Journal of Enginring and Natural Scincs Mühndisli v Fn Bilimlri Drgisi Sigma 4/ Invitd Rviw Par OPTIMAL DESIGN OF NONLINEAR MAGNETIC SYSTEMS USING FINITE ELEMENTS Lvnt OVACIK * Istanbul Tchnical Univrsity, Dartmnt of Elctrical Enginring, Masla-ISTANBUL Gliş Tarihi: 5.6.4 SONLU ÖĞELER İLE DOĞRUSAL OLMAYAN MANYETİK DİZGELERİN ENİYİ TASARIMI ÖZET Manyrti bir cihaz içrisindi blirlnmiş notalarda v doğrultuda arzu diln bir manyti aı yoğunluğu dağılımını yalaşı olara ld tm için, manyti bir cihazın niyi tasarımını ld dbiln bir trs sonlu öğlr yöntrmi gliştirilmiştir. Ağırlaştırılmış Lagrang çaranları yöntrmi ullanılara n üçü arlr fonsiyonundan oluşan bir amaç fonsiyonu il bir dizi ısıt dnlmlrindn oluşan bir niyili roblmi ısıtsız bir niyili roblmin dönüştürülmüştür. Lagrang-Nwton yöntmin dayalı iinci mrtbdn bir yalaşım il dönüştütülmüş roblmin çüzülmsiyl, manyti cihazın tasarımı adım adım iyilştirilmiştir. Manyti malzmd doymanın gözönünd bulundurulması durumunda, iinci mrtbdn duyarlılı analizi için grli türvlrin hsalanması olduça zor bir işlmdir. Bu maald, gliştiriln yni bir yöntm il niyilştirm işlmi il doğrusal olmayan stati manyti alan roblminin çözümün ilşin doğrusal olmayan sonlu öğlr dnlmlri uygun bir biçimd birlştirilr, armaşı duyarlılı analizi vrimli bir şild yaılabilmiştir. Cihazın gomtrisi için aramtri bir modlinin oluşturulması v linr olmayan manyti malzmlrin matmtisl modllrinin urulmasına ilişin yöntmlr araştırılara, gliştiriln bir bilgisayar rogramının içrisind ullanılmasıyla, aralı bir çözüm ld dilbilmiştir. Eltri mainalarının tasarımına ilişin çşitli roblmlr üzrind yaılan dnmlrd, bu maald gliştiriln yöntmlrin sonuçları sunulmuştır. Anahtar Sözcülr: Sonlu lmanlar (öğlr) yöntmi, otimizasyon (niyili), manyti sistmlr (dizglr) ABSTRACT An invrs finit lmnt mthod was dvlod to find otimal gomtric aramtrs of a magntic dvic to aroximat a dsird magntic flux dnsity distribution at crtain tst oints and dirctions slctd in th dvic. Th augmntd Lagrang multilirs mthod was utilizd to transform th constraind roblm consisting of a last-squar objctiv function and a st of constraint quations to th unconstraind roblm. A scond-ordr aroach basd on th Lagrang-Nwton mthod was usd to minimiz th unconstraind roblm to imrov th dsign itrativly. Numrical calculation of drivativs in th scond-ordr dsign snsitivity analysis bcoms a difficult tas if saturation in matrial rortis is accountd. A novl aroach is dvlod to minimiz th comutational ffort by dirctly combining th otimization rocss with th nonlinar finit lmnt quations. Th bst caabilitis to aramtriz th dvic gomtry and to modl th nonlinar matrial charactristics wr incororatd into th otimization rogram for raid snsitivity analysis. Dmonstration of various tst cass arising from otimally dsigning lctrical machinry vrifid th validity of th ovrall thory and dvlomnts. Kywords: Finit lmnt mthod, numrical otimization, magntic systms * -mail:ovaci@l.itu.du.tr, Tl: () 85 6765
L. Ovacı Sigma 4/. INTRODUCTION Th thory of th finit lmnt mthod (FEM) for calculating flux distribution in lctromagntic dvics has bn wll stablishd. Finding a vctor otntial solution for a givn gomtry, matrial rortis, xcitation sourcs, and so on is calld a forward roblm. Du to th comtitiv world mart, dsignrs attmt to rduc cost, wight, and/or to imrov fficincy and rliability of lctromagntic dvics. Thrfor, th dsignr is mor intrstd in finding th gomtry of a crtain ortion of a dvic, such as iron arts, sizs and locations/ositions of xcitation coils to satisfy a givn fild, torqu or forc attrn, rathr than finding how magntic flux is distributd insid th dvic. Sinc this rquirs bac-calculation of dvic-dscritiv dsign aramtrs, ths tys of dsign otimization roblms ar classifid as invrs roblms. Although th thory of mathmatical rogramming has bn widly dvlod sinc th 95 s, th initial rsarch on otimal dsign roblms mrgd from th ara of structural mchanics in th arly 97 s. During th last yars, th otimal dsign tchniqus combining otimization mthods with solution of govrning artial diffrntial quations (PDE) hav bn xtnsivly studid. Th rsults can b found in th txtboos, for xaml, Gallaghr and Ziniwicz [], Haug and Arora [], Kirch [3], Pironnau [4], Vandrlaats [5], and Haslingr and Nittaanmai [6]. In lctromagntics, thr hav rcntly bn divrs alications of dsign otimization roblms aaring in th litratur [7]-[46]. Som alications ar concrnd with otimizing dvic outut forcs (Gitosusastro t al. [9], and Saldanha t al. [], []) whil othrs ar concrnd with minimizing ddy currnt losss in conductiv arts of dvics (Kasr []), or minimizing roduction cost (Arino t al. [3], Albaum t al. [4], [5]). Many of th alications dal with otimizing fild attrns whr an objctiv function in th form of th squard sum of th diffrncs btwn calculatd and dsird valus of fild quantitis at slctd oints is minimizd. Th rsults hav aard in th rcnt ars (Simin and Trowbridg [6], Wbr [7], Wbr and Hool [8], [9], Koh t al. [], Par t al. [], [], Subramaniam t al. [3], and Vasconclos t al. [4]). In this ty of roblms, th objctiv function is xctd to achiv a minimum at zro in th absnc of constraint quations. Howvr, in th rsnc of constraint quations, th otimum is not at zro, but th bst ossibl solution aroaching th dsird critrion is found. In som roblms, a uniqu solution may not xist or thr may b no solution. Thr hav bn diffrnt fild analysis mthods usd in lctromagntic otimization. Initially, intgral mthods wr rformd for fild calculations whil gradints with rsct to dsign aramtrs wr dtrmind by th finit diffrnc mthod (FDM) (Gottvald [44], [45], [46], Simin and Trowbridg [6], and Par t al. []). Prhas, on of th arlist attmts using th finit lmnt mthod (FEM) in an otimization rocss wr rsntd by Salon and Istfan [7], and Istfan and Salon [8]. In thir wor, thy cast a snsitivity analysis basd on dirct diffrntiation of th finit lmnt matrics with rsct to nodal dislacmnts dscribing a dvic s gomtry. Wbr [7], Wbr and Hool. [8], [9], Koh t al. [], Par t al. [], and Hool [5] subsquntly usd this dirct diffrntiation schm for snsitivity analysis. Th boundary lmnt mthod (BEM), dvlod for analysis of on boundary roblms, has rcntly bn usd in snsitivity analysis (Koh t al. [], Par t al. [], Enoizono and Tsuchida [6], and Enoizono t al. [7]) Sinc th objctiv function is a highly nonlinar function of dsign aramtrs, a wid varity of nonlinar otimization stratgis hav bn invstigatd, and tstd on diffrnt roblms for thir sd, convrgnc and fficincy (Pris t al. [8], and Gottvald [44]). Ths otimization mthods ar classifid into two main catgoris: ) dtrministic mthods, and ) stochastic mthods. Th dtrministic mthods includ th first-ordr mthods such as th stst-dscnt, conjugat gradint and quasi-nwton mthods (s,.g., [5], [47] and [48]).
Otimal Dsign of Nonlinar Magntic Systms... Ths mthods ar basd on finding th otimum in th dcrasing gradint dirction, rquiring only th first-ordr drivativs. Th first-ordr drivativs ar calculatd by mans of snsitivity analysis incororating various fild solution tchniqus, such as th finit diffrnc, and th finit lmnt mthods. Stochastic mthods such as simulatd annaling (SA) [6], [49], and th mthods basd on volution stratgis [45], and gntic algorithms (GA) [49] ar zroth-ordr (drivativ-fr) mthods, rquiring valuation of th objctiv function at so many oints that, dnding on th roblm comlxity, thy may bcom comutationally xhaustiv. Howvr, a major advantag is that ths two mthods hav bn shown to b globally convrgnt (Gottvald [46]). Indndntly carrying out th fild analysis and otimization sts has som advantags and disadvantags. On advantag is that commrcially availabl fficint otimization acags sav tim for rogram dvlomnt, scially for rogram modification whn th otimization roblm is changd, (.g., whn th objctiv function is modifid or nw constraint quations ar addd to th roblm). Anothr advantag is that th rogram might hav mor flxibility in choosing anothr otimization algorithm whn on cannot rform as wll as th othr. On th othr hand, th alications in which mor sohisticatd drivativ calculations ar involvd rsult in xcssiv numbr of calls of fild analysis rograms, incrasing th comutational cost. A grat dal of dsign otimization tools dvlod so far convntionally us formal otimization tchniqus using th first-ordr or gradint information to find an otimization dirction to rogrss th dsign towards th otimum solution. Th first-ordr drivativs of th objctiv function as wll as th dsign constraints ar formulatd in a way that th fild otntial is considrd as a function of dsign variabls. In most roblms, this rlationshi is imlicit and th ncssary drivativs ar obtaind by mans of th dsign snsitivity analysis. Following th forward solution to th fild otntials outsid th otimization rocss, th drivativs of th fild otntials with rsct to th dsign aramtrs ar numrically calculatd by rturbing th fild quations. Whil th first-ordr mthods ar comutationally affordabl and can b fficintly solvd for larg-sizd otimization roblms, xtnsion to th scond-ordr otimization mthods bcom cumbrsom, rquiring intnsiv comutational ffort with incrasing numbr of dsign aramtrs and constraint quations. Escially, if nonlinarity of matrial rortis is considrd for mor ralistic dsign roblms, this numrical rocdur bcoms vn mor comlicatd whn forming th larg matrics from th rturbd fild quations.. BASIC CONCEPT OF OPTIMAL DESIGN Th dsign otimization rocss rquirs basically two main moduls: a modul for fild solution which utilizs an analytical (sldom) or numrical aroximation (oftn) basd on diffrntial or intgral aroachs, such as finit diffrnc mthods, finit lmnt mthod, boundary lmnt mthod or hybrid mthods (BEM-FEM); a modul which mloys otimization stratgis ranging from crud trial and rror stratgis to robust mathmatical rogramming tchniqus. Traditional comutr-aidd dsign (CAD) systms intgrat ths two moduls to sarch for an otimal solution basd on siml, trial-and-rror rincils (s, for xaml, Hool [5], Lowthr and Silvstr [5], and Binns t al. [5]). To find an otimal dsign, th otimization aramtrs ar modifid for ach ossibl trial dsign stat, and thn th fild analysis is rformd for th nw dsign. Thn, th objctiv function is valuatd and th constraint quations ar chcd if thy ar satisfid. This rocss is continud until a dsird rformanc is achivd. q This rocss is xctd to grow as n (whr n is th numbr of dsign variabls and q is th numbr of stats for ach dsign variabl). Considr a siml roblm to b otimizd with rsct to thr indndnt dsign variabls, and ach variabl has thr ossibl dsign 3
L. Ovacı Sigma 4/ stats. Also, suos that ach call of fild analysis rogram rquirs on-tnth of a CPU scond on a comutr. Thus, th analysis rogram is calld 3 3 = 9 tims to valuat th objctiv function for all ossibl dsign stats rquiring a total of.9 CPU scond on th comutr. This would robably b considrd as an conomical and fficint solution. Howvr, most ractical dsign roblms tyically rquir as many as variabls and dsign stats for ach variabl. In this cas, a total of analysis calls ar mad to valuat th objctiv for all ossibl dsign stats. Suos a rlativly mor accurat fild analysis is rquird for ths roblms, and ach analysis call rquirs on scond. It will b xctd 3 yars of comutation on th sam comutr! Dsit th fact that ths ossibl trial dsign stats ar liminatd by an xrincd dsign nginr or an xrt systm using som nowldg-basd huristics, th rocss is comutationally xhaustiv and its us is still imractical for comlx nginring roblms. 3. AUTOMATED OPTIMAL DESIGN Figur. Automatd Dsign Procss In rcnt yars, traditional dsign rocdurs hav bn automatd using r- and ost-rocssing moduls coorating with commrcially availabl fild analysis and numrical otimization 4
Otimal Dsign of Nonlinar Magntic Systms... softwar. Th rincil of an automatd otimal dsign (AOD) systm is basd on a tight intgration of th moduls into an itrativ loo in which th unnown dsign aramtrs ar rogrssivly udatd to advanc th dsign towards th otimum solution. Th bloc diagram in Figur shows th rlations among th basic moduls during th itrativ loo is in rogrss. Starting with a givn initial gomtry, th r-rocssor modul gnrats th initial msh data usd in th finit lmnt analysis modul. Aftr th fild otntial is accuratly solvd, th objctiv function is valuatd using th fild or othr quantitis comutd within th ostrocssing modul. If th dsird objctiv is not satisfid, th fild solution is assd to th otimization modul to rform a lin sarch rocdur to obtain an otimum oint along th gradint dirction comutd by th dsign snsitivity analysis. Onc th nw valus of th dsign variabls ar comutd, th msh coordinats ar udatd, and th nw quantitis ar comutd to chc if th dsird rformanc is achivd. If th nw rsult is not satisfactory, thn itrations ar carrid out until a satisfactory rsult is attaind. Whil traditional otimal dsign rocdurs rquir substantial amount of comutational tim and human-comutr intraction, AOD systms using numrical otimization tchniqus offr a logical stratgy to aroach th otimal solution in a systmatic way. Although th discilin of nonlinar rogramming is wll stablishd, alications to ral nginring systms ar quit nw and nginring ingnuity is rquird for adatation of ths tchniqus to a varity of dsign roblms. In th rmaindr of this ar, mthodologis for dvloing a comutr tool for dsign otimization of magntic systms will b discussd. 4. FINITE ELEMENT MODEL OF MAGNETOSTATIC FIELD 4.. Fild Equation Th gnral form of th govrning artial diffrntial quation of a magntostatic roblm is drivd from Maxwll's quations. Nglcting th high frquncy ffcts, considr r r (ν A ) = J, () whr dislacmnt currnts ar nglctd (i.., no nrgy is stord in lctric fild). This quation rrsnts th most gnral cas of magntostatic hnomnon which tas lac in mdiums with nonlinar magntic rluctivity charactristics. Sinc this study focuss strictly on two-dimnsional cass, th currnt dnsity vctor, J r, osssss only th longitudinal (z-dirctd) comonnt. Thus, Equation () is rducd to th scalar nonlinar Poisson's quation (ν A) = J, () whr A ( x, y) and J ( x, y) dnot th z-dirctd comonnts of A r and J r rsctivly. 4.. Local Elmnt Matrics For numrical solution of Equation (), FEM can b furnishd basd on variational rincils by which th corrct otntial minimizs th nrgy functional [53]: B I( A) = Ω ν ( B ) d( B ) J A ds, (3) S ( ) whr Ω is th roblm domain in which otntial fild tas lac. 5
L. Ovacı Sigma 4/ Figur. A triangular lmnt with thr vrtx coordinats First, th roblm rgion is discrtizd into two-dimnsional triangular lmnts (s Figur ) and th otntial function insid ach lmnt is aroximatd by a linar olynomial A( x, y) = α + α x + α y, 3 (4) whr th α s ar th cofficints of this olynomial and dfind as α = ( a A + a A + a A ) 3 3 α = ( b A + b A + b A ) 3 3 (5) α = ( c A + c A + c A ), 3 3 3 whr A, A and A ar th vctor otntials on th vrtics of th triangular lmnt, and is 3 th ara of this lmnt and is dfind by = a + a + a = b c b c. 3 (6) Th gomtrical cofficints, a s b s and c s, ar dfind as a = x y x y b = y y c = x x 3 3 3 3 a = x y x y b = y y c = x x 3 3 3 3 (7) a = x y x y b = y y c = x x, 3 3 3 whr x s and y s ar th coordinats of th lmnt vrtics (s Figur ). Thus, th comonnts of th lmnt flux dnsity ar calculatd by A A B = α + α, whr B = = α ; B = = α 3 x 3 y y x (8) which imlis that th flux dnsity is constant throughout th lmnt. Th nrgy functional givn in Equation (3) can b writtn for a triangular lmnt B I ( A) = ( B ) d( B ) J A ds S ( ) ν. (9) To aly th minimization rincil, th artial drivativs of Equation (9) with rsct to th otntials at th vrtics of this triangular lmnt ar st to zro I ( A) = A for ( =,,3). () Thn, alying th diffrntiation into Equation (9) dirctly yilds th following intgral quation 6
Otimal Dsign of Nonlinar Magntic Systms... S ( B A ν ( B ) dxdy = ) J dxdy. S ( () ) A A Obtaining th closd form valuation of intgrals in Equation () for ach nods (i.., ) rsults in a st of thr nonlinar quations K, () ( A ) A = F 3 3 3 3 3 whr K is th lmnt stiffnss matrix, A is th vctor otntials at th lmnt vrtics, and F is th sourc vctor rrsnting th xcitation sourcs. Th ntris of th lmnt stiffnss matrix in Equation () can b writtn mor xlicitly as K = ν ( B ) P, 3 3 3 3 (3) whr P is th lmnt gomtric cofficint matrix whos ntris ar calculatd by P = ( bb i j + c i c j ) ij (4) 4 and th ntris of th sourc vctor ar calculatd by F i = J, (5) 3 whr J dnots th lmnt currnt dnsity and is assumd constant throughout this lmnt. 4.3. Global Fild Equations Th total nrgy functional is calculatd by summing th individual contributions of NE lmnts insid th roblm rgion. Thus, th total nrgy insid th systm is xrssd by I NE B ( A) = ( B ) d( B ) J A dω, ν (6) Ω = whr Ω dnots th domain of th lmnt- and B dnots th magnitud of th magntic flux dnsity insid this lmnt. Th functional in Equation (6) is a function of all vctor otntials at th nods of th triangular lmnts dfind. Finally, minimization of th nrgy functional with rsct to th nodal vctor otntials (i.., I ( A, A L A N ) / A for i =,, L, N ) rsults in i th wll-nown nonlinar global fild quations of th finit lmnt mthod K N ( ), N A N A = N F N (7) whr K is th global stiffnss matrix, A includs th vctor otntials of N nods, and F is th sourc trm du to alid xcitation currnts to crat th magntic fild in th systm. Th global stiffnss matrix in Equation (7) is a function of th magntic vctor otntials bcaus of th nonlinar rluctivity charactristics of saturabl iron arts. 5. THE OPTIMAL DESIGN PROBLEM This sction is concrnd with rsnting th basic mathmatical sts in formulating th otimization algorithm roosd in [5]. First, th otimization roblm will b dscribd in a standard form. It will b shown how th objctiv function and th constraint quations ar normalizd to imrov th condition of th otimization roblm. Thn, th constraind roblm is transformd to an unconstraind roblm by forming th augmntd Lagrang function as dscribd in [5]. Th otimality conditions ar imosd to th augmntd Lagrang function to find a minimizr for th unconstraind otimization roblm. Th Lagrang-Nwton quations ar obtaind by linarizing th nonlinar quations from th ncssary conditions. 7
L. Ovacı Sigma 4/ Th itrativ otimization rocdur dvlod in this sction altrs th gomtry of crtain dsird arts of th magntic systm until th calculatd flux dnsitis ar aroximatly matchd at slctd locations. Th dsird rformanc is obtaind by minimizing th objctiv function xrssd in th last-squars sns l= [ ] B c ( A, d) B, l s, l M Φ( A, d) =, (8) whr B and s, l B dnot th scifid and calculatd magntic flux dnsitis at M tst oints, c, l d and A dnot th nodal dislacmnts, th unnown dflctions of slctd nods on th otimizd gomtry from th initial dvic gomtry, rsctivly. During th itrativ rocss, it is rquird that th fild quations b satisfid as th gomtry is altrd. By adding th fild quations givn in Equation (7) to th otimization roblm as nonlinar quality constraints, th objctiv function bcoms a function of both th gomtric aramtrs and th magntic vctor otntials. On th othr hand, th gomtric dsign aramtrs ar subjctd to som constraints in th roblm rgion (.g., th xcitation coils cannot b largr than som dimnsions, and th xcitation currnts ar limitd by som magnitud du to thrmal constraints). With ths assumtions, th otimization roblm is xrssd as minimization of an objctiv function subjct to a st of nonlinar constraint quations: l= [ ] B c ( A, d) B, l s, l M minimiz : Φ( A, d) = (9) subjct to : N K ( A, d) A F ( d) = for =, L, N n n () n= L U d d d for =, L, DF, () L U whr d and d ar lowr and ur bounds of th nodal dislacmnt vctor, d, assignd to som slctd nods on th dvic gomtry to b otimizd. 5.. Normalization Of Objctiv Function And Constraints Th objctiv and th constraint functions in Equations (9) and () ar of diffrnt dimnsions. Ths functions dirctly dnd on magnitud of xcitation sourcs (such as alid xcitation currnt dnsity in th fild windings) and th scal in which th hysical dvic dimnsions is dfind (in this study all dimnsions ar dfind in mtr). Th dimnsions of th objctiv function in Equation (9), th rsidual of global finit lmnt quations in Equation (), and th sid constraints in Equation () ar th squar of Tsla (Tsla is th dimnsion of magntic flux dnsity), Amr, and mtr, rsctivly. For xaml, if th alid fild currnt is doubld (ignoring th ffcts of saturation in iron matrials), th objctiv function quadruls and th rsidual of th finit lmnt fild quations doubls, whil th inquality constraints in Equation () will rmain th sam bcaus thy dirctly dnd on th scal of th dvic dimnsions dfind. This unbalanc among th diffrnt functions causs th following undsird ffcts: numrical difficultis in solving th constraind otimization roblm whn a constraint function or th objctiv function dominats th otimization rocss; dndnc of th objctiv function on hysical dvic aramtrs rvnts th usr from corrctly intrrting otimization rformanc comutd for diffrnt xcitation currnts. Objctiv Function. A ror normalization of th objctiv function and th constraint quations is thrfor ncssary to imrov th conditioning of th unconstraind minimization rocss in augmntd Lagrang multilir mthod. On way for normalizing th last-squars rror or th objctiv function in Equation (9) is to divid ach individual contribution by th corrsonding B. This normalization s, l 8
Otimal Dsign of Nonlinar Magntic Systms... mthod is not always numrically suitabl bcaus it is limitd to a scific cas that all valus must b diffrnt from zro. Altrnativly, a bttr normalization is to multily Equation (9) by a normalization factor M [ B ] c ( A, d) B, l s, l Φ( A, d) = Φ () l= M whr th normalization factor for th objctiv is calculatd by = Φ B s. Th objctiv, l function bcoms dimnsionlss sinc it is normalizd rlativ to th squard summation of scifid flux dnsitis. Th form of normalizd squar root of th rror function ( Φ ) indicats th rlativ avrag rror r tst oint and it will b usd in all rsults dmonstratd in this ar. Fild Equations. Th global finit lmnt fild quations in Equation () ar normalizd by multilying th ovrall quations by a constant N K (, ) A F ( ) for,, N f A d d = = L n n (3) n= whr is calculatd by = / max[ F ] (whr i N ) from th assmbld global f f i sourc vctor F. Thrfor, th dndncy of th rsidual of th global finit lmnt N quations on th alid currnts is liminatd. Costraints. To xrss th otimization roblm in th form of a standard constraind otimization roblm, th linar, doubl-sidd inquality constraints in Equation () ar convrtd into a st of quadratic but singl-sidd inqualitis as d This inquality constraint function on th right sid of Equation (4) has two valuabl rortis: th first, it tas a ngativ valu as long as th nodal dislacmnt stays btwn L U U L d and d ; th scond, it is dimnsionlss sinc it is dividd by th trm ( d d ). L 5.. Augmntd Lagrang Function L U ( d d )( d d ) U L ( d d ) U d d g ( d ) (4) Considring th dscrition of th augmntd Lagrang function and th objctiv function and th st of constraint quations, th augmntd Lagrang function is formd by adding th quality and th inquality constraints to th objctiv function as ( M N N L, d, λ, µ, r ) = A B (, ) Φ c A d B, l s, l + l= λ K ( A, d) A F ( d) f n n = n= (5) DF + + µ θ r θ, = whr DF dnots th total numbr of nodal dislacmnts assignd to th gomtry to b otimizd, λ and µ dnot th Lagrang multilirs corrsonding to th quality and inquality constraints, r is th nalty multilir for th inquality constraints usd in th augmntd Lagrang multilir mthod, and θ is th augmntd inquality function xrssd as A [ ] L U ( d d )( d d ) µ ( ), U L d d r θ = max g ( d ). (6) 9
L. Ovacı Sigma 4/ Not that th quality constraints from th finit lmnt fild quations ar not augmntd to avoid comlx scond-ordr drivativ calculations of th squard quality constraints. Thus, th Lagrang multilirs associatd with th quality constraint quations ar calculatd in th itrativ otimization rocss. 5.3. First-Ordr Ncssary Conditions Th augmntd Lagrang function givn in Equation (5) is a nonlinar function of th magntic vctor otntials and th unnown gomtric dsign aramtrs dscribing th otimizd dvic gomtry. This standard otimization roblm with nonlinar constraints is widly studid in th fild of mathmatical rogramming. Th solution is obtaind by stting th otimality conditions on th Lagrangian. Thn, th systm of rsulting nonlinar quations is linarizd, and th unnown variabls dscribing th gomtry and th magntic fild ar obtaind by th Nwton- Rahson mthod. To minimiz th unconstraind otimization roblm in Equation (5), unnown variabls ar comrisd in a vctor X = A, d, λ (7) { } T * * * * Lt th otimal solution b X = { A, d, λ } T *. Thn, th stationary oints, X, ar obtaind by alying th first ordr ncssary conditions. Ths conditions ar thus mt by taing th first artial drivativs with rsct to th unnown variabls and stting thm qual to zro * (i.., L ( X ) = ): f L N A = = λ K A f (8) n= i L M B A c, l N = = i B ( A, d) B N K m Φ c, l s, l + λ = A l= A K + A (9) f i m = n= i i Ai (, d) A F ( d) =, i n n f, [ ] L M B A c, 3 = = [ ] i B ( A, d) B Φ c, l s, l d l= d f, L U U L + d d d d d d d i i i i i i i + δ µ + r = i i ( ) ( ) (3) U L U L d d d d i i i i Th N + DF nonlinar xrssions obtaind in Equations (8)-(3) ar xlicitly xrssd in trms of th finit lmnt matrics, th magntic fild quantitis, and thir drivativs as: LA = for i =, L, N λ i LA = for i =, L, N A i i i l ( )( ) N N K + λ f = n= di LA = di for i =, L, DF, whr an indicator flag is dfind as µ if g ( d ) > δ = r. othrwis (3) m F A m d ( ) i (3)
Otimal Dsign of Nonlinar Magntic Systms... Th valu of this flag tas a valu of ithr or dnding on th stat of th augmntd inquality constraint givn in Equation (6). If th inquality constraint bcoms activ, its drivativs with rsct to th dsign aramtrs ar nonzro [7]. Th rsulting nonlinar quation systm can b uniquly solvd for indndnt variabls for an initial starting oint { } T X = A, d, λ. It should b notd that th initial stimat for th otimization variabls may b vry critical dnding on th roblm. In th cas of multil solutions, convrgnc to th global otimum solution is not guarantd unlss th initial guss is clos nough to th otimal oint. Othrwis, th solution may b trad at a local otimum solution if multil solutions xist. 5.4. Lagrang-Nwton Equations Th multi-dimnsional systm givn in Equations (8)-(3) is nonlinar and it is ncssary to linariz it for th itrativ solution. Givn th initial solution X, th solution for th nxt itration is dtrmind from th multi-dimnsional Taylor's xansion by ing only th linar trms L ( X ) = L ( X ) + L ( X ) X =, A A A (33) + whr th rsidual trm is dfind as X = X X, and obtaind from th solution of th linar systm L ( X ) X = L ( X ). A A (34) Th right hand sid matrix, L ( X ), is calld th Hssian of th Lagrangian and th A right hand sid is obtaind by valuating th gradints of th augmntd Lagrang function as dfind in Equations (8)-(3). If th maximum ntry of th rsidual trm is gratr than a + tolranc, th solution for th th -th itration is udatd by X = X + X. If th st of quations is arrangd and th ncssary diffrntiations ar obtaind, th rsidual vctor X for th otimization variabls is calculatd by solving th non-symmtric sars linar systm S D A f N N N N N DF N N E F G λ = f, (35) N N N N N DF N N T H D J d DF N DF N DF DF DF f 3 N Whr, th bloc matrix form in Equation (35) is th Hssian matrix which includs th information for th siz and th dirction of th dcrasing gradint vctor in th otimization rocdur. Th bloc matrics ar sars and stord by an uncomrssd ointr storag schm in on-dimnsional arrays. Dtaild formulations for bloc matrics can b found in Ovaci [54]. 6. PARAMETRIZATION OF DEVICE GEOMETRY Th tas of otimization is to itrativly calculat th nw valus of th slctd otimization aramtrs to modify th actual sha of th dvic, and thrfor to minimiz th objctiv function in ordr to achiv a dsird rformanc. Onc th valus of th dsign aramtrs ar calculatd, th gomtry is modifid by small incrmnts from its rvious sha to a nw sha, and nw itrations ar rformd until ths incrmntal changs do not significantly ffct th variation of th objctiv function. Prhas, on of th most difficult roblms in sha otimization roblms is to lin th gomtric otimization aramtrs to th hysical dvic gomtry dscribd by th crtain hysical dimnsions and th contours of th dvic (th
L. Ovacı Sigma 4/ matrial intrfacs, such as iron-to-air or cor-to-air). This sction will xlain th aramtrization mthod usd in th dvlomnt of th otimization algorithm. Figur 3. Modling th otimizd contour: a) triangular msh and dvic contour; b) numbring nods on th dvic contour and th usr dfind nots As th nw valus of th gomtric aramtrs ar calculatd, th dvic gomtry changs and thus th nw finit lmnt msh has to b gnratd for th nxt itration. For this uros, an automatic msh gnration algorithm is calld whnvr th dvic gomtry is modifid (Subramaniam t al. [3]). This howvr has crtain drawbacs, scially whn th gradint rror du to th discrtization is significantly larg in rgions containing coars lmnts (Wbr and Hool [8]). In this cas, th convrgnc bhavior of th otimization algorithm is significantly influncd by th discrtization rror du to discrtization of th roblm domain. This may joardiz obtaining smoothly convrging rsults sinc sis on th objctiv function gradint changs th dcrasing dirction of otimization. Th rvious wor showd that this roblm can b gratly circumvntd by maintaining th sam msh toology during th itrativ modification of th gomtry (Pironnau [4], Haslingr and Nittaanmai [6], Wbr and Hool [8], and Wbr [7]). Th nodal coordinats of th finit lmnt msh ar mad to th gomtric aramtrs. Thus, msh nods ar smoothly movd from on osition to anothr nsuring that th discrtization rror smoothly changs during this rocdur. During th itrativ modification of th dvic gomtry, four tys of msh nods ar considrd in trms of rstrictions on thir moving abilitis: Princial nods: nods on th dvic contour which dscribs th otimizd gomtry of th dvic. Princial nods ar allowd to mov only in th dirction of assignd dislacmnts; Associatd nods: msh nods which ar slctd by th usr and ar critical for th gomtry modification; thy mov along with th dflctd surfac in any dirction (unlss th rstrictions ar scifid by th usr) so as to rvnt any ossibl xcssiv lmnt dformation or ovrlaing in th finit lmnt rgion; Constraind nods: nods which ar constraind ithr horizontally or vrtically to avoid any ossibl violation of dvic's hysical gomtry. Fixd nods: nods which ar ithr far from th dflctd surfac (not quit influncd by th surfac dflction) or on th fixd dvic boundaris which ar not allowd to mov in any dirction to avoid any unwantd altration of th roblm gomtry. 6.. Otimizd Dvic Contour To xlain how th otimization gomtry is modld, considr th otimizd gomtry and th finit lmnt msh nods shown in Figur 3(a). Thr ar SN nods on th otimizd ortion of
Otimal Dsign of Nonlinar Magntic Systms... th dvic gomtry, and som of thm ar dirctly lind to th sha otimization aramtrs (th nodal dislacmnts as dfind arlir). Th dislacmnts dirctly control th gomtry of th dvic whil altring th gomtry (s Figur 3(b)). Th dislacmnts ar alid to som of th usr-slctd SK nots (rincial nods) on th surfac. Sinc th otimizd surfac may b comlicatd in most roblms, ths nots ar chosn from ths nods, dfining th otimizd gomtry. Th rmaining SN SK nods on th gomtry surfac ar simly th associatd nods whos dislacmnt wights ar dtrmind from th wights of th nots by cubic-slin introlation. Onc th incrmntal dislacmnts alid to th surfac nods th surfac is dflctd. Th total dflction in th -dirction is xrssd by SK ( s) = Q d i ( ) i ( ) i= D (36) whr s is th aramtric distanc on th surfac, Q ( s), i is th basis of th dislacmnt wight associatd to th i-th not on th surfac, and D (s) is th total dflction of th surfac aftr alying th incrmntal nodal dislacmnts d to th nots. Figur 4 shows th basis functions associatd with 5 nots slctd out of 4 surfac nods on an otimizd surfac. Th wights ta unit valus at th associatd not and vanish on th othr nots. For th surfac nods, th dislacmnt wights associatd to ach dislacmnt is calculatd from th basis functions as β = Q( s ) i ( ) i (37) Hr, using cubic slin introlation is found to b a good choic to smooth th jaggd contours on th otimizd gomtry. As th nw nodal dislacmnts ar udatd th nots ar movd. Thrfor, th indndnt nodal dislacmnts on th otimizd surfac ar lind to th intrnal msh nods in such a way that any dislacmnts of th surfac nods also ffct th nods of th sub-rgion. 6.. Msh Coordinats Th finit lmnt matrics and th magntic flux dnsitis insid th lmnts ar functions of som gomtric cofficints xrssd in trms of th vrtx coordinats of lmnts. If th otimization surfac is dflctd, th lmnts insid a crtain rgion ar dformd. Thrfor, th lmnt matrics and th magntic flux dnsitis chang with th dflction aramtrs. This sction rovids an stimatd analytical xrssion btwn th msh coordinats and th nodal dislacmnt aramtrs alid to th otimizd surfac. Th rlationshi btwn th dislacmnts and th coordinats of th finit lmnt msh is nonlinar sinc th nods narby th dflctd rgion mov mor than th ons far from th surfac. Th xrssion suggstd is usd for closd form drivativ calculations and is valid for small dislacmnts. Th dscribd msh dformation mthodology uss a fixd msh toology during th itrativ modification of th dvic gomtry. As th nw valus of nodal dislacmnts ar calculatd, th otimization surfac is dflctd, and th intrnal nods in th usr-dfind subrgion mov along with th surfac nods. To avoid any ovrlaing lmnts, valus btwn and ar assignd to th dislacmnt wights, in such a way that th wights smoothly dcras as th distanc of th nods to th dflctd surfac incras (whil th wights of nods outsid th sub-rgion rmain zro). Th msh coordinats of th -th itration ar calculatd from th coordinats of th initial msh lus th sum of th roducts of nodal wights and corrsonding dislacmnts in that dirction: DF x = x + d β i ( ) i ( ) i=, (38) 3
L. Ovacı Sigma 4/ whr DF is th numbr of nodal dislacmnts in th -dirction, β is th dislacmnt i ( ) wight of th i-th nod. In vry itration th dislacmnt wight, and d i, th comutd ( ) incrmnt for th dislacmnt associatd with ach nodal dislacmnt, is first dtrmind, and thn th rlation btwn th nw msh nods and th old msh nods is xrssd in trms of th nw coordinats of th nw msh. Thrfor, in calculation of th drivativ trms in th Hssian from Equation (35), th artial drivativ of a gomtry-dndnt function f ( d, x ) with rsct to th dislacmnt d is obtaind by th chain-rul diffrntiation [55]-[57]: i ( ) Figur 4. Modling a fiv-not dvic contour using basis functions: (a)-() th cubic-slin basis functions of ach not; (f) th initial contour (dashd lin) is dflctd aftr alying incrmntal dislacmnts to ach not df dd f x NL j ( ) = + j= d x d i ( ) i ( ) j ( ) i ( ) f, (39) N whr th vctor x contains N lmnt nods from x (i.., x R L x ), and f is any L function which may b th flux dnsity or th local finit lmnt matrix ntry of an lmnt. Th trm x / d can b obtaind by dirctly diffrntiating Equation (38). j ( ) i ( ) Th nodal dislacmnt wights ar unnown for ach moving nod in th finit lmnt rgion. Thy should b assignd bfor ach Nwton-Rahson itration is rformd. A quic way to aroximatly dtrmin ths wights is to dfin a stratgy such that th dislacmnt wights assignd on th otimization surfac and th rstrictions ar alid on th nods in th rgion and th dislacmnt wights on th associatd nods ar dtrmind from basis functions alid to th nods on th otimizd gomtry. Th following dscribs th outlin of this algorithm. 4
Otimal Dsign of Nonlinar Magntic Systms... St. Rad th coordinats of th usr dfind fram, th global nod numbrs of th surfac nods which dscrib th dflctd surfac, dirction of th dislacmnts, indics of th slctd nots from th surfac nods, th not indics assignd to th dislacmnts and th global nod numbrs dscribing th fixd gomtris in th usr dfind gomtris. St. Initializ th dislacmnt wights of th nods. St 3. Assign unit valu of dislacmnt wights to th nots which ar associatd with th dislacmnts, and introlat th dislacmnt wights for th othr surfac nods which ar not slctd as nots. St 4. Itrativly calculat th dislacmnt wights of th othr nods which ar nithr a surfac nod nor a nod dscribing th fixd gomtris. Itrat for th nodal dislacmnt wights which do not blong to th dislacd surfac as wll as th nods which blong to th fixd gomtris. In St 4, th dislacmnt wights of th associatd nods ar dtrmind from th dislacmnt wights of its nighboring nods as illustratd in Figur 5. Ths valus ar calculatd by wightd avrags considring th gomtric distancs of th nighboring nods NG β w β, o ( ) i ( ) i ( ) (4) i= whr th trm w dnots th gomtric wight and is calculatd by i ( ) l i ( ) w i = ( ) NG l i ( ). (4) i= Th itrativ rocdur in St 4 is continud until a minor chang occurs btwn th dislacmnt wights obtaind in two succssiv itrations. Figur 5. Itrativ schm to calculat dislacmnt wights of fr-moving nods from thos of thir nighbors 6.3. Udating Msh Coordinats Th robustnss and fficincy of numrical sha otimization algorithms strongly dnd on th gomtric maing of lctromagntic roblm (Wbr [7] and Wbr and Hool [8]). Onc 5
L. Ovacı Sigma 4/ th incrmntal dislacmnts ar comutd within th Nwton-Rahson rocdur, th nxt st is to rocd towards th otimal dsign by alying ths changs to th currnt dvic gomtry. In this st, car has to b tan that th nw coordinats of th nods on th otimizd dvic contour and th associatd nods in th finit lmnt msh ar rorly dflctd without violating th constraints for th dvic's hysical dimnsions as wll as without forming any ovrlaing or xcssivly distortd lmnts. Thn, this dflctd msh structur is usd in subsqunt Nwton-Rahson itrations; thrby maintaining a constant msh toology throughout th otimization rocss to avoid discontinuitis in drivativs of both objctiv function and constraints causd by discrtization rror. Although limitd but somwhat quit attractiv for siml gomtris, analytical maing xrssd in trms of nodal dislacmnt wights was usd arlir by Marrocco and Pironnau [59], and Istfan and Salon [8]. Howvr, satisfying all constraints of comlx gomtris is avoidably difficult for gnral alications. A numrical maing tchniqu handling mor comlicatd gomtris is usd in th DOPTD rogram. Th msh structur is dflctd basd on structural laws of lasticity using an lastic body analogy to th structural subdomain including th otimizd dvic contour and its nighborhood whr th sha of lmnts may b critically distortd du to changs in gomtric dsign. Sinc th dflctd gomtry in th x-y lan dos not vary along th z-dirction (th axial dirction of th otimizd dvic), th lasticity roblm can b tratd as two-dimnsional. Thrfor, th structural dflction is dscribd by dislacmnt vctor U ( x, y) with its comonnts u ( x, y) and v ( x, y) in th lanar coordinat dirctions x and y, rsctivly. Th analysis of th lan strss roblm is carrid out using th finit lmnt mthod. Th total incrmntal dflction of th otimizd contour is alid as boundary conditions to obtain dislacmnts of associatd nods in th structural sub-roblm rgion. Figur 6. Dformation of an lastic lmnt subjct to alid oint forcs Bcaus of thir nodal comatibility, th first-ordr triangular lmnts idntical to thos for th lctromagntic modl ar also usd in th finit lmnt modl of th structural sub-roblm. Considr th triangular lmnt shown in Figur 6. Th rlation btwn th alid oint forcs and th rsulting dislacmnts at th lmnt vrtics is xrssd by S U = F 6 6 6 6 (4) whr S is th lmnt stiffnss matrix, U is th dislacmnt vctor and F is th nodal oint sourc vctor including th comonnts in th dirctions of and U = { u v u v u } T ; F = { f f f f f f } T v 6 3 3 (43) 6 x y x y x3 y 3 Th drivation of th lmnt matrics is not ursud in this ar. For th thory and a dtaild formulation, th radr should rfr to [53] and [6]. 6
Otimal Dsign of Nonlinar Magntic Systms... Th msh data including th vrtx coordinats of triangular lmnts and connctivity indics ar xtractd from th global msh data usd for th lctromagntic modl. To solv th structural sub-roblm having N s nods, all th lmnt matrics ar calculatd and thn assmbld to form th global systm quations S U F. (44) Ns Ns Ns N = s Thn, th boundary conditions ar alid as ithr th scifid oint forcs or th scifid dislacmnts. Using th lattr is mor advantagous bcaus th dvic dscritiv aramtrs rgarding th dflctd and th constraind gomtry ar dfind as dislacmnts. In this mannr, th U vctor of dislacmnts is artitiond into th vctor of scifid dislacmnts U and th vctor of unnown dislacmnts U so that th global quations ar s u writtn as S S U uu us u =, (45) S S U su ss s whr th subscrits u and s rrsnt th nods of unnown and scifid dislacmnts. Thus, th stat of th structural sub-roblm is dtrmind from Equation (45) S uu U S U =. u us s (46) Th structural finit lmnt analysis adotd hrin is usd for gomtric maing uross and no mhasis has bn lacd on accurat solution of th lasticity roblm. It should b ointd out that th lmnt stiffnss matrix in Equation (45) is drivd basd on variational rincils with infinitsimal small dislacmnts, assuming constant straindislacmnt rlation insid th lastic lmnt. In actuality, howvr, th lasticity roblm is matrially nonlinar, i.., lmnt strain is not constant for larg dislacmnts. With this assumtion, if larg dislacmnts ar alid, srious numrical rrors may occur in solution: unralistically high strain nrgy is stord in th lmnts narby th boundaris of dislacd surfac, causing unvn dislacmnt distribution ovr th structural domain. This vntually will rsult in vry distortd lmnts aftr som numbr of succssiv gomtry modifications. To avoid distortd lmnts, th lasticity roblm is linarizd by alying a fraction of th total dflction (say on art in ). Thn, th algbraic systm in Equation (46) is assmbld and solvd using th boundary condition. Tyically, th alid boundary conditions U includ: s a fraction of th total dflction of rimary nods dtrmind from Equation (36); th dislacmnts of th nods constraind by hysical dvic gomtry (dislacmnt comonnts in constraind dirctions ar forcd to b zro); dislacmnts of th boundary nods (fixd nods) of th structural subroblm domain (dislacmnt comonnts in both dirctions ar forcd to b zro). Onc Equation (46) is solvd for th unnown dislacmnts U of th associatd u nods, th obtaind rsult is dividd by this fraction and thn alid to th msh coordinats. Th DOPTD rogram dscribd in this ar utilizs th subroutins of th MODEL library (Ain [6]). Th siz of structural subroblm rgion is dtrmind by th usr basd on his/hr rvious xrinc. It should b t in mind howvr that this siz should b chosn as small as ossibl to minimiz th additional comutational cost. 7
L. Ovacı Sigma 4/ 7. MODELING NONLINEAR MATERIAL CHARACTERISTICS Sinc most dvics ar dsignd to orat in saturation, accuratly modling rluctivity charactristic of nonlinar frromagntic matrials lays an imortant rol in synthsis of magntic roblms. In most cass, vn undr normal orating conditions, magntic dvics ar dsignd to orat in saturation rgion. Thrfor, considring a linar magntization curv in such dvics is unralistic. Mathmatical modling of such charactristics should b tan car of by an aroriat mthod. Magntic saturation may b undrstood grahically by xamining a curv of magntic flux dnsity B vrsus magntic fild intnsity, H as shown in Figur 7. In gnral, magntic charactristics of matrials ar rrsntd by magntization curvs rlating th magntic fild intnsity H, to that of magntic flux dnsity. Ths curvs of diffrnt magntic matrials ar xrimntally dtrmind and tabulatd as a st of B-H curvs in th manufacturr s catalogu. Forming th finit lmnt stiffnss matrix (Equation (3)) rquirs matrial rluctivity valus. Th rluctivity charactristics usd in th formulations ar xrssd in trms of th squar of th magnitud of th flux dnsity in ach lmnt as ν ( B ). During Nwton-Rahson itrations, th rluctivity and its drivativs ar ratdly valuatd whil assmbling th local lmnt matrics for comuting comlx calculations of th Jacobian and th Hssian matrics. Efficintly comuting th rluctivity charactristic and its drivativs is a y lmnt to rduc th comutational cost. Figur 7. A Tyical B-H charactristic curv of a nonlinar magntic matrial with low- and high-saturation rgions Figur 8. Variation of th nonlinar rlativ rluctivity in low-saturation (Rgion I) and highsaturation (Rgion II) rgions To guarant a quadratic, smooth convrgnc to a uniqu solution of th otimization roblm, rluctivity charactristic curvs must b at last twic diffrntiabl. Thr ar svral aroximation mthods offrd in th litratur. Using ic-wis cubic olynomials (cubic slin mthod) usd by Silvstr t al. [6] has bcom oular in solution of th forward 8
Otimal Dsign of Nonlinar Magntic Systms... roblm. Sinc th charactristic curv is rrsntd by cubic olynomials within givn sgmnts, th drivativs may b oscillatory whn larg sgmnts ar slctd. Whn a larg numbr of smallr sgmnts ar usd to rrsnt th sam curv, howvr, mor ffort is ndd to find th associatd sgmnt for a givn magntic flux dnsity. Sris xansion mthods usd by El-Shrbiny [63] suggst a rlativly mor accurat, but mor xnsiv-to-valuat, analytical aroach using summation of xonntial functions. Using a singl xonntial function suggstd in Hool and Hool [64] is rlativly low in cost to valuat but it can rrsnt th curv in a limitd rang of saturation. A tyical rluctivity charactristic of a frromagntic matrial is shown in Figur 8. Th rlativ rluctivity is xrssd as a function of B bcaus it rovids an asy diffrntiation to avoid rlativly xnsiv squar root oration. Th rlativ rluctivity curv varis vry slowly in th linar rgion, and thn linarly incrass in th modrat saturation lvl (Rgion I). In th high-saturation lvl (Rgion II), howvr, th linarly incrasing curv slightly bnds and s slowly incrass to th limit valu of ν = / µ. o o Thr ar svral good rasons for using squard valus of ths indndnt variabls B or H, rathr than thir magnituds. First, ths variabls ar usually drivd from otntials in vctor comonnt form, so that finding th magnitud involvs first finding th squars of th comonnts and thn xtracting th squar root of thir sum. Th rlativly xnsiv squar root oration is avoidd in this way. Th scond, rhas mor imortant, is th stability roblm: woring with th squar tnds to mhasiz th bhavior of th curv at high flux dnsitis or filds whr highr rcision is usually rquird. Th suggstd modl in this ar can only accuratly rrsnt th rluctivity charactristic u to a crtain saturation lvl of in th low-saturation rgion. Th fitting valus bgin to dviat from th actual valus byond a crtain lvl of magntization B. At this oint, m th rluctivity function ν ( B ) = ( c + c B )x( c B ) r 3 (47) is usd for rrsntation at high-saturation lvls in Rgion II. Hr, th cofficints c, c, and c ar dtrmind by imosing th continuity conditions of th rluctivity function itslf and its 3 first and scond drivativs at th intrsction of th two modls ν ν ( B ) m r Bm = ( c c B )x( c3b ) ν ( B ) r ν = m 3 m B Bm m [ c + c ( c + c B )] x( c3b ) [ c c c ( c + c B )] x( c3b ) 3 m m ν ( B ) r ν = m 3 ( B ) B m m whr th unnown cofficints c, c, and c 3 ar dtrmind by simultanously solving Equation (48): = ν x( c B c B c ) m 3 m c ν + ν 3 m m c = x( c B ) c 3 m c3 c ν = ν m m m. m ν ν m m B ν + ν m m m (48) (49) 9
L. Ovacı Sigma 4/ Thrfor, a mor ralistic modl rrsnting th nonlinar rluctivity charactristics of iron arts in both low- and high-saturation cass may b utilizd in th solution of th fild otntials. 8. OUTLINE OF THE DOPTD PROGRAM Th flow diagram in Figur 9 shows th basic sts of th otimal dsign rocss controlld by th main rogram (DOPTD). Th main rogram coorats with thr moduls: th rrocssor modul radily availabl in th MICROFLUX rogram; th lctromagntic finit lmnt analysis modul (FORWARD); th structural finit lmnt analysis modul in th MODEL rogram library. Th MICROFULX rogram is usd for gnrating th data including: msh, matrial, xcitation currnt and matrial rortis for th lctromagntic finit lmnt modl of th magntostatic dvic to b otimizd. Th data gnratd ar writtn in a fil latr rad by th DOPTD rogram. Th FORWARD rogram is usd to obtain a finit lmnt solution for th magntic vctor otntials in th nonlinar magntostatic roblm: it is calld rior to th otimization rocss for roviding an initial solution to th vctor otntials for th initial gomtry. Th MODEL rogram is usd to solv structural subroblm dscribd in Sction 6.3 to comut th nw coordinats of th moving nods whnvr a nw dvic sha is calculatd. All routins dscribd hrin ar imlmntd using th standard FORTRAN-77 languag. Th outlin of th ovrall otimization rocss is as follows. Th ncssary data fils ar rad from th DOPTD rogram. Ths includ th msh coordinats, scifid flux dnsitis, thir scifid dirctions and th matrial rortis (ithr linar or nonlinar). Th rliminary comutations ar thn carrid out. In this st, lmnt connctivity indics including th indics of nighboring lmnts and nods ar dtrmind and stord in arrays. Th aramtrs for nonlinar saturation charactristics of nonlinar matrials in low- and high-saturation rgions ar comutd. Ths aramtrs ar latr usd in dirct diffrntiation of th lmnt matrics in forming th Lagrang-Nwton quations. In th nonlinar otimization rocss, th augmntd Lagrang function is linarizd and to obtain th systm of quations to calculat th udat of unnown variabls including th dsign aramtrs, vctor otntials and th Lagrang multilirs for th quality constraints (th fild quations). Onc th udats for th unnown dislacmnts ar comutd, thy ar assd to th MODEL rogram to obtain nw coordinats of th msh nods of th finit lmnt modl. Th nonlinarity of th fild quations ar dtrmind calculating th maximum rsidual of th global fild quations R. For svrly saturatd magntostatic fild, th comutd udats for max th vctor otntials ar not rojctd corrctly using th udats calculatd by th Nwton- Rahson rocss. If th nonlinarity is svr (i.., R > ε ), th gomtry is udatd and thn max R th global finit lmnt quations ar solvd using th FORWARD roblm. Ths comutd vctor otntials ar usd to udat th vctor otntials. Onc th vctor otntials ar udatd for th nw gomtry, th flux dnsitis at th tst oints ar valuatd and th last-squars rror is comutd for th nxt itration. 9. RESULTS AND DISCUSSUIONS This sction is concrnd with th otimal dsign of synchronous machinry by using th DOPTD rogram dvlod and imlmntd in this ar. Th dsign objctiv usd for th roblms ariss from th rquirmnt of th air-ga flux dnsity to vary sinusoidally along outr rihry of th airga rgion. Th objctiv function is minimizd, in th sns of last-squars, with rsct to th most snsitiv gomtric aramtrs subjct to gomtric constrains scifid. Th cas studis undrtan for unsaturatd and saturatd salint-ol, and saturatd round-rotor