LARGE SCALE THERMAL-SOLID COUPLING ANALYSIS USING INEXACT BALANCING DOMAIN DECOMPOSITION

Transcription

1 RAC Unversy Journal, vol. VI, no. & 2, 200, pp. -7 LARGE SCALE THERMAL-SOLID COUPLING ANALYSIS USING INEXACT ALANCING DOMAIN DECOMPOSITION Abul Mukd Mohammad Mukaddes Deparmen of Indusral and Producon Engneerng Shah Jalal Unversy of Scence and Technology, angladesh emal: and Ryuj Shoya Faculy of Informaon Scence and Ars, Toyo Unversy, Japan ASTRACT In hs research, a sysem of hermal-sold couplng analyss s developed wh he mplemenaon of Inexac alancng Doman Decomposon wh a dagonal scalng (IDD-DIAG n boh hermal and sold analyss. The IDD-DIAG s an mproved verson of alancng Doman Decomposon (DD, where an ncomplee facorzaon based parallel drec mehod s employed o solve a coarse space problem, and he dagonal-scalng s employed o precondon local fne space problems nsead of he Neumann-Neumann precondoner. The developed sysem performed hea conducve analyss o have emperaure dsrbuons n sold models and hen performed he srucural analyss o see deformaon or expanson due o emperaure dfferences. oh of he analyses employed he Herarchcal Doman Decomposon Mehod (HDDM wh parallel IDD-DIAG. I s shown ha he erave procedure converges rapdly and he convergence s ndependen of he number of subdomans, namely, numercal scalably s sasfed. The presen sysem s mplemened on massvely parallel processors and succeeds n solvng a hermal-sold couplng problem of 2 mllons of nodes. Keywords: Fne elemen analyss, hermal-sold couplng problems, Doman Decomposon Mehod, Incomplee alancng Doman Decomposon. INTRODUCTION Engneerng producs ha have complex geomeres may crack durng he heang process or fal durng operaons. To avod such falures and reduce he cos of producon, he behavor ha occurs n he producs durng manufacurng or whle operang mus be predced n advance. These behavors nclude he sress from exernal loads as well as he hermal sresses from he emperaure dfference n he producs. The purpose of hs research s o develop a sysem ha could be used o analyze hea ransfer problems ha have complex geomeres for he emperaure dsrbuon. The predced emperaure combned wh he appled exernal loads s hen used o compue he deformaon and hermal sresses of he producs. The paral dfferenal equaon descrbed he hermal and srucural problems can be coupled when he sress n he srucural problem s a funcon of emperaure from he hermal problem []. The movaon of he couplng analyss comes from hermal-flud couplng problem [] and he large scale flud-srucure couplng problem [2]. Large scale problems need o be solved for he mprovemen of accuracy. The convenonal algorhm lke Doman Decomposon Mehod (DDM needs much me o solve he large scale problems and moreover s no scalable [3]. A precondoner should be used o reduce he compuaon me and he number of eraons. A suable precondoner mgh make he DDM scalable. The presen research aemps o develop a s y s e m o f h e r ma l -sold couplng analyss o address hese requremens. y usng he Herarchcal Doman Decomposon Mehod (HDDM [4] wh a precondoned erave solver n order o perform boh hermal and sold analyses, s hoped ha he developed sysem would perform he hea conducve analyss o

2 Abul Mukd Mohammad Mukaddes and Ryuj Shoya acheve emperaure dsrbuons n sold models and hen perform he srucural analyss o see deformaon or expanson due o emperaure dfferences. The HDDM employs a precondoned erave solver n order o perform boh hermal and sold analyses. Moreover alancng Doman Decomposon (DD [5] has receved much aenon n he las few years. T h e ma n r e a s o n f o r h e p o p u l a r y o f h s mehod s undoubedly, he need o ake he advanage of parallel compuers. DD s close n spr o mulgrd mehods and s a varaon of Neumann-Neumann algorhm. I nvolves soluon of a coarse problem n each eraon of erave DDM. Moreover o effcenly solve a coarse space problem derved from equlbrum condons for sngular problems assocaed wh a number of subdomans appeared n he DD algorhm, an Inexac alancng Doman Decomposon wh Dagonal Scalng (IDD-DIAG s proposed for srucural problem [6]. In hs paper, hs nexac balancng doman decomposon mehod s nvesgaed n he analyss of hermal-sold couplng problem. In he IDD-DIAG formulaon, a coarse space problem s approxmaed by an ncomplee facorzaon coarse operaor based parallel drec mehod, and he dagonal scalng s employed o precondon local fne space problems nsead of he Neumann-Neumann precondoner. The hermal-sold couplng problem of a nuclear pressure vessel model wh 2 mllon nodes s successfully analyzed wh hs IDD-DIAG. The numercal resuls show beer performance of IDD-D I A G. 2. THERMAL-SOLID ANALYSIS y consderng a hea conducon equaon on a domanω, defnng f as nernal hea generaon, T emperaure appled on he boundaryγ T, Q he hea flux appled on he boundary Γ, he Q fundamenal equaons of hs hea conducon problem s gven by: q λ gradt dv q f q n Q T T Q T ( where Ts emperaure, q he hea flux, λ he hermal conducvy and n an ouer normal un vecor, respecvely. The fne elemen (quadrac erahedral dscrezaon of ( yelded a lnear sysem of he for Ax f (2 where As he global sffness marx, x s an unknown vecor of emperaure and fs a known vecor. Agan we consdered a srucural problem concernng a domanω. Hence, F s he racon force appled on he boundaryγ F, he body force appled n he doma and u he prescrbed dsplacemen on he boundary Γ u. Fundamenal equaons of hs srucural problem are summarzed as follows: τ ε u, j τ n + 0 τ D j u, j ( u + u, j mn ε e mn F 0 j, / 2 u F (3 where, ake he value o 3, m, n ake he value o 3, u s dsplacemen, ε a sran ensor, τ sress e n s o r, D a coeffcen ensor of he mn Hooke s law and n an ouer normal vecor on he j boundary Γ, respecvely. Here, f we consder he hgh emperaure dsrbuon on he whole domanω, elasc sran as e ε, oal sran ε and hermal sran ε hen we have: e ε ε ε. (4 Agan, he hermal s consdered as follows: ε ( T T ( m n α 0 ε (5 0 ( m n where T 0 s he reference emperaure, α he hermal expanson coeffcen and T he emperaure whch s he oupu of he hermal 2

3 Large Scale Thermal-Sold Couplng Analyss analyss. The value of hermal sran ε n he equaon (5 was used n he equaon (3 hrough he equaon (4. The hermal sran ε ook he value α( when mn and oherwse ook T T 0 0. The fne elemen (quadrac erahedral dscrezaon of (3 yelded a lnear sysem of he form: Ku b (6 K, u where and bare, respecvely, he sffness marx, he dsplacemen vecor and he force vecor. 3. THERMAL SOLID COUPLING ANALYSIS The presen sysem s conduced o predc emperaure dsrbuons n sold models and hen o nvesgae he hermal expanson or deformaon due o he emperaure dfference. Analyss seps are as follows: Read he npu daa for he hea conducve analyss and decompose he model by ADVENTURE_Mes [6]. 2 Analyze hea conducve problems wh IDD-DIAG based on he HDDM sysem 3 Gaher emperaure nformaon of all nodes of he model from oupus of hea conducve problems. 4 Read emperaure nformaon of all nodes and oher npu daa for srucural analyss and hen decompose he model by ADVENTURE_Mes. 5 Analyze srucural problems wh IDD-DIAG based on he HDDM sysem. Inpu daa for hermal analyss Doman decomposon (ADVENTURE_Mes Thermal analyss Temperaure nformaon Oupu Inpu daa for sold analyss Doman decomposon (ADVENTURE_Mes Srucural analyss Fg-. Flow char of hermal-sold couplng problems Fg. shows he flow char of hermal-sold couplng analyss wh he developed sysem. The name of he ADVENTURE module used n each analyss s shown n parenheses. 4. DOMAIN DECOMPOSITION METHOD In hs mehod he doma was decomposed no N non-overlappng subdomans, { Ω },...,N. As usual he sffness marx K ( represensa n equaon 2 and K n equaon 6 could be generaed by subassembly: N T K R K R (7 T where R s he 0- marx whch ranslaes he global ndces of he nodes no local numberng. Denong u as he vecor correspondng o he elemens, can be expressed as T u R u. Each u was spl no degrees of freedom u, whch correspond o Ω, called nerface degrees of freedom and remanng neror degrees of freedom u I. The subdoman marx K, vecor u and 0- marxes were hen spl accordngly K K II I K, (8 T KI K u u I, (9 u and ( R,R ( R. (0 I Afer elmnang he neror degrees of freedom, problem (2 was reduced o a problem on nerface Su g ( N T where S R S R s assumed o be posve defne, u s he vecor of he unknown varables on he nerface, gs a known vecor and S are he local Schur complemens of subdoman,..., N, assumed o be posve sem-defne. The problem ( s solved by a precondoned CG mehod whch solves he problem, z M r (2 where r s he resdual of ( and M s a precondoner. When he nerface problem s solved eravely, of course, an effcen soluon of 3

4 Abul Mukd Mohammad Mukaddes and Ryuj Shoya he large scale problems depends on how one chooses an effcen and scalable precondoner. Whole doman Subdoman s Fg. 2 Herarchcal Doman Decomposon Mehod Dsk Par_ Par_2 Par_n Pars Fg.3 Parallel processor scheme Paren Herarchcal Doman Decomposon Mehod (HDDM Consrucng he DDM algorhms for parallel compuers, a good prncple s o dvde he orgnal doman no pars, whch are furher decomposed no smaller subdomans as shown n fg.-2. In hs research, Herarchcal Doman Decomposon Mehod (HDDM [4] whch s a well known parallel DDM s adoped. Fg.3 shows he parallel processor scheme of HDDM sysem. In hs scheme, he Paren processors perform he FEA by hemselves and also coordnae he CG eraon. Here, he number of pars should be he number of processors. 5. ALANCING DOMAIN DECOMPOSITION (DD The DD precondonng echnque proposed by Mandel [5] uses a each CG eraon soluon of he local Neumann-Neumann problems on he subdomans coupled wh a coarse problem n a coarse space. The DD precondoner s of he form: M DD Qc + ( I Qc S Ql ( I SQc (3 whereq s he local level par andq c s he coarse level par of he precondoner. 5. Local level The local level par of he precondoner bascally nvolved he soluon of local problems, where Q s expressed by N ( ( ( + ( T ( T Q R D S D R. (4 l The dagger (+ ndcaes pseudo-nverse, snce he S s sngular for floang subdoman. The DD D mehod uses a collecon marxes ha deermnes paron of uny on nerface [5,9], N ( ( ( T R D R I (5 The smples choce for D s he dagonal marx wh dagonal elemens equal o he recprocal of he number of subdomans wh whch he degree of freedom s assocaed. 5.2 Coarse level The applcaon of he coarse erm T T ( R SR R Qc R amouns o he soluon of a coarse problem whose coeffcen marx s S R T W 0 SR 0. The operaor R 0 ranslaes he coarse degrees of freedom o he correspondng global degrees of freedom and s defned by R 0 ( ( ( ( N [ ( N ( N R D Z,...,R D Z ]. (6 ( For he scalar hea conducve problem, Z s a column consan vecor [8,9] and can be defned by T Z (,..., (7 For he srucural problem ( Z [8] comes from he dof of rgd body of moon. 5.3 Smplfed dagonal scalng (DIAG A dagonal marx s consdered as a precondoner whose dagonal elemens are consruced from he correspondng ones of ( K. The dagonal marx s defned as ( dag( K ( N T QDIAG R R. ( 8 4

5 Large Scale Thermal-Sold Couplng Analyss 6. INEXACT ALANCING DOMAIN DECOMPOSITION To mplemen he coarse grd correcon wh hgh parallel effcency, nexac balancng based on an ncomplee parallel Cholesky facorzaon s employed n he presen DD mehod. In general, such an ncomplee facorzed operaor s ypcally used ogeher wh some erave compuaons o compensae he ncompleeness. In hs paper, however, he coarse problem s approxmaed by he ncomplee facorzed operaor whou eraons. Ths ncomplee balancng process decreases compuaon coss for preprocessng and mproves parallel effcency bu may reduce he convergence rae compared wh exac balancng. However, oal compuaon me s expeced o be reduced. Remarkably, wh he orgnal exac DD precondoner, a coarse grd correcon s performed afer a local subdoman correcon n each eraon. However, wh he presen nxac DD precondoner, a coarse grd correcon s also mplemened o he CG resdual vecor before a local subdoman correcon due o ncomplee deleon of componens of he coarse space. The ncomplee balancng echnque s appled o he DD consderng a dagonal scalng as a local level precondoner. The new precondonng echnque s marked as IDD-DIAG and s defned as: M IDD DIAG Q ~ c + ( I Q ~ c S QDIAG ( I SQ ~ c (9 where Q ~ s consruced from he ncomplee c facorzed coarse operaor. The mplemenaon of he IDD-DIAG precondoner (9 goes as follows: Sep : alance he orgnal resdual by approxmang he coarse problem usng he ncomplee coarse operaor for an unknown vecor ~ N λ R : ~ S ~ T 0 λ R 0 r. (20 Sep 2: Se ~ s~ r SR λ. (2 0 Sep 3: Perform he dagonal scalng and average hese resuls ( ( dag( K N T u~ ( ( R R ~ s. (22 Sep 4: Compue s ˆ r Su~. (23 Sep 5: Approxmae he coarse problem agan for an unknown vecor N μ~ R S ~ T 0 μ~ R 0 ŝ. (24 Sep 6: Fnd he precondoned vecor z u ~ + R0µ ~. (25 The S ~ means he correspondng erm of he coarse 0 marx of RT 0 SR 0, whch s facorzed ncompleely. Hence, s sad ha he resdual s ncompleely balanced n (2. The mplemenaon of ncomplee balancng reduces he compuaon coss for facorzaon of he coarse marx and for forward elmnaon and backward subsuon of he problem (20 and (24 and consequenly he amoun of work of each eraon s reduced. For hs reason alhough IDD-DIAG precondoner may ncrease he number of eraons, a speed up s acheved for large scale problems n he massvely parallel compuer. 7. NUMERICAL RESULTS AND DISCUSSIONS Thermal and srucural analyss on wo models shown n fg. 4 are performed. 7. HTGR Model [8,9] Model descrpon and compuaonal condons The HTGR model shown n Fg. 4 s graphe made, helum cooled reacor core whose hegh was 580 mm. The convergence creron was ha he norm of he relave resdual s reduced o 0-6. The mesh szes of hs model are shown n Table. Compuaonal performances The compuaon performances for boh hermal and srucural analyss of HTGR model are shown n Table 2. Fg. 5 shows he convergence hsory for hermal analy s s o f H T G R model. oh resuls show ha DD and IDD-DIAG converges rapdly. Regardng he memory requremens of he varous precondonng approaches, IDD-DIAG whch employ a dagonal-scalng as a local subdoman correcon, reduce memory requremens by around 40% compared wh DD precondoner. 7.2 Model : AWR [9] Model descrpon and compuaonal condons In hs research, as a large scale and real shape model problem havng a bad convergence, he 5

6 Abul Mukd Mohammad Mukaddes and Ryuj Shoya presen mehod s appled o a 2 mllon node unsrucured mesh for a precse model of Advanced olng Waer Reacor (AWR[9] a s s h o w n n F g. 4. The model s expressed wh 3,000 subdomans, 7,486,792 elemens,,794,506 nodes n he HDDM sysem. T a b l e 3 Performances of DD and IDD-DIAG (AWR for hermal analyss Sub # 3,000 Sub #6,000 # I e r. Tme Mem. I e r. Tme (sec (M (sec DIAG 3,20,259 4,295,03 DD ,758 IDD- DIAG Mem. (M Fg. 4 Par decomposon of HTTR (lef and AWR model (rgh T a b l e : Mesh szes of HTGR model Number of nodes (dof Thermal,893,340 (,893,340 Srucural,893,340 (5,680,020 Number of elemens Number of subdomans,67,268 3,200,67,268 4,000 T a b l e 2 : Performances of DD and IDD-D IA G (HTGR CG resduals Number of subdomans Fg. 5 Convergence hsory of hermal analyss (HTGR Thermal Analyss Srucural Analyss Num. of eraon Compuaon me (sec DIAG DD Memory/ PE (M IDD DIAG DIAG,570 6, DD 8 3, IDD- DIAG 85 2, Fg. 6 Convergence hsory (AWR model 6

7 Large Scale Thermal-Sold Couplng Analyss Speed up Number of processors Fg. 7 Parallel scalably (HTGR Model Compuaonal performances IDD-DIAG s nvesgaed n he hermal analyss of AWR model usng wo dfferen numbers of subdomans. The resuls are shown n Table 3. In 3,000 subdomans, he exac DD ype precondoners reduce he number of eraons o abou 2% and he compuaon me o abou 40%. In 3,000 subdomans, he IDD ype precondoners reduce he number of eraon o abou 4% and he compuaonal me o abou 22%. I s found ha wh almos he same memory sze as DIAG, IDD-DIAG shows he bes performance n compuaon me. Nex n 6,000 subdomans, he exac DD ype precondoners are slower han DIAG. I has he reason ha he compleely Cholesky facorzaon of a coarse grd operaor gves almos all compuaon me. Here, IDD-DIAG shows good performance n compuaonal me wh less memory sze han n he case of 3,000 subdomans. Therefore, he IDD-DIAG s an effecve mehod o analyze large scale hermal-sold couplng problems. The convergence hsory for hermal analyss of AWR model s shown n fg. 5 whch shows he beer convergence of IDD-DIAG. Agan fg. 6 predcs ha IDD-DIAG s parallelly scalable [0]. 8. CONCLUSION In hs sudy a hermal-sold couplng sysem s successfully mplemened on a 2 mllon dof AWR model. The compuaonal speed of he HDDM sysem was mproved dramacally by employng he IDD-DIAG mehod as an effcen precondoner. The IDD-DIAG exhbs an excellen performance n erms of memory requremens, convergence rae and compuaon me. Furhermore, hs sysem has been successfully mplemened on parallel compuer. References. Reddy, J. N. and Garlng, D. K The fne elemen mehod n hea ransfer and flud dynamcs, CRC press, USA. 2. Gang, L., Wenbn, C. and Yaochu, F A flud-srucure couplng analyss of large scale hyperbolc coolng ower subjeced o wnd loads. Proc. of 8 h World Congress on Compuaonal Mechancs (WCCM Farha, C., Chen, P.S. and Mandel, J., 995, A Scalable Lagrange Mulpler ased Doman Decomposon Mehod for Tme-Dependen Problems, Inernaonal Journal for Numercal Mehods n Engneerng, Vol. 38, p Yagawa, G. and Shoya, R Parallel fne elemens on a massvely parallel compuer wh doman decomposon. Compu. Sysems Eng. 4: Mandel, J alancng doman decomposon. Comm. Numer. Mehods Eng. 9: M. Ogno, R. Shoya and H. Kanayam. 2008, An nexac balancng precondoner for large scale srucural analyss, Vol.2. No. 7. hp://advenure.q..u-okyo.ac.jp 8. Shoya, R., Ogno, M., Kanayama, H. and T a g a m, D Large scale fne elemen analyss wh a balancng doman decomposon mehod. Key Eng. Maerals : Shoya, R., Kanayama, H., Mukaddes, A.M.M. and Ogno, M. 2003, Hea conducve analyss wh balancng doman decomposon. Theor. Appl. Mech. 52: Mukaddes, A.M.M., Ogno, M., Kanayama, H. and Shoya, R A scalable alancng Decomposon based precondoner for large scale hea ransfer problems. JSME I n. J. S e r.. 49: