato have bee oosed wth good eslts fo sothemal ewtoa ad owe-law flows. Howeve, fo moe accate soltos ad ode to cosde vscos heatg, mecal aoaches ae eqed. Ths oblem s a ecellet eamle of alcato of RFM olme ocessg comaed wth tadtoal mecal methods. The comlet of the geomet makes a Fte Dffeece mlemetato dffclt. The hgh ato betwee the bak sze o fed-sheet thckess ad the betwee olls makes the geeato of a accetable FEM dffclt. The covectve tasot of eeg dg caledeg eqes wdg techqes fo the solto sg FEM ad FDM. The o-lea behavo of vscost makes the mlemetato of a boda elemets solto dffclt sce t eqes a mesh. RFM s a good alteatve becase t does ot eqe meshes o wdg techqes ad t woks well wth hghl o-lea oblems. h f =h, 5h,1h o 5h Smlated doma =4.77 m =4.77 m R=3mm h =.mm Fge 5: Schematc dagam of a caledeg ocess fed wth a fte sheet. To comae wth lbcato aomato, the same dmesos ad ocess codtos sed b Agassat et al. [1], schematcall dected Fg. 5 ae cosdeed. The oblem was fst solved fo a ewtoa fld µ = 1 Pa-s wth bak o fed-sheet-thckess to atos, h f /h, of, 5, 1 ad 5. The collocato ots fo h f / h = 1 ae show Fg. 6. The boda codtos ae gve b the veloct o the oll sfaces, a zeo esse at the etace ad et sfaces as well as a zeo omal stess at the etace sface gve b / =. Ths boda codto s mosed b settg the veloct of the fst two collocato ots of each ow eqal to each othe. Fthemoe, ths oblem mst be maall teated, sce the fal sheet thckess s ot kow a o. Hece, a sheet seaato ad thckess s assmed fo the fst solto. Ths eslts a esse feld wth ealstc oscllatos close to the ed of the sheet dcatg that the gessed sheet seaato ot does ot 6 11 h 1

13 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. cocde wth the actal oe. Afte the fst solto, the seaato ot s moved lookg fo the edcto of the oscllato. Afte some teatos the coect sheet thckess ad seaato ot ae acheved, alog wth a smooth esse dstbto. mm mm Fge 6: Geomet ad collocato ots fo the caledeg oblem wth a bak to ato of mm mm mm Fge 7: RFM solto of the veloct feld dg caledeg of a ewtoa melt fo a bak to ato of

14 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. Fg. 7 esets a samle veloct feld fo the ewtoa case wth a bak-to- ato of 1. As ca be see, the veloctes look lasble ad eset a ecclato atte that s edcted b lbcato aomato model ad t s ageemet wth eemetal wok doe the ast [4]. Fg. 8 esets the esse dstbto alog the -as fo a ewtoa solto sg seveal bak-to- atos. The soltos ae eseted wth the edctos sg McKelve s lbcato aomato model. Both soltos ae good ageemet. Pesse MPa RFM solto Hdodamc lbcato aomato h f /h = mm Fge 8: Comaso of lbcato aomato solto ad RFM solto of the esse ofles betwee the olls fo seveal vales of bak, o fed sheet, to ato fo ewtoa vscost model. The same collocato ots ad geomet fo a bak-to- ato, h f /h, of 5 wee sed to solve fo the veloct felds ad esse dstbtos fo o- ewtoa shea thg olme melts. A owe law model wth a cosstec de, m, of 1 4 Pa-s ad seveal owe law dces,, of.7,.5 ad.3 wee sed. The esse dstbto alog the -as fo the shea thg melt s eseted Fg. 9 fo vaos owe law dces ad Fg. 1 fo a owe law de of.3. Aga, the RFM eslts wee comaed to soltos sg lbcato aomato. As ca be see, the ageemet s ecellet. The FEM eslts eseted b Agassat [1] ae also ageemet wth the RFM. The FEM eslts edct a slghtl hghe esse tha the lbcato aomato edcto, wheeas the RFM esse edctos ae slghtl lowe. 6 13

15 Pesse MPa Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. Pesse MPa RFM solto Hdodamc lbcato aomato. =.7 =.5 = mm 1 1 Fge 9: Comaso of lbcato aomato solto ad RFM solto of the esse ofles betwee the olls fo a bak-to- ato of 1, ad seveal owe law dces sg owe law vscost model RFM Solto Hdodamc Lbcato Aomato mm 1 1 Fge 1: Comaso of lbcato aomato solto ad RFM solto of the esse ofle betwee the olls sg a owe law vscost model wth a owe law de, of.3. Fall, the o-sothemal case s cosdeed. The same collocato ots ad geomet fo a bak-to- ato, h f /h, of 5 wee sed to solve fo veloct, temeate ad esse dstbtos fo o-ewtoa shea thg olme melts, cldg vscos dssato ad cosdeg the vscost as a fcto of temeate. A owe law model wth a cosstec de, m, of 1 4 Pa-s ad a owe law de,, of.5 was sed. The temeatevscost deedece was modeled sg Ahes eqato, eq.36 wth U = 4 J/mol ad Tef = 473 K. The covectve tem as well as the heat geeated b vscos dssato ae cosdeed fo the solto of the eeg eqato. Fg. 13 dects the temeate feld delveed b RFM fo the osothemal case. De to vscos heatg, the temeate ceases 5.5K the 14 6

16 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. ego of the. The cease of heat edces the vscost magtde ad theefoe, deceases the esse. Fge 14 esets a comaso betwee the o-sothemal ad sothemal soltos fo esse dstbto alog the -as. The effect of vscos heatg o the solto ca ot be eeseted sg the lbcato aomato fo cases whee the vscost s cosdeed as a fcto of temeate. mm mm Fge 11: RFM ate of defomato ofle 1/s ts betwee the olls sg a owe law vscost model wth a owe law de,, of.5. mm mm /s Pa s 6 Fge 1: RFM vscost ofle Pa-s ts betwee the olls sg a owe law vscost model wth a owe law de,, of

17 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. Y mm X mm Fge 13: RFM temeate ofle betwee the olls sg a owe law vscost model wth a owe law de,, of.5. cosdeg vscos heatg. Pesse MPa RFM sothemal solto RFM o sothemal solto mm Fge 14: Comaso of o-sothemal ad sothemal RFM soltos of the esse ofle betwee the olls sg a owe law vscost model wth a owe law de of o C

18 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. 5 COCLUSIOS The eslts of ths wok allow s to cosde RFM as a omsg techqe to solve olme ocessg oblems. The Radal Fctos Method ca solve adeqatel o-sothemal, o-ewtoa flows, cldg the heomeo of heatg b vscos dssato ad the covectve tasot of eeg. Those chaactestcs ae commo olmec flows whch volve hghl o-lea behavo ad stog deedece betwee eeg ad mometm balaces. RFM has the advatage of beg elatvel eas to mlemet, becase t allows fomlatg the goveg eqatos tems of the mtve vaables makg the comtatoal model moe destadable. It does ot eqe the geeato of meshes o gds ad ca se adom dstbto of odes as log as a mmm dstace betwee odes s ket. The method has ove to be stable fo hghl o-lea oblems. The mlemetato s the same deedetl of the mbe of dmesos of the oblem, de to the fact that t s ol based o the dstace betwee odes. The teolato ad etaolato of felds ca be doe wth the same coeffcets delveed b the solto of the PDEs, eldg ease ost-ocessg stes. The bggest dsadvatage of the method s that t solves PDEs wth fll smmetc matces eqg hgh comtg esoces. Seveal alteatves have bee develoed to move ths lmtato, sch as the smmetc fomlato [, 8, 16], the se of sbdomas [7], the se of comact sot adal bass fctos [3] ad the se of secal dstbto of odes that geeates matces that ae eas to smlf [15]. 6 Refeeces [1] Agassat, J. Polme Pocessg: Pcles ad Modelg. Hase Pblshes, ew Yok - USA, 1991, [] Bhma, M. RBFs. I Acta meca, Cambdge Uvest Pess, [3] Che, C., Ho, C., ad Schaback, R. [4] Dg, H., Sh, C., Yeo, K., ad X, D. Scetfc Comtg wth RBFs. Uvest of Las Vegas - evada, 3. mecal comtato of thee-dmesoal Icomessble vscos flows the mtve vaable fom b local mltqadc dffeetal qadate method. Comte Methods Aled Mechacs ad Eg Sbmtted

19 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. [5] Estada, O., ad Lóez, I. [6] Estada, O., Lóez, I., ad Roldá, C. Collocato method solto wth RBFs of the D eeg eqato, ATEC 5. SPE - Socet of Plastcs Egees. Estdo Eemetal Del Comotameto de Algas Fcoes de Base Radal RBF Paa la Solcó de Ecacoes Dfeecales Pacales, Poas de Los Poblemas de Feómeos de Tasote, Emleado Fcoes de Peba. Go de Ivestgacó Alcada de Polímeos - ICIPC, Medellí - Colomba, 5. [7] Fasshae, G. Solvg atal dffeetal eqatos b collocato wth RBFs. I Sface Fttg ad Mltesolto Methods ashvlle - USA., 1996, A.L. Mehate, C. Rabt, ad L.L. Schmake, Edtos. [8] Fasshae, G Solvg dffeetal eqatos wth RBFs: Mltlevel methods ad smoothg. Advaces Comtatoal Mathematcs , [9] Fasshae, G ewto teato wth mltqadcs fo the solto of olea PDEs. Comte ad Mathematcs wth Alcatos 43, [1] Fake, C., ad Schaback, R. Solvg atal dffeetal eqatos b collocato sg RBFs. Aled Mathematcs ad Comtato , [11] Fake, J. Scatteed data teolato: Test of some methods. Mathematcs of Comtato , 181. [1] Kasa, E. Mltqadcs - a scatteed data aomato scheme wth alcato to comtatoal fld damcs -. Comtes ad Mathematcs wth Alcatos , [13] Kasa, E., ad Calso, R. RBFs: A class of g-fee, scatteed data aomatos. Comtatoal Fld Damcs Joal ,

20 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. [14] Kasa, E., Powe, H., Fasshae,G., ad Lg, L. [15] Kaageoghs, A., Che, C., ad Smls, Y. [16] Lasso, E., ad Fobeg, B. [17] Ma-D,., ad Tae, R. [18] Ma-D,., ad Ta- Cog,. [19] Ma-D,., ad Ta- Cog, T. [] Moat, C., ad Beatso, R. [1] Powe, H., ad Baaco, V. A volmetc tegal RBF method fo tme-deedet atal dffeetal eqatos.. fomlato. Eg. Aalss wth Boda Elemets 81 4, A Mat Decomosto Algothm. PART I: Aomato of Fctos ad the Devatves. Uvest of Cs ad Deatmet of Mathematcal Sceces. Uvest of evada., Las Vegas - evada - USA ad Cs - Geca., 4. A mecal std of some RBFs based solto methods fo elltcs PDEs. Comtg o-ewtoa fld flow wth adal bass fcto etwoks. Iteatoal Joal of mecal Methods Flds 48 5, A effcet dect RBF-based method fo mecal solto of PDEs. mecal Methods fo Patal Dffeetal Eqatos 1 5, A effectve RBF-boda tegal aoach fo the aalss of atal covecto flow. Joal mecal Methods Flds 46 4, RBF Collocato. Deatmet of Mathematcs ad Statstcs. Uvest of Cateb.: Cateb - ew Zeelad,. Comaso aalss betwee smmetc ad smmetc RBF collocato methods fo the mecal solto of atal dffeetal eqatos. Comtes ad Mathematcs wth Alcatos 43, [] Sale, B. Mesh - fee comtatoal fld damcs: A RBF collocato method aoach. I XXI U.I.T. atoal Cofeece o Heat Tasfe Ude, Itala, 3, Uoe Italaa Temofldodamca. 6 19

21 Ivá Lóez / Modelg ad Smlato of Polme Pocessg sg RFM 7 Cal Hase Velag, Müche cht z Vewedg Itaet- d Iteet-Agebote sowe elektosche Vetele. [3] Schaback, R., ad Wedlad, H. Boda Elemet Techolog XIII. C.A.B. C. S. Che, ad D. W. Pee eds., Edto., 1999, ch. Usg comactl soted RBFs to solve atal dffeetal eqatos, [4] Uküe, W. Betag z Emttlg des Dckvelafes d de Fleβvogäge m Walzesalt be de Kaledeveabetg vo PVC-hat z Fole. PhD thess, RWTH-Aache, 197. [5] Wedlad, H. Comtatoal Asects of RBF Aomato, Tocs Mltvaate Aomato ad Iteolato. K. Jette ad othes, 5. [6] Zeokat, M., Ddel, K., ad Chaaf, A. [7] Zeokat, M., Powe, H., ad Che, C. A [8] Zhag, X., L, X., Sog, K., ad L, M. Elct ad mlct meshless methods fo lea advecto dffso - te atal dffeetal eqato. Ite. Joal fo mecal Methods Eg. 48, A mecal method fo heat tasfe oblems sg collocato ad RBFs. Joal fo mecal Methods Eg , Least-sqaes collocato meshless method. Ite. Joal mecal Methods Eg. 51 1, Ke wods: Radal bass fctos, smmetc collocato, olea flow oblems, vscos dssato, o-ewtoa flds, Radal Fctos Method, Polme Pocessg. Cotact: Atho: Tm A. Osswald. Polme Egeg Cete, Uvest of Wscos- Madso, Madso-USA [email protected]. 6