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