| Abstract: | A fully implicit numerical method, based upon a combination of
adaptively refined hierarchical meshes and geometric multigrid, is
presented for the simulation of binary alloy solidification in
three space dimensions. The computational techniques are presented for a
particular mathematical model, based upon the phase-field approach,
however, their applicability is of greater generality than for the
specific phase-field model used here. In particular, an implicit
second order time discretization is combined with the use of second order
spatial differences to yield a large nonlinear system of algebraic
equations as each time step. It is demonstrated that these equations
may be solved reliably and efficiently through the use of a nonlinear
multigrid scheme for locally refined grids. In effect, this paper
presents an extension of earlier research in two space
dimensions (J. Comput. Phys., 225 (2007), pp. 1271-1287) to fully
three-dimensional problems. This extension is validated against earlier
two-dimensional results and against some of the limited results
available in three dimensions, obtained using an explicit scheme.
The efficiency of the implicit approach and the multigrid solver are
then demonstrated and some sample computational results for the
simulation of the growth of dendrite structures are presented. |
