Download presentation

Presentation is loading. Please wait.

Published byJocelyn Purcell Modified over 3 years ago

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

Similar presentations

OK

Steady Aeroelastic Computations to Predict the Flying Shape of Sails Sriram Antony Jameson Dept. of Aeronautics and Astronautics Stanford University First.

Steady Aeroelastic Computations to Predict the Flying Shape of Sails Sriram Antony Jameson Dept. of Aeronautics and Astronautics Stanford University First.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on national education days Ppt on kingdom monera definition Ppt on abstract art techniques Ppt on event driven programming visual basic Ppt on interest rate risk management Download ppt on data handling for class 7 Ppt on product advertising images Ppt on waxes and wanes Ppt on panel discussion images Ppt on condition based maintenance of underground cable systems