FDTD from wikipedia.docx

FDTD from wikipedia.docx

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Finite-differencetime-domainmethod

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Finite-differencetime-domain（FDTD）isoneoftheprimaryavailablecomputationalelectrodynamicsmodelingtechniques.Sinceitisatime-domainmethod,FDTDsolutionscancoverawidefrequencyrangewithasinglesimulationrun,andtreatnonlinearmaterialpropertiesinanaturalway.

TheFDTDmethodbelongsinthegeneralclassofgrid-baseddifferentialtime-domainnumericalmodelingmethods.Thetime-dependentMaxwell'sequations（inpartialdifferentialform）arediscretizedusingcentral-differenceapproximationstothespaceandtimepartialderivatives.Theresultingfinite-differenceequationsaresolvedineithersoftwareorhardwareinaleapfrogmanner:

theelectricfieldvectorcomponentsinavolumeofspacearesolvedatagiveninstantintime;thenthemagneticfieldvectorcomponentsinthesamespatialvolumearesolvedatthenextinstantintime;andtheprocessisrepeatedoverandoveragainuntilthedesiredtransientorsteady-stateelectromagneticfieldbehaviorisfullyevolved.

ThebasicFDTDspacegridandtime-steppingalgorithmtracebacktoaseminal1966paperbyKaneYeeinIEEETransactionsonAntennasandPropagation.[1]Thedescriptor"Finite-differencetime-domain"anditscorresponding"FDTD"acronymwereoriginatedbyAllenTafloveina1980paperinIEEETransactionsonElectromagneticCompatibility.[2]

Sinceabout1990,FDTDtechniqueshaveemergedasprimarymeanstocomputationallymodelmanyscientificandengineeringproblemsdealingwithelectromagneticwaveinteractionswithmaterialstructures.CurrentFDTDmodelingapplicationsrangefromnear-DC（ultralow-frequencygeophysicsinvolvingtheentireEarth-ionospherewaveguide）throughmicrowaves（radarsignaturetechnology,antennas,wirelesscommunicationsdevices,digitalinterconnects,biomedicalimaging/treatment）tovisiblelight（photoniccrystals,nanoplasmonics,solitons,andbiophotonics）.[3]In2006,anestimated2,000FDTD-relatedpublicationsappearedinthescienceandengineeringliterature（seePopularity）.Atpresent（2008）,thereareatleast27commercial/proprietaryFDTDsoftwarevendors;8free-software/open-source-softwareFDTDprojects;and2freeware/closed-sourceFDTDprojects,somenotforcommercialuse（seeExternallinks）.

Contents

 [hide] 

∙1WorkingsoftheFDTDmethod

∙2UsingtheFDTDmethod

∙3StrengthsofFDTDmodeling

∙4WeaknessesofFDTDmodeling

∙5Gridtruncationtechniquesforopen-regionFDTDmodelingproblems

∙6HistoryofFDTDtechniquesandapplicationsforMaxwell'sequations

∙7Popularity

∙8Seealso

∙9References

∙10Furtherreading

∙11Externallinks

[edit]WorkingsoftheFDTDmethod

WhenMaxwell'sdifferentialequationsareexamined,itcanbeseenthatthechangeintheE-fieldintime（thetimederivative）isdependentonthechangeintheH-fieldacrossspace（thecurl）.ThisresultsinthebasicFDTDtime-steppingrelationthat,atanypointinspace,theupdatedvalueoftheE-fieldintimeisdependentonthestoredvalueoftheE-fieldandthenumericalcurlofthelocaldistributionoftheH-fieldinspace.[1]

TheH-fieldistime-steppedinasimilarmanner.Atanypointinspace,theupdatedvalueoftheH-fieldintimeisdependentonthestoredvalueoftheH-fieldandthenumericalcurlofthelocaldistributionoftheE-fieldinspace.IteratingtheE-fieldandH-fieldupdatesresultsinamarching-in-timeprocesswhereinsampled-dataanalogsofthecontinuouselectromagneticwavesunderconsiderationpropagateinanumericalgridstoredinthecomputermemory.

IllustrationofastandardCartesianYeecellusedforFDTD,aboutwhichelectricandmagneticfieldvectorcomponentsaredistributed.[1]Visualizedasacubicvoxel,theelectricfieldcomponentsformtheedgesofthecube,andthemagneticfieldcomponentsformthenormalstothefacesofthecube.Athree-dimensionalspacelatticeconsistsofamultiplicityofsuchYeecells.Anelectromagneticwaveinteractionstructureismappedintothespacelatticebyassigningappropriatevaluesofpermittivitytoeachelectricfieldcomponent,andpermeabilitytoeachmagneticfieldcomponent.

Thisdescriptionholdstruefor1-D,2-D,and3-DFDTDtechniques.Whenmultipledimensionsareconsidered,calculatingthenumericalcurlcanbecomecomplicated.KaneYee'sseminal1966paperproposedspatiallystaggeringthevectorcomponentsoftheE-fieldandH-fieldaboutrectangularunitcellsofaCartesiancomputationalgridsothateachE-fieldvectorcomponentislocatedmidwaybetweenapairofH-fieldvectorcomponents,andconversely.[1]Thisscheme,nowknownasaYeelattice,hasproventobeveryrobust,andremainsatthecoreofmanycurrentFDTDsoftwareconstructs.

Furthermore,YeeproposedaleapfrogschemeformarchingintimewhereintheE-fieldandH-fieldupdatesarestaggeredsothatE-fieldupdatesareconductedmidwayduringeachtime-stepbetweensuccessiveH-fieldupdates,andconversely.[1]Ontheplusside,thisexplicittime-steppingschemeavoidstheneedtosolvesimultaneousequations,andfurthermoreyieldsdissipation-freenumericalwavepropagation.Ontheminusside,thisschememandatesanupperboundonthetime-steptoensurenumericalstability.[4]Asaresult,certainclassesofsimulationscanrequiremanythousandsoftime-stepsforcompletion.

[edit]UsingtheFDTDmethod

ToimplementanFDTDsolutionofMaxwell'sequations,acomputationaldomainmustfirstbeestablished.Thecomputationaldomainissimplythephysicalregionoverwhichthesimulationwillbeperformed.TheEandHfieldsaredeterminedateverypointinspacewithinthatcomputationaldomain.Thematerialofeachcellwithinthecomputationaldomainmustbespecified.Typically,thematerialiseitherfree-space（air）,metal,ordielectric.Anymaterialcanbeusedaslongasthepermeability,permittivity,andconductivityarespecified.

Oncethecomputationaldomainandthegridmaterialsareestablished,asourceisspecified.Thesourcecanbeanimpingingplanewave,acurrentonawire,oranappliedelectricfield,dependingontheapplication.

SincetheEandHfieldsaredetermineddirectly,theoutputofthesimulationisusuallytheEorHfieldatapointoraseriesofpointswithinthecomputationaldomain.ThesimulationevolvestheEandHfieldsforwardintime.

ProcessingmaybedoneontheEandHfieldsreturnedbythesimulation.Dataprocessingmayalsooccurwhilethesimulationisongoing.

WhiletheFDTDtechniquecomputeselectromagneticfieldswithinacompactspatialregion,scatteredand/orradiatedfarfieldscanbeobtainedvianear-to-far-fieldtransformations.[5]

[edit]StrengthsofFDTDmodeling

Everymodelingtechniquehasstrengthsandweaknesses,andtheFDTDmethodisnodifferent.

FDTDisaversatilemodelingtechniqueusedtosolveMaxwell'sequations.Itisintuitive,souserscaneasilyunderstandhowtouseitandknowwhattoexpectfromagivenmodel.

FDTDisatime-domaintechnique,andwhenabroadbandpulse（suchasaGaussianpulse）isusedasthesource,thentheresponseofthesystemoverawiderangeoffrequenciescanbeobtainedwithasinglesimulation.Thisisusefulinapplicationswhereresonantfrequenciesarenotexactlyknown,oranytimethatabroadbandresultisdesired.

SinceFDTDcalculatestheEandHfieldseverywhereinthecomputationaldomainastheyevolveintime,itlendsitselftoprovidinganimateddisplaysoftheelectromagneticfieldmovementthroughthemodel.Thistypeofdisplayisusefulinunderstandingwhatisgoingoninthemodel,andtohelpensurethatthemodelisworkingcorrectly.

TheFDTDtechniqueallowstheusertospecifythematerialatallpointswithinthecomputationaldomain.Awidevarietyoflinearandnonlineardielectricandmagneticmaterialscanbenaturallyandeasilymodeled.

FDTDallowstheeffectsofaperturestobedetermineddirectly.Shieldingeffectscanbefound,andthefieldsbothinsideandoutsideastructurecanbefounddirectlyorindirectly.

FDTDusestheEandHfieldsdirectly.SincemostEMI/EMCmodelingapplicationsareinterestedintheEandHfields,itisconvenientthatnoconversionsmustbemadeafterthesimulationhasruntogetthesevalues.

[edit]WeaknessesofFDTDmodeling

SinceFDTDrequiresthattheentirecomputationaldomainbegridded,andthegridspatialdiscretizationmustbesufficientlyfinetoresolveboththesmallestelectromagneticwavelengthandthesmallestgeometricalfeatureinthemodel,verylargecomputationaldomainscanbedeveloped,whichresultsinverylongsolutiontimes.Modelswithlong,thinfeatures,（likewires）aredifficulttomodelinFDTDbecauseoftheexcessivelylargecomputationaldomainrequired.

Thereisnowaytodetermineuniquevaluesforpermittivityandpermeabilityatamaterialinterface.

SpaceandtimestepsmustsatisfytheCFLcondition.

FDTDfindstheE/Hfieldsdirectlyeverywhereinthecomputationaldomain.Ifthefieldvaluesatsomedistancearedesired,itislikelythatthisdistancewillforcethecomputationaldomaintobeexcessivelylarge.Far-fieldextensionsareavailableforFDTD,butrequiresomeamountofpostprocessing.[3]

SinceFDTDsimulationscalculatetheEandHfieldsatallpointswithinthecomputationaldomain,thecomputationaldomainmustbefinitetopermititsresidenceinthecomputermemory.Inmanycasesthisisachievedbyinsertingartificialboundariesintothesimulationspace.Caremustbetakentominimizeerrorsintroducedbysuchboundaries.Thereareanumberofavailablehighlyeffectiveabsorbingboundaryconditions（ABCs）tosimulateaninfiniteunboundedcomputat