1. Towards a New, Accurate, and Efficient Approach for Simulating Grain Growth
Emanuel (Menachem) Lazar, Princeton
Robert D. MacPherson, IAS
David J. Srolovitz, Yeshiva
Computational Materials Science Network
Northeastern University
April 2-3, 2008

2. Background

3. Background
Von Neumann 1952
Mullins 1956

4. Surface Evolver Program
Minimizes surfaces under constraints
Tracks vertices, edges, faces, and bodies
Used for grain growth simulations
Begin with Voronoi tessellations

5. Evolver Data

6. Refining edges and relaxing

7. Evolver Data with Relaxed Edges

8. Algorithm Design Devise efficient algorithm satisfying vNM law
Approach:
Localize LHS
Localize RHS

9. Method Localizing curvature for free body
Integral of curvature is the turning angle, a αi

10. Method Locale area changes for discretized shape
Area change of triangle = area change of body
Move vertex so that v

11. Method αi

12. Method Real grains have neighbors and thus also triple junctions.
Points along edges we can move as above.
How do we move triple junctions?

13. Method Method Move triple points to satisfy No ambiguities
Works even when
angle at triple junction
is not a=p/3 αi

14. Putting this all together…
i – vertices along edges
j – triple points
n – number of neighbors

15. Exact von Neumann Algorithm

17. Generalize vNM to 3d MacPherson and Srolovitz (Nature 2007)
Isotropic boundary properties

18. Mean Width Integral of mean curvature
If we discretize a grain shape in 3d as an arbitrary polyhedron:

19. Localizing Localize LHS Localize RHS
The key is to insure that the angles along the triple lines are properly represented

20. Localizing

21. Localizing

22. Single Step in 3d Grain Growth Simulation using Surface Evolver

23. Single Step in 3d Grain Growth Simulation with Exact von Neumann Algorithm

24. Conclusions
Using exact vNM relation in any dimension allows us to perform simulations on a much coarser mesh but still “exactly” satisfy TJ BCs
Easily implemented in Surface Evolver
Future:
Asymptotic grain structures
Appropriate metrics for describing correlations in microstructure

25. Thank you
Ken Brakke and Dan Lewis