10555 West Flagler Street, Room EC 3474, Miami, Florida 33174, U.S.A. 2 Federal University of Rio de Janeiro, Department of Mechanical Eng.

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1 Iverse Egieerig George S. Dulikravich Helcio R.. Orlade ad ria H. Deis 3 Florida Iteratioal Uiversity Departet of Mechaical & Materials Eg. 555 West Flagler Street Roo EC 3474 Miai Florida 3374 U.S.. Federal Uiversity of Rio de Jaeiro Departet of Mechaical Eg. COPPE Cid. Uiversitaria Cx. Postal 6853 Rio de Jaeiro RJ razil 3 Departet of Mechaical ad erospace Eg. Uiversity of Texas at rligto rligto Texas 787 U.S.. dulikrav@fiu.edu helcio@serv.co.ufrj.br deis@ae.uta.edu Iverse probles are rapidly becoig a ulti-discipliary field with ay practical egieerig applicatios. The objective of this lecture is to preset several such ulti-discipliary cocepts ad applicatios. I soe exaples sophisticated regularizatio forulatios were used. I other exaples differet optiizatio algoriths were used as tools to solve de facto iverse probles. Due to the atheatical coplexity of these ulti-discipliary ad ofte ulti-scale iverse probles the ost widely acceptable forulatios evetually result i a eed for iiizatio of a certai or or a siultaeous extreizatio of several such ors. These sigle-objective ad ulti-objective iiizatio probles are the solved usig appropriate robust evolutioary optiizatio algoriths. Specifically we focus here o iverse probles of deteriig spatial distributio of a heat source for specified theral boudary coditios fidig siultaeously theral ad stress/deforatio boudary coditios o iaccessible boudaries ad deteriig cheical copositios of steel alloys for specified ultiple properties. Itroductio If the iverse proble ivolves the estiatio of oly few ukow paraeters such as the estiatio of a theral coductivity value fro the trasiet teperature easureets i a solid the averagig capabilities provided by the iiizatio of objective fuctios such as ordiary least squares or ca result i stable solutios. Ideed other siilar objective fuctios ca be defied that are based o statistical fudaetals regardig the easureet errors ad the ukow paraeters which are also capable of providig stable solutios for the paraeter estiatio proble []. However if the iverse proble ivolves the estiatio of a large uber of paraeters such as the recovery of the ukow trasiet heat flux copoets f t i fi at ties t i i I excursios ad oscillatios of the solutio ay occur. Therefore the solutio of this type of iverse probles

2 George S. Dulikravich Helcio R.. Orlade ad ria H. Deis kow as fuctio estiatio probles requires special regularizatio stabilizatio techiques. Here we used lifaov s Iterative Regularizatio Method based o the cojugate gradiet ethod of iiizatio as well as o the use of a adjoit proble for the coputatio of the gradiet directio [3-7]. lteratively oe could use boudary regularizatio ethods [8-] with fiite eleets. Cojugate Gradiet Method of Fuctio Estiatio with djoit Proble Forulatio First let us cosider a iverse proble dealig with the estiatio of a distributed heat source ter i a two-regio heat coductio proble. The physical proble of iterest is typical of the aufacturig of Micro-Electro-Mechaical Systes MEMS ad is called Hot Ebossig Microfabricatio Microreplicatio HEMM. Such techique cosists of pressig a old o a substrate uder a prescribed theral history ad ca also be used for the fabricatio of acro-scale parts. I geeral the polyer substrate ad the old are heated to a teperature just above the polyer glass trasitio teperature. The old is the pressed o the polyer util features get replicated ad the both the old ad the substrate are cooled dow to a teperature below the glass trasitio teperature for deebossig [7]. The estiatio of the source ter withi the old will be accoplished by usig the teperature variatio at selected poits withi the substrate. Such a iverse proble ca be aied at the idetificatio of the ukow source ter by usig teperature easureets withi the substrate or alteratively at the desig of the source ter that will result i a prescribed teperature history required for the HEMM techique. The physical proble cosidered i this work cosists of the two-diesioal heat coductio i two cotactig rectagular regios as depicted i Figure. The widths of both regios are a while the thickess of regio is b ad the thickess of regio is c-b. Regios ad are iitially at the uifor teperatures T ad T respectively. For t > the two regios are placed i cotact ad heat is geerated i regio at a voluetric rate gxyt. The cotact coductace at the iterface betwee the two regios h c is supposed to be uifor ad the other boudaries of regios ad are supposed to be isulated. The atheatical odel utilizes the followig diesioless groups T xyt T T θ T xyt T T θ.a-b α t x a a y.c-e a g x y t a G hc a k i k K α Λ k k α T b a where k is the theral coductivity ad α is the theral diffusivity. c C.f-k a

3 Iverse Egieerig 3 y T T x x x x c b g xyt T k y hc TT y b REGION T y y T k h c T T y y b T y y c REGION a T T x x a T x y x y a Fig.. Geoetry coordiates ad boudary coditios. y assuig that the theral properties of regios ad are costat the atheatical forulatio for this proble is give i diesioless for as: x x Regio : θ θ θ + i < < < < for >.a θ θ θ θ at < < for >.b at < < for >.c at < < for >.d i θ θ at < < for >.e θ for i < < < <.f Regio θ θ θ G + + Λ K < < < < C for > 3.a θ θ at < < C for > 3.b at < < C for > 3.c

4 4 George S. Dulikravich Helcio R.. Orlade ad ria H. Deis θ i θ θ K at < < for > 3.d θ at C < < for > 3.e C θ for i < < < < C 3.f θ The proble defied by eqs..a-f ad 3.a-f with kow iitial ad boudary coditios cotact coductace therophysical properties ad source ter costitutes a direct proble that is cocered with the deteriatio of the trasiet teperature fields θ ad θ. For the iverse proble of iterest here the heat source ter G is regarded as ukow while the other quatities appearig i the forulatio of the direct proble are cosidered to be kow. For the solutio of the iverse proble we cosider the available trasiet teperature histories µ at positios locatios..m i regio as well as µ..n at locatios i regio. For the desig iverse proble µ ad µ represet desired teperatures for the fabricatio procedure while for the idetificatio iverse proble µ ad µ represet teperature easureets. The easureets cotai errors which are supposed to be additive ucorrelated orally distributed with kow ad costat stadard deviatio ad zero ea. For the estiatio of the ukow fuctio G we ake o a priori assuptio regardig its fuctioal for except that it belogs to the Hilbert space of square-itegrable fuctios [-6] i the doai < < < < C ad < < f where f is the duratio of the tie iterval of cocer for the iverse aalysis. For the solutio of the preset iverse proble we cosider the iiizatio of the followig fuctioal. M f S[ G ] [ µ θ ; G] d + [ µ θ ; G] d 4 N f For the iiizatio of the objective fuctioal 4 two auxiliary probles are developed aely the sesitivity proble ad the adjoit proble. The sesitivity proble is used to deterie the variatios θ ad θ whe the ukow fuctio is perturbed by G [-6]. The sesitivity proble is derived by substitutig ito the direct proble give by eqs..a-f ad 3.a-f θ by [θ + θ ] θ by [θ + θ ] ad G by [G+ G]. The origial direct proble is the subtracted fro the resultig equatios i order to obtai the followig sesitivity proble: Regio : θ θ θ + i < < < < for > 5.a

5 Iverse Egieerig 5 θ at < < for > 5.b θ at < < for > 5.c θ at < < for > 5.d θ i θ θ at < < for > 5.e θ for i < < < < 5.f Regio : θ θ θ Λ K G + + < < < < C for > 6.a θ at < < C for > 6.b θ at < < C for > 6.c θ i θ θ K at < < for > 6.d θ at C < < for > 6.e C θ for i < < < < C 6.f The adjoit proble is derived by ultiplyig the goverig equatios of the direct proble eqs..a ad 3.a by the Lagrage ultipliers ad respectively. The equatios are the itegrated i the spatial ad tie doais that are valid ad added to the origial fuctioal 4. The directioal derivative of the exteded fuctioal i the directio of the perturbatio of the ukow fuctio is the obtaied ad the resultat expressio after soe legthy

6 6 George S. Dulikravich Helcio R.. Orlade ad ria H. Deis but straightforward aipulatios is allowed to go to zero [-6]. The followig adjoit proble for the deteriatio of the Lagrage ultipliers ad results i Regio : + + M ] [ δ δ µ θ i < < < < < < f 7.a at < < < < f 7.b at < < < < f 7.c at < < < < f 7.d K i at < < < < f 7.e f for f i < < < < 7.f Regio : + + N ] [ δ δ µ θ i < < < < C < < f 8.a at < < C < < f 8.b at < < C < < f 8.c K i at < < < < f 8.d

7 Iverse Egieerig 7 C at C < < < < f 8.e f for f i < < < < C 8.f y applyig the liitig process used to obtai the adjoit proble the directioal derivative of the objective fuctioal alog the directio of the perturbatio G reduces to d d d G K G S C f ] [ 9.a We ow ivoke the hypothesis that the ukow fuctio belogs to the Hilbert space of square itegrable fuctios i the doai < < < < C ad < < f so that we ca write such directioal derivative as d d d G G S G S C f ] [ ] [ 9.b y coparig eqs. 9.a ad 9.b we obtai the gradiet directio as K G S ] [ The iterative procedure of the cojugate gradiet ethod as applied to the estiatio of the fuctio G is give by k k k k G G d β +.a where the superscript k deotes the uber of iteratios ad k is the search step size. The directio of descet d k is obtaied as a liear cobiatio of the gradiet directio at iteratio k with directios of descet at previous iteratios. It is give as [-6] ] [ γ d G S d k k k k +.b Differet expressios are available i the literature for the cojugatio coefficiet γ k. I this work we use the so-called Fletcher-Reeves versio of the cojugate gradiet ethod where the cojugatio coefficiet is give as [-6] ]} [ { ]} [ { k C k C k d d d G S d d d G S f f γ k 3 with γ.c The search step size is obtaied by iiizig the objective fuctioal with respect to k at each iteratio [-6]. The followig expressio results for the preset iverse proble

8 8 George S. Dulikravich Helcio R.. Orlade ad ria H. Deis k β M f [ θ ; G µ ] θ M f [ θ k ; d d + k ; d ] d + N f N f k [ θ ; G µ ] θ ; d d k [ θ ; d ] d.d where θ ;d k ad θ ;d k are the solutios of the sesitivity proble give by eqs. 5.a-f ad 6.a-f obtaied by settig G d k. fter developig the expressios for the search step size ad for the gradiet directio the iterative procedure of the cojugate gradiet ethod give by eqs..a-d ca be applied util a suitable covergece criterio is satisfied. The iiizatio of the objective fuctioal with the cojugate gradiet ethod ad adjoit proble forulatio ca be suitably arraged i a coputatioal algorith which is oitted here for the sake of brevity. The reader should cosult refereces [-6] for further details. I this paper the iterative procedure was stopped whe the objective fuctioal give by equatio 4 becae sufficietly sall. The discrepacy priciple [-6] was used to specify the tolerace for the stoppig criterio if teperature easureets cotaiig rado errors were used for the iverse idetificatio of the source ter. For the iverse desig proble the tolerace was specified as a sall uber. We cosider here the iverse proble of desigig the heat source ter i the old so that the teperature at the old/substrate iterface follows a desired history iposed by the aufacturig process. We assued the substrate ad the old to have the followig properties [7]: k.9 W - K - α.44 x -7 s - k 4.8 W - K - ad α 9.56 x -6 s - that is K ad Λ oth regios ad were iitially at the sae teperature T T 7 o C. The old ad substrate diesios Figure were a.75 b. b-c.65 ad the duratio of the hot ebossig process was take as 4 s that is.3 C ad f.8. Uifor iterface teperature ad perfect cotact betwee regios ad were assued for the results preseted here. Figure presets the desired teperature variatio at the iterface betwee regios ad as well as the calculated teperatures obtaied with the heat source ter that resulted fro the solutio of the preset iverse desig proble. The agreeet betwee desired ad calculated teperatures is perfect withi the graph scale despite the fact that the desired teperature variatio cotais sharp variatios. Figures 3.a-d preset the resultat heat source ter at positios ad. respectively. Each of these figures shows the results for...4 ad.98. We ote i figures 3.a-d that the source ter foud fro the desig procedure does ot vary alog the directio for such test case. I additio periods of heatig positive source ad coolig egative source ca be clearly idetified as required for the sharp icrease ad decrease i teperature at ties ad. respectively see Figure. The obtaied results reveal that the cojugate gradiet ethod of fuctio estiatio is capable of providig sooth ad stable solutios for the source ter. We ote that the preset iverse desig aalysis assues that the source ter is cotiuous withi the old but for practical ipleetatio discrete heat sources ad coolig chaels

9 Iverse Egieerig 9 will be required. The results obtaied with the cojugate gradiet ethod of fuctio estiatio perit the selectio of the uber ad positio of the heat sources ad coolig chaels as well as the heatig power coolig fluid teperature ad covective heat trasfer coefficiet which will result i the desired iterface teperature variatio desired exata calculated calculada.3.5 θ Fig.. Copariso betwee desired ad calculated iterface teperatures G 5 G a.. b G 5 G c d Fig. 3. Iverse solutio at: a.3 b.34 c.67 d.

10 George S. Dulikravich Helcio R.. Orlade ad ria H. Deis 3 Fiite Eleet Forulatio for Fidig Ukow oudary Coditios i 3-D Steady Theroelasticity It is ofte difficult or eve ipossible to place teperature probes heat flux probes or strai gauges o certai parts of a surface of a solid body. This ca be due to its sall size geoetric iaccessibility or exposure to a hostile eviroet. For iverse probles the ukow boudary coditios o parts of the boudary ca be deteried by providig overspecified boudary coditios eforcig both Dirichlet ad Neua type boudary coditios o at least soe of the reaiig portios of the boudary ad providig either Dirichlet or Neua type boudary coditios o the rest of the boudary [-4]. It is possible after a series of algebraic aipulatios to trasfor the origial syste of equatios ito a syste which eforces the overspecified boudary coditios ad icludes the ukow boudary coditios as a part of the ukow solutio vector. This forulatio is a adaptatio of a ethod used by Marti ad Dulikravich [56] for the iverse detectio of boudary coditios i steady heat coductio. Specifically this work represets a extesio of the work preseted by the authors [89]. I the case of steady cobied theral ad elastic probles the objective of the iverse proble is to deterie displaceets surface stresses heat fluxes ad teperatures o boudaries where they are ukow. The proble of iverse deteriatio of ukow boudary coditios i two-diesioal steady heat coductio has bee solved by a variety of ethods [8-5]. Siilarly a separate iverse boudary coditio deteriatio proble i liear elastostatics has bee solved by differet ethods [6]. The iverse boudary coditio deteriatio proble for cobied steady theroelasticity was also solved for several two-diesioal ad three-diesioal probles [8 ]. 3-D fiite eleet forulatio is preseted here that allows oe to solve this iverse proble i a direct aer by over-specifyig boudary coditios o boudaries where that iforatio is available. Our objective is to develop ad deostrate a approach for the predictio of theral ad elastic boudary coditios o parts of a threediesioal solid body surface by usig a fiite eleet approach FE. It should be poited out that the ethod for the solutio of iverse probles to be discussed here is differet fro the approach based o boudary eleet ethod that has bee used separately i liear heat coductio [4] ad liear elasticity [6]. 3. FEM forulatio for theroelasticity The Navier equatios for liear static deforatios u v w i three-diesioal Cartesia x y z coordiates are u v w x xy xz + G G u +

11 Iverse Egieerig u v w + G G v + 3 xy y yz u v w + G G w + Z 4 xz yz z where Eν E G 5 + ν ν +ν Here Z are body forces per uit volue due to stresses fro theral expasio v is the Poisso s ratio G is the shear odulus E is the odulus of elasticity ad is Lae s costat. α θ 3 + G 6 x α θ 3 + G 7 y α θ Z 3 + G 8 z Here α is the coefficiet of three-diesioal theral expasio ad θ is the teperature. This syste of differetial equatios is discretized usig the typical Galerki fiite eleet approach [8 9]. The approach leads to a local stiffess e atrix [ K ] ad a force per uit volue vector { f e } which are deteried for each eleet i the doai ad the assebled ito the global syste [ K ]{} { F} δ 9 fter applyig boudary coditios the global displaceets are foud by solv- σ ca the be foud ig this syste of liear algebraic equatios. The stresses { } i by differetiatig the displaceets { δ }. 3. FEM forulatio for the theral proble The teperature distributio throughout the doai ca be foud by solvig Poisso s equatio for steady liear heat coductio with a distributed steady heat source fuctio Q ad theral coductivity coefficiet k. k θ θ θ + + Q x y z

12 George S. Dulikravich Helcio R.. Orlade ad ria H. Deis pplyig the Galerki fiite eleet ethod over a eleet results i the local e stiffess atrix [ K ] ad heat flux vector { Q e } c which deteried for each eleet i the doai ad the assebled ito the global syste [ K c ]{} { Q} θ 3.3 Direct ad iverse forulatios The above equatios for steady theroelasticity were discretized by usig a Galerki s fiite eleet ethod. This results i two liear systes of algebraic equatios [ K ]{} δ { F} [ K c ]{} { Q} θ The syste is typically large sparse syetric ad positive defiite. Oce the global syste has bee fored the boudary coditios are applied. For a wellposed aalysis direct proble the boudary coditios ust be kow o all boudaries of the doai. For heat coductio either the teperature θ s or the heat flux Q s ust be specified at each poit of the boudary. For a iverse proble the ukow boudary coditios o parts of the boudary ca be deteried by over-specifyig the boudary coditios eforcig both Dirichlet ad Neua type boudary coditios o at least soe of the reaiig portios of the boudary ad providig either Dirichlet or Neua type boudary coditios o the rest of the boudary. It is possible after a series of algebraic aipulatios to trasfor the origial syste of equatios ito a syste which eforces the over-specified boudary coditios ad icludes the ukow boudary coditios as a part of the ukow solutio vector. For exaple cosider the liear syste for heat coductio o a tetrahedral fiite eleet with boudary coditios give at odes ad 4. K K K3 K4 K K K3 K4 K3 K3 K33 K43 K4 θ Q K4 θ Q K34 θ 3 Q3 K 44 θ4 Q4 3 s a exaple of a iverse proble oe could specify both the teperature θ s ad the heat flux Q s at ode flux oly at odes ad 3 ad assue the boudary coditios at ode 4 as beig ukow. The origial syste of equatios 3 ca be odified by addig a row ad a colu correspodig to the additioal equatio for the over-specified flux at ode ad the additioal ukow due to the ukow boudary flux at ode 4. The result is

13 Iverse Egieerig 3 K K3 K4 K K K3 K4 K K3 K33 K43 K3 θ Qs K 4 θ Q K34 θ 3 Q3 K 44 - θ 4 K 4 Q4 Qs The resultig systes of equatios will reai sparse but will be o-syetric ad possibly rectagular istead of square depedig o the ratio of the uber of kow to ukow boudary coditios. 3.4 Regularizatio strategies regularizatio ethod was applied to the solutio of the systes of equatios i attepts to icrease the ethod s tolerace for easureet errors i the overspecified boudary coditios. Differet regularizatio ethods for the 3-D forulatio were give previously []. Here we cosider the regularizatio of the iverse heat coductio proble. The geeral for of a regularized syste is give as [3] K c ΛD {} θ Q The traditioal Tikhoov regularizatio is obtaied whe the dapig atrix [ D ] is set equal to the idetity atrix. Solvig 5 i a least squares sese iiizes the followig error fuctio. θ [ K ]{} θ { Q} Λ[ D]{} θ 4 5 error c + 6 This is the iiizatio of the residual plus a pealty ter. The for of the dapig atrix deteries what pealty is used ad the dapig paraeter Λ weights the pealty for each equatio. These weights should be deteried accordig to the error associated with the respective equatio. The regularizatio ethod uses Laplacia soothig of the ukow teperatures ad displaceets oly o the boudaries where the boudary coditios are ukow. This ethod could be cosidered a secod order Tikhoov ethod. pealty ter ca be costructed such that curvature of the solutio o the uspecified boudary is iiized alog with the residual. θ ub i 7 For probles that ivolve ukow vector fields such as displaceets Eq. 7 ust be odified to the followig.

14 4 George S. Dulikravich Helcio R.. Orlade ad ria H. Deis ˆ { ub} i δ 8 Here the oral copoet of the vector displaceet field {} δ is iiized alog the ukow surface. Eqs. 7 ad 8 ca be discretized usig the ethod of weighted residuals to deterie the dapig atrix [ D ]. [ D] [ ] ub K c θub θ 9 I three-diesioal probles [Kc] is coputed by itegratig over surface eleets o the ukow boudaries. So the dapig atrix ca be thought of as a assebly of boudary eleets that ake up the boudary of the object where the boudary coditios are ukow. The stiffess atrix for each boudary eleet is fored by usig a Galerki weighted residual ethod that esures the Laplacia of the solutio is iiized over the ukow boudary surface. The ai advatage of this ethod is its ability to sooth the solutio vector without ecessarily drivig the copoets to zero ad away fro the true solutio. I geeral the resultig FEM systes for iverse theroelastic probles are sparse usyetric ad ofte rectagular. These properties ake the process of fidig a solutio to the syste very challegig. Oe possible approach is to use iterative ethods suitable for least squares probles. Oe such ethod is the LSQR ethod which is a extesio of the well kow cojugate gradiet CG ethod [3]. The LSQR ethod ad other siilar ethods such as the cojugate gradiet for least squares CGLS solve the oralized syste but without explicit coputatio of [ K ] T [ K ]. These ethods eed oly atrix-vector products at each iteratio ad therefore oly require the storage of [ K ] so they are attractive for large sized odels. However covergece rates of these ethods deped strogly o the coditio uber of the oralized syste that is the coditio uber of [ K ] squared [3]. Therefore solver perforace degrades sigificatly as the size of the fiite eleet odel icreases. Covergece ca be slow whe solvig the systes resultig fro the iverse fiite eleet discretizatio sice they are aturally ill-coditioed probles. 3.5 Nuerical results The accuracy ad efficiecy of the fiite eleet iverse forulatio was tested o several siple three-diesioal probles. The ethod was ipleeted i a object-orieted fiite eleet code writte i C++. Eleets used i the calculatios were of hexahedral shape with tri-liear iterpolatio fuctios. The liear systes were solved with a sparse LSQR ethod [3] with colu scalig. siple test geoetry was chose to clearly deostrate the approach. The test case ivolved a ultiply-coected doai coposed of a outer cylider with legth 5. ad diaeter of.. There are four cylidrical coaxial holes that pass copletely through the cylider each with a diaeter of.5. The

15 Iverse Egieerig 5 aterial doai was discretized usig hexahedral coputatioal esh coposed of 44 eleets ad 98 odes. The ier ad outer boudaries each had 44 odes. For this ultiply-coected geoetry there is o aalytical solutio eve if costat teperature boudary coditios are used o the boudaries. This case cosiders theral ad elastic boudary coditios that vary i all coordiate directios thereby creatig a truly three-diesioal exaple. The all iterior boudary coditios chage liearly alog the z-axis. The exact values used are give i Table. O the outer cylider the displaceet was set to zero ad a fixed teperature of C was specified. diabatic ad stress free coditios were specified at the eds of the cylider. The followig aterial properties were used: E. Pa ν. α. K k.w K. Forward Iverse Forward Iverse E- 5.33E-.73E E- 8.E- 5.33E-.7E- 8.E- 8.E- 8.E- 8.E- 4.E-.33E Fig. 4. Iversely coputed isothers o x - y plae at z Fig. 5. Iversely coputed displaceet agitudes o x - y plae at z.5. Forward Iverse Forward Iverse E- 3.E- 9.3E- 3.E E-.4E-.4E- 6.E-.35E Fig. 6. Iversely coputed isothers o x - y plae at z Fig. 7. Iversely coputed displaceet agitudes o x - y plae at z.5. The iverse proble was geerated by over-specifyig the outer cylidrical boudary with the double-precisio values of teperatures fluxes displaceets ad surface tractios obtaied fro the forward aalysis case. t the sae tie o boudary coditios were specified o the ier cylidrical boudaries. No errors were used i the over-specified boudary data.

16 6 George S. Dulikravich Helcio R.. Orlade ad ria H. Deis Table. Teperature ad pressure boudary coditios for iterior surfaces. Hole T z C T z 5 C P z Pa P z 5 Pa C D Regularizatio was used with Λ Our experiece idicates that a good value for the dapig paraeter Λ is geoetry ad boudary coditio depedet. Curretly the dapig paraeter is chose based o experiece by first choosig a sall value ad gradually icreasig it util the uerical oscillatios i the ukow boudary solutio are reoved. The liear elasticity syste was solved usig the LSQR ethod with colu scalig. The LSQR iteratios were teriated after the Euclidea or of the residual of the oral syste was less tha.. I this exaple 685 LSQR 6 iteratios were required which cosued about iutes of coputig tie o a Petiu 4 workstatio. The average error betwee the iverse ad direct solutios o the ukow boudaries was.% for teperature ad 5.6% for displaceet. The direct ad iverse teperature cotours for two sectios of the doai are show i Figures 4 ad 6. There is good agreeet o all sectios betwee the directly ad iversely obtaied isothers. The direct ad iverse costat displaceet agitude cotours for three sectios of the doai are show i Figures 5 ad 7. For all sectios there is a oticeable differece i the direct ad iverse cotours i the regios far away fro the outer boudary. However the iverse solutio does correctly capture the direct solutio i a qualitative sese. This theroelastic proble was also solved usig the other regularizatio ethods over a wide rage of dapig paraeters. I those cases the error i the iverse solutio was uch higher ad did ot atch the direct solutio eve i a qualitative sese. Iprovig the quality of the dapig atrix for the displaceet field could icrease the accuracy of the displaceet. The curret dapig atrix fro Eq. 8 oly icludes the oral copoet of the displaceet. Soothig the tagetial copoets as well could ake further iproveets. I additio the curret schee depeds o accurate surface uit oral vectors ˆ which are difficult to copute accurately at the odes of flat eleets o curved surfaces. Further reductios i errors could possibly be ade by ipleetig ethods that copute the surface orals with a high degree of accuracy. Reasoable results were obtaied by LSQR with colu scalig i less tha iteratios for displaceets ad 3 iteratios for teperature. lthough ay iteratios are required with the LSQR ethod it requires uch less eory ad is ore robust tha sparse direct solvers such as QR factorizatio for rectagular systes of algebraic equatios.

17 Iverse Egieerig 7 This forulatio ca predict the teperatures ad displaceets o the ukow boudary with high accuracy usig a proper regularizatio. It was show that the FEM forulatio could accurately predict ukow boudary coditios for ultiply-coected doais whe a good regularizatio schee is used. Further research is eeded to develop better regularizatio ethods so that the preset forulatio ca be ade ore robust with respect to easureet errors ad ore coplex geoetries. Further research is also eeded to iprove regularizatio for iverse probles i elasticity over coplicated doais. 4 Iverse Desig of lloys for Specified Properties Our research recetly cocetrated o the iverse ethod i predictig cheical copositio of steel alloys. This forulatio allows a structural desig egieer who desiged a achie part to ask a aterials scietist to provide a precise cheical copositio of a alloy that will sustai a specified stress level σ spec at a specified teperature T spec ad last util rupture for a specified legth of tie θ spec. The iverse proble ca be the forulated as a ulti-objective optiizatio proble where a uber of objectives are siultaeously iiized Table. We have used IOSO ulti-objective optiizatio algorith [33 34] which cosists of two stages. The first stage is the creatio of a approxiatio of the objective fuctios. Each iteratio i this stage represets a decopositio of the iitial approxiatio fuctio ito a set of siple approxiatio fuctios Fig. so that the fial respose fuctio is a ulti-level graph [35 36]. The secod stage is the optiizatio of this approxiatio fuctio. This approach allows for corrective updates of the structure ad the paraeters of the respose surface approxiatio. The distictive feature of this approach is a extreely low uber of trial poits to iitialize the algorith. The result is ot oe but a uber of alloys Pareto frot poits each of which satisfyig the specified properties while havig differet cocetratios of each of the alloyig eleets. This is very practical because the user whe decidig to order a alloy ca use the alloy which is ade of the ost readily available ad the ost iexpesive eleets o the arket at that poit i tie. I particular the objective was to deterie cheical copositios of high teperature steel alloys that will have specified desired physical properties. Desig variables were cocetratios percetages of each of the followig 4 alloyig eleets C S P Cr Ni M Si Mo Co Cb W S Z Ti. Table. Forulatios for te siultaeous objective fuctios i steel alloy desig. Nuber of objectives Operatig Stress σ spec σ Objectives iiize siultaeously Operatig Tie util Low cost alloy teperature rupture iiize T T spec θ θ Ni Cr Nb spec Co Cb W Ti

18 8 George S. Dulikravich Helcio R.. Orlade ad ria H. Deis 5 4 Multicriteria optiizatio of aterial copositio for preset properties iverse proble usig ethod #3 Nuber of variables alloyig eleets: 4. Objectives: Cr ad Ni cocetratio. Costraits: stress4 psi; teperature8 F; tiepreset tie. Ni % 3 tie8 tie7 tie6 tie Cr % This approach allows us to vary the cheical copositio for the sae pre-specified ultiple properties of the alloy. Fig. 8. Iversely desiged Pareto optial cocetratios of Ni ad Cr as a fuctio of tie util-rupture costrait. Fig. 9. llowable variatios of cocetratios of S i ad C o with respect to Cr whe specifyig stress 3 N teperature 975 C ad tie-to-rupture 5 h. Evaluatios of the objectives were perfored usig classical experiets o cadidate alloys. I other words we used a existig experietal database. The results Fig. 8 deostrate that it is possible to create a large uber of alloys havig differet copositios so that each of the will satisfy the specified ultiple properties. This does ot ea that cocetratios vary soothly for these alloys Fig. 9. It should be poited out that these are the visualizatios of oly two of the 4 cheical eleets sice the results of this ultiple siultaeous leastsquare costraied iiizatio proble caot be visualized for ore that two alloyig species at a tie. lso whe T spec ad θ spec progressively icrease the feasible rages for varyig the cocetratios reduce rapidly [35].

19 Iverse Egieerig 9 ckowledgeets The authors are grateful for the partial support provided by a grat fro the US ry Research Office oitored by Dr. Willia M. Mullis ad by a grat fro the US ir Force Office of Scietific Research oitored by Dr. Todd E. Cobs. Refereces. eck JV rold KJ 977 Paraeter Estiatio i Egieerig ad Sciece Wiley Itersciece New ork N US. Kaipio J Soersalo E 4 Statistical ad Coputatioal Iverse Probles. pplied Matheatical Scieces Spriger-Verlag lifaov OM 977 Deteriatio of Heat Loads fro a Solutio of the Noliear Iverse Proble. High Teperature 53: lifaov OM 994 Iverse Heat Trasfer Probles Spriger-Verlag New ork 5. lifaov OM 974 Solutio of a Iverse Proble of Heat-Coductio by Iterative Methods. J. Eg. Physics 64: rtyukhi E Nearokoov V 988 Coefficiet Iverse Heat Coductio Proble. J. Eg. Physics 53: lifaov OM Mikhailov VV 983 Deteriig Theral Loads fro the Data of Teperature Measureets i a Solid. High Teperature 5: lifaov OM Ruyatsev SV 979 O the Stability of Iterative Methods for the Solutio of Liear Ill-Posed Probles. Soviet Math. Dokl. 5: lifaov OM Ruyatsev SV 98 Regularizig Gradiet lgoriths for Iverse Theral-Coductio Probles. J. Eg. Physics 39: lifaov OM Tryai P 985 Deteriatio of the Coefficiet of Iteral Heat Exchage ad the Effective Theral Coductivity of a Porous Solid o the asis of a Nostatioary Experiet. J. Eg. Physics 483: lifaov OM rtyukhi E Ruyatsev 995 Extree Methods for Solvig Ill- Posed Probles with pplicatios to Iverse Heat Trasfer Probles egell House New ork N US. rtyukhi E 98 Recostructio of the Theral Coductivity Coefficiet fro the Solutio of the Noliear Iverse Proble. J. Eg. Physics 44: rtyukhi E 993 Iterative lgoriths for Estiatig Teperature-Depedet Therophysical Characteristics. I Proc. of st Iteratioal Coferece o Iverse Probles i Egieerig Proceedigs Pal Coast FL US: rtyukhi E Ruyatsev SV 98 Descet Steps i Gradiet Methods of Solutio of Iverse Heat Coductio Probles. J. Eg. Physics 39: lifaov OM Matheatical ad Experietal Siulatio i Desigig ad Testig Heat-Loaded Egierig Objects I Proc. of the Iverse Probles i Egieerig: Theory ad Practice Vol. I Orlade H. R.. Editor e-papers Publishig House Rio de Jaeiro razil: 3-6. Özisik MN Orlade HR Iverse Heat Trasfer: Fudaetals ad pplicatios Taylor & Fracis New ork N US 7. Silva PMP Orlade HR Colaço MJ Shiakolas PS Dulikravich GS 5 Estiatio of Spatially ad Tie Depedet Source Ter i a Two-Regio Proble. I Proceedigs of the 5th Iteratioal Coferece o Iverse Probles i Egieerig: Theory ad Practic Lesic D. ed. July -5 5 Cabridge Uited Kigdo

20 George S. Dulikravich Helcio R.. Orlade ad ria H. Deis 8. Deis H Dulikravich GS 999 Siultaeous Deteriatio of Teperatures Heat Fluxes Deforatios ad Tractios o Iaccessible oudaries. SME Joural of Heat Trasfer : Deis H Dulikravich GS 3-D Fiite Eleet Forulatio for the Deteriatio of Ukow oudary Coditios i Heat Coductio. I Proc. of Iterat. Syp. o Iverse Probles i Eg. Mechaics Taaka M Ed. Nagao City Japa. Deis H Dulikravich GS Ha Z 4 Deteriatio of Teperatures ad Heat Fluxes o Surfaces ad Iterfaces of Multi-doai Three-Diesioal Electroic Copoets SME Joural of Electroic Packagig 6 4: Deis H Dulikravich GS oshiura S 4 Fiite Eleet Forulatio for the Deteriatio of Ukow oudary Coditios for 3-D Steady Theroelastic Probles. SME Joural of Heat Trasfer 6: 8. Larse ME 985 Iverse Proble: Heat Flux ad Teperature Predictio for a High Heat Flux Experiet. Tech. report SND Sadia Natioal Laboratories lbuquerque NM US 3. Hesel EH Hills R 989 Steady-State Two-Diesioal Iverse Heat Coductio. Nuerical Heat Trasfer 5: Marti TJ Dulikravich GS 996 Iverse Deteriatio of oudary Coditios i Steady Heat Coductio with Heat Geeratio. SME J. of Heat Trasfer 3: Olso LG Throe RD The Steady Iverse Heat Coductio Proble: Copariso for Methods of Iverse Paraeter Selectio. I Proc. of the 34th SME Natioal Heat Trasfer Coferece-NHTC NHTC- Pittsburg P US 6. Marti TJ Haldera J Dulikravich GS 995 Iverse Method for Fidig Ukow Surface Tractios ad Deforatios i Elastostatics. Coputers ad Structures 56: Marti TJ Dulikravich GS 995 Fidig Teperatures ad Heat Fluxes o Iaccessible Surfaces i 3-D Coated Rocket Nozzle. I 995 JNNF No-Destructive Evaluatio Propulsio Subcoittee Meetig Tapa FL US: Hueber KH Thorto E yro TG 995 The Fiite Eleet Method for Egieers. Joh Wiley ad Sos third editio New ork N US 9. Hughes TJR The Fiite Eleet Method: Liear Static ad Dyaic Fiite Eleet alysis. Dover Publicatios Ic. New ork N US 3. Neuaier 998 Solvig Ill-Coditioed ad Sigular Liear Systes: Tutorial o Regularizatio. SIM Review 4: Paige CC Sauders M 98 LSQR: lgorith for Sparse Liear Equatios ad Sparse Least Squares. CM Trasactios o Matheatical Software 8: Saad 996 Iterative Methods for Sparse Liear Systes. PWS Publishig Co. osto M US 33. Egorov IN Kretii GV 99 Multicriterio stochastic optiizatio of axial copressor. I Proc. of SME COGEN-TURO-VI Housto T US 34. Egorov IN Kretii GV Leshcheko I Kostiuk SS 999 The ethodology of stochastic optiizatio of paraeters ad cotrol laws for the aircraft gas-turbie egies flow passage copoets. SME paper 99-GT-7 Jue 3-7 Idiaapolis IN US 35. Egorov IN Dulikravich GS 4 Iverse desig of alloys for specified stress teperature ad tie-to-rupture by usig stochastic optiizatio. I Proc. of the Iteratioal Syposiu o Iverse Probles Desig ad Optiizatio IPDO; Colaco MJ Dulikravich GS Orlade HR eds.; March 7-9 Rio de Jaeiro razil 36. egorov-egorov IN Dulikravich GS 5 Cheical copositio desig of superalloys for axiu stress teperature ad tie-to-rupture usig self-adaptig respose surface optiizatio. Materials ad Maufacturig Processes 3: