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
Background
Background Von Neumann 1952 Mullins 1956
Surface Evolver Program Minimizes surfaces under constraints Tracks vertices, edges, faces, and bodies Used for grain growth simulations Begin with Voronoi tessellations
Evolver Data
Refining edges and relaxing
Evolver Data with Relaxed Edges
Algorithm Design Devise efficient algorithm satisfying vNM law Approach: Localize LHS Localize RHS
Method Localizing curvature for free body Integral of curvature is the turning angle, a αi
Method Locale area changes for discretized shape Area change of triangle = area change of body Move vertex so that v
Method αi
Method Real grains have neighbors and thus also triple junctions. Points along edges we can move as above. How do we move triple junctions?
Method Method Move triple points to satisfy No ambiguities Works even when angle at triple junction is not a=p/3 αi
Putting this all together… i – vertices along edges j – triple points n – number of neighbors
Exact von Neumann Algorithm
Generalize vNM to 3d MacPherson and Srolovitz (Nature 2007) Isotropic boundary properties
Mean Width Integral of mean curvature If we discretize a grain shape in 3d as an arbitrary polyhedron:
Localizing Localize LHS Localize RHS The key is to insure that the angles along the triple lines are properly represented
Localizing
Localizing
Single Step in 3d Grain Growth Simulation using Surface Evolver
Single Step in 3d Grain Growth Simulation with Exact von Neumann Algorithm
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
Thank you Ken Brakke and Dan Lewis