SimulationofleakyRayleighwaveatairsolidcylindricalinterfacesbyniteelementmethodYanZhao,ZhonghuaShen*,JianLu,XiaowuNiSchoolofscience,NanjingUniversityofScienceandTechnology,Xiaolingwei200#,Nanjing210094,PRChinaAvailableonline9June2006AbstractThe nite element method is used to simulate the laser-excited leaky Rayleigh wave at airsolid cylindrical interfaces. A whole arith-metic of uidsolid interaction is presented, which includes a coupling matrix that describing the process of the interaction between uidand solid, the Arbitrary LagrangianEulerian (ALE) formulation for treating the variation of uid domain, which results from the Ray-leighwavepropagatingonthecylindricalinterface,etc.TypicalcalculationisexecutedandtheresultsshowthatthepolarityofleakyRayleighwaveformgraduallychangesasitpropagatesontheairsolidcylindricalinterface.2006ElsevierB.V.Allrightsreserved.Keywords: Airsolidcylindricalinterface;Finiteelementmethod;LeakyRayleighwave1.IntroductionIn the last few decades, leaky waves at uidsolid planeinterface (leaky Rayleigh and leaky Lamb wave) haveattracted extensive attention for their potential applicationin nondestructive test and characterization of materials [13]. The development of laser ultrasonic technique providesaneectivesolutiontoinvestigatetheleakywavesontheuidsolid interface. In recent years, some researchersbegin to study the leaky waves by laser ultrasonic method.For example, Desmet et al. detected the laser-induced leakyRayleighwavesonwatermetal interfaces, andtheyreal-izedall-optical excitationanddetectionofleakyRayleighwaves by means of the laser ultrasonic technique [3]. How-ever,theseworksmainlyconcentratedontheplanestruc-ture, and less attention was attached to the leakyRayleighwavesonuidsolidcylindrical interfaces[13].As a result, the main objective of this paper is to investigatethe laser-inducedleakyRayleighwaves onthe airsolidcylindrical interface. Except for experiment, numerical cal-culationisanothereectivesolutiontosolvetheproblemofacousticwave,suchasniteelementmethod,nitedif-ferencemethod,andsoon.In this research, the nite element method is introducedtosimulatethelaser-excitedleakyRayleighwaveonairsolidcylindrical interface because it has twoprominentadvantages: one is the ability to model laser-induced ultra-sonic under any conditions. The other is the convenience totake the temperature dependence of physical parametersintoaccount.2.Finiteelementmodel2.1.CongurationConsider a homogenous and isotropic cylinder of radiusa, and density q, which is placed in air. As depicted in Fig. 1,the symmetric axis of the cylinder coincides with thez-axisof ourcylindrical coordinates(r, h, z). Usingacylindricallens,thepulselaserbeamwasfocusedontothesurfaceofsolidcylindertoformalinesource, whoseorientationisalongthezdirection.Owingtothesymmetryimposedbythesourceshape,thisproblemshowsinvariancealongthez direction. As aresult, the nonzerocomponents of the0041-624X/$-seefrontmatter 2006ElsevierB.V.Allrightsreserved.doi:10.1016/j.ultras.2006.05.177*Correspondingauthor.Tel.:+862584315075;fax:+862584315699.E-mail addresses: zhaoyan7906@sina.com(Y. Zhao), shenzh@mail.njust.edu.cn(Z.Shen).Ultrasonics44(2006)e1169e1172http://www.paper.edu.cn displacement vector only depends on two spatial variables r,h and on timet.2.2.HeatconductionFor laser-induced heat transfer, convection can beneglected[4], the classical heat conductionequationforniteelementscanbeexpressedas[5]KfTg Cf_Tg fpqg fpQg; 1where {T} is the temperature vector, f_Tg is the temperatureratevector, [K] istheconductivitymatrix, [C] istheheatcapacitymatrix, {pq}is theheat uxvector and{pQ}istheheatsourcevector.Foranelement,{pq}isRSNTqdS,and{pQ}isRVNTQdV , whereSandVaretheareaandvolumeof anelement, respectively; [N]Tis thetransposeoftheshapefunction; qistheheatuxandQistheheatsource.2.3.WaveequationFor the generation and propagation of ultrasonic, ignor-ing damping, the NavierStokes equation can be expressedasMsolidfUg HsolidfUg fFsolidg; 2where{U}isthedisplacementvector, fUgistheaccelera-tionvector, [Msolid] isthemassmatrix, [Hsolid] isthesti-ness matrixand{Fsolid}istheexternal forcevector. Forultrasonicpropagationsinelasticuidmedium, thegov-erningequationcanbewrittenasMairfPg HairfPg fFairg; 3where{P}istheacousticpressurevector.2.4.SolutionschemesA lumping scheme [5] coupled with the central-dierenceintegrationmethod,whichismostapplicabletoproblemsof largesizeandisacompromisebetweenaccuracyandeciency, isusedtosolvetheequationsfortheuidandsolid domains independently of each other. Then, uidforcesandheatuxesandsoliddisplacements, velocities,and temperatures are transferred across the uidsolidinterface. The algorithmcontinues to loop throughthesolidanduidanalyses until convergence is reachedforthat timestep(oruntil themaximumnumberof staggeriterationsisreached).2.4.1.FluidsolidinteractionThegenerationofleakyRayleighwavecanbeconsid-eredastheresultoftheuidandsolidinteractionattheinterfacebetweensolidcylinderanduid.Theinteractioncauses the acoustic pressure toexert a force appliedtothesolidandthesolidmotionsproduceaneectiveuidload.ThegoverningniteelementmatrixequationsthenbecomeMsolidfUg HsolidfUg fFsolidg RfPg; 4MairfPg HairfPg fFairg q0RTfUg; 5[R] is a coupling matrix that represents the eective sur-face area associated with each node on the uidsolid inter-face. Thecouplingmatrix[R] alsotakesintoaccountthedirection of the normal vector dened for each pair of coin-cident uid and solid element faces that comprises theinterface surface. The positive direction of the normal vec-tor, inourcalculation, isdenedtobeoutwardfromtheuidmeshandintowards thesolid. Boththesolidanduid load quantities that are produced at the uid-structureinterfacearefunctionsofunknownnodal degreesoffree-dom.Placingtheseunknownloadquantitiesonthelefthandsideof theequationsandcombiningthetwoequa-tionsintoasingleequationproducesthefollowing:Msolid0q0RTMair UP( ) HsolidR0 Hair UP FsolidFair ;6This equation implies that nodes on a uid-structure inter-face have both displacement and pressure degrees offreedom.2.4.2.InterfacedeterminationUltrasonic waves propagating inthe surface of solidmakeitssurfacedeform, whichleadtotheuiddomainchange with time. As a result, the nite element mesh mustmovetosatisfytheboundaryconditionsintheniteele-ment calculation. The Arbitrary LagrangianEulerian(ALE) formulation is taken to solve this problem. The gen-eral concept oftheALEformulationisthatanarbitraryreferential domainis dened forthe descriptionof motionthat is dierent from the material (Lagrangian) and spatial(Eulerian)domains.In a pure Lagrangian system, the mesh deforms with thematerial beingmodeledsothat thereisnomaterial owbetweenelements. Themaindisadvantage of theLagrang-ianapproachis that problems developinphysical situa-tions that involve highly deformed surfaces. Otherdisadvantages of theLagrangianapproacharethat onlyone material canbe modeledineachelement andthatnewsurfacescannotbecreated.InanEulerianbasedformulation, themeshisstation-ary, the material ows through the mesh, and new surfacesare automatically created. However, the greatest disadvan-tage of the Eulerian approach is that a ne mesh is requiredSolidcylinder AirLaser beamaFig. 1. Congurationforthelaser-inducedexcitationofleakyRayleighwavesattheairsolidcylindricalinterface.e1170 Y.Zhaoetal./Ultrasonics44(2006)e1169e1172 http://www.paper.edu.cntocapturethematerialresponse,makingthemethodverycomputationallyexpensive.TheArbitraryLagrangianEulerian(ALE)approachisa very eective alternative. Inits most basic sense, theALEmethoddenesthatthemeshmotionisindependentof the motionof the material beinganalyzed. Althoughthe meshmotionmay be arbitrary, it typicallydeformswiththe material innear Lagrangian owelds. Thegreatest advantage of the ALEmethodis that it allowssmoothing of a distortedmeshwithoutperforminga com-plete remesh. This smoothing allows the free surface of thematerial tobefollowedautomaticallywithoutencounter-ing the distortional errors of the Lagrangianapproach.ThemaindicultyoftheALEmethodisthepathdepen-dent behavior of the plastic ow being modeled. Due to thepathdependence, the relative motionbetweenthe meshand the material must be accounted for in the material con-stitutive equations. In addition,the ALE methoddoes notallownewsurfacestobecreatedandislimitedtogeome-trieswherethematerialowisrelativelypredictable.2.4.3.FiniteelementmeshandtimestepThe selection of nite element size and time step are crit-ical for the stability and accuracy of nite element calcula-tion. In our calculation, the nite element sizes Le for bothsolid and uid domain and time step are determined by thecriterionssuggestedbySchubertetal.[6]Le 6110Cfmax; 7Dt 6LeC 3p; 8whereC represents the highest wave speed of the medium,fmaxisthehighestfrequencyintheultrasonicwave,whichcanbeevaluatedby[7]fmax 22pCpr0; 9wherer0istheradiusofthelaserpulsespot.3.Numericalsimulationandresults3.1.LaserandmaterialparametersUsing above nite element arithmetic of uidsolidinteraction, we have simulated the laser-induced leakyRayleighwaveatthecylindricalinterfacebetweenairandaluminumcylinder. Theparametersoflaserandmaterialwerechosenasfollowing: thelaserpulserisetimet0, andthe radius of the laser pulse spot on the interface are10 ns and300 lm, respectively. The radius of aluminumcylinderistakentobe4 mm,andthephysicalparametersof airandaluminumusedinthecalculationarelistedinTable1.3.2.TypicalresultsFig. 2showstheleakyRayleighwaveformrecordedatthe interface 90 away from the source. Compared this cal-culatedwaveformwiththeRayleighwaveformonthefreesurfaceofasolid, asexpected, verygoodagreementsareobservedinthetime, shape, andrelativeamplitude. LR1andLR2represent therst leakyRayleighpulseandthesecondonewhichpropagaterespectively1/4and3/4turnonthe cylindrical interface. Leaky Rayleighwave onauidsolidinterface is anelastic wave corresponding tothe Rayleigh wave on the free surface of a solid thatbecomes dampedinits directionof propagationbecauseof seepageof energyin the liquid. In addition, leakywhis-pering gallery modes (LWG) corresponding to the whisper-ing gallery modes on the free surface of a solid are also veryobvious.Thankstotheadvantageofniteelementmethod, thewholeelddatacanbeobtainedinasimulation, wealsoobtainedthewaveformsofleakyRayleighwaveobtainedat dierent angles hbetweenthe excitedsource andtherecording point, as shown in Fig. 3. Compared those wave-forms in Fig. 3, one can easily nd an interesting result thatTable1Physicalparametersofairandaluminumusedintheniteelementcalculation(T = 300 K)Thermaldiusivity(K1)Specicheat(J kg1K1)Heatconductivity(W m1K1)Absorptioncoecient(m1)Density(kg m3)Velocityoflongitudinalwave(m s1)Velocityoftransversewave(m s1)Al 71.8 106896 237 1032700 6260 3080Air 37.2 1031005 0.0261 1031.21 3430 2 4 6 8-6-4-20246LWGLR2LR1Normal displacement (nm)Time (s)Fig.2. Leaky Rayleigh waveformrecorded attheairaluminumcylindercylindricalinterface90awayfromthesource.Y.Zhaoetal./Ultrasonics44(2006)e1169e1172 e1171 http://www.paper.edu.cnthe polarity of leaky Rayleigh waveform gradually changesas it propagates on the airsolid cylindrical interface. Nearthesource, theleakyRayleighpulseismonopolar(nega-tive). Then, the leaky Rayleigh waveformchanges, itbecomes bipolar at h = 90 andmonopolar (positive) ath = 180. Thechangeof polarityresultsfromthedisper-sion eectof cylindrical interface. In thisgure,the pulsesfor the clockwise and counterclockwise propagation mergeintoa single peakat h = 180 sothat the amplitude isincreasedtwofold.4.ConclusionsA nite element model is developed to simulate the leakyRayleigh wave at the airsolid cylindrical interface inducedby pulse laser. The whole process of uidsolid interactionhas been formulated by nite element equations. Accordingtoniteelementcalculation,typicalleakyRayleighwave-formsareobtainedandtheyindicatethatthepolarityofleakyRayleighwaveformgraduallychanges as it propa-gatesontheairsolidcylindricalinterface.AcknowledgementThis work is supported by the Young Scholars Founda-tionoftheNanjingUniversityofScienceandTechnology(GrantNJUST200503).References[1] C.Mattei,L.Adler,Ultrasonics38(2000)570.[2] V. Gusev, C. Desmet, W. Lauriks,C. Glorieux, J. Thoen,J. Acoust.Soc.Am.100(3)(1996)1514.[3] C. Desmet, V. Gusev, W. Lauriks, C. Glorieux, J. Thoen, Opt. Lett. 22(2)(1997)69.[4] J.B. Spicer, Laser ultrasonics in nite structures: comprehensivemodelingwithsupportingexperiment. Ph.D. thesis, Johns HopkinsUniversity,1991.[5] R.D. Cook, D.S. Malkus, M.E. Plesha, Concepts and Applications ofFiniteElementAnalysis,thirded.,JohnWiley,1989.[6] F. Schubert, B. Koehler, A. Peier, J. Comput. Acoust. 9 (2001) 1127.[7] Y.Sohn,S.Krishnaswamy,Ultrasonics39(2002)543.0 2 4 6 8-0.60.00.61.20 2 4 6 8-0.30.00.30.6-0.30.00.30.60 2 4 6 8-0.30.00.30.60 2 4 6 8-0.6-0.30.00.30.60 2 4 6 8-0.6-0.30.00.30 2 4 6 8-0.6-0.30.00.30 2 4 6 8-0.6-0.30.00.3Time (s) 180 150 135 120 90 60 45 300Normal displacement (nm)0 2 4 6 8Fig. 3. Leaky Rayleigh waveforms recorded on the airaluminum cylindercylindricalinterfaceversusangle hfrom30to180.e1172 Y.Zhaoetal./Ultrasonics44(2006)e1169e1172 http://www.paper.edu.cn