ESTIMATION OF HEAT SOURCES WITHIN TWO DIMENSIONAL SHAPED BODIES

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1 Proceedigs of 3ICIPE 99 3rd It. Coferece o Iverse Problems i Egieerig Jue 13-18, 1999, Port Ludlow, Washito, USA HT11 ESTIMATION OF HEAT SOURCES WITHIN TWO DIMENSIONAL SHAPED BODIES Abou Khachfe R., Jary Y. Laboratoire de Thermociétique de l ISITEM, UMR CNRS 667 Rue Christia Pauc BP 964, 4436 Nates Cedex 3 jary@isitem.uiv-ates.fr ABSTRACT A iterative algorithm based o the cojugate gradiet method is developed i order to estimate simultaeously the spatial locatio ad the stregth of heat sources withi two dimesioal shaped bodies. The case of puctual sources, together with temperature sesors located o the boudary of the body, is studied. The iterative regularizatio pricilple is used to avoid the amplificatio of the measuremet errors o the computed solutio. NOMENCLATURE A streght of the source c heat capacity J(S) fuctioal equatio (8) J S gradiet of J(S) h heat trasfer coefficiet i ormal uit vector o Γ i T temperature (x y ) locatio of the source Γ i boudary surface i λ thermal coductivity δt sesitivity fuctio ψ adjoit fuctio ρ desity INTRODUCTION The umerical resolutio of the Direct Heat Coductio Problem (DHCP) requires the kowledge of a set of data which ivolves : the geometrical data (shape of the body), the medium data (heat capacity, desity, thermal coductivity, heat trasfer coefficiet), the source data (heat flux desity, fixed temperature o the boudary, heat sources withi the body), the iitial coditio (spatial distributio of the temperature field at the iitial time). Whe some part of this set of data is ukow, the DHCP solutio caot be computed, the umerical resolutio of the Iverse Heat Coductio Problems (IHCP) aims to determie the ukow part of these data from additioal iformatio. These ew data are usually give by temperature sesors which are located o the boudary or iside the body. Durig the last decade, due to the tremedous advacemet i scietific computatio, a icreasig attetio has bee devoted to the techiques for solvig multidimesioal IHCP. Numerical resolutios of IHCP are well kow to be ill - coditioed, therefore some regularizig process has to be cosidered to avoid umerical istabilities o the computed solutios. Several approaches are available (Alifaov, 1994), (Jary et al, 1991), (Beck, 1977). A geeral approach cosists i solvig IHCP with umerical optimizatio tools like the cojugate gradiet method. Combied with the use of a Fiite Elemet Library, It is a powerful method which offers a wide field of practical applicatios i iverse thermal aalysis. I this work, some results are preseted o the estimatio of heat sources withi two dimesioal shaped bodies. This problem has bee cosidered by several authors (Silva Neto et al, 1992), (Le Niliot, 1998),(Park H.M. et al, 1999) especially i the case of puctual sources ad assumig the spatial locatio is kow. I this paper it is show how the simultaeous determiatio of the spatial locatio ad the stregth of poctual heat sources ca be achieved from temperature histories delivered by sesors located o the boudary of the body. Numerical experimets show that the ifluece of the error measuremets 1 Copyright 1999 by ASME

2 o the computed solutio ca be strogly reduced accordig to the iterative regularizatio priciple. DIRECT HEAT CONDUCTION PROBLEM (DHCP) The heat coductio process ivolvig heat source is cosidered withi a arbitrary shaped domai with the boudary Γ 1 [Γ 2. The set of equatios of the DHCP are writte i the form of a parabolic problem where P(t) is the itesity (W =m) ad ω is a parameter used to simulate a puctual heat source, it is chose equal to the mesh size. We cosider ad (x y ) is the source locatio. A(t) = P(t) πω 2 (7) ρc T ; λ T = S(x y t) i t > (1) λ T 1 + ht = ht e o Γ 1 t > (2) λ T 2 = oγ 2 t > (3) T = T i t = (4) Solvig the DHCP cosists i the determiatio of the temperature field T (x y t) o the domai, at each time t, whe the medium data ρc, λ, the heat trasfer coefficiet h, the exteral temperature T e, the heat source S(x y t) ad the iitial field T, are assumed to be kow. I this study, the heat source is cosidered with the followig form S(x y t)=a(t):f(x y;x y ) Fd = 1 where A ad F are respectively the stregth ad the spatial distributio of the source. The goal is to determie the stregth A(t), ad the locatio (x y ) for a prescribed spatial distributio F. (5) FORMULATION OF THE IHCP Additioal iformatio are provided by sesors located withi the spatial domai or o its boudary, o the time iterval [ t f ]. Let Y m (t) (m = 1 ::) be the temperature histories delivered at the locatio (x m y m ). Solvig the IHCP cosists i the determiatio of the source S which matches the solutio of the DHCP T (x m y m t;s) with the data Y m (t). Because of illposedess, the IHCP is solved i the least square sese, by miimizig the residual fuctioal J(S) = 1 2 t f jt (x m y m t;s) ;Y m (t)j 2 dt (8) m=1 with T (x y t;s) solutio of equatios (1)-(4) MINIMIATION OF THE FUNCTIONAL J(S) Miimizatio of J(S) is achieved by usig the cojugate gradiet method (CGM). Let S be the ukow vector to be determied. S T =[x y A k (k= ::Nt) ] (9) where Nt is the umber of time step. The (CGM) is iterative, at each iteratio, the previous estimate S is corrected by S +1 = S ; γ d (1) where the search directio d ad the step legth γ are determied i order to have J(S +1 ) < J(S ) (11) PONCTUAL HEAT SOURCE The spatial distributio of the heat source to be determied is described by a gaussia fuctio S(x y t) = P(t) πω 2 exp (x ; ; x ) 2 ; (y ; y ) 2 ω 2 (6) The vector [d x d y d Ak (k= ::Nt)], is chose accordig to the CGM rules d = JS + β ;1 J S β = β = < J S J S ; J S;1 > kjs k 2 (12) 2 Copyright 1999 by ASME

3 where J S is the gradiet of the J(S) at the iteratio J S =[J x J y J A k (k= ::Nt) ] (13) k:k is the vector orm associated to the scalar product <: :> The step legth γ is give by γ = m=1 t f δt (x m y m t;s )[T (x m y m t;s ) ;Y m (t)]dt [δt (x m y m t;s )] 2 (14) where δt is the sesitivity fuctio which results of the variatio δs of the source. SENSITIVITY EQUATIONS Solvig the sesitivity equatios cosists i the determiatio of the temperature variatio δt which results of a variatio δs of the source. The liearity of the equatios (1)-(4) leads to I order to get J S (x y t;s), equatio (19) has to be put i the form of equatio (2). This is doe by makig statioary the Lagragia associated to the optimizatio problem, equatio (8) LAGRANGIAN AND ADJOINT EQUATIONS Let us itroduce the Lagragia L (T S ψ) t f 1 2 m=1 t f + L((T S ψ)) = (T (x y t;s) ;Y m (t)) 2 δ(x ; x m )δ(y ; y m )dtd (ρc T ; λ T ; S)ψdtd (21) where T S ψ ca be idepedet fuctios. The Lagragia multiplier ψ is, as the fuctio T, a fuctio of x y, ad t. ψ beig fixed, the differetial of L satisfies δl = T δt + δs (22) S δs = FδA + A ρc δt ; λ δt = δs! F δx F + δy i t > x y (15) Let us take ψ solutio of the followig equatio (so called adjoit equatio) δt = 8 δt (23) T the by takig T solutio of the equatios (1)-(4), it comes λ δt 1 + hδt = oγ 1 t > (16) δj = δl = δs (24) S λ δt 2 = oγ 2 t > (17) δt = i t = (18) ad the associated variatio of the fuctioal J(S) is t f m δj(s) = (T (x y t;s) ;Y m (t))δ(x ; x m )δ(y ; y m )δtdtd (19) where δ(:) is the Dirac impulse. By defiitio, J S (x y t;s) satisfies t f δj(s) = J S (x y t;s)δsdtd (2) where S δs = ; t f t f = ; ψδsdtd ψfδadtd ; t f ; t f ψa F y δy dtd ψa F x δx dtd (25) ad the gradiet compoets J S (x y t;s) are easily extracted from equatio (25), J A (t) =; ψf d (26) t f J x = ; ψa F dtd (27) x 3 Copyright 1999 by ASME

4 t f J y = ; ψa F dtd (28) y λ ψ 1 + hψ = oγ 1 t > (34) It ca be oted that J A is a fuctio of time, like the fuctio A(t), J x ad J y are real umbers like x ad y. Let us develop equatio (23). From the defiitio of the Lagragia, (21), we have t f m=1 T δt = (T (x y t;s) ;Y m (t))δ(x ; x m )δ(y ; y m )δtdtd t f + itegratio by parts give t f λ δt ψd = (ρc δt ; λ δt )ψdtd δt ψdt =[ψδt ] t=t t f f t= ; (29) ψ δtdt (3) (λ Γ1 ψ + hψ)δtdγ 1 + λ 1 Γ2 ψ δtdγ λ ψ δtd (31) the with the boudary ad iitial coditios (16), (17), (18), equatio (29) ca be writte i the ew form t f + T δt = m=1 t f (;ρc ψ ; λ ψ)δt dtd (T (x y t;s) ;Y m (t))δ(x ; x m )δ(y ; y m )δt dtd + Γ2 t f Γ1 t f (λ ψ 1 + hψ)δt dtdγ 1 + λ ψ δtdtdγ 2 + ψ(t = t f )δt d 2 (32) Therefore equatio (23) is satisfied by takig the lagrage multiplier ψ solutio of the followig set of equatios m=1 ;ρc ψ ; λ ψ = (T (x y t;s) ;Y m (t))δ(x ; x m )δ(y ; y m ) i t > (33) λ ψ 2 = oγ 2 t > (35) ψ = i t = t f (36) Equatios (33)-(36) are liear, the solutio of which ψ is computed backward i time o the iterval [ t f ] NUMERICAL ALGORITHM Each iteratio of the CGM icludes the umerical resolutio of three distict parabolic set of equatios. The iterative algorithm has the followig structure a) Choose a iitial guess S b) Solve the direct problem, equatios (1)-(4), to compute T (x m y m t;s ), ad J(S ), equatio (8) c) Solve the adjoit problem,equatios (33-36) to compute the vector J S, equatios (26-28) d) Solve the sesitivity problem, equatios (15-18), to compute γ, equatio (14), ad the ew vector S +1 equatio (1) e) Stoppig coditio if J(S +1 ) > J(S ) or J(S ) < ε else set = + 1gotob) f) Ed go to f) ε is defied accordig to the iterative regularizatio priciple which is illustrated i the followig sectio. NUMERICAL EXPERIMENTS The spatial domai cosidered is a square, see figure (1), L = :5m. The values of the thermophysical parameters are : λ = :17W =K:m, a = 1:21 ;7 m 2 =s, h = 1W=m 2 :K, the iitial temperature T = 293K, ad the exteral temperature T e = 293K. A regular mesh with = 121 odes is cosidered. ω = 6:1 ;3 m. The parabolic solver of the Fiite Elemet Library MODULEF is used to compute the solutio of the DHCP. Two cases have bee studied : case#1: x = :1m y = :4m case#2: x = :3m y = :1m The temperature resposes delivered by four sesors located at the middle of each side of the domai,to a time varyig stregth A(t) of the source, are show o figures (2) ad (3). 4 Copyright 1999 by ASME

5 y Γ 1 Source S Sesor 1 Γ Case#1 Iitial 1 locatio Γ 2 S 2 Γ 1 S 3 h Case#2 h S4 x h Temperature ( o C) 3 25 S 1 S 2 S 3 S 4 Figure 1. DOMAIN S 1 S 2 S 3 S 4 Figure 3. TEMPERATURE RESPONSES : case#2 Temperature ( o C) x or y (m) σ =.5 y x Figure 2. TEMPERATURE RESPONSES : case#1 These resposes are corrupted by addig a zero mea gaussia oise, to build the simulated additioal data Y m (t) required to solve the IHCP Iteratio umber Figure 4. IDENTIFICATION OF THE LOCATION : case#1, σ = :5 Y k m = T k m exact + σek m m = 1 ::: k = :::Nt (37) where σ is the stadard deviatio of the oise (assumed to be idetical for each sesor) ad e k m is the ormally distributed radum umber. NUMERICAL RESULTS Case#1 The iitial guess of the source locatio is take at the ceter of the square x = :25m y= :25m. The iitial guess of the stregth of the source is take equal to zero. A alterate iterative research process of the cojugate gradiet has bee performed : istead of modifyig the locatio ad the stregth at 5 Copyright 1999 by ASME

6 A(t) W/m Exact solutio Estimated solutio at = 242 σ =.5 A(t) W/m Exact solutio Estimated solutio at = 155 Estimated solutio at = 186 σ = Figure 5. IDENTIFICATION OF THE STRENGTH : case#1, σ = :5 Figure 7. IDENTIFICATION OF THE STRENGTH : case#1, σ = :2.5.4 σ = σ=.2 σ=.5 x or y (m).3.2 y x J(S ) Iteratio umber Iteratio umber Figure 6. IDENTIFICATION OF THE LOCATION : case#1, σ = :2 Figure 8. MINIMIATION OF J(S ): case#1 each iteratio, oe iteratio is used to correct the locatio keepig the stregth umodified, the followig iteratio is used to correct the stregth while the locatio is hold, ad so o. O figure (4) ad (6) are show the computed locatios resultig of this miimizatio process, with a oise level σ = :5 ad σ = :2 respectively. Figure (8) illustrates the covergece of the sequece J(S ) : asymptotic values J as are observed, they are directly re- lated to the oise level σ as idicate i Table 1. The fial umber of iteratios is chose accordig to the followig equatio J as = J(S )=Nt::σ 2 = ε (38) Whe the iterative process is stopped at the iteratio umber, 6 Copyright 1999 by ASME

7 Table 1. RESULTS OF IDENTIFICATION OF (x, y ), case#1 σ ε J as x y Table 2. RESULTS OF IDENTIFICATION OF (x, y ), case#2 σ ε J as x y A(t) W/m Exact solutio Estimated solutio at = 63 σ = σ =.5 Figure 1. IDENTIFICATION OF THE STRENGTH : case#2, σ = :5 x or y (m) x y Iteratio umber Figure 9. IDENTIFICATION OF THE LOCATION : case#2, σ = :5 x or y (m) x σ =.2 y o istability occur o the computed solutio as it is show o figure (5). Figure (7) illustrates the iterative regularizatio priciple : - with σ = :2 ad = 155, the computed stregth A(t) is stable, close to the exact values; if the iterative process is cotiued, for example = 186, oscillatios appear. - with a lower oise level σ = :5, the fial umber = 242 is greater ad the computed stregth is more accurate figure (5) Case#2 The same iitial guess is cosidered. Figures (9) ad (11) illustrates the covergece of the computed source locatios to the exact values. Figures (1) ad (12) show the ifluece of the oise level o the computed stregth A(t). Results are similar to Iteratio umber Figure 11. IDENTIFICATION OF THE LOCATION : case#2, σ = :2 case#1. It ca be observed that the fial umbers of iteratios depeds o the level oise, like i the previous case. The ifluece of the sesor locatios must be also oted. The distace betwee the earest sesor ad the source is lower tha i case#1, the for σ = :5, istead of = 242 oly = 63 iteratios are required, ad for σ=.2, istead of = 155, = 59 are sufficiet table 2. 7 Copyright 1999 by ASME

8 described by a stiff gaussia fuctio i order to approximate a puctual distributio. The iterative regularizatio priciple together with the use of the cojugate gradiet method has bee illustrated. To avoid istabilities o the computed stregth, the iterative process of miimizatio has to be stopped whe the residual fuctioal reaches a critical value which depeds o the oise level. A(t) W/m Exact solutio Estimated solutio at = 59 Estimated solutio at = 78 σ = Figure 12. IDENTIFICATION OF THE STRENGTH : case#2, σ = : σ=.2 σ=.5 REFERENCES Alifaov O.M.,Iverse Heat trasfert problems Spriger- Verlag,1994 Jary Y., Ozisik M.N., Bardo J.P., A geeral optimizatio method usig adjoit equatio for solvig multidimesioal iverse heat coductio, It. J. Heat Mass Tras.,34 (1991) Beck J. V., Arold K.J., Parameter Estimatio i Egieerig Sciece, Joh Whiley ad Sos, New York, 1977 Silva Neto A.J., Ozisik M.N., Two-dimesioal iverse heat coductio problem of estimatig the time-varyig stregth of a lie heat source, Joural of Applied physics 71 (11) (1992) Le Niliot C., Gallet P., Ifrared thermography applied to the resolutio of iverse heat coductio problems: recovery of heat lie sources ad boudary coditios, Revue Geerale de la Thermique,37 (1998), Park H.M., Chug O.Y., Lee J.H., O the solutio of iverse heat trasfer problem usig the Karhue-Loève Galerki method, It. J. Heat Mass Tras.,42 (1999) J(S ) Iteratio umber Figure 13. MINIMIATION OF J(S ): case#2 CONCLUSION The iverse heat coductio problem which cosists i the determiatio of the locatio ad simultaeously the time varyig stregth of a heat poctual source withi a two dimesioal arbitrary shaped body, has bee computed by the cojugate gradiet method, usig a parabolic solver of the Fiite Elemet Library MODULEF. The spatial distributio of the source has bee 8 Copyright 1999 by ASME