An Enthalpy—Level-set MethodVaughan R Voller, University of Minnesota

+ f

)TK(t

Tc

L

1)t,(vn

x

+ speed def.

Lt

f)TK(

t

Tc

Single Domain Enthalpy (1947)

Heat sourceA Problem ofInterest— Track Melting

Melt

Solid

]1,0[f

,0t

ffvn

Narrow band level set form

Diffuse interface 1<f<0

Outline

* Brief Overview of Level sets

T=0f=0

f=1*Diffusive Interface, Enthalpy, and Level Set

f

f*Application to Basic Stefan Problem Velocity and Curvature

*Application to non-standard problems Phase Change Temp and Latent heat a function of space

t1t2

t3

Level sets 101Problem 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 valuesOf (x,t) into physical model—through source tern and/ormodification of num. scheme

time 2

time 3

0t

vn

Evolve the function (x,t) with timetime 1

Problems

*What is suitable “speed” function

vn(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 meltingFrom a line heat source

Assume constant densityGoverning Equations For Melting Problem

TKt

Tc 2

TKt

Tc 2

liquid-solid interface

T = 0

n

Two-Domain Stefan Model

T=0f=0

f=1

Use a Diffusive Interface

Phase change occurs smoothly acrossA “narrow” temperature range

Lt

f)TK(

t

Tc

nLvTK

Lt

f)TK(

t

Tc

Results in a Single Domain Equ.

The Enthalpy FormulationTm

f=0

f=1

2

TTf

m

liquid fraction

0t

vn

General Level Set Enthalpy-Level Set

dist. function

update-eq.

]1,0[f,0t

ffvn

narrow band

“appropriate” choice for vn

f

)TK(t

Tc

L

1)t,(vn

x

recovers governing equation

Lt

f)TK(

t

Tc

5.0f liquid fraction

]1,0[f,0t

ffvn

AND f

)TK(t

Tc

L

1)t,(vn

x

How does it Work—in a time step

1. Solve for new f

Calculated assuming that current time Temp values are given by

Tm

f=0

f=1

2

TTf

m

If explicit time int. is used NO iteration is required

With narrow band constraint

2. Update temperature field by solving

Lt

f)TK(

t

Tc

*As of now no modification of discretization scheme used

*If explicit time intergration NO ITS

L=10

T=1 T=-0.5

f

)TK(t

Tc

L

1)t,(vn

x

Velocity—as front crosses node

Front Movement with time

Application to A Basic Stefan Melt Problem

T=0

c = K = 1t= 0.075, x = .5

p

5.0fp

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

smear

fastslow

Tm

f=0

f=1

Calculation of Curvature

f

f

Melting from corner heat source

diag front pos.

time

50x50, x=0.5, t=0.037L = K=c 1

Curvature as front crossesdiag. node

2x1

Novelty Problem 1—Solidification of Under-Cooled Melt with spacedependent solidification Temperature Tm

T=-0.5Tm=f(x) Liquid at

Temperature Profiles at a fixed point in time

Temperature

under-cooledtemperature

L= c = K = 1t= 0.125, x = 1

Note Heat “leaks”In two-dirs.

Special Case

T=-0.5T=0 Liquid at

Analytical Solution in Carslaw and Jager

Front Movement

Red dotsEnthalpy-level set

Line--analytical

Application growth of Equiaxed dendrite in an under-cooled melt

Liquid at T<Tm

Temp at interface a Function ofSpace and time

0

0.02

0.04

0.06

0.08

0.1

0 5000 10000 15000 20000

Dim. Time

Dim

. Tip

Vel

.

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= 0T= 0 Solid at

c = K = 1t= 0.25, x = 1

q0 = 1

L=0.5x

Latent Heat

temperaturetime

x

Predictions of front movement compared with analytical solution

(analytical solutionFrom Voller 2004)

Application Growth of a Sedimentary Ocean Basin/Delta

sediment

h(x,t)

x = u(t)

0q

bed-rock

ocean

x

shoreline

x = s(t)

land surface

Related to restoring Mississippi Delta20k

“Wax Lake”

Summary

+ f

)TK(t

Tc

L

1)t,(vn

x

+ speed def.

Lt

f)TK(

t

Tc

Single Domain Enthalpy (1947)

Heat source Melt

Solid

]1,0[f

,0t

ffvn

Narrow band level set form

Diffuse interface 1<f<0

Essentially No more than a reworking ofThe 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

2

TTf

m

Tm

f=0

f=1