Artificial Neural Networks Approach for Solving Stokes Problem

Transcription

1 Appled Mahemacs, 00,, 88-9 do:0436/am Pblshed Ole Ocober 00 (hp://wwwscrporg/joral/am) Arfcal Neral Neworks Approach for Solvg Sokes Problem Absrac Modjaba Bama, Asghar Keraecha, Sohrab Effa Deparme of Mahemacs, Ferdows Uvers of Mashhad, Mashhad, Ira E-mal: Receved Jl 0, 00; revsed Ags, 00; acceped Ags 4, 00 I hs paper a ew mehod based o eral ework has bee developed for obag he solo of he Sokes problem We rasform he mxed Sokes problem o hree depede Posso problems whch b solvg hem he solo of he Sokes problem s obaed The resls obaed b hs mehod, has bee compared wh he exsg mercal mehod ad wh he exac solo of he problem I ca be observed ha he crre ew approxmao has hgher accrac The mber of model parameers reqred s less ha coveoal mehods The proposed ew mehod s llsraed b a example Kewords: Arfcal Neral Neworks, Sokes Problem, Posso Eqao, Paral Dffereal Eqaos Irodco CFD sads for Compaoal Fld Damcs, a sbgere of fld mechacs ha ses compers (mercal mehods ad algorhms) o represe, or model, problems ha egage fld flows CFD sofware s sall sed o solve eqaos a dscrezed wa The doma s rasferred o a grd or mesh a reglar/rreglar ad D/3D srface of cells Afer dscrezao, a eqao solver rs o solve he eqaos of fld moo (Eler eqaos, Naver-Sokes eqaos, ec) Algorhms from mercal lear algebra, lke: Gass-Sedel, sccessve over relaxao, Krlov sbspace mehod or algorhms from Mlgrd faml are pcall sed These mehods volve mllos of calclaos, so, as ca be easl observed, compg s me cosmg Also ma problems, eve wh parallel programmg ad spercompers, ol approxmae solos ca be reached There are varos opmzao mehods of comper scece whch ca be sed for CFD Smlaed Aealg, Geec Algorhms, Evolo Sraeg, Feed-Forward Neral Neworks are poplar ad we remplemeed ma projecs I [] a framework s creaed for evoloar opmzao whch s he esed o aerodamc desg example The framework was based o covarace marx adapao, wh he feed-forward eral ework as a approxmae fess fco A aerodamc desg procedre whch combes eral eworks wh polomal fs s rodced [] ad [3] dscssed a arfcal eral ework whch s a approxmae model ha s sed for opmzao of he blade geomer b smlaed aealg mehod Parallel sochasc search algorhm s rodced [4] ad esed o defg a shape of wo arfols I hs work, a performace eral ework for solvg Sokes eqaos s preseed Lagars, e al [5] sed arfcal eral eworks (ANN) for solvg ordar dffereal eqaos ad paral dffereal eqaos for boh bodar vale ad al vale problems Cah ad Cog [6] preseed a ew echqe for mercal calclao of vscoelasc flow based o he combao of eral eworks ad Browa damcs smlao or sochasc smlao echqe (SST) aa ad Karam [7] sed a modfed eral ework o solve he Berger s eqao oe-dmesoal qaslear paral dffereal eqao The Sokes eqaos descrbe he moo of a fld R ( or3) These eqaos are o be solved for a kow veloc vecor x (, ) ( ( x, )) R ad pressre px (, ) R We resrc or aeo here o compressble flds fllg all of R ( = ) as follow: p f R x p f 0 x Coprgh 00 ScRes

2 M BAYMANI ET AL 89 wh bodar codos: (, ) (, ), o ere, s gve, C dvergece-free vecor feld o, f, f are he compoes of a gve, exerall appled force ( eg grav) The frs ad secod eqaos of are js Newo s law f ma for fld eleme sbjec o he exeral force f ( f, f) ad o he forces arsg from pressre ad frco The hrd eqao of sas ha he fld s compressble For phscall reasoable solos, we wa o make sre (, ) does o grow as ( x, ) ece we wll resrc or aeo o he force f ad al co- 0 do ha sasf 0 K x ( x) C K x, o R, for a ad some K, K x f( x) C K x, o R, for a ad some K We accep a solo of as phscall reasoable ol f sasfes p, C ( R ) ad x ( ) dx C (boded eerg) Descrpo of he Mehod The sal proposed approach for problem wll be llsraed erms of he followg geeral paral dffereal eqao: Gx, x, x, x0, xd x Gx, x, x, x0, xd () x G3 x, x, x0, xd x sbjec o cera bodar codos (BC s) (for example Drchle ad/or Nema), where x ( x,, x ) R, D R deoes he defo doma ad ( x), ( x), ( x) are he solos o be comped If ( x, P), ( x, P), ( x, P3) deoe ral solos wh adjsable parameers P, P, P 3, he problem () s rasformed o,, 3 3 m G x G x G x P P P x D (3) sbjec o he cosras mposed b he BC s I he proposed approach, he ral solos ( x, P), ( x, P), ( x, P3) emplo a feed forward eral ework ad parameers P, P, P 3 correspod o he weghs ad bases of he eral archecre We choose ral fcos ( x, P), ( x, P), ( x, P3) sch ha b cosrco sasf he BC s Ths acheves b wrg as a sm of wo erms where x A xfx, Nx, P x AxFx, Nx, P x A3xF3x, N3x, P3 N ( x, P) ( z ) ad z w j jxj k ( k,,3) are sgle-ops feed forward eral ework wh parameers P, P, P 3 ad p s fed wh he p vecor x The erms A ( x) (,,3) coa o adjsable parameers ad sasf he bodar codos The secod erms F( x, N( x, P))(,,3) s cosrced so as o o corbe o he BC s, sce ( x, P), ( x, P), ( x, P3) ms also sasf he BC s These erms emplo a eral ework whose weghs ad bases are o be adjsed order o deal wh he mmzao problem Noe a hs po ha he problem has bee redced from he orgal cosraed opmzao problem o a cosraed oe (whch s mch easer o hadle) de o he choce of he form of he ral solo ha sasfes b cosrco he BC s I he ex seco we prese a ssemac wa o cosrc he ral solo, e, he fcoal forms of boh A ad F We rea several commo cases ha oe freqel ecoers varos scefc felds As dcaed b or expermes, he approach based o he above formlao s ver effecve ad provdes reasoable compg me accrae solos wh mpressve geeralzao (erpolao) properes 3 Neral Nework for Solvg Sokes Eqaos To solve problem wh [0,] [0,], we appl he operaors ad o he frs ad secod eqa- x os respecvel The we oba: x x p p ( f ) x ( f) (4) (5) Usg he hrd eqao, he Eqao (5) ma be wre as: p x p ( f ) x ( f) hs s he Posso eqao, ad has fel ma solos B mposg some bodar codos, we are gog o oba a approprae solo for Eqao (6) (6) Coprgh 00 ScRes

3 90 M BAYMANI ET AL b he eral ework The ral solo s wre as p ( x, ) A( x, ) x( x) ( ) N( x,, P) (7) where A( x, ) s chose so as o sasf he BC, amel Ax (, ) ( xh ) 0( ) xh( ) ( ) g0( x) [( x) g0(0) xg0] (8) g ( x) [( x) g (0) xg ] where h0( ) p(0, ), h( ) p(, ), g0( x) p( x,0) ad g ( x) p( x,) Noe ha he secod erm of he ral solo does o affec he bodar codos sce vashes a he par of he bodar where Drchle BC s are mposed I he above PDE problems he error o be mmzed s gve b p( x, ) p( x, ) E( p) F( x, ) (9) x where F ( f) x ( f) ad ( x, ) s a po Ω B solvg he opmzao problem (9), he weghs v, wj, are obaed ad he a ral solo for p s obaed B sbsg he ral solo p he frs eqao of we oba: ( p ) f (0) x whch s a Posso eqao for, b sg (9) ad b ( p ) sbsg ad f for or p ad F, res- x pecvel, we ca oba a ral solo for To oba, we ca sbse he ral solo p he secod eqao o oba: ( p ) f () I a smlar maer we ca oba 4 Nmercal Examples I hs seco we prese oe example o llsrae he mehod We sed a mllaer percepro havg oe hdde laer wh fve hdde s ad oe lear op For a gve p vecor x ( x, x) he op of he ework s 5 N( x, P) ( z ) where z w j jxj ad ( x) The exac aa- x e lc solo s kow advace Therefore we es he accrac of he obaed solo Example Cosder he problem wh Ω = [0,] [0,] We choose f ad f sch ha he exac solo for, ad p be as follows: 0 x ( x) ( 3 ) 0 x( ) ( 3x x ) p 5( x ) The doma Ω s frs dscrezed b form mesh of sze h /3 (4 pos) Ths al mesh s sccessvel refed o prodce meshes wh szes h /4 ad h /5 (respecvel 9 ad 6 pos) Table repors he maxmm error a odal pos (Maxmm error) a he rag se pos ad he dsaces p p ad p p L bewee he exac solo ad he rag solo I Fgre he error fco p p for N = 5 s depced whch shows he solo s ver accrae We sed he Eqao (0) ad obaed he solo Table repors he maxmm error a he rag se pos ad he dsaces ad bewee he exac solo ad he rag solo I a smlar maer we obaed Table 3 repors he maxmm error a he rag se pos ad he ds- Table a rag se pos ad he dsaces p p ad p p ( ) Maxmm Error p p p p N = e e e-4 757e-8 750e-6 N = e e-5 490e-3 Table a rag se pos ad he dsaces ad ( ) e N = 6 073e e e-5 N = 5 733e-8 576e-7 63e-5 Table 3 a rag se pos ad he dsaces ad ( ) e N = 6 64e-05 7e e-008 N = e e-4 700e- Coprgh 00 ScRes

4 M BAYMANI ET AL 9 aces ad bewee he exac solo ad he rag solo I Fgres ad 3 he error fcos ad for N = 5 s depced whch shows he solos are ver accrae (eve bewee rag pos) I Fgres 4-7 dffereal of he error fcos ad respec o x ad, for N = 5 are depced, respecvel, whch show he solos are dffereable Ama ad Keraecha [8] sed lear programmgmehods o solve he above problem The covered he Sokes problem o a mmzao problem, he b dscrezg he mmzao problem The obaed a lear programmg problem ad solved Table 4 prese he comparso bewee he proposed mehod ad wh Ama Keraecha mehod for 5 pos Fgre 3 The error fco Fgre Error fco p p Fgre 4 The dffereal of error fco o x respec Fgre The error fco Table 4 The comparso of or proposed mehod wh Ama Keraecha mehod Errors The proposed mehod Ama- Keraecha mehod 7335e e e e p p 639e p p 448e Coprgh 00 ScRes

5 9 M BAYMANI ET AL Fgre 5 The dffereal of error fco o respec Fgre 7 The dffereal of error fco o respec oba a approxmaed fco for he solo ad so we ca calclae he aswer a ever po mmedael 6 Refereces Fgre 6 The dffereal of error fco o x 5 Coclsos respec I hs paper a ew mehod based o ANN has bee appled o fd solo for Sokes eqao The solo va ANN mehod s a dffereable, closed aalc form easl sed a sbseqe calclao The eral ework here allows s o oba fas solo of Sokes eqao sarg from radoml sampled daa ses ad refed who wasg memor space ad herefore redcg he complex of he problem If we compare he resls of he mercal mehods (see [8]) wh or mehod, we see ha or mehod has some small error Oher advaage of hs mehod, he solo of he Sokes problem s avalable for each arbrar po rag erval (eve bewee rag pos) Ideed, afer solvg he Sokes problem, we [] Y C J, M Olhofer ad B Sedhoff, A Framework for Evoloar Opmzao wh Approxmae Fess Fcos, Evoloar Compao, IEEE Trasacos, Vol 6, No 5, 00, pp [] M Ma, R Naer ad K Madava, Aerodamc Desg Usg Neral Neworks, Amerca Ise of Aeroacs ad Asroacs Joral, Vol 38, No, 000, pp 73-8 [3] S Perre, Trbo Macher Blade Desg Usg a Naver-Sokes Solver ad Arfcal Neral Nework, ASME Joral of Trbomach, Vol, No 3, 999, pp [4] T Ra, Tsa ad C Ta, Effecs of Solver Fdel o a Parallel Search Algorhm s Performace for Arfol Shape Opmzao Problems, 9h AIAA/ISSMO Smposm o Mldscplar Aalss ad Opmzao, Alaa, Georga, 00, pp [5] I E Largrs ad A Lkas, Arfcal Neral Neworks for Solvg Ordar ad Paral Dffereal Eqaos, IEEE rasaco o eral eworks, Vol 9, No 5, 998, pp [6] D Tra-Cah ad T Tra-Cog, Compao of Vscoelasc Flow Usg Neral Neworks ad Sochasc Smlao, Korea-Asrala Rheolog Joral, Vol 4, No 4, 00, pp 6-74 [7] M aa ad B Karam, Feedforward Neral Nework for Solvg Paral Dffereal Eqaos, Joral of Appled Sceces, Vol 7, No 9, 007, pp 8-87 [8] M Ama ad A Keraecha, Solvg he Sokes Problem b Lear Programmg Mehods, Ieraoal Mahemacal Joral, Vol 5, No, 004, pp 9- Coprgh 00 ScRes