APPLICATION OF GALERKIN FINITE ELEMENT METHOD IN THE SOLUTION OF 3D DIFFUSION IN SOLIDS

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1 Cênca/Scnc APPLICATION OF GALERKIN FINITE ELEMENT METHOD IN THE SOLUTION OF D DIFFUSION IN SOLIDS E C Romão a, M D d Campos c, J A Martns b, and L F M d Moura a Unvrsdad Estadual d Campnas Faculdad d Engnhara Mcânca a Dpartamnto d Engnhara Térmca Fludos b Dpartamnto d Engnhara d Matras ABSTRACT Ths papr prsnts th numrcal soluton by th Galrkn Fnt Elmnt Mthod, on th thr-dmnsonal Laplac and Hlmholtz quatons, whch rprsnt th hat dffuson n solds For th two applcatons proposd, th analytcal solutons found n th ltratur rvw wr usd n comparson wth th numrcal soluton Th rsults analyss was mad basd on th th L Norm (avrag rror throughout th doman) and L Norm (maxmum rror n th ntr doman) Th two applcaton rsults, on of th Laplac quaton and th Hlmholtz quaton, ar prsntd and dscussd n ordr to to tst th ffcncy of th mthod Kywords: Fnt Elmnt Mthod, Galrkn Mthod, Dffuson, Sold, Posson Equaton, Hlmholtz Equaton Rua Mndlyv, 00 Cdad Unvrstára "Zfrno Vaz" Dstrto d Barão Graldo CEP: Campnas SP Brasl stanr@yahoocombr c Unvrsdad Fdral d Mato Grosso Insttuto d Cêncas Exatas da Trra Av Govrnador Jam Campos, 690 CEP Barra do Garças MT Brasl NOMENCLATURE f sourc trm k thrmal conductvty N ntrpolaton functon N nods numbr of nods n ach fnt lmnt T tmpratur Tˆ tmpratur approxmaton n th fnt lmnt v wght functon v wght functon n th lmnt Grk symbols Γ Γ Subscrpts thr-dmnsonal doman thr-dmnsonal doman n th lmnt contour of a doman contour of an lmnt nod dntfcaton of a nod 1 INTRODUCTION Th frst publcatons n fnt lmnt mthod appard n 1950 s wth th works wrttn by Turnr t al (1956), Clough (1960) Argyrs (196) Ths wr usd to solv problms n structural analyss Som dcads latr, Znkwcz and Chung (1965), Odn and Wllford (197), Chung (1978) Bakr (198), among othr publcatons, tratd th hat transfr and flud flow problms solutons Th classcal fnt lmnt mthod s known as Bubnov-Galrkn Fnt Elmnt Mthod (GFEM) Thr ar othr varants of th fnt lmnt such as thos by Ptrov- Galrkn Fnt Elmnt Mthod and th Last- Squars Fnt Elmnt Mthod (LSFEM), both dvlopd n ordr to compnsat th lmtatons of th GFEM whn appld for hat transfr and flud flow problms Th GFEM appld for ths knd of problms gnrally producs oscllatng solutons for hgh Péclt and Rynolds numbrs Rcntl svral authors hav prsntd applcatons of th fnt lmnt mthod for two and trdmnsonal problms, among thm Camprub t al (000), Romão t al (008a), Romão t al (008b) Hannukann t al (010) Engnhara Térmca (Thrmal Engnrng), Vol 8 N o 0 Dcmbr 009 p

2 Cênca/Scnc Romão t al Applcaton of Galrkn Fnt Elmnt In ths work t s prsntd an applcaton of Galrkn Fnt Elmnt Mthod to th numrcal soluton of Laplac trdmnsonal quatons for dffuson n solds and th Hlmholtz s quaton for dffuson wth gnraton dpndabl of tmpratur n solds In ths stud analytcal solutons valdat th numrcal rsults by th analyss of th L Norm of th rror that rprsnts an avrag of th rror n th soluton and L Norm that rprsnts th maxmum rror n th soluton MODEL EQUATION It s prsntd th trdmnsonal dffuson quaton wth gnraton dpndabl of th tmpratur n closd lmtd sold domans dsgnd as R Th modl quaton has th followng form: k + k + k + B T + f = 0 (1) whr k s a postv constant; T = T ( x, y), B = B( x, and f = f ( x, ar functons of th spac x, z R Th boundary condtons ar of th frst scond knd 1 Dscrtzaton Galrkn Mthod Th GFEM s appld for dscrtzaton of th ntgral quatons In ths mthod an approxmaton of unknown varabl s a functon Tˆ that whn substtutd n th Eq (1) producs a null rsdual So, th approxmaton form s: N nods T Tˆ = N Tˆ () = 1 whr N nods s numbr of nods nsd a fnt lmnt, N ar th ntrpolaton functons for th lmnt and Tˆ ar nodal valus of T n th lmnt Th rsdual s dtrmnd by substtutng th approxmaton Tˆ n Eq (1) and s dfnd as: R = k + k + k + B Tˆ + f () Th soluton s found by forcng th pondrd rsdual to b null In othr words, t must b found as a functon of T ˆ V, V C ( ), such as: R v d = 0, v V, = 1,,,N nods (4) whr R s a lmtd and closd doman In th Galrkn Fnt Elmnt Mthod, th wght functon s th sam ntrpolaton functon,, v = N, = 1,,, Nnods Aftr ntgraton of Eq (4) th rsult s an algbrac systm of quatons wrttn n matrcal form as follow: [ K ]{ T} = { F} ˆ (5) n whch th matrx coffcnts ar; K = k d k d + k BN N d d (6a) F = f N d (6b) wth, = 1,,, Nnods NUMERICAL APPLICATIONS Th matrx coffcnts ar obtand by numrcal ntgraton usng Gauss Mthod (Rdd 199) and mappng th ral lmnts n th mastr lmnt n th local coordnats ξ, η ζ (-1 ξ,η,ζ 1) Th ntrpolaton functons and thr drvatvs for th hxahdral lmnt can b found towards Dhatt t al (1984) Th systm of algbrac quatons rprsnt by Eq (5) was solvd by th Gauss-Sdl mthod and crtra of stop wth maxmum rror E max Th computatonal cod was dvlopd n FORTRAN languag Th mshs wr rfnd untl th lmt of th computr s mmory capacty Both lnar (ght nods) and quadratc hxahdra (twnty svn nods) wr usd, wth h rprsntng th sz of th lmnt (cubc lmnt) Th L norm of th rror was dfnd lk n 1/ Nnost / Nnost = 1 (Zlhmal, 1978): = In ths quaton, Nnost s th total numbr of nods n th msh and = T T, whr ( num) ( an) T (num) s th rsult from th numrcal soluton and T (an) s th rsult form th analytcal soluton rspctvly Applcaton 1 Posson Equaton Dffuson n Solds In ths applcaton th coffcnt B n Eq () s null and th doman s an untary cub = [0,1] Th govrnng quaton s rducd to; 80 Engnhara Térmca (Thrmal Engnrng), Vol 8 N o 0 Dcmbr 009 p 79-8

3 Cênca/Scnc Romão t al Applcaton of Galrkn Fnt Elmnt + + = 0 (7) whr T = T ( x, Th analytcal soluton of Eq (7) s of th form: sn( π y)sn( π T ( x, = snh( π ) [ snh( x) + snh( (1 x) ] π π (8) Fgur Tmpratur profl n plan xz wth y = 0,5 by LSFEM from a msh wth 491 nods (h = 1/16) usng hxadrals wth 8 nods, Applcaton 1 In Fgur s prsntd th tmpratur profl n a transvrsal scton of th doman Fgur 1 L and L Norms for mshs wth hxahdral of 8 nods applcaton 1 Applcaton Hlmholtz Equaton Dffuson wth Gnraton n Solds In ths applcaton th coffcnt B n Eq () s non null and th doman s an untary cub = [0,1] So th govrnng quaton s of th form: T = 0 (9) whr T = T ( x, Fgur L and L Norms for mshs wth hxahdral of 7 nods applcaton 1 Ths boundary condtons wr chosn to satsfy th analytcal soluton of th proposd problm Th rsults for th mdum and maxmum rrors by usng lnar and quadratc lmnts ar prsntd n Fgurs 1 and rspctvl whr h rprsnts th rfnmnt of th msh It s obsrvd that th rrors ar hghr to gross mshs, as xpctd Fgur 4 L and L Norms for mshs wth hxahdral of 8 nods applcaton Th Eq (9) has an analytcal soluton, ths soluton s: T( x, = sn x + sn y + sn z (10) Engnhara Térmca (Thrmal Engnrng), Vol 8 N o 0 Dcmbr 009 p

4 Cênca/Scnc Romão t al Applcaton of Galrkn Fnt Elmnt h = 1/4 th quadratc lmnt bttr rsults wr rachd than th lnar lmnt For pur dffuson th quadratc msh wth h = 1/4 prsnts bttr rsults than th mor rfnd msh of lnar lmnts wth h = 1/ Th sam bhavor was obtand n th Applcaton, whr nthr th mor rfnd msh of lnar lmnts prsnts bttr rsults than th h = 1/4 msh of quadratc lmnts 5 ACKNOWLEDGMENTS Fgur 5 L and L Norms for mshs wth hxahdral of 7 nods applcaton As sm n th Applcaton 1, th boundary condtons wr chosn to satsfy th analytcal soluton n th proposd problm Th rsults for th mdum and maxmum rrors by usng lnar and quadratc lmnts ar prsntd n Fgurs 4 and 5 rspctvl whr h rprsnts th rfnmnt of th msh Smlar to thos rsults of Applcaton 1, t s obsrvd that th rrors ar hghr to gross mshs, as xpctd Fgur 6 Tmpratur profl n plan xz wth y = 0,5 by LSFEM from a msh wth 491 nods (h = 1/16) utlzng hxadrals wth 8 nods, Applcaton In Fgur s prsntd th tmpratur profl n a transvrsal scton of th doman 4 CONCLUSION In th proposd applcatons, th Galrkn Fnt Elmnt Mthod shown good rsults, manly whn quadratc lmnts wr usd, vn for th msh wth Th prsnt work was supportd by th Natonal Councl of Scntfc Dvlopmnt and Tchnology CNPq Brasl 6 REFERENCES Argyrs, J H, 196, Rcnt Advancs n Matrx Mthods of Structural Analyss, Prgamon Prss, Elmsford, Nw York Bakr, A J, 198, Fnt Elmnt Computatonal Flud Mchancs, Nw York: Hmsphr, McGraw-Hll Camprub, N, Colomnas, I, Navarrna, F and Castlro, M, 000, Galrkn, Last-Squars and GLS numrcal approachs for convctv-dffusv transport problms n ngnrng, Europan Congrss on Computatonal Mthods n Appld Scncs and Engnrng Chung, T J, 1978, Fnt Elmnt Analyss n Flud Dynamcs, Nw York: McGraw-Hll Clough, R W, 1960, Th Fnt Elmnt Mthod n Plan Strss Analyss, Procdngs of nd Conf on Elctronc Computaton, Amrcan Socty of Cvl Engnrs, Pttsburgh, Pnn, pp Dhatt, G, and Touzot, G, 1984, Th Fnt Elmnt Mthod Dsplayd, John Wly & Sons Hannukann, A, Korotov, S, and Krzk, M, 010, Nodal O(h 4 )-Suprconvrgnt n D by avragng pcws lnar, blnar, and trlnar FE approxmatons, Journal of Computatonal Mathmatcs, Vol 8, No1, pp 1-10 Odn, J T, and Wllford Jr, L C, 197, Analyss of vscous flow by th fnt lmnt mthod, AIAA J, Vol 10, pp Rdd J N, 199, An Introducton to th Fnt Elmnt Mthod, Scond Edton, McGraw- Hll Romão, E C, Moura, L F M, and Slva, J B C, 008a, Analyss of Error n th Soluton of th -D Dffuson Equaton by Fnt Elmnt Mthods TEMA Tndêncas m Matmátca Aplcada Computaconal (n portugus), Vol 9, No, pp Romão, E C, Moura, L F M, and Slva, J B C, 008b, Hat Transfr n Mult-Connctd and Irrgular Domans wth Non-Unform Mshs, Thrmal Engnrng, Vol 7, No, pp Engnhara Térmca (Thrmal Engnrng), Vol 8 N o 0 Dcmbr 009 p 79-8

5 Cênca/Scnc Romão t al Applcaton of Galrkn Fnt Elmnt Turnr, M J, Clough, R W, Martn, H C, and Topp, L P, 1956, Stffnss and Dflcton Analyss of Complx Structurs, J Aron Sc, Vol, No 9, pp Znkwcz, O C and Chung, Y K, 1965, Fnt lmnts n th soluton of fld problms Th Engnr, Vol 10, pp Zlhmal, M, 1978, Suprconvrgnc and rducd ntgraton n th fnt lmnt mthod, Math and Comput, Vol, p 66 Engnhara Térmca (Thrmal Engnrng), Vol 8 N o 0 Dcmbr 009 p