Appld Physcs Frontr Novmbr 013 Volum 1 Issu 4 PP.3-44 A Novl Fnt Volum Schm wth Gomtrc Avrag Mthod for Radatv Hat Transfr Problms * Cunyun N 1 Hayuan Yu 1. Dpartmnt of Mathmatcs and Physcs Hunan Insttuton of Engnrng Xangtan Hunan 411104 Chna. Hunan Ky Laboratory for Computaton & Smulaton n Scnc and Engnrng and Ky Laboratory of Intllgnt Computng & Informaton Procssng of Mnstry of Educaton Xangtan Unvrsty Hunan 411105 Chna Emal: ncy108@gmal.com n7@alyun.com Abstract W construct a novl fnt volum schm by nnovatvly ntroducng th wghtd gomtrc avrag mthod for solvng thr mult-matral radatv hat transfr problms and compar t wth th wghtd arthmtc and harmonc avrag mthods rspctvly. W also put forward th ffct of th convxty of nonlnar dffuson functons. Thn w prsnt a cylndr symmtrc fnt volum lmnt (SFVE) schm for th thr-dmnsonal problm by transfrrng t to a two-dmnsonal on wth th axs symmtry. Numrcal xprmnts rval that th convrgnt ordr s lss than two and numrcal stmulatons ar vald and ratonal and confrm that th nw schm s agrabl for solvng radatv hat transfr problms. Kywords: Fnt Volum Schm; Wghtd Gomtrc Avrag Mthod; Radatv Hat Transfr Problms; Convxty of Dffuson Functons 1 INTRODUCTION Numrcal approxmatons of scond ordr llptc or parabolc problms wth dscontnuous coffcnts ar oftn ncountrd n matral scncs and flud dynamcs. Dscontnuous dffuson coffcnts corrspond to multmatral hat transfr problm whch s on of mportant ntrfac problms [134]. Jafar prsntd a -D transnt hat transfr fnt lmnt analyss for som mult-physcs problm n [1]. P and hs collagus obtand satsfactory numrcal rsults n th ntgratd smulaton of gnton hohlraum by Lard-H cod n th ltratur [] and th Arbtrary Lagrangan-Eulran mthod was usd to trat th larg dformaton problm and mult-matral clls wr ntroducd whn th matral ntrfac s svrly dstortd. Jams and Chn dscussd fnt lmnt mthods for ntrfac problms and obtand th optmal L -norm and nrgy-norm rror stmats for rgular problms whn th ntrfacs ar of arbtrary shap but ar smooth n [34]. Th harmonc avrag was promotd by Patankar [5]. It was oftn dsgnd to solv hat transfr problms nvolvng multpl matral proprts. Rcntly n [6] Kadoglu and hs workrs has takn a comparatv study of th harmonc and arthmtc avragng of dffuson coffcnts for nonlnar hat conducton problms and rvald that th harmonc avrag s not always bttr than th arthmtc on. Th wght harmonc avrag fnt volum and fnt dffrnc schms wr dscussd for som ntrfac problms n som othr ltraturs [78910]. It s wll-known that th ngnrs ar usually zalous for th arthmtc and harmonc avrag mthods for ntrfac problms but gnor th gomtrc avrag on. In ths papr w construct a novl fnt volum schm for solvng thr mult-matral radatv hat transfr problms. On nnovatv da of our work s that w ntroduc th * Ths work was partally supportd by NSFC Projct (Grant No. 11031006 9113000 1117181) th Ky Projct of Scntfc Rsarch Fund of Hunan Provncal Scnc and Tchnology Dpartmnt (Grant No. 011FJ011) Hunan Provncal Natural Scnc Foundaton of Chna (Grant No. 1JJ3010). - 3 -
wghtd gomtrc avrag mthod for dsposng th mult-matral lmnt (Also t s calld as th mxd lmnt) Whn computng th dffuson coffcnt. Hnc w compar t wth arthmtc and harmonc avrag mthods. Numrcal rsults vrfy that th nw schm s vald and agrabl. Anothr good da of our work s that w consdr th ffct of th convxty of nonlnar dffuson functons n mxd lmnts and prsnt two ways: on way s to valuat th dffuson functon valus for two dffrnt matrals frstly and to tak som avrag on two computd functon valus scondly; th othr way s n th oppost ordr. Numrcal rsults show that th strongr th convxty of dffuson functon s th mor mportant th ordr of frst avragng and scond valuatng s also show that th gomtrc avrag mthod s bttr than th harmonc on and n ln wth th arthmtc on for th strong convxty. Th rsult abov can b gnralzd to th hghr dmnsonal problms although th dscusson s confnd to on-dmnsonal cas. Th thrd fn da of our work s that w dsgn a novl SFVE schm wth th gomtrc avrag mthod basd on th work n [8] for a two-dmnsonal mult-matral radatv hat transfr problm. Numrcal xprmnts show that th convrgnt ordr s lss than two and oscllats gong wth th oscllatng proporton of th volum of som matral n mxd lmnts. Basd upon th abov schm w construct a cylndr SFVE schm for a thrdmnsonal hat transfr problm by transfrrng t to a two-dmnsonal on wth th axs symmtry. Numrcal xprmnts xhbt that numrcal stmulatons ar vald and nrgy consrvatv rrors ar small and ratonal and that t s convrgnt of ordr on whn th backward Eulr mthod s mployd. Th rmandr of ths papr s organzd as follows. In Scton w dsgn a nw fnt volum schm basd on th gomtrc avrag mthod and dscuss th ffct of th convxty of nonlnar dffuson functons. In Scton 3 w construct a SFVE schm for a two-dmnsonal mult-matral radatv hat transfr problm. In scton 4 w prsnt a cylndr SFVE schm for a thr-dmnsonal problm and carry on som numrcal xprmnts. Fnally w summarz our work n ths papr. THE GEOMETRY AVERAGE FINITE VOLUME SCHEME AND THE CONVEXITY OF DIFFUSION FUNCTION In ths scton w wll consdr th followng nonlnar hat transfr problm T T ( D ) f ( x) a x b0 t t t x x T( a) T1 T( b) T (1) whr D s th nonlnar dffuson coffcnt such as D T. Th varaton of th dffuson coffcnt may brng a fast movng wav front whch s known as th Marskak wav [1]..1 Th Fnt volum Schm Frstly w tak th unform partton of th ntrvals [ab] and [0 t ] rspctvly and dnot a x x... x b0 t t... t t 0 1 N 0 1 N x x x 01... N 1 t t t 01... M 1. 1 1 In th followng th Crank-Ncolson schm and a consrvatv scond ordr fnt volum schm ar appld to modl problm 1 dfnd by Eq. (1). Hnc th numrcal schm s convrgnt of ordr two about spac and tm. Th Jacoban-Fr Nwton Krylov mthod [1113] s mployd to th nonlnar part. Th tm and spac dscrtzatons for Eq. (1) yld to Whr n T T D 1/ ( T 1 T ) D 1/ ( T T 1) t x n1 T: T T(x t )D D can b computd or approxmatd. n1 1/ 1/ f () - 33 -
Th followng drvaton s basd on two assumptons: (1) w assum that th hat transfr coffcnt s pcws constant and contnuous. () W consdr stady stat solutons. W can classfy th lmnts as non-mxd lmnts and mxd lmnts. Th non-mxd lmnt mans that thr only on matral n t and th mxd lmnt mans that thr ar (at last) two matrals n t. For xampl n FIG.1 [ x x 1] s a non-mxd lmnt and [ x 1 x] s a mxd lmnt. For th formr D 3/ can asly and unquly b dtrmnd undr abov two condtons. Howvr for th lattr D 1/ can only b approxmatd.. t nds to b valuatd by som avragng such as th arthmtc harmonc or gomtrc avrags. Now w wll focus on how to us thr avrag mthods for th computaton of D 1/ whr th ntrfac s shown as FIG. 1. FIG. 1 THE INTERFACE AND GRID POINT Th arthmtc and harmonc avrag mthods (S thm n [6]) lad to and rspctvly. D D D D (3) a 1 1/ DD h 1 1/ D D 1 Th ltratur also shows that th truncaton rrors of abov two mthods ar W can ntroduc anothr avrag mthod for computng It s th wll-known gomtrc avrag mthod. From th ltratur [15] on can s that D g D 1/.. g 1/ 1-34 - (4) O(Δx ). D D. (5) D D D h g a 1/ 1/ 1/ and th truncaton rror of t s O( x ) obtand smlarly to th ltratur [6]. Th abov rlaton mans that th gomtrc avrag may b a agrabl on du to ts mmunty to th xtrm valus. In th followng w wll prsnt som numrcal xprmnts to dsplay th charactrstcs of th gomtrc avrag mthod.. Th Convxty of Dffuson Functon and Numrcal rsults In th scton w wll dsplay two numrcal xampls. On s from and mor abundant than that n th ltratur [6]. W not only compar wth thr avrag mthods but also consdr th convxty of th dffuson functon T. Th othr xampl s constructd whr th xact soluton s dsgnd to vrfy th convrgnt ordr of thr avrag mthods. Exampl 1. ([6]) W consdr th modl problm 1 dfnd by Eq. (1) wth [ ] t0 t =[00.08] [ab]=[01] f(x)=0 0 T(xt ) =0.1 T(at) =.0 T(bt) =0.1. α W not only prsnt thr cass of th dffuson coffcnt D T α 136 but also consdr th convxty of th α dffuson functon T. Th approxmaton of D 1/ can b obtand by th followng two ways: (A) Evaluat T T thn tak som avrag for thm to gt 1 D 1/.
(B) Tak som avrag by TT 1 to approxmatt 1/ thn valuat ( T 1/). Numrcal rsults ar shown as th fgurs from FIG. to FIG.4. Th frst two agr wth thos n th ltratur [6]. FIG. COMPARISON FOR 1: (A) ARITHMETIC. (B) HARMONIC. (C) GEOMETRIC. (D) F-ARITHMETIC. (E) F-HARMONIC. (F) F-GEOMETRIC. FIG. 3 COMPARISON FOR 3 : (A) ARITHMETIC. (B) HARMONIC. (C) GEOMETRIC. (D) F-ARITHMETIC. (E) F-HARMONIC. (F) F-GEOMETRIC. In FIG. (A) (B) and (B) ar th rsults for thr avrags (dnotd as Arthmtc Harmonc and Gomtrc) n th way (A) rspctvly; and (d) () and (f) ar also for thr avrags (dnotd as f-arthmtc f-harmonc and f- Gomtrc) n th way (B) rspctvly. From ths fgur on can s that for D T ( 1 ) th dffrncs - 35 -
btwn thr avrag mthods ar vry small. 3 From FIG.3 on can s that for D T ( 3 ) by abov two approxmat ways th rsults ar smlar for harmonc and gomtrc avrag mthods but thr s grat dffrnc for th arthmtc avrag mthod and also can s that th way (B) s bttr than th way (A). 6 From FIG.4 on can s that for D T ( 6 ) th dffrncs ar obvously grat among thr avrag mthods. Frstly th way (B) s bttr than th way (A) spcally for th harmonc avrag mthod. Scondly th arthmtc avrag mthod s th bst th gomtrc on transfrs modratly howvr th harmonc on transfrs too slowly. Hnc as th s bg on should b cautous to choos th harmonc avrag mthod. FIG. 4 COMPARISON FOR 6 : (A) ARITHMETIC. (B) HARMONIC. (C) GEOMETRIC. (D) F-ARITHMETIC. (E) F-HARMONIC. (F) F-GEOMETRIC. Exampl. W consdr modl problm 1 dfnd by Eq.(1) and tak [ab]=[ t 0 t ]=[01] 4 and th xact t x t1 soluton T( x t). In ths xampl w mploy abov thr avrag mthods to comput D 1/. Th L -norms of th rrors ar shown as FIG.5 for approxmat solutons at th momnt t 1. From t on can s that th convrgnt ordrs of thr mthods ar all two whch agrs wth that n th ltratur [6]. Th rror of th arthmtc on s th smallst and that of th harmonc on s th bggst whch also s n accord wth th rlaton (6). FIG. 5 COMPARISONS FOR CONVERGENT ORDERS. - 36 -
3 A SYMMETRIC FINITE VOLUME ELEMENT SCHEME In ths scton w consdr th followng ntrfac modl problm ( u) f x ( x1 x ) 1 u 0 x 0 x 1 whr f(x) L (Ω) and 1 x. (6) 3.1 SFVE SCHEME Assum that h { E1 M} s a quadrlatral partton for th rgon (Shown as FIG.6). In ths fgur th ntrfac (th rd sold ln) lads to som mxd lmnts whr th valus of cannot b dtrmnd accuratly. W wll construct a novl SFVE schm basd on th schm n th ltratur [8] and th wghtd gomtrc avrag mthod. Th ky da of t s that th dffuson coffcnt s approxmatd by som avrag valu n a mxd lmnt. W wll dsplay how th SFVE schm s constructd as follows. FIG. 6 THE PARTITION AND MIXED GRIDS. Stp 1: Obtan th lmnt stff matrx and load vctor for any non-mxd lmnt A ( a ) F ( f ). E E E E lm 44 l 41-37 - E as th ltratur [8] Stp : Comput th approxmat dffuson coffcnt n th mxd lmnt by th wghtd gomtrc avrag mthod (7) 1 1 whr s th proporton of th volum n th mxd lmnt of 1 and 1 s for. Stp 3: Gt th corrspondng lmnt stff matrx and load vctor for th mxd lmnt E a E k m E lm ds f l fdx 1 l m 4 MlOk Ml1 n Dl whr M0 M4 Ok and D l ar th barycntr of th lmnt and th sub-control volum about th vrtx rspctvly (S FIG.7(a)). Stp 4: Assmbl all th lmnt stff matrcs and load vctors to total stff matrx and load vctor.. to obtan th nw schm. On can also comput by th followng wghtd arthmtc and harmonc avrag mthods n Stp to obtan th corrspondng SFVE schms 1 X k l (1 ) (8) 1. (9) / (1 ) / 1 From abov on can s that th total stff matrx s symmtrc. Hnc th nw schm s symmtrc whch s hlpful to ncras th computatonal ffcncy whn solvng th corrspondng dscrt systm. 3. NUMERICAL EXPERIMENTS Now w wll prsnt som numrcal xprmnts for th nw schm.
Exampl 3. In modl problm dfnd by Eq. (6) w choos (01) 1 {( x1 x )0 x1 10 x / 3} x1 x x1 x {( )0 1 / 3 1} th ntrfac x / 3 0 16 and sn(4 x1 )sn(4 x ) x 1 u( x1 x) sn(4 x1 )sn(16 x ) x. FIG. 7 (A) THE SUB-CONTROL VOLUMES. (B) (C) AND (D): THE OSCILLATION OF THE VOLUME PROPORTION. Numrcal rsults ar shown n TABLE. 1 whr N 1 s th partton numbr along th x -axs drcton rspctvly and s th rato of rror of th approxmatons for two partton stp szs h h/ h u uh. From 0 t on can s that th convrgnt ordrs for thr wghtd avrag mthods (Arthmtc Harmonc and Gomtrc) ar all lss than two and th rrors of approxmaton uh sm oscllatng whn th grds ar rfnd. It s du to th oscllaton of th proporton of th volum of som matral n th mxd lmnt. Th oscllaton of th proporton s shown as FIG.7 (b) to (d). In addton th oscllaton of th gomtrc avrag s th smallst among th thr mthods; mayb t s th advantag of ths avrag mmun to th xtrm valu. TABLE 1 THE L -ERROR FOR THREE AVERAGE METHODS. N N 1 Arthmtc Harmonc Gomtrc h h h 16 1.6-1 1.44-1 1.5-1 3 8.61-1.88 9.73-1.48 1.04-1.46 64.43-3.54 1.8-5.34.5-4.6 18 1.44-1.69 1.65-1.11 1.69-1.33 56 4.86-3.96 3.86-3 4.7 4.60-3 3.67 4 A SFVE SCHEME IN THE CYLINDER COORDINATE SYSTEM In ths scton w consdr th followng radatv hat transfr modl problm ([14]) T c ( KT ) W ( T T ) Wr ( Tr T ) t T c ( KT ) W ( T T ) (10) t 3 Tr 4 atr ( KrTr ) Wr ( T Tr ) t whr th rgon {( R Z) :0 R R00 0 Z Z0} (shown as FIG.8 (a)) s dvdd two parts flld wth two dffrnt matrals: plastc foam (CH) and dutrum gas (DT) rspctvly and... Th modl problm 3 dfnd by Eq. (10) approxmatly dscrbs th procss of radant nrgy broadcastng n th quscnt mdum and nrgy xchang of lctrons wth photons and ons whos tmpraturs ar dnotd as T and T rspctvly. T r - 38 -
In ths scton w only consdr th Z-axs-symmtrc problm so T T and Tr ar all rrlatv to th angl varabl. Hnc w nd to solv th modl problm 3 dfnd by Eq. (10) on som scton rgon (also dnotd as ) shown as FIG.8 (b). In abov quatons (10) 3/ 1/ () W AT and Wr ArT ar th nrgy xchang coffcnts btwn lctron and on and btwn lctron and photon rspctvly; () K A T rspctvly; () c 5/ 6 3 ; Kr 310 lrtr 3 1.5 c 1.5 cr 4aTr In abov physcal paramtrs s th dnsty of matrals. 4.1 SFVE SCHEME ar th dffuson coffcnts about lctron on and photon ar th thrmal capacts about lctron on and photon rspctvly. A Ar A A lr a ar dffrnt constants n dffrnt sub-domans and Assum that a quadrlatral partton Ω h s shown as FIG.8 (c). Th ntrfac ln (rd sold ln) lads to som mxd lmnts. A non-mxd lmnt and a mxd lmnt ar shown as (d) and () rspctvly. W tak th backward Eulr mthod for dscrtzng th tm drcton to obtan th nonlnar llptc quatons thn mploy th "fxd-coffcnt" (Pcard) mthod for th lnarzaton to gt th lnar llptc quatons. Now w com to construct a SFVE schm for th lnar llptc quatons on Ω and prsnt how to dsgn th schm. Stp 1: Comput th proportons of th volum and mass about two matrals n mxd lmnts whr th mass proportons ar dnotd as 1 and comput th approxmat physcal paramtrs by th mass proportons. () Th thrmal capacts c th matral dnsts and th nrgy xchang coffcnts ar approxmatd by th wghtd arthmtc avrag mthod. () Th atomc wght A r th Rossland of photon l harmonc avrag mthod. h r A T ar approxmatd by th wghtd Stp : Comput th approxmat dffuson coffcnts by th wghtd gomtrc avrag mthods n th drcton R Z of R and Z aftr obtanng th corrspondng lmtd flux dffuson coffcnts dnotd as K K 1 3 for T T and T r rspctvly. r m n r FIG. 8 (A) THE REGION. (B)THE SECTION REGION (B) MIXED GRIDS. (C) A NON-MIXED ELEMENT. (D) A MIXED ELEMENT. Stp 3: Prsnt th lmnt stff matrx and load vctor for any mxd lmnt E whr E A a lm E ( alm) 4 4 F ( fl ) 41 ( d ( b ) l m l m. - 39 - ll 33 lm ) 33
Whn 1 l m 4 and l m w hav R Z K 0 0 1 K1 k lm m R km ( b ) 33 t{ cos lrds 0 K 0 sn 0 lrds Ll R Ll R Z 0 0 K3 0 whr (cos lsnl ) s th unt outr normal vctor nl on th control-volum boundary bass functon about th vrtx X ; Whn 1 l m 4 and l m w hav whr k m 0 K L l a 0 0 d dr d dr ll ll ( d ) ( b ) { 0 a 0 t d 0 0 33 33 } RdRdZ Dl 0 0 a r dr 0 0 Z 0 k m 0 0 } Z K 3 s th nodal lnar f f l f RdRdZ 1 l 4 D l f r f r ar th rght sds of abov lnarzd llptc quatons about T rspctvly. Stp 4: Comput th lmnt stff matrcs and load vctors for non-mxd lmnts. Stp 5: Assmbl all th lmnt stff matrcs and load vctors to obtan th total stff matrx and load vctor. From abov on can s that th total stff matrx s symmtrc. Hnc w hav obtand a nw SFVE schm basd on th wghtd gomtrc avrag mthod for ths thr-dmnsonal modl problm. Th symmtry s also hlpful to ncras th computatonal ffcncy whn solvng th corrspondng dscrt systms. R Z On can also comput K K 1 3 by th wghtd arthmtc and harmonc avrag mthods n Stp to obtan th corrspondng SFVE schms. 4. NUMERICAL EXPERIMENTS In ths subscton w carry on som numrcal xprmnts. Exampl 4. In th modl problm 3 dfnd by Eq. (10) w choos ( 030) (095) th ntrfac : Z=74.1 th propr physcal paramtrs and th propr boundary condtons T n 0 T ; 0 ; r 4 T 1 r n 1 4 and th ntal-valu condton T t 0 310 r; and [ t 0 ] [030] th unt of tm s 0.1 ns. W tak th unform partton Ω h th partton numbrs along R-axs and Z-axs drctons ar 30 and 96 rspctvly. W rfn a ln of grds nghborng to th radatv tmpratur boundary (Drchlt boundary on 1) to dcras th nrgy consrvatv rror. Hnc stp szs of R-axs and Z-axs ar almost on unt. t - 40 -
FIG. 9 THE TEMPERATURE DISTRIBUTIONS FOR GEOMETRIC AVERAGE METHOD AT SIX MOMENTS: TOP FOR TE. MIDDLE FOR TI. BOTTOM FOR TR. Frstly w consdr th wghtd gomtrc avrag mthod. Numrcal rsults ar shown n FIG.9. From FIG.9 (top) to FIG.9 (bottom) on can s that th transfrs of lctron ron and photon agr wth th ral physcal phnomna. Smultanously th nrgy consrvatv rror s lss than 3% (Shown as th rd sold curv n FIG.1 (a). Th followngs ar also th rd sold curv). Thn w consdr th wghtd harmonc avrag mthod. Numrcal rsults ar shown n FIG.10. From FIG.10 (top) to FIG.10 (bottom) on can s that th transfrs of lctron ron and photon also agr wth th ral physcal phnomna. Smultanously th nrgy consrvatv rror s lss than 3% (Shown as FIG.1 (b)) - 41 -
FIG. 10 THE TEMPERATURE DISTRIBUTIONS FOR HARMONIC AVERAGE METHOD AT SIX MOMENTS: UPPER FOR TE. MIDDLE FOR TI. BOTTOM FOR TR. Fnally w consdr th wghtd arthmtc avrag mthod. Numrcal rsults ar shown n FIG. 11. In th fgurs for tmpratur dstrbutons th horzontal and vrtcal axs ar Z-axs and R-axs rspctvly. From FIG. 11 (top) to FIG. 11 (bottom) rspctvly on can s that th transfrs of lctron ron and photon agr wth th ral physcal phnomna. Smultanously th nrgy consrvatv rror s lss than 3% (Shown as FIG.1 (c)). Furthrmor w rfn th msh abov obtan th corrspondng rrors and dsplay n FIG.1 (s th blu sold ln). Comparng th rd sold ln wth th blu on on can also s that th nrgy consrvatv rrors all rduc by 50% for thr wghtd avrag mthods. Comparng wth all fgurs abov carfully on can fnd th followng facts. (1) Accordng to th small nrgy consrvatv rror th numrcal smulaton s agrabl for wghtd gomtrc avrag mthod. So do th othr two avrag mthods. () For Tr and T thr ar fw dffrncs btwn th wghtd gomtrc and arthmtc avrags. Th transfrs smulatd by th wghtd harmonc avrag ar fastr than thos of th othr two avrags. It s du to th charactrstc proprty of ths avrag mthod.. th avrag valu s partal to that n th small proporton rgon. In ths xprmnt th hghr tmpratur ls n th small proporton rgon n th mxd lmnts whch lads to th bggr dffuson coffcnts. Mayb t s th dfct of th harmonc avrag mthod not mmun to th xtrm valu. (3) For T thr ar fw dffrncs among thr avrags. It can b undrstood by th fact that th procss of nrgy xchang s: from T r to T thn from T to T togthr wth th charactr of th nrgy xchangs' coffcnts. - 4 -
FIG. 11 THE TEMPERATURE DISTRIBUTIONS FOR ARITHMETIC AVERAGE METHOD AT SIX MOMENTS: UPPER FOR TE. MIDDLE FOR TI. BOTTOM FOR TR. FIG. 1 THE COMPARISON FOR ENERGY CONSERVATIVE ERRORS OF THE THREE AVERAGE METHODS. 5 CONCLUSIONS In ths papr w construct a novl fnt volum schm for solvng thr mult-matral radatv hat transfr problms. W nnovatvly ntroduc th wghtd gomtrc avrag mthod for dsposng th dffuson coffcnt n th mxd lmnt and tak th ffct of th convxty of nonlnar dffuson functons nto account. Numrcal rsults confrm that th nw schm s vald and agrabl. ACKNOWLEDGMENT Th authors thank Prof. Sh Shu and Dr. Chunshng Fng for many hlpful dscussons. REFERENCES [1] A. Jafar S.H. Sydn and M. Haghpanah Modlng of hat transfr and soldfcaton of droplt/substrat n mcrocastng SDM procss Intrnatonal Journal of Engnrng Scnc18(008): 187-198 [] C. L. Zha W.B. P Q.H Zng. D LARED-H smulaton of gnton hohlraum In Proc. th World Congrss on Engnrng 010(III) WCE 010 London U.K [3] H. B. Jams J.T. Kng A fnt lmnt mthod for ntrfac problms n domans wth smooth boundars and ntrfacs Adv. Comp. Math. 6(1996): 109-138 [4] Z. M. Chn J. Zou Fnt lmnt mthods and thr convrgnc for llptc and parabolc ntrfac problms Numr. Math. 79(1998): 175-0 [5] S.V. Patankar. Numrcal hat transfr and flud flow. Hmsphr Nw York (1980) [6] S.Y. Kadoglu R. R. Nourgalv and V. A. Moussau A comparatv study of th harmonc and arthmtc avragng of dffuson coffcnts for nonlnar hat transfr problm Idaho Natonal Laboratory March 008 [7] S.B. Yust Wghtd avrag fnt dffrnc mthods for fractonal dffuson quatons Journal of Computatonal Physcs 16(006): 64-74 [8] C. Y. N S. Shu Z.Q. Shng A Symmtry-prsrvng fnt volum lmnt schm on unstructurd quadrlatral grds Chns J. Comp. Phys. 6(009): 17- [9] A. G. Hansn M.P. Bndso and O. Sgmund. Topology optmazaton of hat transfr problms usng th fnt volum mthod. Struct.Multdsc. Optm 31(006): 51-59 - 43 -
[10] G. L. Olson and J. E. Morl. Soluton of th radaton dffuson quaton on an AMR ulran msh wth matral ntrfac. In Tchncal Rport LA-UR-99-949 1999. Los Alamos Natonal Laboratory [11] D. A. Knoll and D. E. Kys. Jacobn-fr nwton krylov mthods: a survy of approachs and applcatons. J. Comput. Phys. 4(004): 357-397 [1] R.E. Marshak Effct of radaton on shock wav bhavor Phys. Fluds 1(1958): 4-9 [13] J.Y Yu G.W. Yuan Z.Q. Shng Pcard-Nwton tratv mthod for multmatral nonqulbrum radaton dffuson problm on dstortd quadrlatral mshs n: Proc. th World Congrss on Engnrng (II) WCE 009 London U.K [14] C.Y. N S. Shu X.D. hang and J. Chn SFVE schms for radatv hat transfr problms n cylndrcal coordnats and numrcal smulaton Journal of Systm Smulaton 4(01): 75-83 [15] S. Toadr and G. Toadr Symmtrs wth Grk mans Cratv Math. 13(004): 17-5 AUTHORS 1 Cunyun N (1974- ) Doctor Dgr assocat profssor major n computatonal mathmatcs major fld of study: numrcal computaton and numrcal stmulaton for hat transfr and flud dynamcs. Hayuan Yu (196- ) Doctor Dgr profssor major n computatonal mathmatcs major fld of study: numrcal computaton and numrcal stmulaton and supr-convrgnc for FEM. - 44 -