Perfectly matched layers for transient elastodynamics of unbounded domains

INTERNATIONALJOURNALFORNUMERICALMETHODSINENGINEERINGInt.J.Numer.Meth.Engng2004;59:1039–1074(DOI:10.1002/nme.896)

Perfectlymatchedlayersfortransientelastodynamicsofunboundeddomains

UshnishBasuandAnilK.Chopra?,?

DepartmentofCivilandEnvironmentalEngineering,UniversityofCalifornia,Berkeley,CA94720,U.S.A.

SUMMARY

Oneapproachtothenumericalsolutionofawaveequationonanunboundeddomainusesaboundeddomainsurroundedbyanabsorbingboundaryorlayerthatabsorbswavespropagatingoutwardfromtheboundeddomain.Aperfectlymatchedlayer(PML)isanunphysicalabsorbinglayermodelforlinearwaveequationsthatabsorbs,almostperfectly,outgoingwavesofallnon-tangentialangles-of-incidenceandofallnon-zerofrequencies.Inarecentwork[ComputerMethodsinAppliedMechanicsandEngineering2003;192:1337–1375],theauthorspresented,interalia,time-harmonicgoverningequationsofPMLsforanti-planeandforplane-strainmotionof(visco-)elasticmedia.Thispaperpresents(a)correspondingtime-domain,displacement-basedgoverningequationsofthesePMLsand(b)displacement-based?niteelementimplementationsoftheseequations,suitablefordirecttransientanalysis.The?niteelementimplementationoftheanti-planePMLisfoundtobesymmetric,whereasthatoftheplane-strainPMLisnot.Numericalresultsarepresentedfortheanti-planemotionofasemi-in?nitelayeronarigidbase,andfortheclassicalsoil–structureinteractionproblemsofarigidstrip-footingon(i)ahalf-plane,(ii)alayeronahalf-plane,and(iii)alayeronarigidbase.TheseresultsdemonstratethehighaccuracyachievablebyPMLmodelsevenwithsmallboundeddomains.Copyright?2004JohnWiley&Sons,Ltd.

KEYWORDS:perfectlymatchedlayers(PML);absorbingboundary;scalarwaveequation;elasticwaves;transientanalysis;?niteelements(FE)

1.INTRODUCTION

Thesolutionoftheelastodynamicwaveequationoveranunboundeddomain?ndsapplicationsinsoil–structureinteractionanalysis[1]andinthesimulationofearthquakegroundmotion[2].Theneedforrealisticmodelsoftencompelsanumericalsolutionusingaboundeddomain,alongwithanarti?cialabsorbingboundaryorlayerthatsimulatestheunboundeddomainbeyond.Correspondenceto:AnilK.Chopra,DepartmentofCivilandEnvironmentalEngineering,707DavisHall,UniversityofCalifornia,Berkeley,CA94720,U.S.A.?E-mail:chopra@ce.berkeley.edu

Contract/grantsponsor:WaterwaysExperimentStation,U.S.ArmyCorpsofEngineers;contract/grantnumber:DACW39-98-K-0038

Copyright?2004JohnWiley&Sons,Ltd.Received18December2002Revised25April2003Accepted28May2003

1040U.BASUANDA.K.CHOPRA

Ofparticularimportanceareabsorbingboundariesthatallowtransientanalysis,facilitatingincorporationofnon-linearitywithintheboundeddomain.

Classicalapproximateabsorbingboundaries[3–6],althoughlocalandcheaplycomputed,mayrequirelargeboundeddomainsforsatisfactoryaccuracy,sincetypicallytheyabsorbinci-dentwaveswellonlyoverasmallrangeofangles-of-incidence.Forsatisfactoryperformance,approximateabsorbinglayermodels[7,8]requirecarefulformulationandimplementationtoeliminatespuriousre?ectionsfromtheinterfacetothelayer.Thesuperpositionboundary[9]iscumbersomeandexpensivetoimplement,andin?niteelements[10,11]typicallyrequireproblem-dependentassumptionsonthewavemotion.Rigorousabsorbingboundariesaretyp-icallyformulatedinthefrequencydomain[12–14];correspondingtime-domainformulations[15–17]maybecomputationallyexpensiveandmaynotbeapplicabletoallproblemsofinterest.Thedif?cultyinobtainingasuf?cientlyaccurate,yetnot-too-expensivemodeloftheun-boundeddomaindirectlyinthetimedomainhasledtotheuseoftraditionalfrequency-domainmodelstowardstime-domainanalysis.Onesuchmethoduseshybridfrequency–time-domainanalysis[1,18],iteratingbetweenthefrequencyandtimedomainsinordertoaccountfornon-linearityintheboundeddomain;thiscomputationallydemandingmethodrequirescare-fulimplementationtoensurestability.Anotherapproachreplacesthenon-linearsystembyanequivalentlinearsystem[19]whosestiffnessanddampingvaluesarecompatiblewiththeeffectivestrainamplitudesinthesystem.Athirdapproach[20–22]approximatesthefrequency-domainDtNmapofasystembyarationalfunctionandusesthisapproximationtoobtainatime-domainsystemthatistemporallylocal.Althoughthisapproachisconceptuallyattractive,computationofanaccuraterational-functionapproximationmaybeexpensive.

Aperfectlymatchedlayer(PML)isanabsorbinglayermodelforlinearwaveequationsthatabsorbs,almostperfectly,propagatingwavesofallnon-tangentialangles-of-incidenceandofallnon-zerofrequencies.Firstintroducedinthecontextofelectromagneticwaves[23,24],theconceptofaPMLhasbeenappliedtootherlinearwaveequations[25–27],includingtheelastodynamicwaveequation[28,29].Inarecentwork[30],theauthorshavedevelopedtheconceptofaPMLinthecontextoffrequency-domainelastodynamics,utilisinginsightsobtainedfromPMLsinelectromagnetics,andillustrateditusingtheone-dimensionalrodonelasticfoundationandtheanti-planemotionofatwo-dimensionalcontinuum,governedbytheHelmholtzequation.ExtendingthePMLconcepttothedisplacementformulationofplane-strainandthree-dimensionalmotion,theyhavealsopresentedanoveldisplacement-based,symmetric?niteelementimplementationofsuchaPML.

Theobjectiveofthispaperistopresent(a)time-domain,displacement-based,equationsofthePMLsforanti-planeandforplane-strainmotionofa(visco-)elasticmedium,and(b)displacement-based?niteelement(FE)implementationsoftheseequations.Thefrequency-domainPMLequationsfromReference[30]are?rsttransformedintothetimedomainbyaspecialchoiceoftheco-ordinate-stretchingfunctions,andthenthesetime-domainequationsareimplementednumericallybyastraightforward?niteelementapproach.Time-domainnumericalresultsarepresentedfortheanti-planemotionofasemi-in?nitelayeronrigidbaseandfortheclassicalsoil–structureinteractionproblemsofarigidstrip-footingon(i)ahalf-plane,(ii)alayeronahalf-plane,and(iii)alayeronarigidbase.Additionally,theadequacyofthespecialchoiceofthestretchingfunctionstowardsattenuatingevanescentwavesisinvestigatedthroughnumericalresultsinthefrequencydomain.ThispaperpresentsonlyabriefexplanationoftheconceptofaPML;adetaileddevelopment,andthederivationofthefrequency-domainequationsarepresentedinReference[30].

Copyright?2004JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng2004;59:1039–1074

TRANSIENTELASTODYNAMICSOFUNBOUNDEDDOMAINS1041

Tensorialandindicialnotationwillbeusedinterchangeablyinthispaper;thesummationconventionwillbeassumedunlessanexplicitsummationisusedoritismentionedotherwise.Anitalicboldfacesymbolwillrepresentavector,e.g.x,anuprightboldfacesymbolwillrepresentatensororitsmatrixinaparticularorthonormalbasis,e.g.D,andasans-serifboldfacesymbolwillrepresentafourth-ordertensor,e.g.C;thecorrespondinglightfacesymbolswithRomansubscriptswilldenotecomponentsofthetensor,matrixorvector.Anoverbarover

¯,denotesatime-harmonicquantity;suchdistinguishingnotationwasnotasymbol,e.g.u

employedinReference[30]becausetheentireanalysiswasinthefrequencydomain.

2.ANTI-PLANEMOTION

2.1.Elasticmedium

Consideratwo-dimensionalhomogeneousisotropicelasticcontinuumundergoingonlyanti-planedisplacementsintheabsenceofbodyforces.Forsuchmotion,ifthex3-directionistakentopointoutoftheplane,onlythe31-and32-componentsofthethree-dimensionalstressandstraintensorsarenon-zero.Thedisplacementsu(x,t)aregovernedbythefollowingequations(i∈{1,2}):

??*??i=??u¨ii

??i=??εi

εi=*u

i(1a)(1b)(1c)

where??istheshearmodulusofthemediumand??itsmassdensity;??iandεirepresentthe3i-componentsofthestressandstraintensors.

Onanunboundeddomain,Equation(1)admitsplaneshearwavesolutions[31]oftheform

u(x,t)=exp[?iksx·p]exp(i??t)

whereks=??/csisthewavenumber,withwavespeedcs=denotingthepropagationdirection.

2.2.Perfectlymatchedlayer

ThediscussionofPMLpresentedhereisasynopsisofthecorrespondingdevelopmentinReference[30].Thesummationconventionisabandonedinthissection.

ConsiderawaveoftheforminEquation(2)propagatinginanunboundedelasticdomain,thex1–x2plane,governedbyEquation(1).Theobjectiveofde?ningaperfectlymatchedlayer(PML)istosimulatesuchwavepropagationbyusingacorrespondingboundeddomain.

ThegoverningequationsofaPMLaremostnaturallyde?nedinthefrequencydomain,throughfrequency-dependent,complex-valuedco-ordinatestretching.Assumingharmonictime-dependenceofthedisplacement,stressandstrain,e.g.u(x,t)=u(¯x)exp(i??t),with??theCopyright?2004JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng2004;59:1039–1074(2)√andpisaunitvector

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frequencyofexcitation,thegoverningequationsofthePMLforanti-planemotionare

??

i1*??¯i=???2??u¯i(xi)i

¯i??¯i=??ε

ε¯i=¯1*u

i(xi)*xi(3a)(3b)(3c)

where??iarenowhere-zero,continuous,complex-valuedco-ordinatestretchingfunctions.Ifthestretchingfunctionsarechosenas

??i(xi):=1?ifi(xi)

ks(4)

intermsofreal-valued,continuousattenuationfunctionsfi,thenEquation(3)admitssolutionsoftheform??????u(¯x,t)=exp?Fi(xi)piexp[?iksx·p](5)i

where

Fi(xi):=??0xifi(??)d??(6)

Thus,ifFi(xi)>0andpi>0,thenthewavesolutionadmittedinthePMLmediumisoftheformoftheelastic-mediumsolution[Equation(2)],butwithanimposedspatialattenuation.Thisattenuationisoftheformexp[?Fi(xi)pi]inthexi-direction,andisindependentofthefrequencyifpiis.

Considerreplacingthex1–x2planeby BD∪ PM,asshowninFigure1,where BDisa‘bounded’(truncated)domain,governedbyEquation(1),and PMisaPML,governedbyEquation(3),with??1oftheforminEquation(4),satisfyingf1(0)=0,and??2≡1.Themediumin BDbeingaspecialPMLmedium[??i(xi)≡1],thematchingofstretchingfunctionsatthe BD– PMinterfacemakesthePML‘perfectlymatched’to BD:wavestravellingoutwardfromtheboundeddomainareabsorbedintothePMLwithoutanyre?ectionfromthe BD– PMinterface.AnoutgoingwaveenteringthePMLisattenuatedinthelayerandthenre?ectedbackfromthe?xedendtowardstheboundeddomain.Iftheincidentwavehasunitamplitude,thentheamplitude|R|ofthere?ectedwaveasitexitsthePMLisgivenby

|R|=exp[?2F1(LP)cos??](7)

Thisre?ected-waveamplitudeiscontrolledbythechoiceoftheattenuationfunctionandthedepthofthelayer,andcanbemadearbitrarilysmallfornon-tangentiallyincidentwaves.Becausesuchoutgoingwavesinsuchasystemwillbeonlyminimallyre?ectedbacktowardstheinterface,thisbounded-domain-PMLsystemisanappropriatemodelfortheunboundedx1–x2plane.

Copyright?2004JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng2004;59:1039–1074

TRANSIENTELASTODYNAMICSOFUNBOUNDEDDOMAINS

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Figure1.APMLadjacenttoa‘bounded’(truncated)domainattenuatesandre?ectsbackanoutgoingplanewave.

2.3.Time-domainequationsforthePML

ConsidertworectangularCartesianco-ordinatesystemsfortheplaneasfollows:(1)an{xi}system,withrespecttoanorthonormalbasis{ei},and(2)an{xi??}system,withrespectto??},withthetwobasesrelatedbytherotation-of-basismatrixQ,anotherorthonormalbasis{ei??.Equation(3)canbere-writtenintermsoftheco-ordinatesx??withcomponentsQij:=ei·ejibyreplacingxibyxi??throughout,representingamediumwhereinwavesareattenuatedinthe??ande??directions,ratherthanintheeandedirectionsasinEquation(3).Thisresultante1122equationcanbetransformedtothebasis{ei}toobtain[30]

????????·(??¯)=???2??[??1(x1)??2(x2)]u¯(8a)

(8b)

(8c)??¯=??(1+2ia0??)??¯??¯=??(?u)¯

where

??

??¯:=??¯1

??¯2??,??¯:=??ε¯1ε¯2??,??*??????*x??1?:=?*???????*x2(9)

Copyright?2004JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng2004;59:1039–1074