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School of somethingFACULTY OF OTHERSchool of Computing
“An Adaptive Numerical Method for Multi-Scale 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
Pure Materials
Temperature
Alloys
Concentration
Driving force for Solidification
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
Phase Equation
Properties:
▪ highly nonlinear
▪ noise introduced by anisotropy function A(Ψ)
▪ where
Jimack, Mullis and Rosam
)arctan( xy
Thermal-solute Phase-field model• Karma’s Phase-field Model
Concentration EquationTemperature Equation
▪ highly nonlinear
Jimack, Mullis and Rosam
Thermal-solute Phase-field model• Karma’s Phase-field Model (multiple time scales)
Concentration EquationTemperature Equation
▪ highly nonlinear
D
Jimack, Mullis and Rosam
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
Further mesh refinement on
coarser levels to represent the
temperature field accurately !!
The sharp interfaces of the phase and
concentration fields lead to large gradients so fine
mesh resolution is essential !!
• 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
Jimack, Mullis and Rosam
Fully Implicit Temporal Discretisation• Explicit time integration methods
• explicit methods are `easy` to apply
but impose a time step restriction
2
2hCt
• 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
2
2hCt
• very fine mesh resolution is
needed to resolve the large
gradients in the interface region,
so the time steps become
excessively small
C
Jimack, Mullis and Rosam
Fully Implicit Temporal Discretisation• Influence of the Multiscale problem
• STRONG dependence on material
parameters, LE = D/α :
2),(
2hLECCt
• 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 solved for the new values: , and .
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
1nij 1n
ijU1n
ij
Nonlinear Multigrid solver• the FAS scheme
Jimack, Mullis and Rosam
The FAS scheme, for solving A(U)=f, has the following features…
•Smoother is nonlinear -- we use a pointwise weighted G-S scheme:
•The correction step requires an approximation to the full coarse grid problem but with a modified right-hand-side:
)(
))((1
111
kij
ij
jkijk
ijkij
uuA
fuAuu
)())(( 22
22 h
hhhhhh
hhh uIAuAfIf
Nonlinear Multigrid solver• the FAS scheme
Jimack, Mullis and Rosam
•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:
Nonlinear Multigrid solver• the MLAT scheme
Jimack, Mullis and Rosam
For the MLAT scheme the nodes at the interface between refinement levels are treated as a Dirichlet boundary by the smoother…
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:
Jimack, Mullis and Rosam
Iteration form Conv. rate No. of cycles Time (s)
V(1,1)
V(2,1)
V(2,2)
W(1,1)
W(2,2)
0.00879
0.00087
0.00010
0.00879
0.00010
5
4
3
5
3
2.39
2.47
2.30
3.32
3.10
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
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
Numerical method:
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