A Finite Difference Delay Modeling Approach to the

AFiniteDifferenceDelayModelingApproachtotheDiscretizationoftheTimeDomainIntegralEquationsofElectromagneticsXiaoboWang1,RaymondA.Wildman1,DanielS.Weile1,andPeterMonk21UniversityofDelawareDepartmentofElectricalandComputerEngineering140EvansHallNewark,[e-mail],[e-mail],[e-mail]1UniversityofDelawareDepartmentofMathematicalSciences513JohnEwingHallNewark,[e-mail]—Anewmethodforsolvingthetime-domainintegralequationsofelectromagneticscatteringfromconductorsisintroduced.Thismethod,calledfinitedifferencedelaymodeling,appearstobecompletelystableandaccuratewhenappliedtoarbitrarystructures.Thetemporaldiscretizationusedisbasedonfinitedifferences.Specifically,basedonamappingfromtheLaplacedomaintothez-transformdomain,first-andsecond-orderunconditionallystablemethodsarederived.Spatialconvergenceisachievedusingthehigher-orderdivergence-conformingvectorbasesofGragliaetal.Lowfrequencyinstabilityproblemsareavoidedwiththeloop-treedecompositionapproach.Numericalresultswillillustratetheaccuracyandstabilityofthetechnique.IndexTerms—Finitedifferences,integralequations,transientelectromagneticscattering,methodofmoments.2I.INTRODUCTIONThepastdecadehaswitnessedthegrowthoftimedomainintegralequation(TDIE)basedmethodsforthesolutionofelectromagneticscatteringproblems[1-3]fromcuriousitiestonearlypracticalmethods.Forsomevarietiesofproblems,TDIEmethodsofferimportantadvantages:Asintegralequationmethods,theyneedonlydiscretizesurfacesforhomogeneousscatterers,andastimedomainmethodstheyworkfornonlinearproblemsandcananalyzewholebandsoffrequencyinasinglesimulation.DespitethepotentialadvantagesofTDIEmethods,stabilityandaccuracystillseemelusiveasapair.Theoldestapproachestosolvingtheseproblemsusespatialandtemporalaveraging[4-10].Unfortunately,thesemethodsgenerallycompromiseaccuracyandaredifficultifnotimpossibletoapplytocurvedscatterermodels.Mostrecentworkhasfocusedonaccuratespatialtestinginthefaceoftheabruptshadowregioncreatedbythesuddenonsetofradiation[11-20].Thisisusuallyaccomplishedbyparticularchoicesoftemporalbasisfunctions,butotherapproachesarebothpossibleandimportant.Twoofthesemorerecentapproachesrequirefurtherdiscussion.Thefirstoftheseusessimplebasisfunctions,sothatitcanperformanexactspatialintegrationovertheilluminatedregionsofthetestingregion[18].Thismethodappearsstableinallcases,butishardtoputintopracticeevenforflatpatchesandshouldbemoredifficultifnotimpossibleforcurvedpatches.Thesecondapproachusesbandlimitedinterpolationfunctions(BLIFs)forthetemporaldiscretization[19-25].Inthiscase,accurateintegrationsareachievedduetothegradualtaperofthesefunctions.BecausetheBLIFsareinterpolatoryandsymmetricabouttheirinterpolationpoint,extrapolationmustbeused.Thisapproachhasno3problemswitharbitrarygeometryandisextremelyaccurate.However,becauseofthenecessityofextrapolation,itonlyworksforsmallstepsizesandcanbemadeunstableinrarecases.Moreover,themethodisincapableofproducingroughapproximationsforlowercomputationalcost;itisstrictlyahighaccuracymethod.Inthispaper,anewmethodthatappearsabsolutelystableforanystructureandforanytimestepsizeisintroduced.Whilehigherorderbasisfunctionsof[26]arestillusedforspatialdiscretization,thenewmethodemploysatemporaldiscretizationbasedonfinitedifferences.Specifically,amappingfromtheLaplacedomaintothez-transformdomainbasedonafinitedifferenceapproximationaccomplishesthediscretizationinthetransformdomain,andtheresultisinversetransformedtocreateatimedomainmethod.Temporalconvergenceisgovernedbytheorderofthefinitediffernceapproximation,andstabilitypropertiesaregoverned(inlargepart)bythedispositionofthelefthalfplaneoftheLaplacedomainaftertransformationtothez-domain.Thebasicideaisveryold,andgoesbythename“convolutionquadrature”inthemathliterature[27],and“filterdesignbyapproximationofderivatives,”or“thebilineartransformation”inthesignalprocessingliterature[28].In[29],LubichanalyzesthestabilityofconvolutionquadratureappliedtoaTDIEfortheHelmholtzequation.Numericalresultsandfurtheranalysisiscontainedin[29,30]wherefastmethodsareconsidered.TheseresultsarenotdirectlyapplicabletoTDIEforMaxwell’sequationsandeffortstoextenttheconvergenceresultstoTDIEinelectromagnetismarecurrentlyunderway.Nevertheless,theseresultsfortheHelmholtzequationsuggestthatthemethodhaspromiseinelectromagneticapplications.Aslongasthespatialdiscretizationdoesnotcreateunstable4eigenvalues,thisnewfinitedifferencedelaymodeling(FDDM)methodhascompletelypredictablestabilityandaccuracyproperties.Unfortunately,whendealingwiththeelectricfieldintegralequation(EFIE)implementationofthismethod,thereisstillaslowlygrowing,lowfrequencyinstabilityaftermanytimesteps[31-34].Asdetailedin[21],staticsolenoidalcurrentsgeneratenoelectricfield,andtheEFIEiscompletelyblindtothem.Toovercomethisproblem,thesamestabilizationtechniquedetailedin[21]isemployed.Inthisapplication,themethodworksperfectly,andthelowfrequencyinstabilityiscompletelyeradicated.Therestofthepaperisorganizedasfollows:SectionIIdetailstheproposedFDDMschemeafterintroducingtheintegralequati