Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T21:26:59.708Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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
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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ben Amar, M. and Brener, E. A., Theory of pattern selection in 3-dimensional nonaxisym-metric dendritic growth, Phys. Rev. Lett., 71 (1993), pp. 589–592.Google Scholar
[2]Athreya, B. P. and Dantzig, J. A., in Solidification Process and Microstructures: A Symposium in Honour of Wilfred Kurz, edited by Rappaz, M., Beckermann, C. and Trivedi, R., TMS, Warrendale PA, (2004), pp. 357–368.Google Scholar
[3]Baines, M. J., Hubbard, M. E., Jimack, P. K. and Mahmood, R., A moving-mesh finite element method and its application to the numerical solution of phase-change problems, Commun. Comput. Phys., 6 (2009), pp. 595–624.Google Scholar
[4]Barbieri, A. and Langer, J. S., Predictions of dendritic growth-rates in the linearized solvability theory, Phys. Rev. A, 39 (1989), pp. 5314–5325.Google Scholar
[5]Brandt, A., Multi-level adaptive solutions to boundary value problems, Math. Comput., 31 (1977), pp. 333–390.Google Scholar
[6]Briggs, W. L., A Multigrid Tutorial, second edition, SIAM, 2000.CrossRefGoogle Scholar
[7]Browne, D. J. and Hunt, J. D., A fixed grid front-tracking model of the growth of a columnar front and an equiaxed grain during solidification of an alloy, Numer. Heat Trans. B, 45 (2004), pp. 395–419.Google Scholar
[8]Caginalp, G., Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations, Phys. Rev. A, 39 (1989), pp. 5887–5896.CrossRefGoogle ScholarPubMed
[9]Chen, S., Merriman, B., Osher, S. and Smereka, P., A simple level set method for solving Stefan problems, J. Comput. Phys., 135 (1997), pp. 8–29.Google Scholar
[10]Ciobanas, A. I., Fautrelle, Y. and Batraretu, F., Bianchi, A. M. and Noppel, A., in Modelling of Casting Welding and Advanced Solidification Processing XI, edited by Gandin, Ch. A. and Bellet, M., TMS, Warrendale PA, (2006), pp. 299–306.Google Scholar
[11]Dorr, M. R., Fattebert, J. L., Wickett, M. E., Belak, J. F. and Turchi, P. E. A., A numerical algorithm for the solution of a phase-field model of polycrystalline materials, J. Comput. Phys., 229 (2010), pp. 626–641.CrossRefGoogle Scholar
[12]Emmerich, H., Advances of and by phase-field modelling in condensed-matter physics, Adv. Phys., 57 (2008), pp. 1–87.CrossRefGoogle Scholar
[13]Gandin, Ch.-A. and Rappaz, M., A coupled finite-element cellular-automaton model for the prediction of dendritic grain structures in solidification processes, Acta Mater., 42 (1994), pp. 2233–2246.Google Scholar
[14]George, W. L. and Warren, J. A., A parallel 3D dendritic growth simulator using the phase-field method, J. Comput. Phys., 177 (2002), pp. 264–283.CrossRefGoogle Scholar
[15]Gibbs, J. W., A method of geometrical representation of the thermodynamic properties of substances by means of surfaces, T. Conn. Acad., 2 (1873), pp. 382–404.Google Scholar
[16]Goodyer, C. E. and Berzins, M., Adaptive timestepping for elastohydrodynamic lubrication solvers, SIAM J. Sci. Comput., 28 (2006), pp. 626–650.Google Scholar
[17]Green, J. R., Jimack, P. K., Mullis, A. M. and Rosam, J., An adaptive, multilevel scheme for the implicit solution of three-dimensional phase-field equations, Numer. Meth. Partial Differential Eq., 27 (2011), pp. 106–120.Google Scholar
[18]Hou, T. Y., Li, Z., Osher, S. and Zhao, H., A hybrid method for moving interface problems with application to the Hele-Shaw flow, J. Comput. Phys., 134 (1997), pp. 236–252.Google Scholar
[19]Hundsdorfer, W. and Verwer, J. G., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer-Verlag, 2003.Google Scholar
[20]Rosam, J., A fully Implicit, Fully Adaptive Multigrid Method for Multiscale Phase-Field Modelling, PhD Thesis, University of Leeds, (2007).Google Scholar
[21]Jeong, J. H., Goldenfeld, N. and Dantzig, J., Phase field model for three-dimensional dendritic growth with fluid flow, Phys. Rev. E, 64 (2001), 041602.Google Scholar
[22]Karma, A. and Rappel, W.-J., Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics, Phys. Rev. E, 53 (1996), pp. R3017–R3020.CrossRefGoogle ScholarPubMed
[23]Karma, A. and Rappel, W.-J., Numerical simulation of three-dimensional dendritic growth, Phys. Rev. Lett., 77 (1996), pp. 4050–4053.CrossRefGoogle ScholarPubMed
[24]Karma, A. and Rappel, W.-J., Quantitative phase-field modeling of dendritic growth in two and three dimensions, Phys. Rev. E, 57 (1998), pp. 4323–4349.Google Scholar
[25]Karma, A., Phase-field formulation for quantitative modeling of alloy solidification, Phys. Rev. Lett., 87 (2001), 115701.Google Scholar
[26]Kessler, D. A., Koplik, J. and Levine, H., Pattern selection in fingered growth phenomena, Adv. Phys., 37 (1988), pp. 255–339.Google Scholar
[27]Kim, Y. T., Goldenfeld, N. and Dantzig, J., Computation of dendritic microstructures using a level set method, Phys. Rev. E, 62 (2000), pp. 2471–2474.Google Scholar
[28]Kobayashi, R., Modeling and numerical simulations of dendritic crystal-growth, Phys. D, 63 (1993), pp. 410–423.CrossRefGoogle Scholar
[29]Langer, J. S., Directions in Condensed Matter Physics (eds. Grinstein, G. and Mazenko, G.), World Scientific Publishing: Singapore, 1986, pp. 164–186.Google Scholar
[30]Lin, H. K., Chen, C. C. and Lan, C. W., Adaptive three-dimensional phase-field modeling of dendritic crystal growth with high anisotropy, J. Cryst. Growth, 318 (2011), pp. 51–54.Google Scholar
[31]Macneice, P., Olson, K. M., Mobarry, C., Defainchtein, R. and Packer, C., PARAMESH: A parallel adaptive mesh refinement community toolkit, Comput. Phys. Commun., 126 (2000), pp. 330–354.Google Scholar
[32]Martorano, M. A., Beckermann, C. and Gandin, Ch.-A., A solutal interaction mechanism for the columnar-to-equiaxed transition in alloy solidification, Metall. Mater. Trans. A, 34 (2003), pp. 1657–1674.Google Scholar
[33]Martorano, M. A. and Biscuola, V. B., Columnar front tracking algorithm for prediction of the columnar-to-equiaxed transition in two-dimensional solidification, Modell. Simul. Mater. Sci. Eng., 14 (2006), pp. 1225–1243.Google Scholar
[34]Meca, E. and Plapp, M., Phase-field study of the cellular bifurcation in dilute binary alloys, Metall. Mater. Trans. A, 38 (2007), pp. 1407–1416.Google Scholar
[35]Merriman, B., Bence, J. K. and Osher, S. J., Motion of multiple junctions-a level set approach, J. Comput. Phys., 112 (1994), pp. 334–363.Google Scholar
[36]Mullis, A. M., Rosam, J. and Jimack, P. K., Solute trapping and the effects of anti-trapping currents on phase-field models of coupled thermo-solutal solidification, J. Cryst. Growth, 312 (2010), pp. 1891–1897.CrossRefGoogle Scholar
[37]Olson, K. and Macneice, P., An over of the PARAMESH AMR software and some of its applications, in Adaptive Mesh Refinement-Theory and Applications, Proceedings of the Chicago Workshop on Adaptive Mesh Refinement Methods, Series: Lecture Notes in Computational Science and Engineering, eds. Plewa, T., Linde, T. and Weirs, G. (Berlin: Springer), (2005).CrossRefGoogle Scholar
[38]Olson, K., Paramesh, : A Parallel Adaptive Grid Tool, In Parallel Computational Fluid Dynamics 2005: Theory and Applications, ed. Deane, A.et al. (Elsevier), 2006.Google Scholar
[39]Penrose, O. and Fife, P. C., Thermodynamically consistent models of phase-field type for the kinetics of phase-transitions, Phys. D, 43 (1990), pp. 44–62.CrossRefGoogle Scholar
[40]Pomeau, Y. and Ben-Amar, M., Solids far from equilibrium, Cambridge University Press, 1992.Google Scholar
[41]Provatas, N., Goldenfeld, N. and Dantzig, J., Adaptive mesh refinement computation of solidification microstructures using dynamic data structures, J. Comput. Phys., 148 (1999), pp. 265–290.Google Scholar
[42]Provatas, N., Greenwood, M., Athreya, B., Goldenfeld, N. and Dantzig, J., Mul-tiscale modelling of solidification: phase-field methods to adaptive mesh refinement, Int. J. Mod. Phys. B, 19 (2005), pp. 4525–4565.Google Scholar
[43]Pusztai, T., Tegze, G., Toth, G. I., Kornyei, L., Bansel, G., Fan, Z. and Granasy, L., Phase-field approach to polycrystalline solidification including heterogeneous and homogeneous nucleation, J. Phys. Condens. Matter, 20 (2008), 404205.Google Scholar
[44]Qiao, Z., Zhang, Z. and Tang, T., An adaptive time-stepping strategy for the molecular beam epitaxy models, SIAM J. Sci. Comput., 33 (2011), pp. 1395–1414.Google Scholar
[45]Ramirez, J. C. and Beckermann, C., Examination of binary alloy free dendritic growth theories with a phase-field model, Acta Mater., 53 (2005), pp. 1721–1736.Google Scholar
[46]Ramirez, J. C., Beckermann, C., Karma, A. and H.-J. DIEPERS, Phase-field modeling of binary alloy solidification with coupled heat and solute diffusion, Phys. Rev. E, 69 (2004), 051607.Google Scholar
[47]Rosam, J., Jimack, P. K. and Mullis, A. M., A fully implicit, fully adaptive time and space discretisation method for phase-field simulation of binary alloy solidification, J. Comput. Phys., 225 (2007), pp. 1271–1287.Google Scholar
[48]Rosam, J., Jimack, P. K. and Mullis, A. M., An adaptive, fully implicit multigrid phase-field model for the quantitative simulation of non-isothermal binary alloy solidification, Acta Mater., 56 (2008), pp. 4559–4569.Google Scholar
[49]Shampine, L. F., Implementation of implicit formulas for the solution of ODEs, SIAM J. Sci. Stat. Comput., 1 (1980), pp. 103–118.Google Scholar
[50]Tan, Z., Lim, K. M. and Khoo, B. C., An adaptive mesh redistribution method for the incompressible mixture flows using phase-field model, J. Comput. Phys., 225 (2007), pp. 1137–1158.CrossRefGoogle Scholar
[51]Touheed, N., Selwood, P., Jimack, P. K. and Berzins, M., A comparison of some dynamic load-balancing algorithms for a parallel adaptive flow solver, Para. Comput., 26 (2000). pp. 1535–1554.CrossRefGoogle Scholar
[52]Trottenberg, U., Oosterlee, C. and Schuller, A., Multigrid, Academic Press, 2001.Google Scholar
[53]Tsai, Y. L., Chen, C. C. and Lan, C. W., Three-dimensional adaptive phase field modeling of directional solidification of a binary alloy: 2D-3D transitions, Int. J. Heat Mass Trasf., 53 (2010), pp. 2272–2283.Google Scholar
[54]Vanherpe, L., Wendler, F., Nestler, B. and Vandewalle, S., A multigrid solver for phase field simulation of microstructure evolution, Math. Comput. Simul., 80 (2010), pp. 1438–1448.Google Scholar
[55]Wang, C. Y. and Beckermann, C., Prediction of columnar to equiaxed transition during diffusion-controlled dendritic alloy solidification, Metall. Mater. Trans. A, 25 (1994), pp. 1081–1093.Google Scholar
[56]Wang, H., Li, R. and Tang, T., Efficient computation of dendritic growth with r-adaptive finite element methods, J. Comput. Phys., 227 (2008), pp. 5984–6000.Google Scholar