generaleaccelerator

LeonidG.
hTorobievandRichardC.
YorkNationalSuperc~nduct~ingCy~lot~ronLaboratory,Michigan%ateIhiversity,EastLansing,MI48824-1321,US4{Vorobiev,York}@nscl.
msu.
eduhttp://www.
nscl.
msu.
eduAbstract.
Amethodforobtainingself-potentialsandfieldsofchargedparticlebeamscalledthetemplatepotentidalgorithmhasbeendevel-oped.
Theapproachst,emsfromtheanalyticalGreen'sfiinct,ionforrnu-&ionandisbasedonadiscreterepre~entat~ionbyauxiliarymacro-e1ernent)sort(ernp1at)esoft,hechargedensit,ydi~t~ribut~ion.
S~perposit~ionofthepot,ent,ialsandfieldsincludingimageforcesforeacht,ernplat,eisusedtoreproducethetot,alpotent,ialandfieldoftheoriginalchargedistri-butionwit,hinacondu~t~ingboundary.
ThetechniqueisespeciallyusefulwhenthePoissonsolverisbeingusedrepeatedlyaswiththesimulationofchargedparticlebeamdynamicsinaccelerators.
Numericalresultsarepresentedandlimitationsofthemethodarediscussed.
1IntroductionThephrasefastPoissonSolverismostoftenusedinrelationtogridmethodsforsolvingthePoissonequationwithinaregion'R,d,u(x)=-47rp(x)forx=(x,y,z)E'Rwithspecifiedconditionsontheboundary3RusingproceduresbasedontheFastFourierTransform(FFT)orcyclicreductionforsimpleregionsandmulti-gridalgorithmsformorecomplicatedboundaryshapes.
Thesetechniquesderivethegridpotentialu(x)(x=(x,y.
z))andfieldE,,,,,fromagriddensityp(x)assumedknownaprioribysolvingasetoffinite-differenceequations.
Theresultsusuallyhaveanaccuracyof1:3(h)=O(h"11:+h#+11:)(where/I,.
,.
,arethemeshsizes).
Thesesolversareincommonuse.
See,e.
g.
,[l]andreferencestherein.
However,forchargedparticlebeamsimulation,ateachstepoftheintegrationofthemotionequationsonlythecoordinatesofATpmacro-particlesareknown.
Thespatialgridchargedensityp(x)isderivedfromthepositionofthebeampar-ticles.
Hence,reproductionofthegriddensityinfluencesboththespeedandtheaccuracyy9c=~)(12.
ATp)ofthecalculatedpotential[2,3].
Increasingthenumberofmacro-particles,a%-p,reducesthecomputationalnoiseofthegriddensityatP.
M.
A.
Slootetal.
(Eds.
):ICCS2002,LNCS2331,pp.
315324,2002.
Springer-VerlagBerlinHeidelberg2002FastPoissonSolverforSpaceChargeDominatedBeamSimulationBasedontheTemplatePotentialTechniquetheexpenseofcomputationalspeed.
Theuseoffewerparticleswithnumericalsmoothingtoreducethenoise.
risksthemaskingofrealphysicalphenomena.
ThispaperdescribesfastPoissonsolversforderivingself-potentialsandfieldsfrommacro-particlescoordinatesforparticle-in-cell(PIC)codesimulation.
Wedonotpurposetoimprovestandardproceduressuchasdensityblockorgridpotentialsolvers.
Instead,weintroduceanewformulationbasedonthetem-platepotentialconcept[4,5]forreconstructionofthetotalpotentialofchargedparticlebeaminthepresenceofboundaries.
Thisnewapproachallowsasignifi-cantreductioninthenumberofmacroparticles.
I\$,andasparsergridwithoutconcomitantlossofaccuracy.
ThetechniquemaybeusedforeitherenvelopeorPICmodelsineithertwo-dimensional(2D)orthree-dimensional(3D)georne-tries.
Thetemplatetechniquehasbeenverifiedandshowntobeappropriateformanypracticalspacechargerelatedapplications.
2MomentMethodandTemplates4potentialu(x)generatedbyachargedensitydistributionp(x)insideavolumeRwithspecifiedboundaryconditionsonaRcanberepresentedviatheGreen'sfunction[GI.
Forfreespace,thepotentialu(x)canbederivedfromasimpleGreen'sfunctionoftheformGfTe,(x,xl)=l/lx-x'l.
Difficultiesariseinthepresenceofaconductingboundary.
4sapracticalmanner,theGreen'sfunctionapproachislimitedtosimplebeamdistributionsandsurfaceaRgeometries.
SeverthelesstheideaoftheGreen'sfunctioncanbeemployedfornumericalPoissonsolvers,appropriateforrathergeneralbeamdistributionsandboundaryshapes.
Inthemomentmethod[7].
theGreen'sfunctionformulationisusedasapartofcomputationaltechnique.
wherethetotalpotentialisrepresentedasutotal(x)=u,.
,,(x)+uanlnge(x),xERsatisfyingontheboundaryutotal18~=0:Thepotentialuf,,,isproducedbythechargedensitypinfreespacewithGfT,,(x,x').
Thecorrespondingimage-potential,uZ,,,,,(x).
definedbyo,,,,,,,-isfoundfromasetofequationsIIAlloZ,ufTeethatsatisfytheconstraintux)I~~~~=0ontheconductingsurface.
Thus,thediffucultyinthecon-structionoftheproperGreen'sfunctionforcomplexgeometriesisreplacedbyfindingtheimagedensity.
Sowthemomentmethodwithsomemodificationsisknownasthechargedensitymethodwithnumerousapplicationsin,e.
g.
,ionoptics[8].
Bothmethodsareslowerthancontemporarygridmethods,butcanbeusedinsituationsinwhichrapidgridalgorithmsmaybedifficulttoemploy.
Inapreviouspaper[4],weintroducedanumericalmethod,calledtheslicealgorithm,basedontheuseofthetemplatepotentialconceptforspacechargecalculationsofa3Dbunchedbeam.
Thebeambunchisrepresentedbycharged,infinitesimallythindisks,orslices,andthetotalbeampotentialisfoundbythesuperpositionofthepotentialsfromallslices.
Thespacechargepotentialofan316L.
G.
VorobievandR.
C.
YorkindividualsliceofradiusRslicewithinaconductingboundaryisfoundbythemomentmethod(1).
Then,thematrix1lAlliscalculated,aimageisderivedfromthesetofequations,andthetotalpotentialut,talobtainedfrom:whereS(z)isashapefunctionwhichdefinesthelongitudinalz-profileofthebeamforthecaseofroundslices.
Forellipticalslices,therearetwoshapefunc-tionsS,,,,whichdeterminethez-profiles.
Fig.
1illustratesthemomentmethod.
Fig.
1.
Templatepotentialsproducedbythreechargedslicesofdifferentradiiwithinaconductingbeampipe4cmindiameterrespectively.
Left:ufVee(x)(positive)anduimage(x)(negative)withx=(0,0,z),plottedwithdotted,dashedandsolidlinesforslicesof1,2and3cmindiameter.
Right:ThecorrespondingtotalpotentialsUtotat=ufVee+uimageforeachslicerespectively.
Fig.
2.
Twopossible3Dbeamswithinaconductingbeampipe4cmindiameterrepresentedbyN,=50chargedslices.
Left:Ellipsoid-likebeambunchwithsemiaxes&xRoxz,=1cmxlcmxlOcmandtheshapefunctionS(z)=~odi--(zlz,)2.
Right:Beambunchwithmoregenerallongitudinalvariation.
Forthisspecificcaseallsliceswereassumedtohaveconstanttransversedensityaslice(r)=c$lice.
Thepotentialforeachtemplatesatisfiesthezeropotentialboundaryconditionsandtherefore,thesuperpositionofallN,slicesrepresentingthetotalbeambunchsatisfiesthePoissonequation.
InFig.
2areshowntwopossiblebunchedbeamgeometries.
Inbothcases,thebeamwas317FastPoissonSolverforSpaceChargeDominatedBeamSimulationassumedtohaveatotalchargeof10-'ICandbecontainedinaconductingchamber4cmindiameter.
Thepotentialsu:,"zhshowninFig.
3wereobtainedbysuperpositionofthepotentialsofN,=50slices,representingthebunch.
TheseN,slicepotentialswerefoundbyappropriatescalingandinterpolationofthetabulatedtemplatepotentials.
Fig.
3.
Potentialsu(z,y,z)ofbunchedbeamsfromFig.
2(leftandrightcorespond-ingly)asafunctionoflongitudinalposition,z,fordifferentradii05r=5Rcyllwith2R,,~=4cm.
(See[9],p.
407forcomparison).
3TemplatesforArbitraryBeams.
Thetemplateapproachcanbeusedforvariabletransversechargedistributionsandellipticalbeamshapescommontoquadrupolefocusingchannels.
However,thebeamisassumedcenteredwithintheconductingchamber.
Ageneraltwo-dimensionalmodelofthechargedensity,oSlice,fortemplatesisbasedontheconceptofequivalentbeams[9-111:wherex,(p),y,(p)aremaximalslicecoordinatesdependentontheparameterp.
Thermssizemaybefoundfrom[ll]:/z,yx20sLice(x,y,p)dxdy(p)=S,,,4x,Y,P)~X~YInFig.
4,rms-matched2Dchargedensitydistributionsareplotted.
Forp>0thedensitiesaremaximuminthecentergoingtozeroneartheedge.
Forp=0,thechargedensityisconstant,andforp0.
-YIE6t2mkTransverseDimensionXFig.
4.
Left:Rms-matchedchargedensitieso(r,p)asfunctionsofr=fordifferentp.
Right:Slicewithinellipticalconductingboundaries.
ForellipticalsymmetrytherewillbehorizontalandverticalshapefunctionsS,,,(z).
Seesection4.
thetemplatepotentialsalongthreerays(4=0,~/4,~/2)tobesufficienttointerpolatepotentialsforintermediateq5values.
Further10-15differentaspectratios(K)werefoundtoprovidegoodaccuracyforpossibletransversebeamconfigurations.
4BeamSimulation,UsingTemplatesForthesimulationofchargedparticlebeamswithsignificantspacechargeef-fects,eitherrmsenvelopeequationsorstep-by-stepPICcodesmaybeused.
Theenvelopeformalismthoughcomputationallyfast,isappropriateonlyforbeamswithellipticalsymmetrypropagatingthroughalinearfocusingchannelintheabsenceofimageforces.
ThePICmethodsaresignificantlyslower,butacco-modatearbitrarybeamparticledistributions,conductingchambergeometries,andfocusingstructures.
ThetemplatetechniquemaybeappliedwithdifferentdegreesofgeneralitytobridgethegapbetweenrmsenvelopesandgeneralPICformulations:4.
1Extensionof2Dand3DRMSEnvelopeEquations.
In[ll]itwasshownhowusingthetemplatetechniquethe2Drmsenvelopeformalism[9,12]canbeextendedtoincludetheeffectsofaconductingellip-ticalchamber.
Inthiscontext,chargedcylinders,insteadofdisks,areusedastemplates.
ThedifferencebetweenthemorecompletetemplateapproachandthatpresentedbythefreespaceKVformalismismostpronouncedforellipticalbondarieswithlargeaspectratios.
Thermsenvelopeequationsfora,,,,,,may319FastPoissonSolverforSpaceChargeDominatedBeamSimulationbealsogeneralizedfor3Dellipsoid-likebeambunch:whereE~,,,,arermsemittances,KX,,,,isthelinearfocusingandF,S,;,,thespacechargeforce(see[12],p.
278).
However,withtheinclusionaconductingboundary,equation(5),assuminglinearspacechargeforcesinfreespace,isnotvalid.
Inthepresenceofconduct-ingwalls,thebehaviorofthespacechargeforcesbecomesstronglynon-linearevenforanidealellipsoid[9,13]ornon-ellipsoidalbeam,asshowninFig.
5.
However,thetemplatepotentialmethodmaybeusedtocorrectlyobtaintheFig.
5.
LongitudinalspacechargefieldsE,(z,y,z)=-du/dzasafunctionofz,atdifferentradii05r=5RcVl.
Left:E,fortheellipsoid-likebeam.
Right:E,forthearbitrarybeamfromFig.
2.
requiredfields,evenforrathergeneralbeamdistributionsandboundaryshapes.
BothlongitudinalFiotalandtransversefieldsF,t;,talmaybeobtainedfromthepotentials,byaveragingandlinearizationofPi;;$.
SubstitutionofthisresultinlieuofP,S,$,,in(5),providesamoreself-consistentmodel.
4.
23DNon-EllipsoidalBeam.
Arbitrarybeams,likethatofFig.
2(right),maynotbeappropriatelyacco-modatedbyanenvelopemodel.
Weneedtoincludemacro-particles{xi),i=1,.
.
.
,N,inthemodel.
Inthiscase,thetemplateapproachmayagainbeappliedtogeneralbeamdistributionsandconductingboundaries.
Thetransversermsbeamdimensions'I2(z),asfunctionsofthelongi-tudinalcoordinate,z,arecalculated.
TheshapefunctionS,,,(z)isthenfoundfrom(4)foraspecificp.
Previousanalysis[ll]hasshownthatfor0,E.
t,,E;1'3DmEi,j:zFig.
7.
Spacechargecalculationsbydifferentmethodsforonestepofintegrationofthe3Dbeammot,ionequat,ions.
Left,:Adet,hodfromsection4.
2.
Middle:Sub-3DPoissonSolverfromsection4.
3.
R.
ight:Regular3DSolver.
Tote,thatthegenerationofthetransverseforcesisseparatedfromthelongi-tudinalforces.
ThederivativedE,/daparticipatesineachofthe2DSolversbycorrectingthe2Dchargedensity.
Withoutinclusionofthedrivingtermd%/Dz2thecalculationofEx,,wouldbeinerror[5].
Fig.
7illustratesastepofintegrationusingthesub-3Dandageneral3DPoissonSolvers.
5TemplatesasSpecialFunctionsThetabulationofthetemplatepotentialspriortobeamsimulationfollowstheprincipleusedforotherspecialfunctionsthatdonothaveasimpleanalyti-calrepresentation.
Duringtheactualspacechargesimulations,thetemplatedataisextractedfromthetablewithappropriatescalingandinterpolation.
Thestoragerequirementsforthetemplatetablecanbeminimized.
Sinceeachtem-platepotentialrepresentsanevenfunction,itisnecessarytostoreonlyhalfofthem,sayu(x,y,a)forz>0.
SeeFig.
1.
Inaddition,thetemplatedatacouldbeparametrizedandonlytheresultingcoefficientsstored.
Thenumberoftemplatesrequireddependsonthegeometryoftheproblem.
FortheaxiallysymmetricbeamofFig.
2.
only10differentradiiwererequired.
Foreachsuch322L.
G.
VorobievandR.
C.
Yorktemplate,10off-axispotentialsshouldbecalculated,resultinginatotalof100templates.
Forcaseswithlesssymmetry.
e.
g.
,forbeamandboundarywithellip-ticalsymmetryofFig.
4aboutof3000templatesarerequiredwiththeincreasefromnecessarydifferentazimuthalpositions(3x)andaspectratios(10x).
Thus,thetableoftemplatesisofamoderatesize.
Allintermediatequantitiesareob-tainedbyinterpolationandscalingofthetabulateddata.
For2Dcases.
whenthereisnodependenceonz,thememorydemandsareanorderofmagnitudeless.
Theaccuracyofthesub-3DPoissonsolverapproachesisthatofthegen-eral3DPICmodels.
Nevertheless,thetemplateformalismisnotcompletelyself-consistentbecausethepre-calculateddatamaynotadequatedlyreflectallpossibleevolutionsoftheparticledistribution.
Sincethetransverse2Danaly-siscanbeusedforarbitraqdensitiesP~D(X,y-)-the2Dproblemsaresolvedwithallpossiblegenerality.
Thelackofself-consistencyisinthereplacementof32u/3a2by3E,/3ain(9).
Nonetheless.
thequantitativeanalysisin[lo,111showedthatforalargeclassofdensitydistributionso(x.
y,p)thereisarela-tivelyweaksensitivityofthelongitudinalE,fieldstothedetailsoftransversechargedensities.
Astheresult.
thespacechargefields:E,,,foundfromaseriesof2Dproblems(8)andE2,suppliedbytheslicealgorithm,provideanaccuraterepresentationofspacechargeforces.
Forbeams.
whosetransversedensitiesmaybedescribedanalytically(3).
thetemplateformalismisappropriate.
However,forcaseswherethebeambunchhas,e.
g.
,isolatedoff-axisclusters,thetemplateformalismisnotappropriateandageneral3DPICmethodshouldbeemployed.
6DiscussionandConclusionsThetemplatetechiqueisorientedtowardrepeatedcalculations.
e.
g.
,forchargedparticlebeamdynamicssimulation.
Forthosesituations,wherethebeampipesizesarefixedoronlyslightlyvaxyingandthebeamis'well-behaved"inthesensediscussedabove,thetemplatesproceduresignificantlyreducessimulationcomputationaltime.
Theverificationofthemethodhasshownagoodagreementwithgeneral3Dgridsolversforalargeclassofchargedensitydistributions.
Theproperinclusionofchangingboundarieswouldrequireadditionalpre-calculatedtemplatedata.
Thismightbejustifiedifonlyafewpossiblegeometriesarerequired.
However.
whentheboundariesarecomplicatedand/orchangingsig-nificantly.
andifthebeamdistributionisarbitrary(i.
e.
off-setfromtheaxis,disintegratedintoclusters,etc.
)thetemplatetechniqueislikelyinappropriateandconventionalgridmethodsarerecommended.
Preliminarynumericalstudiesprovideconfidencethatthetemplateformal-ismisanefficientmethodforfastspacechargecalculationsinthepresenceofconductingboundaries.
Itallowstheextentionof2Dand3Dbeamrms-envelopeequationsincludingconductingboundaries.
Itprovidesatransitiontoself-consistentrathergeneralsub-3DPIC,thatissignificantlyfaterthancon-ventional3DPICformulations.
323FastPoissonSolverforSpaceChargeDominatedBeamSimulationAcknowledgmentThisworkwassupportedbytheU.
S.
DepartmentofEnergyunderContractNo.
DE-FG02-99ER41118.
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