Pecahan Persamaan 2-D Diffusi Secara Paralel menggunakan Metoda Modified Gauss (Mike Susmikanti)

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1 Pechn Persmn -D Dffus Secr Prlel menggunn Metod Modfed Guss (Me Susmnt) ABSTRAK PEMECAHAN PERSAMAAN -D DIFFUSI SECARA PARALEL MENGGUNAKAN METODA MODIFIED GAUSS SEIDEL DALAM KOMPUTER KLUSTER Me Susmnt * PEMECAHAN PERSAMAAN -D DIFFUSI SECARA PARALEL MENGGUNAKAN METODA MODIFIED GAUSS SEIDEL DALAM KOMPUTER KLUSTER Prlelss penyelesn dffus -D dlm omputer luster dbhs dlm tulsn n Perhtungn nl rterst dn dstrbus flus neutron dlh pust dr nlss fss retor nulr Sebg pendetn persoln trnsport neutron, metod dffus merupn metod stndrd yng dgunn dlm nlss n Dlm pemechn persmn dffus, prosedur n berhdpn dengn persoln mtrs Dlm penyelesn dffus -D, mtrs membentu mtrs pent-dgonl Untu membut perhtungn dffus lebh efetf, proses lgortm Modfed Guss Sedel dfousn dlm unsur-unsur pent-dgonl Metod Modfed Guss Sedel dlm pemechn dffus -D dengn mtrs yng besr n dpresentsn Sebgn besr opers dlm metod modfs Guss Sedel dpt derjn secr prlel Komputs prlel dr pent-dgonl untu pemechn model dfuss -D dmplementsn secr ntegrs menggunn progrm C dn Messge Pssng Interfce (MPI) dlm omputer luster Kt-t unc : Model Dffus, Sstem Pent-Dgonl, Komputs Prlel, Modfed Guss Sedel, Komputer Kluster ABSTRACT PARALLEL -D DIFFUSION EQUATION SOLVER USING MODIFIED GAUSS- SEIDEL METHOD IN COMPUTER CLUSTER Prllelzton of -D dffuson soluton n Computer Cluster reported n ths pper Neutron flux dstrbuton nd egen vlue clculton s the centrl of nucler rector physcs nlyss As n pproxmton to the neutron trnsport problem, dffuson method s nowdys stndrd method used n ths nlyss In solvng the neutron dffuson equton the procedure wll end up wth the mtrx problem For -D dffuson soluton the mtrx would be pent-dgonl mtrx To me the -D dffuson clculton more effectve, the process of ths lgorthm s focused n elements pent-dgonl The Modfed Guss Sedel method n solvng the -D dffuson equton wth the bg mtrx wll be presented Most of the operton n the modfed Guss- Sedel method cn be done n prllel The prllel computng of pent-dgonl for solvng the -D dffuson model re mplementton wth ntegrted used progrm C nd Messge Pssng Interfce (MPI) n computer cluster \ Keywords : Dffuson Model, Pent-Dgonl System, Prllel Computng, Modfcton Guss Sedel, Cluster Computer * Pust Pengembngn Informt Nulr (PPIN) BATAN, e-ml: me@btngod 45

2 Rslh Lory Komputs dlm Sns dn Tenolog Nulr, Otober (45-58 ) PENDAHULUAN Slh stu persoln dlm bdng retor nulr dlh pemechn trnsport neutron Perhtungn nl rterst dn dstrbus flus neutron dlh pust dr nlss fss retor nulr Dlm perhtungn dstrbus neutron dlm retor hrus dperhtn proses trnsport neutron, ytu pergern lur neutron Bny stud retor mempeljr perlun pergern neutron sebg sutu proses dffus Pr ten nulr mempeljr bgmn mendesn retor demn sehngg terdpt esembngn ntr produs neutron dlm res fs dn hlngny neutron ren hmburn tu dserp Slh stu persoln trnsport neutron dlh penyelesn model dffus -D Sebg pendetn persoln trnsport neutron, model dffus merupn model stndrd yng dgunn dlm nlss n Dlm pemechn persmn dffus, prosedur n berhdpn dengn persoln mtrs Dlm penyelesn dffus -D, mtrs membentu mtrs pent-dgonl Model mengsumsn bhw semu neutron dpt drterst dengn perlun energ pd du dmens Model dffus -D mempuny sstm persmn lner pentdgonl Prlelss penyelesn dffus -D dlm omputer luster dbhs dlm tulsn n Beberp lgortm telh dembngn untu dembngn untu penyelesn prlel pd sstem pent-dgonl Sstem pent dgonl dlh sstem persmn lner yng memerlun penyelesn husus Untu membut perhtungn dffus lebh efetf, proses lgortm Modfed Guss Sedel dfousn dn dterpn dlm unsur-unsur pent-dgonl Metod Modfed Guss Sedel dlm pemechn dffus -D dengn mtrs yng besr n dpresentsn Metod n sngt sesu dterpn dlm sstem omputer luster Sebgn besr opers dlm metod modfs Guss Sedel dpt derjn secr prlel Komputs prlel dr pent-dgonl untu pemechn model dfuss -D dmplementsn secr ntegrs menggunn progrm C dn Messge Pssng Interfce (MPI) dlm omputer luster Dperoleh wtu yng sngt sngt dlm menyelesn sstem persmn Dffus -D untu uurn mtrs besr dengn metod modfed Guss Sedel secr prlel dlm omputer luster METODOLOGI Perhtungn dffus neutron menghususn perlun -D tu -D Pd retor nt dhtung freuens hmburn neutron tu berp bny neutron dserp Rso jumlh neutron dlm stu geners dn jumlh neutron pd geners sebelumny dsebut dengn ftor multpls J =, jumlh neutron dlm geners fs sebelumny dn sesudhny dlh sm dn res rnt td bergntung wtu Krterst sstm dnytn dengn = menunjun retor rts Persmn dffus dlh persmn dferensl prsl geners fs yng 46

3 Pechn Persmn -D Dffus Secr Prlel menggunn Metod Modfed Guss (Me Susmnt) menytn flutus erptn mterl selm dffus Solus numer persmn dffus lebh umum dnytn pd persmn () [], D( r) + Σ ( r) ( r) = S( r) () (r) erptn mterl dffus pd los r nd D(r) dlh umpuln oefsen dffus untu erptn pd los r Smbol merepresentsn smbol opertor dfferensl vetor pd oordnt bdng S(r) dlh sumber pd los r dn Σ dlh probblts res neuron-nulr yng terjd yng dnytn dengn nl penmpng lntng Geometr yng ngn dseld ddsrtss edlm mesh dr sel-sel sepert grd-grd segempt yng dnytn pd Gmbr Gmbr Grd seg empt -D J ddefnsn D j dlh oefsen dffus pd mesh tt dn j, D j ( D + D j ) () M persmn () dpt dnytn dlm bentu persmn () J D J j Dj () j + ( + ) = S j j= j j = j dmn j = jr dntr mesh tt dn j dengn berjln pd semu tt-tt mesh J dperlus persmn bed untu persmn dffus -D dengn med mesh sergm dlm geometr segempt dnytn pd persmn (4), D D ( x, y) = S( x, y) x y (4) 47

4 Rslh Lory Komputs dlm Sns dn Tenolog Nulr, Otober (45-58 ) Tt-tt mesh dnytn dlm rung mesh dengn pendetn persmn (5), x, j, j +, j ( x) + x, y, j, j +, j + y x, y ( y) (5) J nl pd persmn (5) dsubttus pd persmn (4), evlus terhdp tt mesh ( x, y j ) dperoleh dr persmn (6) [], D D ( + + ) ( + + ) +, j, j, j, j ( x) ( y) D + Σ + ( + ) j S, = j, =,, N-,j=,,M- (6) ( x) ( y), Syrt bts yng dgunn,, j, N, j,, dn, M Pd penyelesn persmn dffus -D, pertm-tm dlh membentu mtrs oefsen dfus dengn member tnd ndes tunggl untu tp tt mesh Sebg contoh dlm mesh -D, tt mesh dber lbel (,j) = + (j-)(n-) Untu persoln -D, strutur mtrs dengn jjrn mesh x j, dperoleh prmeter dstrbus flus sebny = x j, ytu,, S 5 S 6 S 45 4 S 4 = S A = S Sebg contoh, Pd brs e- = 4, ( = 7 dn j =), hsl perln mtrs dts dnytn pd persmn (7), D7, D7,6 D7, D7,6 D7,8 D7, ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ) 7 ( 7, 7,6 7, 7,6 7,8 7, 48

5 Pechn Persmn -D Dffus Secr Prlel menggunn Metod Modfed Guss (Me Susmnt) D7,8 D7, ) ( ) ( 7 ) = ( S ) ( ) ( ) (7) ( 7 7,8 7, Persmn (7) dgunn dlm perhtungn rtlts pd persmn (8) [], M = F (8) Penyelesn dengn metode ters membern tebn sumber wl dn -effetf ( eff ), dnytn pd persmn (9), () S( r) F S ( r) () (9) Kemudn dhtung flus () berdsrn persmn (), M ( = S ( ) () () D + Σ r) () Sumber bru dn eff dpt dhtung menggunn persmn (), () () () S = F = υσ f () () () Prosedur ters untu menghtung eff dpt dsederhnn berut n [, 5], Tebn wl untu = eff () dn () Kemudn pechn untu (n+) = / (n) eff x F x (n), F = ν f ( N M Htung eff bru : (n+) (n) eff = eff ( M dr dr Ulng ()-() smp eduny eff (n+) dn (n+) onvergen, Dgrm lur perhtungn eff dnytn pd gmbr + ) N ) 49

6 Rslh Lory Komputs dlm Sns dn Tenolog Nulr, Otober (45-58 ) Input geometr dn omposs Input sumber fs wl (S () ) dn () Iters dlm M = F ( n + ) ( n ) ( n ) S ( n + ) ( n+ ) = F d ( n ) rs d ( n+ ) ( n + ) rs ( r ) ( n ) ( r ) Iters lur ( n) ( n) ( n ) Y ( n) ( n ) S S < ε < ε ( n) S Td eff Gmbr Dgrm Alur Perhtungn Krtlts Sstem persmn A = S membern nl flus dlm vetor Mtrs A dlh mtrs oefsen dffus dn S dlh vetor nl sumber Nl unsur-unsur mtrs A dlh bentu sederhn dr persmn bed hngg Penyelesn dsrtss flus dperoleh dr penyelesn persmn berut = = = S S S NN N + NNN = S N 5

7 Pechn Persmn -D Dffus Secr Prlel menggunn Metod Modfed Guss (Me Susmnt) Prnsp lgortm untu pemechn sstm persmn lner Ax = b dterpn secr prlel dengn memodfs metod Itertf Guss-Sedel dengn Modfed Guss-Sedel Persmn dts dpt dnytn sebg persmn (), A = E + D + F () dmn E, D dn F dlh mtrs n x n dengn msng-msng unsur e j, d j dn f j berut; e j = > j untu j untu lnny d j = = j untu j untu lnny f j = < j untu j untu lnny Persmn Ax = b dpt dnytn dlm bentu persmn (), (E + D + F)x = b () tu dlm bentu persmn (4) Dx = b Ex Fx (4) Dmul dengn vetor x ( dugn sembrng untu x wl), penyelesn vetor dperoleh mellu proses tertf dmn ters e- dbern pd persmn (5), D x = b E x F x (5) Iters dtn onvergen j untu sutu, memenuh persmn (6), n = bs + ( x x ) < c, (6) dmn c dlh nl tolerns eslhn Proses dts dpt dmodfs secr umum menjd lgortm prlel Modfed Guss Sedel yng dpt dterpn pd sstem omputer luster, ren ddesn untu sstem prlel Multple Instructon Multple Dt Berut lgortm prlel dr metode Modfed Guss Sedel [], Prosedur MIMD MODIFIED GAUSS SEIDEL Thp : for = to n do () old = x () new = x Htung Proses end for 5

8 Rslh Lory Komputs dlm Sns dn Tenolog Nulr, Otober (45-58 ) Thp : Proses () repet () old = new () new = ( b ( x old ) ( x old )) / n untl () x = new = = n = + bs( new old ) < c Thp, menytn stu dr n proses dlm Thp Msln untu n = 4 (empt vrbel yng ngn dcr) dn ters wl, =, mulmul dbern nl x, x, x, x4 wl dn nl tolerns c Selm ters e- (=) pendetn nl x yng td dethu, dsubsttus edlm rus nn persmn (5), yng menghsln nl pendetn bru berut n, x = b ( x + x + ) 4 x4 x = b ( x ) ( x 4 x4 ) + x = b ( x + x ) ( 4 x 4 ) 44 x4 = b4 ( 4x + 4 x + 4x ) tu x = ( b ( x + x + x )) 4 4 / = ( b ( x ) ( x 4 x 4 )) / x + x 4 = b4 ( 4x + 4 x + 4 x ) ) / ( Ilustrs peerjn untu msng-msng prosesor sebg berut, prosesor membern nl old = x dn new = x Secr smultn prosesor membern nl old = x dn new = x Demn pul untu prosesor dn prosesor 4 Nl x, x,, x4 dpt dhtung dr sstm persmn dts yng menjd nl new, new,, new4 pd ters pertm, sedngn nl old = x, old = x,, old 4 = x4 Kemudn dhtung new old + + new4 old 4 < c J syrt onverven belum tercp m ters dlnjutn dengn menympn nl old = new,, old 4 = new4 dn menghtung new, new,, new4 dengn menghtung x, x,, x4 Demn selnjutny untu ters e- sehngg syrt onvergen terpenuh 44 5

9 Pechn Persmn -D Dffus Secr Prlel menggunn Metod Modfed Guss (Me Susmnt) HASIL DAN PEMBAHASAN Proses smuls prlel dlun dengn beberp prosesor Proses n ntegrs dengn progrm C, onsep MPI (Messge Pssng Interfce) pd sstem Open Source LINUX dlm sstem luster [,4] Geometr nput dbg dlm pnjng mterl 5 cm dn jumlh prts 4 per cm (n =, delt() =, =, ) Input mterl melput penmpng lntng sgmbs(), onstnt dffus d(), produtvts per fss nusgm(), tebn wl () = dn eff wl ( eff ()) = Pertm-tm dhtung unsur-unsur mtrs oefsen dffus A, dr persmn A = S Untu perhtungn dstrbus flus, dgunn metod Modfed Guss Sedel yng dterpn dlm omputs prlel pd sstem luster untu penyelesn sstm persmn pent-dgonl A = S dlm bentu mtrs x Sstem pent-dgonl dperoleh dr proses pembentun mtrs oefsen dffus A Vetor S menytn nl sumber yng dnytn pd rus nn sstm persmn Vetor menytn nl flus yng n dhtung dlm bentu mtrs berut n, = Nl flus,,, 4, 5, 6, 7, 8, 9,,,,,, 998, 999, dhtung dr proses Modfed Guss Sedel yng msng-msng dperoleh, 58666; 4864; 55549; ; 954; 99648; ; 49966; ; ; ; Dperoleh nl -eff =,45544 Proses prlel dlun dengn fslts MPI pd tg omputer Dul Core dlm lol jrngn (LAN) dengn sstem luster Knerj prlel perhtungn rtlts dn dstrbus flus menggunn metod Modfed Guss Sedel dnytn pd tbel pd sstem luster untu penyelesn sstm persmn pent-dgonl dengn uurn mtrs x 5

10 Rslh Lory Komputs dlm Sns dn Tenolog Nulr, Otober (45-58 ) Tbel Knerj Prlel Perhtungn Krtlts dn Dstrbus Flus Jumlh Prosesor Wtu (det) Wtu Per Prosesor (det) 6,64, 6,6, 4,4,8 5,4,69 6 7,97, 7 6,8,4 8,,88 Flutus nerj prlel perhtungn rtlts dn dstrbus flus dengn memperhtn jumlh prosesor terhdp wtu dtmpln pd Gmbr Gmbr Knerj Prlel Perhtungn Krtlts Flutus nerj prlel per-prosesor terhdp wtu dtmpln pd Gmbr 4 54 Gmbr 4 Knerj Prlel Per-Prosesor Secr umum sstm persmn n vrbel dpt dproses secr prlel menggunn lgortm Modfed Guss Sedel yng drncng pd sstem prlel MMID dlm sstem luster yng mengjnn proses ddstrbus pd node-node dengn pembgn bebn yng dserhn pd sstem Seln hl dts secr onsep omputer dul core msng-msng berfungs sebg omputer dengn du prosesor

11 Pechn Persmn -D Dffus Secr Prlel menggunn Metod Modfed Guss (Me Susmnt) sehngg proses dpt ddstrbusn pd mnml du node tu lebh untu msngmsng omputer Sehngg tg omputer dul core berfungs sebg mnml enm prosesor Smuls prlel pd sstem luster yng berntegrs dengn fslts MPI, dperennn dproses pd beberp node tu prosesor Dlht dr grf nerj prlel per prosesor bhw nerj prlel yng optml dperoleh pd tg node wlupun proses dmungnn dlun untu enm smp delpn node, tetp dperoleh hsl yng td optml wlupun dlm jrngn lol selpun Hl n dsebbn wtu untu mendstrbusn bebn tu mengumpuln hsl perhtungn n menmbh wtu proses Td menutup emungnn pul dny jrngn omuns yng td lncr wlupun dlm jrngn lol selpun KESIMPULAN Model dffus -D secr numer dnytn dlm sstem persmn pentdgonl Metode Modfed Guss Sedel dpt dterpn dlm penyelesn sstem persmn pent-dgonl Algortm Modfed Guss Sedel dpt dplsn untu sstem persmn dengn uurn mtrs oefsen dfus yng besr dlm wtu sngt dengn sstm prlel Multple Instructon Multple Dt pd sstem luster Dlm smuls n, pd tg omputer Dul Core dengn jrngn lol ternyt dperoleh nerj prlel yng efsen dbebnn hny terhdp tg prosesor, wlupun dpt dproses pd lebh dr tg prosesor Hl n dmungnn, dlm mendstrbusn peerjn dn mengumpuln hsl n menmbh wtu proses DAFTAR PUSTAKA SELIM G, AKL (989), The Desgn nd Anlyss of Prllel Algorthms, Prentce Hll, Inc, New Jersey DUDERSTADT, JJ, HAMILTON, L J, (976), Nucler Rector Anlyss, John Wley & Sons, Inc, (976) GRAMMATIKAKIS, MILTOS, D, FRANK HSU, D, nd KRAETZL, MIRO (), Prllel System Interconnectons nd Communctons, CRC Press LLC, New Yor 4 KARNIADAKIS, GEORGE, E, KIRBY, LL, ROBERT, M (), Prllel Scentfc Computng n C++ nd MPI, Cmbrdge Unversty Press, Frst Publshed 55

12 Rslh Lory Komputs dlm Sns dn Tenolog Nulr, Otober (45-58 ) 5 SUSMIKANTI M, SETIDIPURA, T, ADHIYAKSA, A (9), The Prllel Computton for Trdgonl System In One-Dmenson Dffuson Model, Journl of Theortcl nd Computtonl Studes Volume 8 Number 46, ISSN , LIPI KARSONO DISKUSI Srn : Mlh sehrusny menjelsn lgortm prlel yng dgunn dn jug penjelsn pembgn bebn erj e prosesor-prosesor Aspe prlelss n dlh nt mlh (yng belum djelsn) Mohon penjelsn urv nerj prosesor flututf; menmbh jumlh prosesor td sellu mempercept proses /wtu htung MIKE An dtmbhn dn dmsun dlm mlh sebg perbn Sstem prlel n derjn pd empt omputer Dul Core, tetp dengn emmpun berbed, sehngg setelh dlun smuls dperoleh wtu yng mnml pd tg prosedur JAd pd prnspny d emungnn bhw nerj slh stu omputer td b dn emungnn ln bhw perjlnn untu melun proses pd 8 prosesor (4 omputer dengn Dul Core) lebh memn wtu DAFTAR RIWAYAT HIDUP Nm : Me Susmnt Tempt & Tnggl Lhr : Jrt, November 956 Penddn Rwyt Peerjn Kelompo : S- Mtemt, S- Mgster Mnjemen : 98 sd serng d BATAN : Mtemt dn Sttst Terpn 56

13 Pechn Persmn Persmn -D Dffus -D Dffus Secr Secr Prlel Prlel menggunn menggunn Metod Metod Modfed Modfed Guss (Me Guss (Me Susmnt Susmnt) Mlh : Pemechn Persmn -D Dffsus Secr Prlel menggunn Metod Modfed Guss Sedel dlm Komputer Kluster 57