Volume 4, Issue 6
On the Fully Implicit Solution of a Phase-Field Model for Binary Alloy Solidification in Three Dimensions

Christopher E. Goodyer, Peter K. Jimack, Andrew M. Mullis, Hongbiao Dong & Yu Xie

Adv. Appl. Math. Mech., 4 (2012), pp. 665-684.

Published online: 2012-12

Preview Full PDF 104 1037
Export citation
  • 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.

  • Keywords

Phase-field simulations binary alloys mesh adaptivity implicit methods nonlinear multigrid

  • AMS Subject Headings

65M06 65M22 65H10 65M55

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
Copy to clipboard
The citation has been copied to your clipboard