On the Fully Implicit Solution of a Phase-Field Model for Binary Alloy Solidification in Three Dimensions (Advances in Applied Mathematics and Mechanics, 2021)
A fully implicit numerical method, based upon a combination ofadaptively refined hierarchical meshes and geometric multigrid, ispresented for the simulation of binary alloy solidification inthree space dimensions. The computational techniques are presented for aparticular mathematical model, based upon the phase-field approach,however, their applicability is of greater generality than for thespecific phase-field model used here. In particular, an implicitsecond order time discretization is combined with the use of second orderspatial differences to yield a large nonlinear system of algebraicequations as each time step. It is demonstrated that these equationsmay be solved reliably and efficiently through the use of a nonlinearmultigrid scheme for locally refined grids. In effect, this paperpresents an extension of earlier research in two spacedimensions (J. Comput. Phys., 225 (2007), pp. 1271-1287) to fullythree-dimensional problems. This extension is validated against earliertwo-dimensional results and against some of the limited resultsavailable in three dimensions, obtained using an explicit scheme.The efficiency of the implicit approach and the multigrid solver arethen demonstrated and some sample computational results for thesimulation of the growth of dendrite structures are presented.