1. Materials Process Design and Control Laboratory MULTISCALE COMPUTATIONAL MODELING OF ALLOY SOLIDIFICATION PROCESSES 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 lt69@cornell.edu URL: http://mpdc.mae.cornell.edu/ Lijian Tan and Prof. Nicholas Zabaras

2. Materials Process Design and Control Laboratory OUTLINE OF THE PRESENTATION  Brief review of alloy solidification process  Brief review of macro-scale model Review of meso-scale model  Multi-scale modeling  Adaptive meshing  Database approach

3. Materials Process Design and Control Laboratory solid Mushy zone liquid ~10 -1 - 10 0 m (b) Meso/micro scale ~ 10 -4 – 10 -5 m solid liquid (a) Macro scale (mushy zone is represented by volume fraction) q g MULTISCALE NATURE OF SOLIDIFICATION

4. Materials Process Design and Control Laboratory  Governing equations  Volume fraction is related with microstructure.  Lever rule or Scheil rule is used to obtain volume fraction.  Meso scale modeling is required to obtain volume fraction more accurately. MACRO SCALE MODELING

5. Materials Process Design and Control Laboratory MESO SCALE MODELING  A moving solid-liquid interface  Presence of fluid flow  Heat transfer  Solute transport Although equations are even simpler than macro-scale model, numerically it is very hard to handle due to the moving solid-liquid interface.

6. Materials Process Design and Control Laboratory ISSUES RELATED WITH MOVING INTERFACE  Jump in temperature gradient governs interface motion  No slip condition for flow  Gibbs-Thomson relation  Solute rejection flux Requires curvature computation at the moving interface!

7. Materials Process Design and Control Laboratory  Cellular automata  Phase field method  Front tracking  Level set method Techniques for handling moving interface Interface motion (level set equation) Ref. S. Osher 1997, Devised by Sethian&Osher Signed distance Level set method MESO SCALE MODELS

8. Materials Process Design and Control Laboratory OUR WORK WITH LEVEL SET METHOD  L. Tan and N. Zabaras, "A level set simulation of dendritic solidification of multi-component alloys", Journal of Computational Physics, in press  N. Zabaras, B. Ganapathysubramanian and L. Tan, "Modeling dendritic solidification with melt convection using the extended finite element method (XFEM) and level set methods", Journal of Computational Physics, in press.  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 method in literature, our method has better convergence, less computational requirement. Our method can also be easily extended to 3D.

9. Materials Process Design and Control Laboratory COVERGENCE BEHAVIOR Our method Osher (1997) Different results obtained by researchers suggest that this problem is nontrivial. All the referred results are using sharp interface model. Triggavason (1996) Benchmark problem: Crystal growth with initial perturbation.

10. Materials Process Design and Control Laboratory COMPUTATIONAL REQUIREMENT Our diffused interface model with tracking of interface Phase field model without tracking of interface

11. Materials Process Design and Control Laboratory EXTEND TO 3D

12. Materials Process Design and Control Laboratory MULTISCALE MODELING WITH ADAPTIVE MESH Solution changes rapidly only near solid-liquid interface. Still solve the meso-scale model, but with refined element only near interface. When combine adaptive meshing with parallel computing, we may be able to solve some macro-scale problems using the meso-scale model.

13. Materials Process Design and Control Laboratory Le=10 (boundary layer differ by 10 times) Micro-segregation can be observed in the crystal; maximum liquid concentration about 0.05. (compares well with Ref Heinrich 2003) SIMPLE BINARY ALLOY GROWTH EXAMPLE

14. Materials Process Design and Control Laboratory Domain decomposition Colored by process id COMBINE WITH PARALLEL COMPUTING

15. Materials Process Design and Control Laboratory APPLY TO MACROSCALE PROBLEM

16. Materials Process Design and Control Laboratory APPLY TO MACROSCALE PROBLEM

17. Materials Process Design and Control Laboratory IDEA OF DATA BASE APPROACH Even with adaptive meshing and parallel computation, solving very large problem with a meso scale model is not realistic. Runs in meso- scale Data base Build data Solve the interested problem Run in macro- scale Results with statistical properties. Model verification (the same problem) Results with details in mesoscale Run in meso- scale Results with statistical properties. Statistics Compare

18. Materials Process Design and Control Laboratory DETAILS OF THE DATA BASE APPROACH  Macro scale: Model the evolution of T (temperature), fl (liquid volume fraction), and two state variables s1 (grain size), s2 (grain type).  Model the unknowns (fl, s1 and s2) with the following form. R,G are the cooling rate and temperature gradient, defined at each point with value equal to the temperature rate and gradient when fl reaches 0.

19. Materials Process Design and Control Laboratory WHY LINK THE TWO SCALES IN THIS WAY  Volume fraction is mainly related with temperature and microstructure type.  Studies (experimental and numerical) have shown that microstructure type is determined by cooling rate and temperature gradient for a given alloy.  All variables (T, fl, R, G, s1, s2) have direct physical meaning, so data can be easily obtained from meso scale simulation results. No optimization is required to fit parameters.  In our numerical example, the following simplification is used.

20. Materials Process Design and Control Laboratory LINK BETWEEN TWO SCALES Linking hypothesis: Relations between macroscopic variables follow the same relations between the averaged microscopic variables in meso scale e.g. Macro scale Meso scale func(.)  Macro scale and meso scale are decoupled.  Accuracy of this approach relies on the accuracy of data obtained from meso scale simulation and the way for modeling and analyzing the data.

21. Materials Process Design and Control Laboratory PROBLEM CONSIDERED Domain: 10cm by 10cm Alloy: Al - 0.3%Cu Superheat: 30K Nucli density: 10^6/m^3 Cooling rate: 0.5K/s Want to know the solidification time, and the final microstructure. Use symmetry, around 2 million elements are required to solve microstructure evolution using the meso scale, which takes 24 hours.

22. Materials Process Design and Control Laboratory DATA GENERATION Adiabatic Cooling rate 0.4K/s, 0.6K/s Heyn’s interception measure mean of interception length is a function of angle s1 (grain size): mean of interception length for all angles s2 (structure type): ratio of smallest mean interception length to largest mean interception length For each point, extract information of (R, s1, s2) For each point every 100 time step, extract information of (T, fl). Remember from which run case and which position the data is generated. It will be useful to give an idea what (s1, s2) means. 2 hours for each case

23. Materials Process Design and Control Laboratory OBTAINED MICROSTRUCTURE INFORMATION Microstructure type (s2) Volume fraction Grain size (s1) Solid Liquid Fine Coarse Columnar Equiaxed

24. Materials Process Design and Control Laboratory COMPARE WITH MESO SCALE SIMULATION Volume fraction Predicted structure at P Q

25. Materials Process Design and Control Laboratory COMPARE WITH STATISTICAL RESULT Microstructure type (s2) Grain size (s1) Fine Coarse Columnar Equiaxed Meso scale modelData base approach

26. Materials Process Design and Control Laboratory Lever Rule Scheil Rule COMPARE WITH MACRO SCALE MODEL Solidification time ~420s Data base approach: ~423s Both macro modeling (with Lever rule or Scheil rule) and the data base approach gives approximate total solidification time. The “true” solution (meso-scale simulation result) is about 428s. Liquid fraction Lever Rule Database Volume fraction

27. Materials Process Design and Control Laboratory COMPUTATION REQUIREMENT StrategyComputational workAccuracy Meso scale model (Adaptive meshing 24 hours Supposed to be the most accurate Data base approach Data: 4 hours Macro scale simulation : 5min Close to the meso scale model Macro scale model 5 min Total solidification time correct. No information about microstructure can be given.

28. Materials Process Design and Control Laboratory THANK YOU FOR YOUR ATTENTION