A NEW MIXED HEXAHEDRAL FINITE ELEMENT IN HEAT TRANSFER ANALYSIS

Transcription

1 A NEW MIXED HEXAHEDRA FINIE EEMEN IN HEA RANSFER ANAYSIS Dubrvk MIJUCA, An ŽIBERNA & Bojn MEDJO Fulty of Mthmtis, Univrsity of Blgrd, P.O.Box 55, Srbi nd Montngro -orrsondn: Abstrt. A nw originl riml-mixd finit lmnt roh nd rltd hxhdrl finit lmnt HC: for th nlysis of bhvior of solid bodis undr thrml loding is rsntd hr. h ssntil ontributions of th rsnt roh r th trtmnt of tmrturs nd ht fluxs s fundmntl vribls, furthr th solution of tmrturs nd ht fluxs from th sm systm of linr utions, nd initil/rsribd tmrtur nd ht flux bility. In ordr to minimiz ury rror nd nbl introdutions of flux onstrints, th tnsoril hrtr of th rsnt finit lmnt utions is fully rstd. h roosd finit lmnt is subjtd to th stndrd bnhmrk tst in ordr to tst onvrgn of th rsults, whih nlightn th fftivnss nd rlibility of th roh roosd. Kys-words: stdy st ht, finit lmnts, mixd formultion Introdution In th rsnt r nw originl finit lmnt roh for th solution of th stdy stt ht trnsfr in th solid body is rsntd. h min motiv for th rsnt invstigtion is found in lk of hxhdrl finit lmnt whih is rlibl [] nd robust in ordn to hng of its st rtio, nd simultnously, finit lmnt rodur whih trts both vribl of intrst, tmrtur nd ht flux, s fundmntl ons []. Furthr, th motiv is lso found in th known roblm of onnting th finit lmnts of diffrnt dimnsionlity, i.. whn modl roblm hs gomtril trnsitions from solid to thik or thin. Furthr, th min objtiv of th rsnt invstigtion is to show tht nw rlibl [] mixd hxhdrl (brik) finit lmnt HC: [], my b usd in th nlysis nginring onstrutions of rbitrry sh, without nd for ostriori lultion of ht fluxs. hus, fort th diffrn to th riml rohs, rsnt finit lmnt roh hs two fundmntl vribls: tmrtur nd ht fluxs. Consuntly, th min gol of th rsnt r is to vlidt th us of finit lmnt HC: in th stdy stt ht nlysis of isotroi, orthotroi or multi-mtrils solid bodis undr th diffrnt thrml or mhnil loding snrios. Nvrthlss, it is lnnd to imlmnt th rsnt roh in th xisting in-hous rimlmixd lstiity od for th mor urt dtrmintion of thrml strsss, whr no onsistny roblms will our in lultion of thrml nd mhnil dformtions [3, 4]. Wk form of th stdy stt ht fild utions Considr body whih ouis th losd nd boundd domin Ω of th Eulidin s E n ( n =,,3). h innr rt of Ω is dnotd by Ω nd its boundry by, Ω =Ω. h boundry is subdividd into four rts,,, r suh tht r =. h stt of th body is dsribd by th tmrtur nd by th ht flux vtor. Contt thrml flux through n intrf is n, whr n is th unit outwrd norml vtor on th intrf.

2 t us onsidr omlt systm of th fild utions in th stdy-stt ht trnsfr whr, div + f = in Ω, () = k in Ω, () r rstivly th utions of thrml bln tht stts tht th divrgn of th ht flux is ul to th intrnl ht sour f, Fourir lw of ht ondution whih ssums tht th ht flux is linrly rltd to th ngtiv grdint of th tmrtur, whr k is sond ordr tnsor of thrml ondutivity. hs two utions r subjtd to th nxt boundry onditions: = on, (3) n = = h on, (4) h n = = h ( ) on, (5) n = = h ( ) on, (6) 4 4 r r r Firstly, boundry ondition r tmrtur (3) whih is rsribd to b ul to ovr ortion of th boundry nd to ht flux. Sondly, boundry onditions du to th rsribd ht flux on boundry (4), nxt du to th onvtion (5), whr h is th onvtiv offiint nd is th tmrtur of th surrounding mdium, nd finlly, du to th rdition (6),whih will not b onsidrd bus thy r sour of nonlinrity. t us suos tht ll boundry onditions (3), (4), (5) r ssntil, nd hn xtly stisfid by th tril funtions of roblm. hn w nd to onsidr only th wk forms of th utions () nd (). By th us of th Glrkin rodur, on n sk th wk solution of () ( div + f ) ΘdΩ = (7) Ω whr Θ is tkn from th Hilbrt s of ll rl msurbl sur intgrl slr funtions. Furthr, w will onsidr th wk form of invrtibl onstitutiv utions (.), whr Q is th tst funtion tkn from th s of ll msurbl sur intgrbl vtor filds: ( k + ) QdΩ= (8) Ω By th siml summtion of (7) nd (8) w obtin th xrssion whih rrsnts symmtri wk formultion of mixd roblm. Howvr, it is rsntly ommon sns tht symmtri formultions r imrtil from th omuttionl oint of viw. Intgrting by rts nd lying divrgn thorm on th first trm on th lft sid yilds symmtri wk form of mixd roblm: Find{, } H ( Ω ) ( Ω) suh tht = nd Q Q Ω Ω Ω Ω k dω+ dω= ΘdΩ ΘfdΩ Θhd Θd () for ll { θ, Q } H ( Ω ) ( Ω) suh tht θ =, whr H is th s of ll slr filds whih r sur intgrbl nd hv sur intgrbl grdint.

3 Finit lmnt roximtions of th fild utions t C h b th rtitioning of th domin Ω into lmnts Ω i nd lt us dfin th finit lmnt subss for th tmrtur slr, th ht flux vtor nd th rorit wight funtions, rstivly s: _ h = H ( Ω ): =, = P ( ), Ωi Ωi Ch Θ = Θ Ω Θ = Θ=Θ Ω Ω M { H ( ):, P ( ), C } { ( ): n, n ( ), ( ), } M { Q H ( ): Q n, V ( ), C } Q Q h M i i h Q = H Ω = h = h = V Ω Ω C h i i h ϑ = Ω = = Ω Ω h M i i h In ths xrssions nd r th nodl vlus of th tmrtur slr nd flux vtor, rstivly. Aordingly, P nd V r orrsonding vlus of th introltion funtions, onnting tmrturs nd fluxs t n rbitrry oint in Ω i (th body of n lmnt), nd th nodl vlus of ths untitis. h omlt nlogy holds for th tmrtur nd flux wight funtions Θ nd Q. It should b notd tht whn rsnt finit lmnt roh is lid on modl roblms with brut mtril hngs whr lol ht flux disontinuity is ossibl to xist, th rsnt rul tht lol strss roximtion funtion r from ontinuous funtion s ( H ) n n is to hrd. It is lft for th futur invstigtion to rlx strss ontinuity on th intrf surf(s), whr fluxs will b hosn from s ( Ω ) formultion. n n sym (), s in originl riml-mixd 3 Numril imlmnttion By nlogy with finit lmnt roh in lstiity [5], ftr disrtiztion of th strting roblm by finit lmnt mthod, rsnt shm n b writtn s th systm of linr utions of ordr n= n + n, whr n is th numbr of tmrtur dgrs of frdom, whil n is th numbr of flux dgrs of frdom: A vv B vv v A v B v B = + vv Dvv v Bv Dv F H K () + In this xrssion, unknown (vribl) nd known (initil, rsribd) vlus of th fluxs nd tmrturs, dnotd by th indis v nd rstivly, r domosd. h nodl fluxs nd tmrturs omonnts r onsutivly ordrd in th olumn mtris nd rstivly. h homognous nd nonhomognous ssntil boundry onditions r tmrturs nd fluxs r introdud s ontribution to th right-hnd sid of th xrssion (). h mmbrs of th mtris A, B nd D, of th olumn mtris F, H, nd K (disrtizd body nd surf fors) in (), r rstivly: A = g V r g V dω ; B = g V P dω b Mr ( ) b ( M ) r M M ( ) M, Ω Ω D = h P P ; F = P fdω M M M M Ω H = P hd ; K = P h d M M h M M h ()

4 In th bov xrssions, r b r th omonnts of th invrs of th sond ordr tnsor of b th thrml ondutivity givn by th k = k g g b, whil th Eulidin shifting ortor g ( ) i j z z i is givn by g ( ) = δ ij g ( ), whr, z (, i j, k, l =,,3) is th globl Crtsin oordint ξ y ( r ) systm of rfrn. Furthr, y (,, r s t =,,3) is oordint systm t h globl nod, r ht flux. Furthr, th lol nturl (onvtiv) oordint systms r finit lmnts r dnotd by ξ ( bd,,, =,,3), whil, g b nd g ( ) mn r th omonnts of th ontrvrint fundmntl mtri tnsors, th first on with rst to nturl oordint systm of finit M lmnt ξ, nd th sond to y ( ) n M P t globl nod. Furthrmor, P,. For th rson ξ tht tnsoril hrtr is fully rstd, on n sily hoos rorit oordint systm t h globl nod for th introdutions of known fluxs nd/or tmrturs, or intrrttion of th rsults. 4 Finit lmnt HC8/7 h finit lmnt HC8/7 is shown in Figur. Its ronym is tkn from [4], whr th first lttr H stnds for hxhdrl lmnt gomtril sh, whil th lttr C indits th us of ontinuous roximtion funtions. h tmrtur nods r dnotd by shrs, whil ht flux nods r dnotd by ttrhdrons, s shown in Figur ζ ξ η Fig.. Finit lmnt HC8/7 h tmrtur nd ht flux fild is roximtd by tr-linr sh funtions, onntd to ight ornr nods. In ddition, stbiliztion of finit lmnt is hivd by full or rtil hirrhi introltion of ht flux on ordr highr thn tmrtur onntd to th dditionl u to nintn nods. hus, twnty-svn strss nods r vilbl to ommodt full triudrti xnsion in nturl oordints ξ, η nd ζ. Consuntly, th nxt multifild ombintions of th tmrtur nd flux nods r vilbl to usr: HC8/9, HC/, HC8/7 nd HC/7. Morovr, r h ornr nod thr is mximum four dgrs of frdom n = 4 (on tmrtur dgr of frdom n = nd thr dgrs of frdom r ht flux n = 3 ), whil in hirrhil nods (f ntr, mid-sid nd bubbl) thr r only dgrs of frdom onntd to ht flux. 5 Solution of th rsulting systm of linr utions In th rsnt r, th mthod bsd on th multifrontl roh, on of th min tgoris of dirt mthods for th solution of th rsulting systm of linr utions, is usd. h or of tht mthod is tkn from th od MA47 [6], rrsnting vrsion of srs Gussin limintion whih is imlmntd using multifrontl mthod

5 6 Numril xrimnt 6. Stdy Stt Ht rnsfr in Solid Stl Billt In th rsnt xml [6] th stdy stt ht trnsfr in solid stl billt, shown in Fig., A is nlyzd. h trgt vlu is tmrtur t oint A: = 3.8 C. h mtril rortis r 5 W / m o C nd W / m o C for thrml ondutivity nd ht trnsfr offiint, rstivly. Ambint bulk fluid tmrtur is, lik in th first modl roblm, of o C.h modl is disrtisizd by inrsing sun of th rfinmnt ftor, N 3N, whr N = 4, 6,8,,,4, nd = 3, 4,5,6,7,8, in ordr to hk th onvrgn of th finit lmnt solutions. Figur : Solid stl billt From th tmrtur rsults t th trgt oint shown in Figur 3, w my s tht both finit lmnt rohs, rsnt nd riml, onvrg uniformly to th sm vlu whih is littl bit lowr thn trgt vlu. [ o C] rgt 33 rltiv bsolut rror r (%) 4 3 Numbr of Elmnts 3 HC8/9 Strus7 HEXA Numbr of Elmnts Figur 3: Solid billt: h onvrgn of tmrtur t trgt oint h min diffrn is in lultion of ht flux fild, s Figur 4. It is lultd ostriori in th xmind riml roh (Strus7) rsulting with bnorml disontinuity long lmnt intrfs, whih ris th nd for th us of som rovry or smoothing thniu of th ht flux (dul) vribl [8]. On th othr hnd, rsntly ht flux fild is lultd s th fundmntl vribl, nd it is ontinuous s it is xtd to b.

6 y Figur 3: h ht flux lultd by th finit lmnts HEXA nd HC8/9 Conlusion From th stndrd bnhmrk xml in stdy st ht nlysis of solid bodis solvd by th rsnt hxhdrl finit lmnt HC8/9, w my rliminry onlud tht it hs good onvrgn. Nvrthlss, th dtild invstigtion of ll sts of onvrgn, s onsistny nd stbility ruirmnts r [], is lft for furthr invstigtion. Morovr, w my mhsiz tht on of th min otntil of th rsnt finit lmnt is tht it us ovroms known trnsition roblm of onnting finit lmnts of diffrnt ty nd dimnsions. Consuntly, w my nd with th onlusion tht rsnt finit lmnt roh giv us grtr dsign frdom thn stndrd riml rohs tht us diffrnt kind of finit lmnts: solid, lt/shll, bm... In ddition, th rsnt finit lmnt roh will b usd in onntion with th xisting in-hous softwr [5], bsd on th originl rlibl mixd u σ finit lmnt roh in lsti nlysis, for dtrmintion of th thrml strsss, whih is lft for furthr rort, lso. Aknowldgmnts h uthor is grtful to th Ministry of Sin, hnologis nd Dvlomnt of Rubli of Srbi for roviding finnil suort for this rsrh, undr th rojts No.IO865 nd EE97b. Rfrns [] Bth KJ () h inf-su ondition nd its vlution for mixd finit lmnt mthods. Comutrs & Struturs 79: 43-5 [] Arnold DN (99) Mixd finit lmnt mthods for lliti roblms. Comut. Mth. Al. Mh. Eng. 8: 8-3 [3] Mirnd S, Ubrtini F () On th onsistny of finit lmnt modls, Comut. Mthods Al. Mh. Engrg., 9, 4-4 [4] Cnnrozzi AA, Ubrtini F () A mixd vritionl mthod for linr ould thrmolsti nlysis, Intrntionl Journl of Solids nd Struturs 38, [5] Miju D (4) On hxhdrl finit lmnt HC8/7 in lstiity, Comuttionl Mhnis 33 (6), [6] Duff IS, Gould NI, Rid JK, Sott JA, nd urnr K (99) Ftoriztion of srs symmtri indfinit mtris, IMA J. Numr. Anl., 8-4 [7] G+DComuting, Strus7, Finit lmnt nlysis systm softwr kg Vrifition Mnul, Austrli [8] Hinton E., Cmbll J.S (974) ol nd globl smoothing of disontinuous finit lmnt funtions using lst surs mthod, Int.J.Num.Mths. Eng. 8, 46 48