OPTIMAL EXPERIMENTAL DESIGN WITH INVERSE PROBLEMS

Transcription

1 Inverse Problems n Engneerng: Theory and Pracce rd In. Conference on Inverse Problems n Engneerng June -8, 999, Por Ludlow, WA, USA EXP5 OPTIMAL EXPERIMENTAL DESIGN WITH INVERSE PROBLEMS Mkhal Romanovsk CAD/CAE Deparmen, POINT Ld /, ave. Andropova, Moscow, Russa E-mal: ponld@glasne.ru ABSTRACT The man goal s a heory developmen of mal expermenal desgn for he hea properes denfcaon of a sold. Sequenal and mehodcal sudy of denfcaon of a specfc hea and hermal conducvy are carred ou. The wde range of mahemacal models are consdered. The model choce s deermned by possbles of analycal analyss of desgn condons. The nvesgaon answers o a number of prncpal quesons of hermophyscal expermenal realzaon and desgn. Ther praccal sgnfcance s expressed n dscoverng he exsence of he expermenal nformably upswng even of very small sample volume. NOMENCLATURE x Spaal coordnae Tme u(x,) Temperaure feld f Volume hea sources u Inal emperaure v, Boundary emperaures a Specfc hea coeffcen a Thermal conducvy coeffcen ε Nose of observaons Absolue error of observaons u Sample of observaons u Prooype sae ν Absolue error of denfcaon Relave error of denfcaon Ξ Opmal desgn Ω Sablzng funconal θ, Form-facors of denfcaon errors mode R Indeermnacy power of model denfcaon INTRODUCTION A he momen, despe a sgnfcan hsory of expermenal hea exchange researches, he heory of s mal desgn has no answers o he whole seres of basc n essence problems. The known soluons [-] were consdered for cases, whch don allow o receve a full pcure of mal desgn condons. We shall delver and sudy he followng quesons. Wha s he essence of mal observaons desgn: n searchng of locally-mal pons, ndvdual for each specmen and condons of s hermal loadng or n exsence of he common crcu of observaons, dencal for any expermen, bu updaed on a number of condons? How o fnd denfcaon errors dependence vs observaons allocaon for any knd of an expermen? Whch measuremens guaranee a mnmum level of denfcaon errors of hea ransfer properes? Whch facors and condons of expermenal realzaon ensure a decrease of an denfcaon errors level? Are sgnfcan funconal feaures lke symmery of observaons, heerogeney of emperaure dsrbuon, raos beween objec properes, raos beween boundary condons and ohers for an mal denfcaon? Wheher sascal ndeermnacy of observaons has an nfluence o an denfably of an expermen? Is possble o fnd as hea properes of specmen and s boundary condons usng only one pon of observaon? These quesons on he whole brng ou he prncpal characer of desgn peculares. Ther soluons are carred ou below. The ones demonsrae he ools for he nformably analyss of a complex expermen. Copyrgh 999 by ASME

2 EXPERIMENTAL DESIGN METHOD The deermnaon of mal observaons mus be carred ou n general wh allowance for he mehodologcal characerscs of nverse problem. Ths can be accomplshed o he fulles exen by means of he Tkhonov regularzaon prncple []. We use such regularzaon below, nvokng a specal procedure [5]. The one allows o acheve he bes f o he observaons and doesn requre a regularzaon parameer. Essenal feaure of he desgn mehod s a ype and characer of obaned esmaons. Ths nvesgaon s based on dea of a guaraneed error [6]. In hs case a mnmzaon of denfcaon error ν rms n ν / n. s carred ou. In addon o he rms norm, whch provdes a means for analyzng observaons from he sandpon of oal error, oher forms of esmaon error can be consdered. The mos praccal form here could be he absolue-error esmaor. The one requres a mnmzaon of he error ν abs max ν. Le us defne he facors of expermenal realzaon, whch nfluences o he accuracy soluon of nverse problem. Defnon. An ndeermnacy power of mahemacal model denfcaon s a facor deermnng a level of denfcaon error n accordance o he measuremens errors and he ype and srengh of a drvng force. The one s expressed a man condons due o denfcaon errors s decreased o zero. To separae he condons of expermenal realzaon whch has an nfluence o he sensors allocaons s necessary o nroduce Defnon. A form-facor of denfcaon error mode s a facor deermnng he characer of denfcaon errors dsrbuon vs sensors allocaons. A sgnfcance of hese wo ype of facors and her comparson wll be shown below. ONE UNKNOWN COEFFICIENT We specfy he mahemacal model du au ( am u), > ; d u u () wh an unknown coeffcen a cons>. The observaon fng equaon [6] for he model () s expressed n he fnal form max ( u u ) exp( a)[exp( a) ] + ε am where ( aa)/ as he relave denfcaon error, a s he rue value of he requred coeffcen. The denfcaon error s deermned by he expresson ln[ + R exp( a)], a The correspondng graphs are show n Fg.. Here R u u am s he ndeermnacy power of he model () denfcaon. The one deermnes man facors and condons due o an denfcaon error level s esablshed and reduced. For any nose level ε < he coeffcen a wll be (C) deermned wh mal error mn ( ) a he me ln C C a ( ) ( ) The R-mal and C-mal desgns [6] for he model () denfcaon are equvalen. The fnal expresson for he mnmum guaraneed denfcaon error s represened n mplc form ( C ) ( C) ( C) ( ) R The behavor of he guaraneed error and he characer of he mal observaon me have he nex peculares. Frs, he denfcaon error of he model () depends on he nose level and he dfference beween nal and amben emperaures. Consequenly, he condon for offseng growh of he observaon errors and dmnshng her nfluence s o ncrease he emperaure dfference. Second, he upper bound of he ndeermnacy power of he model () denfcaon s R max /. Expermens, whch have R R max, canno o guaranee an denfcaon accuracy. Thrd, he lower bound of he mal measuremen me s he value T mn / a. Ths value esablshes a ceran barrer, below whch observaons are no recommended,.e., >T mn. Le he sae of an objec be descrbed by followng model u u a, < x <, > ; x u u < x <, ; u v u x, v x, > () ( C ) 5 6 Fgure. IDENTIFICATION ERROR OF MODEL (6) -R., - R.,-R., - R.,5-R.,6-R.9 Copyrgh 999 by ASME

3 In case v, cons he observaon fng equaon s expressed as k ( ) ( v u ) ( v u ) k max sn x x, k k exp a k exp a k + ε () where ( a a)/ a. From Eq.() follows ha for any a and ε < he mnmum denfcaon error s aaned for he mal observaon allocaon x /. The me dependence of he denfcaon error s expressed by he equaon where.5.5 Fgure. C-OPTIMAL IDENTIFICATION OF MODEL () ( ) ( ) exp k k a k k ( k exp a ) R, R v + v u s he ndeermnacy power of he model () denfcaon. The approprae mal me s expressed as local-mal Arg mn ( ) The dependence of C-mal denfcaon error from he condons of expermenal realzaon s represened n Fg.. The model () has followng denfcaon peculares. The lower bound of he mal measuremen me s Tmn / a. The upper bound of he ndeermnacy power of he model () denfcaon s R max /. TWO UNKNOWN COEFFICIENTS Le us specfy he nex mahemacal model u u a a + f, < x <, > ; x u u x, < < ; u v, u v, > x x () I s requred o fnd such par of pons Ξ { x, }, n whch known sample of observaons u u( x, ) + ε,, can o defne a specfc hea a and a hermal conducvy a wh a mnmum guaraneed error on rms or mnmax ( R ) mn x, + ( C ) mn max, x, crera. Here, ( a, a, )/ a, are relave errors of denfcaon, a, are unknowns defned he rue sae u, ε s measuremens nose. We shall accep a number of he supposons. A frs, we shall consder a, cons. Besdes a specmen densy s ncluded as known consan n coeffcen a. Secondly, we shall lm an aspec of hermal loadng condons o values f, u,v, cons. These resrcons wll allow o smplfy he analyss of mal desgns dependence from condons of expermenal realzaon. A he same me hese resrcons do no reduce number of raos beween boundary condons. They envelop a broad band of praccal cases and all characersc raos beween hermal loadng are refleced. Noe, ha he volume of observaons{ u }, s seleced from a condon of supporng of s admssble mnmum. Ths resrcon of sample volume allows o analyze he achevemen of maxmum expermenal nformabaly, resed mnmum nal nformaon [8]. The condon of he model sae fng wh observaons s gven by sysem max F (, ) s x Fs( x, ) + ε,,, x, s s (5) where f F a k k ( ) a k ( ) k exp sn ( ) x a F [ u v v ( + )] k k a k ( ) exp a k sn ( ) x ( v v ) a k F k exp sn x k k a The funcons F s are represened by rue values a of unknowns, and he funcons F s are expressed by values a, obaned n an oucome of denfcaon. Copyrgh 999 by ASME

4 Opmal desgns Le us sudy a behavor of funcons, a varaons of her varables x,. A erm by erm comparson of expresson, obaned by a dfference for any ' < " one of he equaons (5) allows o fnd he followng condons E E >, E E >, where a k E ( ) exp a ( ) From hem follows ha <. By vrue of for any ' < " and < a condon > akes place. I means, ha he funcon s monooncally decreasng on me. Then he leas denfcaon error of a hermal conducvy can be reached only a he saonary sae observaon,.e., T. Accordance o hs fac for he mal me of measuremens he spaal dependence of he funcon from a sensor coordnae s expressed as mn T f ( x x ) ± a (6) The one s obaned n case of absolue norm usng for he model sae fng wh observaons. From expresson (6) we defne he leas guaraneed error mn. I s reached n he pon x x T / and maers 6R, 6R (7) where R a f s he ndeermnacy power of he model () denfcaon. Any expermens canno o guaranee an denfcaon accuracy n case R R max /, because >. We shall remark ha he funcon T has he second mnmum 6R +. 6R + The one for R> does no express guaraneed level of hermal conducvy denfcaon, bu hs error s necessary for akng no accoun herenafer a error mnmzaon. The characer of he error s deermned by a sage of he hermal process and raos beween a source funcon and boundary condons {f,u,v, }. On an nal sage of he process, when ~, sharp growh of values s supposed, snce he exponenal erms n funcons F,, lnearly depend from. Then he resrcon of growh of exponenal erms of he seres (5) s reached by a choce such x, for whch as he funcon sn(k-) x/, and sn(k x/ ) approach o zero. I means, ha on an nal sage of he hermal process he error s descreased when observaons asympocally approach o he boundares of a specmen, x, x. In hemselves boundary x,x, he funcon, as follows from (5), suffers a dsconnuy of he second knd. Accordng o hs fac he desgns Ξ u v Ξ u v x x,,, x T,, x T are as R-, and C-mal f he condon for example s fulflled u ( v + v ) v v and he funcon f does no exceed a value, whch nsalls a sgnfcance of erms of funcon F n (5). In case >> he resrcon of growh a he expense of parameers varaons n exponenal erms (5) s unsuffcen. Then an denfcaon error decreasng can be reached, f he observaon s fulflled n a pon, where here s he maxmum of harmoncs of he seres (5). In vew of he seres (5) convergence only frs erms as sgnfcan can o ensure he resrcon of growh. Therefore he exreme of s localzed n areas, whch ceners should place n one of pons {.5;.5;.75}. A devaon of mal coordnae from ndcaed pons wll be less sgnfcan wh hermal dffusvy aa /a growh because a convergence velocy of he seres (5) depends from a value of a hermal dffusvy. The me for he second mal observaon n hese cases s deermned as locally-mal Arg mn max(, ) - + x, Characersc cases of rms error dependence vs sensor allocaon mnmzed on me are shown n Fg. and. I should be menoned, ha all mal desgns are obaned despe essenal nonlneary of a model sae dependence from unknowns, and n he supposon of any naure of measuremens nose. Ths resul shows ha even essenally nonlnear desgn problem supposes s aggregae and has srcly defned crcu of measuremens. Copyrgh 999 by ASME

5 Fgure. RMS ERROR DEPENDENCE: v - var, v <u, v () <v () <v () <..., -u v,8-u (v +v )/ Fgure. RMS ERROR DEPENDENCE: v - var, v >u, v () <v () <v () <..., 5-u (v +v )/, -u v. Le us consder mal desgn of he conrolled expermens. I should be assumed, ha he hermal loadng s no fxed, and one s modfed n desrable drecon. There s he hermal loadng condon v v (8) whch may be o defne as he bes guaraneed condon. The one means ndependence of a poson of he mal pon x from rao beween facors {f, u, v, }. For a deermnaon of an mal observaon a realzaon of he condon (8) we shall subrac one from oher he equaon (5) for dfferen values x' and x". We oban a ( ) exp sn ( ) k k x sn ( k ) x a a ( ) ( k ) exp sn ( k ) x a ( ) a ( ) ( k ) exp sn ( k ) x a ( ) From follows sn ( k ) x sn ( k > ) x Then a any {f, u,v, } for he error mnmzaon he observaons would be seleced when sn( k ) x / have a maxmum. From all such pons only one x / ensures a requred maxmum n each erm of he seres (5). Therefore, f he condon (8) ake place hen he second mal observaon concdes wh frs pon. The mddle of a specmen appears he unque pon, measuremens n whch guaranee he denfcaon of hea properes wh a mnmum error. The approprae mal desgn s represened as Ξ v v T { /,, } Thus he analyss of exreme properes of he observaon fng equaon (5) allows o specfy n analycal form he wde range of desgn characers. Basc properes of mal desgn The model () nvesgaon has allowed o place he followng feaures of hea properes denfcaon.. The mal desgn s expressed by he srcly defned crcu of measuremens of a specmen emperaure feld.. In general he represenaon of a volume hea source, boundary emperaures and wo neror observaons are enough for denfcaon as consan specfc hea and hermal conducvy.. The consderaon of ndvdual properes of a specmen and s hermal loadng does no change hs crcu, bu requres o make more accurae he poson of one of he sensor and me of a measuremen. A nose level does no nfluence o an aspec of he mal desgn, bu defnes a magnude of he denfcaon errors.. The mal poson of he frs neror observaon does no depend on a characer of hermal loadng of a specmen and s hea properes. A sensor should be nsalled n he mddle of a specmen. 5. The mal observaon me no he mddle of a specmen for any expermens s a measuremen of a saonary emperaure of a specmen. 6. The mal allocaon of he second neror observaon s localzed n fve lmed areas. The sgnfcance any of hem depends on condons of expermenal realzaon. 7. There s he broad band of boundary condons varaons, a whch he poson of he second neror observaon s mum only near o one of specmen boundares, and he bes me of measuremens s deermned 5 Copyrgh 999 by ASME

6 by he begnnng of he hea process. The smlar expermens are characerzed by low dfference beween nal and one of he boundary emperaures. 8. Opmal observaons may be carred ou n one of he pons {.5;.5;.75} or her neghborhoods. The range of a devaon from hese pons s deermned by magnude of a specmen hermal dffusvy and raos beween boundary condons. The mal me of second measuremen can be found as locally-mal. 9. A desgn of he condons of expermenal realzaon allows o reduce volume of he observaons o s mnmum. The requremen of he boundary emperaures equaly among hemselve s he mporan condon for he measuremens decreasng. In hs case he number of observaons for he model () denfcaon s defned only as one neror pon (no counng observaons for dervng boundary condons). EXPERIMENTAL PECULIARITES Now we shall gve aenon o he nfluence of he expermenal condons on an denfcaon accuracy. Idenfably volaon The unqueness soluon of he sysem (5) requres a samples geng, whch does no reduce equaons o he lnear dependence n pons {x, },. The one leads o he followng condon sn exp x k a k ( ) λ a sn x k, k,,..., λ. (9) Then he model () sae s an undenfable on dscree sample u * only when he observaons wll be carred ou n any wo pons x*, such, ha x* + x*, and he measuremens wll be execued a he same momen,. The exsence of undenfable sae of he model () was shown earler n []. The condon (9) expresses he addonal undenfably, when he one-o-one correspondence s away only n separae pons Fgure 5. THRESHOLD LEVEL OF NOISE -f 5, -f,-f,-f,5-f,6-f 5 Threshold level of nose As s possble o see f hen he sysem (5) s degeneraed. Then he usage of observaons, for whch >*, s mean a volaon of doman of admssble values of sysem (5). By vrue of he hreshold level of nose * s exsed. Exceedng of hs level wll no allow o solve nverse problem. The reason of he denfably volaon n hs case s a degeneraon of sample approxmaon by he model sae. The one appears ndependen from coeffcen a. A behavor of R- mal error a varaons of a nose level s represened n Fg.5. The exsence of a hreshold * esfes, ha here s a value f *, lower from whch f < f * s mpossble o guaranee a sasfacory denfcaon. The condons of expermenal realzaon, ncludng funcon f *, whch caused a loss of an denfably due o exceedng of some level of measuremens nose, we shall defne as hreshold condons. Sngular observaons From (6) follows, ha for any f s possble o specfy such pons < x * < n whch he mnmzaon of an denfcaon error on me does no allow o ge resrcon of s values,.e., x *, T. The deermnaon of hea properes on such observaons s mpossble, and emperaure measuremens n pons x* ± 6R canno o denfy unknowns a,. Measuremens n he mal pon x /, fulflled n expermen whch sasfes o he condon R/6, no allow o fnd hea properes. Such volaon of denfcaon dfference from as an undenfably n a whole and n a small [7,8]. As appears he exsence of some errors corrdor reduces a choce of hermal conducvy o arbrary value. I s possble o * specfy such f 6a /, for whch he requred fng beween observaons and model saes n lms of a specfc errors corrdor s reached, f a lne, f he boundary emperaures s conneced by lne. In an oucome he funconal represenaon of a model sae s degeneraed and doesn' depend from a coeffcen a.if f f *, hen he approxmaon of observaons depends on a. We shall mark also exsence of he low bound of a hea * source srengh. As s follows from (7) he use f < f reduces he denfcaon error o values >>. Consequenly s mpossble o guaranee sasfacory denfcaon. For he deermnancy, and also underlnng he specal nfluence of measuremens error o full loss of an denfably, s offered o dsngush observaons as an -undenfable. Condons of expermen, and n parcular funcon f *, generang unlmed growh of denfcaon error under degeneraon of a sample approxmaon, we shall defne as a sngular. 6 Copyrgh 999 by ASME

7 Self-compensang condons of hermal loadng I s always can o specfy he value f F*, for whch he sum of negave erms of F wll be of one order of he sum of posve erms F (F ). Then a values f varaons he consderable growh of exponenal erms of he sysem (5) s possble n consequence of self-compensang of facors {f, u,v, }. In hs connecon s necessary o mean, ha n general here s no monoone decrease of a mnmum of he error a magnfcaon of a value f. In case {v +v < u, v v, f >} he evaluaon s a F*~ ( u v, ) The one expresses condons of emergng of he funcon maxmum a varaons f > f, *. The unque facor, permng o lm a growh of he error s varaons of observaon me near he nal sae. The hermal loadng deecon, reducng o he denfcaon accuracy loss n consequence of selfcompensang of facors {f, u,v, }, s necessary o consder as he mporan desgn feaure. We shall defne smlar condons as self-compensang loadngs. Facors of denfcaon errors reducon There are he followng facors, whch allow o decrease he denfcaon errors. A frs, he decrease s reached a he expense of a dmnuon of nose level. From (5) mples, ha f, hen,. Secondly, from (7) s followed, ha for a dmnuon of magnude s necessary o ncrease a srengh of hea source. In accordng o he exsence of -undenfable and he self-compensangs condons s necessary o requre, * * ha f > max( f, f ) and f F*. There s he value v v f 6 + g ( u max ) a / snce whch C-mal denfcaon error of hermal conducvy wll no exceed of a relave level of measuremens nose / u max, u max max( u, v, ). f f g Thrdly, a rao beween nal and boundary emperaures {u, v, } nfluences o he mode of denfcaon errors. Magnfcaon of a dfference beween u and v +v dmnshes funcon.ifv,, hen we oban a ( ) ( ) exp k exp a ( ) a ( k ). a The one shows, ha a magnfcaon of emperaure shock beween he nal and boundary emperaures he funcon ends o he value. From follows ha a emperaure shock doesn allow o acheve absolue decreasng of denfcaon errors o zero. Ths resul esablshes, ha for he model () he emperaure shock as a facor of an denfcaon errors decreasng has only local sgnfcance. Wherea such funconal feaures of emperaure felds as he pons of her maxmum, nflecon and ohers don' explcly deermne an mal allocaon of measuremens. Le us express an operaon of he ndcaed facors as some generalzed complexes. The sysem (5) has he dmensonless varables τa /( a ) and ξx/. Therewh dmensonless groups θ u ( v + v ) a f, θ v v f a can be nroduced. The ones deermne a mode of denfcaon errors (Fg.,). By vrue of he facors θ, may be defned as form-facors of denfcaon error mode. A varaon of he mahemacal model parameers accordng o he condons R em, θ, em () and a choce of approprae τ and ξ don change a soluon of he sysem (5). I means, ha he errors, are nvaran relave o he condons () of expermenal realzaon. Sngulares of expermenal realzaon. Major facors of he denfcaon error decreasng are he rase of he hea source srengh and he magnfcaon of he dfference beween nal emperaure on he one hand and boundary emperaures wh oher.. From wo called facors he mos sgnfcan s he magnfcaon of a hea source srengh. For any specfc nose level he hea source can be ndcaed, snce whch s guaraneed ha an denfcaon error always wll be less hen a nose level.. A he same me boundless magnfcaon of a dfference beween nal and boundary emperaures doesn' allow o reduc he denfcaon error o zero. In hs case an asympoc approach o he guaraneed denfcaon error of hermal conducvy s reached only.. A varaons of condons of expermenal realzaon and n parcular magnfcaon of a hea source srengh s necessary o ake no accoun a presence of some sngulares cases. A hreshold level expresses a lmng measuremens nose. A mahemacal model s undenfable when a sample has measuremens error hgher hen hreshold level. A sngular observaons reduce o an denfably loss hereof an arbrary choce of unknowns a a ceran breadh of a measuremens error corrdor. A self-compensang loadngs represens a sgnfcan worsenng of he denfcaon accuracy because of a couneracon each oher of a specmen hea loadng facors. 5. Alongsde wh unqueness of a smulaneous deermnaon of all hea properes an undenfably emperaure felds as a whole, and n a small are exsed. 7 Copyrgh 999 by ASME

8 SIMULTANEOUS IDENTIFICATION OF HEAT PROPERTIES AND BOUNDARY CONDITIONS As s proved above, he desgn of condons of expermenal realzaon allows o reduce a volume of sample o he mnmum number of observaons. The smlar fndng are characersc for he approach of observaons processng, based on nverse problems mehodology [8]. Ths noe from he vewpon of expermenal nformably pus a queson on a furher research of unqueness of smulaneous denfcaon as hea properes and boundary condons. Mahemacal model We specfy he mahemacal model () and shall consder he measuremens crcu wh one pon of observaon. I s requred o fnd such desgn Ξ { x, },..., for whch known sample of observaons u u( x, ) + ε,,..., can o defne smulaneous a specfc hea a, a hermal conducvy a and boundary emperaures v, wh a mnmum guaraneed error on rms ( R ) mn x, Here, ( a, a, )/ a, and, ( v, v, )/ v, are relave denfcaon errors. The soluons of observaon fng equaons are shown n Fg.6. The one s expressed he rms error dependence from sensor allocaon mnmzng on observaon me. There are hree mal allocaon of a sensor. The global mnmum has allocaon n he mddle of a specmen. Oher mal allocaon s near from he specmen boundares. The varaon of observaons error doesn change hs allocaon. Thus he usng only one nernal observaon allows o fnd as specmen properes and s loadng facors. Ths resul shows ha nverse problems can be consdered as a powerful exrapolaon ool for expermenal daa processng. The praccal realzaon of such sandpon s repored n [9]. CONCLUSIONS Among of observaons s possble o defne he mos nformably sample. A small volume of observaons can o ensure he denfcaon of a sgnfcance number of unknowns. The necessary volume of observaon s defned by denfably condons. If nverse problem peculares are akng no accoun hen s possble o denfy as phenomenologcal objec properes and s loadng facors. The hea properes denfcaon has srcly defned crcu of measuremens of a specmen emperaures. Opmal desgn mus be carred ou smulaneous a several drecon. Specmen loadng, sensors allocaon and measuremens me are he man characers of desgn Fg.6. SIMULTANEOUS IDENTIFICATION OF HEAT PROPERTIES AND BOUNDARY TEMPERATURES -., -.5,.6,.8, 5. A deermnaon of he facors srucure of denfcaon errors reducon has a doubless neres. These facors defne he condons of he denfcaon error decreasng o zero and s dependence mode from sensors allocaon. The analyss shows he exsence of sascal ndeermnacy of nverse problem soluon. The one s descrbed as a correspondng creron, whch has an upper bound of s admssble value. The furher drecon of nvesgaon s a generalzaon on a number of unknowns, nonlnear hea properes denfcaon and oher ype of boundary condons. REFERENCES. Uspensk A. B. and Fedorov V. V., Approxmae Mehods for he Soluon of Opmal Conrol Problems and Ceran Inverse Problems, MGU, Moscow, pp.8-97 (97).. Sun N.Z., Yeh W.W.G., Waer Resour. Res., Vol.6, No., pp.57-5 (99).. Takak R., Beck J.V., Sco E., In. J. of Hea Mass Transfer, v.6, No., pp.977 (99).. Tkhonov A.N. and Arsenn V. Ya., Soluons of Ill- Posed Problems, Halsed Press, New York (977). 5. Romanovsk M.R., Hgh Temperaure, Vol.8, No. (98) 6. Romanovsk M.R., Indusral Laboraory, Vol.59, No., pp (99). 7. Romanovsk M.R., J. of Engneerng Physcs and Thermophyscs, Vol., No., pp.5-57 (98) 8. Romanovsk M.R., J. of Engneerng Physcs and Thermophyscs, Vol.57, No. (989) 9. Romanovsk M.R., J. of Engneerng Physcs and Thermophyscs, Vol.5, No., pp.76-8 (98) 5 8 Copyrgh 999 by ASME