In this report we will present detailed 64257niteelement formulations and resulting eigenvalue problems for lossless cavities cavities with lossy materials cavities with imperfectly conducting surfaces and cavities with waveguide coupling We will di ID: 24013
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Work supported in part by US Department of Energy contract DE-AC02-76SF00515 Omega3P:AParallelFinite-ElementEigenmodeAnalysisCodeforAcceleratorCavitiesLie-QuanLee,ZenghaiLi,ChoNg,andKwokKoSLACNationalAcceleratorLaboratory,2575SandHillRoad,MenloPark,CA94025—Omega3Pisaparalleleigenmodecalculationcodeforacceleratorcavitiesinfrequencydomainanaly-sisusingnite-elementmethods.Inthisreport,wewillpresentdetailednite-elementformulationsandresultingeigenvalueproblemsforlosslesscavities,cavitieswithlossy @tr�!�! @t�!r�! r�!�!=0r r�!�!=0istheangularwavenumber speedoflight.Eitherequationcanbeusedinthenumericalsimulation.Eq(8)referstoE-formulation SLAC-PUB-13529 February 2009 andEq(9)H-formulation.Withoutlossofgen-erality,wewilluseE-formulationfortherestofIn1980,Nedelecdiscussedtheconstructionofedgeelementsontetrahedraandrectangularbricks[1].Edgeelementsprovidetangentially-continuousbasisfunctionsfordiscretizingelectriceld.Theuseofedgeelementsnotonlyleadstoconvenientwayofimposingboundaryconditionsatmaterialinterfacesaswellasatconductingsurfaces,butalsotreatsconductinganddielectricedgesandcornerscorrectly.InOmega3P,asetofhierarchicalhigh-orderNedelecbasisfunctions[2]areusedtodiscretizeelectriceld.�!�!B.LosslessCavitiesAtaperfectlyconductingsurface,theboundaryconditionforelectriceldcanbeexpressedas:�!�!=0�!isthesurfacenormal.Ifthereisasymmetryinthecavitytobesimulated,onlyapartofgeometryneedstobemodeledwhilethefollowingboundaryconditioncanbeimposedonthesymmetricplane.�! r�!)=0�!isthesurfacenormalonthesymmetricWithniteelementdiscretizationinEq(10),thevectorwaveequation(8)alongwiththeboundaryconditions(11)and(12)becomesageneralizedeigenvalueproblemforalosslesscavity:wherethematrices r�!r�!;and�!�!Herewedenote�!�!tobeaninnerproduct,whichistheintegraloverthedomain�!�!andcanbenumericallyevaluatedwithGaussianintegralrules.Notethatmatrixissymmetricpositivedeniteandmatrixissymmetricpositivesemi-denitewithalargenullspace.Oncetheeigenvalueproblemissolved,theelectriceld�!recoveredwithEq(10)whilethemagneticeld�!iscomputedwith�!jkcr�!isthesquarerootofthespeedofC.CavitieswithLossyMaterialsTheniteelementanalysisintheprevioussectioncanstillbeappliedtocavitieswithlossymaterialswiththeextensionofusinggeneralizedvariationalprinciple[3].Insuchcases,relativeelectricpermit-tivityand/ormagneticpermeabilitybecomecom-plexinpartorallofthedomain.ThatmakesmassmatrixinEq(14)and/orstiffnessmatrixinEq(15)complex.Notethattheeigenvaluealsobecomecomplex.Anphysicalquantity,qualityfactor,isdenedastherealpartoftheeigenvaluedividedbytwotimestheimaginarypartoftheeigenvalue.Namely,real Thequalityfactorcanbeviewedasameasureoftheloss.ThehightheQvalue,thelesstheloss.D.CavitieswithImperfectlyConductingSurfacesWhenacavitywallisanimperfectconductor,itcanbeshownthattheelectricandmagneticlesatthesurfaceoftheconductorcanbeexpressedasthefollowingimpedanceboundarycondition:�! r�! �!�!�!)=0istheelectricalconductivityofthecavitywall.Withtheimpedanceboundarycondition(18),thevectorwaveequation(8)canbedescretizedasaquadraticeigenvalueproblem: 3 Fig.1.Acavityconnectedwithawaveguide.Whenthewaveguideislongenough,theelectriceldinsidethewaveguideisacombinationofthewaveguidemodesthatcanpropagate.E.CavitieswithWaveguideLoadingAnacceleratingcavityisoftenconnectedwithwaveguidestoinputpowerortodampthehighordermodes.Fig1showsacavitythatisconnectedwithawaveguide.Whenthewaveguideislongenough,theelectriceldinsidethewaveguidecanbeexpandedtoasetofwaveguidemodesthatcanpropagateinsidethewaveguidewithoutattenuation.Therefore,aspointedinRef[4],theboundaryconditiononthewaveguideportcanbeexpressedasfollows: TEMTEMTETE TMTM=0 isthecutoffwavenumberofthethwaveguidemode.TEMTE,andTMarethenormalizedwaveguideTEMthTEmode,andthTMmode,respectively.Theniteelementdiscretizationofthevectorwaveequation(8)alongwithelectricboundarycondition(11),magneticboundarycondition(12)andwaveguideboundarycondtion(20)leadstoacomplexnonlineareigenvalueproblem(NEP): k2�(kc)2WTEMx+iXmp k2�(kcm)2WTEmx+iXmk2 p wherematrices,andTEMTEMdSTEMdSdSdSdSdSNotethatallthethreetypesofmatricesaresymmetricbuthavedenseblocks.Ifthefrequencyofinterestisabovetherstwaveguidecutoffbutbelowthesecondwaveguidecutoff,i.e.,onlyonewaveguidemodecanpropagateinthewaveguide,asimplerboundaryconditioncanbeused: )=0 .Weoftenrefer(25)absorbingboundarycondition(ABC).NotethatABCisaccurateifthereisonlyonewaveguidemodepropagatinginthewaveguide.Otherwise,itisjustanapproximation.WithABC,thediscretizedeigensystembecomes: wherematrixisasparsematrixandisdenedIII.ALGORITHMSFORIGENVALUEROBLEMSTosolveeigenvalueproblem(13),weuseeithertheImplicitRestartedArnoldimethodthroughARPACK[5]oranexplicitrestartedArnodimethod[6]implementedinOmega3P.Ashiftandinvertspectraltransformation[7]isappliedtoEq(13)intheprocessofsolvingtheeigenvalueproblemssincetheinterioreigenvaluesareofinterestintheacceleratorcavitymodeling. =(isaprescribedshiftclosetotheeigenvaluesofinterest.Theabovespectraltransformationrequiresasolutionofahighlyindenitelinearsystemineveryeigenvalueiteration,whichisnotoriouslydifculttosolvewithiterativemethods. TosolvetheshiftedlinearsystemEq(29),weoftenusedsparsedirectsolvers[8],[9],[10]orKrylovsubspacemethodswithmulti-levelprecondition-ers[11],[12],[13].Sparsedirectsolversrequirealargeamountofmemorytostorethefactorofthethustheirusageislimited.Thenonlineareigenvalueproblem(26)canbetransformedintoaquadraticeigenvalueproblemby .WeimplementedtheSecondOrderArnoldimethod[14]inOmega3Pforsolvingthequadraticeigenvalueproblem(19)andthenonlineareigenvalueproblem(26).Forsolvingthenonlineareigenvalueprob-lem(21),weimplementedaself-consistentitera-tion[15],anonlinearJacobi-Davidsonmethod[16],[17],andanonlinearRayleigh-Ritziterativeprojec-tionalgorithm,NRRIT[18].Intheself-consistentiterationmethod,werstcalculateaninitialguessofaneigen-pairbyignoringallthewaveguidetermsin(21).Thatinitialguessshallbeagoodapproximationifthelossduetowaveguideisnotstrong(qualityfactorislargerthan10).Weusethattoevaluatethewaveguidetermsandaddthemintothestiffnessmatrixandrecalculatetheeigen-pair.Thisloopterminatesuntiltheeigen-pairisself-consistent.Itcanbeshownthatthemethodwillyieldconvergedeigen-pairsaslongasthelossisnotstrong.IV.PARALLELIZATIONTRATEGYANDOFTWAREESIGNOmega3PiswritteninC++andusesMPIforinterprocesscommunication.IttakesatetrahedralmeshinNetCDFformatasinputforthegeometryofthecavity.DomaindecompositionisusedforparallelizationinOmega3Pandthemeshisparti-tionedintosubdomainsusingParMetis[19]orZoltan[20]whereisthenumberoftheMPIprocessors.Ameshregionaroundprocessboundaryisreplicatedineachprocessortoreducecommuni-cation.Thehierarchicalbasisfunctiondescribedin[2]isemployedforthediscretizationoftheelectriceldsinthedomain.Theedge,faceandvolumedegreesoffreedomarelocatedinparallelandmatricesareassembled.Afterthat,anappropriateeigensolverisinvokedtondspeciceigen-pairs,whicharesavedintolesforfurtherpost-processingorforvisualizationpurpose.Weusemany3rdpartylibrariesinOmega3PandtakeamodulardesignasshowninFig2.For Fig.3.ThestrongscalabilityofOmega3PonaCrayXTcomputer(Jaguar)atOakRidgeNationalLaboratory.example,forlinearalgebraoperations,weusethegenericlibrarydescribedin[21].Thatmakesusveryeasytoaddnewsolvercomponentssuchasprecondtionersforiterativelinearsolvers.Fig3showsthestrongscalabilityofOmega3PrunningonaCrayXTcomputerwithcatamountkernelatOakRidgeNationalLaboratory.Thetest-ingproblemwasforsolvingthemodeoftheRFgundesignedfortheLinacCoherentLightSource.Thecomputermodelhas1.5milliontetrahedralelements.Itresultedinarealeigenvalueproblemwith9.6milliondegreesoffreedomand506millionnon-zeroentriesinthematrixoftheeigen-system.Asacomparison,theperfectlinearscalabilityisalsoplottedastheblackline.ItisevidentthatOmega3Pscalesverywellforthisproblemupto4096pro-cessors.Inthecaseusing4096processors,eachprocessoronaveragehaslessthan400tetrahedralV.EXAMPLESA.ASphericalCavityInthisexample,wetakeaquarterofunitspherecavityshowninFig4andsetallthethreesymmetricplanestobemagneticboundary.Wegenerateaserialofquadratictetrahedralmeshesforaconver-gencestudy.WeuseOmega3Ptocomputetherst8non-zeromodesofthislosslesscavitywith2ndorderiso-parameticelements.Theeigen-frequencyresultsarelistedinTableI.B.ADampedDetunedStructure(DDS)CellInthisexample,wetestedOmega3PwithacavitywithimperfectlyconductingsurfacesshownFig5. 5 Fig.2.ThelibrarydependencyforOmega3Pversion6.Greyboxesrepresent3rdpartylibrariesandwhiteboxesareOmega3PinternalTABLEIHECONVERGENCESTUDYOFAQUARTEROFUNITSPHERE.ALLTHETHREESYMMETRICPLANESARESETTOBEMAGNETICBOUNDARY.FOURSETOFTETRAHEDRALMESHESAREGENERATED.MESHHASMESHSIZEOF0.4AND76ELEMENTS.MESHHASMESHSIZEOF0.2AND494ELEMENTS.MESHHASMESHSIZEOF0.1AND4173ELEMENTS.MESHHASMESHSIZEOF30883ELEMENT.MSSTANDSFORMESHSIZE.NESTANDSFORTHENUMBEROFELEMENTS.THEUNITOFFREQUENCIESLISTEDIS ModeNo. Mesh1(MS=0.4,NE=76) Mesh2(MS=0.2,NE=494) Mesh3(MS=0.1NE=4173) Mesh4(MS=0.05,NE=30883) 0 184710699.0458458 184668561.5453410 184662807.1252133 184662469.11162961 184746123.6983275 184670011.8803936 184662831.0783886 184662470.16621602 289565891.0071810 289275483.5347070 289240006.0647894 289236779.69979213 289925940.4188058 289289885.9817063 289240233.4329534 289236802.73800724 290260435.5049497 289296420.3611495 289240375.0801812 289236806.82836375 333965112.5260288 333626751.6019411 333428670.6436561 333419095.14529136 355348302.7894213 355359797.5528432 355149284.9449404 355136370.79737987 355970365.9768929 355395570.6044523 355149826.1846064 355136415.2978936 Theelectricalconductivityofthecoppersurfaceis5.8.Weuseameshwith19788quadratictetrahedralelementsandsecondorderbasisfunc-tions.WecomputebothwalllossqualityfactorforacavitywithperfectlyconductingsurfacesandthequalityfactorfromEq(17)foracavitywithimpedanceboundarycondition(18).TheresultsfromtheformerhasbeenvalidatedbymicrowaveQCofthefabricatedcellsshowingmeasuredfre-quencieswithin0.01%oftargetvalue[22],[23].TableIIliststheresultsofthefrequenciesandqualityfactorscomputedfrombothways.TheyareTABLEIIREQUENCIESANDQUALITYFACTORSCOMPUTEDFORADDSCELL Method Frequency Factor cavity 11.4459GHz vitywithimpedanceBC 11.4468GHz 6564 goodagreement. 6 Fig.4.Acomputermodelforaquarterofsphericalcavity. Fig.5.AcomputermodelforoneeighthofthevacuumpartofaDDScell.C.ACavityCoupledwithRectangularWaveguidesFig6showsacomputermodelofacavitycoupledwithtwoidenticalrectangularwaveguides.Therstwaveguidecutoffis5.2597GHz.Thefrequencyofthemodeisaround9.4GHz,whichisabovetherstcutoffbutbelowthesecondcutoff.Thus,wecansolvetheeigenvalueproblem(26)tocomputethefrequencyandthequalityfactorofthemode.Withameshof8378elementsforaquartergeometry,thecomputedfrequencyis9.3992GHzandthequalityfactor177.98.Alternatively,wecanuseS-paramtercalculationstodecideresonancefrequencyandthequalityfactorbyttingthetransmissioncoefcientsinordertoverifytheresultsoftheeigenvalue Fig.6.Acomputermodelforacavitycoupledwithtwoidenticalrectangularwaveguides. Fig.7.TransmissioncoefcientsversusoperatingfrequenciesinSparametercalculationsforthecavitycoupledwithrectangularwaveguides.Thettingcurveisalsoplotted.Thettingfunction 1+Q2(f f0�f0 .ThettedqualityfactorQis177.81andresonancefrequency9.40GHz.computations.Weusedameshwith19818elementsforahalfgeometryintheS-parametercalculations.Fig7showstransmissioncoefcientswithrespecttooperatingfrequenciesinthecavity.Wettedthedataandgottheresonancefrequency9.40GHzandthequalityfactor177.81,inremarkableagreementwiththosefromeigenvaluecomputations.VI.SUMMARIESWepresentedOmega3P,aparalleleigenmodecal-culationcodeforacceleratorcavitiesinfrequencydomainanalysisusingnite-elementmethods.Wedescribedthedetailednite-elementformulationsandresultingeigenvalueproblemsforlosslesscav-ities,cavitieswithlossymaterials,cavitieswithimperfectlyconductingsurfaces,andcavitieswithwaveguidecoupling.Wediscussedtheparallelal-gorithmsforsolvingthoseeigenvalueproblems anddemonstratedmodelingofacceleratorcavitiesthroughdifferentexamples.CKNOWLEDGMENTThisworkissupportedbytheU.S.Depart-mentofEnergyundercontractnumberDE-AC02-76SF00515.PartoftheworkusedresourcesoftheNationalCenterforComputationalSciencesatOakRidgeNationalLaboratoryandresourcesoftheNationalEnergyResearchScienticComputingCenter,whicharesupportedbytheOfceofSci-enceoftheDepartmentofEnergyunderContractDE-AC05-00OR22725andDE-AC02-05CH11231,respectively.EFERENCES[1]J.C.Nedelec,“Mixedniteelementsinr,”Numer.Metho.vol.35,pp.315–341,1980.[2]D.-K.Sun,J.-F.Lee,andZ.Cendes,“Constructionofnearlyorthogonalnedelecbasesforrapidconvergencewithmultilevelpreconditionedsolvers,”SIAMJ.SCI.COMPUT,vol.23,no.4,pp.1053–1076,2001.[3]J.Jin,TheFiniteElementMethodinElectromangetics,secondeditioned.JohnWiley&Sons,INC,2002.[4]Z.LouandJ.Jin,“Anaccuratewaveguideportboundaryconditionforthetime-domainnite-elementmethod,”AntennasandPropagationSocietyInternationalSymposiumvol.1B,pp.117–120,2005.[5]R.B.Lehoucq,D.C.Sorensen,andC.Yang,ARPACKUsers'Guide:SolutionofLargeScaleEigenvalueProblemswithImplicitlyRestartedArnoldiMethods.SIAM,1997.[6]R.B.Morgan,“Onrestartingthearnoldimethodforlargenon-symmetriceigenvalueproblems,”MathematicsofComputationvol.65,no.215,pp.1213–1230,1996.[7]Z.Bai,J.Demmel,J.Dongarra,A.Ruhe,andH.vanderVorst,Eds.,TemplatesforthesolutionofAlgebraicEigenvalueProblems:APracticalGuide.Philadelphia:SIAM,2000.[8]A.Gupta,G.Karypis,andV.Kumar,“Ahighlyscalableparallelalgorithmforsparsematrixfactorization,”IEEETransactionsonParallelandDistributedSystems,vol.8,no.5,pp.502–520,May1997.[9]P.Amestoy,I.Duff,J.-Y.LExcellent,andJ.Koster,“Afullyasynchronousmultifrontalsolverusingdistributeddynamicscheduling,”SIAMJournalonMatrixAnalysisandApplica-,vol.23,no.1,pp.15–41,2001.[10]X.S.LiandJ.W.Demmel,“SuperLU 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