3D Transient Thermal Solver using Non-conformal Domain Decomposition Approach

Transcription

1 3D raet hermal Solver ug o-coormal Doma Decompoto Approach Jayog Xe, Madhava Swamatha Itercoect ad Pacagg Ceter (IPC), School o Electrcal ad Computer Egeerg Georga Ittute o echology, Atlata, GA 3033 Emal: ayog.xe@gatech.edu, madhava@ece.gatech.edu Abtract 3D tegrato become promg to be able to cotue the ytem tegrato tred due to hort SV tercoecto ued or taced de. h paper propoe a ecet traet thermal modelg method ug o-coormal doma decompoto approach or 3D taced IC ad ytem. o allevate the problem arg rom the eature cale derece betwee taced de a well a pacage ad PCB, the 3D ytem dvded to may ubdoma. Each ubdoma (de, pacage or PCB) ca be mehed depedetly ug deret grddg baed o t eature ze ad thereore the requred mehg cell are greatly reduced compared to covetoal method uch a te elemet or te volume method. he heat low cotuty betwee ubdoma captured ug the troduced terace ba ucto. I addto, the propoed compact mcro-ludc model baed o te volume method proved to be compatble wth the te elemet model or old medum baed o troduced orced covecto boudary ad eergy coervato. he expermetal reult how the propoed method oer up to 5x uow reducto ad 9x peed-up compared to covetoal te elemet method. eyword - Doma decompoto, traet, ludc coolg, multcale I. IRODUCIO Compared to the covetoal D tegrato approach, 3D tegrato become promg to be able to cotuouly creae tegrato dety ad mprove ytem perormace ug advaced tercoect cheme uch a hrough Slco Va (SV) [-]. Wth the SV abrcato proce maturatg ad IC cotug to cale toward m ode ad beyod, the power dety o 3D tegrated ytem expected to creae dramatcally, leadg to the pot where thermal eect o ytem perormace uch a thermal duced tre, electro-mgrato become mportat or chp ad pacage deg. o mtgate the thermal problem, or ecet coolg o hgh perormace 3D taced IC, thermal maagemet approache uch a mcro-ludc coolg become attractve [3-5]. For 3D ytem, whch compre o 3D taced de, pacage ad PCB board, the thermal couplg betwee taced chp ca caue harp temperature creae o Permo to mae dgtal or hard cope o all or part o th wor or peroal or claroom ue grated wthout ee provded that cope are ot made or dtrbuted or prot or commercal advatage ad that cope bear th otce ad the ull ctato o the rt page. o copy otherwe, to republh, to pot o erver or to redtrbute to lt, requre pror pecc permo ad/or a ee. ICCAD 0, ovember 5-8, 0, Sa Joe, Calora, USA Copyrght 0 ACM //... $5.00 the taced chp a well a create pacage hot pot. hereore, thermal modelg approache, whch are able to mulate 3D ytem cludg taced IC, pacage ad PCB ecetly, are requred. For 3D taced IC, each de uually ha t ow archtecture ad uctoal bloc baed o deg. hereore, each de uually requre deret mehg grd baed o t eature ze a compared to other de thermal modelg. Modelg ad grddg the 3D IC or 3D ytem a a whole ug covetoal te elemet or te derece method (FDM) ca reult mllo o uow due to mehg grd propagate rom oe rego to aother. he great umber o uow ca aect the thermal modelg ececy, whch a mportat apect or 3D chp deger. o overcome the eect o creag problem ze o mulato peed, thermal modelg approach whch ca tae to accout the pecal tructure o 3D ytem eeded, whch the ubect o th paper. h paper propoe a ecet 3D thermal modelg approach ug o-coormal doma decompoto or mulatg 3D taced IC or ytem. Itead o modelg the 3D IC or ytem ug covetoal te elemet method (FEM) [6] or te volume method (FVM) [7], whch ue coormal mehg grd, the propoed method allow dvdg the complex 3D ytem to may ubdoma wth o-matchg grd at the terace. Each ubdoma ca be mehed depedetly baed o t eature ze. A a reult, the total requred mehg cell are greatly reduced ad mulato ececy mproved compared to thermal modelg approach ug coormal FEM. he maor cotrbuto o th paper are lted a ollow: a) h paper rt propoe a ew 3D traet thermal modelg method baed o o-coormal doma decompoto. It eable ecet traet thermal mulato o complex 3D ytem wth o-coormal mehg grd o doma terace. he ytem uow ad computatoal tme are greatly reduced compared to covetoal FEM. he accuracy o the propoed method ha bee valdated by comparg wth reult rom coormal FEM ad meauremet (the temperature error le tha 5%). b) he propoed modelg method provde a modelg approach whch ca combe the FEM model or old chp ad compact FVM model or mcroludc coolg

2 to a hybrd model baed o eergy coervato rule ad troduce orced covecto boudary codto. he accuracy o the model valdated agat FEM ad meauremet. (chp temperature error le tha 5% ad chael outlet temperature error le tha %). c) he propoed method allow modelg 3D ytem cludg taced de a well a IC pacage ad PCB a a whole. he cluo o pacage ad PCB provde better boudary codto or 3D taced IC a compared to ug equvalet thermal retace to repreet the thermal covecto eect o pacage ad PCB. I addto, the temperature prole o IC pacage ad PCB ca alo be obtaed. hu, t ca be ued or thermal duced tre ad electro-mgrato aaly o pacage level, whch alo a mportat apect or 3D ytem deg [8]. d) Lat, the propoed method, whch baed doma decompoto, provde lexblty to ue deret mehg ze or deret doma. h eature eable certa ubdoma to ue er meh grd, whch wll ot aect the grddg o other doma amog a 3D ytem. he orgazato o the ret o the paper a ollow. I Secto II, the prevou wor o the thermal modelg o IC chp ad pacage cludg mcro-ludc coolg dcued. Secto III preet cell baed thermal modelg o old ad FVM baed lud model or mcro-chael. he approach or combato o the FEM model o old ad te volume model o lud alo troduced. Secto IV dcue the 3D ytem thermal modelg ug o-coormal doma decompoto method (DDM). Secto V preet the mplemetato detal or the propoed method. Secto VI how expermetal example ad dcuo. Fally, the cocluo ummarzed Secto VII. II. PREVIOUS WOR I the pat, coderable amout o approache ad eort have bee devoted or both teady tate ad traet thermal modelg o IC chp ad pacage [6, 7, 9-5]. Amog the method or thermal modelg wth covetoal heat coolg, [6, 9] are baed o te elemet method ad the approache [0, ] are baed o te derece method. he te volume baed thermal modelg method ha bee propoed [7, ]. I addto, the oule heatg (elheatg) eect wa codered [7] ad []. o allevate the eect o creag problem ze o computatoal cot, the thermal modelg ug 3D geometrcal multgrd approach ha bee propoed or IC chp thermal mulato [3] ad chp-pacage mulato [4]. For traet thermal modelg o IC chp, a 3D olver baed o alteratg drecto mplct (ADI) method ha bee propoed [5] or obtag traet repoe ecetly. o capture the mcroludc coolg eect, everal approache have bee propoed [7, 6-0] or teady tate thermal aaly. For traet aaly, a compact traet thermal modelg approach ha bee propoed or 3D taced IC wth ter-ter lqud coolg []. Fg.. 3D tegrated ytem tructure. Fg.. Dcretzato o old wth homogeeou materal tacg. 8- ode hexahedral elemet (cell). For the above thermal modelg approache, the method have ocued o modelg o IC chp, pacage ad 3D taced IC, whch are modeled a a gle doma. Coormal mehg grd are ued or mehg the ytem a a whole. h caue the e mehg grd to propagate through the 3D multcale tructure. A a reult, or modelg o multcale 3D ytem cludg taced IC, pacage ad prted crcut board (a how Fg. ), the problem ze ca creae dramatcally leadg to log mulato tme. he o-coormal DDM [], whch allow the dvdg o a large complex problem to may ubdoma that are maller ad eaer to hadle, become tractve. o tacle problem arg rom multcale 3D taced IC or 3D ytem, th paper propoe a traet thermal modelg method ug doma decompoto ad compact model or mcro-ludc coolg. It mportat to ote that th paper ocue o ytem level thermal modelg ad thu modelg the teral e detal o chp ot the ocu. III. HERMAL MODELIG OF SOLID AD FLUID 3. Cell Baed hermal Modelg o Sold For thermal modelg o 3D problem cot o old medum, the goverg heat equato decrbed a: c p [ r, ) ] P( r, () where, ) ad repreet the thermal coductvty ad temperature dtrbuto, repectvely. P deote the total heat exctato whch are rom the chp power map. o mply the problem, the thermal coductvty codered wthout temperature varato a (r). For modelg o lco chp cotg o homogeeou materal tacg (Fg. a) due to BEOL proce, the cell baed te elemet modelg

3 ca be appled. I th paper, the 3D problem mehed ug 3D o-uorm 8-ode hexahedral elemet (Fg. b) wth trlear ba ucto [9]. he method or hadlg materal homogeety how Appedx A. For modelg o realtc problem, the covecto boudary codto hc ( a ) () c eed to be codered at the perphery o chp ad pacage to mmc the ar coolg eect. a ad h c repreet the ambet temperature ad ar covecto coecet, repectvely. Followg the FEM modelg proce [9], by multplyg tetg ucto at both de o equato () ad tegratg over the volume, ater ug the dvergece theorem, the ew orm o the heat equato ca be obtaed a c ds PdV (3) t For mplcty, the ame ba ucto ca alo be ued a tetg ucto. he temperature ca be expreed a a lear combato o ba ucto: 8 (4) A a reult, wth total o mehed cell, the ytem equato wth the appled covecto boudary codto ca be wrtte a C (5) where, ad b C c( g( e e e e c c, c c p a D g, P b h d e h d, I equato (5-6), D dv, D( P( ) e e e PdV dv (6) C repreet the elemetary capactace matrx. ad repreet the elemetary te matrx or each elemet (cell) due to heat coducto ad heat covecto. ad P g b repreet the heat exctato ad temperature gradet due to covecto, repectvely. For thermal modelg wth homogeou euma boudary codto (atural boudary codto) [4], we ca mply let b ad g equal to zero. 3. Compact hermal Modelg o Flud Becaue o the large umber o mcro-chael ued or coolg o 3D IC, to obta at temperature etmato at early deg tage, compact thermal modelg o the lud requred to overcome the mulato ecet ug detaled computatoal lud dyamc (CFD) modelg []. For coolat low mcro-chael o IC chp, the Reyold umber uually le tha 300 ad the low lamar [5]. Sce the logtudal drecto o the chael much larger tha the lateral drecto, the mcro-ludc low ca be treated a ully developed lamar low. h property allow the developg o the compact lud model. he goverg equato or thermal aaly o mcro-ludc coolg expreed a: c p ( v ) ( ) P (7) Compared to equato () or old medum, t ha a extra term due to the coolat low. Here, v repreet the velocty dtrbuto o the lud. the thermal coductvty o the lud [6]. For heat traer coolat low, t cot o heat coducto due to te thermal coductvty ad heat traportato due to low velocty. I equato (7), except the ecod term o the let-had de related to low velocty, other term ca be modeled a old. Sce the coolat ha certa low drecto, pecal coderato requred. Fg. 3. Dcretzato o mcro-chael to cell. (Oly bottom hal part o the mcro-chael how o the let gure) For ludc coolg a how Fg. 3, ce the mcrochael cro-ectoal dmeo much maller tha t legth, the low velocty alog the logtudal drecto much larger tha the lateral drecto. hu, t ca be aumed that the coolat oly low the logtudal drecto ad low velocty thereore cotat. he average low velocty v alog y drecto (Fg. 3) ha bee ued or mulatg the lud low th paper. A a reult, equato (7) ca be traormed a: c p ( v ) ( ) P (8) y By tegratg equato (8) over the dahed te volume cell (Fg. 3) ad applyg the dvergece theorem, equato (8) ca get rd o the d order dervatve ad become c p c pv y ds ds P dv t ˆ ˆ ˆ (9) cell S where, S the urace o dahed cell (Fg. 3). It mportat to ote that or the ecod term o the let had de, ce the cetral te derece cheme ca geerate tablty certa cae [7, 30], the bacward derece approxmato ued. By applyg the te derece approxmato to equato (9) ad applyg the covecto boudary codto (aumg covecto coecet o h at 4 de o the S cell

4 chael), the te volume cheme or lud low at ode (,, ) ca be expreed a:,,,, W / LH h HL,,,,,,, W / LH h HL,,,,,,,, H / WL h WL,,,,,,,, H / WL h WL,,,, cp m(,,,, ) cpvc P Vc (0) where, V c WHL the cell volume ad m vwh the volumetrc low rate. Baed o cheme derved (0), the equvalet crcut repreetato o the ludc cell how Fg. 4b. Sce the old part ca be modeled ug FEM ad the ludc coolg modeled ug FVM, the combato o thee two model requred. A how Fg. 4a, the coecto betwee the FEM model o old ad FVM model o lud ormed ug the orced covecto boudary codto. he orced covecto dcated ug arrow Fg. 4a. Sce the covecto boudary capture the heat traer rom chp to mcro-chael eectvely, the combato o thee two model become eable by ollowg the eergy coervato rule. he covecto tregth, amely the average covecto coecet h at the 4 de o the mcro-chael ca be obtaed aalytcally rom the uelt umber a: h u / () D h where, Dh the hydraulc dameter o the mcro-chael [7]. he eect o th boudary codto mportat, ce elmatg t ca caue correct chp temperature [7, 8]. For ully developed lamar low de rectagular mcrochael wth cotat heat lux, the uelt umber ca be expreed a [5]: u 8.35 () where, W / H the apect rato o the rectagular chael. Ater ug the uperpoto rule or tampg the te elemet old model ad FVM baed lud model baed o global ode umber, the al te matrx ad ytem equato ca be obtaed a equato (3). Fg. 5. o-coormal grddg o 3D ytem to doma, heat low cotuty llutrato. (c ad b repreet the vector o coecet o temperature ba ucto) IV. C g g g (3) 3D SYSEM MODELIG USIG DOMAI DECOMPOSIIO 3D ytem cludg 3D taced IC, pacage, PCB ad bump layer how a Fg.. Sce each chp ha t ow uctoal bloc, t thereore requre deret mehg grd a compared to other chp. I addto, due to the eature cale derece betwee chp, pacage ad PCB rego, mllo o mehg cell are requred ug coormal FEM or FVM baed mehg grd. o allevate th problem, the problem ca be dvded to eparate ubdoma cludg eparated chp doma, pacage ad PCB doma, a how Fg. 5a. Each doma ca be mehed depedetly ug 3D o-uorm grd. A a reult, the mehg grd rom oe chp doma do ot terere wth grd rom other doma ad thereore the requred mehg cell ca be greatly reduced. For mplcty, the doma decompoto method explaed wth D rectagular grd (Fg. 5a). For traet thermal modelg, at each tme tep, the heat traer out o oe doma equal the heat low to aother doma through the commo terace. At the terace, the cotuty o heat traer eed to be eured to capture the couplg betwee eparated doma. For two adacet ubdoma wth a commo terace (Fg. 5b), by aumg / (,) () (), we have the relatohp o []. he the wea cotuty o heat low acro the terace ca be etablhed. I th ecto, the thermal modelg baed o-coormal DDM dcued or traet aaly. By troducg (the Lagrage multpler) or each terace, the couplg betwee doma ca be captured ug the couplg matrx (Fg. 5b). For mplcty, we ca aume the ytem oly ha two eparated doma. he al ytem equato ug DDM ca be obtaed a: Fg. 4. Forced covecto boudary betwee old ad lud, equvalet crcut model o ludc cell. C C 0 B B B B 0 (4)

5 he detaled dervato o equato (4) rom equato (3) lted Appedx B. For ytem wth total ubdoma, the geeralzed ytem equato ca be derved ad obtaed ealy ug the uperpoto rule baed o ode order. By ug the bacward tme derece approxmato ( ) ( ) (5) Eq (4) ca be coverted to a lear equato a: ( ) ( ) A B ( ) ( ) ( ) ( ) A A B (6) B B 0 ter 0 where, C ( ) C ( ) A, (=, ) Here, the upercrpt (+) ad repreet the tme tep. he C ad repreet the capactace ad mpedace matrce or -th doma. ( ) repreet the temperature () vector o -th doma. deote the heat exctato -th doma calculated at tme tep. he umercal cheme baed o Cra colo (C) method [30], whch ha ecod order accuracy tme, ca alo be obtaed by ug ( ) ( ) ( ) / to approxmate the term Eq (4). However, ce the C cheme ha tme tep lmtato ad ca reult temperature ocllato or large tme tep [5], the cheme o Eq. (6), whch baed o bacward Euler method, ued our mulato. For mulatg 3D taced IC ug DDM, t mportat to ote that the coecto betwee chp doma through the tratoal ubdoma o bump layer (Fg. 5a). Sce the doma o bump layer ca be mehed ug much coarer grd a compared to the chp, the requred terace ba ucto ca be greatly reduced ad thu B matrx ha mall dmeo. o obta the te matrx or each doma, the aocated boudary codto eed to be ued or the correpodg ubdoma. I addto, the homogeeou euma boudary codto eed to be aged at the commo terace. V. OOL IMPLEMEAIO AD FEAURES A thermal mulato tool baed o the propoed modelg method wa mplemeted ug Matlab. he put o the tool clude: a) 3D geometrc tac-up ormato ad materal dex, b) materal ormato cludg heat capacty, thermal coductvty, etc. It accept homogeeou materal wth deret thermal coductvty x, y ad z drecto, c) tme tep ad tme depedet power exctato, whch allow to dee IC blocwe heat exctato, d) doma ad terace parameter cludg doma volume, doma umber ad terace area, e) uer deed mehg parameter or each doma cludg maxmum meh ze x, y ad z drecto ad meh reemet level. Wth hgher meh reemet level, er meh grd ca be geerated, ) boudary codto cludg cotat temperature ad covecto boudare. It allow pecyg atural ar or orced covecto boudary codto at the perphery o IC (except the let ad outlet o mcro-chael), pacage ad PCB urace. h tool accept tructure wth homogeeou materal tac-up. he method or hadlg materal homogeety how Appedx A. h tool ca geerate 3D o-uorm 8-ode hexahedral cell or each doma cludg mcro-ludc cell baed o the put le. Wth the geerated hexahedral cell, the global ode order aged or both old medum ad ludc cell. he global te matrx ad capactace matrx C are ormed ug the uperpoto rule [3, 9] ad tored a pare matrce. For 3D problem, the terace become a urace. Sce the terace urace ca have everal thouad o ode, the 4-pot Gaua quadrature or rectagular elemet ued to calculate the couplg matrx B eectvely. o olve the lear equato A ( ) ( ), the pare matrx olver Matlab ued or both covetoal FEM ad the propoed method or ar comparo. A PC wth a 3.9 GHz Xeo(M) CPU ad 3.0 GB memory ued or mulato. VI. EXAMPLES AD DISCUSSIO 6. Model Vercato Example (et Set ) o evaluate the accuracy o the propoed method, a expermetal example wth both covetoal heat ad ludc coolg mulated, a how Fg. 6. he tet vehcle Fg. 6b cot o a lco chp wth ludc coolg ug mcro-chael. he chp ze cm cm ad the uorm power coumpto 45 W. A total umber o 5 mcro-chael are dtrbuted uormly o the chp a decrbed [3]. he cro-ectoal dmeo o each mcrochael 0. mm 0. mm. he put water temperature at the let o mcro-chael et to be C a meauremet. Becaue o the cale derece betwee chp ad pacage, the example Fg. 6a ad Fg. 6b are both dvded to two doma cludg chp doma ad pacage doma. he detaled materal properte ad geometrcal ormato are lted able I. Sce the meauremet [3] wa carred out at atural covecto codto, the covecto coecet o 5 W /( m ) appled to both the top ad bottom urace o the pacage our mulato. Fg. 6. et et wth heat coolg ad mcro-ludc coolg.

6 ABLE I. MAERIAL PROPERIES AD GEOMERICAL IFORMAIO et et umber o layer 4 (de: 0 mm x 0 mm) 4 (pacage: 4 cm x 4 cm) Chael wdth * heght * legth 0. mm x 0. mm x 0 mm Chael ptch 96 mcro Bottom ad top lco heght 00 mcro, 0 mcro Flud low rate 65 ml/ m Pyrex gla heat capacty, thermal coductvty 80 J/g-,. W/m- IM heat capacty, thermal coductvty 60 J/g-,.6 W/m- Heat boudary temperature 5 Celu et et umber o layer 4 (de: 0 mm x 0 mm), 4 (pacage: 3 cm x 3 cm ) 0 (board: 0 cm x 0 cm) Chael wdth * heght * legth 0. mm x0. mm x 0 mm Chael ptch 500 mcro op ad bottom lco heght 50, 50 mcro Flud low rate (per chp) 6 ml/m Commo Parameter hermal coductvty o lud, lco, BEOL 0.6, 0, (W/m-) Heat capacty o lud, lco, BEOL layer 487, 700, 50 (J/g-) Ambet temperature 5 Celu h tet vehcle wth heat coolg ad ludc coolg ha bee mulated ug the propoed method ad covetoal FEM olver. he o-coormal ad coormal rectagular mehg grd have bee ued or the propoed method ad FEM olver, repectvely. ote that the ludc model combed to the FEM olver. he comparo o mulated chp temperature ad chael outlet temperature how Fg. 7b. A ee rom Fg. 7, the reult rom the propoed model ug DDM agree very well wth reult rom the covetoal FEM olver. he maxmum temperature derece le tha 0. ad 0.5 Degree or heat ad ludc coolg, repectvely. For ludc coolg, the meaured teady tate chp temperature ad outlet temperature are 40.8 ad 3. Celu [3]. he derece betwee the mulated coverged chp, chael outlet temperature ad meauremet [3] are.7 C ad 0.4 C, repectvely. he relatve error are about 4.% ad.% wth repect to the meauremet, repectvely. he uow (cludg uow or terace) ad mulato tme ug the propoed method ad FEM are how able II. It how that the propoed method ca reduce uow about -4 tme. Due to the reduced uow, the mulato tme peed up ca reach 35x or mulato wth ludc coolg. Fg. 7. emperature waveorm comparo ug propoed method ad covetoal FEM wth heat coolg, ludc coolg. ABLE II. PROBLEM SIZE AD SIMULAIO IME COMPARISO DDM Solver FEM Solver me Sze () me () Sze () me () Speed-up et Fludc coolg x35 Set Heat x4 et 3D ytem x9 Set 3D IC oly x7 (c) Fg. 8. 3D ytem wth mcro-ludc coolg, layer cro-ecto o taced chp, (c) power map o chp. 6. 3D Stacg Example (et Set ) A 3D ytem wth ter-ter mcro-ludc coolg alo mulated ug the propoed method. he 3D ytem clude three taced chp, 4-layer pacage ad 0-layer PCB, a how Fg. 8a. Each chp ha 0 mcro-chael. he layer tac-up or the taced chp ad mcrochael how Fg. 8b. he geometry ad materal parameter are ummarzed able I. Ar covecto wth heat traer coecet o 0 W /( m ) appled to the top urace o the pacage ad both de o the PCB. 3 chp are uppled wth ame water low rate. o-uorm heat dpato ued or chp, a how Fg. 8c. Uorm power coumpto o 40 W/cm ued or chp ad chp3. h tet et dvded to 7 doma cludg 3 chp doma, doma or mcrobump layer, oe pacage doma ad oe PCB doma ug DDM. he three chp are mehed depedetly ug deret meh ze. h example mulated or.. I the rt ecod, the 3 chp operate wth uorm power dety o 40 W/cm. From.0-.0, the 3 mm wde mddle rego o chp (Fg. 8c) wtched betwee 70 W/cm ad 40 W/cm perodcally. he mulated temperature ug the propoed method ad covetoal FEM how Fg. 9. It how the mulated reult ug the propoed method agree very well wth the FEM. he maxmum temperature derece le tha 0.5 degree ad the temperature error about %. From Fg. 9, t alo oberved that due to the covecto o pacage ad PCB, the bottom chp temperature about 3 Celu lower tha the top chp. o how the capablty or mulatg oly 3D IC, the example (Fg. 8a) alo mulated wthout IC pacage ad PCB. he requred uow ad mulato tme comparo are how able II. A ee rom able II, the propoed method ca reduce uow about tme or mulatg 3D IC ad 3D ytem. Due to the reduced uow, a mulato peed up to 9x ca be obtaed, dcatg the ececy o the propoed method. he de vew ( yz plae) o temperature dtrbuto o 3D IC at t =.9 how Fg. 0 wth ocoormal grddg ad hotpot llutrated.

7 Fg. 9. emperature waveorm comparo ug propoed method ad FEM. Fg. 0. Sde vew ( yz plae) o temperature dtrbuto at t =.9. VII. COCLUSIOS I th paper, we propoed a ecet 3D traet thermal modelg method or mulatg 3D ytem ug ocoormal doma decompoto approach. he propoed modelg method ca capture the traet behavor o 3D IC wth mcroludc coolg. Detaled aaly o o-coormal doma decompoto or 3D ytem ad 3D taced IC oreted thermal modelg preeted. he method or compact modelg o mcroludc coolg dcued. he accuracy ad mulato ececy o the propoed method have bee valdated agat covetoal FEM olver a well a meauremet. Baed o the expermetal reult, the propoed method how the maxmum temperature error o 5%. Due to the o-coormal doma decompoto approach ued, the propoed method ca reduce the mehed cell up to 5x, whch ca reult up to 9x peed-up a compared to the covetoal FEM olver, whch eable at temperature etmato o 3D ytem or 3D IC ther early deg tage. REFERECES []. 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8 Appedx A. Hadlg Ihomogeeou Materal 3D ytem ca cot o homogeeou materal, a how Fg. a. o hadle the homogeety, cell baed FEM ormulato eed to be ued [9]. Wth o-uorm mehg, each hexahedral cell oly cota a gle materal wth 8-ode, a how Fg. b. Due to o-uorm mehg, the cell legth deret x, y ad z drecto. he elemetary te matrx or oe cell due to heat coducto ca be ormed, decrbed by: yz xz xy D x x y y z z (7) x y z where, x, y ad z repreet the 8 8 tadard elemetary te matrx x, y ad z drecto wth ormalzed cell thermal coductvty ad x y z [9]. Baed o the relatohp betwee elemetary cell ode umber ad t global ode order, the global te matrx ca be ormed ug the uperpoto rule, decrbed a: D e D x (8) B. Formulato ug o-coormal DDM Aumg the ytem dvded to two ubdoma, the two ubdoma ow a commo terace, a how Fg.. By aumg / (,), we have the relato- ( ) ( ) hp o []. he the wea cotuty acro the terace ca be etablhed. Baed o equato (3) ad troduced Lagrage multpler, the ollowg equato or doma ad terace ca be derved a t er c c p p ( ) dt 0 dxdy dt P dv P dxdy t er dt y z dt t er dt (9) where,, ad repreet the ba ucto or doma, doma ad Lagrage multpler pace, repectvely [, 3]. Wth temperature beg expreed a lear combato o ba ucto ad b, the ytem equato or the problem wth two ubdoma (Fg. ) ca be wrtte a equato (4) a C B C B B B where, the matrx etre or -th doma ca be expreed a C B t er c p ( ) ( ) ( ) dt ( ) ( ) dv ) ( ) ( ) (,) (0) PdV ( ) I the above equato,, B ad C repreet the coducto mpedace matrx, couplg matrx ad capactace matrx or -th doma, repectvely. repreet the exctato vector or -th doma. For Lagrage multpler o the terace, t ba ucto ca be cotructed bae o the terace grd rom ether de. o reduce the umber o uow or the terace, t ca be cotructed baed o a coarer mehg grd. o eure the couplg matrx B or the terace ha ull ra, the ba ucto or the terace caot be radomly elected. For D problem wth 4-ode (blear) elemet (Fg. ), ce the terace become a le, the terace ba ucto (Fg. ) ca be cotructed baed o lear hape ucto, expreed a ( ) ( ) () ( ) where, the lear hape ucto aocated wth ode. For 3D problem, the terace become a urace coectg two ubdoma. For D terace wth ode, the terace ba ucto ca be obtaed baed y o D blear hape ucto. For eaer repreetato, t ca be decrbed baed o D ba ucto two drecto a: ( x, y ) () x y x y dt x Fg.. wo doma wth o-matchg grd at terace. Fg.. Ba ucto or D terace.