Discontinuous Galerkin Methods Applied to Shock an

DOI:10.1007/s10915-004-4138-4JournalofScientificComputing,Volumes22and23,June2005(©2005)DiscontinuousGalerkinMethodsAppliedtoShockandBlastProblemsN.Chevaugeon,1J.Xin,1P.Hu,1X.Li,1D.Cler,2J.E.Flaherty,1andM.S.Shephard1ReceivedJanuary12,2004;accepted(inrevisedform)May5,2004WedescribeprocedurestomodeltransientshockinteractionproblemsusingdiscontinuousGalerkinmethodstosolvethecompressibleEulerequations.Theproblemsaremotivatedbyblastflowssurroundingcannonswithperforatedmuzzlebrakes.Thegoalistopredictshockstrengthsandblastoverpressure.Thisapplicationillustratesseveralcomputationaldifficulties.Thesoftwaremusthandlecomplexgeometries.Theproblemsfeaturestronginteractingshocks,withpressureratiosontheorderof1000aswellasweakerprecursorshockstravelingrearwardthatalsomustbeaccuratelycaptured.Theseaspectsareaddressedusinganisotropicmeshadaptation.Ashockdetectorisusedtocon-troltheadaptationandlimiting.Wealsodescribeprocedurestotrackprojectilemotionintheflowbyalevel-setprocedure.KEYWORDS:DiscontinuousGalerkinmethods;hyperbolicconservationprob-lems;blast;shockdetection;anisotropicmeshrefinement;adaptivemethods.1.INTRODUCTIONWeproposetosolveblastproblemsmotivatedbyflowssurroundingcannonswithperforatedmuzzlebrakesusingadiscontinuousGalerkinmethod(DGM),asdescribedinSec.2.Detectionofshockstructuresappearscrucialtodrivebothadaptivemeshrefinementandlimitingandwedescribesuchdetectionscheme(Sec.3)andpresentsomeone-dimen-sionalexamples.Wealsodescribeananisotropicadaptivemeshrefinementprocedure(Sec.4),thatallowsustosolvelargeproblems,inanefficient1ScientificComputationResearchCenter,RensselaerPolytechnicInstitute,Troy,NY12180,USA.E-mail:fl[e-mail]2BenetLaboratories,TACOM/ARDEC,U.S.Army,Watervliet,NY12189,USA.2270885-7474/05/0600-0227/0©2005SpringerScience+BusinessMedia,Inc.228Chevaugeonetal.manner.Weapplythistotwoexamplesinvolvingblastsimulation.InSec.5,wedescribeprogresstoincludethepresenceofsolidobjectsmovingintheflowfieldbyusingalevel-set/ghost-fluidapproach,andpresentsomeapplications.2.DISCONTINUOUSGALERKINMETHODSFOREULEREQUATIONSThediscontinuousGalerkinmethodwasintroducedbyReedandHillin1973[23]asatechniquetosolveneutrontransportproblems.LesaintandRaviart[18]presentedthefirstnumericalanalysisofthemethodforalinearadvectionequation.However,thetechniquelaydormantforsev-eralyearsandhasonlyrecentlybecomepopularasamethodforsolvingfluiddynamicsandelectromagneticproblems[6].Aswithallmeshbasedprocedures,DGMusesadoublediscretization.First,thephysicaldomainΩisdiscretizedintoacollectionofNeelementsTe=Ne
e=1e(2.1)calledamesh.Then,thefunctionspaceV(Ω)containingthesolutionoftheproblemisapproximatedoneachelementeofthemeshbyafinite-dimensionalspaceVe(Te).Theaccuracyofthedoublediscretizationdependsongeometricalandfunctionaldiscretization.ClassicalGalerkinfiniteelementapproximationusesconformingmesheswhereelementsshareonlycompleteboundarysegments.ThespaceVeisalsoconstrainedtobeasubspaceofacontin-uousfunctionspace,forexample,H1,withabasisthatistypicallyasso-ciatedwithelementvertices,edges,faces,orinteriors.ThesesimplifytheimpositionoftheC0continuityrequirementsofH1butlimitchoices.TheDGMallowsmoregeneralmeshconfigurationsanddiscontinuousbasesthatsimplifybothh-andp-refinement.Forexample,non-conformingmeshesandarbitrarybasesforfunctionalapproximation[32]maybeused.Herein,weuseaL2-orthogonalbasisasaproductofJoacobipolynomials[8]thatyieldsadiagonalmassmatrix[24].TheDGMcaneasilysupportarbitraryordersofspatialdiscretizationaccuracywithouttheneedtoconstructcomplexstencilsforhigh-orderreconstruction[24].Indeed,theDGMstencilremainsinvariantforallpolynomialdegrees.Thisgreatlysimplifiesparallelimplementation.TheDGMhasadditionalflexibilityinspecifyingfluxesacrosselementsfacesandpermitstheuseoffluxesbasedonexactorapproximatesolutionsofRiemannproblems[5].ShockandBlastProblems229WeapplytheDGMtohyperbolicconservationlaws(Sec.2.1)usingaspatialdiscretizationofp-degreeorthogonalpolynomials.Timediscret-izationisperformedbyanexplicittotalvariationdiminishingRunge–Kuttamethod[14].Toimprovetheperformanceoftheexplicitintegrationonirregularandunstructuredgrid,weuseanewlocaltimesteppingpro-cedure[11,26].2.1.DiscontinuousFiniteElementFormulationforConservationLawsConsideranopensetΩ⊂R3withboundary∂Ω.Weseektodeter-mineu(Ω,t):R3×R→L2(Ω,t)m=V(Ω,t)asthesolutionofasystemofmconservationlaws∂tu+divF(u)=r.(2.2)Herediv=(div,...,div)isthevectorvalueddivergenceoperatorandF(u)=(F1(u),...,Fm(u))(2.3)isthefluxvectorwiththeithcomponentFi(u):(H1(Ω))m→H(div,Ω),whereH(div,Ω)consistsofsquareintegrablevectorvaluedfunctionswhosedivergenceisalsosquareintegrable,thatis,H(div,Ω)=v|v∈L2(Ω3),divv∈L2(Ω).(2.4)WiththeaimofconstructingaGalerkinformofEq.(2.2),let(·,·)Ωand·,·∂Ω,respectively,denotethestandardL2(Ω)andL2(∂Ω)scalarproducts(u,w)Ω=Ωuwdω(2.5)andu,w∂Ω=∂Ωuwdσ.(2.6)MultiplyEq.(2.2)byatestfunctionw∈V(Ω),integrateoverΩandusethedivergencetheoremtoobtainthevariationalformulation(∂tu,w)Ω−(F(u),gradw)Ω+F(u)·n,w∂Ω=(r,w)Ω,∀w∈V(Ω).(2.7)230Chevaugeonetal.WiththeDGM,Veisa“broken”functionspacethatconsistsofthedirects