Boundary-Condition-Independent Reduced-Order Modeling of Complex 2D Objects by POD-Galerkin Methodology

Transcription

1 Boundary-Condton-Independent Reduced-Order Modelng of Complex 2D Objects by POD-Galerkn Methodology Arun Prakash Raghupathy, Urmla Gha, Karman Gha Computatonal Flud Dynamcs Research Laboratory Unversty of Cncnnat, Cncnnat, OH 4522 Wllam Maltz Electronc Coolng Solutons Inc Mountan Vew, CA 9534 Abstract he objectve of the current work s to ntroduce the concept of boundary-condton-ndependent (BCI) reducedorder modelng (ROM) for complex electronc packages by the POD-Galerkn methodology. Detaled models of complex electronc packages are used wthn system-level models n Computatonal Flud Dynamcs (CFD)-based heat transfer analyss. At tmes, these package-level models are complcated, and ther smulaton tends to consume large amounts of computatonal resources. hs problem s compounded further f multple nstances of these models are used wthn the system. If a package-level model that reduces computatonal resources (reduced-order model), and provdes results n many dfferent flow stuatons (boundary-condtonndependent model) can be deployed, t wll accelerate the desgn and analyss of the end products that make use of these components. hs work focuses on how the Proper Orthogonal Decomposton (POD)-Galerkn methodology can be used wth the Fnte Volume (FV) method to generate reducedorder models that are boundary-condton-ndependent. hs method s successfully used n the present study to generate boundary-condton-ndependent reduced-order models for D and 2D objects. Successful mplementaton of the method s also shown on 2D objects made of multple materals and multple heat generatng sources. Also, the fnal BCI ROM n each case s found to work extremely well (errors less than %) even for boundary condtons outsde the range n whch t was generated, makng t a truly boundary-condtonndependent model. Keywords POD, Galerkn, Compact hermal Model, Boundary- Condton Independent Models. Introducton Network-based Compact hermal Models (CMs) were ntroduced by the DELPHI (DEvelopment of Lbrares of PHyscal models for an Integrated desgn envronment) project [, 2] for accurate package characterzaton and to replace computatonally ntensve detaled CFD package-level models. hs project created a revoluton n the feld of model development for thermal analyss of electronc packages. o overcome the lmtatons of the DELPHI method, such as XXXX-X/06/$ IEEE nablty n modelng packages wth multple heat sources and nablty n predctng tangental gradents n temperature, alternate approaches [3, 4, 5] have been proposed. hese approaches cannot be drectly nterfaced wth CFD software that s currently used n the electroncs coolng ndustry. he present work addresses ths ssue by demonstratng the use of the POD-Galerkn methodology for developng BCI ROMs. Shapro [6] suggested the applcaton of POD to develop reduced-order models of complex electronc systems. He has employed a reduce-and-nterconnect n hs work [6, 7], where reduced-order models for components are generated ndependently, and then connected usng a frequency-response approach. Further applcaton of POD-Galerkn approach n the area of electroncs coolng can also be found n the work of Rambo [8]. However, a revew of the lterature shows lmted applcaton of POD along wth commercal software for heat transfer analyss. On ths front, most of the work [9, 0] nvolves ntegraton wth the Fnte Element Method (FEM) as t s commonly used for heat conducton analyss. Also, FEM-based commercal software allows easy access to the mass and stffness matrces whch are used for generatng the reduced-order model. Currently, software used n the electroncs coolng ndustry s based on FV solvers. herefore, t s mportant to dentfy methods that can be easly mplemented n the exstng codes wth mnmal modfcatons or by the addton of user-defned subroutnes. he work by Astrd [] addresses the ssue of ntegratng the POD-Galerkn methodology wth FV method, and hence, forms the bass for the proposed methodology. he present work dentfes POD-Galerkn as a vable technque to generate boundary-condton ndependent reduced-order models for complex packages. 2. POD-Galerkn Methodology Introducton to reduced-order modelng usng POD- Galerkn methodology s avalable n the work of Gha [2] and Shroon [3] where ths method s employed to generate reduced-order models of turbulent flow n an axsymmetrc combustor. Also, detaled ntroductons are avalable n the work of Rambo [8], Astrd [] and Atwell [5]. POD s a statstcal method of examnng the characterstcs of a partcular data set. Projecton of the governng equatons onto a subspace, that defnes the

2 characterstcs of the data, creates the reduced-order model. he applcaton of POD to obtan a reduced-order model for the transent heat equaton s dscussed n detal by Astrd [], and s summarzed here n the followng steps: Step : A complete relevant data set s constructed. hs s acheved by constructng a matrx composed of snapshots of the soluton over tme. For example, the matrx of snapshots, snap, for the D transent heat equaton s shown below: snap (, tmental _ )... (, Notmesteps )..... = ( No_ grdpo _ nt, stmental _ )... ( No_ grdpo _ nt, s Notmesteps ) () For the 2D case, the same structure as that of snap matrx s used. Step 2: he covarance matrx s calculated as C = snap snap (2) No where No s the number of tme steps. Step 3: Sngular-Value-Decomposton of C s performed to dentfy the egenvalues and ther correspondng egenvectors. Step 4: Few of the largest egenvalues and ther correspondng egenvectors ( φ ) are selected and arranged n a matrx. hs matrx forms the POD bass ( Φ ), whch s the requred subspace. Step 5: he governng equaton s formulated n the dscrete form. For the D case, the equaton s: 2 ρ c p = k (3) 2 t x Equaton 3 s dscretzed by the fully mplct method for reasons of stablty; the method s uncondtonally stable. he dscretzed form s as below: 0 A ( t + t) = A ( t) + Bu( t) (4) Coeffcents of the unknown soluton are grouped together n the A matrx and those of the known temperatures are grouped n the A 0 matrx. Sources or nformaton on the boundary condtons s stored n the B matrx. Coeffcent matrces hold nformaton on the geometry and materal propertes. he A matrx s a sparse matrx consstng of three dagonals for the D case and fve dagonals for the 2D case. Step 6: Governng equaton n the dscrete form s now projected onto the subspace, Φ, by Galerkn projecton. he Galerkn projecton yelds: 0 Φ AΦa t + t = Φ A Φa( t) + Φ B u( t (5) ( ) ) Equaton 5 s the reduced-order model of the heat equaton. Step 7: he soluton s now represented by n ( x, t) a ( t) ( x) = φ (6) = where as are the POD coeffcents, and φ s are the orthonormal POD bass vectors. he reduced-order model governs the evoluton of the POD coeffcents. he POD coeffcents are determned by solvng the followng equaton. a ( t) = ( x, t) (7) φ he number of POD coeffcents s of the order of the number of the chosen POD bass vectors. hus, computaton of the soluton on a number of grd ponts s now reduced to computng the evoluton of a few POD coeffcents. hs provdes a reducton n the amount of calculaton, whle the accuracy of the calculated soluton varable s mantaned at a hgh level because the φ s contan maxmum nformaton on the heat transfer process n the D or 2D object. Step 8: he mportant contrbuton of the current work s to show that a BCI ROM can be obtaned for complex objects usng the above methodology. It s possble to obtan such a model by applyng the prncple of superposton of select boundary condtons. By appendng solutons of dfferent types of boundary condtons, the snapshot matrx s constructed. he snapshot matrx for a BCI ROM s of the form: snap [... ]... where = (8) BC BC BC 2 BCn s a soluton matrx of sze P x Q, P s the number of grd ponts and Q s the number of tme steps. Snce transent solutons capture a majorty of the characterstcs of the object, transent solutons of select boundary condtons are used n the constructon of the snapshot matrx. he absolute necessty of usng transent solutons for snapshots was also determned durng the course of ths research. Also, snce the method s based on the behavor of solutons, non-lnearty n materal propertes can also be captured and modeled. 3. Error Analyss In order to understand performance of the BCI ROM, t s necessary to determne all the relevant errors. he BCI ROM that s generated can have two man types of errors. he frst type of error results from usng napproprate boundary condtons to generate the snapshot matrx. Once the reduced-order model s generated, f t does not produce an accurate soluton even wth reasonable number of POD bass vectors, then the error les n the constructon of the snapshot matrx. On ncreasng the number of POD bass vectors, the error wll not decrease n magntude; nstead t reaches a plateau. If ths happens, the error s reduced by modfyng the snapshot matrx wth approprate solutons for the snapshot matrx. he second type of error results from nsuffcent number of POD bass vectors. he number of POD bass vectors s determned by analyzng the spectrum of egenvalues and selectng the ones whch are of order 0. and hgher. he followng heurstc approach cannot be used as a hard rule for judgng the number of POD bass vectors for the BCI ROM. Raghupathy, BCI ROM usng POD-Galerkn Methodology

3 n = N λ λ (9) = N where λ s the egenvalue, n s the selected number of egenvalues and N s the total number of egenvalues. Equaton (9) does not show how close to should the rato be. he smaller egenvalues hold nformaton lke temperature gradent n corners, varatons n the ntal tme step, etc. Error plots and the obtaned soluton are used as gudelnes to select the rght number of POD bass vectors. By ncreasng the number of POD bass vectors, the amount of error contnues to decrease. hs decrease n error occurs only when the rght snapshot matrx s used for developng the BCI ROM. An mportant observaton about the errors obtaned from the applcaton of the current methodology s that the error s always undrectonal. he BCI ROM always underpredcts the soluton when compared to the soluton obtaned by CFD. hus, f the generated reduced-order model has a fxed level of error, a hghly accurate fnal soluton can be predcted by addton of ths error to the obtaned soluton. Errors are analyzed at the ntal and fnal tme step by the followng relaton Error = abs max( ( t) max( ( t)) (9) tmestep ( ) CFD For a complete perspectve over all tmesteps, tmeaveraged error s used and s computed as N Error = tme averaged ( abs( max( CFD ( t) max( POD ( t)) )) (0) N t = where N s the number of tme steps. Both these errors are analyzed n order to understand the performance of the BCI ROM. 4. Results and Dscusson 4.. D Object he methodology and results for the D has been dscussed n detal n an earler work [7]. A BCI ROM was generated for a rod wth a sngle heat source at ts center, and asymmetrc boundary condtons on ts ends. he tmeaveraged error n all cases was found to be less than C (range of 0 C to 00 C). Also, computaton of 400 equatons was reduced to calculaton of 2 equatons. Cases ( BC ) Left Rght Ins 0 C 2 Ins 80 C 3 Ins Ins 4 0 C Ins 5 80 C Ins Valdaton Case 0 C 0 C Valdaton Case -20 C C able : Boundary condtons formng the snapshot matrx for the generaton of the POD bass vectors for the BCI ROM of the D case. Fve snapshots were suffcent to generate a BCI ROM of the heat conducton process n a rod. Wth a fnal set of 2 POD Raghupathy, BCI ROM usng POD-Galerkn Methodology POD bass vectors, heat conducton n a rod wth a sngle heat source can be smulated for any type of boundary condton on ts extremtes. It was expected that, based on the type of solutons used n the constructon of the snapshot matrx, a BCI ROM can be obtaned wth 9 snapshots for the 2D case, and 3 snapshots for the 3D case. he cases were based on: two snapshots applyng extreme boundary condtons for each sde, and a sngle snapshot wth nsulated boundary condton on all sdes. Another mportant result was establshng the valdty of the BCI ROM for boundary condtons outsde the range of temperatures used for generatng the BCI ROM. he methodology and results obtaned for the D case requres further nvestgaton before t can be used for complex packages. he sectons that follow present the result of ths nvestgaton for smple and complex 2D objects D Object wth a Sngle Heat Source ransent conducton s modeled n a copper 2D plate wth thermal conductvty k= 385 W/m 2 K, densty ρ = 8930 kg/m 3 and specfc heat capacty c p = 385 J/kg K -. he problem confguraton s shown n Fg.. Number of grd ponts n the x-drecton s 40 and 50 n the y-drecton. A heat-generatng pont source s located at the center of the copper plate. Conducton n the plate s smulated untl steady state s reached. me step sze of t = 0 sec s used, and the smulaton s performed for 0 seconds (steady state s reached by ths tme). _Left _op Source _Bottom L = m _Rght H = 0.5 m Fgure : Problem confguraton for the 2D case wth a sngle pont heat-generatng source Cases Boundary ( BC ) op Rght Bottom Left 0 C Ins Ins Ins 2 00 C Ins Ins Ins 3 Ins 0 C Ins Ins 4 Ins 00 C Ins Ins 5 Ins Ins 0 C Ins 6 Ins Ins 00 C Ins 7 Ins Ins Ins 0 C 8 Ins Ins Ins 00 9 Ins Ins Ins Ins able 2: Boundary condtons formng the snapshot matrx for the generaton of the POD bass vectors for the BCI ROM of the 2D case

4 Valdaton Boundary cases op Rght South Left 0 C 0 C 0 C 0 C 2 0 C 30 C 5 C 80 C able 3: Valdaton cases for verfyng the boundarycondton-ndependence of the reduced-order model Based on the conclusons of the D case, the snapshot matrx for the 2D object was constructed usng the cases n able 2. Soluton for the nne cases s obtaned by applyng the respectve boundary condton to each of the four sdes. hese solutons are then arranged n the form of the snapshot matrx shown n Eq. (8). wo valdaton cases, shown n able 3 are used to prove that the BCI ROM developed can be used to predct solutons for boundary condtons that were not a part of the generatng set. Valdaton Case : A boundary condton of 0 C s appled on all four sdes. he reduced-order model s generated wth 70 POD bass vectors. he errors at both early (t = 0s) and fnal (t = 0s) tmesteps are analyzed. he error at t=0s s n the order of e-7 C. he dstrbuton of error n the fnal tme step s shown n Fg. 2. Here t has grown to a maxmum of 2.5 C. he tme-averaged maxmum error s.39 C (for a range of 0 C to 00 C). Also, the maxmum error s observed to occur near the boundary. Heght (m) (b) POD soluton Fgure 3: Comparson of soluton at steady state for valdaton case 2 Valdaton Case 2: Second set of boundary condtons shown n able 3 are appled on the respectve sdes. hese values are arbtrary n nature. emperatures are chosen n such a way that none of them are equal to the values used n generatng the reduced-order model. Also, these temperatures offer strong and weak gradents from the source n the respectve drectons. It s observed from Fgs 3a and 3b that the POD soluton at steady state s notceably dfferent from the CFD soluton. he tme-averaged maxmum error s 4.6 C (for a range of 0 C to 00 C). he maxmum error durng t = 0s s 8 C and 4 C at t = 0s. hese can be seen n Fgs. 4a and 4b Fgure 2: Errors at the fnal tme step over the 2D grd for Valdaton Case (a) Intal tmestep error Heght (m) (a) Detaled CFD Model soluton (b) Fnal tmestep error Fgure 4: Errors at varous tme steps over the 2D grd for Valdaton Case 2 Raghupathy, BCI ROM usng POD-Galerkn Methodology

5 he hgh value of errors shows that the BCI ROM s not capturng the dynamcs of the transent heat flow n the desred way. Snce ths error does not reduce by ncreasng the number of POD bass vectors, and the maxmum error appears close to the boundares n both valdaton cases, the fundamental snapshot matrx from whch the POD bass s generated has to be revsed. he maxmum error near the boundary ndcates that the BCI ROM s not able to capture strong gradents occurrng near boundares. herefore n order to capture these gradents, the snapshot matrx s modfed to nclude solutons of cases 0 through 3, shown n able 4. he BCI ROM s constructed from 70 POD bass vectors obtaned from the revsed snapshot matrx. hs model s used to predct soluton for the two valdaton cases. Performance of the BCI ROM wth dfferent number of POD bass vectors s shown n able 5 for both valdaton cases. he errors durng the ntal and steady state tmesteps reduced consderably when hgher numbers of POD bass vectors are used. he trend n error shows that the snapshot matrx constructed from cases presented n able 4 s ndeed the rght one. Cases Boundary ( BC ) op Rght Bottom Left 0 C Ins Ins Ins 2 00 C Ins Ins Ins 3 Ins 0 C Ins Ins 4 Ins 00 C Ins Ins 5 Ins Ins 0 C Ins 6 Ins Ins 00 C Ins 7 Ins Ins Ins 0 C 8 Ins Ins Ins 00 C 9 Ins Ins Ins Ins 0 00 C 0 C 0 C 0 C 0 C 00 C 0 C 0 C 2 0 C 0 C 00 C 0 C 3 0 C 0 C 0 C 00 C able 4: Boundary condtons formng the revsed snapshot matrx for the generaton of the POD bass vectors for the BCI ROM of the 2D case. Although the error from a BCI ROM constructed from 50 POD bass vectors s less than C, the BCI ROM constructed from 70 POD bass vectors s used because of the expectaton that, wth strong gradents n unknown boundary condtons, ths model wll be able to perform better than one constructed usng 50 POD bass vectors. Fgure 5 shows the POD soluton for valdaton case 2 generated wth a BCI ROM constructed usng 70 POD bass vectors of the revsed snapshot matrx. In comparson wth Fgs. 3a and 3b, t can be seen that the present soluton provdes excellent agreement wth the soluton of the detaled CFD model. In comparson wth Fg. 4a, error dstrbuton n Fg. 6 shows a unform dstrbuton of a low level of error across the 2D object. Confdence n the boundary-condton-ndependence of the reduced-order model can only be obtaned by employng the reduced-order model for multple arbtrary boundary condtons. Instead of performng an arbtrary analyss, aguch s Desgn of Experments (DOE) [8] approach s employed. he Orthogonal Array method uses par-wse combnatons of the ndependent varables. Heght (m) Fgure 5: POD Soluton from BCI ROM constructed wth 70 POD bass vectors of the revsed snapshot matrx Valdaton Max Error of POD Bass vectors case Intal Steady-State me-averaged Intal Steady-State me-averaged able 5: Errors from the BCI ROM generated usng the revsed snapshot matrx Fgure 6: Error dstrbuton at the ntal tme step for Valdaton Case 2 obtaned from BCI ROM constructed wth 70 POD bass vectors of the revsed snapshot matrx he mnmum number of experments to be conducted s gven by the followng relatonshp of Experments = + ( ) () M = L Raghupathy, BCI ROM usng POD-Galerkn Methodology

6 where M = of ndependent varables L = Level of values the varable can take op Rght Bottom Left able 6: L9 Orthogonal Array for verfyng boundarycondton-ndependence of the reduced-order model he L factor s defned by boundary condtons mposed on the reduced-order model. In ths case, the L9 Orthogonal Array s used for defnng the experments. he number of factors (4) corresponds to the number of sdes. Levels are the temperatures that are appled to the sdes. hree temperature levels, namely, 25 C, 50 C and 75 C are used. None of the temperatures used for generatng the reduced-order model are used. able 6 shows the experments used for valdaton of the reduced-order model. In all cases, the BCI ROM generated from 70 POD bass vectors of the revsed snapshot matrx s used. able 7 shows the comparson of the solutons predcted by the BCI ROM aganst the solutons of the detaled CFD model. he maxmum error occurs at lower tme steps and s n the order of 0. deg C (over a range of 0 to 00 C). Subjectng the BCI ROM to such an experment provdes confdence n the boundary-condton-ndependence of the reduced-order model. Max Error Steady- me Intal State Averaged able 7: Errors from the BCI ROM for the experments conducted wthn the temperature range used for generatng the reduced-order model In able 6, the three levels of temperature (25 C, 50 C and 75 C) used le wthn 0 and 00 deg C (extreme values used Raghupathy, BCI ROM usng POD-Galerkn Methodology whle generatng the ROM). able 7 showed that the reducedorder model can ndeed be used n an envronment that mposes any temperature between 0 and 00 deg C on any sde of the 2D object. he nature of reduced-order model generated usng POD s such that t captures the characterstcs of the data and does not confne tself to any partcular boundary condton. he premse that such a reduced-order model should be able to provde accurate results for any boundary condton, even for those that le outsde ts generatng range, was vald for the D case [7]. hs s tested agan wth the 2D object usng the DoE approach. Experments used for testng of ths premse are shown n able 8. For the experments n able 8, the relatvely low errors presented n able 9 show that the reduced-order model generated usng the POD-Galerkn method s truly boundarycondton-ndependent. Although the BCI ROM was generated wth nsulated cases n ts snapshot matrx, t faled to accurately predct certan combnatons of nsulated and fxedtemperature boundary condtons. By ncreasng the number of POD bass vectors, those combnatons of nsulated and fxed-temperature boundary condtons could be predcted wth errors of approxmately 0%. op Rght Bottom Left able 8: L9 Orthogonal Array for verfyng truly boundarycondton-ndependent nature of the reduced-order model Max Error Steady- me Intal State Averaged able 9: Errors from the BCI ROM for the experments conducted outsde the temperature range used for generatng the reduced-order model

7 4.3. Complex 2D Object he concept of BCI ROM s now extended to an object wth ncreased complexty. A reduced-order model s generated for the 2D object shown n Fg. 7. he object has the same dmensons as earler but now t s constructed from two dfferent materals. Lower half of the 2D plate s made of copper (k =385 W/m 2 K, ρ= 8930 kg/m 3 and c p =385 J/kgK - ) whle the upper half s made of alumnum (k =20 W/m 2 K, ρ = 270 kg/m 3, c p =93 J/kgK - ). he object also has two heat-generatng pont sources (S and S2) of equal strength located asymmetrcally. hs arrangement of more than one materal and multple sources makes ths a complex 2D object. he changes n materal propertes and the addton of sources are reflected n the co-effcent matrces A, A0 and B of Eq. (4). he BCI ROM for the above object s obtaned usng 70 POD bass vectors generated from the snapshot matrx constructed accordng to able 4. hs BCI ROM s tested for boundary condtons wthn the range used for generatng the reduced-order model as well as those outsde that range. he experments are dscussed n able 6 and able 8. _Left S2 Al _op S Cu _Rght H = 0.5 m y =0.00 Max Error Steady- me Intal State Averaged able : Errors from the BCI ROM for the experments conducted outsde the temperature range used for generatng the reduced-order model Errors correspondng to the experments n able 6 are presented n able 0, and errors correspondng to experments n able 8 are shown n able. It can be seen that the errors n all cases are less than C (over a range of - C to 400 C). Fgures 8a and 8b show good agreement between the results from the detaled CFD model and the BCI ROM. he maxmum error, of the order of 0. C, between the solutons from the detaled and reduced-order model stll occurs only durng the fnal tme step. CFD Soluton _Bottom L = m x =0.00 Fgure 7: Problem confguraton for the 2D case wth multple materals and multple heat-generatng sources Heght (m) Max Error Steady- me Intal State Averaged able 0: Errors from the BCI ROM for the experments conducted wthn the temperature range used for generatng the reduced-order model Heght (m) (a) Detaled CFD Model soluton POD Soluton (b) POD Soluton Fgure 8: Comparson of solutons from detaled CFD model and BCI ROM for boundary condtons of 8 n able Raghupathy, BCI ROM usng POD-Galerkn Methodology

8 5. Conclusons he usage of POD-Galerkn methodology to generate BCI ROMs has been successfully demonstrated for both D and complex 2D objects. Major advantages of ths methodology are as follows: It s possble to predct the complete characterstcs of the system wth a hgh level of accuracy. Dfference between the detaled CFD model and the BCI ROM was always mantaned less than C over a range of - C to 400 C for the complex 2D object. It s possble to obtan large reducton n computatonal resources and tme consumed. hs was shown by reducng the computaton of 0 equatons to 70 whle preservng a hgh level of accuracy for the complex 2D object. he methodology s capable of generatng reduced-order models for any complex electronc package wth multple heat-generatng sources. he methodology s capable of accurately predctng solutons over a wde range of boundary condtons because of the truly boundary-condton ndependent nature of the obtaned reduced-order model. he methodology s capable of relable and accurate predcton of transent solutons Also, the methodology, by ts nature, s capable of modelng the non-lnear characterstcs of complex electronc packages Future efforts wll concentrate on extendng ths methodology to 3D objects and also on nterfacng the methodology wth exstng commercal codes through userdefned functons or macros. Acknowledgments he authors acknowledge the support of Electronc Coolng Solutons, Csco Systems and Avago echnologes, Calforna, for fundng ths research. Also, communcaton wth Dr. Attla Aranyos s gratefully acknowledged. References. Lasance, C., Vnke, H., Rosten, H., and Wener, K.L., 995, A Novel Approach for the hermal Characterzaton of Electronc Parts, Proc. of SEMIHERM XI, San Jose, CA, pp Lasance, C. J. M., 8, en Years of Boundary- Condton-Independent Compact hermal Modelng of Electronc Parts: A Revew, Heat ransfer Engneerng, Vol. 29, 2, pp Sabry, M. N., 7, Flexble Profle Compact hermal Models for Practcal Geometres rans. of the ASME, pp Vol Codecasa, L., D Amore, D., Maffezzon, P., and Batty, W., 2, Mult-Pont Moment Matchng Reducton of Dstrbuted hermal Networks, 8th Internatonal Workshop on hermal Investgaton of ICs and Systems, HERMINIC, Madrd, Span, pp Augustn, A., Hauck,., Maj, B., Czernohorsky, J., Rudny, E.-B., and Korvnk, J.-G., 6, Model Reducton for Power Electroncs Systems wth Multple Heat Sources, Raghupathy, BCI ROM usng POD-Galerkn Methodology Proc. of 2th Internatonal Workshop on hermal nvestgatons of ICs, HERMINIC, p Shapro, B., 3, Creatng Compact Models of Complex Electronc Systems: An Overvew and Suggested Use of Exstng Model Reducton and Expermental System Identfcaton ools, IEEE rans. on Comp., Parts and Manf. echnology, vol 26, no, pg Matha, P., and Shapro, B., 6, Interconnecton of Subsystem Reduced Order Models n the Electro-thermal Analyss of a Large System, IEEE rans. on Components and Packagng echnologes., Vol 30, pp Rambo, J., 6, Reduced-order Modelng of Multscale urbulent Convecton: Applcaton to Data Center hermal Management, Doctoral Dssertaton, Georga Insttute of echnology. 9. Fc, A., Bałeck, R. A., and Kassab, A.J., 4, Solvng ransent Nonlnear Heat Conducton Problems by Proper Orthogonal Decomposton and FEM, ICHM Internatonal Symposum on Advances n Computatonal Heat ransfer, Norway, Books of Abstracts, CH Rudny, E. B., Lenemann, J., Grener, A., and Korvnk, J.G., 4, mor4ansys:generatng Compact Models Drectly from ANSYS Models echncal Proceedngs of the 4 Nanotechnology, Conference and rade Show, Nanotech 4, Boston, Massachusetts, USA, vol. 2, p , March 7-.. Astrd, P., 4, Reducton of Process Smulaton Models: A Proper Orthogonal Decomposton Approach, Doctoral Dssertaton, echnsche Unverstet Endhoven. 2. Gha, U., Shroon, S., Gha, K., and Osswald, G., 99, Examnaton of a Vortex-Rng Interacton Phenomenon n an Axsymmetrc Flow, AIAA Aerospace Scences Meetng, 29th, Reno, NV. 3. Shroon, S., 994, Analyss of Flow hrough Combustor Geometry usng Numercal Smulaton and Dynamcal System Approach, Doctoral Dssertaton, Unversty of Cncnnat. 4. Versteeg, H.K., and Malalasekera, W.K.,995, An Introducton to Computatonal Flud Dynamcs, he Fnte Volume Method, Pearson Prentce Hall, Essex. 5. Atwell, J. A., 0, Proper Orthogonal Decomposton for Reduced Order Control of Partal Dfferental Equatons, Doctoral Dssertaton, Vrgna Polytechnc Insttute and State Unversty. 6. Srovch L., 987, urbulence and Dynamcs of Coherent Structures, Part I: Coherent Structures, Quarterly of Appled Mathematcs, XLV: Raghupathy, A.P., Gha, U., Gha, K., and Maltz, W., 8, Boundary-Condton-Independent Reduced-Order Modelng of Heat ransfer n Complex Objects by POD- Galerkn Methodology: D Case Study, ASME Journal of Heat and Mass ransfer, n revew. 8. G. aguch and Y. Yokoyama, 994, aguch Methods: Desgn of Experments, Amercan Suppler Insttute, Dearbom MI.