1. School of something FACULTY OF OTHER School 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

2. 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

3. Introduction Real Solidification Structures Dendritic microstructure formation in Xenon systems J.H. Bilgram, ETH Zurich Jimack, Mullis and Rosam

4. Introduction Thermodynamic Background Pure Materials Temperature Alloys Concentration Driving force for Solidification Dilute Alloys Temperature + Concentration Jimack, Mullis and Rosam

5. Thermal-solute Phase-field model Basic Idea of Phase-Field Phase-field variable describes microstructure with diffuse interface approach Jimack, Mullis and Rosam

6. Thermal-solute Phase-field model Karmas Phase-field model Phase Equation Properties: highly nonlinear noise introduced by anisotropy function A(Ψ) where Jimack, Mullis and Rosam

7. Thermal-solute Phase-field model Karmas Phase-field Model Concentration Equation Temperature Equation highly nonlinear Jimack, Mullis and Rosam

8. Thermal-solute Phase-field model Karmas Phase-field Model (multiple time scales) Concentration Equation Temperature Equation highly nonlinear D Jimack, Mullis and Rosam

9. 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

10. 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

11. Fully Implicit Temporal Discretisation Explicit time integration methods explicit methods are `easy` to apply but impose a time step restriction 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

12. Fully Implicit Temporal Discretisation Explicit time integration methods explicit methods are easy to apply but impose a time step restriction 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

13. Fully Implicit Temporal Discretisation Influence of the Multiscale problem STRONG dependence on material parameters, LE = D/α : increasing ratio of diffusion coefficents leads to a further drop in the time steps size! Jimack, Mullis and Rosam

14. 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

15. 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

16. 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

17. 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

18. 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:

19. 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:

20. 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…

21. 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

22. 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

23. 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 formConv. rateNo. of cyclesTime (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 5435354353 2.39 2.47 2.30 3.32 3.10

24. Simulation results Dilute binary alloy solidification simulation with Lewis number 500 Lewis number 500 Jimack, Mullis and Rosam

25. Simulation results Progression of the adaptive mesh Jimack, Mullis and Rosam

26. 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

27. Future Work Beginning with an explicit solver in 3-d An example of solidification with a six-fold symmetry… Jimack, Mullis and Rosam

28. 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

29. 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

30. 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