School of something
School
of Computing
FACULTY OF OTHER
“An Adaptive Numerical Method for MultiScale Problems Arising in Phase-field
Modelling”
Peter Jimack, Andy Mullis and Jan Rosam
BICS Numerical Analysis Conference, 7 September 2007
Outline
1. Introduction
2. Coupled Phase-Field model for Modeling Alloy Solidification
3. Adaptive Spatial Discretisation
4. Fully Implicit Temporal Discretisation
5. Nonlinear Multigrid solver
6. Results
Jimack, Mullis and Rosam
Introduction
• Real Solidification Structures
Dendritic microstructure formation in Xenon systems
J.H. Bilgram, ETH Zurich
Jimack, Mullis and Rosam
Introduction
• Thermodynamic Background
Driving force for Solidification
Pure Materials
Alloys
Temperature
Concentration
Dilute Alloys
Temperature + Concentration
Jimack, Mullis and Rosam
Thermal-solute Phase-field model
• Basic Idea of Phase-Field
Phase-field variable describes microstructure
with diffuse interface approach
Jimack, Mullis and Rosam
Thermal-solute Phase-field model
• Karma’s Phase-field model
     
1
 
A( ) 2   Mc 1  (1  k )U 
 A( ) 2  2  2 A( ) A' ( ) 


 Le
 t
 x x y y 

   
 
  A( ) A' ( )    A( ) A' ( )  
x 
y  y 
x 
   3   (1   2 ) 2 (  McU )
Properties:
▪ highly nonlinear
Phase Equation
▪ noise introduced by anisotropy function A(Ψ)
▪ where
Jimack, Mullis and Rosam
  arctan( y x)
Thermal-solute Phase-field model
• Karma’s Phase-field Model
 1   U  U  1   2 
 1  k 1  k  U


 D  


 U  


2
2

t
2

x

x

y

y
2






     x      y  
1 
 1  (1  k )U  
 
 
2 2
 x  t |  |  y  t |  |  
 U    x  U    y   
 (1  k )



  
 x  t |  |  y  t |  |   
1
 
  (1  (1  k )U ) 
2
t 
Concentration Equation
▪ highly nonlinear
Jimack, Mullis and Rosam

1 
  2 
t
2 t
Temperature Equation
Thermal-solute Phase-field model
• Karma’s Phase-field Model (multiple time scales)
 1   U  U  1   2 
 1  k 1  k  U


 D  


 U  


2
2

t
2

x

x

y

y
2






D
     x      y  
1 
 1  (1  k )U  
 
 
2 2
 x  t |  |  y  t |  |  
 U    x  U    y   
 (1  k )



  
 x  t |  |  y  t |  |   
1
 
  (1  (1  k )U ) 
2
t 
Concentration Equation
▪ highly nonlinear
Jimack, Mullis and Rosam


1 
  2 
t
2 t
Temperature Equation
Thermal-solute Phase-field model
• Multiscale Problem
Cross-section of typical solution
Large ratios of the diffusion coefficients lead to a
multiscale problem that is highly stiff
Jimack, Mullis and Rosam
Adaptive Spacial Discretisation
• Adaptive mesh refinement
• second or fourth order Finite Difference method
• based upon quadrilateral meshes (non-uniform)
• adaptive remeshing controlled by a gradient criterion
• compact stencils used to reduce grid anisotropy
The sharp interfaces of
the phase and
concentration fields lead
to large gradients so fine
mesh resolution is
essential !!
Jimack, Mullis and Rosam
Further mesh
refinement on
coarser levels to
represent the
temperature field
accurately !!
Fully Implicit Temporal Discretisation
• Explicit time integration methods
• explicit methods are `easy` to apply
but impose a time step restriction
h2
t  C
2
• very fine mesh resolution is
needed to resolve the large
gradients in the interface region,
so the time steps become
excessively small
Jimack, Mullis and Rosam
Fully Implicit Temporal Discretisation
• Explicit time integration methods
• explicit methods are easy to apply
but impose a time step restriction
h2
t  C
2
• very fine mesh resolution is
C
needed to resolve the large
gradients in the interface region,
so the time steps become
excessively small
Jimack, Mullis and Rosam
Fully Implicit Temporal Discretisation
• Influence of the Multiscale problem
• STRONG dependence on material
parameters, LE = D/α :
2
h
t  C  C  ( LE ,  )
2
• increasing ratio of diffusion
coefficents leads to a further drop
in the time steps size!
Jimack, Mullis and Rosam
Fully Implicit Temporal Discretisation
• Fully implicit BDF2 method
• fully implicit second-order BDF method is used to overcome these time
step restrictions
Jimack, Mullis and Rosam
Fully Implicit Temporal Discretisation
• adaptive time step control
• the BDF2 method is combined with variable time stepping based upon a
local error estimator
The adaption of the time
step leads to a much
larger time step than the
maximum stable time step
for the explicit Euler
method !!
Jimack, Mullis and Rosam
Fully Implicit Temporal Discretisation
• convergence of solution as maximum mesh level is increased
• The BDF2 method allows sufficiently fine spatial meshes to be used:
Jimack, Mullis and Rosam
Nonlinear Multigrid solver
• nonlinear Multigrid solver for adaptive meshes
At each time step a large nonlinear algebraic system of equations must be
n 1
n 1
solved for the new values: ij , U ij and ijn1 .
Unless this can be done efficiently the method is worthless…
• A fully coupled nonlinear Multigrid solver is used to achieve this:
 based upon the FAS (full approximation scheme) approach to resolve
the non-linearity
 and the MultiLevel AdapTive (MLAT) scheme of Brandt to handle the
adaptivity
 a pointwise weighted nonlinear Gauss-Seidel iterative scheme is
seen to be an adequate smoother.
• Excellent, h-independent, convergence results are obtained.
Jimack, Mullis and Rosam
Nonlinear Multigrid solver
• the FAS scheme
The FAS scheme, for solving A(U)=f, has the following features…
•Smoother is nonlinear -- we use a pointwise weighted G-S scheme:
k 1
(
A
(
u
ij )  f j )
k 1
k 1
uij  uij  
k 1
A
(u )
uij ij
•The correction step requires an approximation to the full coarse grid
problem but with a modified right-hand-side:
f 2 h  I ( f h  Ah (uh ))  A2 h ( I u )
2h
h
Jimack, Mullis and Rosam
2h
h
h
Nonlinear Multigrid solver
• the FAS scheme
•In this implementation:
Interpolation from coarse to fine grids is bilinear
Restriction from fine to coarse grids is simple injection
•We use a full multigrid (FMG) version because of the locally refined
meshes:
Jimack, Mullis and Rosam
Nonlinear Multigrid solver
• the MLAT scheme
For the MLAT scheme the nodes at the interface between refinement
levels are treated as a Dirichlet boundary by the smoother…
Jimack, Mullis and Rosam
Nonlinear Multigrid solver
• nonlinear Multigrid solver for adaptive meshes
• Here we see the mesh-independent convergence rate of the nonlinear
multigrid solver:
Jimack, Mullis and Rosam
Nonlinear Multigrid solver
• nonlinear Multigrid solver for adaptive meshes
• Here we see the optimal solution time for the nonlinear multigrid solver:
Jimack, Mullis and Rosam
Nonlinear Multigrid solver
• nonlinear Multigrid solver for adaptive meshes
• The specific choice of multigrid cycle can be selected for the best
performance, as shown in the following table:
Iteration form
Conv. rate
No. of cycles
Time (s)
V(1,1)
0.00879
5
2.39
V(2,1)
0.00087
4
2.47
V(2,2)
0.00010
3
2.30
W(1,1)
0.00879
5
3.32
W(2,2)
0.00010
3
3.10
Jimack, Mullis and Rosam
Simulation results
• Dilute binary alloy solidification simulation with Lewis number 500
Lewis number 500
Jimack, Mullis and Rosam
Simulation results
• Progression of the adaptive mesh
Jimack, Mullis and Rosam
Future Work
• Physical solidification occurs in 3-d:
these 2-d simulations have limited quantitative predictive
capabilities.
•We must generalize the approach to a fully coupled 3D phase-field model:
adaptivity on hexahedral meshes,
fully implicit time-stepping with 3d nonlinear multigrid,
parallel implementation likely to be essential.
• So far we have only just begun this process by looking at a pure thermal
problem (so explicit time-stepping is feasible)…
Jimack, Mullis and Rosam
Future Work
• Beginning with an explicit solver in 3-d
• An example of solidification with a six-fold symmetry…
Jimack, Mullis and Rosam
Future Work
• Beginning with an explicit solver in 3-d
• A snap shot of the mesh with at most 8 refinement levels…
Jimack, Mullis and Rosam
Future Work
• Beginning with an explicit solver in 3-d
• A snap shot of the mesh with at most 9 refinement levels…
Jimack, Mullis and Rosam
Summary
Numerical method:
1. Finite Difference approximation on adaptively refined meshes
2. Fully implicit second order BDF time integration
3. Variable time step control based upon a local error estimator
4. Fully coupled nonlinear Multigrid solver
Phase-field:
1. The first time a multiscale Phase-field model has been solved fully
implicitly
2. Quantitative simulation of dilute binary alloys with high Lewis number
Now trying to extend everything to 3-dimensions in parallel…
Jimack, Mullis and Rosam