Formulation of the extended Finite Element and the matched asymptotic development methods applied to the transient heat transfer in brazed assembly

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1 6th IASME/WSEAS Intrnatonal Confrnc on HEA RASFER, HERMAL EGIEERIG and EVIROME (HE'8) Rhods, Grc, August -, 8 Formulaton of th Xtndd Fnt Elmnt and th matchd asymptotc dvlopmnt mthods appld to th transnt hat transfr n brazd assmbly GUYE D.H.,, LAZARD M., BILERYS F., LAMESLE P., DOUR G. GIP InSIC, ERMP 7 ru d Hllul, 88 Sant-Dé-ds-Vosgs, Franc Ecol Mns Alb, CROMP Campus Jarlard, 83 Alb C Cédx 9, Franc mlazard@nsc.fr Abstract: h transnt hat transfr n th on-dmnsonal modl of brazd assmbly s nvstgatd by couplng th Xtndd Fnt Elmnt Mthod (X-FEM) and th matchd asymptotc dvlopmnt mthod (DAR). h approach proposd by th DAR mthod s basd on to construct th nrchd functons n th X-FEM framwork. h accuracy of th rsults obtand by th couplng s valuatd by comparng thm wth th solutons of th quadrupol mthod and of th commrcal computatonal cod ABAQUS. Ky-Words: Xtndd Fnt Elmnt Mthod (X-FEM), matchd asymptotc dvlopmnt (DAR), hat transfr, transnt rgm, quadrupol mthod, brazd assmbly Introducton In th most rcnt yars, th brazng has bn usd as an ffcnt and approprat tchnqu of assmblag, spcally n th sctor of toolng such as th fabrcaton of moulds []. In ths applcatons, th brazd assmbly must wthstand th thrmomchancal solctatons. So ts thrmomchancal bhavor mrts a profound rsarch. At th frst stp, w consdrd th modllng of th mchancal bhavor of th brazd assmbly takng nto account th brazd jont as a sngularty by usng th couplng of th Xtndd Fnt Elmnt Mthod (X- FEM) and th matchd asymptotc dvlopmnt mthod (DAR) []. hs tndncy of couplng gvs us th promsng sgnals. h DAR mthod provds an approach of th bhavor of a sngular doman at two scals: macroscopc and mcroscopc [3]. On th othr hand, th X-FEM s basd on th da that th sngularty s modld by usng th approxmaton of standard fnt lmnt and th nrchd functons [4]. hs nrchd functons ar chosn to rprsnt as closly as possbl th bhavor of th sngularty. In othr words, w bas on th nformaton whch s known a pror about th bhavor of th doman whch contans th sngularty to construct th nrchd functons. In our da of th couplng, ths nformaton s xplotd from th solutons of th DAR mthod. In th scond stag, w contnu to apply th couplng of X-FEM and DAR to trat th transnt thrmal bhavor of a brazd assmblag. In ths papr, th X-FEM formulaton and ts couplng wth th DAR mthod ar prformd. In ordr to valuat th fasblty and accuracy of th couplng n ths problm, w compar ts rsults wth thos obtand through th quadrupol mthod [5] and wth th fnt lmnt commrcal cod ABAQUS. Problm Formulaton h outln of ths paragraph s as follows: n sctons. and., lt us consdr th formulaton of th fnt lmnt mthod (FEM) and thn of X-FEM whch srv th thrmal problm. In scton.3, w nd to construct th formulaton of X-FEM by couplng wth th soluton obtand by th DAR n th on dmnsonal thrmal problm. In scton.4, w prsnt gnrally about th quadrupol formulaton as a rfrnc.. Formulaton of FEM = mp ϕ = ϕ mp Γ Fg. Doman wth th boundary condtons W hav a body wth th boundary condtons as n th fgur. h strong form of th thrmal transnt problm whch s wrttn for th tmpratur ( x, t ) s: uuur ρc dv( λ ) = ω dans ], t[ t ( x,) = x () = mp sur Γ ], t[ uuur r λ n. = ϕmp sur Γ ], t[ Γ ISS: ISB:

2 6th IASME/WSEAS Intrnatonal Confrnc on HEA RASFER, HERMAL EGIEERIG and EVIROME (HE'8) Rhods, Grc, August -, 8 whr ρ s th mass dnsty, c s th hat spcfc capacty, ω s th ntrnal sourc and λ s th thrmal conductvty. h wak form of th problm () s gvn by: uuuruur ρc vd λ. vd = ωvd ϕmpvdγ t () Γ whr v s th tst functon. o dtrmnat th tmpratur fld unknown soluton of (), w partton th doman by a st of sublmnts. hs sub-lmnts ar connctd by n nods. h tmpratur s approachd by th dscrtzd form: n ( x, t) = ( x) ( t) (3) = whr s th shap functon and s th nodal tmpratur of th -th nod. By substtutng (3) n th wak form () and applyng a fw combnatons, w obtan th matrx form: [ C] [ K]{ } = { Q} (4) t whr { } s th vctor of unknowns, [ C ] s th matrx of capacty, [ K ] s th matrx of conductvty, { Q } s th vctor of charg. h componnts of ths matrx and vctors ar calculatd as follows: [ C] ρ [ ][ ] = c d C = ρc d j j = = uuuuruuuur = d K = λ d [ K] [ B][ D][ B] j j = = { Q} = ω [ ] d ϕmp [ ] dγ = Γ Q = ωd ϕmpdγ = Γ o approach th soluton of th systm (4), w must mplmnt a schm of tmporal drvatv. In ltratur, thr ar svral schms whch gv us th approxmaton n varaton of th tm. Hr, w choos th schm of two stps tm of Crank-cholson. h quaton (4) s translatd nto th rlaton of rcurrnc: [ C] m [ C] m [ K] { } = [ K] { } t t (5) m m { Q} { Q} { } m can b calculatd by assumng that { } m s known. In th abov formula, th stp tm t must satsfy th condton of stablty: t a (6) 6 ( x) whr a s th thrmal dffusvty ( a= λ / ρc), x s th dstanc btwn nods for lmnt nar surfac wth hghst tmpratur gradnt.. Formulaton of X-FEM From th xprsson of FEM n th prvous scton, w nd to th formulaton of th X-FEM for th thrmal problm. Lt s consdr now th doman whch contans a sngularty, a brazd jont n our study. Bcaus th poston of th jont s fxd n th doman, w accpt that t s consdrd as a sngularty wthn th manng "spatal" and "no tmporal". Instad of usng a msh whch conforms to th brazd jont as n FEM, thanks to X-FEM, w can apply an arbtrary msh. Morovr, w ntroduc th partcular functons calld nrchd functons to captur th charactrstc of th brazd jont. hs functons ar mbddd for th nods of msh whch ar n th vcnty of jont. W call thm th nrchd nods. h dscrtzd approxmaton of th tmpratur fld s gvn by: n nr ( x, t) = ( x) ( t) ( x) ψ ( x) b ( t) k k k = k= cla nr whr nr s th numbr of nrchd nods, ψ k s th nrchd functon and b k s th paramtr of nrchmnt. In th formula (7), w dstngush two dffrnt parts: cla s th classcal part whch prsnts th bhavor of doman wthout th sngularty lk th fld (3), whl nr s th nrchd part that s addd to rprsnt th bhavor of th sngularty. By substtutng (7) n wak form (), w gt th sam matrx form as (4): C K { } = { Q } t (8) In th abov formula, X rprsnts a quantty whch s calculatd wthn X-FEM to dstngush ths magntud X n FEM. h formulaton of quantts n th quaton (8) s gvn by: { } = b (7) ISS: ISB:

3 6th IASME/WSEAS Intrnatonal Confrnc on HEA RASFER, HERMAL EGIEERIG and EVIROME (HE'8) Rhods, Grc, August -, 8 C = ρc d = = ψ ψ j j j = [ ] K = B D B d B =, ( ψ ),, ( ψ ) x x jx j j, x { Q } = ω d ϕ mp dγ = Γ In th sam way of FEM, to ovrcom th tmporal drvat n (8), w apply th schm of Crank- cholson. h rsoluton of (8) bcoms to dtrmn { } m through { } m by th rlatonshp of rcurrnc: C m C m K { } = K { } t t (9) m m { Q } { Q }.3. Approach of DAR mthod W consdr our D modl of a brazd assmbly n th thrmal problm (Fg. 3). In addton to assumng that th thcknss of th brazd jont s small, w can dfn a charactrstc paramtr ε = / L whr L dnots th thcknss of a consttutv sht. So, w dfn, as usual n matchd asymptotc mthods, th followng nnr varabl y = x/ ε n ordr to strtch th boundary layr contanng th brazd jont. W ar thn lookng for outr and nnr asymptotc xpansons of th tmpratur and hat flux flds out of ths layr and n ths layr. Morovr, by assumng that both xpansons hold tru n an ntrmdat zon, matchng condtons ar addd to ths two xpansons..3 Couplng of X-FEM and DAR n th D thrmal problm Fg. 3 On-dmnsonal thrmal modl of brazd assmblag Fg. Ida of couplng X-FEM and DAR h dtrmnaton of nrchd functons n (7) plays th most ssntal rol wthn X-FEM. hs functons hav to rprsnt n th approprat mannr th bhavor of whol doman takng nto account th sngularty. For ths purpos, t s prfrabl that w know a pror th nformaton about th bhavor of th body whch contans th sngularty. In our work, w ar confdnt of usng th DAR mthod to acqur ths nformaton. h da of couplng X-FEM and DAR s prsntd n th fgur abov. h ror tmpratur flds n ( ) and 3 ( ) vald far from th brazd jont ar rspctvly gvn by: ε m ( x) = m ( x) εm ( x)... () whr th ror trms ar dfnd by: λ3( s ) λx s λ3x s ( x) = x c c λ( s ) λx s λ3x s ( x) = x c c L λ3( s) c =. ( xx ) λc L λ( s) c =. ( xx ) s λc wth c = λλ3 λλ3 λλ ; c = λxs λ3x h ntror tmpratur flds vald n th nghborhood of th brazd jont ar rspctvly gvn n and by: ISS: ISB:

4 6th IASME/WSEAS Intrnatonal Confrnc on HEA RASFER, HERMAL EGIEERIG and EVIROME (HE'8) Rhods, Grc, August -, 8 ε m ( x) = τ ( y) ετm ( y)... () whr th ntror trms ar dfnd by: λx s λ3x s τ = c Lλ3( s ) λ y xc τ = ( λ λ) λc L c Lλ( s ) λ3y xsc τ = ( λ λ3) λc L c.3. Constructon of th nrchd functons of X- FEM by usng th solutons of th DAR mthod From th solutons of DAR mthod obtand n th abov scton, w construct th nrchd functons of X- FEM. h nrchd doman contans all th nrchd nods around th brazd jont (on th lft and on th rght). In ths doman, w dstngush thr typs of lft rght nrchd functons: th ntror ons ψ nt, ψ nt ar lft rght vald nsd th jont; th ror ons ψ, ψ ar usd outsd th lmnt whch contans th jont and th lft rght functons of transton ψ trans, ψ trans ar ntrpolatd btwn th two prcdnt typs. Fg. 4 Rlaton btwn th solutons of DAR mthod and th nrchd functons of X-FEM h dtrmnaton of ths nrchd functons from th solutons of DAR mthod bass on th condtons blow: lft ψ ( x = xd ) = lft ψ ( x = x) = ε ( x = x) rght ψ ( x = xf ) = rght ψ ( x = x) = ε ( x = x) lft ψ nt = ετ () rght ψ nt = ετ lft lft ψtrans ( x= x) = ψ ( x= x) lft lft ψtrans ( x = xj) = ψnt ( x= xj) rght rght ψtrans ( x = x) = ψ ( x = x) rght rght ψtrans ( x = xj ) = ψnt ( x = xj) whr xd, xf ar two nods of rmty of th nrchd doman, x, x ar two nrchd nods th closst to th jont, xj, x j ar two nds of th brazd jont..4 Quadrupol mthod o valdat th rsults w obtand, svral mthods could hav bn usd such as th consrvatv avragng mthod [6] or th mthod basd on th Papouls Brg mthod [7] but w usd th quadrupol mthod. hs mthod has bn dvlopd and usd succssfully to solv transnt thrmal problms such as for nstanc hat transfr n cuttng tools [8] or n stratfd moulds [5]. In ths work, w xplot t to obtan th rsult as rfrnc for our couplng of X-FEM and DAR. h formulaton of quadrupol mthod provds us th rlaton btwn th nput tmpratur-hat flux vctor at th front sd and th output vctor at th back sd through a transfr matrx M : θ( x, p) a b θ( xs, p) = φ φ ( x, p) c d 3 ( xs, p) M (3) whr p s th Laplac varabl, a, b, c and d ar th componnts whos forms ar ntrprtd blow, θ and φ ar tmpratur and hat flux rspctvly n th Laplac s spac. In our cas, th brazd assmbly contans thr layrs: plat - jont - plat and th hat flux s orthogonal to th layrs. hn th transfr matrx s th product of thr matrcs: 3 M = M (4) = Each matrx M coshu M s calculatd by: snhu = snh u coshu whr u = kl, = λk, k = p/ a λ, a ar rspctvly th thrmal conductvty and dffusvty of th th layr. In (3), two trms ar normally dtrmnd from th boundary condtons. hn w calculat th trms of rst. Whn ths two vctors ar known, t s asy to dduc th vctor of tmpratur and hat flux at any pont n th doman. Wth th functons that dpnd on ISS: ISB:

5 6th IASME/WSEAS Intrnatonal Confrnc on HEA RASFER, HERMAL EGIEERIG and EVIROME (HE'8) Rhods, Grc, August -, 8 th Laplac varabl p, w hav svral formulatons (Sthfst [9], Hoog [], Fourr ) to rturn to tmporal spac to gt th functons that dpnd on tm t. 3 Problm Soluton ow w consdr th modl as Fg. 3. h plats ar mad of stl and th brazd jont s formd by th alloy of nckl and slvr wth th thcknss, th conductvts and th dffusvts summarzd n th tabl blow: Plat Jont hcknss (mm) Conductvty ( W / mm. C ) Dffusvty ( mm / s ) ab. Data usd for th smulatons W assum that th ntal tmpratur s qual to zro for th whol doman: = n = C (5) h boundary condtons ar th Havsd tmpratur at th front sd and th mposd tmpratur at th back sd: C t = - ( x= x ) = C t > - x ( = xs ) = C t h tmpratur dstrbuton s dtrmnd n th ntrval of tm t = [, s ] wth th constant stp tm t = 5s. h msh contans 5 nods, so = x 7.69 ( mm) h charactrstc tm of th problm s calculatd by: a t a t tc = = = =.336 (6) x x 7.69 hs valu s qut satsfd wth th condton of stablty (6). h profls of tmpratur at t = 5,,,5,, s for th whol doman ar prsntd n th fgur 5. XFDA and Quad dnot th tmpraturs whch ar calculatd by th couplng of X-FEM and DAR and th quadrupol mthod rspctvly. In th fgur 6, w compar th voluton of th tmpratur as a functon of tm at th pont P whch ar obtand by two abov mthods and by th computatonal cod ABAQUS ( ABA ). Fg. 6 Evoluton of tmpratur as a functon of tm at th pont P h symbols XFDA_P and Quad_P dnot th volutons of tmpratur at th pont = (,,3) whch ar obtand by th couplng of X-FEM and DAR and th quadrupol mthod rspctvly. ( C) 8 Evoluton of tmpratur Error ( C) XFDA_P Quad_P XFDA_P Quad_P XFDA_P3 Quad_P3 Err_P Err_P Err_P3-7 5 t (s) Fg. 5 Profls of tmpratur at th tm t = 5,,,5,, s Fg. 7 Evoluton of tmpratur as a functon of tm at svral ponts and th rror btwn th couplng of X-FEMDAR and th quadrupol mthod ISS: ISB:

6 6th IASME/WSEAS Intrnatonal Confrnc on HEA RASFER, HERMAL EGIEERIG and EVIROME (HE'8) Rhods, Grc, August -, 8 Furthrmor, w can s th rror for ach coupl of curvs: Err_P = XFDA_P - Quad_P In th fgur 8, w can s th voluton of th functon hat flux dnsty tm at thsl ponts (P, P and P 3 ). Idntcally wth th tmpratur, th appllatons FluXFDA_P and FluQuad_P dnot th volutons of tmpratur at th pont = (,,3) whch ar obtand by th couplng of X-FEM and DAR and th quadrupol mthod rspctvly. And w hav also th rror for ach coupl of curvs: Err_P = FluXFDA_P - FluQuad_P,35,3,5,,5,,5 φ (W/mm ) Evoluton of hat flux dnsty Error (W/mm ) -,5 5 t(s) 5,5 FluXFDA_P FluQuad_P FluXFDA_P FluQuad_P FluXFDA_P3 FluQuad_P3 Err_P Err_P Err_P3 Fg. 8 Evoluton of hat flux dnsty as a functon of tm at svral ponts and th rror btwn th couplng of X-FEMDAR and th quadrupol mthod 4 Concluson h couplng of X-FEM and DAR s appld to rsolv th thrmal transnt problm of th on-dmnsonal modl of brazd assmbly. h nspraton of th solutons of th DAR mthod n th constructon of nrchd functons n X-FEM provd th fasblty and th accuracy. h profls of tmpratur n th whol doman and th voluton of tmpratur and hat flux dnsty n functons of th tm at svral ponts of doman whch obtand by ths couplng ar vry clos to th rsults of th quadrupol mthod and of ABAQUS modl. hs rsults ncourag us to go on prformng a twodmnsonal formulaton and obtanng fnally thrmomchancal rspons of th brazd assmbly thanks to th couplng of ndd fnt lmnt and matchd asymptotc dvlopmnt mthods. Rfrncs: [] J. haboury, C. Barlr, F. Bltryst, M. Lazard, J.L. Batoz, Rapd lamnatd toolng n d castng: dsgn, modlng and xprmntal tsts, Advancs n Producton Engnrng and Managmnt Journal ISS , undr prss. [] D. H. guyn, F. Bltryst, M. Lazard, P. Lamsl, G. Dour, Couplng of th Xtndd Fnt Elmnt Mthod and th matchd asymptotc dvlopmnt n th modllng of brazd assmbly, Procdngs of th th Intrnatonal ESAFORM Confrnc on Matral Formng, 8. [3] D. Lgullon, R. Abdlmoula, Mod III nar and far flds for a crack lyng n or along a jont, Intrnatonal Journal of Solds and Structurs, Vol.37,, pp [4]. Moës, J. Dolbow,. Blytschko, A fnt lmnt mthod for crack growth wthout rmshng, Intrnatonal Journal for umrcal Mthods n Engnrng, Vol.46, 999, pp [5] M. Lazard, ransnt thrmal bhavor of multlayr mda: modlng and applcaton to stratfd moulds, Journal of Engnrng Physcs and hrmophyscs, Vol.79, o.4, 6, pp [6] R. Vlums, A. Buks, Consrvatv Avragng Mthod and ts applcaton for on hat conducton problm, Procdngs of th 4 th WSEAS Int. Conf. on Hat ransfr, hrmal Engnrng and Envronmnt, Elounda, Grc, August -3, 6 (pp 6-3). [7] L. Garba t al, ransnt hat conducton n compost systms, Procdngs of th 4 th WSEAS Int. Conf. on Hat ransfr, hrmal Engnrng and Envronmnt, Elounda, Grc, August -3, 6 (pp ). [8] M. Lazard, P. Corvsr, Modllng of a tool durng turnng. Analytcal prdcton of th tmpratur and of th hat flux at th tool s tp, Appld hrmal Engnrng, Vol. 4, Issus 5-6, 4, p [9] H. Sthfst, Rmarks on Algorthm 368. umrcal Invrson of Laplac ransform, Com. A.C.M, Vol. 64, o.3, 97, p [] D. Hoog t al., SIAM J. Sc. Stat. Comput., Vol. 3, 98, p ISS: ISB: