Inverse Problem of Finding an Unknown Parameter for One- and Two-dimensional Parabolic Heat Equations

Transcription

1 Iverse Problem of Fdg a Ukow Parameter for Oe- ad Two-dmesoal Parabolc Heat Eqatos Mohamed Elmadob Problem Report sbmtted to the Statler College of Egeerg ad Meral Resorces at West Vrga Uversty partal flfllmet of the reqremets for the degree of Master of Scece Chemcal Egeerg Charter D. Stesprg Ph.D. Char Ferado V. Lma Ph.D. Yog Yag Ph.D. Departmet of Chemcal Egeerg Morgatow West Vrga 5 Keywords: Iverse problem cotrol parameter o-classcal bodary codtos Temperatre overspecfcato Copyrght 5 Mohamed Elmadob

2 ABSTRACT Iverse Problem of Fdg a Ukow Parameter for Oe- ad Two-dmesoal Parabolc Heat Eqatos Mohamed Elmadob I may traset heat trasfer problems accrately measrg thermal propertes has prove to be a mportat ad dffclt feld of stdy. It s possble to fd the temperatre dstrbto as well as the cotrol parameter p that smltaeosly satsfy the goverg partal dfferetal eqato. The aalyss of smltaeosly recoverg the heat sorce cotrol parameter ad the solto of the parabolc partal dfferetal eqato s referred to as a verse partal dfferetal eqato IPDE. I ths problem report verse problems of fdg a kow tme-depedet parameter oe- ad two-dmesoal Cartesa coordates are cosdered. The Crak Ncolso fte dfferece method ad the predctor corrector method are sed to estmate the tme-depedet cotrol parameter ad the parabolc solto. The secod part of the problem report s devoted to mercal soltos of oe- ad two-dmesoal verse parabolc heat eqato cyldrcal coordates. The comptatoal models created ths work are valdated wth a exact solto for Cartesa problems real expermetal data for oe-dmesoal cyldrcal problem ad MATLAB PDE toolbox solto for two-dmesoal cyldrcal problems. Nmercal smlatos demostrated that oe- dmesoal Cartesa comptatoal model s accrate stable ad less tme expesve tha the two- dmesoal Cartesa comptatoal model. However the real applcato of the scheme the reslts obtaed for oe-dmesoal cyldrcal problem are accrate for short tmes acceptable for moderate tmes ad accrate aga for large tmes. I geeral the model prodces relable reslts ad the smlated temperatre measremets were cosstet wth the expermetal data. I the two-dmesoal cyldrcal comptatoal model the drect problem solto s the fodato of the verse problem. The drect problem s solved by MATLAB PDE toolbox ad the overspecfed bodary codto E whch s oe solto of the drect has bee chose at the mdpot of the of r ad z coordates. The model prodce acceptable reslts at pots ear the bodary where z s wth terval of < z <. or.8 < z < bt the solto dverges tl t reaches ts maxmm at the mdpot of z.

3 ACKNOWLEDGEMENTS The completo of ths work cold ot have bee possble wthot the help cotrbtos ecoragemet ad spport of so may people. I wold frst lke to thak my Prof. Charter Stesprg for hs stadg besde me drg my stressfl tme ad gve me hope to move ahead. Bt also for hs ecoragemet ad fredly gdace drg the corse of ths work. Hs valable advce great gdace ad cotrbtos wrtg ths work are always grateflly ackowledged. Commttee members Dr. Ferado Lma ad Dr. Yog Yag have provded a great gdace ad helpfl sggesto ad formatve dscsso. My thaks go to all Chemcal Egeerg staff ad colleages who helped me throghot my program. Frthermore I wold also lke to thak Dr. Hamd Bdms for provdg hs expermetal data from hs stdy. Last bt ot least I wsh to express my apprecato to my wfe Eas for her patece her derstadg ad her ever-edg spport. Fally I scerely thak my parets brothers ad ssters for the costat ecoragemet they have gve me. Apologes advace to all others whom I may forget to meto here.

4 TABLE OF CONTENTS Ttle Page... Abstract... Ackowledgemets... Table of Cotets... v Lst of Tables... v Lst of Fgres... v Nomeclatre... v Chapter : Itrodcto.... Obectves of the Stdy.... Scope of ths Work... 3 Chapter : Lteratre Revew...4. Drect ad Iverse Problem Overvew of Prevos Work D Parabolc Iverse Problem D Parabolc Iverse Problem Applcato of Cotrol Parameter Iverse Problem... Chapter 3 : The Nmercal Techqe The Crak Ncolso Fte Dfferece Method D Iverse Problem wth Pot Overspecfcato The Predcto-Correctg Mechasm for D Problem D Iverse Problem wth Pot Overspecfcato The Predcto-Correctg Mechasm for D Problem Applcato of Iverse Problem to Cyldrcal Coordates... 3 v

5 3.6. D Cyldrcal Coordate Batch Vessel wth Wall Coolg D Cyldrcal Coordates Usg PDE Toolbox...6 Chapter 4 : Model Valdato Ad Dscssos Model Nmercal Test D Cartesa Coordates Model Nmercal Test D Cartesa Coordates Model Nmercal Test D Cyldrcal Coordates Model Nmercal Test D Cyldrcal Coordates... 4 Chapter 5 : Coclsos ad Ftre Work Coclsos Ftre Work... 5 Refereces...5 Appedx A...55 Appedx B...6 v

6 LIST OF TABLES 3. Radal locato of thermocoples the batch vessel Sample reslts of x for the frst model The RMSE MAE ad CPT tme for both x ad p Sample reslts of xy at t = T the secod model The RMSE MAE ad CPT tme for both xy ad p Sample reslts of r for TC ad TC6 oly The RMSE MAE ad CPT tme for both r ad p Sample reslts of rz at t = T the forth model The RMSE MAE ad CPT tme for rz B. The otpt reslts of x for the frst model h= k =/ s = ad T =... 6 B. the otpt reslts of p for frst model h= k =/ s = ad T =... 6 B.3 the otpt reslts of xy for the secod model h= /5 k =/... 6 B.4 the otpt reslts of p for the secod model h= k =/ s =5 ad T = B.5 the otpt reslts of r or TC for the thrd model h= k =/ B.6 the otpt reslts of r or TC 3 for the thrd model h= k =/ B.7 the otpt reslts of r or TC 4 for the thrd model h= k =/ B.8 the otpt reslts of r or TC 5 for the thrd model h= k =/ B.9 the otpt reslts of r or TC 6 for the thrd model h= k =/ B. the otpt reslts of p for the thrd model h= k =/ B. the otpt reslts of rz for the forth model h= / k =/... 7 B. the lsts otpt reslts of p for forth model h= k =/ s = ad T =... 7 v

7 LIST OF FIGURES 3. The Crak Ncolso comptatoal molecle for D Flow chart of the mercal rote wrtte MATLAB code for D Cartesa The Crak Ncolso comptatoal molecle for D Flow chart of the mercal rote wrtte MATLAB code for D Cartesa Batch vessel for deposto wth cooled vessel wall Flow chart of the mercal rote wrtte MATLAB code for D cyldrcal Flow chart of PDE Toolbox rote wrtte MATLAB for D cyldrcal Srface plot of the mercal solto x for the frst model The mercal solto p for the frst model ADE ad PE of x at t = T ADE ad PE of p at all x Srface plot of the mercal solto xy for the secod model at y = The mercal solto p for the secod model ADE ad PE of xy for the secod model ADE ad PE of p at all x ad y Srface plot of the mercal solto r for the thrd model The mercal solto p for the thrd model The mercal solto r for the TC The mercal solto r for the TC ADE ad PE of r for the TC ADE ad PE of r for the TC Srface plot of the mercal solto rz for the forth model The mercal solto p for the forth model Comparso betwee the Crak-Ncolso ad PDE toolbox soltos ADE ad PE of rz for the forth model v

8 NOMENCLATURE f a C p g E x y t T h O k N l M xx yy p R b q Heat sorce term Thermal coeffcet Heat capacty term cotrol parameter Parabolc solto Bodary codto Overspecfcato bodary codto Overspecfcato bodary codto coordate x-axs Overspecfcato bodary codto coordate y-axs Tme Fal tme Step sze x drecto Trcato error Step sze t drecto Mesh grd the t drecto Iterato mber Mesh grd the x drecto Secod dervatve of wth respect to x Secod dervatve of wth respect to y Ital gess of p Trdagoal matrx coeffcet Kow system of eqatos coeffcet Heat sorce Greek Symbols ρ α ϕxx ϕyy Ω Δ ϕ Desty Thermal codctvty Secod dervatve of the tal codto wth respect to x Secod dervatve of the tal codto wth respect to y Doma Amot of chage Ital codto v

9 CHAPTER INTRODUCTION Over the past years heat trasfer parabolc verse problems ad mercal techqes fte dfferece methods sed to solve them have bee creasg. I fact ths has bee oe of the fastest growg areas varos applcato felds. The stdy of verse problems plays a mportat role today appled mathematcs ad physcs. Ths kd of problem also arses may other mportat applcatos areas sch as mathematcal models for poplato dyamcs qasstatc theory of thermoelastcty medcal scece electrochemstry cotrol theory bochemstry ad certa bologcal processes []. For stace t s challegg to perform accrate measremet of the tme-depedet blood perfso throgh a certa rego of tsse der vestgato []. Ths models are ofte developed ad ted sg expermetal data to detfy the tme-depedet perfso from the verse problem to overcome ths dffclty. Addtoal measremets are of corse ecessary to reder a qe solto sch as heat flx teror temperatre or mass measremets [3]. Several physcal pheomea are modeled by a parabolc verse problem wth oclasscal bodary codtos. Sch a bodary codto may appear as a temperatre at a gve pot x the spatal doma at tme t; whch case the bodary codto s called pot overspecfcato bodary codto [4]. If the goverg partal dfferetal eqato s sed to descrbe a heat trasfer process where a sorce parameter exsts the the tegral bodary codto ca be terpreted as a weghted thermal eergy cotaed a porto of the spatal doma [5]. Whle may soltos for drect problems wth stadard bodary codtos have bee sed sch as fte dfferece fte elemet fte volme ad bodary elemet methods

10 there has bee less research to the mercal approxmato of verse parabolc partal dfferetal eqatos IPDEs wth overspecfed bodary data. Fte dfferece methods are kow as the frst techqes for solvg IPDEs. Eve thogh these methods are very effectve for solvg varos kds of PDEs some fte dfferece methods are kow as stable ad are restrcted by the stablty crtera. However mplct schemes sch as the Crak Ncolso are cosdered codtoally stable based o the vo Nema stablty aalyss. Moreover the Crak Ncolso has secod order accracy whch meas less trcato error assocated wth ths method [6]. Few vestgatos are kow the lteratre that volve the verse problem cyldrcal coordate systems. Addtoally after a extesve of the research there s o kow valdato for the exstg comptatoal models based o real expermetal data or valdato data agast other mercal methods sch as the fte elemet method. Therefore the overall focs of ths problem report was to model a parabolc verse problem ad valdate the reslts wth a exact solto for selected examples. I addto the comptatoal model created ths work was tested o cyldrcal coordate system.. Obectve of the Stdy The obectves of ths stdy were frst to solve a PDE as a verse problem wth se of codtoally stable ad secod-order accracy scheme sch as the Crak Ncolso method. Secod developg a advaced MATLAB comptatoal code for approxmato soltos of oe-dmesoal D ad two-dmesoal D problem wth heat sorce volved. To acheve these goals the followg steps were prsed:

11 Use a mplct fte dfferece scheme for D ad D of heat eqato wth heat sorce term volved. The Crak Ncholso method s sed for ths prpose. I addto se o-classcal bodary codtos: pot overspecfcato codtos. Develop MATLAB code for both D ad D problems. 3 Valdate the reslts wth the aalytcal solto Cartesa problems. 4 Apply the developed code to cyldrcal geometry. 5 Valdate the reslts wth a expermetal data ad PDE toolbox cyldrcal problems.. Scope of ths Work I ths problem report Chapter revews the prevos work o the developmet ad applcato of a dfferet mercal scheme for solvg the PDE ad smltaeosly determes of two gredets as a verse problem. Chapter 3 descrbes the se of the Crak Ncolso fte dfferece method. I ths chapter the predctor-corrector method that s sed to predct the tme-depedet heat sorce parameter s vestgated. Chapter 4 presets the mercal expermets ad reslts of the Crak Ncholso method for D ad D problems wth o-classcal bodary codtos. I addto the Cartesa problems MATLAB codes wll sed for cyldrcal problems wth ecessary adstmets o the bodary codtos of cyldrcal geometry. All reslts ad graphs are show ths chapter. Chapter 5 smmarzes the coclsos ad sggestos for ftre work. Some dervatos are appeded Appedx A. Appedx B lsts all otpt reslts of the code to llstrate the mercal reslts. The accompayg CD-ROM cotas the sorce codes for D ad D solver for for models. 3

12 CHAPTER LITERATURE REVIEW May mathematcas ad egeers have bee terested solvg parabolc verse problems. Coseqetly several fte dfferece schemes for soltos to ths type of problem have bee establshed. Wth the developmet of hgh-speed persoal compters t has become more coveet to se mercal techqes to solve heat trasfer verse problems. Ths s especally tre for problems wth o-classcal bodary codtos. Ths lteratre revew s teded to provde a geeral backgrod of some verse problems ad partclar the parabolc heat eqato verse problem for determato of the tme-depedet heat sorce cotrol parameter D ad D verse problems. For refereces regardg specfc techqes readers shold look at cted refereces to detfy frther relevat stdes.. Drect ad Iverse Problem A complete mathematcal descrpto of a physcal system allows the otcome of some measremets to be predcted. The estmato of the measremet reslts s referred to ether as the modelzato problem the smlato problem the forward problem or the drect problem [7]. There are may well-kow methods to solve drect problems. For stace the PDEs descrbg the physcal pheomea of heat codcto ca be solved sg exact ad mercal methods. The exact methods clde the classcal examples of separato of varables ad Laplace trasforms. Frthermore drect problems are cosdered well-posed problems ad more typcal whe modelg a physcal system where the model parameters ad materal propertes are kow. Hadamard sggested that a problem s well-posed f ad oly f the followg propertes hold [8]: 4

13 A solto exsts at least oe solto exsts exstece; The solto s qe at most oe solto exsts qeess; The solto depeds cotosly o the data stablty; that s to say do ot prodce a wldly dfferet reslt for very small chage the pt data. The verse problem cossts of sg the actal reslt of some measremets to fer the vales of the parameters that characterze the system. Accordg to may refereces the recet lteratre t s commoly thoght that most verse problems are cosdered ll-posed [9 ]. Problem s sad to be ll-posed f t fals to meet the propertes provded by Hadamard. The llposed problems cota errors. Ths meas that a small error of measred data may reslt a stable predcto whch reslts estmates rather tha actal reslts for a target property that eeds to be estmated. Ths maks the solto extremely sestve to measremet errors.. Overvew of Prevos Work Mch research has bee codcted o fdg a cotrol parameter a oe-dmesoal D parabolc verse sg varos mercal methods sch as secod-order explct ftedfferece FTCS the secod-order mplct fte-dfferece BTCS procedre Cradall s formla the Salyev's frst ad secod schemes etc. the explct schemes are very easy to mplemet for ths type of problems bt t wll be restrcted by the stablty crtera ad the step sze mght be ot good eogh to archve good accracy. I cotrast the mplct schemes are dffclt to mplemet bt they are ofte codtoally stable where the step sze ca be chose wthot ay lmtato. 5

14 .. D Parabolc Iverse Problem There are may examples of verse problem of detfyg dfferet gredets of the parabolc PDE sch as detfyg kow sorce term fxt [4 5 6] kow thermal coeffcet a [7 8] ad kow capacty term C [9]. Ths revew s teded to lmt to models wth a estmato of cotrol parameter p. Some of them are dscssed below C x a x p x f x t x x. x t T sbected to the gve tal ad bodary codto x x x. t t T g t t t T g t.3 wth overspecfcato at a pot the spatal doma x E t T.4 or tegral overspecfcato over a porto of the spatal doma s x dx E t T s.5 Where C a f g g ϕ ad E are kow fctos whle ad p are kow soltos. The problem..4 ca model certa types of physcal problems where. ca be sed to descrbe a heat trasfer process wth a sorce parameter preset. As a example.4 represets the temperatre measred by a actal sesor at a gve pot x a spatal doma at tme t. Ths the prpose of solvg ths verse problem s to detfy the sorce parameter that wll prodce at each tme t a desred temperatre at that pot [4 ]. 6

15 Cao ad hs co-workers pad a lot of atteto to the mercal treatmet of ths problem. They demostrated the exstece ad qeess of a smooth global solto par p whch depeds cotosly po the data der some certa assmptos []. Cao ad L exteded the vestgato to qaslear parabolc eqatos []. However becase of the restrcto of the method they sed oly local soltos were obtaed. Later they preseted a ew approach to demostrate the exstece of the global solto of. by trasformg the olear eqato. They also stded sg a backward Eler scheme va a trasformato of p to v r to elmate the term p whch led to trasfer the sem-lear PDE to a lear PDE. I addto they vestgato of the covergece of wth the covergece order of ad of p wth the covergece order of / whe [3]. A mercal scheme for a smlar problem whch the pper lmt of the tegrato s s a fcto of tme has bee stded by Cao ad Matheso ad they have also dscssed the covergece as well []. Recetly Dehgha has doe extesve research o verse problem parameter estmatos ad preseted several mercal methods for the verse problems smlar to..5 D ad D problems. He trodced the 3 ad 5 FTCS the 3 BTCS procedre 33 Cradall s formla ad the Salyev's frst ad secod kd formla. Hs algorthms were tested o two problems ad were see to prodce good reslts ad sggest covergece to exact solto whe h goes to zero [6 4 5]. A more geeral ad complex mercal treatmet of the cotrol parameter estmato has bee developed by the same athor [6]. He trodced the θ-method or weghted fte dfferece formla whch was based o the modfed eqvalet PDE as descrbed by Warmg ad Hyett [7]. Methods based o the meshless property of mltqadrc MQ qas-terpolato ad movg least-sqare MLS approxmato are fod to be a alteratve to the tradtoal mesh 7

16 depedet techqes sch as FTCS BTCS Cradall ad Salyev etc. M ad Zog-M proposed the MQ qas-terpolato method for solvg D parabolc eqato wth both pot overspecfed data ad tegral overspecfcato. I ther scheme the spatal dervatves of the eqatos were approxmated by the dervatves of MQ qas-terpolato whle a smple forward dfferece to the depedet varable dervatves. They also trodced a polyomal as a effectve techqe MQ qas-terpolato schemes. Later ther paper t was otced that wth the trodced polyomal some terms of the parabolc eqato dsappeared ad ther roles are represeted mplctly by the polyomal [8]. Cheg preseted a techqe based o the movg MLS approxmato for fdg the solto of problem.-.4. He sed MLS approxmato for dscretzato of both tme ad space varables. Several mercal examples were trodced ths paper showg that the methods are coverget wth good accracy. The athor metoed that meshless methods allow to solve problems wth o-reglar geometry as compared to other mercal methods based o meshes whch the reglarty of the geometry s ecessary [9]... D Parabolc Iverse Problem The D parabolc cotrol parameter verse problem wth oe demsso space x ca be exteded to D verse problem as follows: C x y a x y x y p x y t x y x y t T sbected to the gve tal ad bodary codto f x y.6 x y x y x y.7 8

17 t y t g t t.8 wth a addtoal overspecfcato codto x y E t T t.9 or tegral overspecfcato over a porto of the spatal doma ab x y dxdy E t T a b. Several mercal methods have bee developed to deal wth problems smlar to.6-.9 that may ot have aalytcal soltos or statos drg whch sch soltos become dffclt to obta. The theoretcal dscsso s flly addressed early work by Wag. He vestgated the solvablty of parameter estmato p by trodcg two dfferet o-classcal bodary codtos. He also trodced a par of trasformatos that led to trasfer the sem-lear PDE to a lear PDE order to overcome the dffclty of p estmato ad to elmate the term p whch s mplctly composed r fcto. A trdagoal system was prodced as reslt of sg a fte dfferece approach. Wag metoed that the pot overspecfcato s mch slower tha eergy overspecfcato. Ths s de to the fact that the terato process was qte oscllatory. However the mercal expermets show satsfactory reslts comparso to the exact solto for selected problems [3]. 9

18 Varos mercal methods sch as the Sc-Collocato Method [3] the 5 flly explct scheme the Noye Hayma 5 5 flly mplct method the 3 9 ADI method [3] were appled to compte the cotrol parameter the D verse problem. However each method metoed above has some sort of advatage ad dsadvatage. For stace explct methods are cosdered smple to mplemet bt de to ther stablty reqremet the tme cremet wll be restrcted by the stablty crtera. O other had mplct methods are cosdered stable these techqes se a extesve amot of comptato tme compared to the explct methods for the same selecto of vales s ad h. For more detal o these methods we refer the reader to the above sorces ad the refereces there..3 Applcato of Cotrol Parameter Iverse Problem May physcal statos mght be modeled by.-.3 wth.4 or.5. Problems of these types ca also arse from laser materal treatmets. HÖmberg ad Yamamoto vestgated the cotrollablty o a crve for a sem-lear parabolc eqato of a laser heat treatmet der observed temperatre. Therefore they proposed the heat eqato smlar to.-.4 order to evalate the temperatre ad the laser power p. They also showed that ther theory cofrms the applcato of PID-cotrol to ther expermet ad provded mercal smlatos for a PID cotrol of laser hardeg. Moreover they tested ther approach o a dstral case stdy whch was preseted cofrmg the practcal applcablty of sg verse problem modelg [33]. Frthermore applcato of cotrol parameter verse problem ca also arse the medcal feld. For stace t s mportat to mata a accrate estmato of both the temperatre ad the blood perfso of tsse der vestgato ad ths task cold be

19 performed before or drg a srgcal terveto as well as other termo-reglatory tests. Therefore these types of tests cold dstrb the tsse to be measred ad allow for the possblty of fecto [3]. Prevosly the blood perfso was assmed a costat ad has already bee vestgated for both mercal ad aalytcal aalyses. However the blood perfso coeffcet s the fcto of tme all the regos of the body ad for ths reaso treatg ths physcal pheomea as a verse problem wll lead to more accrate estmatos. Trc et al has vestgated the detfcato of the tme-depedet perfso coeffcet the traset bo-heat codcto eqato. They scceeded developg a geeral mercal method that wold estmate both the temperatre ad the blood perfso for dfferet types of bodary codtos ad measremets [].

20 CHAPTER 3 THE NUMERICAL TECHNIQUE Now cosder the mercal solto of the verse problems..4 for D ad.6.9 for D. The approxmato for the fcto of x ad xy s attempted D ad D respectvely. I addto the tme-depedet fcto p for both problems wll be evalated based o predcto-correcto method that has bee descrbed detal elsewhere [5 6]. The overestmato bodary s ecessary for these problems. Therefore those models wll exame pot overspecfcato bodary codto as a addtoal bodary codto. Ths Chapter provde sghts detals of the algorthm procedre ad applcato of the Crak Ncolso method for solvg ths type of problems. 3. Crak Ncolso Fte Dfferece Method The explct methods are cosdered to be a comptatoally smple ad easer to program bt de to the stablty reqremet the tme cremet wll be restrcted by the stablty crtera. Alteratvely the Crak Ncolso scheme s cosdered mplct ad stable. It ca be prove that the Crak-Ncolso method s stable ad coverges for all vales of [Δt/Δx ]. I addto t has secod order accracy ad ts trcato error s of the order of [Δx + Δ ] [34]. However the Crak Ncolso scheme volves addtoal comptato. If N represets the teral grd pots over a rego the ths method volves the solto of N smltaeos algebrac eqatos for each tme step. The Crak Ncholso scheme has bee chose as the comptatoal scheme ths work de to propertes of the stablty ad the accracy of the scheme. Todays compters are mch powerfl so comptatoal tme wll be mmzed sgfcatly.

21 3. D Iverse Problem wth Pot Overspecfcato The followg secto cosders the mercal solto of the verse problem..4. Sppose the approxmato of x ad p at the th level = are defed the comptato procedre starts by assgg a approprate tal gess to p for the +st level. If the solto satsfes the overspecfed bodary codtos.4 wth a prescrbed tolerace the the correspodg vales of x ad p wll be accepted ad move to the ext tme cremet level. Otherwse p wll be cotosly pdated as a ew gess. Ths the comptato wll be repeated wth the ew gess pt +l l = where l s the terato mber. The workg doma s defed by [ ] [ T]. If by represetg the mber of mesh grd the x drecto as M ad the t drecto as N the step sze wold be h = /M ad k = T/N respectvely. The grd pots x are defed by Applyg the mplct Crak Ncolso scheme to. the followg fte dfferece workg formla reslts: [ k h ] p p f f M N Where The comptatoal molecle of the Crak Ncolso comptatoal molecle for D s gve fgre 3.. 3

22 4 Fgre 3. The Crak Ncolso comptatoal molecle for D. A M- x M- lear system of eqatos s obtaed by rewrtg the resltg system to matrx form. All detals cocerg the dervatos of R ad b to bm- are appeded appedx A. B U AU 3.4 M M M M b b b b R R R R The Predcto-Correctg Mechasm for D Problem Up to ths pot t ca be otced that B vector s a fcto of oly. Ths every term the rght had sde of 3.5 shold be kow. The A matrx has oe term p whch s kow ad that eeds to be gve as a tal gess to start the comptato. Therefore the

23 predcto-correcto mechasm for p s demostrated ths secto. Notce that f x ad p are a solto for..4 the E or xx x p E f x 3.6 E p xx x E f x 3.7 A few assmptos eed to be addressed before precedg to the ext eqato. It was assmed that f = fx t x q s a smooth fcto wth respect to all of ts varables ad f. The compatblty codto s satsfed o Ω x {} by the data ad ϕx = E > [5]. Ths the fte dfferece form of 3.7 ca be rewrtte as p E h E k k k f k 3.8 The p o ad x at = level gve by the tal codto provdes a startg pot for the comptatos. Sbstttg the compatblty codtos to 3.7 reslts p E xx x f x 3.9 x Notce that the step sze s very small. Therefore ths wll lead to to assme that p + s ot far from p. Ths t s reasoable to choose of the tal gess for p + = p = N [6]. The p ad p + ca be sbsttted to 3.5 whch wll make the lear system ready to solve. The above eqatos have a coeffcet matrx that s trdagoal; therefore Thomas algorthm ca be sed for block trdagoal matrces to get the soltos [35]. The reslts that obtaed are + = M- correspodg to p +. If p +l represets the lth gess for p at + level the +l represets the correspodg vales 5

24 obtaed by sg p +l = N- l =. As reslt the 3.8 ca be sed to costrct p +l+ as follows l l l l E h k k k p l E f k The p +l wll be adsted cotosly tl t coverges at a prescrbed tolerace. The the correspodg vales +l = M- ad p +l as + = M- ad p + are accepted. Ths completes comptatos from level to +. The followg chapter presets a mercal example to show how ths algorthm works ad to valdate t wth the exact solto for a selected problem. I addto ths algorthm wll be tested o cyldrcal coordate system by sg expermetal data provded by Bdms [36].The mercal rote based o MATLAB R4a code as oe descrbed the flow chart of fgre 3. ad the pertet detals of each block of the flowchart are provded below. 6

25 Start Ipts: M N T L x Calclate: h k s k BC BC IC f E Ital gess: p = p+ l sg Eq. 3.9 M N T L x h k s k BC BC IC f E Total mber of h x cremet Total mber of k t cremet Total Tme of t Total legth of x Overspecfcato grd pot x cremet t cremet Stablty codto the step sze at gve x Bodary codto at x = Bodary codto at x = Ital codto at t = Heat sorce E fcto Eq..9 Calclate; R b b M- b. See Appx. A For the lear system Eq. 3.5 Update p+m = p+ m+ Solve the lear system Eq. 3.5 sg Thomas algorthm + Calclate; p + l+ sg Eq. 3. pl+-pl h 3 No Yes Save + ad p+ l+ Post the reslts Ed Fgre 3. Flow chart of the mercal rote wrtte MATLAB code for D Cartesa. 7

26 8 3.4 D Iverse Problem wth Pot Overspecfcato The prpose of solvg D verse problem mercally s to recover the sorce parameter that wll prodce a desred temperatre at each tme t at a specfc locato x y. I ths secto the Crak Ncolso s sed for comptg the mercal vales of xy ad p smltaeosly. The procedre that was metoed [3] wll be followed to solve the D model eqato.6.9. The frst step s to dvde the workg doma to [ ] [ T] to M N mesh wth step sze h = Δx = Δy = /M ad tme step sze k = Δt = T/N. The grd pots x y are defed by The drect applcato of Crak Ncolso scheme to D problem.6 leads to the followg fte dfferece workg formla: ] [ h k 4 p p f f M N Where 4 Eqato 3.3 also ca be rewrtte as: b R

27 9 Fgre 3.4 shows the comptatoal molecle of the Crak Ncolso comptatoal molecle for D problem. Fgre 3.3 The Crak Ncolso comptatoal molecle for D. To beg the resltg system of eqatos descrbed by 3.4 ca be wrtte to matrx form. Ths reslts a M- M- lear system of eqatos ad cota kows... M. All detals cocerg the dervatos of R ad b to bm- are appeded appedx A. U B AU M M M M M M M M M M b b b b b b R R R R R R R R R 3.6

28 The lear system of eqatos s dffclt to solve for ths problem. The coeffcets matrx A prodced by 3.5 are sally a large ad sparse matrx wth -M whch caot be hadled easly. However the sccessve over-relaxato SOR method wll be sed to solve the system of eqatos. For more detals abot ths method readers are referred to. 3.5 The Predcto-Correctg Mechasm for D Problem The predctor-corrector method s qte smlar to what have doe for D problem secto 3.3. The predcto-correcto mechasm for p s demostrated ad how to se t wth the lear system 3.6. If xy ad p are a solto for.6.9 the yy xx yy xx E t y x f t y x t y x E t p or t y x f t E t p t y x t y x t E ] [ Aga the assmptos that have bee trodced secto 3.3 hold for ths problem. Ths the fte dfferece form of 3.7 ca be rewrtte as: k l k E f h E p 3.9 If the compatblty codtos sbsttted to 3.7 t reslts ] [ y x t y x f y x y x E p yy xx 3. Both p o ad xy at = level s gve by the tal codto provde a startg pot of the comptatos. For practcal comptato however the step sze s very small. Therefore

29 ths wll lead to assme that p + s close to p. Ths t s reasoable to choose the tal gess for p + = p = N. Sbsttte p ad p + to 3.5 t wll make the lear system ready to solve. The SOR method descrbed earler wll be sed to solve the system of eqatos. The reslts obtaed are + = M- correspodg to p +. If p +l represets the lth gess for p at + level the +l represets the correspodg vales obtaed by sg p +l = N- l =. As reslt the 3.9 ca be sed to costrct p +m+ as follows p E l... h l l E f 3. Where 4 The algorthm wll adst p +m cotosly tl t coverges ad satsfes the prescrbed tolerace. The the correspodg vales +m = M- ad p +m as + = M- ad p + are accepted respectvely. Now the comptaos are completed from level to +. The reslts of the D algorthm are dscssed wth the exact solto for a selected problem the chapter 4. Moreover ths algorthm s tested o a cyldrcal coordates system by sg solto data geerated by the PDE Toolbox R4a. The mercal rote based o MATLAB R4a code as oe descrbed the flow chart of fgre 3. s smmarzes the algorthm. The pertet detals of each block of the flow chart are provded below.

30 Start Ipts: M N T L x y Calclate: h k s BC BC BC3 BC4 IC f E Ital gess: p = p+ l sg Eq. 3. Calclate; R b J. See Appx. A For the lear system Eq. 3.5 M N T L x y h k s k BC BC BC3 BC4 IC f E Total mber of h x cremet Total mber of k t cremet Total Tme of t Total legth of x Overspecfcato grd pot x drecto Overspecfcato grd pot y drecto x cremet t cremet Stablty codto The step sze at gve x Bodary codto at x = Bodary codto at x = Bodary codto at y = Bodary codto at y = Ital codto at t = Heat sorce E fcto Eq..9 Update p+m = p+ m+ Solve the lear system Eq. 3.5 sg SOR method + Calclate; p + l+ sg Eq. 3. pl+-pl h 3 No Yes Save + ad p+ l+ Post the reslts Ed Fgre 3.4 Flow chart of the mercal rote wrtte MATLAB code for D Cartesa.

31 3.6 Applcato of Iverse Problem to Cyldrcal Coordates I ths secto the verse problem for a geeralzed heat eqato cyldrcal coordates s cosdered. The ma goal ths secto s a exteso of the D verse problem model Cartesa coordates secto 3. to D verse problem cyldrcal coordates. Ths ca be doe by adstg the bodary codtos so that the model otpt reslts ca satsfy ad mmc the real expermetal data of a selected cyldrcal problem. Recall that the smplest form of the D heat eqato cyldrcal coordate r s t r r r p r f r r t T 3. r r r sbected to the gve tal ad bodary codto r r r 3.3 t t T g t t t T g t 3.4 wth overspecfcato at a pot the spatal doma r t E t t T D Cyldrcal Coordate Batch Vessel wth Wall coolg [36] Bdms expermet Batch Vessel wth Wall coolg s a good example to valdate the D verse problem cyldrcal coordates. Ths expermet was sed to vestgate the wax deposto that occr der statc coolg codtos. It cossted of a cyldrcal vessel made of copper wth a 4 ch sde dameter ad 6 ches heght. I addto there were two temperatre-reglated baths for matag the temperatres of the wax-solvet mxtre ad the coolat. The hot medm was a wax-solvet mxtre ad the cold srface was the vessel wall. Therefore thermal eergy was radally dspersed otward to the cold vessel wall. I order to 3

32 measre the temperatre hstory drg the expermet 7 pre-calbrated thermocoples were sed sde the cyldrcal vessel at dfferet radal locatos [36]. Fgre 3.5 shows the schematc for the vessel for deposto wth statc coolg. The thermocoples are labeled TC l to TC7 based o ther dstaces from the srface or the vessel wall. I addto the Fractoal radal dstace of the thermocoples are lsted table 3.. Table 3. Radal locato of thermocoples the batch vessel [36]. Fractoal radal dstace Thermocople mber dstaces from vessel wall dstaces from vessel ceter TCl TC.88.8 TC TC TC TC TC7.. Fgre 3.5 Batch vessel for deposto wth cooled vessel wall [36]. 4

33 Data Preparato Start Ipts: M N T L x Load the Bdms s Exp. Data TC-TC7 [C o ] Normalze the data -. Calclate: h k s k BC BC IC f E Ital gess: p = p+ l sg Eq. 3.9 Use scaledata M-fle. Choose TC5 = E as Overspecfcato B.C Fd a fttg fcto for E Use createft_tc5 M-fle Calclate; R b b M- b. See Appx. A For the lear system Eq. 3.5 Solve the lear system Eq. 3.5 sg Thomas algorthm + Calclate; p + l+ sg Eq. 3. Update p+m = p+ m+ pl+-pl h 3 No Yes Save + ad p+ l+ Deormalzed Data TC-TC7 [C o ] Compare to Bdms s Exp. Data Post the reslts Ed Fgre 3.6 Flow chart of the mercal rote wrtte MATLAB code for D cyldrcal. 5

34 3.6. D Cyldrcal Coordates Usg PDE Toolbox The PDE Toolbox s a tool for the stdy ad solto of PDEs two space dmesos ad tme. The PDE toolbox solver ses a algorthm based o the Fte Elemet Method for problems defed o boded domas the plae. I addto PDE toolbox s capable of solvg the drect heat trasfer PDE a cyldrcal system as a fcto of tme. Ths the ma goal of sg the PDE toolbox s to geerate data whch ca be sed wth the D verse problem model. Recall that the heat eqato PDE a cyldrcal coordate system rz s T T T r C kr kr rp t T rq Ω 3.6 t r r z z where ρ C ad k are the desty specfc heat ad thermal codctvty of the materal respectvely T s the temperatre p s the cotrol parameter as fcto of tme oly ad q s the heat geerated the cyldrcal. The PDE toolbox accepts the eqatos a Cartesa system. Ths to solve the parabolc eqato a cyldrcal system the PDE eeds to be expressed a Cartesa form so that PDE Toolbox solver ca recogze. d t c p t f Ω 3.7 The eqato 3.6 ca be coverted to a form that spports the PDE Toolbox after mltplyg by r defg r as y ad z as x. Ths rewrtg 3.6 eqato gves: y t y yp yf Ω 3.8 The ma steps for solvg the drect problem 3.8 sg the PDE toolbox are metoed detal ad appeded appedx A. Fgre 3.7 shows a block dagram of the PDE 6

35 Toolbox code ad smmarze all steps eed to be doe order to geerate the data that later wll be sed to solve the D verse problem. Start The PDE Toolbox PDE Toolbox Code Defe a -D Geometry Defe Bodary Codtos Defe PDE Coeffcets Geerate Mesh Defe Ital Vales & Total Tme Solve the PDE Post Reslts Traglar-Rectaglar Grd Iterpolato Code Save the PDE Toolbox Solto Ed Iterpolate to Rectaglar Grd Solto Choose x o y o o The Rectaglar Grd Solto Pass the E to D Iverse Problem Model Fg. 3.4 Fttg Code Save The Solto at x o y o Save the Best Fttg Model E as Overspecfcato B.C Call the Fttg Toolbox Fgre 3.7 Flow chart of PDE Toolbox rote wrtte MATLAB for D cyldrcal. 7

36 CHAPTER 4 MODEL VALIDATIONS AND DISCUSSIONS I ths Chapter for mercal tests based o the Crak-Ncolso method are provded. A selecto of sample reslts of the mercal expermets wth those models are gve the form of some fgres ad tables. For terested readers the complete reslts data are also provded Appedx B. I ths comptatoal model sch a scheme wold be evalated based o some error crtera. The root mea sqared error RMSE the absolte error dfferece AED the percetage error PE ad the maxmm absolte error MAE are sed for both ad p order to assess the effectveess of each model ad ts ablty to make precse predctos. The RMSE calclated by RMSE I Ua I U 4. the AED calclated by AED Ua U 4. also the PE s defed by PE Ua Ua U 4.3 where Ua s the aalytcal solto at ode ad tme U s the mercal solto at ode ad tme ad I s the mber of er odes ot cldg the bodary odes. The MAE s defed as the maxmm vale of the AED betwee the exact soltos ad mercal soltos at all er odes. 8

37 4. Model Nmercal Test D Cartesa Coordates I ths secto the solto to the D verse problem s solved o the terval Ω = [ ]. The followg example llstrates the reslt obtaed sectos 3. ad 3.3 t xx p [ t ]exp t cos x s x 4.4 Wth bodary codtos g g exp t exp t tal codto x cos x s x ad overspecfed codto E exp t 4.7 The exact solto for x ad p are: x exp t p t cos x s x 4.8 The parabolc solto x s show fgre 4. ad the fgre 4. shows the mercal solto of p to the frst example..5 x t x Fgre 4. Srface plot of the mercal solto x for the frst model. 9

38 Fgre 4. The mercal solto p for the frst model. The sample reslts obtaed for at the fal tme T =. compted for step sze h= k =/ ad s = sg the Crak-Ncolso methods are lsted table 4.. Table. 4. Sample reslts of x for frst model. x Ex. Theo. AED PE As t s llstrated table 4. the RMSE ad MAE dcate the dscrepacy betwee the exact ad mercal vales. The lower the RMSE ad MAE the more accrate the predcto. These reslts show that the D verse problem model s able to prodce a good predcto. The 3

39 far rght colm of table 4. represets the comptatoal process tme CPT tlzed determg the mercal solto. Table 4. The RMSE MAE ad CPT tme for both x ad p RMSE of MAE of RMSE of p MAE of p CPT[secods] Fgre 4.3 dsplays the ADE o the left y-axs ad PE o the rght y-axs for the mercal solto x at the fal tme T. As t s observed the msmatch betwee the exact ad the mercal starts at zero ad the creases for the terval of < x <.4. The absolte dfferece error decreases for x >.4 tl t reach zero aga. Ths s de to the fact that at x = ad x = the same bodary codtos are sed for solvg the exact ad mercal problems. The percetage error for ths test s fod to be wth the rage -.4 to x ADE of PE of x Fgre 4.3 AED ad PE of x at t = T. Fgre 4.4 represets the AED o the left y-axs ad PE o the rght y-axs for the mercal solto p for all vales of x. 3

40 x ADE of p PE of p t Fgre 4.4 ADE ad PE of p at all x. 4. Model Nmercal Test D Cartesa Coordates The D verse problem s solved o the terval Ω = [ ] [ ]. I order to llstrate the reslt obtaed sectos 3.4 ad 3.5 cosder the followg example 5 t xx xx p t 5t exp t s x y 6 4 Wth bodary codtos y g y exp s g y exp s y 4 4 x g x exp s g x exp s x 4 4 tal codto x y s x y 4 ad overspecfed codto E t exp t s. x y.4. The exact solto for x ad p are: x y exp s x y 4 p 5t

41 The algorthm obtaed sectos 3.4 ad 3.5 for the D verse problem s mplemeted. Fgre 4.5 shows the otpt xy prodced by the Crak-Ncolso scheme wth a step sze h= /5 k =/ ad s = 5. The parabolc solto obtaed for at y =.5 ad at fal tme t = T xy x...4 t Fgre 4.5 Srface plot of the mercal solto xy for the secod model. The mercal solto of p s plotted graphcally fgre 4.6. As t s show ths s a lear eqato betwee the p ad tme t whch cofrms the exact solto of p p p t Fgre 4.6 the mercal solto p for the secod model. 33

42 The sample reslts obtaed for at the fal tme T =. compted for a step sze h= /5 k =/ ad s = 5 sg the Crak-Ncolso methods are lsted table 4.3. Table 4.3 Sample reslts of xy at t = T the secod model x y Ex. Theo. AED PE..... NaN Table 4.4 shows the RMSE ad MAE for both xy ad p of the secod model. Althogh the secod model ca be able to prodce a good predcto based o the lower vales of RMSE ad MAE the frst model s more accrate estmatg the mercal solto. Ths s de to the fact that the local trcato error assocated wth the approxmato of D model Δx + Δ s less tha D model Δx + Δy + Δ. The far rght colm of table 4.4 represets the CPT tlzed determg the mercal solto. Table 4.4 The RMSE MAE ad CPT tme for both xy ad p RMSE of MAE of RMSE of p MAE of p CPT [secods]

43 The plots a b ad c fgre 4.7 show AED ad PE for a selected y coordates. As t s show both AED ad PE are zero at the bodares where x = ad x =. Ths s a atral reslt de to se the same bodary codtos both exact ad mercal soltos. However the maxmm absolte error that ca be defed by the peak pot o the ble crve s creased o fgre 4.7 b at y =.5 the decreased o fgre 4.7 c at y =.8. Ths s de to the mercal error occrs most ofte at pots far from the bodary y =.5..5 x x ADE of P E of ADE of PE of x a ADE ad PE for xy. y =. ad t = T.5 x -3 ADE of x b ADE ad PE for xy. y =.5 ad t = T P E of x c ADE ad PE for xy. y =.8 ad t = T Fgre 4.7 The ADE ad PE of xy for the secod model. 35

44 Fgre 4.8 represets the ADE o the left y-axs ad PE o the rght y-axs for the mercal solto p for all vales of x. Notce that the error grows as the tme creases de to accmlato of the error drg the mercal comptatos ADE of p PE of p t Fgre 4.8 The ADE ad PE of p at all x ad y. 4.3 Model Nmercal Test D Cyldrcal Coordates For the prpose of examg the D cyldrcal model valdty the mercal techqes otled secto 3.4 are ow appled to solve a specfc problem of Bdms expermet. Ths problem s solved o the terval Ω = [ ] [ ] for r ad z. The axsymmetrc heat eqato s gve by r r r p r f r r t T 4.4 t r r r sbected to the gve tal ad bodary codto r.6 r 4.5 t TC 7 t T t TC t T

45 wth overspecfed bodary codto at a pot the spatal doma E r TC5 t T 4.7 f r t T 4.8 where TC TC5 ad TC7 are the temperatres measred by the thermocoples Bdms expermet as fcto of tme t. The r s the coordate locato of TC5. Note that the tal codto ad the heat sorce fr are chose after may trals of dfferet vales of those parameters to observe a good matchg betwee the model reslts ad Bdms expermetal data. Fgre 4.9 s the srface plot prodced by the otpt of the Crak-Ncolso scheme wth a step sze h= / k =/ ad s = r t...4 r Fgre 4.9 Srface plot of the mercal solto r for the thrd model. All the solto vales of p obtaed by the predctor-corrector formla trodced secto 3.3 ca be represeted graphcally as below: 37

46 p t Fgre 4. The mercal solto p for the thrd model. Table 4.5 presets sample reslts obtaed throgh the applcato of the proposed algorthm for. The step sze s h= / ad k =/ r ad z drectos respectvely. The Stablty codto s =k/h = 5. For terested readers the reslts data of TC3 TC4 ad TC5 are also provded appedx B. Table 4.5 Sample reslts of r for TC ad TC6 oly. TC TC6 t Exp. Theo. AED PE t Exp. Theo. AED PE

47 Table 4.6 shows the RMSE MAE for r of the D verse cyldrcal model. Note that for ths partclar problem the p fcto was kow the Bdms expermet. Therefore oe goal of solvg ths problem s to fd ths kow cotrol parameter. The D verse cyldrcal model was able to prodce a good predcto based o the lower vales of RMSE ad MAE. The CPT tlzed determg the mercal solto s also provded the far rght colm as show below. Table 4.6 The RMSE MAE ad CPT tme for r. RMSE of MAE of CPT Tme [secods] Fgre 4. ad 4. show the valdato reslts for a selected data of TC ad TC6 respectvely. Both fgres show a excellet agreemet betwee the expermetal ad smlato data. It s also worth to ote that all Bdms data were ormalzed betwee vales ad. The after the model completed the smlato all otpt reslts were deormalzed back. Fgre 4. The mercal solto r for the TC. 39

48 6 g p [ ] Tc6 & T Theo.[C] Exact solto Theoretcal solto t Fgre 4. The mercal solto r for the TC6. Fgre 4.3 represets the ADE ad PE of the mercal solto for TC at r =.8 from the ceter. The model prodced a hgher error at small vales of t. Ths s lkely to be de to the tal gess. However as the tme progresses the error becomes small tl reaches zero at t =. Ths s lkely de to the tal gess s pdated drg the comptato presses ADE.4 PE t Fgre 4.3 ADE ad PE of r for the TC. 4

49 Fgre 4.4 represets the ADE ad PE of the mercal solto for TC6 at r =.5 from the ceter. Ths fgre also shows good reslts for TC6 ad both ADE ad PE become almost steady at terval of. < t >. It was observed both cases TC ad TC6 that the ADE ad PE of the mercal solto are hgher at t =. Ths s de to the fact the bodary codtos for ths partclar problem have bee kept the same wthot ay adstmet bt the tal codto ad the heat sorce were adsted order to prodce a reasoable reslts. The goal was to se the Cartesa problem model to solve the cyldrcal verse problem wth adstg the bodary codto tal codto or the heat sorce parameter order to mmc a acceptable reslts..5 g g 3 ADE PE t Fgre 4.4 ADE ad PE of r for the TC6. 4

50 4.4 Model Nmercal Test D Cyldrcal Coordates For the prpose of llstratg the algorthm obtaed secto 3.6. cosder the followg example where the axsymmetrc the D cyldrcal coordates problem s gve by t r z r r z r r r z r z p r z f r z 4.9 r z t T sbected to the gve tal ad bodary codto r z r z e r z 4. z z t e z t T z r z t e r t e z t T r t T 4. r r e ad heat sorce parameter r t T f r z t e z The overspecfed at a pot the spatal doma E r t T bt dt ae ce z r t T where a b c d are overspecfed bodary codto coeffcets ad they are eqal to ad -5.4 respectvely. Note that the frst step of ths algorthm s to beg wth solvg 4.9 as drect problem sg the PDE toolbox to geerate data. Ths these data ca be sed wth the D cyldrcal coordate model. Fgre 4.5 s the srface plot prodced by the otpt of rz sg the Crak- Ncolso scheme wth a step sze h= / k =/ ad s =. 4

51 rz z r Fgre 4.5 Srface plot of the mercal solto rz for the forth model. The solto vales of p prodced by the predctor-corrector method secto 3. s represeted graphcally fgre p t Fgre 4.6 The mercal solto p for the forth model. The sample reslts obtaed throgh the applcato of the proposed algorthm for at the fal tme T =. s lsted table 4.7. The reslts s compted for a step sze h= / k =/ ad s =. 43

52 Table 4.7 Sample reslts of rz at t = T the forth model r z PDE. Theo. AED PE The RMSE ad MAE for rz are lsted table 4.8. The D model of the cyldrcal coordates prodces a larger error tha that prodced by the D model of the Cartesa coordates. I addto the tme for comptato ad the memory sage are hgher as well. Table 4.8 The RMSE MAE ad CPT tme for rz RMSE of MAE of CPT [secods] Fgre 4.7 ses a selected data at three dfferet r coordates ad the fal tme t =T to show vsally the valdato reslts ad compare the Crak-Ncolso solto wth the PDE toolbox solto. The model prodce qte acceptable reslt at pots ear the bodary where z ca be wth terval of < z <. or.8 < z < bt hgher msmatch at the mddle as show. 44

53 CN at r =. PDE at r =. CN at r =.5 PDE at r =.5 CN at r =.8 PDE at r = z Fgre 4.7 Comparso betwee the Crak-Ncolso ad PDE toolbox soltos. The plots a b ad c fgre 4.8 represet AED ad PE for a selected r coordates. As t s show both AED ad PE are zero at the bodares where z = ad x =. Ths s expected reslt de to se the same bodary codtos both PDE toolbox ad the Crak-Ncolso scheme. However the MAE s creased o fgre 4.8 b at r =.5 ad the creased o Fgre 4.8 c at r =.8 as well. For ths partclar problem both AED ad PE errors domated by creasg the step mber creasg the r drecto. Ths s de to the trcato error trodced by applyg the Crak Ncolso fte dfferece scheme where t s drectly proportoal to the spatal step sze. 45

54 Fgre 4.8 ADE ad PE of rz for the forth model. 46

55 CHAPTER 5 CONCLUSIONS AND FUTURE WORK I ths chapter some cocldg remarks are gve abot the work the problem report. Followg ths there s a dscsso abot some potetal extesos for ths ftre work. 5. Coclsos Ths problem report descrbes mercal methods for verse heat problems ad estmates the tme-depedet heat sorce cotrol parameter for oe- ad two-dmesoal problems both Cartesa ad cyldrcal coordates. For comptatoal models were stded ths work. I the frst two models the mercal reslts were smlated throgh the exact soltos of selected example. I the thrd model the raw expermetal data were employed for valdato of the otcome reslts. The forth model was valdated sg the PDE toolbox geerated data. I all for models the Crak-Ncolso fte dfferece method was appled to solve the verse heat codcto problems ad the predctor-corrector method was sed to recover the tme-depedet heat sorce cotrol parameter p. The frst model was for D verse problem Cartesa coordate wth overspecfed bodary codto where what may appear as a temperatre at a gve pot x the spatal doma at tme t. The mercal reslts for the predcted parabolc solto x ad the tmedepedet heat sorce cotrol parameter p demostrated excellet agreemet betwee the exact solto ad smlato data. The Crak Ncolso scheme s cosdered mplct ad codtoally stable. Therefore the step sze was chose to be a relatvely small of. both x ad t drecto. Ths showed sgfcat mprovemet whe compared wth several mercal tests where a larger step sze was sed. Moreover the approxmato reslt for the tme-depedet heat sorce cotrol parameter p demostrated accrate reslts ad provded a 47