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### Grain growth via the Potts model

The Potts model is essentially a many-spin Ising model, where one can
represent a multi-granular material as a large 2-d or 3-d lattice of
spin values. A group of conjoined same-spin pixels is a grain. The
grain "boundaries" between different-spin pixels are a high-energy
state that can be relaxed over time via the usual Monte Carlo spin
update rules to simulate grain-growth, grain boundary migration, or a
variety of other interesting physical effects. The energetics for all
these models are captured in the Hamiltonian one writes for the
collection of lattice points and their spins.

We worked with Liz Holm at Sandia to parallelize her Potts-model
grain-growth code. The basic idea is similar to how one would
parallelize an Ising model on a regular lattice. Each processor owns a
sub-section of the lattice and updates its spins. Care is taken to
insure Monte Carlo updates are made independently by each processor
without violating the detailed-balance principle that two (or more)
spins cannot be updated simultaneously if they interact with each
other. This requires "coloring" the lattice into sets of
non-interacting points. Then all lattice points of the same color can
be updated simultaneously on different processors. For example, in 2-d
on a square lattice with an 8-neighbor stencil (a lattice point has
energetic interactions with all 8 points surrounding it), 4 colors is
sufficient to enable 4 sweeps thru the lattice to update every lattice
point. Similary, for a 26-point stencil on a 3-d lattice, 8 colors are
used.

Collaborators on this project:

- Richard Fye, Sandia
- Liz Holm, Sandia

This paper describes the details of our parallelization approach, some
enhancements we made to the basic Metropolis update algorithm to
speed-up the calculations, and some applications of the Potts model to
grain-growth problems of interest to Sandia.

**Potts-model Grain Growth Simulations: Parallel Algorithms and
Applications**, S. A. Wright, S. J. Plimpton, T. P. Swiler, R. M. Fye,
M. F. Young, E. A. Holm, SAND Report 97-1925, August
1997. (abstract)