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On the Fully Implicit Solution of a Phase-Field Model for Binary Alloy Solidification in Three Dimensions

Published online by Cambridge University Press:  03 June 2015

Christopher E. Goodyer,
Peter K. Jimack,
Andrew M. Mullis,
Hongbiao Dong and
Yu Xie
Christopher E. Goodyer*
Affiliation:
School of Computing, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT, UK
Peter K. Jimack*
Affiliation:
School of Computing, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT, UK
Andrew M. Mullis*
Affiliation:
School of Process, Environmental and Materials Engineering, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT, UK
Hongbiao Dong*
Affiliation:
Department of Engineering, University of Leicester, University Road, Leicester, LE1 7RH, UK
Yu Xie*
Affiliation:
Department of Engineering, University of Leicester, University Road, Leicester, LE1 7RH, UK
*
URL: http://www.engineering.leeds.ac.uk/people/speme/staff/a.m.mullis, Email: c.e.goodyer@leeds.ac.uk
Corresponding author. Email: p.k.jimack@leeds.ac.uk
Email: a.m.mullis@leeds.ac.uk
Email: h.dong@le.ac.uk
Email: yx38@leicester.ac.uk
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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

65M0665M2265H1065M55Phase-field simulationsbinary alloysmesh adaptivityimplicit methodsnonlinear multigrid

Information

Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 6 , December 2012, pp. 665 - 684
Copyright
Copyright © Global-Science Press 2012

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