1. Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 169 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://mpdc.mae.cornell.edu/ Nicholas Zabaras and Lijian Tan Multiscale modeling of solidification

2. Materials Process Design and Control Laboratory  Mathematical model & two main difficulties  Applying boundary conditions  Level set method  Present model  Analysis and numerical studies  Multiple moving interfaces  Multiple signed distance functions  Single signed distance function with markers  Multi-scale modeling  Adaptive meshing, domain decomposition  Database approach Outline Of The Presentation

3. Materials Process Design and Control Laboratory Two main difficulties Mathematical Model  Applying boundary conditions on interface for heat transfer, fluid flow and solute transport.  Multiple moving interfaces (multiple phases/crystals).

4. Materials Process Design and Control Laboratory  Jump in temperature gradient governs interface motion  Gibbs-Thomson relation  No slip condition for flow  Solute rejection flux Complexity Of The Moving Interface

5. Materials Process Design and Control Laboratory History: Devised by Sethian and Osher (1988) as a mathematical tool for computing interface propagation. Advantage is that we obtain extra information (distance to interface). This information helps to compute interfacial geometric quantities, define a novel model, doing adaptive meshing, etc. Level Set Method We pay additional storage and extra computation time to maintain the above signed distance by solving Level set variable is simply distance to interface

6. Materials Process Design and Control Laboratory Assumption 1: Solidification occurs in a diffused zone of width 2w that is symmetric around the zero level set. A phase volume fraction can be defined accordingly. This assumption allows us to use the volume averaging technique. (N. Zabaras and D. Samanta, 2004) Present Model Don’t need to worry about boundary conditions of flow and solute any more!

7. Materials Process Design and Control Laboratory Unknown parameter k N. How will selection of k N affect the numerical solution? Assumption 2: The solid-liquid interface temperature is allowed to vary from the equilibrium temperature in a way governed by Gibbs-Thomson condition has to be satisfied (one of the major difficulties) Extended Stefan Condition Do not want to apply this directly, because any scheme with essential boundary condition is numerically not energy conserving. Introduce another assumption: Temperature boundary condition is automatically satisfied. Energy is numerically conserving!

8. Materials Process Design and Control Laboratory In the simple case of fixed heat fluxes, interface temperature approaches equilibrium temperature exponentially. Stability requirement for this simple case is Although this is only for a very simple case, we find that selection of is stable for all problems we have considered. Stability Analysis

9. Materials Process Design and Control Laboratory Our method Osher (1997) Different results obtained by researchers suggest that this problem is nontrivial. The literature results shown are based on a sharp interface model. Triggavason (1996) Benchmark problem: Crystal growth with initial perturbation. Convergence Behavior Energy conserving makes the difference!

10. Materials Process Design and Control Laboratory Our diffused interface model with tracking of interface Phase field model without tracking of interface Computation Requirement Tracking of the interface makes the difference!

11. Materials Process Design and Control Laboratory  L. Tan and N. Zabaras, "A level set simulation of dendritic solidification of multi-component alloys", Journal of Computational Physics, Vol. 221, pp. 9-40, 2007.  L. Tan and N. Zabaras, "A level set simulation of dendritic solidification with combined features of front tracking and fixed domain methods", Journal of Computational Physics, Vol. 211, pp. 36-63, 2006. Comparing with the other methods in literature, the method presented has much better convergence properties, and much less computational requirement for the same level of accuracy. Related publications But a single crystal is too far away from reality! We want to handle multiple phases/crystals.  L. Tan and N. Zabaras, "Modeling the growth and interaction of multiple dendrites in solidification using the level set method”, Journal of Computational Physics, in press.

12. Materials Process Design and Control Laboratory Handling of multiple interfaces Method 2: Markers to identify different solid regions Method 1: A signed distance function for each phase. Each color (orientation of the crystal) is used as a marker. Efficient, appropriate for hundreds of crystals.

13. Materials Process Design and Control Laboratory  Stable growth with 4 seeds  Unstable growth with 2 seeds  Unstable to stable growth with 10 seeds Compute Eutectic Growth with Multiple Level Sets Parameters of the alloy taken from Apel, Boettinger, Dipers, and Steinbach, 2002.

14. Materials Process Design and Control Laboratory Solute concentration for peritectic growth of Fe – 0.3wt% C alloy at time 0.6s, 1.5s, 1.8s, and 2.4s. Compute Peritectic Growth with Multiple Level Sets

15. Materials Process Design and Control Laboratory Interaction of multiple crystals: level set method with markers

16. Materials Process Design and Control Laboratory The other way is to explore common features in the solution. What if number of crystals goes to thousands One way is to stretch the computation limit by using Adaptive Meshing and Domain decomposition.

17. Materials Process Design and Control Laboratory  Tree type data structure for mesh refinement Coarsen Refine Adaptive Meshing

18. Materials Process Design and Control Laboratory Adaptive Domain Decomposition (Mesh Partition) 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Mesh Dual graph

19. Materials Process Design and Control Laboratory Demonstration of Adaptive Domain Decomposition

20. Materials Process Design and Control Laboratory Application of Adaptive Domain Decomposition

21. Materials Process Design and Control Laboratory A numerical example Material properties: Boundary conditions: Initial condition: 010203040 0 10 20 30 40

22. Materials Process Design and Control Laboratory Computational results using adaptive domain decomposition Computation time: 2 days with 8 nodes (16 CPUs). Cannot wait so long! Can we obtain results in a faster way (multi-scale modeling)?

23. Materials Process Design and Control Laboratory What we can expect from multi-scale modeling  Microstructure features are often of interest, e.g. 1 st /2 nd arm spacing, Heyn’s interception measure, etc. Let us denote these features as: Of course, we cannot expect microscopic details. But  We want to know macroscopic temperature, macroscopic concentration, liquid volume fraction.

24. Materials Process Design and Control Laboratory Widely accepted assumptions Assumption 1: Without convection, macroscopic temperature can be modeled as Assumption 2: At a reasonably high solidification speed and without fluid flow, macroscopic concentration constant. Assumption 4: Volume fraction only depends on microstructure, and temperature. Assumption 3: Microstructure depends on macroscopic cooling history and thermal gradient history.

25. Materials Process Design and Control Laboratory Macro-scale model Temperature Liquid volume fraction Microstructure features Unknown functions: First two equations coupled. Microstructure features determined as a post-processing process. Solve sample problems using the fully- resolved model (micro-scale model) to evaluate them!

26. Materials Process Design and Control Laboratory Relevant sample problems Infinite number of sample problems can be selected. How to select the ones related to our problem of interest is the key! Use a very simple model to find relevant sample problems. Model M: (1) treat material as pure material (sharp and stable interface) (2) do not model nucleation

27. Materials Process Design and Control Laboratory Comparison of three involved models

28. Materials Process Design and Control Laboratory Solution features of model M Define solute features of model M to be the interface velocity and thermal gradient in the liquid at the time the interface passes through.

29. Materials Process Design and Control Laboratory Given any solution feature of model M, we can find a problem, such that features of model M for this problem equals to the given solution feature. Selection of sample problems Chose a domain (rectangle is used) with initial and boundary condition form the following analytical solution: Sample problem:

30. Materials Process Design and Control Laboratory Multi-scale framework

31. Materials Process Design and Control Laboratory Solve the previous problem Material properties: Boundary conditions: Initial condition: 010203040 0 10 20 30 40

32. Materials Process Design and Control Laboratory Step 1: Get solution features of model M Plot solution features of model M for all nodes in the feature spaces

33. Materials Process Design and Control Laboratory Step 2: Fully-resolved solutions of sample problems

34. Materials Process Design and Control Laboratory Obtained liquid volume fraction

35. Materials Process Design and Control Laboratory Use iterations to obtain temperature, volume fraction, microstructure features

36. Materials Process Design and Control Laboratory Temperature at time 130 Macro-scale model result with Lever rule Fully-resolved model results with different sampling of nucleation sites. Average Data-base approach result

37. Materials Process Design and Control Laboratory Liquid volume fraction at time 130 Left: temperature field and volume fraction contours (0.95 and 0.05) Right: volume fraction contour on top of fully-resolved model interface position

38. Materials Process Design and Control Laboratory Predicted microstructure features Results in rectangle: predicted microstructure Results in the middle: fully-resolved model results Black solid line: predicted CET transition location

39. Materials Process Design and Control Laboratory Solidification of Al-Cu alloy

40. Materials Process Design and Control Laboratory Step 1: Solution features of model M 1 2 3 4 56 7 8 9 10 11

41. Materials Process Design and Control Laboratory Step 2: Fully-resolved solution of sample problems

42. Materials Process Design and Control Laboratory Periodic boundary condition for the sample problem Top half: results copied from below Bottom half: Computational domain Periodic boundary condition to minimize effects of boundary on directional solidification solution

43. Materials Process Design and Control Laboratory Lquid volume fraction for different microstructure features

44. Materials Process Design and Control Laboratory Iterative process for convergence Left half (black points): results after iter 0. Right half (green points): results after iter 3.

45. Materials Process Design and Control Laboratory Comparison with Lever rule (temperature at t=12.7s) Left: Lever rule Right: Database approach

46. Materials Process Design and Control Laboratory A B C D A (95mm,75mm) B (90mm,75mm) C (75mm,75mm) D (60mm,80mm) Microstructure in the domain 1 2 3 4 56 7 8 9 10 11 E F G H E (90mm,10mm) F (80mm,20mm) G (65mm,35mm) H (50mm,50mm) A B C D E F G H

47. Materials Process Design and Control Laboratory A B C D Fine columnar  coarse columnar  Equiaxed Microstructure from side to center A B C D

48. Materials Process Design and Control Laboratory Microstructure from corner to center E F G H Fine equiaxed  Coarse equiaxed E F G H

49. Materials Process Design and Control Laboratory Conclusions A micro-scale energy-conserving level set model combining features of front tracking method and fixed domain methods was introduced. This model is efficient, accurate and applicable to multi- component and multi-phase solidification systems. A database approach is introduced to allow rapid implementation of a multiscale approach using fully- resolved microstructure evolution results available off-line from a database. The database approach uses a `trivial model’ for identifying `relevant sample problems’ to our problem of interest. Iterations are used to further improve this idea. Interpolation is performed in the microstructure feature space – thus further mimimizing the needed number of sample problems.