Identification of the thermo-physical parameters of an anisotropic material by inverse problem

Size: px

Start display at page:

Download "Identification of the thermo-physical parameters of an anisotropic material by inverse problem"

Hilary Doyle
5 years ago
Views:

1 Amrcan Journal of Engnrng Rsarch (AJER) E-ISSN: p-issn: Volum-8, Issu-3, pp Rsarch Papr Opn Accss Idnfcaon of h hrmo-physcal paramrs of an ansoropc maral by nvrs problm H. Arjdal a, J. Chaouf a, J.C. Dupr b, A. Grmanau b, S. Ouhn b a. Laboraory of lcroncs, Sgnal Procssng and Physcal Modlng (LESPPM,) Unvrsy Ibn Zohr, Agadr Morocco. b. Unvrsy of Pors, Insu P ', UPR 3346 CNRS, Franc. Corrspondng Auhor : H. Arjdal ABSTRACT: Th purpos of hs work s o drmn h hrmo-physcal paramrs of an ansoropc maral. Th mhod consss n lookng for hs paramrs from h knowldg of h mpraur fld. Th rsoluon of h problm s basd on h fn lmn mhod. Th drc problm has yldd convncng rsuls. Th lar hus found ar n agrmn wh h xprmnal rsuls. Subsqunly, w approach h oppos problm whou apprhnson by proposng an opmzaon mhod basd on h conjuga gradn algorhm. KEYWORDS: Idnfcaon; Thrmophyscal paramrs; Invrs problm; Fn Elmn Mhod; Infrard Camra; Infrard Thrmography; Projcd Conjuga Gradn Mhod Da of Submsson: Da of accpanc: I. INTRODUCTION Th good knowldg of h hrmophyscal proprs of h marals, mad possbl o prdc hrmal phnomna. In h fld of mchancal and dsgn knowldg, h hrmal proprs of marals ar ndd for mor ralsc modlng. I s also crucal for h dsgn of phoovolac clls. Th xploaon of xprmnal mpraur flds consus h bass of h approach framng h prsn sudy. A rcangular pla s had [1, 2] on on sd and h mpraur fld s capurd by an nfrard camra. Th Invrs Problm s solvd by crossng back h quaons oband by h Fn Elmn Mhod for solvng h Drc Problm. Dong so, w labora ffcn algorhms abl o accuraly dnfy h hrmal paramrs of polymhylmhacryla. II. DIRECT PROBLEM 2.1 Poson of h problm In our ndavour o dnfy h hrmal characrscs of a maral w wll undrak a procdur ha s boh xprmnal and numrcal. W consdr a rcangular hn sold pla of lngh L, wdh h and hcknss. Th pla s homognous; whl s hcknss s small, s lngh s vry clos o s wdh. W wll assum ha h mpraur dsrbuon s wo-dmnsonal. 2.2 Problm formulaon In ordr o xnd h dscrpon o srongly ansoropc marals whr h conducvy marx s sold, an arlr sudy was carrd ou for h dagonal conducvy marx [1]. Th problm o b solvd s dncal o h prvous on xcp ha h numbr of paramrs o b drmnd s grar. Ths par s dvod o solvng a problm of ha ransfr [5]. L us consdr ha a rcangular homognous sold pla, wh lngh l, wdh h and of vry small hcknss n fron of s ohr dmnsons. Suppos hn ha h pla occups h nrval [, L] of h, Ox axs, [, h] of h, Oy axs and ha a m =, h dsrbuon of h mpraur s known o all M (x, y) of h fld, and qual o, T( x, y, ). A consan ha flux φ 1 s mposd on h sd boundd by x = (dnod Γ 1 ), and a consan φ 2 s mposd on h sd w w w. a j r. o r g Pag 7

2 boundd by y = (dnod Γ 2 ). Th ohr sds (dnod Γ 3 and Γ 4 ) ar wll procd agans any convcv, radav or conducv currns [2, 4]. L FIG. 1 Rcangular pla and boundary condons c b h ha capacy pr un volum ( bng h spcfc mass and c h ha capacy pr un mass). L 12 b h hrmal conducvy nsor. Th drc problm for calculang h mpraur T( x, y, ) s hrfor dfnd by h followng Paral Dffrnal Equaon (PDE) sysm: T c dv(. gradt ) n (1) Th abov boundary and nal condons rad as follows: T T gradt n s x y on Γ 1 (2a) x T T gradt n 12 2 on 2 x y (2b) T T on x y y (2c) T T 12 on (2d) 4 x y T ( x, y,) T ( x, y) (2) Equaons (1), (2a), (2b), (2c), (2d) and (2) dfnng h drc problm can b solvd numrcally by h fn lmn mhod. 2.3 Solvng h drc problm by h fn lmn mhod Wh h fn lmn approach, w wll dvlop a calculaon cod wh quanav nformaon on h hrmo-physcal proprs of h marals, h boundary condons and ha fluxs appld, as wll as h characrscs of h chosn dscrzaon (yp of lmn, msh sz). w procd o h dscrzaon of h doman Ω n sub domans Ω calld lmns. Th gomry of hs lmns s quadrangular lmn wh lnar nrpolaon funcons N x, y. Th oal numbr of nods s N, whl h oal numbr of lmns s N Local rprsnaon In hs sudy, h approxma soluon s h mpraur funcon T(x,y,) wh h form: T ( x, y, ) 4 T ( ) N ( x, y) 1 1 f j As nrpolaon funcons: N ( x j, y j ) j ls For ach lmn doman, Afr smplfcaon of h calculaons, w g: dt C K T F d (4) w w w. a j r. o r g Pag 8

whr pr lmn : K C c F K, C and ( grad N ) ( grad N ) dxdy [ N ] d ( N ) ( N ) dxdy F ar rspcvly conducanc marx, h capacanc marx and h ha load vcor 2.3.

Th assmblag sysm quaon aks h marx form dt C K T F (6) d whr h conducanc marx, h capacanc marx and h ha load vcor ar rspcvly : n n n K A K A, C A C A and F A F 1 1 1 In solvng a problm of ransn

3 whr pr lmn : K C c F K, C and ( grad N ) ( grad N ) dxdy [ N ] d ( N ) ( N ) dxdy F ar rspcvly conducanc marx, h capacanc marx and h ha load vcor Global rprsnaon and assmbly W dploy w o rfr o h cross sconal ara and [J] o sand for h Jacoban of h gomrc ransformaon by adopng an soparmrc lmn. Ingrang h Equaons (5). Th assmblag sysm quaon aks h marx form dt C K T F (6) d whr h conducanc marx, h capacanc marx and h ha load vcor ar rspcvly : n n n K A K A, C A C A and F A F In solvng a problm of ransn conducon, w ar gudd no solvng a sysm of frs ordr dffrnal quaon wh rspc o m (Equaon (6)) for whch h nal condon s: T T T 1( ) T2 ()... TN () Th drmnaon of h mpraur fld n h maral rurns o fnd h mpraur valus a h nods ovr m. Th numrcal soluon of h prvous sysm drmns h voluon of h mpraur n h maral for hrmo-physcal paramrs mposd. 2.4 Exprmnal dvc Th goal s o cra a mpraur fld, w wll hav o ha h sampl sudd o dnfy varaons n mpraur. A har s confnd bwn wo plas, and h lcrcal powr s suppld by a gnraor. Undr hs condons, s assumd ha h mposd flow s dvdd qually bwn h wo plas. Th har s conrolld by a volmr and an ammr. I rqurs vry low currn nnss. To accss h mpraur flds of h maral, w us an nfrard hrmography. Fnally, a vdo monor conncd o can follow h voluon of h hrmal mappng (5) FIG. 2 Fng h xprmnal masurmn of h mpraur. Fgur 3 blow llusas h voluon of xprmnal or smulad mpraurs of h pla, wh rspc o m w w w. a j r. o r g Pag 9

4 Tmpraur ( C) Tmpraur ( C) Tmpraur ( C) Tmpraur ( C) Tmpraur ( C) Tmpraur ( C) Tmpraur ( C) Tmpraur ( C) Exprmnal Tmpraur a = Smulad mpraur a = Abscca (mm) Ordna (mm) 2 5 Abscca (mm) Ordna (mm) Exprmnal Tmpraur a = 4 s Smulad Tmpraur a = 4 s Abscca (mm) Ordna (mm) Abscca (mm) Ordna (mm) 2 Exprmnal Tmpraur a = 8 s Smulad mpraur a = 8 s Abscca (mm) Ordna (mm) 2 5 Abscca (mm) Ordna (mm) Exprmnal mpraur a = 1 s Smulad mpraur a = 1 s Abscca (mm) Ordna (mm) Abscca (mm) Ordna (mm) FIG. 3 Tmpraur dsrbuon of h pla wh rspc o m w w w. a j r. o r g Pag 1

5 No ha h progrsson of boh mpraurs,.., numrcally calculad and xprmnal, ar vry clos. Ths comparson also valdas h drc approach dvlopd. Nx, n h nvrs problm, and o undrak h xprmnal condons, a nos wh sandard dvaon.2 C s mposd on h mpraur accordng o h prcson currn nfrard camras III. INVERSE PROBLEM Th xprmns wr conducd a h Insu P' of h Unvrsy of Pors From h masurd mpraur flds, w ry o dnfy h hrmal conducvy nsor 12 and h spcfc ha c of an ansoropc maral. Onc h mpraur funcon T(x, y, ) s rcordd n a srs of pons on h surfac of h pla, a svral ms, w apply h las squars mhod for smang h hrmophyscal paramrs. W mad m xprmns ndxd from 1o m.th m duraon of h h xprmn s rfrrd o by. Th las squars mhod brngs abou h consrand opmzaon procss: Mnmz h objcv funcon dt 2 m 1 J(ρc, ) 2 C B A B A T F d 1 d (7) undr h posv consran for c, and h symmrc posv-dfn consrans for. To mnmz hs funcon by a sps dscn mhod, w nd o xprss s gradn wh rspc o h conducvy nsor : To mnmz hs funcon, w calcula for h conducvy nsor J c J c f,,, B A U T B A d wh dt U C K T F d. Snc s symmrc, h gradn R of h of h cos funcon J, c vrsus s qual o h marx zros mpls: f R B A U T T U B A d J W wll also nd s drvav r. ( c) 3.1. Idnfcaon algorhm Th conjuga gradn algorhm s nroducd n dal and appld n hrmophyscal paramrs dnfcaon. As h sps dscn mhod o mnmz h funcon J( c, ), w mplmn h projcd conjuga gradn mhod, whch consss n consrucng ravly a squnc convrgng o h mnmum. Th algorhm of hs mhod can b summarzd as follows 1. Inalz by and c by ( c), Dduc h nal valus r and R of r and R, Inalz a squnc of scalars d by d r and a squnc of drcons D by D R, 2. A raon calcula and whch mnmz J( c d, D ) wh rspc o and ( c) 1 ( c) d 1 D f 1 proj, f r 1 and R 1 sop, ohrws w w w. a j r. o r g Pag 11

6 d r 1 1 ( R R ) R 1 RR D 1 R 1 D 1 and rurn o sp 2. Th abbrvaon (Proj) s a subroun ha xcus h projcon algorhm of Hgham [8]. Indd, h projcon of a no posv symmrc marx aks plac orhogonally o h dg of h posv con marx Idnfcaon Rsuls Smulaons for ansoropc marals Th Projcd Conjuga Gradn mhod dvlopd n h las scon s appld o h smulad mpraur flds oband by solvng quaon (6) by FEM. Th maral s supposd o b ansoropc. Th rsuls from our dnfcaon algorhm ar shown n h abl blow. Paramrs Valus usd n h smulaon dnfd Valus λ 11(W/m/ C) ±.17 λ 12(W/m/ C).2.34 ±.1 λ (W/m/ C) ±.14 ρc(j/m3/ C) ± Tabl 1: Idnfd Valus for an ansoropc maral from smulad mpraur flds Exprmns for soropc marals Th xprmnal dvc was appld o an soropc polymr (polymhylmhacryla, PMMA), wh hrmophyscal paramrs [1]: Kg m c 14 J Kg C,17W m C Th rsuls from our dnfcaon algorhm ar shown n h abl blow. Paramrs λ 11(W/m/ C) ρc(j/m 3 / C) Manufacurr valus [1] Valus dnfd Tabl 2: Idnfd Valus for PMMA from xprmnal mpraur flds IV. CONCLUSION Th fn lmn mhod ms h rqurmns mposd by h sampl gomry and h boundary condons. Is applcaon on a homognous ansoropc maral nabld us o ransform h Fourr s ha conducon quaon n a frs ordr ordnary dffrnal quaon. Thrfor, h rsoluon of h drc problm nds solly a m ngraon algorhm. Th dvlopd algorhm allows us o smula h mpraur fld n h bdmnsonal cas. Th accuracy of h smulaons nsurd h valdy of our approach. Morovr, our cod provd o b fas handlng boh for vard gomrc dmnsons and for vard boundary and nal condons. Th dnfcaon algorhm s basd on h Projcd Conjuga Gradn mhod. I allows characrzng h hrmal conducvy nsor and h spcfc ha of polymrs. Th dnfcaon rsuls ar dmonsrad o b n good agrmn wh h manufacurr valus. REFERENCES [1]. K. Achonouglo, Idnfcaon ds Paramèrs Caracérsqus d un Phénomèn Mécanqu ou Thrmqu Rég par un équaon dfférnll ou aux dérvés Parlls, Thès d Docora, Unvrsé d Pors, 27. [2]. B. Aouragh, J. Chaouf, H. Famaou, J.C. Dupré, C. Vallé and K. Achonouglo, Smulaon of hrmo mchancal bhavor of srucurs by h numrcal rsoluon of drc problm, Appld Mchancs and Marals, vol. 61, pp , 211. [3]. J.F. Sacadura : Inaon aux ransfrs hrmqus, Edur: Tchnqu documnaon, [4]. J.M. Brghau, R. Forunr, Smulaon numérqu ds ransfrs hrmqus par élémns fns, Edons Hrmès, Pars, 24. [5]. N. E. Aksan : A numrcal soluon of Burgrs quaon by fn lmn mhod consrucd on h mhod of dscrzaon. Appl. Mah. Compu, Vol. 17, pp , 25. w w w. a j r. o r g Pag 12

7 [6]. K. Achonouglo, M. Banna, C. Vallé and J.C Dupré, Invrs Transn Ha Conducon Problms and Applcaon o h Esmaon of Ha Transfr Coffcns, Ha and Mass Transfr, Vol. 45, Numbr 1, pp. -29, 28. [7]. E. Polak, Compuaonal Mhods n Opmzaon: A Unfd Approach, R. Bllman, Ed., Mahmacs n Scnc and Engnrng, Vol. 77, Nw York, NY: Acadmc Prss, [8]. N.Hgham: Compung a nars symmrc posv sm dfn marx. Lnar Algbra and s Applcaon, Vol. 13, pp , [9]. R W Lws, K Morgan, H R Thomas, K N Sharamu, Th fn lmn mhods n ha ransfr analyss, Nw York: John Wly (1996) [1]. A. Ghafr, J. Chaouf,, C. Vallé,H. Arjdal, J.C. Dupré, A. Grmanau, K. Achonouglo and H. Famaou: Idnfcaon of Thrmal Paramrs by Trang h Invrs Problm Inrnaonal Journal of Compur Applcaons ( ) Volum 87 No.11, Fbruary 214 E. Arjdal" Idnfcaon of h hrmo-physcal paramrs of an ansoropc maral by nvrs problm" Amrcan Journal of Engnrng Rsarch (AJER), vol.8, no.3, 219, pp.7-13 w w w. a j r. o r g Pag 13

Consider a system of 2 simultaneous first order linear equations

Consider a system of 2 simultaneous first order linear equations Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm