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{\large \bf Mesoscale simulation of grain growth}\\

David Kinderlehrer\\
Department of Mathematical Sciences\\
Carnegie Mellon University\\
Pittsburgh, PA  15213\\

Jeehyun Lee\\
Mathematics Department\\
Yonsei University\\
134 Sinchon-Dong, Seodaemum-Gu\\
Seoul, 463-928, South Korea\\

Irene Livshits\\
Department of mathematics
University of Central Arkansas
UCA Box 4912\\
201 Donaghey Avenue\\
Conway, AR  72032\\

Anthony Rollett\\
Department of Materials Science and Engineering\\
Carnegie Mellon University\\
Pittsburgh, PA  15213\\

and\\

Shlomo Ta'asan\\
Department of Mathematical Sciences\\
Carnegie Mellon University\\
Pittsburgh, PA  15213

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\noindent {\bf Abstract}: Simulation is becoming an increasingly
important tool, not only in materials science ina general way, but in
the study of grain growth in particular.  Here we exhibit a
consistent variational approach to the mesoscale simulation of large
systems of grain boundarie subject to Mullins Equation of curvature
driven growth.  Simulations must be accurate and at a scale large
enough to have statistical significance.  Moreover, they must be
sufficiently flexible to use very general energies and mobilities.  We
introduce this theory and its discretization as a dissipative system
in two and three dimensions.  The approach has several interesting
features.  It consists in solving very large systems of nonlinear
evolution equations with nonlinear boundary conditions at triple points
or on triple lines.  Critical events, the disappearance of grains and
the disapperance or exchange of edges, must be accommodated.  The data
structure is curves in two dimensions and surfaces in three
dimensions.  We discuss some consequences and challenges, including
some ideas about coarse graining the simulation.

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