Evolutionary, Iterative Optimum-Optimorum Theory

Transcription

1 Evoltoar, Iteratve Opt-Optor Theor Adraa ĂSTASE* *Correspodg athor RWTH-AACHE, Aeroda des Flges Teplergrabe 55, 56 Aache, Gera, DOI:.3/ Abstract: The perforg of the aerodacal, global optzed (GO) shape of flg cofgratos (FCs) leads to a elarged varatoal proble th free bodares. The opt-optor theor as developed b the athor order to solve ths elarged varatoal proble, sde of a class of FCs, th soe chose coo propertes. Ths theor as sed for the vscd GO shape of three odels th hgh aerodacal perforaces, ael: ADELA (a delta g aloe) ad of to tegrated g-fselage FCs FADET I ad FADET II, flg spersoc flo. The refeet of the optzato strateg, for of a evoltve, teratve opt-optor theor, s here preseted. The vscd GO shape of FC, represets o ol the frst step of ths teratve ethod. A coptatoal checg of ths shape s ade b sg e hbrd aaltcal-ercal soltos for the aver-stoes laer. The total drag coeffcet (cldg frcto) s copted ad a ea teracto aerodacs/ strctre, va e ad/or odfed costrats s proposed. Up the secod step of terato process, a grato the drag fctoal ad the costrats s perfored. Ke Words: elarged varatoal ethod, aver-stoes laer, spersoc flo, ea teracto aerodacs/strctre, deterstc optzato, geetc algorths propertes.. ITRODUCTIO The classcal varatoal proble, cocerg the aerodacal optzato of the shape of a FC th a gve plafor, leads to a classcal varatoal proble th fed bodares. The athor has to tes elarged ths varatoal proble, order to be able: to detere the GO shape of the FC ad to clde the frcto effect the coptato of the total drag fctoal ad the optal desg. The frst elargeet cossts the deterato of the vscd GO shape of the FC (ael, the sltaeos optzato of ts caber, tst, thcess ad also of ts slart paraeters of the plafor), hch leads to a elarged varatoal proble th free bodares. A o opt-optor (OO) theor as developed order to solve ths elarged varatoal proble. The vscd GO shape of FC s chose the frae of a class of adssble FCs th good sted coo propertes. The OO-theor as sed b the athor for the aerodacal GO of the shapes of three aerospace odels, ael: of ADELA (a delta g aloe) ad of to tegrated g-fselage FCs FADET I ad FADET II, optzed, respectvel, for the crsg Mach bers.,. ad 3.. The secod elargeet of the varatoal ethod, sed here, cossts the developet of a teratve OO-theor, hch taes the vscd GO of the FC's shape, prevosl deter-ed, as frst step of terato. Ths shape s checed b sg the e developed hbrd aaltcal-ercal soltos for the aver-stoes laer (SL), hch se the aaltcal hperbolcal potetal soltos of the flo over the sae FC tce, ael frstl as oter flo ad secodl the veloct s copoets are cosdered betee the correspodg copoets of the potetal veloct ad poloal epasos th ICAS BULLETI, Vole, ber 4/, pp. 53-6

2 Adraa ĂSTASE 54 arbtrar coeffcets, as []. The frcto drag coeffcet s copted ad ths FC's shape s also cotrolled for the strctre pot of ve. Up the secod step of terato, the total drag coeffcet s the e fctoal ad the e ad/or odfed set of costrats, resltg after a ea teracto aerodacs/strctre, are cosdered. The evoltve, teratve opt-optor theor ca be sed for the refeet of GO shape of FC.. THE OPTIMUM-OPTIMORUM THEORY The frst elargeet cossts the vscd GO shape of the FC (ael, the optzato of ts caber, tst ad thcess dstrbtos ad also of ts slart paraeters of the plafor), hch leads to a elarged varatoal proble th free bodares. A o OO-theor as developed order to solve ths elarged varatoal proble. The global optzed FC's shape s chose sde of a class of adssble FCs, hch s defed b soe coo propertes. To FCs belog to the sae class, f: ther srfaces-epasos are pecese epressed for of sperpostos of poloes th the sae aal degree (th o coeffcets), ther plafors are polgoes (th o slart paraeters of plafor), hch ca be related throgh affe trasforatos ad the flfll () the sae costrats. A loer-lt hpersrface of the drag fctoal C d as fcto of the slart paraeters s defed, ael, (C ) ( ) d opt f,,..., ) () ( Each pot of ths hpersrface s obtaed b solvg a classcal varatoal proble th gve bodares (.e. a gve set of slart paraeters). The posto of the of ths hpersrface, hch s ercall detered, gves s the best set of the slart paraeters ad the FC's optal shape, hch correspods to ths set, s the sae te the global optzed FC's shape of the class. The deterato of the GO shape of the odel FADET I, va OO-theor, s preseted here, as eeplfcato. 3. THE GLOBAL OPTIMAL DESIG OF THE MODEL FADET I A g-fselage FC s here cosdered as a dscotos g alog the jcto les g/fselage. If, addtoall, the g ad the fselage have the sae ea srface ad the sae taget plae, each pot of ther jcto les, the evalet g of the gfselage cofgrato s here called tegrated g-fselage. Its ea srface Z (, ) s spposed to be cotos, bt, for the sae of geeralt, the thcess dstrbtos Z (, ) o the lateral sdes, correspodg to the g ad Z '(, ) o the cetral part, correspodg to the fselage zoe, are dfferet. The coptato s ade a desoless sste of coordates O 3, as [], [ ], t s: h h (,,,, 3 3 h ) (a-c) ICAS BULLETI, Vole, ber 4/

3 55 Evoltoar, Iteratve Opt-Optor Theor The doashes o the th copoet ad ad o the g ad o the fselage of the thc-setrcal copoet of the tegrated FC are epressed for of sperposto of hoogeeos poloes (th o coeffcets), ael:,,, (3a-c) The jcto les g/fselage are cosdered as to artfcal rdges of the tegrated g. The correspodg aal dstrbace veloctes ad o the th ad thcsetrcal copoets of the FC, obtaed b the athor, b sg the hdrodac aalog of Carafol, are:, cosh C E, A E (4), cosh C E, D E cosh ) ( cosh G. (5) ( ) ( ) )( (, M, ) ( ) )( (, ) The paraeters of optzato are the coeffcets of the doashes, j j ad j ad the slart paraeters of the plafor of the etre FC ad of the plafor of the f-selage ad, respectvel,. The otet /, hch depeds o the prpose of the FC, s tae costat, drg the optzato process. For a gve vale of the slart paraeter of the FC s plafor, the optzato of the shapes of ts th ad thc-setrcal copoets ca be separatel treated. The tal costrats for the th FC copoet are: gve lft ad ptchg oet coeffcets ad also the o trodced Ktta codto o leadg edge order to cacel the dced drag ad to avod the cotreet of flo o leadg edge, at crse. ICAS BULLETI, Vole, ber 4/

4 Adraa ĂSTASE 56 The tal costrats for the thc-setrcal copoets are: the gve relatve voles of the fselage ad of etre FC, the closg codto o leadg edges ad also the o -trodced tegrato codtos alog the jcto les g/fselage. For a gve vale of the slart paraeter of the plafor of the FC the correspodg optal vales of the coeffcets of doashes are aaltcal, el detered b solvg of to lear algebrac sstes, as [], []. If the slart paraeter of the plafor of FC s seetall vared a loer lt-le of the vscd drag fctoal of optal FCs, as fcto of ths slart paraeter s obtaed b solvg the varatoal probles for the correspodg vales of the slart paraeter (for FCs th sbsoc leadg edges s ). The posto of the of ths les-le gves the optal vale of the slart paraeter opt ad the correspodg optal FC s, the sae te, the global optzed FC of the class. The GO shape of the fll-tegrated g-fselage odel FADET I, desged b the athor, for the crsg Mach ber M., obtaed b sg the OO-theor, s preseted the Fg. ad loos, trasversal sectos, le a flg brd! Fg. The global optzed shape of the fll-tegrated g fselage odel FADET I The aerodacal characterstcs of the tegrated, GO shape of the odel FADET I as checed the trsoc d tel of the DLR-Cologe, the frae of the athor s research project, sposored b the DFG. A ver good agreeet betee the theoretcal ad eperetal reslts, obtaed b sg vscd aaltcal, hperbolcal soltos for ts lft ad ptchg oet coeffcets, s preseted the Fg. a,b. I the Fgs. 3a-c s copared the theoretcal predcted dstrbto of pressre coeffcet th eperetal reslts, the logtdal cetral secto of the pper sde of odel FADET I, at the agles of attac 8,, 8 ad for the rage of Mach bers M (.5.4). For ths rage of Mach bers, the odel FADET I has sbsoc leadg edges. ICAS BULLETI, Vole, ber 4/

5 57 Evoltoar, Iteratve Opt-Optor Theor Fgs. a, b The lft ad the ptchg oet coeffcets of the global optzed odel FADET I ICAS BULLETI, Vole, ber 4/

6 Adraa ĂSTASE 58 Fg. 3a-c The dstrbto of pressre coeffcet o the cetral ct at the pper sde of the odel FADET I, for the agles of attac 8,, 8 A ver good agreeet betee eperetal ad theoretcal reslts s obtaed b sg of the o aaltcal hperbolcal potetal soltos for the lft, ptchg oet ad pressre coeffcets o FCs th sbsoc leadg edges, at oderate agles of attac, spersoc flo, as t ca be see the Fgs. a,b - Fgs. 3a-c. Ths agreeet leads to the rears: - The valdt of the three-desoal aaltcal hperbolcal potetal soltos for the aal dstrbace veloctes th the chose balaced sglartes ad the correspodg developed softare for the above aerodacal coeffcets are cofred; - The flece of frcto po these coeffcets s eglectable; - The flo s laar, as spposed here ad reas attached spersoc flo, for larger agles of attac tha b sbsoc flo; - Hbrd soltos for the aver-stoes laer (SL) th portat aaltcal propertes are proposed here, for the coptato of the total drag (cldg frcto) ad of all the aerodac characterstcs, at hgher agles of attac. 4. HYBRID SOLUTIOS, FOR AVIER-STOKES LAYER The aaltcal, hperbolcal, potetal soltos are sed the frst, vscd step of the evoltoar, teratve OO-theor. These soltos are replaced b o developed reforced hbrd soltos for the SL, obtaed b crossover of aaltcal th ercal soltos ICAS BULLETI, Vole, ber 4/

7 59 Evoltoar, Iteratve Opt-Optor Theor order to obta taleted hbrd SL s soltos, hch have the geeralt of the ercal soltos, flfll the o-slp codtos ad have also portat aaltcal propertes. Let s trodce a spectral varable, t s: 3 Z(, ). ( ) (, ) Hereb Z (, ) s the eato of the srface of the flatteed FC ad (, ) s the SL s thcess dstrbto. The spectral fors of the aal, lateral ad vertcal veloct s copoets, ad, the dest fcto R l ad the absolte v teperatre T are here proposed, as []-[3], ael: e, v v e v, e, R R ( Re R) r, (6a-e) T T ( T T ) e Here R ad are the gve vales of ad T at the all,,,, ad T R e ve e Re Te are the vales of, v,, R ad T at the SL s edge, obtaed fro the oter vscd hperbolcal potetal flo, ad, v,, r ad t are ther free spectral coeffcets, hch are obtaed b flfllg of the SL s partal dfferetal eatos (PDEs). The phscal eato of deal gas for the pressre p ad a epoetal la of the vscost verss T are sed: p R T R e g g R t T T. T (7a-b) Here are: R ad T the versal gas costat ad the absolte teperatre of the g dstrbed flo ad s the vscost epoet. The se of the dest fcto R l (stead of the dest ), proposed here, cobed th the forlas for R, T,, v,, p ad, ad th the collocato ethod allo the deterato of the spectral coeffcets of all the phscal ettes, ael: R, T, p ad ol as fctos of the spectral coeffcets of the veloct s copoets, as []-[3]. The spectral fors (6a-e) atoatcall satsf the bodar codtos at all ( ). The bodar codtos at the SL s edge are elated b fg seve spectral coeffcets of the veloct s copoets. If the spectral fors gve (6a-c) are trodced the SL s PDEs of plse ad the collocatos ethod s sed, the spectral coeffcets of the veloct s copoets, v ICAS BULLETI, Vole, ber 4/

8 Adraa ĂSTASE 6 ad are obtaed b the teratve solvg of a lear algebrac sste th slghtl varable coeffcets, hch vales are tae for the precedet terato. The aaltcal propertes of the hbrd ercal SL s soltos proposed here are the follog: the have correct last behavors, the have correct jps alog the sglar les (le sbsoc leadg edges, jcto les g/fselage) ad the sglartes are bala-ced), are accrate becase the are eshless ad the dervatves ca be eactll copted. Frther, b sg of a logarthc dest fcto R l, a splttg of the SL s PDEs s realzed, hch speeds p the coptato te. 5. EVOLUTIVE, ITERATIVE OPTIMUM-OPTIMORUM THEORY The secod elargeet of the varatoal ethod cossts the developet of a teratve OO-theor, order to trodce also the flece of frcto the drag fctoal ad the aerodacal GO shape of FC. A teredate coptatoal checg of the vscd GO shape of the FC s ade th o hbrd solvers, for the three-desoal aver- ( f ) Stoes laer (SL). The frcto drag coeffcet C d of the FC s detered. The teratve OO-theor ses the vscd hperbolcal potetal soltos as start soltos ad the vscd GO shape of the FC, ol ts frst step of terato. The vscd GO shape s checed also for the strctral pot of ve. Addtoal or odfed costrats, trodced order to cotrol the caber, tst ad thcess dstrbtos of the aerodacal, global optzed FC s shape, for strctral reasos, are here proposed. I the secod step of optzato, the predcted vscd optzed shape of the FC s corrected b cldg these sppleetar costrats the varatoal proble ad of the frcto drag coeffcet the drag fctoal. The teratve optzato process s repeated, tl the aal local odfcato of the shape to cosectve optzato steps presets o sgfcat chage. The chart flo of the evoltve, teratve OO-theor s gve the Fg. 4. A ea teracto aerodacs/ strctre, va e ad odfed costrats, trodced for the strctre reasos, the process of the deterato of aerodacal GO shape, s proposed. Fg. 4 The chart flo of the evoltve, teratve opt-optor theor ICAS BULLETI, Vole, ber 4/

9 6 Evoltoar, Iteratve Opt-Optor Theor 6. COCLUSIOS The evoltoar, teratve opt-optor theor s a deterstc theor, hch has alost all the attrbtes of the geetc algorths le gratos, tato, crossover, ltple selectos, allos the ltdscplar optal desg b sg addtoal ad/or odfed costrats, reest fro the strctre prposes, allos the ltpot desg b orphg, s fleble, accrate, ecooc ad copettve. REFERECES [] A. astase, Coptato of spersoc flo over flg cofgratos, Elsever, Oford, UK, 7. [] A. astase, The elarged varatoal ethod as strateg for the aerodacal optal shape s desg, MAO Coferece, Alba, USA, AIAA Techcal Paper (4). [3] A. astase, Aerodacal optal shape s desg, copled th strctre costrats, Copled Probles II ECCOMAS Coferece, CIME, Barceloa, Spa, pp (7) ICAS BULLETI, Vole, ber 4/