See [BNP98A]
for a description of these
crystalline evolution examples.
ODE type solver
A first set of simulations where obtained by using
the crystalline evolution law first given by
J. Taylor,
under the assumption that faces do non
break.
Of course such evolution can be adapted even in case of
breaking if the breaking lines are known in advance.
Reaction diffusion approximation
A second set of simulations where obtained (oct 1998- ) by
using the Allen-Cahn type approach described in
[Pao95]
in 3D.
The code is written in Fortran90 by F.
Pasquarelli.
The space discretization is obtained by a structured tetraedral
grid of a cubic domain large enough to contain the initial
surface.
Graphics obtained by postprocessing output data with Geomview and Povray.
Cubic Wulff shape
- Ode solver.
Example:
evolution of an L-shape with a predefined breaking.
- Allen Cahn approx.
This example shows a comparison
between a simulation using the Allen-Cahn approach with the exact
solution (obtained with the ode solver.
The initial L-shaped surface is described in
[BNP98A].
The domain for the Allen-Cahn solver is [-1.4,-0.1]x[-1.4,1.4]x[-1.4,0.4]
with 70, 150, 95 subdivisions
in the three coordinate directions, time step is 0.00001,
The Allen-Cahn singular perturbation parameter is 0.02.
Hexagonal Wulff shape
- The Wulff shape
- Ode solver.
- Allen-Cahn approximation.
- Example 1:
Preliminary simulations of
evolution of the prism as described in
[BNP98A].
The domain for the Allen-Cahn approximation
is [-1.2,1.2]3 with 80, 80, 100 subdivisions
in the three coordinate directions, time step is 0.00005,
Allen-Cahn singular perturbation parameter is 0.05.
- Example 2:
In this second simulation we fully exploit the reflection
symmetries of the initial surface, which is now changed to
be symmetric with respect to the (x-y) plane.
Moreover, the Wulff shape
is also modified
in order to emphasize motion in the y direction.
The resolution domain is
[0,1.5]x[0,1.5]x[0,2.2] with 16, 159, 239 subdivisions,
time step is 1.08e-6,
Allen-Cahn singular perturbation parameter is 0.02.
Pictures obtained with the Ray-tracing free software
POV-RAY.
