To Luca, and to Michele, who knows about EEG and MEG MS&A Series Editors: Alfio Quarteroni (Editor-in-Chief ) • Tom Hou • Claude Le Bris • Anthony T. Patera • Enrique Zuazua Ana Alonso Rodríguez and Alberto Valli Eddy Current Approximation of Maxwell Equations Theory, algorithms and applications AnaAlonsoRodríguez AlbertoValli DepartmentofMathematics DepartmentofMathematics UniversityofTrento UniversityofTrento Trento,Italy Trento,Italy ISBN978-88-470-1505-0 e-ISBN978-88-470-1506-7 DOI10.1007/978-88-470-1506-7 LibraryofCongressControlNumber:2010929481 SpringerMilanDordrechtHeidelbergLondonNewYork ©Springer-VerlagItalia2010 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialiscon- cerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,re- productiononmicrofilmsorinotherways,andstorageindatabanks.Duplicationofthispublicationor partsthereofispermittedonlyundertheprovisionsoftheItalianCopyrightLawinitscurrentversion,and permissionforusemustalwaysbeobtainedfromSpringer.Violationsareliabletoprosecutionunderthe ItalianCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. 9 8 7 6 5 4 3 2 1 Theimageonthecovershowstheeddycurrentinatrefoilknot(realandimaginarypart) Cover-Design:BeatriceB,Milano TypesettingwithLATEX:PTP-Berlin,ProtagoTEX-ProductionGmbH,Germany(www.ptp-berlin.eu) PrintingandBinding:GrafichePorpora,Segrate(Mi) PrintedinItaly Springer-VerlagItaliasrl–ViaDecembrio28–20137Milano SpringerisapartofSpringerScience+BusinessMedia(www.springer.com) Preface Continuamentenasconoifatti aconfusionedelleteorie1 CarloDossi2 Electromagnetismiswithoutanydoubtafascinatingareaofphysics,engineeringand mathematics.SincetheearlypioneeringworksofAmpère,Faraday,andMaxwell,the scientificliteratureonthissubjecthasbecomeimmense,andbooksdevotedtoalmost allofitsaspectshavebeenpublishedinthemeantime. However,webelievethatthereisstillsomeplacefornewbooksdealingwithelec- tromagnetism,particularlyiftheyarefocusedonmorespecificmodels,ortrytomix differentlevelsofanalysis:rigorousmathematical results,soundnumericalapproxi- mationschemes, real-lifeexamplesfromphysicsandengineering. The complete mathematical descriptionofelectromagnetic problemsisprovided by the celebrated Maxwell equations, a system of partial differential equations ex- pressedintermsofphysicalquantitiesliketheelectricfield,themagneticfieldandthe currentdensity.Maxwell’scontributiontotheformulationoftheseequationsisrelated totheintroductionofaspecificterm,calleddisplacementcurrent,thatheproposedto addtothesetofequationsgenerallyassumed toholdatthattime,inordertoensure theconservationoftheelectriccharge. Thepresence ofthedisplacementcurrentpermitstodescribeoneofthemostim- portant phenomenon in electromagnetism, namely, wave propagation; however, in many interesting applications the propagation speed of the wave is very high with respecttotheratioofsometypicallengthandtimescaleoftheconsidereddevice,and thereforethedominantaspectbecomesthediffusionoftheelectromagneticfields. Whenthefocusisondiffusioninsteadofpropagation,fromthemodelingpointof viewthiscorrespondstoneglectingthetimederivativeoftheelectricinduction(i.e., thedisplacementcurrentintroducedbyMaxwell)or,alternatively,neglectingthetime derivativeofthemagneticinduction. 1Constantlyfactsarisetomuddletheories. 2CarloDossi,1849–1910,Italianwriter. VI Preface Thisbookisdevotedtotheformermodel.Theresultingequationsareusuallycalled magneto-quasistaticequations, or else eddy current equations,and can be seen as a low-frequencyapproximationofthefullMaxwellsystem.Inthefollowingwearein- deedconcernedwiththetime-harmoniccase,inwhichthedataandtheelectromagnetic fieldsareassumedtobesinusoidalintime.Thismodelisveryoftenusedinelectrical engineering(forsomeexamples,see Section1.2andChapter9).Indeed,forthetyp- icalproblemsinthisfieldalternatingcurrents are applied,the electromagneticwave propagationcanbeneglected,butthevariationofthemagneticfieldisstillsignifica- tive:infact,inconductingmediathisvariationgeneratescurrentdensitiesthathaveto betakenintoaccount.Summingup,thetermthatcanbedroppedisthedisplacement current. Inouropinion,thereasonsfortheinterestinthetime-harmoniceddycurrentmodel are manifold.In fact, itis notonlyan importanttopicin electromagnetism, butalso an intriguingmathematical problem in which one has to face some delicate aspects thatcanalsobepresentinothersituations.Therefore,thestudyofthisproblemcanbe usefulforunderstandinggeneral techniquesthatcan be appliedinothercontexts,as well. One of these peculiar aspects is that the time-harmonic eddy current problem presents differential constraints:the magnetic field is curl-free and the electric field isdivergence-freeintheinsulatingregion,andthemagneticinductionisdivergence- freeinthewholephysicaldomain(insulatorplusconductor). Thereareseveralmathematicalapproachesthatallowustotreattheseconstraints. Inthisbookwerefertothefollowing: • saddle-pointformulationswithLagrangemultipliers; • introductionofvectorandscalarpotentials; • penalizationmethods. Eachoftheseapproachesgivesrisetodifferentfiniteelementapproximations:mixed finiteelement methods are used when consideringsaddle-pointformulations,and in thesecasesedgeelementsareneededfordescribingthediscretemagneticandelectric fields; nodal vector elements and nodal scalar elements are used for approximating vectorandscalar potentials,respectively;nodalvectorelements are employedwhen dealingwithpenalizationmethods.Ouraimistogiveapresentationinwhichallthese differentapproachesareconsideredandanalyzed. Onecouldaskwhyitisnecessary tointroducemanydifferentmethodsforsolv- ingthesame problem.Letusquotefromthewell-knownbookbySilvesterandFer- rari[227],p.345:“Inrecentyears,aconsiderableliteraturedealingwiththenumerical solutionof problemsrelatingtoeddy currents has accumulated. Practical configura- tionsareinvariablyirreduciblythree-dimensional.Noclearconsensusappearstohave emergedastothebestmethodofattack,althoughinmanycases somefiniteelement approachorotherisused.” Preface VII Infact,ashopefullyitwillbeclearbytheendofthebook,eachmethodhasassets anddrawbacks: • saddle-pointandLagrangemultipliers.Plus:physicalfieldsasprincipalunknowns; nodifficultywiththetopologyoftheconductingdomain.Minus:manydegreesof freedom;algebraicproblemwithamorecomplexstructure; • magneticscalar potential.Plus:fewdegrees offreedom;“positivedefinite”alge- braicproblem.Minus:somedifficultiescomingfromthetopologyofthecomputa- tionaldomain,inparticularoftheconductor;needtocomputeinadvanceavector potentialofthecurrentdensity; • magnetic vector potential and penalization.Plus: standard nodal finite elements for allthe unknowns;no difficultywiththe topologyof the conductingdomain; “positivedefinite” algebraic problem.Minus:many degrees of freedom; lack of convergenceforre-entrantcornersofthecomputationaldomain. Therefore,itisnotaneasy tasktodevisethebestmethodforallseasons:thisisalso apparent looking at the literature, especially the part related mainly to engineering applications,inwhichnewmethodsareproposedineachissue. Nevertheless, letusnotethat,as faras we know,thereare nobookswhere eddy currentproblemsare widelytreatedfromboththemathematical andtheengineering pointof view. In fact, various monographs are devoted to modeling throughpartial differentialequationsandtheirnumericalapproximation(justtoquoteacoupleofthe mostknown,see Erikssonet al. [102]and Quarteroni[198]), butingeneral theydo notcoverelectromagnetismanditsmathematicaltheory. Ontheotherhand,amongclassical textsonelectromagnetismonlySilvesterand Ferrari[227]andespeciallyBossavit[58],[59]devotesome pagestothistopic.The eddycurrentmodelisalsobrieflypresentedinKˇrížekandNeittaanmäki[158],though onlyforconductivemedia,andinBondesonetal.[55].Finally,achapterinGrossand Kotiuga [115] is concerned with eddy current problems, but more specifically with thosetopologicalissuesthatarerelevantfortheirnumericalapproximation. IntheengineeringliteraturewerecallthebooksbyTegopoulosandKriezis[233] andMayergoyz[173],whereanalyticalmethodsaresystematicallyemployedforde- terminingtheexplicitsolutionofeddycurrentproblems,butonlyinsimplegeometrical configurations,theformerforlinearmaterials,thelatterinthenonlinearcase. This bookis the storyof a fallinginlove. When inthe mid 1990swe started to study eddy current problems, we even did not know the usual way these equations arereferredto(indeed,wewroteapaperon“heterogeneouslow-frequencyMaxwell equations”).However,wewerequicklyattractedbytheirpeculiaraspects: • variational formulations set in somehow unusual spaces like H(div;Ω) and H(curl;Ω), for which some basic results were not completely clarified (for in- stance,thecharacterizationofthespaceoftangentialtracesoffunctionsbelonging toH(curl;Ω)); • thepresenceofdifferentialconstraints,whichgiverisetosomedifficultiesinde- visingefficientfiniteelementnumericalapproximationschemes; VIII Preface • the stronginterplaybetween the topologicalshape of the computationaldomain andthewell-posednessoftheproblem,involvingdelicateargumentsofalgebraic topologynotsurprisinglyalreadyconsideredbyMaxwellhimself,butnotalways addressedinacorrectwayinthemorerecentliterature; • theproblemofdeterminingmeaningfulboundaryconditions,orelserealisticexci- tationtermsassociatedtosignificativephysicalquantitiessuchasvoltageorcurrent intensity; • thebreakingofthesymmetrybetweentheelectricandthemagneticfields,which isspecificinthiscontext,anddoesnottakeplace inthecase ofthefullMaxwell equations; • theunusuallylargenumberofdifferentmethodsproposedforfindingtheapprox- imatesolution,someofthembasedonvariouschoicesofvectorandscalarpoten- tials,mainlyalreadypresentinclassicalworksinelectromagnetismbutnotcom- pletelyunderstoodintheframeworkofeddycurrentproblems; • thelossofconvergenceofnodalfiniteelementapproximationschemesinthepres- enceofre-entrantcornersoredges. This book is the story of an obsession. Having to face such a large number of dif- ferent aspects, and their even larger possible interplays, our research work on eddy current problems has soon become a never-ending wandering among formulations, approximationmethods,analysesofconvergence,topologicalobstructions,choicesof boundaryconditions,andsoon.Tryingtowriteinastructuredwayallthesetopicshas beenawaytoexitthelabyrinthandtostoplookingfora furtherresult.(Asamatter offact,wehaveinmindanotherpossibleapproach,butthemarginofthepageistoo narrowforwritingithere.3)Wehopewesucceeded ingivingamaptopeopleinter- estedinthemathematical theoryoflow-frequencyelectromagnetismand therelated numericalapproximationschemes. Wehavetriedtowriteaself-containedbook.StartingfromtheMaxwellequations wederivetheeddycurrentmodel,andwemake clearinwhichsenseitisanapprox- imationofthefullMaxwellsystem.Theexistenceanduniquenessofthesolutionare provedforallthedescribedformulations,andstabilityandconvergence ofthefinite elementnumericalschemesarepresented.Someusefultoolsfromfunctionalanalysis andfiniteelementtheoryarecollectedintheAppendix. Duetothestructuredescribedabove, thismonographisaddressedtoresearchers andPh.D.studentsinmathematicalelectromagnetism,aswellastoelectricalengineers andpractitioners,whocanfindhereasoundmixtureoftheory,numericalapproxima- tionschemes andimplementationissues,withalimitedneedofprerequisites. Thebookisorganizedasfollows. InChapter1weintroducetheeddycurrentproblemandwepresentitsmathemat- icalformulation,forthetime-harmoniccaseandforthreealternativesetsofboundary conditions.Particularattentionisdevotedtothedescriptionofcertainspaces ofhar- monicfields,whicharerelatedtothetopologicalshapeofthecomputationaldomain andmustbetakenintoaccountinordertodeviseawell-posedproblem. 3Hancmarginisexiguitasnoncaperet. Preface IX The second chapter deals with a mathematical justification of the eddy current modelinadomaincomposedbyaconductorandaninsulator.Itisobtainedthrough twodifferentasymptoticlimitsofthefullMaxwellequations:inthefirstcasetheelec- tricpermittivityvanishes,andinthesecondcasethefrequencyvanishes. Theanalysisofwell-posednessofeddycurrentproblemsisperformedinChapter3: theexistenceanduniquenessofthesolutionisproved,and,moreover,animportantre- markispresented,concerningtheverificationoftheFaradayequationontheso-called “cutting”surfacescontainedintheinsulator.Thisfacthasbeensometimesoverlooked inthe existingliterature,leading toincorrectresults forthe numerical computations basedonformulationswheretheprincipalunknownisthemagneticfield. InChapter4wedescribeandanalyzesomecoupledformulationsthatemployLa- grangemultipliersforimposingthedifferentialconstraintsonthemagneticandelectric fields.Theadvantageoftheseapproachesisthattheyinvolvenorestrictionsoriginat- ingfromthetopologyoftheconductor,andthattheusedmeshesdonotneedtomatch ontheinterface. Totesttheperformance ofthemethodswepresentsome numerical computationsfordomainsofgeneral shape, inparticularsome resultsforproblem7 oftheTEAMworkshopandforaconductingdomaingivenbythetrefoilknot. Twoformulationsbasedontheintroductionofa scalar magneticpotentialinthe insulatorareillustratedinthefifthchapter:theunknownusedintheconductoristhe magneticfieldinthefirstcase,andtheelectricfieldinthesecondcase.Thesemethods useasmallnumberofdegreesoffreedom(theunknownsareavectorfunctioninthe conductorandascalarfunctionintheinsulator,plusafewdegreesoffreedomassoci- atedtothetopologicalshape),butrequiresomepre-processing,likethedetermination ofthe“cutting”surfacesandthatofavectorpotentialoftheappliedcurrentdensity. TheclassicalapproachesusingvectorpotentialsarepresentedinChapter6,mainly forthecase ofamagneticvectorpotential.Thegaugeconditions,neededforfinding a unique potential, are analyzed in depth, in particular in the case of the Coulomb gaugeandtheLorenzgauge.Theadvantageoftheseformulationsliesinthefactthat classicalnodalfiniteelementsareemployed,sothatthesamediscretebasisfunctions canbeusedforalltheunknowns.Moreover,nodifficultycomesfromthetopologyof theconductor. InChapter7wesettheprobleminthewholespaceandweintroducesomecoupled finiteelement/boundaryelementmethods,which,byusingpotentialtheory,allowto reducethedegreesoffreedomintheinsulatortodegreesoffreedomontheinterface. Inparticular,we present inmore detailthecoupledapproach based onthe magnetic vectorpotentialandthescalarelectricpotentialintheconductor:thismethodhasthe characteristic of being stable with respect to the frequency, hence can be also used withoutmodificationforthestaticcase. The eighthg˘¡ chapter deals with the case of excitation terms given by a voltage droporacurrentintensity,asituationthatcanbeinterestingwhenthecouplingwith circuitproblems has to be considered. In order todevise a well-posed problem itis necessarytochoosesuitableboundaryconditions.Forotherboundaryconditionsthe solutioncanbefoundonlyifthevoltageorthecurrentintensityareinterpretedasan excitationtermgivingrisetoaspecificcurrentdensity.