CORE REACTOR CALCULATION USING THE ADAPTIVE REMESHING WITH A CURRENT ERROR ESTIMATOR

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1 007 Inrnaonal Nuclar Alanc Confrnc - INAC 007 Sanos, SP, Brazl, Spbr 30 o Ocobr 5, 007 ASSOCIAÇÃO BRASIEIRA DE ENERGIA NUCEAR - ABEN ISBN: CORE REACTOR CACUATION USING THE ADAPTIVE REMESHING WITH A CURRENT ERROR ESTIMATOR Rnaldo J. Jospn, uz O. d B. Aghna, Marjn W. Hndrkx and Dongos E. d S. Nry Insuo d Engnhara Nuclar (IEN / CNEN - RJ) Rua Hélo d Alda 75 Cd. Unrsára Ilha do Fundão CxP: Ro d Janro, RJ rj.jospn@n.go.br laghna@n.go.br dongos@n.go.br Unrsy of Twn Calslaan MD Enschd, Th Nhrlands.w.hndrkx@sudn.uwn.nl ABSTRACT Wh h objc o pro h racor physcs calculaon on a D and 3D nuclar racor a h Dffuson Equaon, an adap auoac fn ln rshng hod, basd on h lnary ara (D) or olu (3D) consrans, has bn dlopd. Th adap rshng chnqu, gudd by a posror rror saor, aks us of wo xrnal sh gnraor progras: Trangl and TGn. Th us of hs fr xrnal fn ln sh gnraors and an adap rshng chnqu basd on h currn fld connuy show ha hy ar powrful ools o pro h nuron flux dsrbuon calculaon and by consqunc h powr soluon of h racor cor n hough hy ha a nor nflunc on h crcal coffcn of h calculad racor cor xapls. Two nurcal xapls ar prsnd: h D IAEA racor cor nurcal bnchark and h 3D odl of h Argonaua rsarch racor, bul n Brasl.. INTRODUCTION Ths arcl prsns a posror rror saor o buld n h dffuson hory, an adap auoac rshng sragy. Ths rror saor calculas nw characrsc lngh, ara or olu consran n ach ln o hoognz h rror n all h doan akng us of wo xrnal sh gnraor progras: Trangl, dlopd for D analyss by Jonahan Shwchuk [] and TGn, dlopd for 3D analyss by Hang S []. Ths sofwars can b usd o procd h adap rshng for an unld nubr of rgons wh dffrn proprs. Th rror aluaon dpnds on h knd of condons o b fulflld on h nrlns rgons. In h parcular cass of srss and hral analyss, and nuron dffuson, whch nol connuy rqurns n nrfac rgons for srss, hral flux and nuron currn, rspcly, an rror saor basd n a polynoal nrpolaon of h srss, hral flux or nuron currn can b usd. Ohr knds of rror saor could b dlopd, for xapl, n h lcroagnosac probls whr dsconnus n h angnal currn a h nrfac rgons ak plac [3]. In hs arcl, h arg s o nsur ha for arous yps of probls such as srss, ha ransfr and nuron dffuson, h connuy n h srss flds, hral flux and nuron currn, rspcly, ar sasfd.

2 . FUX ERROR ESTIMATOR En n so ral probls whr h ru flux fld us b connuous, h us of a connuous pcws lnar funcon such as a lnar fn ln o rprsn h fld F prsns a calculad flux h ha has nr-ln dsconnus. Ths calculad flux s dfnd by: h = λ F () whr λ and F rprsn quans corrspondng o h probl a hand. For nuron dffuson probls hy rprsn h dffuson coffcn D and h nuron flux φ, and for h ha ransfr probls hy rprsn h ha ransfr coffcn k and h praur T, rspcly. Th flux fld rror, dfnd as h dffrnc bwn h ru and h calculad flux fld can b akn as an saor for h rror producd by h dscrzaon. Th coarsr h sh, h or pronouncd h dsconnus of h gradn fld wll b. As h ru flux fld h s unknown, w rprsn by an nrpolaon funcon ha producs a connuous flux fld: h = n = N h () whr n s h nubr of ln nods. Ths nrpolad flux fld s or prcs han h on oband fro h gradn of h fld slf. Thrfor, h rror can b sad by h dffrnc bwn h nrpolad flux fld approxaon and h calculad flux alu fro h fld soluon. To oban h unknowns h, a an rror of h flux fld or h nr doan can b sad for xapl, by h followng roo an squar (RMS) adjusn: n ( h h ) d = = N h h d (3) = Mnzng n rlaon of h nodal arabl parars h on can dducs h rlaon: h = n = N N j h j N h d = 0 (4) ha can b pu n a arx for: Mh = f (5) whr M s h global arx of h lnary M arx and f s h global cor of lnary f cor. Th lnar quaon sys prsnd n h quaon (5) for D or 3D cas, can b sold by h ra procss dfnd n [4]: k k h = h k k k M r whr r = M h (6) ( ) ( ) ( ) f INAC 007, Sanos, SP, Brazl.

3 whr M s a pr-condonng arx dfnd as h condnsd for (lupd arx) of k h conssn arx M and r s h rsdu of h sys soluon a sp k. Fro h quaons (6) w can wr an ra procss dfnd by h quaon: [ f ] k ( ) = ( M ) ( M M)( h ) k h + Whn h conrgnc crru s basd on a rlaon bwn currn flux fld and s araon, h procss wll rna hr whn h rsuls conrg o h rqurd olranc, or whn h axu nubr of raons s rachd. (7) 3. THE ADAPTIVE REFINEMENT As sn bfor, h an squar roo of h dffrnc bwn h nrpolad currn fld coponns h and h calculad flux fld h dfns an rror asurn for h soluon n h sub-doan of a gnrc ln and n h doan ar gn by h rlaons: d and = d whr = = = (8) Th an rror pr ln can b dfnd by: = (9) Th adap rshng s basd on h da ha h nw sh gnrad has a unfor dsrbud rror or h nw lns. Ths arg rror lads o a rfnn sragy. Unforunaly, hs sragy dos no prod any nforaon abou h nubr of lns gnrad, whch could xcd h ls of copuaonal fasbly. Anohr sragy, dlopd by Sapao al. [5], prs o oban for a cran pr-dfnd nubr of lns, a br rror dsrbuon: = (0) whr s h usr prscrbd axu dsrd nubr of lns, s h nubr of lns n h currn sh and s h arag rror pr ln asurd on h currn sh. For lnar lns as ployd hr, nddual ln rrors ar assud o b proporonal o h corrspondng ln sz. Thus, h nw ln sz dsrbuon rqurd o aan h arg rror s dfnd on ry ln of h nw sh. A arabl ha wll dcd whr o r-dd h sh o dnsh h flux fld nr-ln dsconnus has o b dfnd. Dpndng on h probl dnson h nw ln sz can b xprssd n rs of h characrsc lngh (GnMsh), h ara A (Trangl) or h olu V (TGn). Assung ha h nw ln sz dsrbuon s xprssd n INAC 007, Sanos, SP, Brazl.

NUMERICA RESUTS: In hs scon, aong a lo of nurcal rsuls prsnd bfor [7, 8, 9], h rsuls of h D-IAEA bnchark nurcal sulaon of a hypohc racor s chosn and copard wh h soluon oband by h prsn work ha was

4 rs of h ln szs rror ad for h nw sh as [6]: k, h rrors on h currn sh and h unfor arg I k k + = () hn, h nw ln sz o procss a nw rshng sp could b gn n rs of h ara (D) or olu (3D) by: A k + k k + k 3 k ( ) A or V = ( ) V = () k + + ha ar h forulas usd n h cas of h xrnal progras Trangl(D) or TGn(3D). 4. NUMERICA RESUTS: In hs scon, aong a lo of nurcal rsuls prsnd bfor [7, 8, 9], h rsuls of h D-IAEA bnchark nurcal sulaon of a hypohc racor s chosn and copard wh h soluon oband by h prsn work ha was ralzd wh a squnc of sx adap rshng sps [0]. Fgur. Th frs basc sh and h nw sh Fgur. Thral Flux for Basc and Nw Msh. In fgur., only h basc and h fnal adap sh ar prsnd. Th hral flux soluon oband for h basc and h las adap sh ar gn n fgur. No h sgnfcan pron n h hral nuron flux calculaons, whch wll pro h INAC 007, Sanos, SP, Brazl.

rsuls of a subsqun local racor powr calculaon as wll. Tabl shows h coparson bwn h kff rsuls oband by hs work and ha oband by h nurcal bnchark calculaon. Tabl. Crcaly Coffcn k ff Rqurd Msh Oband sh Prsn work IAEA Nurcal Bnchark Error(%) Basc sh 45.

Mddl plan (a) (b) (c) Fgur 4. Th aral (a), h basc sh (b) and h las nw sh (c) 0 l 36 l 644 l 6497 l (a) (b) (c) Fgur 5. Thral Flux (a and b) and Thral Flux Graph (c) for arous shs.

5 rsuls of a subsqun local racor powr calculaon as wll. Tabl shows h coparson bwn h kff rsuls oband by hs work and ha oband by h nurcal bnchark calculaon. Tabl. Crcaly Coffcn k ff Rqurd Msh Oband sh Prsn work IAEA Nurcal Bnchark Error(%) Basc sh Hr, a splfd 3D Argonaua Racor odl s prsnd usng, by syry, an ocan of h racor. Thr rgons ar dfnd: h cor (grn) and h rflcors: graph (yllow) and war (blu) [0]. Mddl plan (a) (b) (c) Fgur 4. Th aral (a), h basc sh (b) and h las nw sh (c) 0 l 36 l 644 l 6497 l (a) (b) (c) Fgur 5. Thral Flux (a and b) and Thral Flux Graph (c) for arous shs. Th flux corrspondng o h wo nuron nrgy groups and h crcaly coffcn s oband for a squnc of adap rshng sps. Fgur 4 prsns h aral rgons (a), h basc (b) and h las adap sh (c) oband usng h fr sofwar TGn and h rror saor prsnd bfor n hs work. Th hral nuron fluxs ar prsnd n h fgur 5 for h coars sh (a) and for h las rfnd sh (b). Th hral flux graphs n h z-drcon passng on h cor racor cnr ar prsnd n fgur 5 (c) for arous shs. Whn an ncrasng nubr of fn lns ar usd, an pron has bn obsrd n h hral nuron flux shap coparng wh h xprnal rsuls []. Ths pron wll rflc drcly n h local powr calculaon of h racor. If s possbl o oban hs shap wh a gnrc rfnn, h rror saor ha h adanag o INAC 007, Sanos, SP, Brazl.

6 concnrad h lns, as can b obsrd fro fgur 4 (c), o rndr h currn fld connuous aodng wasng cpu. Th sh rfnns can b basd on hr h rsuls for h fas or hral nuron flux dsrbuon, as spcfd by h usr. 5. CONCUSIONS Th plnaon of Trangl and TGn, xrnal fr fn ln sh gnraors, assocad wh h currn rror saor, shows ha hy ar powrful ools o pro h nuron flux dsrbuon and by consqunc h powr soluon of h racor cor n hough hy ha a ll nflunc on h crcal coffcn of h racor cor. Th nor drawback of hs plnaon, s h rqurn o g h gory odl n a spcal way: h Planar Sragh n Graph (PSG) daa fl n h D cas and h Pcws nar Coplx (PC) daa fl n h 3D cas. Th plnaon of hs chnqu o h srss and hral analyss s sraghforward. In h cas of lcroagnosac probl nw rror saor ha o b dlopd [3]. REFERENCES. J. Shwchuk, Trangl Manual, jrs@cs.brkly.du, U.S.A, (976). H. S, TGn-A Qualy Trahdral Msh Gnraor and Thr-Dnsonal Dlaunay Trangulaor - Usr's Manual, Wrsrass Insu for Appld Analyss and Sochascs - Rsarch Group of Nurcal Mahacs and Scnfc Copung - hp://gn.brlos.d, (006). 3.. bnszajn and M. R. Hrann and J. R. Cardoso, Esadors d Erro para Procssos Auo-Adapaos para Cálculos d Capos Elroagnécos, Procdngs do Congrsso Braslro d Cálculo d Capos Elroagnécos - CBMag'95l, ol., pp 36-39, Brasl, (995) 4. O. C. Znkwcz and R.. Taylor, Th Fn Eln Mhod - 4h don, McGraw Hll Inrnaonal Edons - Physcs Srs, ol., pp.-807, (99). 5. P.A.B. d Sapao and A..G.A. Counho, Sulaon of Fr and Forcd Concon Incoprssbl Flows Usng an Adap Paralll/Vcor Fn Eln Procdur, Inrnaonal Journal for Nurcal Mhods n Fluds, ol. 9, pp , (999) 6. D. E. d S. Nry and R. J. Jospn, Adap Rshng wh Gradn Error Esaor n MEF Progra, Rlaóro Técnco IEN -006, Ro d Janro - Brasl, (006). 7. A. Corra Flho, Dffuson Probls usng Fn Eln Mhod, M Sc hss a U.F.P. - Prnabuco - Brasl, january, (976). 8. D. A. Bolho, Muldnsonal Fn Eln Cod, PHD hss a M.I.T. - U.S.A., january, (976). 9. M. W. Hndrkx, Exnson of a Forran Fn Eln Cod for Nuron Dffuson probls n Nuclar Racors, Rlaóro Técnco IEN RT-IEN , Ro d Janro - Brasl,.w.hndrkx@sudn.uwn.nl, (006). 0. R. J. Jospn and. O. B. Aghna and P. A. B. Sapao, Nuron Dffuson Equaon Soluon usng Fn Eln Mhod, Rlaóro Técnco IEN 0-006, Ro d Janro - Brasl, (006).. uz O. d B. Aghna and all, Rlaóro d Anals d Sgurança do Raor Argonaua IEN, Vol. I, IEN/DERE/DIFIR (S/n), Brasl, (990). INAC 007, Sanos, SP, Brazl.

Consider a system of 2 simultaneous first order linear equations

Consider a system of 2 simultaneous first order linear equations Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm