Mesh Free Parallel Programming for Electromagnetic Problems

Transcription

1 Joal of Mcowaves Optoelectocs ad lectomagetc Applcatos ol. No. Je 9 S Mes Fee Paallel Pogammg fo lectomagetc Poblems Aleade R. Foseca Mgel L. Medes Reato C. Mesqta lso J. Slva Depatameto de geaa létca Uvesdade Fedeal de Mas Geas {afoseca lmamedes eato elso}@cpdee.fmg.b. Abstact Mes fee paallel mplemetatos sg mlt-coe pocessos ae dscssed. Patclal paallelzato of te MLPG ad SPM ae sow. Te appopate stateges to coectl paallelze te mes fee metods ae dscssed. Paallel vesos p to.9 tmes faste o dal-coe pocesso ad.7 o qad-coe pocesso we compaed to te seqetal oe. Ide Tems Paallel Pogammg Mes Fee Metods MLPG SP. I. INTRODUCTION Pscal poblems descbed b patal dffeetal eqatos ae sall solved sg mecal metods. Tadtoal metods deped o a mes o gd dscetzato of te doma wc s obtaed tog te coectvt amog odes. Te most commo metods ae te Fte lemet Metod FM ad Fte Dffeece Metod FDM. Te mes geeato mposes some lmtato to te class of poblems tese metods ca solve. Fo stace FDM eqe a stcted gd mes wc does ot ft well to comple geometes sc as tose wt cvates o wt ve dffeet geometc feate szes. Tese caactestcs ma lead to dscetzato eos o to meses tat ae too efed de to small feates te geometc pt. O te ote ad FM ses a stcted mes wc s able to ft comple geometes. oweve te tas of mes bldg s ot alwas smple especall fo comple D geometes wee geeatg a mes wt satsfacto qalt s stll a eseac poblem. Ule mes-based metods mes fee metods MFM do ot eed te se of a mes to defe te base fctos. Istead MFM se a set of scatteed odes wt te poblem doma ad defe appopate base fctos o eac ode Te wa tose fctos ae defed vaes fo eac metod ad s beod te scope of ts eseac. Tese metods ae a ve attactve coce especall we solvg poblems tat volve movemet boda ad sape defomato. oweve te absece of a mes todces a few dawbacs sc as egbo seac local mat vesos ad ote comptatos tat ltmatel cease te comptatoal cost. Coseqetl ode to cease te metods effcec specal tecqes sold be emploed []. Amog seveal mesless metods some of te most sed b te electomagetc commt ae: lemet Fee Gale FG [] [] [] [] Mesless Local Petov-Gale MLPG [5] [] [] [7] ad Smoot Patcle dodamcs SP []. All tese metods solve te poblem costctg te appomato sg patces to cove te doma. ac patc as a assocated ode wee te Bazla Mcowave ad Optoelectocs Socet-SBMO eceved 5 Nov 9; evsed Mac 9; accepted Mac 9 Bazla Socet of lectomagetsm-sbmag 9 SBMO/SBMag ISSN 5-799

2 Joal of Mcowaves Optoelectocs ad lectomagetc Applcatos ol. No. Je 9 S ow vale s defed. SP was oe of te fst mes fee metods to be poposed. It s based o a tegal fomlato of a fcto. Te Dac s delta fcto s appomated b a eel fcto ad te tegal s pefomed ove a set of patcles []. Itall SP was fomlated to solve astopscs poblems [9] bt latel t s mostl appled fld damcs []. Recetl te SP as bee adapted to solve tme doma electomagetc poblems esltg te metod ow as Smooted Patcle lectomagetcs SPM []. I SPM te patcles of te SP ae dvded to two gops: electc ad magetc patcles. Te eale sall eep te fomato abot te electcal feld compoets wle te latte eeps te fomato abot magetc feld compoets. Te SP tepolato s te appled to te tme depedet Mawell eqatos []. SP as ma comptatoal advatages: t s smple to pogam fll paallelzed etc. FG was developed b Beltsco et al. [] 99. Te metod s based o te Dffse lemets Metod DM wc was te fst Gale tpe of mesless tecqe. Itall bot metods wee developed to solve mecacal poblems ad ecetl o appled to comptatoal electomagetc [] []. FG s cosdeed a mesfee metod te sese tat o coectvt amog odes s ecessa. Neveteless to pefom te mecal tegato FG eqes a ala gd tat mst volve te ete doma. Ts gd does ot deped po te ode dstbto ad ca be cose to be egla; teefoe t s fa smple ta te tpcal FM mes. Le FM becase of te Gale metod te tal ad sape fctos mst be fom te same space. Movg Least Sqae MLS appomato tecqe s commol cose fo costctg tal ad sape fctos []. As t wll be peseted secto II te MLPG ses a pocede amed local wea fom. I ts tecqe eac ode defes ts ow sb-doma tat s depedet fom te emag. Tose sbdomas ae allowed to ave a sze ad sape ad te elmate te eed of a gd fo mecal tegato. Ftemoe te Petov-Gale metod allows te tal ad sape fctos to be cose fom dffeet spaces. Tese two feates mae ts metod ve fleble ad dstgses MLPG as a tl mesfee metod. Te mesfee metods dscssed te pevos paagaps ave te advatage of beg easl paallelzed. Ts s especall mpotat te cet sceao of pocessos evolto. Te fast wdespead of te mlt-coe pocessos became otable. Oe of te ma mpovemets te ea fte wll be te ceasg mbe of coes [] so mlt-teaded codes ae mpeatve to pefomace eacemet. Fo a lage mbe of mecal metods tat solve dffeetal patal eqatos te pocess ca be dvded to two dstct pases: te costcto of a lea sstem ad ts mecal solto. Ts s te fo FM FG ad MLPG fo stace. To solve te lea sstem a teatve metod s sall sed. Te paallelzato of te solto of a lea sstem wt a teatve metod ca be coveted to a gap patto poblem [] []. I [] a algotm fo spase sstem of lea eqatos s poposed ad appled fo a dstbted veso of FG. Bazla Mcowave ad Optoelectocs Socet-SBMO eceved 5 Nov 9; evsed Mac 9; accepted Mac 9 Bazla Socet of lectomagetsm-sbmag 9 SBMO/SBMag ISSN 5-799

3 Joal of Mcowaves Optoelectocs ad lectomagetc Applcatos ol. No. Je 9 S Tee ae ote metods to solve dffeetal patal eqatos tat do ot lead wt te costcto of a lea sstem le RKPM SP ad SPM. I tese metods te solto s calclated dectl fo eac ode sg ts egbog. I [5] DDM Doma Decomposto Metod s mplemeted fo D RKPM sstem. Te pape focses te patto metods ad data stctes fo comple commcatos amog domas. Te ode pocessg o mes fee metods s ofte based pocessg appomatos o te egbog odes. Ts pocede s ve tme cosmg we compaed fo stace wt te Fte lemet Metod FM pocede bt ts paallelzato as sowed to be a smple wa to speedp ad t stfes te seac fo ote paallel alteatves. OpeMP s a wdel sed API fo paallelzato [] [] becase t eeds st a few cages o a seqetal code. A attactve alteatve s te fee C++ Boost Tead lba [7] wc s based o a obect oeted appoac. Fo tose wo ae accstomed to pogam sg te obect oeted paadgm te last opto wll be qte famla ad eas to se. OpeMP s also eas to se bt eqes atve comple sppot to te OpeMP API wc s ot alwas avalable. I ts wo te MLPG ad SPM fomlatos ae peseted ad some eas appoaces to paallelze tose metods sg te Boost Tead Lba mlt-coed pocessos ae poposed. Te esltg pocessg tmes ae compaed wt te oes fom seqetal vesos. We focs o a mltcoe CPU wt saed memo model. I sc scee commcato amog domas s ot a poblem ad te paallel pocessg of odes becomes te mao coce. I SPM calclatos te pocessg s totall depedet o eac ode at a gve tme step so te metod paallelzato s doe mal based o ode dstbto to pocesso coes ad scozato of tme steps. I MLPG te solto of te lea sstem cold be pefomed sg gap pattog tecqes le MTIS [] bt tat s ot te am of ts wo. ee we lead wt te paallelzato of te costcto of te lea sstem. De to te decomposto to depedet sbdomas te paallelzato of te lea sstem assemblg of MLPG s atal ad t s a eas wa to cease ts speedp. II. T MLPG MTOD Te Mesless Local Petov Gale MLPG metod dffes fom te ote mesless tecqes based o te Gale metod becase te Petov-Gale appoac ses tal ad test fcto fom dffeet spaces [5] []. B sg ts appoac t s possble to solve a boda vale poblem as seveal localzed poblems stead of a sgle global poblem as te FG metod []. To llstate te Mesless Local Petov Gale MLPG metod fomlato we wll cosde a two dmesoal electostatc poblem a abta doma wt Dclet boda codto fo te electc potetal o Fg.. Bazla Mcowave ad Optoelectocs Socet-SBMO eceved 5 Nov 9; evsed Mac 9; accepted Mac 9 Bazla Socet of lectomagetsm-sbmag 9 SBMO/SBMag ISSN 5-799

4 Joal of Mcowaves Optoelectocs ad lectomagetc Applcatos ol. No. Je 9 S q q q q q q q Fg.. MLPG doma epesetato. Te appomated solto boda codto s mposed sg te pealt metod as s sogt sg te wegted esdal metod ad te essetal W d W d wee W s te test fcto s te electc pemttvt of te medm ad s te pealt paamete wt tpcal vales fom to []. Te appomated solto at te ode s defed as: wee s te vale of te sape fcto of te ode at te ode s te mbe of odes te egbog of tat ae sed to bld te appomato ad s te vale of at te ode. Te test fcto W s cose to ave compact sppot wc defes a closed sb-doma ceted at eac ode as sow Fge. Coseqetl ca ow be solved locall [5] eac q : q q W d W d Te geealzed local wea fom obtaed ca be tasfomed to: q q W d W d W d q I mesless metods te sape fctos ca be defed sg dffeet pocedes sc as te Movg Least Sqaes MLS te Keel metod Radal Bass Fctos RBF o polomal appomatos amog otes. Accodg to Atl [5] a of te above metods ca be sed te MLPG. Atl also sggests possble fctos tat ca be sed to defe W sc as te eavsde step fcto o te MLS wegt fcto. I ts wo W s cose to be te eavsde fcto.e. W o q ad W otsde q. It meas tat W o q ad ol te tegato o te segmets of te boda of te sb-domas Bazla Mcowave ad Optoelectocs Socet-SBMO eceved 5 Nov 9; evsed Mac 9; accepted Mac 9 Bazla Socet of lectomagetsm-sbmag 9 SBMO/SBMag ISSN q q eeds to be evalated. Ts leads to []:

5 Joal of Mcowaves Optoelectocs ad lectomagetc Applcatos ol. No. Je 9 5S q ˆd d q Wee ˆ s te omal ta vecto ove q. I fact o coce of wegt fcto s based o te Atl s wo [5] wee t s sow tat amog seveal wegt fctos te eavsde fcto povdes accate eslts. Applg te gadet opeato we obta Replacg ad 5 ad dog ˆ leads to 5 N d d d 7 q q q fo te t ode ad N s te mbe of odes belogg to te flece doma of ts ode. I mat fom we ave: wee v s te vecto of ows ad K K q f Kv f q d d 9 f d Te tegato of a sbdoma defed b te ode cotbtes ol to te K mat ad to te t q t le of te posto of te f vecto. ac ode eeds to ow te egbo odes wc flece ts doma. Ts s doe b sg a seac tee []. If a ode does ot affect te ode s sbdoma te te elemet K otewse K Fg.. K f Fg.. Node cotbtos fo te lea sstem. Te te tegato of dstct odes fleces dffeet les of K ad ca be calclated a ode. So te paallelzato of te assemblg pocede s a atal wa to edce comptg tme Bazla Mcowave ad Optoelectocs Socet-SBMO eceved 5 Nov 9; evsed Mac 9; accepted Mac 9 Bazla Socet of lectomagetsm-sbmag 9 SBMO/SBMag ISSN 5-799

6 Joal of Mcowaves Optoelectocs ad lectomagetc Applcatos ol. No. Je 9 S as sow at Fg.. Note tat tee s o eed of a scozato metod le a mte o semapoe [9] sce eac posto o te mat s depedet. Load data ad Italze seac tee Assemblg Tead Assemblg Tead Assemblg Tead N Jo Teads Solve Lea Sstem ad Wte Solto Fg.. Paallel assemblg fo MLPG. Ol te assemblg pase fo te MLPG was paallelzed. Te mfpac lba was sed to solve te smmetc spase lea sstem [] []. III. T SPM MTOD Te Smooted Patcle dodamcs SP [] s based o te appomato of a fcto f b ts tegal epesetato: f f d wee s te doma of fcto f ad s te Dac s delta fcto. Nmecall we solve te tegato o sg te patcles appomato. Te delta fcto s appomated b a fcto W sall efeeced as eel fcto. Te eel fcto defto depeds o a smoot ads. Fo smplct we wll assme tat all patcles ave te same smoot ads. Te elato betwee ad te dest of patcles te poblem s doma sall be teated a caefl mae []. A too small combed wt a low dest of patcles leads to bad appomatos wle te se of a too lage leads to te loss of local popetes ad also eslts bad appomatos []. Te tegal s pefomed ove a set of patcles. Deotg te SP appomato b te spescpt te vale of fcto ca be appomated b: N W wee deotes te posto of patcle s a measg of te sodg meda at te posto of patcle ad N s te mbe of patcles belogg to te sppot doma of patcle. Te measg fcto ca be obtaed b te followg: N W Te eel appomato s of secod ode accac. It s poved tat ts appomato ca also be appled to devates of fcto : Bazla Mcowave ad Optoelectocs Socet-SBMO eceved 5 Nov 9; evsed Mac 9; accepted Mac 9 Bazla Socet of lectomagetsm-sbmag 9 SBMO/SBMag ISSN 5-799

7 Joal of Mcowaves Optoelectocs ad lectomagetc Applcatos ol. No. Je 9 Bazla Mcowave ad Optoelectocs Socet-SBMO eceved 5 Nov 9; evsed Mac 9; accepted Mac 9 Bazla Socet of lectomagetsm-sbmag 9 SBMO/SBMag ISSN S N z z W Tee ae ma possble coces fo W. Ts fcto mst ol espect some basc eqemets: ave compact sppot ave te t popet be a mootocall deceasg fcto amog otes []. I ts wo te coose eel fcto s te epoetal: ' ' ' ep d f d f d d W 5 wee ' d s te dstace dvded b te smootg legt ad s set accodg to te mbe of dmesos []. Te Smooted Patcle Itepolato SPI s a geealzed veso of SP. Te SPI was poposed fo te solto of geeal patal dffeetal eqato sstems. I [] [] te SPI metod s appled to te tme depedet Mawell eqatos esltg te metod amed as Smooted Patcle lectomagetc Metod SPM. We appled to electomagetsm te patcles of te SP ae dvded to two gops of patcles: electc ad magetc patcles. Te eale sall eep te fomato abot te electcal feld compoets wle te latte eep te fomato abot magetc feld compoets. I ts wo patcles ae assmed to be alwas mmoble. Te SP tepolato s appled to te tme depedet Mawell eqatos []. Te SP s sed to solve te spatal devates ad te leap-fog sceme s sed to solve te tempoal devates. Te tme step of te esltat sceme s sbected to te Coat-Fedcs-Lev CFL stablt codto wc eqes te tme step to be popotoal to te smallest smootg legt []. Also we sg eve patcle dstbto te cosstec codto mst be sed to pevet degade te eslts []. I ts wo te PML Pefectl Matced Laes meda eqatos ae sed to smlate absobg codtos []. Te fal sceme eqatos ae: / / / z N z z W t / / / z N z z W t 7 * / / N z z W t * / / N z z W t 9 wee te vaos ae omogeeos electc ad magetc codctvtes s te magetc feld s te electc feld ad ae te vacm electc pemttvt ad magetc pemeablt. qatos tog 9 ae sed to pdate te patcle vaables o te ete meda. Te patcles dstbted ove te o-absobg meda ae set wt wle te absobg patcles as appopate vales [5].

8 Joal of Mcowaves Optoelectocs ad lectomagetc Applcatos ol. No. Je 9 S B eqatos to 9 we ca see tat electc patcles pdate depeds ol o te magetc patcles te egboood ad vce vesa. Fg. sows a eample of ts scema. Te ccles epeset magetc patcles ad te coss smbols epesets electc patcles. Te glgted magetc patcle as fo electc glgted teacto pas. Notce tat magetc patcles te egboood of ts patcle ae ot glgted becase te ae ot teacto pas. Fg.. Iteacto betwee patcles SPM. It mst be also otced tat magetc feld pdates mst be doe befoe te begg of pdates of electc feld. Ts s epessed te eqatos b te dstcto of steps to two dffeet sb-steps. We pdatg electc felds we sa tat we ae advacg fom step / to step / wle we pdatg magetc felds we ae advacg fom step to step. A complete step of SPM s scema wold volve bot electc ad magetc feld pdates. As pdate of patcles of te same tpe do t deped o eac ote tese patcles ca be depedetl pdated. Te ol estcto s tat all electcal patcles mst be pdated befoe te magetc patcles pdate begs as sow Fg. 5. Notce tat fo SPM te ode pocessg coespods to te solto at eac tme step. Italze odes Update lectc Feld Tead Update lectc Feld Tead Update lectc Feld Tead N Jo Teads Update Magetc Feld Tead Update Magetc Feld Tead Update Magetc Feld Tead N Jo Teads Fg. 5. Paallel loop fo SPM. I. NOD PROCSSING DISTRIBUTION STRATGIS To avod effcet mplemetatos te ode dstbto to te teads mst be caefll doe. Some smple appoaces ave bee cosdeed ts wo. A fst ad mmedatel appoac wold be to smpl dvde te odes to gops accodg to te ode s dees. If we ave a pocesso wt coes ad m odes we pt te odes deed fom to m / to be pocessed b te fst tead te odes deed wt m/ to m / to te secod Bazla Mcowave ad Optoelectocs Socet-SBMO eceved 5 Nov 9; evsed Mac 9; accepted Mac 9 Bazla Socet of lectomagetsm-sbmag 9 SBMO/SBMag ISSN 5-799

9 Joal of Mcowaves Optoelectocs ad lectomagetc Applcatos ol. No. Je 9 9S tead ad so o. I Fg. tee s a eample of sc stateg fo dvg eqall spaced ad seqetal deed odes to two coes. We wll efe to ts as Seqetal Dstbto SD. Fg.. Node dstbto sg seqetal appoac. Tag a fst loo to Fg. we cold t tat SD wold be eog to mae a eqall dstbted cage of odes to teads. Bt pelma tests ave sow tat ts s ot te ad te easo s qte obvos: odes tat ae ea to te bodaes ave less egbo odes ta tose fa awa fom t. If we ad fo coes we wold ave te coes two ad tee ovecaged we compaed to coes oe ad fo. Tat wll eslt a geate pocesso dle tme. A alteatve to SD tat ovecomes ts poblem s te appoac we wll efe as Alteated Dstbto AD. Istead of dstbtg te m/ odes seqetall fo eac ode we ca dstbte tese odes a alteate mae amog coes. All odes tat ae deed b a tege wose tege dvso b te mbe of pocesso coes as modls zeo wll be allocated to te fst pocesso. To te odes wee ts dvso s oe te pocessg coe wll be te secod oe. Te dstbto to te ote coes follow te same stateg. If we ae dealg wt a poblem wee te odes ae eqall spaced ad seqetall dstbted te we wll obvosl obta a alteate dstbto of odes to te coes. Fg. 7 sows a eample of ts appoac beavo sc stato. Fg. 7. Node dstbto sg alteated appoac =. Te poblem of cage dstbto ma loo completel teated at ts pot. Bt te AD appoac fals oe mpotat aspect. We ad to mae a ve stog assmpto o te poblem s defto specall we we ae dealg wt mesless metods: we spposed eqall dstbted ad seqetall mbeed odes. Bt as dscssed eale oe of te geatest advatages of mesless Bazla Mcowave ad Optoelectocs Socet-SBMO eceved 5 Nov 9; evsed Mac 9; accepted Mac 9 Bazla Socet of lectomagetsm-sbmag 9 SBMO/SBMag ISSN 5-799

10 Joal of Mcowaves Optoelectocs ad lectomagetc Applcatos ol. No. Je 9 S metods s tat odes ma be dstbted a eve wa stogl depedet o te poblem beg solved. To emove tese stog sppostos we ave to et o appoaces. Oe alteatve wold be to se a adom dstbto of odes as sow Fg.. I sc case we sot a eqal mbe of odes to eac coe. Ts sotg s made accodg to odes deg bt ow we do ot mae a eqemet o te ode dstbto o mbeg. We amed ts appoac as Radom Dstbto RD a b c a - Node d b - Radom allocato c - Tead d c = modb Fg.. Node dstbto sg adom appoac =. Te RD wll ot gaatee te best cage dstbto bt wll be mc moe applcable sce we do ot mpose a eqemets o ode dstbto. Also t wll be capable of dog a easoable dstbto of cage. It s possble to optmze te pefomace of RD wt a pe-pocessg. Oce ode dstbto s ow a gve poblem t s possble to cot te mbe of egbog odes o eac ode. Wt ts fomato we ca te dstbte te odes to gops of eqall caged odes.e. gops of odes wt te same mbe of egbos. Afte tat we st dstbte all odes te same gop a adom faso to all coes. We epeat ts last acto to all gops of odes ad obta a balaced dstbto. Fg. 9 sows a eample of ow ts appoac wold wo. We efe to ts appoac as Caged Radom Dstbto CRD. De to te pe-pocessg te CRD s sefl ol we ma teatos ae to be pefomed le SPM a b c d 5 7 e f ac - Node d bd - Nmbe of egbos f - Tead d f = mode e - Radom allocato Fg. 9. Node dstbto sg caged appoac =.. RSULTS I ode to valdate te MLPG metod te poblem sow Fg. a s cosdeed. Fg. b sows te eslt sg a egla 55 gd of odes. Te aaltcal solto fo ts poblem s gve b []: Bazla Mcowave ad Optoelectocs Socet-SBMO eceved 5 Nov 9; evsed Mac 9; accepted Mac 9 Bazla Socet of lectomagetsm-sbmag 9 SBMO/SBMag ISSN 5-799

11 Joal of Mcowaves Optoelectocs ad lectomagetc Applcatos ol. No. Je 9 S s s 5... s spaces a Poblem defto b MLPG eslt. Fg.. MLPG poblem defto ad eslt. To mease te aveage pecetage eo fo ts poblem we cosde: eo eact % ma wee s te total mbe of odes scatteed ove te doma. Usg te aveage eo was.7%. I SPM we cosde te poblem of te fte paallel codctve plates. Te plates ae paallel to te as wt a plate ove te le = ad te ote ove te le = a =.5m. Bot ae lmted to te left b te le = m ad ae lmted to te gt. A soce wave s efoced to te left of te ego at =m. To tcate te doma te PML absobg codto s mposed at te le = l =.m ad.m tc as sow Fg.. Addtoall we mpose te omogeeos Dclet boda codto o te lmts defed b te codctve plates. A soce wave s foced te left boda wt ampltde =./m ad feqec.f c wee f c s te ctoff feqec of te TM mode. A dest of patcles pe wavelegt was sed te smlato. Fg. sows te compaso betwee te aaltcal ad mecal soltos obtaed afte teatos tog te le =.m. Te aveage eo dsegadg te eslts at te absobg lae was.9%. a Poblem defto b Smlato eslt at tme t=.57-9 Fg.. Poblem defto ad mecal eslt fo SPM. Bazla Mcowave ad Optoelectocs Socet-SBMO eceved 5 Nov 9; evsed Mac 9; accepted Mac 9 Bazla Socet of lectomagetsm-sbmag 9 SBMO/SBMag ISSN 5-799

12 Joal of Mcowaves Optoelectocs ad lectomagetc Applcatos ol. No. Je 9 S Commo pesoal comptes wt dal ad qad-coe pocessos wee sed to test te paallel mplemetatos. Table I sows te obtaed speedps fo SPM ad MLPG. Te mplemetatos se eqall dstbted odes ad te deg s seqetal. De to tese smplfcatos te AD appoac s te optmal cage dstbto ode dstbto stateg. We ca se ts fact to mease ow good RD wll pefom. Bot appoaces wee sed SPM ad MLPG mplemetatos. Table I sows te eslts fo AD appoac. Secod colm sows te elatve tmes fo te lea sstem assemblg pase ol. As te ote pases fo te MLPG wee ot paallelzed et te total pefomace does ot ave te same speedp as we ca see colm tee. I all smlatos sg RD fo MLPG ad SPM te spet eta tme was less ta % we compaed to AD. Ts sows tat RD s a good solto fo geec ode dstbtos. TABL I. SPDUP FOR PARALLL IMPLMNTATIONS Nmbe of Pocessos MLPG Assemblg MLPG Total SPM I SPM te ode pocessg coespods to oe tme step solto. I tat wa te cosdeed tmes sowed te last colm ae tose obtaed fo eac teato. Scozato betwee te pdate of te dffeet ds of patcles cases a small oveead ad leads to some teads to be dle we t s ot possble to mae a pefect cage dstbto amog pocesso coes. I. CONCLUSIONS A smple wa to pefom te paallelzato of te lea sstem assemblg pocess was peseted fo te MLPG. Tat pocess s te most tme cosmg pase of te metod. oweve t s possble to eted te paallelzato to te ote pases as te solto of te lea sstem b sg paallelzed solves ad we obtag te solto appomato b eqato a postpocessg pase. I te same wa a smple paallelzato fom fo SPM was peseted. I bot metods tee s o eed of scozato fo a ode pocessg bt SPM te ecesst of scozg betwee pdates of electc ad magetc felds leads to g depedec o a good cage dstbto amog coes to obta bette speedps. Reslts sow ow mesless metods ca tae advatage of mlt-coe pocessos wtot mao cages o adaptatos o seqetal codes. Wt te ceasg mbe of coes commo pocessos t s epected tat tese metods become eve moe attactve. I fte wos te mplemetatos wll be eted to dstbted sstems. ACKNOWLDGMNT Bazla Mcowave ad Optoelectocs Socet-SBMO eceved 5 Nov 9; evsed Mac 9; accepted Mac 9 Bazla Socet of lectomagetsm-sbmag 9 SBMO/SBMag ISSN 5-799

13 Joal of Mcowaves Optoelectocs ad lectomagetc Applcatos ol. No. Je 9 S Te atos wold le to ta te Bazla agec CNPq fo facal sppot. RFRNCS [] G. F. Paea A. R. Foseca A. C. Lsboa. J. Slva ad R. C. Mesqta ffcet algotms ad data stctes fo elemet-fee gale metod. Magetcs I Tasactos o vol. o. pp. 59. [] G. F. Paea. J. Slva A. R. Foseca ad R. C. Mesqta Te elemet-fee gale metod teedmesoal electomagetc poblems. Magetcs I Tasactos o vol. o. pp [] G. R. L Mes Fee Metods: Movg Beod te Fte lemet Metod. CRC Pess. [] T. Beltsco Y. Y. L ad L. G lemet-fee Gale metods Iteatoal Joal fo Nmecal Metods geeg vol. 7 o. pp [5] S. N. Atl ad T. Z A ew mesless local Petov-Gale MLPG appoac comptatoal mecacs Comptatoal Mecacs vol. o. pp [] S. A. aa D. Rodge ad. C. La Mesless Local Petov-Gale Metod wt Radal Bass Fctos Appled to lectomagetcs Scece Measemet ad Tecolog I Poceedgs vol. o. 5 pp [7] A. R. Foseca S. A. aa. J. Slva ad R. C. Mesqta Imposg boda codtos te mesless local petov gale metod IT Scece Measemet & Tecolog vol. o. pp [] G. R. L ad M. B. L Smooted Patcle dodamcs: A Mesfee Patcle Metod. Wold Scetfc Pblsg Co. Pte. Ltd. [9] L. B. Lc A mecal appoac totestg te fsso potess Te Astoomcal Joal vol. pp [] G. Ala. Facomao A. Totoc. Toscao ad F. ola Smooted patcle electomagetcs: a mes-fee solve fo tasets Joal of Comptatoal ad Appled Matematcs vol. 9 o. pp [] C. Bscof D. a. Me C. Tebove ad S. Saolz Paallel Comptes vewee t Iteatoal Cofeece o te Comptato of lectomagetc Felds Je 7 pp [] G Kaps. Kma MTIS: Ustcted Gap Pattog ad Spase Mat Odeg Sstem Uvest of Mesota Depatmet of Compte Scece 995 [] G. Kaps. Kma Mltlevel -wa Patotog Sceme fo Iegla Gaps Joal of Paallel ad Dstbted Comptg vol. pp [] I.. Sga P.K. Ja "Paallel FG algotm fo eat tasfe poblems" Advaces geeg Softwae vol pp [5] L. Wag. L. Gagao a X. Z. Z. a "Developmet of paallel D RKPM mesless bl fomg smlato sstem" Advaces geeg Softwae vol pp 7 7. [] S. Io T. aawa T. Taaama ad A. Kamta valato of paallelzed mesless appoac: Applcato to seldg cet aalss ts Magetcs I Tasactos o vol. o. pp.. [7] BOOST Boost.Tead lba ttp:// 7. [] S. Io K. Taaa ad A. Kamta Iflece of metod fo mposg essetal boda codto o mesless Gale/Petov-Gale appoaces Magetcs I Tasactos o vol. o. pp [9] W.R. Steves. UNIX Netwo Pogammg olme Secod dto: Itepocess Commcatos Petce all 999. [] T. A. Davs "A colm pe-odeg stateg fo te smmetc-patte mltfotal metod" ACM Tasactos o Matematcal Softwae vol o. pp [] T. A. Davs "Algotm : UMFPACK a smmetc-patte mltfotal metod" ACM Tasactos o Matematcal Softwae vol o. pp [] G. Ala. Facomao A. Totoc. Toscao ad F. ola A smooted patcle tepolato sceme fo taset electomagetc smlato Magetcs I Tasactos o vol. o. pp [] G. Ala. Facomao A. Totoc. Toscao ad F. ola Coectve mesless patcle fomlatos fo tme doma mawell s eqatos Joal of Comptatoal ad Appled Matematcs vol. pp. 7. [] J. P. Beege A Pefectl Matced Lae fo te Absopto of lectomagetc Waves Joal of Comptatoal Pscs vol. pp [5] M. L. Medes L. C. A. Pmeta R. C. Mesqta. J. d. Slva ad T. C. Sataa Smooted patcle electomagetcs wt boda absobg codto sg pefectl matced laes Te IT 7t Iteatoal Cofeece o Comptato lectomagetcs - CM pp. 5. [] M. N. O. Sad. lemets of lectomagetcs Ofod Uvest Pess t ed.. Bazla Mcowave ad Optoelectocs Socet-SBMO eceved 5 Nov 9; evsed Mac 9; accepted Mac 9 Bazla Socet of lectomagetsm-sbmag 9 SBMO/SBMag ISSN 5-799