Finite Element Formulation of Exact NonRe ecting B

FiniteElementFormulationofExatNon-ReetingBoundaryConditionsfortheTime-DependentWaveEquationLonnyL.ThompsonandRunnongHuanyAbstratAmodiedversionofanexatnonreetingboundaryondition(NRBC)rstderivedbyGroteandKellerisimplementedinaniteelementformulationforthesalarwaveequation.TheNRBCannihilatetherstNwaveharmonisonaspherialtrunationboundary,andmaybeviewedasanextensionoftheseond-orderloalboundaryonditionderivedbyBaylissandTurkel.Twoalternativeniteelementformulationsaregiven.Intherst,theboundaryoperatorisimplementeddiretlyasa‘natural’boundaryonditionintheweakformoftheinitial-boundaryvalueproblem.Intheseond,theoperatorisimplementedindiretlybyintroduingauxiliaryvariablesonthetrunationboundary.Severalversionsofimpliitandexpliittime-integrationshemesarepresentedforsolutionoftheniteelementsemidisreteequationsonurrentlywiththerst-orderdierentialequationsassoiatedwiththeNRBCandanauxiliaryvariable.NumerialstudiesareperformedtoassesstheaurayandonvergenepropertiesoftheNRBCwhenimplementedintheniteelementmethod.Theresultsdemonstratethattheniteelementformulationofthe(modied)NRBCisremarkablyrobust,andhighlyaurate.KeyWords:NonreetingBoundaryConditions,WaveEquation,FiniteEle-mentMethod,WavePropagation.1IntrodutionWhentheniteelementmethodisusedtomodelthewaveequationininnitedo-mains,aurateabsorbingboundaryonditions,inniteelements,orabsorbinglayers,arerequiredonanartiialtrunationboundarythatsurroundsthesoureofra-diationorsattering[1℄.Iftheformoftheboundarytreatmentisover-simplied,AssistantProfessorofMehanialEngineeringandEngineeringMehanis,Diretor:AdvanedComputationalMehanisLaboratory,ClemsonUniversity,Clemson,SC,29634(USA).yGraduateResearhAssistant,DepartmentofMehanialEngineering,ClemsonUniversity,Clemson,SC.29634(USA).1L.L.ThompsonandR.Huan2spuriousreetedwavesanbegeneratedattheartiialboundary,whihansub-stantiallydegradetheaurayofthenumerialsolution.Forexample,astandardapproahistoapplyloal(dierential)boundaryoperatorsonwhihannihilateleadingtermsintheradialexpansionforoutgoingwavesolutions.Awellknownsequeneofboundaryonditionsappliedtoaspherialboundaryaretheloalop-eratorsderivedbyBaylissandTurkel[2℄.Thersttwoboundaryonditionsinthissequeneare:B1=r+1t+1r=0(1)B2=r+1t+3rr+1t+1r=0(2)where(x;t)isthesolutiontothesalarwaveequation,isthewavespeed,andrisevaluatedattheradiusR,ofaspherialartiialboundary.Theboundaryondition(1)isexatforauniformspherialwave,while(2)isexatforboththerst(spherial)andseondharmoniforoutgoingwaves.Whiletheseboundaryonditionsareexatforlowermodes,theyexhibitsigniantspuriousreetionforhigher-orderwaveharmonis,espeiallyasthepositionofapproahesthesoureofradiationorsattering[3℄,andforlowfrequeny(longwavelength)omponents[4℄.Reently,GroteandKeller[5,6℄havederivedasequeneofnonreetingboundaryonditionsforthewaveequation:B1+1R1Xn=1nXm=nenznm(t)Ynm(;’)=0(3)B21R1Xn=2nXm=n~enznm(t)Ynm(;’)=0(4)whereYnmarespherialharmonis,en,~en,arevetorsofoeÆients,andznm(t)aresolutionstoarstordersystemofordinarydierentialequationsdrivenbytheradialharmonisnm=(;Ynm).Thesummationovertheseriesin(3)and(4)maybeviewedasextensionsoftheloalB1andB2operatorsofBaylissandTurkel[2℄respetively.Inomputation,thesumovernin(3)and(4)istrunatedatanarbitraryvalueN1,andN2,respetively.BothboundaryonditionsareexatformodesnN.FormodesnN,(3)reduesto(1)while(4)reduesto(2).Therefore,whentrunatedatanitevalueN,boundaryondition(4)isexpetedtoapproximatethemodesnNwithgreaterauraythan(3).Theimplementationof(3)and(4)usingnitedierenemethodsisdisussedin[6℄.In[7,8℄weimplementedamodiedversionof(3)inastandardsemidisreteniteelementformulationwithseveralalternativeimpliitandexpliittime-integrators.InthispaperweshowhowtoimplementtheNRBC(4)intheniteelementmethod.Touse(4)inaniteelementformulation,wereformulatetheboundaryonditionandderiveanequivalentbutmoretratableform,whihdoesnotinvolvehigh-orderInternationalJournalforNumerialMethodsinEngineering,45,1607-1630(1999)L.L.ThompsonandR.Huan3radialderivatives.Twoalternativeformulationsaregiven.Intherst,theboundaryoperatorisimplementeddiretlyasa‘natural’boundaryonditioninthevariationalequation.Intheseond,theoperatorisimplementedindiretlybyintroduingauxil-iaryvariablesonthenonreetingboundary.Theindiretimplementationleadstoasymmetrisystemofmatrixequations,andavoidstheunsymmetridampingmatrixpresentinthediretimplementation,albeitattheexpenseofsolvingforadditionalunknownson.Severalversionsofimpliitandexpliittime-integrationshemesarepresentedforsolutionoftheniteelementsemidisreteequationsonurrentlywiththerst-orderdierentialequationsassoiatedwiththenonreetingboundaryonditionandanauxiliaryvariable.Numerialstudiesareperformedforahallengingtestproblemofradiationfromapistononasphere,toassesstheaurayandonvergenepropertiesofthenon-reetingboundaryonditionswhenimplementedintheniteelementmethod.Weomparetheevolutionofthediretandindiretapproahestoformulating(4)intheniteelementmethodusinganL2errornorm.WethenomparethebehavioroftheniteelementformulationsoftheNRBC(3)and(4)asafuntionofth