材料加工中的数值模拟方法-微观组织数值模拟(8)

《材料料加工过程的数值模拟》——微观组织数值模拟(VII)任课教师:王锦程Office：凝固技术国家重点实验室403室Tel：029-8840650(O)Email：[e-mail]•LevelSet方法是由Sethian和Osher于1988年提出，最近十几年得到广泛的推广与应用。简单的说来，LevelSet方法把低维的一些计算上升到更高一维，把N维的描述看成是N＋1维的一个水平。举个例子来说，一个二维平面的圆，如x^2+y^2=1可以看成是二元函数f(x,y)=x^2+y^2的1水平，因此，计算这个圆的变化时就可以先求f(x,y)的变化，再求其1水平集。•这样做的好处是，第一，低维时的拓扑变化在高维中不再是一个难题；第二，低维需要不时的重新参数化，高维中不需要；第三，高维的计算更精确；第四，LevelSet方法可以非常容易的向更高维推广；最后，也是非常重要的一点就是，上升到高维空间中后，许多已经成熟的算法可以拿过了直接用，并且在这方面有非常成熟的分析工具，譬如偏微分方程的理论及其数值化等。当然，这种方法最为诟病的就是他增加了计算量，但新的快速算法不断出现，使得这也不是个大问题。CellularautomataPhasefieldmethodFronttrackingLevelsetmethodTechniquesforhandlingmovinginterfaceLEVELSETMETHODInterfacemotion(levelsetequation)Ref.S.Osher1997,||0tV(,)(,)0(,)dxtxxtxdxtxDevisedbySethian&OsherAdvantagesInterfacegeometriescanbeeasilyandaccuratelycomputed.Levelsetequationwellstudied(FDMwithhigherorderaccuracy,FEM)Disadvantage:Applicationofboundaryconditionsstillnoteasy(mostapplicationsarerestrictedtopurematerialswithoutmeltconvection).SigneddistanceIntroducedtothisareabyJ.Dantzig2000,R.Fedkiw2003etc.利用φ=0描述界面，定義φ0為液相且其值為至界面的距離，φ0為固相且其絕對值為至界面的距离。W為界面移動速度向量*00300.55/0.65:400400,115cos(4),0.5,0.05,1CrystalwithinitialradiusgrowingindomainwithinitialundercoolingdomainTddotherparametersnormalizedto..,.(2002)YTKimNGoldenfield&(1998)KarmaRapel0100200300400050100150200250300350400~3000hoursonDECAlpha:~20MeshelementsizeCPUT12~minuteonaGHzPCOurdiffusedinterfacemodelwithtrackingofinterfacePhasefieldmodelwithouttrackingofinterface:1Meshelementsize:~270nodeno:~160000nodenoComputationRequirementTrackinginterfacemakesthedifference!Stablegrowthwith4seedsUnstablegrowthwith2seedsUnstabletostablegrowthwith10seedsComputeEutecticGrowthwithMultipleLevelSetsParametersofthealloytakenfromApel,Boettger,Dipers,andSteinbach,2002.SoluteconcentrationforperitecticgrowthofFe–0.3wt%Calloyattime0.6s,1.5s,1.8s,and2.4s.ComputePeritecticGrowthwithMultipleLevelSetsComputeInteractionofMultipleCrystalswithMarkersApplicationofAdaptiveDomainDecompositionComputationalresultsusingadaptivedomaindecompositionComputationtime:2dayswith8nodes(16CPUs).Cannotwaitsolong!Canweobtainresultsinafasterway(multi-scalemodeling)?ABCDFinecolumnarcoarsecolumnarEquiaxedMicrostructurefromsidetocenterABCDMicrostructurefromcornertocenterEFGHFineequiaxedCoarseequiaxedEFGH3DCRYSTALGROWH(Ni-CuAlloy)3millionelements(withoutadaptivemeshing200millionelements)3DCRYSTALGROWTHWITHCONVECTIONComparingwiththepurematerialcase,thegrowthforalloyismuchmoreunstableduetotherejectionofsolution.