Inverse Heat Conduction in a Finite Slab with Measured. Back Surface Temperature and Heat Flux

Transcription

1 Inverse Hea Conducion in a Finie Slab wih Measured Bac Surace Temperaure and Hea Flux Jianhua Zhou, Yuwen Zhang, J. K. Chen 3, and Z. C. Feng 4 Universiy o Missouri, Columbia, MO 65, USA In High-Energy Laser (HEL) heaing o arge, he emperaure and hea lux a he heaed surace is no direcly measurable, bu can be esimaed by solving an inverse hea conducion problem (IHCP) based on measured emperaure or/and hea lux a he accessible (bac) surace. In his sudy, he one-dimensional (-D) IHCP in a inie slab is solved by he conjugae gradien mehod (CGM) using measured emperaure and hea lux a he accessible (bac) surace. Simulaed measuremen daa are generaed by solving a direc problem where he ron surace o he slab is subjeced o high inensiy periodic heaing. Two cases are simulaed and compared, wih he emperaure or hea lux a he heaed ron surace chosen as he unnown uncion o be recovered. The resuls showed ha he laer choice, i.e., choosing bac surace hea lux as he unnown uncion, can give beer esimaion accuracy in he IHCP soluion. The ron surace emperaure can be compued wih a high precision as a byproduc o he IHCP algorihm. The robusness o his IHCP ormulaion is esed by dieren measuremen errors and requencies o he inpu periodic heaing lux. Nomenclaure c p = speciic hea d () = direcion o descen a ieraion level h = convecion hea ranser coeicien = hermal conduciviy L = hicness o -D slab '' q = inensiy o heaing source a ron surace q () = hea lux a ron surace q[;t ()] = compued hea lux a he bac surace S = objecive uncions S[ T ] = gradien direcion o objecive uncional a ieraion level Δ S[ T ] = objecive uncion variaion = ime = inal ime T = emperaure T = iniial emperaure T = ambien emperaure T () = ron surace emperaure T[;T ()] = compued hea lux a he bac surace Δ T[ ; T ] = emperaure variaion x = spaial coordinae variable Posdocoral Research Associae, Deparmen o Mechanical and Aerospace Engineering, AIAA Member. Associae Proessor, Deparmen o Mechanical and Aerospace Engineering, AIAA Associae Fellow. 3 William and Nancy Thompson Proessor, Deparmen o Mechanical and Aerospace Engineering 4 Proessor and Direcor o Graduae Sudies, Deparmen o Mechanical and Aerospace Engineering American Insiue o Aeronauics and Asronauics

2 Y() = measuremen daa (emperaure or hea lux) wih errors a bac surace obained by numerical simulaions Y exac () = measuremen daa (emperaure or hea lux) wihou errors a bac surace obained by numerical simulaions Y ql () = measuremen hea lux a he bac surace Y TL () = measuremen emperaure a he bac surace Gree symbols β = search sep size a ieraion level χ = olerance used o sop he CGM ieraion procedure δ = sandard deviaion o he measuremens ε = surace emissiviy γ = conjugae coeicien a ieraion level λ ( ) = Lagrange muliplier ρ = densiy σ = Sean-Bolzmann consan, σ = W/(m K 4 ) ω = random variable beween and Superscrips = ieraion level Subscrips = iniial = inal q = hea lux T = emperaure. Inroducion igh-energy Laser (HEL) weapons oer he advanages o remoe delivery a he speed o ligh ono a small spo on miliary arge. During laser irradiaion, i is criical o now he emperaure a he heaed ron arge H surace in order o accuraely undersand damage mechanism. However, he heaed ron-surace is eiher inaccessible or oo ho so ha i is unsuiable or aaching a sensor. Similar problem can also be encounered in some laser manuacuring processes []. Under hese circumsances, he ron surace emperaure can be deermined indirecly by solving an inverse hea conducion problem [-4] based on he ransien emperaure or/and hea lux measured a he bac surace. In he mahemaical ormulaion o he inverse problem, eiher emperaure or hea lux can be measured a he bac surace. Mos previous researchers preer emperaure measuremens because emperaure can be measured wih ewer uncerainies compared o hea lux measuremens [5-8]. However, recen sudies have showed ha using measured hea lux as addiional inormaion in he ormulaion o an IHCP can increase he sabiliy o he soluion and is less prone o he inheren insabiliy o he ill-posed problem o inverse hea conducion [9, ]. Alhough he inverse hea conducion problems have been exensively sudied or dieren applicaions in he pas (e.g., [-9]), lile wor has been done or he inverse problem relaed o laser irradiaion o a remoe surace. The laser energy is delivered o he arge surace in a periodic way because o laser or amosphere variaions. The periodic hea lux may pose exra diiculies in he soluion o he inverse problems. Since he ormulaion o he IHCP is quie subjecive, i is necessary o deermine which ormulaion is mos appropriae or hese applicaions. In his sudy, a -D inverse hea conducion problem in a inie slab is ormulaed and solved using he conjugae gradien mehod (CGM) wih adjoin problem or uncion esimaion [3, 4]. The inverse soluions or he cases where he ron surace o he slab is subjeced o high inensiy periodic heaing is illusraed o ideniy he mos robus IHCP ormulaion or he laser manuacuring applicaions. Boh he emperaure and hea lux are measured a he bac surace. The ormer is used as he bac surace boundary condiion while he laer is adoped in he objecive uncion o be minimized. Two cases are examined in deail where he emperaure or hea lux a he heaed ron surace is chosen as he unnown uncion o be recovered. The mos robus and error-insensiive IHCP ormulaion will be proposed. American Insiue o Aeronauics and Asronauics

3 . Model Descripion For he case ha he laser beam size is much larger han he hicness o he wall, one can rea he inverse hea ranser problem as one-dimensional. To illusrae he mehodology o he inverse hea ranser algorihms used in his sudy, a inie slab wih a hicness o as shown in Figure is considered. Iniially, he slab is uniormly a he emperaure o T. From >, he ron surace o he slab is subjeced o a high inensiy laser heaing. The purpose o he inverse problem is o reconsruc he heaing condiion a he ron surace based on he measured emperaure and hea lux a he bac surace. Due o he ac ha emperaure measuremen daa conain much less errors compared o hea lux measuremen [5-8], he emperaure Y TL () is used as he boundary condiion and he hea lux Y ql () is adoped in he objecive uncion. I is assumed ha he densiy, speciic hea and he hermal conduciviy o he slab are consans. Two cases, i.e., Case I and Case II as shown in Fig. (a) and Fig. (b), respecively, are examined o es which quaniy (emperaure or hea lux) is more appropriae o be employed as he unnown o be recovered such ha he bes accuracy can be achieved. In he ollowing, he mahemaical ormulaions or dieren cases will be described. s B.C. T ()=? O x (a) s B.C. Y TL () Y ql () is used in objecive uncion nd B.C. q ()=? x= x=l O x (b) s B.C. Y TL () Y ql () is used in objecive ouncion x= x=l Figure Physical models. (a) Case I: recovering he ron-surace emperaure T ( ). (b) Case II: recovering he ron-surace hea lux q ( ).. The direc problem.. Case I For his case, he ron surace emperaure is chosen as he unnown o be recovered (Figure (a)). The direc problem or his case can be expressed as ollows: T T ρ c p = ( ) in < x < or > () T ( ) = T in < x < or = () T (,) ( ) a x =, or > (3) T( ) = YTL a x = or > (4) where T ( ) is he ron surace emperaure; TL ) is he measuring emperaure a he bac surace. In he direc problem associaed wih he physical problem described above, he ron-surace emperaure, T ( ), and he bac surace emperaure, Y ( TL ), are considered o be nown. The objecive o he direc problem is hen o deermine he ransien emperaure disribuion in he slab... Case II 3 American Insiue o Aeronauics and Asronauics

4 For his case, he ron surace hea lux is chosen as he unnown o be recovered (Figure (b)). The mahemaical ormulaions is almos he same as hose in Eqs. () ~ (4) excep ha he boundary condiions a x = is replaced by: T (, ) = q ( ) a x =, or > (5) where q ( ) is he ron surace hea lux.. The inverse problem For he inverse problem, he boundary condiion a x = is unnown, bu everyhing else in he corresponding direc problem is nown. The addiional inormaion needed in recovering he ron surace boundary condiion is available rom he readings o a hea lux sensor insalled a he bac surace... Case I The inverse problem or his case can be expressed as ollows: T T ρ c p = ( ) in < x < or > (6) T ( ) = T in < x < or = (7) T( ) = YTL a x = or > (8) where he ron surace emperaure, T ( ), is regarded as unnown. The hea lux readings a he bac surace, Y ql (), are considered available as addiional inormaion. The soluion o he inverse problem will be obained in such a way ha he ollowing objecive uncion is minimized (or coninuous measured daa): = S[ T ] { YqL q[ ; T ]} d (9) where Y ql () and q [ L, ; T ( )] are he measured and compued hea luxes a he bac surace, respecively... Case II The governing equaions o he inverse problem or his case are he same as hose in Case I (i.e., Eqs. (6) ~ (8)). The only dierence is ha he hea lux a he ron surace, q ( ), is regarded as unnown and he ollowing objecive uncion is minimized: = S[ q ] { YqL q[ ; q( )]} d ().3 Conjugae gradien mehod or minimizaion The ieraive process based on he CGM [3, 4] is now derived or he esimaion o unnown emperaure ( T ( ) ) or hea lux ( q ( ) ) by minimizing he objecive uncion S. The ieraive process or Case I is as ollows: T + = T β d () where β is he search sep size in going rom ieraion o ieraion +, and d () is he direcion o descen (i.e. search direcion) given by: d = S[ T ] + γ d () which is a conjugaion o he gradien direcion S[ T ] a ieraion and he direcion o descen d ( ) a ieraion -. The conjugae coeicien γ is deermined rom: S[ T ] { S[ T ] S[ T ]} d γ = wih γ = (3) { S[ T ]} d The ieraive process or Case II can be obained by replacing T ( ), T ( ) and T + ( ) wih q ( ), q ( ) and q + ( ), respecively in Eqs. () ~ (3). 4 American Insiue o Aeronauics and Asronauics

5 To perorm he ieraions according o Eq. (), we need o compue he sep size β and he gradien o he uncional S[ T ]. In order o develop expressions or he deerminaion o hese wo quaniies, a sensiiviy problem and an adjoin problem or each case are consruced below..4 Sensiiviy problem and search sep size.4. Case I The sensiiviy problem is obained rom he original direc problem (Eqs. () ~ (4) or Case I) in he ollowing manner: I is assumed ha when T ( ) undergoes a variaion Δ T, T is perurbed by Δ T. Then replacing in he direc problem T by T + ΔT and T by T + ΔT, subracing rom he resuling expressions he direc problem and neglecing he second-order erms, he sensiiviy problem or he sensiiviy uncion Δ T can be obained. The sensiiviy problem or Case I can be expressed as: ΔT ΔT ρ c p = ( ) in < x < or > (4) ΔT ( ) = in < x < or = (5) Δ T(, ) = ΔT a x =, or > (6) ΔT ( ) = a x = or > (7) The objecive uncion or ieraion + is obained by re-wriing Eq. (9) as: + S[ T ] = { YqL q[ ;( T β d )]} d (8) + where T is replaced by he expression given by Eq. (). I q[ ;( T β d )] is linearized by a Taylor expansion, Eq. (8) aes he ollowing orm: + S[ T ] = { YqL q[ ; T ] β Δq( d )} d (9) where q [ ; T ] is he soluion o he direc problem (Eqs. () ~ (4)) by using esimaed T a x= and ime. The sensiiviy uncions Δ q( d ) are aen as he soluions o Eqs. (4) ~ (7) a he measured posiions x=l and ime by leing ΔT = d. The search sep size β is deermined by minimizing he uncion given by Eq. (9) wih respec o β. The ollowing expression resuls: { q[ ; T ql Δ ( )] Y } q[ ; d ] d β = () { Δq[ ; d ]} d.4. Case II For his case, he mahemaical expression o he sensiiviy problem is almos he same as ha in Eqs. (4) ~ (7) excep ha he boundary condiions a x= is changed o: ΔT (, ) = Δq a x =, or > () The search sep size β or his case can be obained by replacing T ( ) wih q in Eq. ()..5 Adjoin problem and gradien equaion.5. Case I To obain he adjoin problem, Eqs. () ~ (4) is muliplied by he Lagrange muliplier λ ( ) and he resuling expression is inegraed over he corresponden space and ime domains. Then he resul is added o he righ-hand side o Eq. (9) o yield he ollowing expression or he uncion S[ T ] = { YqL q[ ; T ]} d () L T T + λ( )[ ρc p ] dxd The variaion Δ S is obained by perurbing T by Δ T, T by Δ T and q by Δ q in Eq. (), subracing rom he resuling expression he original Eq. () and neglecing he second-order erms, one inds 5 American Insiue o Aeronauics and Asronauics

6 L ΔS[ T ] = { YqL q[ ; T ]} Δq( ) δ ( x L) dxd (3) L ΔT ΔT + λ( )[ ρc p ] dxd where δ ( ) is he Dirac dela uncion. In Eq. (3), he domain inegral erm is reormulaed based on he Green s second ideniy; he boundary condiions o he sensiiviy problem give by Eqs. (6) and (7) are uilized and hen Δ S is allowed o go o zero. The vanishing o he inegrands conaining Δ T leads o he ollowing adjoin problem or he deerminaion o λ ( ) : λ λ Δq[ ; T ] ρ c p + + { q[ ; T ] Y ( )} ( x L) = ql δ in < x < or > (4) ΔT[ ; T ] λ ( ) = in < x < or = (5) λ (, ) = a x =, or > (6) λ ( ) = a x = or > (7) The adjoin problem is dieren rom he sandard iniial value problems in ha he inal ime condiions a ime = is speciied insead o he cusomary iniial condiion. However, his problem can be ransormed o an iniial value problem by he ransormaion o he ime variables as τ =. Finally, he ollowing inegral erm is le: λ(, ) ΔS[ T ] = ΔT d (8) From deiniion [], he uncional incremen can be presened as: ΔS[ T ] = S[ T ] ΔT d (9) A comparison o Eqs. (8) and (9) leads o he ollowing expression or he gradien o uncional S [ T ] : λ(, ) S[ T ] = (3).5. Case II The adjoin problem or his case is almos he same as ha in Eqs. (4) ~ (7) excep or he boundary condiion a x = : (, ) λ = a x =, or > (3) The gradien o uncional S [ q ] is he same as Eq. (3)..6 Sopping crierion Following he recommendaions o he Re. [3, 4], he discrepancy principle is used as he sopping crierion (or Case I): S [ T ] < χ (3) where χ denoes he olerance. I is assumed ha he absolue value o he hea lux residuals may be approximaed by: Y ( ) [, ; ( ql )] δ (33) where δ is he sandard deviaion o he measuremens. Subsiuing Eq. (33) ino Eq. (9), he ollowing olerance χ is obained: χ = δ (34) The sopping crierion is hen given by Eq. (3) wih χ deermined rom Eq. (34). The sopping crierion or Cases II can be obained in a similar way. 6 American Insiue o Aeronauics and Asronauics

7 3. Compuaional Procedure Tae Case I as he example, he compuaional procedure or he soluion o his inverse problem using CGM may be summarized as ollows. Suppose an iniial guess T is available or he uncion T ( ). Se = and hen: Sep. Solve he direc problem given by Eqs. () ~ (4) and compue T ( ) based on T. Sep. Chec he sopping crierion, Eq. (3). Coninue i no saisied. Sep 3. Solve he adjoin problem given by Eqs. (4) ~ (7) or λ ( ). Sep 4. Compue he gradien o he uncional S[ T ] rom Eq. (3). Sep 5. Compue he conjugae coeicien γ and direcion o descen d () rom Eqs. (3) and (), respecively. Sep 6. Se Δ T = d and solve he sensiiviy problem given by Eqs. (4)~(7) or Δ T ( ). Sep 7. Compue he search sep size β rom Eq. (). Sep 8. Compue he new esimaion or T + rom Eq. () and reurn o Sep. One o he advanages o using he conjugae gradien mehod o solve he inverse problems is ha he iniial guess (i.e. T or case I) o he unnown quaniies can be chosen arbirarily. In all he simulaion cases considered in his sudy, he iniial guess is aen as zero or convenience. 4. Resuls and Discussion 4. Generaion o measuremen daa In he inverse problems considered in his sudy, he esimaion o he ron surace heaing condiion is based on he measured daa a bac surace. The measured emperaure and hea lux a bac surace are generaed based on he numerical simulaion o he ollowing direc problem: T T ρ c p = ( ) in < x < or > (35) T ( ) = T in < x < or = (36) T ( ) '' 4 4 = q h( T T ) εσ ( T T ) a x =, or > (37) T ( ) 4 4 = h( T T ) + εσ ( T T ) a x = or > (38) '' where q is he periodic hea lux imposed a ron surace; h is he convecion hea ranser coeicien a ron and bac suraces, T is he ambien emperaure, ε is he surace emissiviy, and σ is he Sean-Bolzmann consan. This direc problem is numerically solved by he compuer code ha has been veriied in our previous wor []. By solving Eqs. (35) ~ (38), he hea luxes and emperaures a boh ron and bac suraces can be obained. The ron surace emperaure and hea luxes will be used as exac soluions o examine he accuracy o he inverse hea conducion algorihm. The bac surace emperaure and hea luxes will be used as boundary condiion and employed in objecive uncion, respecively. Real measuremens generally conain errors. We assume normally disribued uncorrelaed errors wih zero mean and consan sandard deviaion. Thereore, measuremens conain random errors are simulaed by adding a random noise (error erm) in he orm: Y = Yexac + ωδ (39) where Y () is he simulaed experimenal daa ha are obained rom he soluion o he direc problem, Eqs. (35) ~ (38); δ is he sandard deviaion o he measuremens. In his sudy, δ is se as a percenage o he highes measuremen value a he bac surace; ω is a random variable beween and generaed by he Mersenne Twiser mehod []. 7 American Insiue o Aeronauics and Asronauics

8 4. Resuls o direc problem In he ollowing simulaions, he ollowing hermal properies will be used: ρ = 76 g/m 3, c p = 55 J/(g K), = 8 W/(m K). Oher parameers are: L =.5 mm, T = 3 K, T =3 K, h =5 W/(m K), and ε =. 9. '' The ron surace hea lux is assumed o be q = qc +.q c sin(π) (W/m ), where q c is a consan hea lu which represens he hea lux inensiy level a he ron surace, and is he requency o he sinusoidal componen. This ype o hea lux is inenionally used o represen a general periodic heaing condiion alhough radiional lasers usually have a Gaussian or la beam proile in ime. Unless speciied oherwise, he ollowing simulaion parameers will be used: hea lux inensiy level q c = W/cm, requency o he sinusoidal componen =. Hz, sandard deviaion o he hea lux measuremens δ = % [ Y ql ] max. I is assumed ha here are no errors in he emperaure measuremens ( Y TL () ). 4 9 Fron surace hea lux (W/cm ) 8 Fron surace emperaure (K) (a) (b) 3 9 Bac surace hea lux (W/cm ) Bac surace emperaure (K) (c) (d) Figure Calculaing resuls or he direc problem described by Eqs.(35)~(38). (a) Fron surace hea lux; (b) Fron surace emperaure; (c) Bac surace hea lux; (d) Bac surace emperaure. Figure shows he resuls or he direc problem described by Eqs. (35) ~ (38). I can be seen rom Fig. (b) ha here are some lucuaions in he ron surace emperaure because he ron surace is subjeced o a sinusoidal hea lux heaing. However, no lucuaions are observed in he bac surace hea lux (Fig. (c)) and emperaure (Fig. (d)) due o damp and delay eecs in hea diusion phenomena. In Fig., i seems ha he inal simulaion ime is 3. s. Bu he acual inal ime is aen as = 3.6 s. The reason is as ollows. For he conjugae gradien mehod, he gradien equaion is null or some cases a he inal ime. Under his siuaion, he iniial guess used or T ( ) or q ( ) is never changed by he ieraive procedure, generaing insabiliies on he soluion in he 8 American Insiue o Aeronauics and Asronauics

9 neighborhood o. One approach o overcome such diiculies is o consider a inal ime longer han ha o ineres [4]. Thereore, we use =3.6 s or he simulaions in Fig Comparison beween Cases I and II Measuremen hea lux a bac surace (W/cm ) Recovered emperaure a ron surace (K) Exac soluion obained rom Eqs. (35)~(38) Recovered value using CGM mehod (a) (b) Hea lux a ron surace as by-produc (W/cm ) Exac soluion obained rom Eqs. (35)~(38) By-produc value o CGM mehod (c) Figure 3 Calculaing resuls or Case I. (a) Bac-surace hea lux measuremens (wih % measuremen errors); (b) Esimaed ron-surace emperaure by CGM inverse mehod; (c) Fron-surace hea lux as a by-produc. Recovered hea lux a ron surace (W/cm ) Exac soluion obained rom Eqs.(35)~(38) Recovered value using CGM mehod Temperaure a ron surace as by-produc (K) Exac soluion obained rom Eqs.(35)~(38) Recovered value using CGM mehod (a) (b) Figure 4 Calculaing resuls or Case II. (b) Esimaed ron-surace hea lux by CGM inverse mehod; (c) Fron-surace emperaure as a by-produc. 9 American Insiue o Aeronauics and Asronauics

10 Figure 3 shows he resuls or Case I. Fig. 3 (a) gives he measuremen hea lux a he bac surace obained by adding % random error via Eq. (39). As is seen in Fig. 3(b), he recovered emperaure a ron surace agrees very well wih he exac soluion obained rom he soluion o Eqs. (35) ~ (38). Bu he esimaion o he ron surace hea lux as a byproduc o he CGM algorihm is no saisacory since here are very large lucuaions around ime =. Figure 4 shows he calculaing resuls or Case II. The measured hea lux a bac surace has he same error (%) as in Fig. 3 and hereore is no shown here. I is seen rom Fig. 4 ha boh he recovered emperaure and he byproduc hea lux a ron surace are in excellen agreemen wih he exac soluions. Thereore, compared o Case I, Case II provides he mos robus numerical scheme or he inverse esimaion o he ron-surace heaing condiion based on he measuremen daa a he bac surace. I should be poined ou ha he no prior inormaion abou he ron heaing condiion is required in he CGM algorihm used in his sudy. 4.4 Parameric Sudy or Case II A powerul numerical scheme or inverse hea ranser should be insensiive o he random measuremen errors since he real measuremen uncerainies may become very large. Figure 5 presens he eec o hea lux measuremen error on he accuracy o he IHCP ormulaion described by Case II. The simulaion parameers are almos he same as hose in Fig. 4 excep or he random error on he measured hea lux. I can be seen rom Fig. 5 (b) ha when he hea lux measuremen error is increased o %, he ron surace hea lux can sill be reconsruced wih a good accuracy. The ron surace emperaure can be recovered wih excellen accuracy (Fig. 5(c)). Clearly, he IHCP ormulaion o case II proposed in his sudy is insensiive o measuremen errors. Measuremen hea lux a bac surace (W/cm ) Recovered hea lux a ron surace (W/cm ) Exac soluion obained rom Eqs.(35)~(38) Recovered value using CGM mehod (a) (b) Temperaure a ron surace as by-produc (K) Exac soluion obained rom Eqs.(35)~(38) Recovered value using CGM mehod (c) Figure 5 Eec o hea lux measuremen error on he recovery accuracy or Case II. (a) Bac-surace hea lux measuremens (wih % measuremen errors); (b) Esimaed ron-surace hea lux by CGM inverse mehod; (c) Fron-surace emperaure as a by-produc. American Insiue o Aeronauics and Asronauics

11 Figure 6 presens he eec o he requency o he sinusoidal componen on he recovery accuracy o he IHCP ormulaion involved in Case II. The simulaion parameers are almos he same as hose in Fig. 4 excep or he requency o he sinusoidal componen. I is ound rom Fig. 6(a) ha when he requency is increased o 5 Hz, he phase o he esimaed hea lux agrees well wih ha o he exac soluions, and here is a lile decrease in he accuracy o he magniude o he recovered hea lux signal. Again, he ron surace emperaure can be recovered wih excellen accuracy as a by-produc o he inverse algorihm (Fig. 6(b)). Recovered hea lux a ron surace (W/cm ) 5 Exac soluion obained rom Eqs.(35)~(38) Recovered value using CGM mehod Temperaure a ron surace as by-produc (K) 7 Exac soluion obained rom Eqs.(35)~(38) Recovered value using CGM mehod (a) (b) Figure 6 Eec o requency o sinusoidal componen on he recovery accuracy or Case II ( = 5 Hz). (a) Esimaed ron-surace hea lux by CGM inverse mehod; (b) Fron-surace emperaure as a by-produc. 5. Conclusions A conjugae gradien mehod (CGM) algorihm is proposed o reconsruc he hea lux and emperaure a he ron surace o a inie slab which is subjeced o high-inensiy heaing. To achieve high recovery accuracy, boh emperaure and hea lux are measured a he bac surace. The emperaure measuremen daa is used as he bac surace boundary condiion whereas he hea lux is employed in he objecive uncion. Two cases are examined in deail: I) he ron surace emperaure is chosen as he unnown o be recovered; II) he ron surace hea lux is chosen as he unnown o be recovered. The sensiiviy problems and adjoin problems or he wo cases are derived and ormulaed. A CGM ieraive numerical procedure is esablished aiming o obain a convergen IHCP soluion. The resuls showed ha a beer accuracy on he recovered ron surace boundary condiion can be obained when he ron surace hea lux is chosen as he unnown o be recovered (Case II). Consequenly, he ron surace hea lux insead o ron surace emperaure should be recovered when boh bac surace emperaure and hea lux are nown. The robusness o he suggesed IHCP ormulaion (Case II) is esed or many dieren cases and i is ound ha he accuracy o he suggesed IHCP ormulaion is insensiive o hea lux measuremen error and can handle periodic ron-surace heaing luxes wih dieren requencies. The wor presened in his paper suggess a robus IHCP ormulaion and provides a guideline or he opimal experimenal design in manuacuring and hea reamen. Acnowledgmens The wor presened in his aricle was unded by he US Army Program Execuive Oice or Simulaion, Training, & Insrumenaion under Projec No. W9KK-8-C- direced by Ami Kapadia and Minh Vuong. The auhors would lie o express heir graiude o Dr. James L. Griggs or his valuable discussions. American Insiue o Aeronauics and Asronauics

12 Reerences [] dell Erba, M., Galanucci, L. M., and Migliea, S., An Experimenal Sudy on Laser Drilling and Cuing o Composie Maerials or he Aerospace Indusry Using Excimer and CO Sources, Composies Manuacuring, Vol.3, No., 99, pp [] Bec, J. V., Blacwell, B., and S-Clair, C. R., Inverse Hea Conducion: Ill Posed Problems, Wiley, New Yor, 985. [3] Alianov, O. M., Inverse Hea Transer Problems, Springer-Verlag, Berlin/Heidelberg, 994. [4] Özisi, M. N., and Orlande, H. R. B., Inverse Hea Transer: Fundamenals and Applicaions, Taylor & Francis, New Yor,. [5] Diller, T. E., Advances in hea lux measuremens, Advances in Hea Transer, Vol. 3, 993, pp [6] Childs, P. R. N., Greenwood, J. R., and Long, C. A., Hea lux measuremen echniques, Proceedings o he Insiue o Mechanical Engineers. Par C: Journal o Mechanical Engineering Science, Vol. 3, No. C7, 999, pp [7] Tong, A., Improving he accuracy o emperaure measuremens, Sensor Review, Vol., No. 3,, pp [8] Childs, P. R. N., Advances in emperaure measuremen, Advances in Hea Transer, Vol. 36,, pp [9] Saidi, A., and Kim, J., Hea lux sensor wih minimal impac on boundary condiions, Experimenal Thermal and Fluid Science, Vol. 8, 4, pp [] Loulou, T., and Sco, E. P., An inverse hea conducion problem wih hea lux measuremens, Inernaional Journal or Numerical Mehods in Engineering, Vol. 67, 6, pp [] Sparrow, E. M., Haji-Sheih, A., and Lundgren, T. S., The inverse problem in ransien hea conducion, ASME Journal o Applied Mechanics, Vol.86, 964, pp [] Jarny, Y., Özisi, M. N., and Bardon, J. P., A general opimizaion mehod using adjoin equaion or solving mulidimensional inverse hea conducion, Inernaional Journal Hea and Mass Transer, Vol.34, 99, pp [3] Pasquei, R., Nilio, C. L., Boundary elemen approach or inverse hea conducion problems: applicaion o a bidimensional ransien numerical experimen, Numerical Hea Transer, Par B, Vol., 99, pp [4] Yang, C. -Y., and Chen, C. K., The boundary esimaion in wo-dimensional inverse hea conducion problems, Journal o Physics D: Applied Physics, Vol. 9, 996, pp [5] Huang, C. H, and Wang, S. P., A hree-dimensional inverse hea conducion problem in esimaing surace hea lux by conjugae gradien mehod, Inernaional Journal o Hea and Mass Transer, Vol. 4, 999, pp [6] Emery, A. F., Nenaroomov, A. V., and Fadale, T. D., Uncerainies in parameer esimaion: he opimal experimenal design, Inernaional Journal o Hea and Mass Transer, Vol. 43,, pp [7] Monde, M., Arima, H., and Misuae, Y., Esimaion o surace emperaure and hea lux using inverse soluion or one-dimensional hea conducion, ASME Journal o Hea Transer, Vol. 5, 3, pp [8] Xue, X., Luc, R., and Berry, J. T., Comparisons and improvemens concerning he accuracy and robusness o inverse hea conducion algorihms, Inverse Problems in Science and Engineering, Vol. 3, No., 5, pp [9] Franel, J. I., Osborne, G. E., and Taira, K., Sabilizaion o ill-posed problems hrough hermal rae sensors, AIAA Journal o Thermophysics and Hea Transer, Vol., No., 6, pp [] Alianov, O. M., Soluion o an inverse problem o hea conducion by ieraion mehods, Journal o Engineering Physics, Vol. 6, 974, pp [] Zhou, J., Zhang, Y., Chen, J. K., and Smih, D. E., A nonequilibrium hermal model or rapid heaing and pyroloysis o organic composies, ASME Journal o Hea Transer, Vol. 3, 8, 645. [] Masumoo, M., and Nishimura, T., Mersenne Twiser: a 63-dimensionally equidisribued uniorm pseudorandom number generaor, ACM Transacions on Modeling and Compuer Simulaion, Vol.8, No., 998, pp American Insiue o Aeronauics and Asronauics