An Enthalpy—Level-set Method

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

2. f = 0 f =-1 f = 1 f=1 f=0 T=0 Outline * Brief Overview of Level sets *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

3. Level sets 101 f = 0 f =-1 f = 1 Incorporate values Of f(x,t) into physical model— through source tern and/or modification of num. scheme Problem Melting around a heat source- melt front at 3 times t1 t2 t3 Define a level set function f(x,t) - where The level set f(x,t) = 0 is melt front, and The level set f(x,t) = c is a “distance” c from front

4. Problems *What is suitable “speed” function vn(x,t) time 1 Evolve the function f(x,t) with time time 2 time 3 *Renormalize f(x,t) to retain “distance” property

5. Problems can be mitigated by Using a “Narrow-Band” Level set Essentially “Truncate” so that -0.5 <f < 0.5 f=.5 f=-.5 Results – For two-D melting From a line heat source

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

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

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

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

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

11. e= 0.015 e= 0.15 time diag front pos. Curvature as front crosses diag. node Calculation of Curvature 50x50, Dx=0.5, Dt=0.037 L = K=c 1 Melting from corner heat source

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

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

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

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

16. Predictions of front movement compared with analytical solution time temperature Latent Heat (analytical solution From Voller 2004) Novelty Problem 2—Melting by fixed flux with space dep. Latent heat c = K = 1 Dt= 0.25, Dx = 1 x q0 = 1 T= 0 T= 0 Solid at L=0.5x

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

18. Summary Narrow band level set form Single Domain Enthalpy (1947) + speed def. Land Growth Melt Heat source Solid 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

19. f=1 -e f=0 Tm