Simulation of coupled nonlinear electromagnetic heating with the Green element method

Transcription

1 Advacd Copuaoal Mhods Ha rasfr IX 77 Sulao of coupld olar lcroagc hag wh h Gr l hod A. E. agbu School of Cvl ad Evroal Egrg, Uvrs of h Wwarsrad, Johasburg, Souh Afrca Absrac h olar coupld dffral uaos ha govr h probl wh arals ar lcrcall had ar solvd wh a flux-basd Gr l forulao ha has sgfcal hacd copuaoal faurs coparso o prvous Gr l forulaos. h flux-basd Gr l forulao as advaag of h abl of h boudar l hor o corrcl calcula h oral drvav of h prar varabl b plg h hor a f l ss so ha hacd accurac s achvd wh coars dscrzao. h copl soluo forao (praur, lcrc poal ad hr oral drvavs) ar ad avalabl b h flux-basd forulao ach l so ha rfd soluos a a po, wh dd, ar calculad usg ol h l whch h po s locad. h closur probl assocad wh havg or uows ha dscrzd uaos a ral ods s addrssd a ovl ar b a copabl rlao for h oral drvavs of h prar varabl ha has uvrsal appal. h copuaoal accurac of h flux-basd Gr l forulao s dosrad wh a urcal xapl of olar lcroagc hag probl. Kwords: lcroagc hag, olar dffuso-advco, olar Posso uao, Gr l hod. Iroduco h food ad rlad dusrs ar vr rsd addrssg h probls assocad wh h hag of food subsacs b lcrcal currs. Of 006 WI Prss WI rasacos o Egrg Sccs, Vol 53, ISS (o-l) do:0.495/h06008

2 78 Advacd Copuaoal Mhods Ha rasfr IX parcular rs s h dsrbuo of h praur fld (dcav of h ffcvss of h appld ha) ha s grad for a gv lcrc poal ha s appld o h food aral. For h praur fld, h govrg uao closl rsbls h dffuso-advco-raco uao ha has a applcaos a grg produco procsss. Is olar aur poss a challg spcall wh h advcv r s doa ad h uao h has a prdoal hprbolc aur. hs uao s coupl wh h olar lcroagc uao whch ssall has a llpc characr. hs uaos wr radoall solvd b h f dffrc ad f l hods. A boudar l soluo has b provdd b Cra al. [] who usd h Laplac rasfor forulao o splf h poral drvav ad h dual rcproc hod o sur ha h soluo s carrd ou ol o h boudar. ha approach, rfrrd o as h Laplac rasfor dual rcproc hod (LDRM), s closl slar o ha of Saravaha ad Zhu [] for h soluo of olar ha coduco probls. Larzao of h dffral uaos has o b do o b abl o appl h Laplac rasfor dual rcproc BEM. Hr h Gr l hod (GEM), whch ras h oral drvavs a vr odal po, s usd o solv h lcroagc hag probl [3 5]. I s hrfor rfrrd o as h flux-basd GEM. h gral uaos ha rsul fro h applcao Gr s d ar solvd ach l. h graos ar valuad aalcall, ad h ol approxao ha s rurd s do wh rpolag h prar varabl ad s oral drvav h l. Hgh lvl of accurac s hus achvd wh coars dscrzao of h copuaoal doa ad hs copsas for h larg ubr of dgrs of frdo a ach od. h closur probl a h ral ods ha s as a rsul of a fwr ubr of gral uaos ha uows s rsolvd a ovl ar b grag a addoal uao fro urcall plg h grao of h oral fluxs aroud h ral od. A xapl solvd b h LDRM of Cra al. [] whch plod 40 ods s solvd b h flux-basd GEM wh 0 ods o achv coparabl accurac. Govrg uaos h lcroagc hag probl ha s addrssd hs papr s govrd b h coupld olar uaos ha ar gv b Plas al. [6] ( ) ( α ) v ( α ) σ φ () ad ( σ φ) 0 () whr ad φ rprs h praur ad lcrc poal flds of h du o whch ohc hag s appld. h aral proprs of h du ar: s h hral coducv ad σ s h lcrcal coducv whch ar boh dpd o h praur fld, α s h ha capac, 006 WI Prss WI rasacos o Egrg Sccs, Vol 53, ISS (o-l)

3 Advacd Copuaoal Mhods Ha rasfr IX 79 v u v s h vloc fld wo dsos. h soluo o s () ad () a coupld ar ca b achvd wh h boudar ad al codos ar spcfd. h frs-, scod- ad hrd-p boudar codos ar adssbl, whl h codo for h praur ad poal flds ar assud ow a h al 0. Es () ad () ar rwr as Posso-p uaos: K ( α ) v ( α ) σ φ (3) φ Φ (4) whr K l ad Φ lσ. h soluos o s (3) ad (4) ar achvd a Gr l ss. 3 Gr l forulao h Gr l forulao ha s plod for h soluo of h coupld uaos (3) ad (4) uss h fudaal soluo G l( r r ) o h Laplaca opraor G ( r ) h f spac o drv gral r uaos wh a spaal l Ω wh closd boudar Γ ha cosus o of h ls usd dscrzg h r copuaoal rgo Ω. h gral uaos ar gv b [7] λ Γ ( G G ) ds, ( α ) G K v Ω ( α ) σ φ da 0 (5) λφ Γ ( φg Gφ ) ds G( Φ φ ) da (6) whr λ s h odal agl fro grag h dpd varabl wh h Drac dla fuco a Cauch ss, ad s h ouward pog oral o h lal boudar. I should b od ha so far o approxao whasovr has b roducd h forulao arrvg a s (5) ad (6). Approxaos ar roducd b prscrbg a rpolao for h dsrbuo of h dpd varabl ad hr oral drvavs h l. h Lagrag-p rpolao fucos ar usd. Usg hr lar ragular or lar rcagular ls ad carrg ou h graos rsuls h arx uaos Ω 006 WI Prss WI rasacos o Egrg Sccs, Vol 53, ISS (o-l)

4 0. ) (, Υ Χ x x v u d d S K v K u L R φ σ α α α α α (7) 0 Φ Υ Φ Χ x p p p L R φ (8) whr ( ).,,,, da G S da G da x G G ds L ds G R Υ Χ Ω Ω Ω Γ Γ λ δ (9) x,, (0a) φ σ φ σ φ σ p x p p,, (0b) I (9), rprss h rpolao fuco wh rspc o od. I should b od ha h graos (9) ar do ach l, ad s for splc ha h dx has b xcludd. All h graos ar do aalcall for h wo ps of ls, al rcagular ad ragular ls. Carrg ou h arx apulaos s (7) ad (8) rsuls wo uows a vr odal po for ach of h uaos. hs ar h praur ad s oral flux for h frs dffral uao, ad h lcrc poal φ ad s oral flux p. Es (7) ad (8) ar aggrgad for all ls usd dscrzg h copuaoal doa ad splfd o 0 d d S B A () 0 p E φ C () W hav roducd a w dx, h rao ubr, o s () ad () o dca ha h ls of h arcs ar valuad wh ow ras, whl h uow uas ar o b copud a h curr rao lvl of. Essall, hs rao procss s h Pcard algorh. h drvav () s splfd b h gralzd f dffrc sch wh wghg facor θ o bco ( ) ( ) 0,,,, S B B A A ω θ ω θ (3) whr h dcs ad, rspcvl, ar dcav of h curr ad prvous, ad θ ω. Iroducg h al 80 Advacd Copuaoal Mhods Ha rasfr IX 006 WI Prss WI rasacos o Egrg Sccs, Vol 53, ISS (o-l)

5 Advacd Copuaoal Mhods Ha rasfr IX 8 codos o (3), ad h ow boudar codos o xral boudars o s () ad (3) splf h coupld uaos o,, F H R (4) whr F C φ E p M (5) S θ, θ (6) A h vcors o h rgh had sd of s (4) ad (5) ar ow; hr valus co fro h prscrbd boudar ad al codos. h soluo procdur wh ach sp s hr ould. () For h ow cosuv rlaos for ad σ, ad a ow dsrbuo,, of h praur fld {, },, oba updad valus {, } H B, h arx uao (4) s solvd o. () hos updad soluos fro () ar usd (5) o solv for h lcrc φ, p. poal fld { } () h a dvao of h ras fro h praur ad poal flds s calculad ad copard wh a prscrbd accurac olrac ε. Covrgc s sad o b achvd wh h a dvao s lss ha ε. If covrgc s o aad, sps () hrough () ar pld usg h rfd soluo ras. (v) Wh covrgc wh a sp has b, aohr cr s ad ad h abov hr sps ar h rpad. 4 Copabl of h ral fluxs Bcaus h curr flux-basd Gr l forulao calculas h dpd varabl ad as wll as s oral drvav (rfrrd o as h flux) a ach odal po, h ubr of grad gral uaos s shor b o h ubr of dgrs of frdo a ral ods. hs s o h cas wh h xral ods whr h prscrbd boudar codos a up for h shorfall h ubr of grad dscr uaos. hs s h closur probl ha has b rcogzd boudar l crcls. I h pas o approach of rsolvg h closur probl has b o arfcall cra addoal ods ad rloca h b sall dsacs fro h orgal locao of h ral od alog h ral sgs [8]. Such a approach s a urcal arfac ha usuall rcofgurs h gor a h ral ods o su h urcal forulao. ha approach s o followd hr bu rahr a addoal uao s grad o a up for h shorfall. wo prvous prsaos o hs addoal uao dcad ha hs addoal uao s sa of h cou of h oral fluxs a h ral od [4, 5]. W hav ow foud hs o b rror, ad spcal acowldg gos o Dr. Ello who propd us o r-xa ha sa. W hav ow carrd 006 WI Prss WI rasacos o Egrg Sccs, Vol 53, ISS (o-l)

6 8 Advacd Copuaoal Mhods Ha rasfr IX ou a rgorous proof of hs copabl codo of h ral oral fluxs. I s oo lgh o prs hr. h proof wll b prsd dals auscrps ha ar currl bg prpard. Suffc o sa ha h proof s basd o grag h oral drvav of h prar varabl aroud a crcl of sall radus crd a h ral od. I h l as h radus ds o zro, h gral uals h valu of dscou of h prar varabl a h ral po. Whr o dscou xss, h gral uals zro. hs ca b sad as d 0 (7) o pl (7) urcall, π M d L π dβ β 0 (8) ζ 0 0 whr M s h ubr of ls a h ral od, ζ s h radus of h crcl a h ral od (S Fgur ). O a as wh xcll rsuls wr obad h prvous publcaos wh a wrog copabl codo. h raso s ha all h sulaos wr carrd ou wh rcagular ls ad, as ca b obsrvd fro (8), β π /,,, 3, 4. W hav sc sd h codo of (8) wh ragular ls whr dffr agls of β ar courd ad xcll rsuls wr obad β 3-5- ζ Fgur : oral fluxs a a ral od. wo uu characrscs of h flux-basd forulao ar ha provds h copl soluo forao for ach l, ad hgh accurac s achvd wh coars dscrzao. h lar copsas for h scalao h ubr of dgrs of frdo du o h valuao of h prar varabl 006 WI Prss WI rasacos o Egrg Sccs, Vol 53, ISS (o-l)

7 Advacd Copuaoal Mhods Ha rasfr IX 83 ad h flux a vr od. h forr allows for calculaos of h soluo a a po o volv ol graos wh h l whch h po s locad. o rfrc s ad o ohr ls h copuaoal rgo. ha rprss orous copuaoal savgs wh soluos ar rurd a pos of rs ohr ha grd pos. 5 urcal xapl h urcal xapl usd b Cra al. [] hr Laplac rasfor dual rcproc boudar l forulao s also usd hr o valda h curr flux-basd Gr l forulao. h xapl s ssall o dsoal h spaal dso for whch h aalcal soluos: ( x x / )( ) ad φ x ( x x ), 0 x (9) ar proposd for h dffral uaos ( ) ( α ) v ( α ) σ φ f( x, ) (0) ad ( σ φ) f ( x, ) () h choc of h fucos f ( x, ) ad f ( x, ) has b ad so ha h proposd soluos sasf h dffral uaos. h boudar codos ar: ( x, ) ( x 0, ) 0, 0, φ ( x 0, ) 0 ad φ ( x, ) () x h parars for h du ad h advco vloc fld ar:, σ, α ad v. Usg ol four lar rcagular ls, h Gr l sulaos ar carrd ou a wo dsoal doa of so ha h sz of ach l s 0.5. Effcvl, 0 ods ar usd o dscrz h copuao doa, rprsg a coars dscrzao coparso o h 40 ods usd b Cra al. []. h full plc sch wh θ s usd for h dffrcg, whl h sp s vard sarg wh 0.05 for 0 0., h 0. for 0. ad 0.5 for 5. h accurac olrac valu usd s ε 0 5 ad covrgc was achvd wh 3 raos ach sp. h spaal dsrbuos of h praur ad lcrc poal flds for s of 0., 0.5,, ad 5 ar prsd Fgurs ad 3, whl hr poral dsrbuos a x 0. 5, x 0. 5, ad x 0.75 ar prsd Fgurs 4 ad 5. hr s good agr bw h Gr l soluos ad h aalcal soluo. I should b pod ou ha h soluos a pos ohr ha grd pos wr grad usg ol h soluos obad for h l whch h grd pos ar locad. For sac, h soluo a x 0. was carrd ou o h frs l {( x, ) : x [0,0.5], [0,]}. h hgh lvl of accurac ha s achvd wh such a coars grd s as a rsul of h fac ha h ol approxao hs forulao arss fro h rpolao of h prar varabl ad s oral drvav. 006 WI Prss WI rasacos o Egrg Sccs, Vol 53, ISS (o-l)

8 84 Advacd Copuaoal Mhods Ha rasfr IX 0.8 (x,) Exac GEM(0.) GEM(0.5) GEM() GEM(5) x Fgur : Aalcal ad GEM soluos for h spaal praur dsrbuo a varous s. 0.8 φ(x,) Exac GEM(0.) GEM(0.5) GEM() GEM(5) x Fgur 3: Aalcal ad GEM soluos for h spaal lcrc poal dsrbuo a varous s. 6 Cocluso h olar coupld dffral uaos whch govr hag of food arals b lcrcal curr hav b solvd usg a flux-basd Gr l forulao ha solvs o ol for h praur ad lcrc poal flds bu also hr oral drvavs. h hgh lvl of accurac achvd b h forulao arss o ol bcaus h soluo procdur ras h olar aur of h dffral uaos, bu also du o h fac 006 WI Prss WI rasacos o Egrg Sccs, Vol 53, ISS (o-l)

9 Advacd Copuaoal Mhods Ha rasfr IX 85 ha h oral drvavs of h praur ad lcrc poal ar drcl calculad. h crasd ubr of dgrs of frdo a ach od s copsad b h coars grd wh whch hgh accurac s achvd. x x0.5 (x,) x0.5 Exac GEM (x0.5) GEM (x0.5) GEM (x0.75) Fgur 4: Aalcal ad GEM soluos for h poral praur dsrbuo a varous posos. 0.8 Exac GEM (x0.5) GEM (x0.5) GEM (x0.75) φ(x,) 0.6 x0.75 x x Fgur 5: Aalcal ad GEM soluos for h poral lcrc poal dsrbuo a varous posos. h ovl faur of h curr Gr l forulao s ha h lal soluo s copl h ss ha h soluo a a po a l s obad b carr ou h boudar ad doa graos wh ha l. o rfrc s ad o ohr ls ha hav b usd o dscrz h rgo. hos soluos a pos ohr ha h grd pos ar uall scod-ordr accura as hos a h odal pos. I hs papr, h 006 WI Prss WI rasacos o Egrg Sccs, Vol 53, ISS (o-l)

10 86 Advacd Copuaoal Mhods Ha rasfr IX curr flux-basd GEM has usd 5% of h grd pos of LDRM forulao of Cra al. []. ha s pcal of h flux-basd GEM whch grall provds accura soluo wh vr coars dscrzao. Acowldg Spcal has o Dr. L. Ello of Lds Uvrs who propd a rh of copabl uao of h fluxs a h ral od a h d of h 5 h UK cofrc o Boudar Igral Mhods Lvrpool Spbr 005. Rfrcs [] Cra, D., Davs, A.J. & Chrsaso, B., h Laplac rasfor dual rcproc boudar l hod for lcroagc hag probls. Proc. I Advacs Boudar El chus VI, Eds. A.P. Slvadura, C.L. a & M.H. Alabad, EC Ld, UK, pp. 9-34, 005. [] Saravaha, P. & Zhu S., A applcao of h LDRM o ras dffuso probls wh olar aral proprs ad olar boudar codos, App. Mahs Cop., 87, pp. 7-60, 997. [3] Pchr, R., Harrs, S.D., Kp, R.J., Ello L. & Igha D.B., w forulao of h Gr l hod o aa s scod-ordr accurac D/3D, Egrg. Aal. wh Boudar Els, 5, pp. -9, 00. [4] agbu, A.E., h flux-corrc Gr l hod for lar ad olar poal flows, Proc. I Advacs Boudar El chus VI, Eds. A.P. Slvadura, C.L. a & M.H. Alabad, EC Ld, UK, pp , 005. [5] agbu, A.E., Iprovs ha coduco calculaos wh fluxbasd Gr l hod, Proc. Of h 5 h UK cof. o boudar gral hods, Ed. K Ch, h Uvrs of Lvrpool Prss, pp , 005. [6] Plas, C.P., Schwda, D.W. & Haga, P.S., Ohc hag of foods durg aspc procssg, IMA J. of Maag Mahs., 5(), pp , 993. [7] agbu, A.E. h Gr El Mhod, Kluwr, Boso, USA, 999. [8] Lgg, J.A. & Lu, P.L-F, h boudar Igral Euao Mhod for Porous Mda Flow, Gorg All & Uw, WI Prss WI rasacos o Egrg Sccs, Vol 53, ISS (o-l)