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<CENTER><A HREF = "main.html">Return to Steve Plimpton's home page</A>
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<H3>Grain growth via the Potts model
</H3>
<P>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.
</P>
<P>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.
</P>
<P>Collaborators on this project:
</P>
<UL><LI>  Richard Fye, Sandia
<LI>  Liz Holm, Sandia
</UL>
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<P>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.
</P>
<P><B>Potts-model Grain Growth Simulations: Parallel Algorithms and
Applications</B>, S.  A. Wright, S. J. Plimpton, T. P. Swiler, R. M. Fye,
M. F. Young, E. A. Holm, SAND Report 97-1925, August
1997. (<A HREF = "abstracts/sand97.html">abstract</A>)
</P>


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