Alternative Finite Element Formulations for Transient Thermal Problems. Application to the Modelling of the Early Age Thermal Response of Concrete

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1 Intrnal Rport, DECvl, March Altrnatv Fnt Elnt Forulatons for ransnt hral Probls Applcaton to th Modllng of th Early Ag hral Rspons of Concrt João A. xra d Frtas, Cuong. Pha and Vasco Mano chncal Unvrsty of sbon, Insttuto Supror écnco Dpartnt of Cvl Engnrng and Archtctur Av. Rovsco Pas, 49- sboa, Portugal -al: frtas@cvl.st.utl.pt. Introducton hs rport addrsss th odllng of th arly ag thral rspons of concrt usng dffrnt fnt lnt forulatons, naly th convntonal and hybrd forulatons. h govrnng quatons ar statd frst, to dntfy th assuptons ad and to stablsh th notaton. h ncrntal for (n t of ths quatons s statd nxt and th t dnson of th probl s dscrtzd usng a trapzodal ntgraton rul. h altrnatv fnt lnt forulatons ar drvd n th scond part of th rport. h convntonal (sngl-fld, confor approach s prsntd frst. It s basd on th drct approxaton of th tpratur fld n soparatrc lnts. h xd vrson s prsntd nxt ntroducng th ndpndnt approxaton of th hydraton dgr fld. h conforty condton s thn rlaxd nxt to ntroduc th hybrd forulaton, basd on th ndpndnt approxaton of th tpratur fld n th doan of th lnt and of th hat flux on ts boundary. h scond part of th rport closs wth th prsntaton of two varants of th hybrd forulaton, th hybrd-xd and th hybrd-rfftz varants. h thrd part of th papr addrsss th nurcal plntaton of th fnt lnt forulatons. h dfnton of th approxatons functons usd n th plntaton of ach forulaton s prsntd for on-dnsonal, two-dnsonal (and axsytrc probls and thr-dnsonal probls and th algorth usd n th soluton of th transnt thral probl s outlnd for ach forulaton.

2 Intrnal Rport, DECvl, March. Basc quatons h syst of quatons govrnng th transnt thral rspons of a concrt spcn, wth doan V, boundary Γ and unt outward noral n, can b statd as follows, usng a t fra t and a Cartsan rfrnc syst x = ( x yz : σ + Q = ρ c nv ( ε = nv ( σ = kε nv (3 n σ = t on (4 Γ σ = onγ (5 n σ = h ( onγ (6 cr a q In th quatons abov, s th tpratur fld, and and ε dfn ts t and spac gradnts, rspctvly, wth = { } x y z. In th thral qulbru condton (, vctor σ dfns th vlocty of hat flow, ρ c s th volutrc spcfc hat and Q s th hat rlas rat du to hydraton. It s dfnd by th Arrhnus law, whr Ea Q = A f( xp (7 R E a s th actvaton nrgy ( J / Mol, R s th unvrsal gas constant ( A J / Mol K, s th axu valu of th hat producton rat ( J / s and th functon f ( dfns th voluton of th noralsd hat producton rat as a functon of th hydraton dgr,. h hydraton dgr at nstant t s dfnd as th rato of hat rlasd up to that nstant and th total hat xpctd upon coplton of cnt hydraton, Q : Qt ( ( t = (8 Q In th last doan condton, th consttutv condton (3, k s th thral conductvty of concrt. It s dfnd as follows [4], k = k ( (9 whr k s th valu of th thral conductvty of concrt n ts hardnd stat. In th Nuann-typ boundary condton (4, n s th unt outward noral vctor and t dfns th prscrbd hat flux. In th Drchlt-typ boundary condton (5, s th

3 Intrnal Rport, DECvl, March prscrbd tpratur. h Robn-typ boundary condton (6, whr a s th ar tpratur, odls radaton and convcton usng th Nwton s coolng law usng a sngl convctonradaton coffcnt: hcr = hc + h r ( h convcton coffcnt s gvn by, v, v < 5 /s hc = ( v, v > 5 /s whr v (/s s th wnd spd at th surfac of th concrt [3]. h radaton coffcnt s gvn by, h r ( ε , a> 78.5 K = ( 4.8 ε, a< 78.5 Kawhr ε s ssvty of concrt, whch s usually n th rang of [,5].ratur. Cobnaton of quatons ( to (3 lads to Fourr s law of hat consrvaton, whch can b statd as follows: k + Q = ρc nv (3 h odl dfnd abov, takn fro Rf. [], dos not contan a watr consrvaton quaton, t bng assud that watr s prsnt at vry pont n quantts suffcnt to nsur full aturaton. 3. Incrntal forulaton It s assud that all varabls ar wrttn n ncrntal for, say, v( x, t = v ( x + δ v( x (4 for a varabl v at nstant t, whr v dfns ts valu at th bgnnng of th t ncrnt, t. Assung that all condtons ar t at nstant t, th rsultng ncrntal for of syst (-(5 s, usng dfnton (8, δ Q δ ρcδ nv (5 σ + = δε = δ nv (6 δσ = k δε + R nv (7 k n δσ = δ Γ (8 t on σ δ = δ onγ (9 h n δσ = h δ + R onγ ( q 3

4 Intrnal Rport, DECvl, March whr, accordng to condton (3 and dfnton (9: k = k ( ( R k =.33 k δ( ε + δε ( h ncrntal for of dfnton ( for th convcton coffcnts s, 3.95δ v, v < 5 /s δ hc = ( ( v + δ v 7.6 v, v > 5 /s t bng gnrally assud that th wnd spd s constant throughout th aturng procss, δ hc =. As th ncrntal for of dfnton ( for th radaton coffcnts s,.75 εδ a, a > 78.5 K δ hr = (4, a < 78.5 K th coffcnts prsnt n th ncrntal for ( of th convcton-radaton condton ar, h = + hcr δ hcr (5 R = h δ + δ h ( (6 h cr a cr a whr a s th ar tpratur at th nd of th ncrnt: a = a + δa and δ h = δh + δ h Dfnton (7 of th hat rlas rat du to hydraton s wrttn n for, cr c r E a Q = A f( xp (7 R s quaton (8, t bng assud that th voluton of th noralsd hat producton rat as a functon of th hydraton dgr, dfnd by functon f (, s known and dfnd n pc-ws lnar for: f ( = f + f δ (8 Undr ths assupton, th xprsson s found for th ncrntal for of quaton (7 s: δ f δ = β + δ + β R (9 f E a β = (3 R f f δ R = + δ R + δ f f (3 δ δ R = ( β ( β+ 6β + (3 4

5 Intrnal Rport, DECvl, March Equaton (9 s drvd assung that and f. hrfor, t s not vald at nstant t =, whn f = and = = rplacd by th followng:, accordng to dfntons (7 and (8, and should b δ Q δ = A f xp( β + β + βr δ (33 It s rcalld that th ntal tpratur fld,, s known at nstant t =, as t dfns th ntal condton of probl (-(6. 4. Dscrtzaton n t Dffrnt procdurs can b usd to dscrtz syst (5-( n th t dnson,.g. Rf. [6]. rapzodal ruls ar dfnd n th gnral for, for a gnral varabl v, v= v + γ δtv + γδtv (34 whr v and v rprsnt th valus of th t drvatv of th varabl at nstant t (start of t stp and at nstant t = t + δt (nd of t stp, rspctvly, and γ and γ ar t ntgraton factors (.g., γ = and γ = for th backward-eulr ntgraton rul and γ = γ = for th Crank-Ncholson thod. h ncrntal for of th t ntgraton rul (34, γ δtδv = δv ( γ + γ δtv (35 s usd to obtan th dscrtzaton n t of th thral qulbru condton (5, and of dfnton (9: γδ δσ + δ= ρ δ + (36 t Q c R nv ( R = ( γ + γ δt Q ρc (37 δ = A δ + R + A R (38 R A A A = (39 fβγδt = f f γ δt f( γ + γ δt = f f γ δt Accordng to rsult (33, quaton (38 holds f th coffcnts ar rdfnd as follows: (4 (4 A = (4 5

6 Intrnal Rport, DECvl, March R 5. Dscrtzaton n spac A = (43 ( ( β A f γδtxp β δ = + R β δ Q A f γδtxp hs scton s usd to forulat th altrnatv fnt lnt forulatons that can b usd to solv th odllng of th thral rspons of arly ag concrt. hr applcaton to on-, twoand thr-dnsonal probls s addrssd n Scton 6. o support th drvaton of th fnt lnt quatons, for a typcal lnt wth doan V, t s convnnt to rcall frst th doan quatons of th probl, th thral qulbru condton (36, th tpratur gradnt dfnton (6 and th consttutv rlaton (7, t Q c R nv (44 γδ δσ + δ= ρ δ + (45 δ = δ nv ε (46 and th Arrhnus law (38: δ δ σ = k δε + R k nv (47 = A δ + R+ A R nv (48 Four parts ust b dntfd n th boundary of th lnt, Γ, naly ts ntrlnt boundary, Γ, and th parts of th Nuann, Drchlt and convcton-radaton boundars of th sh th lnt ay contan, Γ = Γ Γ Γ Γ (49 σ q on whch condtons (8 to ( hold, stll undr condtons (6 and (7: δσ = δt onγ σ n (5 δ = δ onγ (5 n δσ = hδ + Rh onγ q (5 It s assud that th ntrlnt contnuty condton s nforcd on th tpratur fld, anng that condton (5 holds, δ = δ onγ (53 whr δ rprsnts th tpratur fld on th boundary of th connctng lnt. 6

7 Intrnal Rport, DECvl, March 5. Sngl-fld forulaton h convntonal lnt s drvd approxatng th tpratur fld n th doan of th lnt, whr th (row- vctor ( x =Ψ Τ nv (54 Ψ lsts th ntrpolaton functons typcal of soparatrc transforatons and vctor Τ dfns th tpratur at th nods of th lnt. h thral contnuty condtons (5 and (53, on th Drchlt boundary and on ntrlnt boundars, ar locally nforcd through th usual nodal ncdnc condtons, whch can b drctly nforcd whn soparatrc lnts ar usd. It s assud, also, that th tpratur gradnt condton (6 s locally nforcd: ε( x = Β n V (55 Β = Ψ (56 Dffrnt tchnqus can b usd to drv th fnt lnt quatons. h Galrkn vrson of th wghd rsdual thod can b suarzd n th followng stps:. h tpratur approxaton functons ar usd to nforc on avrag th thral qulbru condton (45: ( γδ δ δ ρ δ Ψ t σ Q + c + R dv =. h frst tr s ntgratd by parts to forc rndr xplct th boundary tr ncssary to nforc th boundary condtons: ( Ψ σ Ψ σ Ψ ( γδt δ dv + γδt n δ dγ Q δ ρcδ R dv = 3. Condtons (46 and (47 ar nforcd, undr dfnton (56, to obtan th stffnss atrx of th lnt, K, and th rsdual tr assocatd wth th consttutv rlaton, ( k Ψ σ Ψ ( γδt δ γδt δ dγ Q δ ρcδ R dv K R + n = R k : K B k B dv (57 = R dv k B R k (58 = 4. Approxaton (54 s nsrtd to obtan th lnt spcfc hat atrx, H, and th qulbru rsdual tr, R : ( ( γδt K + H δ + R γδt R + γδt Ψ n δσ dγ Ψ Q δdv = k H = Ψ ρ c Ψ dv (59 7

8 Intrnal Rport, DECvl, March R Ψ RdV (6 = 5. h boundary tr s uncoupld to nforc condtons (5 and (5 to dfn th lnt convcton-radaton atrx, C, th (prscrbd nodal hat flux vctor, quvalnt nodal hat vctor, Q, as wll as th assocatd rsdual trs: ( ( ( Q σ, and th (fr γδt K + C + H δ + γδt Rh Rk + R + γδt Qσ + Q Ψ Q δdv = C = Ψ h Ψ dγ (6 q Q = Ψ δ tdγ (6 σ σ Q = Ψ n δ σ dγ (63 R = Ψ R dγ (64 h h q 6. h Harrhnus law (48 s nsrtd to dfn th lnt hydraton atrx, A, and th assocatd rsdual tr: ( ( ( γδt K + C + H A δ + γδt Rh -Rk + R R R + γδt Qσ + Q = A = Ψ Q A Ψ dv (65 R Q R dv Ψ (66 = = h copact for of th lnt solvng syst s: R Q A R dv Ψ (67 ( t ( S δ = R γδt Qσ + Q (68 S = H A+ γδ K + C (69 ( R = R + R R γδt R -R (7 h k Syst (68 s assbld for th fnt lnt sh usng th tchnqus adoptd n convntonal lnts, naly nforcng nodal connctvty (now n trs of nodal tpraturs and th qulbru of th quvalnt nodal forcs (now n trs of th quvalnt nodal hat flux. h drvaton suarzd abov s partcularzd for on-dnsonal probls n Appndx A. Aftr syst (68 s assbld and solvd, th tpratur fld s updatd usng approxaton (54 wth, 8

9 Intrnal Rport, DECvl, March = + δ (7 accordng to quaton (4. In ordr to nsur that th doan condtons of th probl ar satsfd at th nd of th t stp, th hydraton dgr and ts t drvatv ar coputd usng quatons (48 and (7, rspctvly. h t rat of th tpratur fld s dtrnd fro th Fourr quaton (3. 5. Mxd forulaton By dfnton, a xd forulaton s charactrzd by th ndpndnt approxaton of two (or or flds n th doan of th lnt. In th prsnt contxt, th xd forulaton s dsgnd to allow th usr to adopt ndpndnt approxatons for th thral and hydraton dgr flds. Approxatons (54 and (55 stll hold but th hydraton dgr fld s no longr approxatd nsrtng th tpratur approxaton n th Arrhnus law (48. It s nstad ndpndntly approxatd n for, ( x =Ψ n V (7 whr th (row- vctor Ψ lsts th soparatrc ntrpolaton functons and vctor dfns th hydraton dgr at th nods. Dffrnt dgrs of approxaton, that s, lnts wth dffrnt nubr of nods, can b usd to st up approxatons (54 and (7. As th Arrhnus law (48 s no longr locally nforcd for th assud tpratur fld, ths condton s nforcd ndpndntly n wak for to yld, n th Galrkn vrson of th wghd rsdual thod: ( δ δ Ψ Q A A R A R dv = (73 h way th Arrhnus law s rwrttn nsurs th sytry of th fnt lnt solvng syst, as shown blow. h frst fv stps dscrbd n Scton 5. stll apply to th forulaton of th xd lnt. Approxaton (7 s plntd n th sxth stp, nstad of th drct substtuton of th Arrhnus law, to yld, t ( σ ( S δ S δ = R γδ Q + Q (74 S = H + γδt K + C (75 S (76 Q dv = Ψ Ψ ( R = R γδt R R (77 h k 9

10 Intrnal Rport, DECvl, March Substtuton of approxatons (54 and (7 n quaton (73 lads to th wak for of th Arrhnus law, whr us s ad of rsult (76: S δ + S δ = R (78 S = Ψ Q A Ψ dv (79 R = R + R (8 R Ψ Q A RdV (8 = R Ψ Q A A R dv (8 = h xd lnt solvng syst s obtand cobnng quatons (74 and (78: S S δ R Qσ + Q γδt = S δ S R ( Hybrd forulaton h couplng of th approxaton of th gotry and thral flds that typfs soparatrc lnts coplcats th coputaton of th spac gradnts prsnt n dfnton (56. Morovr, t s not spl to plnt procdurs for p-, h-, or hp-rfnnt of th soluton, that s, rfnnt of a soluton obtand avodng th r-valuaton of th altrnatv systs (68 or (83. Both ltatons can b ovrco by sparatng th gotry and thral fld approxatons and, spcally, by dfnng th lattr n th global syst of rfrnc of th sh, nstad of usng th co-ordnat syst of th astr lnt. Consquntly (s Scton 6, th functons n approxatons (54 and (7 ar no longr nodal, and thr wghts, and, rprsnt gnralzd apltuds nstad of nodal tpraturs and hydraton dgrs, rspctvly. Whn ths gnralzaton of th convntonal fnt lnt forulaton s plntd, t s no longr possbl to nforc n strong for th boundary thral contnuty condtons (5 and (53. As t shown blow, ths lads to ncssty of coplntng th doan approxaton (54 wth th approxaton of th hat flux on th boundars of th lnt whr t s not prscrbd. hs approxaton s wrttn n for, t( x = n σ = Ψ t onγ = Γ Γ (84 t t and th lnt s classfd as hybrd as (at last on boundary fld s drctly assud.

11 Intrnal Rport, DECvl, March h drvaton prsntd n Scton 5. for th sngl-fld lnt stll appls, wth th dffrnc that, n stp 5, th hat flux on boundary Γ t s drctly approxatd usng quaton (84, nstad of bng tratd as a fr tr. hrfor, quaton (68 xtnds nto for: S δ + S δt = R γδ Q (85 t t σ S = γ δt Ψ Ψ dγ (86 t t t R = R + R R + γδtr k h wak for of th thral contnuty condtons (5 and (53 s stll statd n wghd rsdual for, for th assud tpratur fld (54, to yld: ( Ψ γδ t t δ δ d Γ t = (87 S δ = R (88 t t t R = γ δt Ψ δ dγ (89 It s rcalld that th tr δ n ths dfnton rprsnts th tpratur fld on a connctng lnt on ntrlnt boundars, as statd by quaton (53. h hybrd lnt solvng syst s obtand cobnng quatons (85 and (88: 5.4 Hybrd-xd forulaton S St δ R Q σ γδt = S δ t O t R h hybrd-xd forulaton s basd on th ndpndnt approxaton of th tpratur and hydraton dgr flds n th doan of th lnt and of th hat flux on ts boundary. It can b vrfd that th cobnaton of th procdurs dscrbd n Sctons 5. and 5.3 lad to th followng solvng syst: It s rcalld that tr S S S t δ R Qσ S S O δ = R γδt t δ S O O t R (9 (9 R s dfnd by quaton (7 for th sngl-fld and hybrd forulatons and by quaton (77 for th xd and hybrd-xd forulatons. 5.5 Hybrd-rfftz forulaton As shown blow, th solvng syst (9 for th rfftz varant of hybrd fnt lnt forulaton, as th forulaton s stll basd on th tpratur and hat flux approxatons

12 Intrnal Rport, DECvl, March (54 and (84. h ssntal dffrnc s that th tpratur approxaton s now dntfd wth th foral solutons of th (lnarzd Fourr quaton (3. 6. Approxaton functons h functons usd to st up th doan approxatons (54, (55 and (7 and th boundary approxaton (84 ar dfnd nxt for soparatrc and hybrd lnts. Du to ts partcular natur, th hybrd-rfftz bass s tratd sparatly n Scton 7. h approxaton functons ar squntally dfnd for on-, two- and thr-dnsonal applcatons. In all nstancs th approxaton bass tak two of th followng gnral fors, whr N rprsnts th dnson of th bass. [ N ] [ ] Ψ = Ψ Ψ Ψ n V or on Γ (9 B = Ψ Ψ Ψ N n V (93 6. On-dnsonal probls h typcal lnt usd to solv on-dnsonal probls s shown n Fgur. Functons (9 ar dfnd and rprsntd n Fgur for lnar (two-nod, N = and quadratc (thrnod, N = 3 lnts, rspctvly: [ ] [ ] hy ar soparatrc lnts n th sns that: Ψ = Ψ Ψ x (94 Ψ = Ψ Ψ Ψ 3 x (95 Ψ f = j = at nod j f j (96 nx = nx = + x Fgur : On-dnsonal lnt h gradnt splfs to = x n on-dnsonal applcatons, yldng th followng xprssons for th tpratur gradnt approxaton functons (93: B = [ B B] = [ + ] (97 B = [ B B B3] = [ 3+ 4x/ + 4x/ 4( x/ ] (98

13 Intrnal Rport, DECvl, March x x Ψ ( x = x / Ψ ( x = ( x / ( x / Ψ ( x = x / Ψ ( ( / ( / x = x x Ψ ( x = 4( x / ( x / 3 Fgur : Approxaton functons for on-dnsonal soparatrc lnts h nod concpt s abandond n hybrd forulatons. gndr polynoals ar usd as approxaton functons to xplot thr orthogonalty (s Fgur 3, + PPd j ξ f = j + = f j whr, accordng to th notaton dfnd n Fgur : x ξ = (99 P ( ξ = P ( = Fgur 3: gndr functons h gndr functons ar so scald as to nsur unt (or null valus for th ntgrals nvolvng th fnt lnt approxaton functons: Ψ ( x = ( / P ( ξ ( ξ ξ P ( ξ = ( 3ξ 3

14 Intrnal Rport, DECvl, March ΨΨ j f = j dx = f j h corrspondng gradnt functon s, accordng to th coordnat transforaton (99: P B ( x = ( / ( / ξ ( 6. wo-dnsonal probls (NO YE h typcal four-nod lnt usd to solv two-dnsonal probls s shown n Fgur 4. Whn ths astr lnt s th appng of th gotry of s dfnd n for, 4 x = Ψ( ζ, ζc ( = Ψ = ( ζ ( ζ 4 Ψ = ( + ζ ( ζ 4 Ψ = ( + ζ ( + ζ 3 4 Ψ = ( ζ ( + ζ 4 4 (3 whr vctors c dfn th coordnats of nod n th global syst of rfrnc of th sh, x= ( x x : c c = (4 c h sa functons ar usd to dfn th tpratur (and, vntually, hydraton dgr approxaton bass (9. h gradnt functons (93 ar dtrnd fro th plct drvaton rul, Ψ Ψ ζ Ψ ζ x Ψ (, j ξ η = = + (5 x ζ x ζ x j j j whr: Ψ = ( ζ ; Ψ = ( ζ ζ 4 ζ 4 Ψ =+ ( ζ ; Ψ = ( + ζ ζ 4 ζ 4 Ψ =+ ( + ζ Ψ =+ ( + ζ ζ 3 4 ζ 3 4 Ψ = ( + ζ Ψ =+ ( ζ ζ 4 4 ζ 4 4 (6 o obtan th ranng trs, t s ncssary to coput th Jacoban atrx of th transforaton, 4

15 Intrnal Rport, DECvl, March whr, x ζ x ζ J = (7 x x ζ ζ x Ψ x Ψ ζ ζ ζ ζ x ζ ζ ζ ζ N N = c; = c = = N N Ψ x Ψ = c ; = c = = ts dtrnant, th Jacoban of th transforaton, x x x x J = J = ζ ζ ζ ζ and th nvrs th of Jacoban atrx, to yld: x ζ ζ x ζ J = J (8 x x ζ ζ x ζ x = + J ; = J x ζ x ζ ζ x ζ x = J ; = + J x ζ x ζ (9 hs procss of coordnat transforaton can b avodd usng hybrd lnts, by dfnng th approxaton functons (9 drctly n th global syst of rfrnc. ttng, x ξ = ( x ξ = ( whr and ar typcal dnsons of th lnts n drctons x and x, rspctvly, th approxaton functons ar dfnd by orthogonal products of gndr functons, n for, qu, para a udança d coordnadas Error! Rfrnc sourc not found.: 5

16 Intrnal Rport, DECvl, March ar dfnd and rprsntd n Fgur for lnar (two-nod, N = and quadratc (thr- ζ (, + (+, ζ (, (, + Fgur 4: Mastr quadrangular lnt. 6

17 Intrnal Rport, DECvl, March Rfrncs. Branco F, Mnds P, Mrabll E. Hat of hydraton ffcts n concrt structurs, ACI Matr J. 99; 89(: Fara R, Aznha M, Fguras JA. Modllng of concrt at arly ags: Applcaton to an xtrnally rstrand slab, Cnt & Concrt Coposts 6; 8: Jonasson JE. Modllng of tpratur, ostur and strsss n young concrt, PhD thss, ula Unvrsty of chnology, ula, Ruz J, Schndlr A, Rasussn R, K P, Chang G. Concrt tpratur odlng and strngth prdcton usng aturty concpts n th FHWA HIPERPAV softwar. 7 th Int. Conf. Concrt Pavnts, Orlando, USA,. 5. Slvra A. h nflunc of thral solctaton on th bhavour of rnforcd concrt brdgs, PhD thss, Unvrsty of Porto, Porto, 996 (In Portugus. 6. aa KK, Zhou X, Sha D. h t dnson: A thory towards th voluton, classfcaton, charactrzaton and dsgn of coputatonal algorths for transnt/dynac applcatons, Arch Coputatonal Mthods n Engnrng ; 7:

18 Intrnal Rport, DECvl, March Appndx A: On-dnsonal probls h basc quatons (45 to (53 splfy as follows for on-dnsonal applcatons (.g. concrt slabs, whr now all varabls ar scalar: γ δt ( δσ + Q δ = ρcδ + R x ( x x δε = x ( δ x (3 δσ = k δε + R x (4 k δ = A δ + R + A R x (5 n δσ = δ t at x = and / or x = (6 x δ = δ at x = and / or x = (7 n δσ = h + R at x = and or x = (8 δ h / It s assud that th ntrlnt contnuty condton s nforcd on th tpratur fld, anng that condton (5 holds, δ = δ at x = and / or x = (9 h procdur suarzd n Scton 5 spcalzs as follows for on-dnsonal probls, for a gvn approxaton functon, Ψ :. h tpratur approxaton functon s usd to nforc on avrag th thral qulbru condton (: ( ( Ψ γδt x δσ Q δ + ρcδ + R dx=. h frst tr s ntgratd by parts to forc rndr xplct th boundary tr ncssary to nforc th boundary condtons: ( ( x γδ t x Ψ δσ dx + γδ t Ψ n δσ Ψ Q δ ρ c δ R dx = 3. Condtons (3 and (4 ar nforcd, undr dfnton (56, to obtan th stffnss atrx of th lnt, K, and th rsdual tr assocatd wth th consttutv rlaton, R k : N γδt Kn δn Rk t ( + γδ Ψ nx δσ Ψ Q δ ρcδ R dx= n= K n R = B k B dx ( k k = B R dx ( 8

19 Intrnal Rport, DECvl, March 4. Approxaton (54 s nsrtd to obtan th lnt spcfc hat atrx, H, and th qulbru rsdual tr, R : N ( ( n= γδtk + H δ + R γδtr + γδt Ψ n δσ Ψ Q δdx= n n n k x n Ψ ρ Ψ H = c dx ( Ψ R = R dx (3 5. h boundary tr s uncoupld to nforc condtons (6 and (8 to dfn th lnt convcton-radaton atrx, C, th (prscrbd nodal hat flux vctor, th (fr quvalnt nodal hat vctor, N n= ( γδt Kn + Cn + Hn δ n + ( ( σ Q, as wll as th assocatd rsdual trs: γδt Rh R k + R + γδt Q + Q Ψ Q δdx= ( ( n = xψ Ψ = xψ Ψ + x xψ Ψ = x= Q σ, and C n h n h n h (4 Qσ = Ψ δt whr prscrbd ( x = and / or x = or non (5 Q = Ψ δσ whr not prscrbd (6 h x h R = n Ψ R (7 6. h Harrhnus law (5 s nsrtd to dfn th lnt hydraton atrx, A, and th assocatd rsdual tr: N n= ( γδt Kn + Cn + Hn An δ n + ( h k γδ ( σ γδt R R + R R R + t Q + Q = n Ψ Ψ A = Q A dx (8 Ψ R = Q R dx (9 Ψ R = Q A R dx (3 It can b radly vrfd that th trs prsnt n th xd lnt solvng syst (83 ar, ( n S = Ψ Q Ψ dx (3 9

20 Intrnal Rport, DECvl, March and, for th hybrd lnt syst (9: ( Ψ S = Q A Ψ dx (3 n ( St γδt n n xψ Ψ t = (33