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-164--123143007424300743430073;;;1[1-3][4-5][67]ALE-165-21t0XΩv123OXXXP123(),,XXXX=v(123),,iXi=PtxΩvoxxx123Pp),,(321xxxx=ϖ)3,2,1(=ixiPχΩv-oχχχ123Pq),,(321χχχχ=ϖ123(),,iiχ=P1LXΩvXv(),Xtxx=vvvXvExΩvxϖ(),XXxt=vvvxϖ-ALEχΩvχϖ-166-()txx,χϖϖϖ=χϖ(),Xtχχ=vvvXv3Xϖuϖ),(tXxxϖϖϖ=tXttXxuϖϖϖϖ∂∂=/),(XϖXϖXϖvϖ),(tXϖϖϖχχ=tXttXvϖϖϖϖ∂∂=/),(χXϖXϖχϖwϖ),(txxχϖϖϖ=tχχϖϖϖϖttxw∂∂=/),(χϖχϖLEALEALEALELEuwϖϖ=L0=wϖE0≠≠uwϖϖALEXϖXϖXϖALE),(tXϖϖϖχχ=)),,((ttXffϖϖχ=ft-167-/////XDfDtftftftχ∂∂∂∂∂∂χ∂χ∂==+vvvvgXϖXϖχϖχϖ)),,((ttXffjϖχ=/////jjjXDfDtftftftχ∂∂∂∂∂∂χ∂χ∂==+vg)),,((ttXxjiϖχ(Xϖiu)()(,),////jiijiijjXuxXtttxtxtχχχχ=∂∂=∂∂+∂∂∂∂vvg//iiijjuwxtχχ−=∂∂∂∂giiicwu=−/iijjcxvχ=∂∂gic/iijjcxvχ=∂∂g//jijitcxχχ∂∂=∂∂gXϖ///jiiXftftcfxχ∂∂=∂∂+∂∂vgXϖ)),,((ttXujiϖχXϖ//jiikikautcuxχ=∂∂+∂∂gjtuiχ∂∂/iujχkckukwLjtuiχ∂∂/Xituϖ∂∂/iukc0kkuw=Xiituaϖ∂∂=/Ejtuiχ∂∂/jxitu∂∂/iukcku0=kw//jiikikxautuux=∂∂+∂∂g4[8][9-10]NS12-[11]-168-NS-NS-3ALE[11]ABCDBC51231AKovacs,MKawahara.Afiniteelementschemebasedonthevelocitycorrectionmethodforthesolutionofthetime-dependentincompressibleNavier-Stokesequations,Intjnumer.methodsfluids,1991,13:403–423.2GRen,TUtnes.Afiniteelementsolutionofthetime-dependentincompressibleNavier-Stokesequationsusingamodifiedvelocitycorrectionmethod,Intjnumer.methodsfluids,1993,17:349–364.3TNakayama,MMori.AnEulerianfiniteelementmethodfortime-dependentfreesurfaceproblemsinhydrodynamics,Intjnumer.methodsfluids,1996,22:175–194.4TOkamoto,MKawahara.Two-dimensionalsloshinganalysisbyLagrangianfiniteelementmethod,Intjnumer.methodsfluids,1990,11:453–477.5MHayashi,KHatanaka,MKawahara.LagrangianfiniteelementmethodforfreesurfaceNaver-Stokesflowusingfractionalstepmethods,Intjnumer.methodsfluids,1991,13:805–840.6RamaswamyB,KawaharaM.ArbitraryLagrangian-Eulerianfiniteelementmethodforunsteady,convective,incompressibleviscousfreesurfacefluidflow.InternationalJournalforNumericalMethodsinFluid.1987,7:1053–1075.7HuretaA,LiuWK.Viscousflowwithlargefreesurfacemotion.ComputMethodsApplMechEngrg,1988,69:277–324.-169-8.[].,,1998.9CodinaR,VazquezM,ZienkienwiczOC.Ageneralalgorithmforcompressibleandincompressibleflows.PartIII:thesemi-implicitform.InternationalJournalforNumericalMethodsinFluid.1998,27:12–32.10SungJ,ChoiHG,YooJY.Time-accuratecomputationofunsteadyfreesurfaceflowsusinganALE-segregatedequal-orderFEM.Comput.Methods.Appl.Mech.Engrg.,2000,190:1425–1440.11.:[],:,2006.ComparisonsoftheLAGRANGIAN,EULERIANandarbitraryLAGRANGIAN-EULERIANdescriptionsforfiniteelementmethodSUNJiang-long1YANGWen-yu2YANGXia3(1SchoolofNavalArchitecture&OceanEng.,HuazhongUniv.ofSci.&Tech,Wuhan430074;2SchoolofMechanicalSci.&Eng.,HuazhongUniv.ofSci.&Tech,Wuhan430074;3SchoolofMechanical&ElectricalEng.,WuhanInstituteofTechnology,Wuhan,430073)AbstractTheLagrangian,EulerianandArbitraryLagrangian-Euleriandescriptionsarediscussedforfiniteelementmethodinthispaper.Thecorrespondingreferenceframesareplottedtogetherforourcomprehendingvariousdescriptions.Materialderivativesoftensorsforvelocityandaccelerationarederivedandcomparedwitheachotherbasedonthedefinitionofthematerialderivative.Inordertodistinguishandunderstandtheexistingdescriptionsmoredistinctly,comprehensivecomparisonsofthethreekindsofdescriptionsarelisted.Atlast,thefiniteelementanalysisiscarriedout.Keywords:Lagrangian,Eulerian,ArbitraryLagrangian-Eulerian,Finiteelementmethod