“An Adaptive Numerical Method for Multi-Scale Problems Arising in Phase-field Modelling”

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


Introduction• Thermodynamic Background

Pure Materials

Temperature

Alloys

Concentration

Driving force for Solidification

Dilute Alloys

Temperature + Concentration


Thermal-solute Phase-field model• Basic Idea of Phase-Field

Phase-field variable describes microstructure with diffuse interface approach


Thermal-solute Phase-field model• Karma’s Phase-field model

Phase Equation

Properties:

▪ highly nonlinear

▪ noise introduced by anisotropy function A(Ψ)

▪ where


)arctan( xy

Thermal-solute Phase-field model• Karma’s Phase-field Model

Concentration EquationTemperature Equation



Thermal-solute Phase-field model• Karma’s Phase-field Model (multiple time scales)

Concentration EquationTemperature Equation


D


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


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


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


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


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!


Fully Implicit Temporal Discretisation• Fully implicit BDF2 method

• fully implicit second-order BDF method is used to overcome these time

step restrictions


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 !!


Fully Implicit Temporal Discretisation• convergence of solution as maximum mesh level is increased

• The BDF2 method allows sufficiently fine spatial meshes to be used:


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.


1nij 1n

ijU1n

ij

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:

•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


•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


For the MLAT scheme the nodes at the interface between refinement levels are treated as a Dirichlet boundary by the smoother…


• Here we see the mesh-independent convergence rate of the nonlinear multigrid solver:



• Here we see the optimal solution time for the nonlinear multigrid solver:



• 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)

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


Simulation results• Progression of the adaptive mesh


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)…


Future Work• Beginning with an explicit solver in 3-d

• An example of solidification with a six-fold symmetry…



• A snap shot of the mesh with at most 8 refinement levels…



• A snap shot of the mesh with at most 9 refinement levels…


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…


“An Adaptive Numerical Method for Multi-Scale Problems Arising in Phase-field Modelling”

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