Quasi static field computation by finite elements: Recent developments with respect to the modeling of electrical machines

Transcription

1 999 Computatoal Methods Egeeg'99 Eds.: P. M. Pmeta;. M. L.. F. Basl; E. S. lmeda. Quas statc feld computato by fte elemets: ecet developmets wth espect to the modelg of electcal maches K. Hameye Katholee Uvestet Leuve E.E. Dept. Dv. ELE Kadaal Mecelaa 94 B-3 Leuve Belgum Itoducto ogal desg ad the step by step optmsato of physcal techcal devces s pactce ofte a tal ad eo pocess. Dug the desg ad costucto of a devce seveal expesve pototypes have to be bult to moto ad chec the mathematcal appoxmatos ad the physcal ealty. hs pocedue s tme cosumg ad expesve. Successful dustal developmets demad shote cycle tmes to fx o mpove the ecoomcal competto of patcula compaes. o effectvely compete the maet place owadays developed poducts of hghe qualty mpoved effcecy ad bette fuctoalty ae ecommeded leadg to devces wth vey complex geometes. Futhemoe custom desgs ae becomg vey mpotat. he added value of stadad masspoducto devces s fa lowe. o solve the techo-ecoomcal demads the dea s to eplace the expesve pototypg by umecal smulatos. If a appopate smulato model s foud vaous opeatg pots ca be smulated o a compute. Eve the behavou of the devce fo hazadous stuatos that caot be measued sde a laboatoy ad the use of abtay eve futue mateals ca be studed. he appopate choce of a calculato techque fo a electomagetc devce s always closely led to the complexty of the poblem. o develop a techcal poduct paastc effects such as: feomagetc satuato ceased leaage flux hgh opeatg tempeatues evesble flux losses by usg pemaet maget mateals at elevated tempeatues couplg betwee dffeet effects such as themal-magetc-mechacal-flow feld poblems ad duced cuets due to moto effects have to be cosdeed the calculatos accoutg fo suffcet accuacy. I devces wth complex geometes those effects ca ot be teated by a classcal aalytcal appoach. esults wth a hgh accuacy ae equed to pedct the behavou of the techcal poduct. I ths case the smulato of the electomagetc felds ad the effects by umecal models s sutable as a appopate egeeg tool. Usg compute models ad the appopate umecal algothms solves the physcal poblem. he umecal method has to fulfl specfed demads such as: elablty obustess applcato age accuacy pefomace. o see whee the umecal smulato fds ts place the aalyss of techcal devces Fg. shows the ls betwee the eal techcal devce the classcal physcal theoy ad the umecal smulato. hs fgue maes obvous that the umecal smulato s a coectg elemet betwee ealty measuemets ad theoetcal pedctos. s a cosequece all umecal computatos epeset ealstc actvtes a fctve laboatoy. hs meas that smulato esults should be theoetcally measuable pactce. he umecal smulato s fact a

2 expemet pefomed o the compute as a fcttous laboatoy whee the egee s usg umecal tools to pefom the expemets stead of measuemet devces such as cuet voltage powe tempeatue ad foce metes. eal devce modellg mathematcal model of the devce measuemet compute smulato theoy expemetal data computed data theoetcal pedcto compaso compaso vefcato of the model by smulato vefcato of the model by theoy Fg.. heoy expemet ad smulato. he umecal smulato flueces the aalytcal theoy whee sometmes ough appoxmatos o costats ae used to cosde physcal effects such as feomagetc satuato o hysteess. he vefcato of umecal solutos ad esults obtaed by the aalytcal theoy ca lead to mpoved aalytcal models ad vce vesa. Both umecal smulato ad aalytcal theoy help to udestad the physcal ealty ad to mpove techcal pedctos. umecal soluto pocess ad aalyss I Fg. the soluto pocess fo a system of patal dffeetal equatos s outled. System of patal dffeetal equatos ssumptos Choce of potetals gauges cuts Fomulato Choce of soluto cteo Choce of dscetsatos elemets Fg.. Soluto pocess fo a system of patal dffeetal equatos. he felds ae descbed by dffeetal equatos. ssumptos coceg bouday codtos mateal popetes such as sotopy depedeces tme etc. have to be made befoe a computato of a feld ca be pefomed. Fo example mageto-statc felds the tme devatve s assumed to be zeo ad theefoe o duced cuets ca be cosdeed. he choce of the potetals s based o these smplfcatos. Fo each poblem type the choce of a appopate potetal s dffeet. he choce of a gauge s ecessay to obta a egula system of equatos. Usg the fte elemet method the choce of the gauge also detemes the choce of the elemet type. Howeve the.

3 use of a CD pogam pacage that smulates magetc electc o themal felds s usually ot volved choosg fo such basc umecal popetes. he umecal method to solve the patal dffeetal equato s udestood as a soluto cteo. he appopate soluto method depeds o the type of equato such as paabolc hypebolc o ellptc. Fo example the choce of the elemets fo the fte elemet method depeds o the dffeetal equato the potetal fomulato ad the soluto method. I a two-dmesoal magetostatc poblem the uows ae ode potetals. Hee the magetc vecto potetal s chose because the odal uows have oly a sgle compoet z. I ths two-dmesoal feld poblem the Coulomb gauge s satsfed automatcally. he choce of method fo solvg a system of lea equatos s depedet of the dffeetal poblem ad ts fomulato. Fo example the mageto-statc poblem s a ellptc dffeetal poblem. he Laplace opeato s symmetcally adjot ad postve defte. system of equatos wth such popetes ca be solved by a cojugate gadet method. o focus o the actve pats pefomed by a desg egee pcple feld computato s pefomed thee majo steps: pe-pocessg pocessg ad post-pocessg. Fg..3 shows a typcal patte fo the FEM appoach. he fst step cossts of the defto of the geomety of the electomagetc devce. Mateal popetes electcal cuet destes ad bouday codtos ae defed. ll the actvtes have to be pefomed by the desg egee. heefoe the pe-pocessg s tme cosumg. he estmated tme expedtue fo a twodmesoal poblem s gve Fg. 3. he pocessg.e. the soluto of a vey lage system of equatos s automatcally doe the secod step. Oly paametes to cotol the soluto pocess have to be defed by the desg egee. I the last pat of the FEM pocedue the teestg feld quattes ae computed fom the soluto out of the pocessg. If the geometcal data ca be paametesed the pe- ad postpocessg ca be automated as well. hs epesets a Fg. 3. Soluto pocesses dug a feld computato sesso. mpotat peequste fo the possblty of the combato of feld computato ad umecal optmsato. Fo desgg ad costuctg electomagetc devces a accuate owledge of the feld quattes sde the magetc ccut s ecessay. I may cases the a gap s of patcula mpotace (e.g. motos swtches elays cotactos actuatos). Hee the coveso fom electcal to mechacal eegy ad vce vesa taes place. I the a gap the feld quattes such as flux desty ad feld stegth have to be calculated vey accuately ode to be able coectly to asses the opeatoal behavou of the devce. lthough Maxwell equatos have bee ow fo moe tha a cetuy the past the tas calculatg a magetc ccut was to fd as may assumptos ad smplfcatos as possble. he esults could be obtaed wth athe low umecal effots. Usg ths appoach oly devces o poblems wth a stogly smplfed geomety could be studed. It was a desg followg smple ules foud empcally. Physcal effects wee cosdeed by coecto factos appled to the exstg ules. I the followg peod of tme ths desg though ules has chaged to aothe desg phlosophy: desg aalyss. Hee compute models wee used to solve the feld poblem. alyss meas the teatmet of the feld poblem by umecal smulato. Wth the ogog developmets compute had- ad softwae ad umecal eseach dffcultes coceg computatoal costs ad umecal poblems ae cotuously movg to the bacgoud. oday effcet umecal solutos ca be obtaed fo a wde age of poblems beyod the scope of aalytcal methods. I patcula the lmtatos mposed by the aalytcal methods the estctos to homogeeous lea ad steady state poblems ca be ovecome usg umecal methods..3

4 put data: geomety mateal bouday codtos put data: eo boud max. teato steps put data: evaluato localty fo dagams colou plots... pocessg pepocessg postpocessg dscetsato appoxmato paametsato couplg: felds geomety ccuts moto methods mesh adaptato um. method equato solve optmsato futhe modellg lumped paamete appoxmato of local feld quattes feld couplg I geeal the pocedue fo aalysg a electomagetc devce ca be dvded to thee steps: pe-pocessg pocessg ad the post-pocessg. I the fst step the feld poblem s defed ad pepaed to be solved. he secod step delves the umecal soluto of the physcal poblem. Dug the post-pocessg the obtaed soluto s pepaed to calculate the equed feld quattes o to evaluate foces ad othe macoscopc quattes. hs theefold appoach of defg solvg ad evaluatg s typcal fo evey aalyss pocedue umecal o aalytcal. he dffeet techques data stuctues o algothms used the dvdual steps fluece ad/o lmt the oveall pocedue dug the aalyss of a feld poblem (Fg. 4). o defe a feld poblem the put data descbg the geomety of the doma of teest the mateal epesetato ad the bouday codtos ae always equed. Eve wth ehaced CD dawg techques most of the aalyss tme wll have to be spet o the pe-pocessg. Gve eo bouds wll suppot a desed accuacy of the soluto. Ofte the use ca ot fluece ths step. Dug the post-pocessg the soluto must be pepaed to study the local feld effects. he post-pocessg epesets a ope-eded pocess because the use of the aalyss ca evaluate the calculated soluto vaous ways fo dffeet aspects. he methods ad algothms used the sgle steps of the oveall pocedue ca fom a effcet aalyss o desg tool ad deteme the qualty of the esults of the aalyss. Fo example a use of patcula teal data stuctue ca eable vey quc seach outes to obta a effcet fast ad automated dscetsato wth paametesed geometes ad mateals. he vaous possble couplg mechasms of dffeet felds ccut equatos methods such as FEM/BEM combatos moto tem o geometes yeld to a accuate appoxmato of a ealstc physcal poblem. he popetes of the coeffcet matx decde whch equato solve o algothms must be used to solve the poblem. 3 Desg stateges Fg. 4. Feld aalyss steps. he developmet ad desg of electomagetc devces eflects a complex pocess. Ogatg fom a tal dea the costucto us though dffeet phases. hs pocedue s temated whe a fal cocept s selected ad cosdeed to be desged subject to vaous tagets ad costats. s a whole the tas of the desg egee s to fd solutos fo techcal poblems. O the way to the latte physcal ad techcal poduct ceta.4

5 aspects have to be cosdeed. echologcal ad mateal-depedet questos as well as cost effectveess ad ecologcal costats have to be tae to cosdeato. cut-set of the metoed bouday codtos cotols the feasblty of the fal desg. Wth emphass o electomagetc devces Fg. 5 shows a smplfed scheme of tedepedeces of tagets ad costats. hs smple patte clafes that the desg pocess s stogly depedet o the expeece of the egee ad eflects a optmsato pocedue wth ofte cotadctoy ams. heefoe the ecessty of a systematc ad stategc desg wth egeeg tools s obvous. Hee soluto stateges usg mode umecal methods to acceleate ad esue a hgh-stadad techcal poduct a oveall desg pocess ae dscussed. evometal fluece mateal maufactug codtos wdg el. ccut magetc ccut desed desg assemblage Fg. 5. Itedepedeces the desg of electomagetc devces. Desgg electomagetc devces cludes the calculato ad aalyss of the electomagetc feld dstbuto. Fom the local feld quattes foces toques ad losses ca be deved to mae pedctos coceg global quattes such as coveted powe ad effcecy. Fo complcated geometes aalytcal feld solutos ae o-exstet o vey had to obta. Usg umecal feld computato techques of a geeal applcato age the mcoscopc feld soluto leads va a lumped paamete appoach to the desed tmedepedet behavou of the devce (Fg. 6). he mcoscopc feld soluto tself delves mpotat owledge egadg the mateal utlsato. Such esults offe the oppotuty to educe mateal weght ad the costs of the latte poduct. o acceleate developmet extesve feld computatos wth vaous types of mateal ca be pefomed avodg expesve pototypg. It s eve possble to pedct system behavou befoe ew mateals ae actually avalable o the maet. Wth ths owledge the desg egee ca ode specal mateal to be developed at the mateal maufactue o vce vesa f the mateal supple uses such umecal tools he ca suggest ad offe the ght choce of mateal fo a patcula devce. complcated geomety 3D FEM model Maxwell equatos mcoscopc feld soluto flux dstbuto losses cuets lumped paamete model L Fg. 6. alyss scheme usg the fte elemet method. t.5

6 Lumped paamete models ae essetal fo the developmet of cotol stateges fo electomechacal devces such as electcal dve systems. o be able to pefom eal tme cotol schemes lumped paamete models ae used to fom a obseve cotol. Hee vey accuate feld computatos ae ecommeded to deteme the cocetated elemets of such models. ωt q-axs I d j(x md X σd ) I q j(x mq X σd ) I jx σ3d E δ δ E U I a I d β I I q cos ϕ - d-axs Fg. 7. FEM model of the ed-wdg aea to compute the leaage eactace dagam epesetato. X σ 3 of a sevomoto wth d-q model vecto D Fo example the computato of the leaage eactace of electcal motos ca be pefomed usg a theedmesoal FEM model (Fg. 7). he owledge of ths eactace s essetal fo the optmum cotol of a pemaet maget-excted sevomoto. he vecto dagam Fg. 7 demostates the lage fluece of the leaage eactace σ o the optmum cotol agle of ths pemaet maget mache. X 3 D 4 Specal modelg Patcula poblems such as coupled pheomea the wsh to umecally optmse a devce automatcally ad umecal accuaces eque a specal teatmet of the poblem. I the followg sectos a ovevew s gve to tacle coupled feld effects some aspects wth espect to the optmsato of electomagetc devces ae dscussed ad a method to ehace the accuacy of computed feld quattes/foces s toduced. 4. Coupled felds he tem coupled poblem s used may umecal appoaches ad applcatos. Vaous couplg mechasms a dffeet cotext such as feld poblems wth electcal ccuts methods a geometcal o physcal sese couplgs tme ad/o coupled methods to solve a feld poblem ae meat wth ths tem. Fo a pope classfcato of these poblems ad elated soluto methods a systematc defto s poposed. It ca be used the evaluato ad compaso of soluto methods fo vaous poblems. coupled system o fomulato s defed o multple domas possbly cocdg volvg depedet vaables that caot be elmated o the equato level (Zeewcz []). I the lteatue ths oto s ofte led to a dstgushg cotext of vaous physcal pheomea o methods wthout futhe specfcato. hs pape poposes a classfcato scheme whch the umecal models meetg the poposed deftos ca be put. hs may lead to the defto of a sees of test poblems fo specfed coupled poblems ad soluto algothms. classfcato scheme ca smplfy the compaso of the vaous examples ad appoaches out of the lteatue that solve such coupled poblems..6

7 ext to "coupled poblems" the tems "wea-" o "stog-coupled" wll be dscussed to popose a moe homogeous temology. o stat wth a defto of stadads o a classfcato of techcal physcal poblems the popetes ad the tedepedeces of such pheomea must be cosdeed ad dscussed. geeal ad smplfed stuctue of cosdeed feld poblems s daw Fg. Hee the l betwee the sgle felds s detemed by mateal popetes depedg o the coespodg feld quattes. If the feld blocs epeset umecal methods to solve the sgle poblem two dmesos futhe couplgs to exteal equatos such as electcal ccuts magetc o themal equvalet ccut models ae possble to complete the scheme. he l betwee the daw blocs s the cotext of coupled poblems ad ts umecal soluto a compute model o method. he followg questo s whch way the physcal pheomea have to be cosdeed a oveall soluto. Fom the dea of how to l the effects umecally a classfcato of the methods ths sese ca be pefomed. he couplg of magetc feld equatos descbed by a patal dffeetal equato (PDE) ad the electcal ccut equatos povdg algebac expessos fo the electcal cuet destes ca be cosdeed as a specal type of coupled poblem. dffeetal equato of moto foce toque magetc feld statoay electcal flow feld eddy cuets hysteess ohmc losses themal feld mech. stuctual feld Fg. 8. Smplfed stuctue of coupled felds. I geeal moe tha oe depedet physcal feld vaable s volved. he feld vaables fo statoay poblems ae peset a set of PDEs o the taset case oday dffeetal equatos (ODE). he couplg s ofte o-lea ad ths esults a complcated umecal soluto pocess. Felds ca be descbed by dffeetal equatos. geeal fom of a dffeetal equato has to be studed to udestad the paamete couplg betwee equatos. Equato () epesets the geeal fom of a dffeetal equato wth ts possble coeffcets the patcula tems. I coupled felds poblems such coeffcets ae feld depedet ad epeset the l betwee the vaous feld types such as magetc/themal etc. a f ( λ f ) γν f αf g () t. dffuso. paabolc taset tem 5. souce 4. absopto 3. covecto.7

8 he fst tem chaacteses the equato beg paabolc. Statoay equatos do ot cota ths tem ( t ). I Laplace s equato tems ad 5 ae peset. o obta the Helmholtz equato tem 4 ca be added. Fo these two types of equatos a vaatoal fomulato exsts. he 3 d tem s typcal fo poblems cosdeg moto effects. he coeffcets () ae usually deved fom gve mateal chaactestcs. Fo example tempeatue depedet mateal popetes of pemaet maget mateal ca be used to defe a coupled magetc/themal feld poblem. Wth a feld poblem defto the chaactestcs vay locally. I geeal t s possble to dstgush betwee the coupled poblem two ways ts physcal o ts umecal atue. Vey ofte a coupled poblem s called ethe stog o wea. I the physcal sese the stog couplg descbes effects that ae physcally stogly coupled ad the pheomea ca ot umecally be teated sepaately. If umecal fomulatos exst the couplg ca be foud the goveg dffeetal equatos due to the couplg tems. he wea couplg descbes a poblem whee the effects ca be sepaated. he poblem wth ths defto s obvous: If coupled poblems ae studed t s ot vey well ow how stog o wea they ae physcally coupled; ths s the desed aswe expected fom the aalyss of the oveall poblem. Fo example f the mateal popety descbg paametes ae o-lealy depedet o the feld quattes the couplg (stog/wea) ca eve chage wth vayg feld quattes ad the feld quattes ae the esult of the aalyss. heefoe the defto of stog/wea couplg should be chose accodg to the umecal aspects stead of the physcal atue. Choosg fo the umecal aspects t s possble to have a combed stog/wea couplg of feld poblems. hs meas that the stategy of couplg ca vay ad thus the methods/models whle solvg the poblem. umecal stog couplg s the full couplg of the poblem descbg equatos o matx level. he equatos of all volved ad modelled effects ae solved smultaeously. hs mples that the couplg tems ae etes the coeffcet matx as well. he umecal wea coupled poblem s udestood as a cascade algothm whee the cosdeed feld poblems ae solved successve steps ad the couplg s pefomed by up-datg ad tasfeg the feld depedet paametes to the othe feld defto befoe solvg aga. Sce the poblems caot be dstgushed by meas of elmato a b-dectoal fluece exsts. he sestvty of a sub-poblem to chages of the vaables of the studed poblem ca dffe stogly. It s dffcult to quatfy a theshold fo sepaato of both goups ad theefoe the sepaato may be cosdeed as somewhat subjectve. I ths espect the tme costats of the sub-poblems play a mpotat ole. Usually the themal ad mechacal tme costats ae seveal odes lage tha the electomagetc tme costats. So o a shot tem the poblem wth a lage tme costat ca be cosdeed as wea coupled. But ths s ot tue f the statoay soluto s of teest. I the followg the stog couplg FEM equato system of a magetc/themal poblem s deved. Fo smplcty t s assumed that both feld poblems ae defed o the same mesh. Fo a moe ealstc couplg pojecto methods ca be appled to eable the feld deftos o dffeet meshes. hs appoach esults addtoal couplg tems the fal coeffcet matx. Fo futhe smplcty the mateal s popetes ν ad ae assumed to be depedet of ad espectvely. he couplg of the felds causes the emag o-leaty by the loss mechasm. he magetc/themal coupled poblem s modelled by a set of two equatos: υ jω σ J () q. It s assumed that the souce tem of the themal equato cossts oly of joule losses: ( ) ω q q ρ J ρ J. (3) joule ρ he fst tem wll appea o the ght-had sde of the system. he secod tem the eddy cuet losses have to be leased ad epeset the couplg tem wth a o-lea coeffcet:.8

9 .9 ( ) ( ) m J J q J q eddy ρ σω ρ ρ (4) Wtte matx/vecto otato eq.(4) s ewtte as: J J m j ρ σ ω υ. (5) hee s a couplg peset though the coeffcets although thee s a zeo ety the off-dagoal of the magetc equato. pplyg the Gale appoach esults a tegal pe elemet of the fom: Ω Ω d J J m j e ρ σ ω υ. (6) Fo two-dmesoal fst ode elemets ths yelds sx algebac equatos: 3 3 F F K M K. (7) he fst thee equatos ae complex the last thee eal. he etes maed wth a ae the same tems that would be foud the de-coupled poblem. he tems maed wth a esult fom the eddy cuet heat souce tem. 4. umecal optmzato he desg pocess of electomagetc devces eflects a optmsato pocedue. he costucto ad step by step optmsato of techcal systems pactce s a tal ad eo-pocess. hs desg pocedue may lead to sub-optmal solutos because ts success ad effot stogly depeds o the expeece of the desg egee (Fg. 9). capabltes dvdual ways of thg ad actg value system emotos opeatoal owledge declaatve owledge poblem fomulato motvato ad pefomace capablty avalable fomato avalable wog tools avalable tme wog evomet socal ad ogazatoal tegato exteal decsos desg pocess dvdual paametes exteal paametes Fg. 9. Paametes affectg the desg.

10 o avod such dvdual paametes ad thus to acheve faste desg cycles t s desable to smulate the physcal behavou of the system by umecal methods. I ode to get a automated optmal desg umecal optmsato s ecommeded to acheve a well defed optmum. Optmsato of electomagetc devces tus out to be a tas of ceasg sgfcace the feld of electcal egeeg. he tem of utomated Optmal Desg (OD) descbes a self-cotolled umecal pocess the desg of techcal poducts (Hameye []). ecet developmets umecal algothms ad moe poweful computes offe the oppotuty to attac ealstc poblems of techcal mpotace (Pahe [3]). he dstctve featue of ths type of optmsato poblem s ts complexty whch esults fom a hgh umbe of desg paametes a complcated depedece of the qualty o desg paametes ad vaous costats. Ofte the dect elato of the desed qualty of the techcal poduct o the objectve vaables s uow. Stochastc optmsato methods combato wth geeal umecal feld computato techques such as the fte elemet method (FEM) offe the most uvesal appoach OD. hs secto dscusses methodology chaactestc featues ad behavou of optmsato methods. o be able to select the appopate optmsato algothms to fom a oveall desg tool togethe wth the umecal feld computato the popetes of typcal electomagetc optmsato poblems wll be dscussed (ao [4] Pahe [3]). Electomagetc desg ad optmsato poblems eflect maly the followg categoes: costaed poblem type: paamete- o statc optmsato f ( x) m. tajectoy o dyamc poblem f ( x x) m. o-lea objectve fucto desg vaables: eal mxed eal/tege mult-objectve fucto tedepedeces of the qualty fucto ad the desg vaables ae uow; o devatve fomato avalable the qualty fucto s dstubed by stochastc eos caused by the tucato eos of the umecal feld computato method. I ealty electomagetc optmsato poblems ae costaed due to the vaous easos (Fg. 5). owadays optmsatos ae pefomed maly as statc poblems. umecal optmsatos eque huge amouts of computato tme. heefoe the optmsato as amed at hee combed wth the FEM of the dyamc system behavou s ot yet pefomed. Fo taset poblems a evaluato of the qualty fucto by umecal methods (FEM) s too tme cosumg. Cosdeg mxed eal/tege desg vaables esults log computato tmes as well. he tc boxes the lst that ae ot maed epeset developmets fo the futue. he optmsato poblems that ca be solved wll gow wth ceasg compute pefomace as well. I geeal optmsato meas to fd the best soluto fo a poblem ude the cosdeato of gve costats ad t does ot mea to select the best out of a umbe of gve solutos. I othe wods the defto of a optmum s: Defe a pot x (x x... x ) wth the depedet vaables x x... x such a way that by the vaato sde the admssble space the value of a qualty fucto Z(x ) eaches a maxmum o a mmum. he pot x s descbed as the optmum. hs defto mathematcal tems: Mmse a qualty fucto Z ( x ) Z ( x! x ) m. cosdeg g ( x) j j () m h ( x) j () p. j he g j ae called equalty ad the h j equalty costats. y costat ca be detemed oe of these foms. Costats epeset lmtatos o the behavou o pefomace of the desg ad ae called behavou o (8).

11 fuctoal costats wheeas physcal lmtatos o the desg vaables (e.g. avalablty maufactuablty) ae ow as geometc o sde costats. If a optmsato poblem wth oly equalty costats g j (x) (Fg. ) s cosdeed all sets of values x that satsfy the equato g j (x) fom a (-)-dmesoal hype-suface of the desg suface the costat suface. he costat suface splts the desg suface to two basc egos: the feasble o acceptable ego wth g j (x) ad the feasble o uacceptable ego wth g j (x) >. If dug the pogess of the optmsato a desg vecto les o a patcula costats suface ths costat s called a actve costat. x feasble ego g 3 g 4 feasble ego g g Fg.. Costat sufaces a hypothetcal two-dmesoal desg space wth sde costats (g ad g ) ad behavou costats (g 3 ad g 4 ). he depedet vaables ae the desg paamete o object vaables. Fg. shows the shape of a twodmesoal qualty fucto wth the global optmum ad dffcultes such as saddle pots ad local extemum. 5 Z(x x ) x x global optmum saddle local extemum x 4 Fg.. Qualty fucto wth two object vaables. o obta commesuable ctea fo the geeato of the desg vaatos ad to suppot a smplfed fomulato of the stoppg ctea of the algothm the desg vaables should be tasfeed to a omalsed fom: x x x x j d j l j d j x o x (9) j j x x j j whee x jd s the ogal paamete wth ts gve physcal dmeso x jl the lowe boud of the paamete vaato age whle x j deotes the actual paamete vaato age. If o lowe o uppe boud of the paamete s gve the desg vaable ca be omalsed to ts tal value x j. he appopate fomulato of the qualty fucto epesets a patcula poblem. ll desg ams must be fomulated ths sgle fucto ad all object vaables must be mplemeted. Multobjectve optmsato exteds the optmsato theoy by pemttg multple objectves to be optmsed smultaeously. It s also ow ude dffeet ames such as Paeto optmsato vecto optmsato effcet optmsato multctea optmsato etc. Oe way of fomulatg a sgle objectve fucto s a weghted lea combato of the q dffeet objectve fuctos: f ( x) q γ f ( x) 6 ().

12 whee γ deotes a weghtg facto best fomulated wth the popetes q γ I < γ < γ () ad f (x) ae the dvdual objectve fuctos. I pactce the choce of the weghtg factos may aleady fluece the esult of the optmsato. It s ofte ot staghtfowad to select a sgle fxed weghtg facto fo each objectve especally f the objectve fucto s eoeous o f o patcula pefeece s gve to oe of the objectves. 4.. Methods I geeal umecal optmsato algothms ae teatve methods costucted to each the desed optmum successve steps. hs s pefomed followg patcula ules to vay the object vaables ad to deteme the seach decto. he vaous algothms dffe oly the choce of step-legth detemato of the seach decto ad the choce of a stoppg cteo. geeal fom of a optmsato algothm ca be gve by applyg: step : Choose a stat-vecto x () the admssble space ad set the coute of teato. step : Evaluate the soluto-vecto accodg to a qualty fucto. step : Chec whethe a stoppg cteo s fulflled. If yes stop the optmsato; f ot set. step 3: Geeate a ew soluto-vecto by vaato of the objectve vaables usg a sutable step-legth ad seach decto. Cotue wth step. Wth the gve popetes of the electomagetc optmsato poblems the equemets of the optmsato algothms ca be fomulated. umecal methods have to be examed wth egad to the followg ctea: elablty obustess sesblty to stochastc dstubaces applcato age accuacy stable solutos pefomace. Optmsato algothms ca be classfed to: detemstc o stochastc ad dect o dect methods. Detemstc methods ae bascally local optmsato methods ofte based o the costucto of devatves o appoxmatos of the devatve of the objectve fucto (Fletche [5] Betseas [6]). Such gadet based methods e.g. Cojugate Gadet (CG) ewto Quas ewto Boyde-Fletche-Goldfab-Shao (BFGS) etc. ae vey popula as they ae effectve ad covege to the local optmum a small umbe of steps. hs low umbe of qualty fucto evaluatos would be deal whe applyg a computatoally athe expesve FEM aalyss to evaluate the objectve fucto. If o aalytcal objectve fucto exsts o the devatve s dffcult to obta the use of these methods s ot appopate. Futhemoe these methods ae vey sestve to stochastc dstubaces especally peset the devatve fomato they ae based upo. Most detemstc methods addtoally eque the tasfomato of a costaed optmsato poblem to a ucostaed oe. I the case of a multmodal objectve fucto as s ofte the case multobjectve optmsatos these methods ae uable to fd the global mmum (optmum). effectve appoach to compute the sestvty fomato dug a FE-aalyss s toduced to feld computato by Pa et al. [7 8]: the method of adjot vaables. hs method was pevously successfully appled electoc ccut optmsato. Hee the sestvty of the objectve fucto wth espect to a set of desg paametes ca be computed wth oly two solutos. I geeal the huma teacto volved fomulatg a optmsato poblem patcula fdg the devatves s a cosdeable ecoomcal facto whe evaluatg the effcecy of ay optmsato method. he pepaato fo such a optmsato tas mght eque wees whle the executo of the actual optmsato u s a matte of mutes. Ove the past yeas eseach has bee caed out fo achevg automatc dffeetato of compute codes. he dea s to povde fst ad hghe ode devatves of coded vecto fuctos wthout huma.

13 teacto. vaety of automatc dffeetato softwae s aleady avalable such as DIC (Bschof et al. [9]) DOL-C PCOMP etc. t peset softwae code cotaed a sgle fle wth up to. les of code ca be automatcally dffeetated. Stochastc optmsato methods o the cotay such as smulated aealg evoluto stategy ad geetc algothms do ot eque devatve fomato. y d of desg costat ca be mplemeted a smple mae by just ejectg a desg that volates ay costat o by usg pealty tems combato wth the objectve fucto. hese methods ae capable of hadlg lage dmesoal optmsato poblems ad ae less sestve to stochastc dstubaces of the objectve fucto value. he majo dawbac of these methods s the lage umbe of fucto evaluatos equed whe compaed to detemstc methods. hs fact has a fst vew a eve geate mpact whe cosdeg FEM based objectve fucto evaluatos. he fst combatos of the fte elemet techque ad stochastc optmsato methods cosdeed patal models. Oe of the fst publcatos epotg the applcato of a stochastc method to optmse a ete electcal mache s epoted by Hameye []. Sce the a lage vaety of optmsato poblems have bee solved usg the combato of stochastc methods ad fte elemet fucto evaluato (Palo []). lthough the pla executo tme of such optmsatos s lage whe compaed to detemstc appoaches the smplfed set-up of the optmsato tas ad the ablty to fd the global optmum mae such a oveall optmsato pocedue attactve. he athe hgh computatoal expese of the FEM has always esulted attempts to educe the umbe of fucto evaluatos by applyg statstcal methods to sample the seach space effcetly. vaety of methods ca be ettled as dect as the optmsato algothms ae executed o a appoxmato of the eal objectve fucto. he combato of the espose Suface Methodology (SM) ad Desg of Expemets offes a whole set of statstcal tools ot oly to optmse a desg but also to evaluate the ma ad teactve effects of the desg paametes. Oly a few applcatos of ths method have bee epoted electomagetcs cojucto wth FEM fucto evaluatos. majo dawbac of these methods s the fact that due to the use of fst o secod ode (global) polyomals thee s oly a emote possblty of fdg the global optmum a seach space wth seveal local optma. hs poblem has ecetly bee elaxed by the applcato of adal bass fuctos fo the appoxmato of objectve fuctos. he fst applcatos employg the so-called Geeal espose Suface Method (GSM) have bee toduced to the electomagetcs commuty by lotto et al. []. he expeece has show howeve that these methods ae applcable to athe low dmesoal poblems oly as the pactcal effcecy deteoates wth a hgh umbe of desg vaables. Othe methods utlse the devatve fomato made avalable by the appoxmato based o adal bass fuctos. hese methods cease the pobablty of fdg the global optmum peset the appoxmato. 4.3 Foce computato Usg abtay potetals stead of physcal quattes ad the assocated fuctoals the fomulato of the equatos ases the eed fo a close loo at the post-pocessg. he use of a FEM system deses to aalyse a physcal system tems of feld stegth eeges foces destes etc. he potetal tself does ot ecessaly have a physcal meag. I some cases such as the electostatc ad the themal aalyss the potetal epesets the electc potetal ad the tempeatue espectvely (able ). heefoe most of the teestg quattes the post-pocess ae umecally deved quattes. he type ad ode of the shape fucto of the potetal ove a elemet (lea quadatc etc.) ad the elemet type (odal edge etc.) deteme the achevable elatve accuacy of umecally deved values. he accuacy of the esults s flueced by the dscetsato ad elated to t the choce of the eo estmato fo a adaptve mesh efemet f appled. othe dffculty ases the calculato of lumped paametes (ductaces eactaces etc.) used o-fem aalyss pocedues such as ccut aalyss. Seveal dffeet deftos of these quattes may exst as fo the ductace calculato of lea ad o-lea eegy tasduces. he am of ths chapte s to povde a ovevew of possble deved quattes the ecessay fomulatos ad ways of fluecg the accuacy of the esults. he chose potetals fo the dffeet types of poblems do ot ecessaly dectly epeset a physcal quatty. he fomulatos fo defg these potetals ae chose such that the applcato mght mpose smplfcatos the fomulato of the fuctoals o the choce of the gauges. selecto of poblem types ad the physcal meag of the potetals ae collected able..3

14 able. Physcal meag of selected potetals. ype of aalyss dffeetal equato type of potetal physcal meag electostatc scala electc potetal magetostatc ε V ρ ν J vecto oe B Q jωσ themostatc λ scala tempeatue tme-hamoc ν J vecto duced cuets magetc elated to he achevable accuacy of all deved values caot be bette tha the accuacy of the computed potetals. he latte s detemed by the choce of the elemet type the shape fucto ad the fuctoals used. Most local feld quattes as well as othe deved quattes such as foce eque umecal devatves of the potetals. Usg odal elemets the potetals ae ow at each ode as a esult of the appoxmate soluto of the patal dffeetal equato. he chage of the potetal sde oe elemet s detemed by the choce of the shape fucto: a bx cy. () Kowg the potetals at the odes of the elemets the coeffcets a b ad c ca be calculated usg ths bass fucto. he defto of the potetal ow detemes the equed mathematcal opeatos yeldg the equed local feld value. I two-dmesoal magetostatc poblems the vecto potetal s defed by: B. (3) Usg such lea shape fuctos to appoxmate the vecto potetal the x- ad y-compoets of the flux desty sde a fte elemet ae calculated as follows: 3 B c cost. x y e (4) 3 B b cost. y x e he flux desty B sde a FEM model s pecewse costat (Eo! ogem da efeêca ão fo ecotada.) f a cotuous dstbuto of the vecto potetal s assumed. ccoutg fo ths ad assumg a small value of h as the maxmum chaactestc damete of a fte elemet the FEM s coveget towads the exact soluto of ode q. he costat q descbes the polyomal ode of the elemets used. Wth ε as the global eo the ode of covegece fo the potetal soluto s q ε C h. (5) he facto C s depedet of the sze h of the elemets ad depeds oly o the type of dscetsato choce of shape fucto smoothess of the exact soluto. Equato (5) detfes the covegece poblem tasfeed to the appoxmato poblem (Hameye []). Usg fst ode lea shape fuctos the ate of covegece s of ode O(h ). Devg the feld quattes fom the potetal fomulato umecally esults a ate of covegece O(h) fo those quattes.e. a loss accuacy of oe ode compaed to the potetal soluto. Usg these feld quattes ths heet accuacy flueces the esults of foce calculatos. hs fact detfes the dffculty obtag accuate feld quattes as a poblem of the ode of covegece of the umecal method used. o llustate ths fact cosde a doma cotag a sgle lea mateal. By applyg Dchlet bouday codtos of dffeet values to the left ad the ght doma bode a costat flux s mposed (Fg. )..4

15 - Fg.. a) Cotuous vecto potetal; b) pece wse costat flux desty. he loss of oe ode of accuacy due to the umecal dffeetato s heet ad effects all quattes based o such values. possble way to cease the accuacy of local feld quattes s dscussed hee: ecalculato of the feld dstbuto pats of the doma by a local post-pocess. Hee t wll be focused o the pactcal applcato of the statc electomagetc feld soluto of Laplace s equato a local post-pocess to cease the accuacy of a exstg soluto obtaed by the stadad fte elemet method usg fst ode elemets. dvatages ad dawbacs ae dscussed. he Laplace equato two dmesos expessed pola co-odates () s: ( ). (6) ssumg leaty ad ufomty ad applyg a Foue sees to eq.(6) yelds the hamoc fucto: α ( ) { α cos( ) β s( ) } (7) wth ts coeffcets: π α ( ) cos( ) d π. (8) π β ( ) s( ) d π he pocedue fo solvg eq.(7) descbes the soluto of a Dchlet poblem o a ccle wth gve bouday values at ts ccumfeece. he coeffcets α ad β ca be calculated usg ow potetals ( ) at the ccumfeece of a ccle wth adus. ow a fte umbe of equ-agulaly aaged pots ae appled oto the ccumfeece of the ccle. π ( ) ( ) ( ). (9) Wth bouday potetal values u ow o the ccumfeece ad accodg to the popetes of hamoc fuctos the fst tem eq. (7) ca be wtte by: α. () he Foue coeffcets ae ewtte as follows: α cos( ) () β s( ). Wth the Foue sees (7) ad the coeffcets eq.() the potetal the cete of a ccle ca be computed owg oly the bouday potetal values o the ccumfeece of the ccle. Usg ths appoach sde a fte elemet soluto the value of the potetal of a feld pot ow depeds o the soluto seveal fte elemets. hus local umec eos sgle elemets have a elatvely small.5

16 fluece o the soluto the cosdeed feld pot. pplyg (7) devatves at the cete of the ccle ca be calculated a closed aalytcal fom avodg umecal dffeetato. β B s x y. () α B cos y x he dea s to adapt the descbed pocess of solvg a Dchlet poblem o a ccula suface to deteme the vecto potetal a pot P of a dscetsed fte elemet doma (Fg. 3). s the adus of the cosdeed ccula suface ad the dots at the ccumfeece dcate the pots of ow vecto potetal values computed befoehad. hese pots do ot have to be odes of the actual fte elemet mesh. hs featue maes the techque vey advatageous to automatc ad adaptve meshg schemes whch the use ca ot guaatee the cotol of the mesh ad especally ts symmety. o obta the potetal dstbuto at a gve cotou sde a fte elemet doma multple ccles have to be evaluated. Ovelappg ccles guaatee a cotuous soluto the cosdeed ego afte the post-pocess. he umecal shape of (7) () ad () eables a easy mplemetato of the pocedue a fte elemet pogam pacage. he devatves the cete of the ccle ae epeseted by the Foue coeffcets. hus o addtoal computatoal effot s ecessay to compute the flux desty the cete. p C Fg. 3. Multple ccles to deteme the vecto potetal o a cotou. he local soluto of the Laplace equato sde a a gap of a electomagetc devce usg a Foue sees appoxmato fo the vecto potetal esults a sgfcat cease accuacy of the deved feld quattes. o compae the esults obtaed by the local feld evaluato to the covetoally obtaed feld quattes of fst ode elemets Fg. 4 shows the computed magetc flux desty deved by B ad B x usg the Laplace appoach. Fo ths applcato of the local Dchlet poblem 4 potetal bouday values o the ccumfeece of the ccle wee used. pplyg such feld quattes to the Maxwell stess teso to compute the local foces actg o bodes sde a electomagetc feld yelds values of hghe accuacy. o obta the toque of a electcal mache the local foce values ae tegated alog a cotou the a gap. [] [] Bx Bx x [m] y [m] a) b) Fg. 4. a) Computed vecto potetal sde the ccula FEM doma ad b) the esultg flux desty B x deved by applyg the local post-pocess. othe appoach to compute the toque moe accuately uses the values of the magetc vecto potetal o two cocetc ccles wth ad ad o as bouday codtos (Fg. 5). Local feld values o the ccula cotou C wth adus < < ae calculated. o x [m] y [m].6

17 .7 o C Fg. 5. Local Dchlet poblem fo a cyldcal a gap. If the e adus s tae as a efeece the geeal soluto of Laplace s equato s: ( ) ( ) ( ) ( ) d c b a s cos s cos ) (. (3) he coeffcets a b c ad d ae depedetly detemed fo each ccula hamoc. fast Foue tasfomato (FF) algothm s used to expess the magetc vecto potetal at the boudaes as a sees of such ccula hamocs: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). s cos s cos o o o b a b a (4). o o o o o o b b d b a a c a (5) Oce the magetc vecto potetal at the cotou C s ow the omal ad tagetal compoet of the magetc flux desty ca be detemed: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). s cos s cos cos s cos s d c b a B d c b a B t (6) he tagetal foce compoet Ft esults the toque of the devce. It ca be show (Salo 3 Metes et al. 4) that the value of the toque s gve by ( ) ( ) d a c b µ π (7) beg depedet of the adus of cotou C. It s ot ecessay to calculate the omal ad tagetal compoet of the magetc flux desty o the cotou esultg a faste algothm whe the oveall toque s amed at. he

18 poposed method ca easly be exteded to tme-hamoc poblems. If all values ae ms-values the toque s obtaed by addg the toque calculated usg the eal- ad the magay-compoet of the soluto. eal mag. (8) hs method has ts advatage fo electcal mache aalyss as t s suted fo small a gaps. By usg eq.(8) the toque s evaluated dectly wthout the explct calculato of the flux destes. Fg. 6. Equpotetal plot of the eal compoet soluto of a 4 W ducto moto. he pefomace of ths method s compaed wth the classcal Maxwell stess teso method usg a model of a 4 W ducto mache fo tests. o esue accuate esults a good tade-off betwee mesh efemet ad usg the ehaced post-pocessg methods s ecessay. elatvely coase dscetsato the a gap of the ducto mache was chose (Fg. 7). Wth such a coase dscetsato the Laplace-based method s less sestve to the actual choce of the cotou sde the a gap tha the classcal method. he a gap spas a ego betwee a e adus of.86 m up to a oute adus of.875 m. Fg. 8 shows the vaato of the calculated toque. Cotous wth dffeet ad ae chose. Fo the Laplace-based method the e ad oute ad ae vaed smultaeously. heefoe the value of the toque vaes symmetcally towads the mddle of the a gap. Fg. 7. Detal of the dscetsato the a gap. he vaato of the calculated toques usg the ehaced method s much smalle whe compaed to the classcal method. It must be stated howeve that a appopate mesh efemet scheme would lead to bette esults eve fo the classcal toque computato. he same basc dea as used the ccle appoach yelds the local soluto fo the thee-dmesoal feld. he local feld poblem s ow defed by the ow potetal values equally dstbuted alog the suface of a sphee assumed to be the bouday potetal values of the local feld poblem (Hameye []). s a esult Fg. 9 the quadatc covegece efeed to the chaactestc legth h of a fte elemet of the FEM potetal soluto ad the ate of covegece of the foce computatos usg both the classcal ad the ew post-pocessg appoach s plotted vesus the umbe of tetahedo elemets. he same statemet ca be made fo the two-dmesoal appoach. he tagles Fg. 9 dcate the theoetcal gadet of covegece eq.(5). he gadet-tagles Fg. 9 dcate the theoetcal ate of covegece fo the quadatc ad the lea.8

19 oque [m] K. Hameye covegece case. It ca be see as theoetcally expected that the elatve eo a eegy om of the FEM potetal soluto coveges quadatcally efeed to the specfc damete h of the elemets eq.(5) by ceasg the umbe of fst ode tetahedo elemets. Due to the aalytcally descbed potetal fucto sde the local feld the esultg oveall foce usg ths appoach s of the same ode of covegece. heefoe o loss of accuacy of the deved feld quattes occus. he covegece of the total foces computed by the classcal appoach dcates the expected lea behavou. he accuacy of the computed values s flueced by the umecally-obtaed devatves. hs shows that the esults obtaed by the classcal method ae heetly accuate whe compaed to the accuacy of the potetal soluto theoetcal O(h) total foce dect devato total foce ew FOUIE appoach el. eo FEM potetal soluto local Laplace method II classcal Maxwell stess adus of the cotou [m] Fg. 8. Vaato of the toque calculated alog dffeet cotous sde the a gap... theoetcal O(h) umbe of elemets Fg. 9. Compaso of the covegece behavou of the FEM potetal soluto wth both the dect devato ad the devatve-fee appoach. FEM potetal soluto O(h ) dect devato solvg the DIICHLEpoblem devatve fee appoach O(h) feld quattes: flux desty feld stegth flux O(h ) foce computato: Maxwell stess teso vtual dsplacemet othe methods Fg.. ddtoal step dug post-pocessg to ehace the accuacy of deved feld quattes. he use of the poposed appoach to ehace the accuacy of computed feld quattes statg fom a exstg potetal soluto demads a addtoal step dug the post-pocessg of the FEM aalyss (Fg. ). Havg obtaed a FEM potetal soluto the use oly has to defe the suface of tegato Γ o whch the feld quattes o foces have to be calculated. Defg a abtay cotou allows the computato of feld quattes o foces alog t as well..9

20 5 Coclusos he effectve ew desg of mode ad compettve electomagetc devces such as electcal maches eques state of the at umecal smulato techques. owadays feld smulatos ae equed cosdeg specfc coupled feld pheomea. heefoe patcula atteto must be pad to the umecal stablty of the oveall system of equatos. If the stuctue of the oveall coupled system s ow specfc equato solves ae avalable to obta the desed soluto. umecal computatoal methods such as the fte elemet method combed wth umecal optmsato algothms yeld automated desg pocedues towads techcal poducts havg specfc physcal ad/o ecoomcal popetes. cowledgemets he authos ae gateful to the Belga Fods voo Weteschappelj Odezoe Vlaadee fo ts facal suppot of ths wo ad the Belga Msty of Scetfc eseach fo gatg the IUP o. P4/ o Coupled Poblems Electomagetc Systems. he eseach Coucl of the K.U.Leuve suppots the basc umecal eseach. efeeces [] Zeewcz O.C. aylo.l. he fte elemet method 4 th edto Volume II Sold ad flud mechacs dyamcs ad o-leaty McGaw-Hll Lodo 99. [] Hameye K. Kaspe M. Shape optmzato of a factoal hosepowe dc-moto by stochastc methods ed. : Compute ded Optmum Desg of Stuctues III Optmzato of Stuctual Systems ad pplcatos CMP Elseve ppled Scece Lodo ew Yo pp [3] Pahe U. geeal desg tool fo the umecal optmsato of electomagetc eegy tasduces Ph.D. thess K.U. Leuve 998. [4] ao S.S. Egeeg optmsato Joh Wley & Sos 996. [5] Fletche. Pactcal Methods of Optmzato Volume - Costaed Optmzato Joh Wley & Sos Chcheste 98. [6] Betseas D.P. Costaed Optmzato ad Lagage Multple Methods cademc Pess ew Yo 98. [7] Pa I.H. Lee H.B. Hah S.Y. Pole Shape Optmzato fo educto of Coggg oque by Sestvty alyss COMPEL- he Iteatoal Joual fo Computato ad Mathematcs Electcal ad Electoc Egeeg vol. 9 suppl. pp. -4. [8] Pa I.H. Lee H.B. Kwa I.G. Hah S.Y. Desg Sestvty alyss fo Steady State Eddy Cuet Poblems by Cotuum ppoach IEEE asactos o Magetcs vol. 3 o. 5 pp Septembe 994. [9] Bschof C. oh L. Maue-Oats. DIC: Extesble utomatc Dffeetato ool fo SI-C goe Pept L/MCS-P66-96 evsed May 997. [] Palo S. Stuctual Optmsato of a Iducto Moto usg a Geetc lgothm ad a Fte Elemet Method. cta Polytechca Scadavca o. 84 PhD-thess Hels Uvesty of echology 996. [] lotto P. Kutsevtch.V. Magele Ch. Mola G. Paul C. Pes K. epetto M. chte K.. Multobjectve Optmzato Magetostatcs: Poposal fo a Bechma Poblem IEEE asactos o Magetcs vol. 3 o. 3 pp [] Hameye K. Metes. Pahe U. Belmas ew techque to ehace the accuacy of -D/3-D feld quattes ad foces obtaed by stadad fte-elemet solutos IEE Poc.-Sc. Meas. echol. vol. 45 o. pp Mach 998. [3] Salo S.J. Fte elemet aalyss of electcal maches Kluwe cademc Publshe Bosto Lodo 995. [4] Metes. Pahe U. Hameye K. Belmas. De Weedt. Foce calculato based o a local soluto of Laplace s equato IEEE-asactos o Magetcs pat II vol. 33 o. pp