Phase-field simulations of solidification
structures
Denis Danilov
HS Karlsruhe
HS Karlsruhe
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 1/20
Introduction: Real Structures
Experimental solidification structures: Snowflakes and Frost
http://www.its.caltech.edu/˜atomic/snowcrystals/
HS Karlsruhe
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 2/20
Real Structures
Experimental solidification structures: Dendrites in transparent materials
M. E. Glicksman
organic material
HS Karlsruhe
J. H. Bilgram, ETH Zürich
Xenon
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 3/20
Diffuse solid-liquid interface
SOLID
LIQUID
ϕ=1
ϕ=0
interface thickness (MD) ∼ 10Å
W. J. Boettinger, J. A. Warren, C. Beckermann, and A. Karma. Annu. Rev.
Mater. Res. 32 (2002) 163
HS Karlsruhe
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 4/20
Phase-field model
– entropy functional
S(e, c, ϕ) =
Z 1
s(e, c, ϕ) − εa(∇ϕ) +
ε
w(ϕ)
dx,
0.06
|∇ϕ| = 0
w(ϕ)
|∇ϕ| ≠ 0
0.04
ϕ=1
0.02
|∇ϕ| = 0
ϕ=0
a(∇ϕ) = γ(∇ϕ)2 ,
HS Karlsruhe
w(ϕ) = 9γg(ϕ),
0
0.2
0.4
0.6
0.8
1
ϕ
g(ϕ) = ϕ2 (1 − ϕ)2 .
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 5/20
Phase-Field Model
– evolution of phase field
δS
,
∂t ϕ ∼
δϕ
2
2ωε∂t ϕ = 2εγ∇ ϕ −
9γ
ε
g,ϕ (ϕ) −
1
T
f,ϕ (c, ϕ),
phase-field parameters: ε, γ, ω.
– free energy density (T = const)
2
X
2
T − Ti
RT X
f (c, ϕ) =
ci Li
h(ϕ) +
ci ln ci ,
Ti
vm i=1
i=1
h(ϕ) = ϕ2 (3 − 2ϕ).
HS Karlsruhe
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 6/20
Phase-Field Model
– chemical potential
µi = f,ci
T − Ti
RT
= Li
h(ϕ) +
(ln ci + 1),
Ti
vm
– mass fluxes
Ji =
2
X
j=1
Lij (c, ϕ) =
vm
R
Di ci
Lij ∇
−µj
T
,
δij − P2
Dj cj
k=1
Dk ck
!
,
c1 + c2 = 1.
H. Garcke, B. Nestler, B. Stinner, A diffuse interface model for alloys with
multiple components and phases, SIAM J. Appl. Math. 64(3) (2004) 775–799.
HS Karlsruhe
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 7/20
Binary alloy
Equilibrium phase diagram
1728 K
liquid phase
cL
cS
solid phase
1358 K
Ni
HS Karlsruhe
0.2
0.4
0.6
0.8
Cu
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 8/20
Binary alloy: Evolution equations
– phase field (nonconserved field)
∂ϕ
∂t
Λ(c, u) = (1 − c0 c)
=
ε2
L2
2
∇ ϕ−
9 dg
2 dϕ
−
ε
2γ
LA u + (T0 − TA )/TQ
TA
u + T0 /TQ
Λ(c, u)
+ c0 c
dh
dϕ
LB u + (T0 − TB )/TQ
TB
u + T0 /TQ
– concentration field (conserved field)
∂c
∂t
Θ(u) =
HS Karlsruhe
= ∇ D̄∇c − ∇ D̄c(1 − c0 c)∇(Θ(u)h(ϕ))
vm
R
D ε2
D̄ =
ν L2
LA u + (T0 − TA )/TQ
TA
u + T0 /TQ
−
LB u + (T0 − TB )/TQ
TB
u + T0 /TQ
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 9/20
Different length scales
Concentration profiles at different growth velocities
0.5
1.0
0.4
0.8
a
0.3
0.6
0.2
0.4
b
0.1
Phase field ϕ(0)
Concentration c(1)
ϕ(0)
0.2
c
0
HS Karlsruhe
-2
-1
0
1
2
3
Coordinate ξ
4
5
6
0
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 10/20
Adaptive grid spacing
E=
1
X
α
phase−field
0.8
0.6
+ EC
Non−uniform
adaptive mesh
∆ x2<∆ x1
∆ x2
0
HS Karlsruhe
−5
|∇cj |d2
+ ET |∇T |d2 .
∆ x1
−10
X
j
0.4
0.2
|∇φα |d2
0
x/ε
5
10
The cell will be refined if
E > Emax and coarsened
if E < Emin . √ By this
choice, d ∼ 1/ E.
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 11/20
FEM: phase field and concentration field
phase field
HS Karlsruhe
concentration field
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 12/20
FEM: Adaptive grid
| ←−
HS Karlsruhe
L
−→ |
L
∆xmin
= 2048
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 13/20
Anisotropy of the interface
gradient energy a(∇ϕ) = γ(1 + δγ cos β)(∇ϕ)2
δγ = 0
HS Karlsruhe
δγ = 0.04
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 14/20
Noise
driving force → (1 + αr)× driving force, −1 < r < +1
δγ = 0, α = 0.25
HS Karlsruhe
δγ = 0.04, α = 0.25
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 15/20
Further examples
∼80 h
(AMD Opteron)
10000 time steps
637 → 1140883
active cells
L
∆xmin
HS Karlsruhe
= 8192
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 16/20
Further examples
1 805 380 active cells
9000 time steps,
HS Karlsruhe
1 862 719 active cells
L
∆xmin
= 4096
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 17/20
Further examples
70
% of computational time
60
50
40
30
20
Matrix & RHS
Linear Solver
Grid Refinement
10
0
0
5e+05
1e+06
1.5e+06
2e+06
Number of active cells
HS Karlsruhe
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 18/20
Further examples
4600 time steps
1303 → 347992
active cells
L
∆xmin
HS Karlsruhe
= 256
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 19/20
Features request
■
■
■
Periodic boundary conditions.
Output to binary OpenDX-files.
Platform-independent binary format in
Vector<Number>::block_write/block_read.
HS Karlsruhe
Deal.II User Workshop, 3-5.01.2006, Heidelberg, p. 20/20