An Enthalpy—Level-set Method Vaughan R Voller, University of Minnesota + + speed def. Single Domain Enthalpy (1947) Heat source A Problem of Interest— Track Melting Melt Solid Narrow band level set form Diffuse interface 1<f<0

Outline * Brief Overview of Level sets    T=0 f=0 f=1 *Diffusive Interface, Enthalpy, and Level Set *Application to Basic Stefan Problem Velocity and Curvature *Application to non-standard problems Phase Change Temp and Latent heat a function of space

   t1t1 t2t2 t3t3 Level sets 101 Problem Melting around a heat source- melt front at 3 times Define a level set function  (x,t) - where The level set  (x,t) = 0 is melt front, and The level set  (x,t) = c is a “distance” c from front Incorporate values Of  (x,t) into physical model— through source tern and/or modification of num. scheme

time 2 time 3 Evolve the function  (x,t) with timetime 1 Problems *What is suitable “speed” function v n (x,t) *Renormalize  (x,t) to retain “distance” property

Problems can be mitigated by Using a “Narrow-Band” Level set Essentially “Truncate” so that -0.5 <  < 0.5 Results – For two-D melting From a line heat source  

Assume constant density Governing Equations For Melting Problem liquid-solid interface T = 0 n Two-Domain Stefan Model T=0 f=0 f=1 Use a Diffusive Interface Phase change occurs smoothly across A “narrow” temperature range Results in a Single Domain Equ. The Enthalpy Formulation TmTm f=0 f=1 liquid fraction

General Level Set Enthalpy-Level Set dist. function update-eq. narrow band “appropriate” choice for v n recovers governing equation liquid fraction

AND How does it Work—in a time step 1. Solve for new f Calculated assuming that current time Temp values are given by TmTm f=0 f=1 If explicit time int. is used NO iteration is required With narrow band constraint 2. Update temperature field by solving *As of now no modification of discretization scheme used *If explicit time intergration NO ITS

L=10 T=1 T=-0.5 Velocity—as front crosses node Front Movement with time Application to A Basic Stefan Melt Problem T=0 c = K = 1  t= 0.075,  x =.5 p

L=.1 T=1 T=-0.5 A Basic Stefan Problem Intro smear  = 0.1 Front Movement velocity as front crosses node sharp front smear fast slow TmTm f=0 f=1 

Calculation of Curvature Melting from corner heat source diag front pos. time 50x50,  x=0.5,  t=0.037 L = K=c 1 Curvature as front crosses diag. node  

Novelty Problem 1—Solidification of Under-Cooled Melt with space dependent solidification Temperature T m T=-0.5 Tm=f(x) Liquid at Temperature Profiles at a fixed point in time Temperature under-cooled temperature L= c = K = 1  t= 0.125,  x = 1 Note Heat “leaks” In two-dirs.

Special Case T=-0.5 T=0 Liquid at Analytical Solution in Carslaw and Jager Front Movement Red dots Enthalpy-level set Line--analytical

Application growth of Equiaxed dendrite in an under-cooled melt Liquid at T<T m Temp at interface a Function of Space and time

Enthalpy-Level Set predictions Enthalpy predicted dendrite shape at t =37,000, ¼ box size 800x800,  t = 0.625, Tip Velocity

Novelty Problem 2—Melting by fixed flux with space dep. Latent heat T= 0 Solid at c = K = 1  t= 0.25,  x = 1 q 0 = 1 L=0.5x Latent Heat temperature time x Predictions of front movement compared with analytical solution (analytical solution From Voller 2004)

Application Growth of a Sedimentary Ocean Basin/Delta sediment h(x,t) x = u(t) bed-rock ocean x shoreline x = s(t) land surface   Related to restoring Mississippi Delta 20k “Wax Lake”

Summary + + speed def. Single Domain Enthalpy (1947) Heat source Melt Solid Narrow band level set form Diffuse interface 1<f<0 Essentially No more than a reworking of The basic 60 year old Enthalpy Method But--- approach could provide insight into solving current Problems of interest related to growth processes, e.g. Crystal Growth Land Growth

TmTm f=0 f=1 