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Phase Field Models Versus Parametric Front Tracking Methods: Are They Accurate and Computationally Efficient?

Published online by Cambridge University Press:  03 June 2015

John W. Barrett ,
Harald Garcke  and
Robert Nürnberg
John W. Barrett*
Affiliation:
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK
Harald Garcke*
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
Robert Nürnberg*
Affiliation:
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK
*
Email:j.barrett@imperial.ac.uk
Corresponding author.Email:harald.garcke@mathematik.uni-regensburg.de
Email:robert.nurnberg@imperial.ac.uk
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Abstract

We critically compare the practicality and accuracy of numerical approximations of phase field models and sharp interface models of solidification. Here we focus on Stefan problems, and their quasi-static variants, with applications to crystal growth. New approaches with a high mesh quality for the parametric approximations of the resulting free boundary problems and new stable discretizations of the anisotropic phase field system are taken into account in a comparison involving benchmark problems based on exact solutions of the free boundary problem.

Keywords

35K5535R3535R3753C4465M1265M5065M6074E1074E1574N2080A2282C26Phase field modelsparametric sharp interface methodsStefan problemanisotropysolidificationcrystal growthnumerical simulationsbenchmark problems

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Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 2 , February 2014 , pp. 506 - 555
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Al-Rawahi, N. and Tryggvason, G., Numerical simulation of dendritic solidification with convection: Three-dimensional flow, J. Comput. Phys., 194 (2004), 677–696.CrossRefGoogle Scholar
[2]Alikakos, N. D., Bates, P. W. and Chen, X., Convergence of the Cahn-Hilliard equation to the Hele–Shaw model, Arch. Rational Mech. Anal., 128 (1994), 165–205.Google Scholar
[3]Almgren, R. F., Variational algorithms and pattern formation in dendritic solidification, J. Comput. Phys., 106 (1993), 337–354.CrossRefGoogle Scholar
[4]Almgren, R. F., Second-order phase field asymptotics for unequal conductivities, SIAM J. Appl. Math., 59 (1999), 2086–2107.Google Scholar
[5]Baňas, L’. and Nürnberg, R., Finite element approximation of a three dimensional phase field model for void electromigration, J. Sci. Comp., 37 (2008), 202–232.Google Scholar
[6]Bänsch, E. and Schmidt, A., A finite element method for dendritic growth, in Taylor, J. E., editor, Computational Crystal Growers Workshop, pages 16–20, AMS Selected Lectures in Mathematics (1992).Google Scholar
[7]Barrett, J. W., Blowey, J. F. and Garcke, H., Finite element approximation of the Cahn–Hilliard equation with degenerate mobility, SIAM J. Numer. Anal., 37 (1999), 286–318.CrossRefGoogle Scholar
[8]Barrett, J. W., Garcke, H. and Nürnberg, R., On the variational approximation of combined second and fourth order geometric evolution equations, SIAM J. Sci. Comput., 29 (2007), 1006–1041.Google Scholar
[9]Barrett, J. W., Garcke, H. and Nürnberg, R., A parametric finite element method for fourth order geometric evolution equations1, J. Comput. Phys., 222 (2007), 441–462.Google Scholar
[10]Barrett, J. W.,H. Garcke and Nürnberg, R., Numerical approximation of anisotropic geometric evolution equations in the plane, IMA J. Numer. Anal., 28 (2008), 292–330.Google Scholar
[11]Barrett, J. W., Garcke, H. and Nürnberg, R., On the parametric finite element approximation of evolving hypersurfaces in R3, J. Comput. Phys., 227 (2008), 4281–4307.Google Scholar
[12]Barrett, J. W., Garcke, H. and Nürnberg, R., A variational formulation of anisotropic geometric evolution equations in higher dimensions, Numer. Math., 109 (2008), 1–44.Google Scholar
[13]Barrett, J. W., Garcke, H. and Nürnberg, R., On stable parametric finite element methods for the Stefan problem and the Mullins–Sekerka problem with applications to dendritic growth, J. Comput. Phys., 229 (2010), 6270–6299.CrossRefGoogle Scholar
[14]Barrett, J. W., Garcke, H. and Nürnberg, R., Numerical computations of faceted pattern formation in snow crystal growth, Phys. Rev. E, 86 (20121), 011604.Google Scholar
[15]Barrett, J. W., Garcke, H. and Nürnberg, R., Finite element approximation of one-sided Stefan problems with anisotropic, approximately crystalline, Gibbs–Thomson law, Adv. Differential Equations, 18 (2013), 383–432.CrossRefGoogle Scholar
[16]Barrett, J. W., Garcke, H. and Nürnberg, R., On the stable discretization of strongly anisotropic phase field models with applications to crystal growth, ZAMM Z. Angew. Math. Mech., (2013), (DOI: 10.1002/zamm.201200147, to appear).CrossRefGoogle Scholar
[17]Barrett, J. W., Garcke, H. and Nürnberg, R., Stable phase field approximations of anisotropic solidification, IMA J. Numer. Anal., (2013), (DOI: 10.1093/imanum/drt044, to appear).Google Scholar
[18]Bartels, S. and Müller, R., A posteriori error controlled local resolution of evolving interfaces for generalized Cahn-Hilliard equations, Interfaces Free Bound., 12 (2010), 45–73.Google Scholar
[19]Bates, P. W., Chen, X. and Deng, X., A numerical scheme for the two phase Mullins–Sekerka problem, Electron. J. Differential Equations, 1995 (1995), 1–28.Google Scholar
[20]Ben-Jacob, E., From snowflake formation to growth of bacterial colonies. Part I. Diffusive patterning in azoic systems, Contemp. Phys., 34 (1993), 247–273.Google Scholar
[21]Blank, L., Butz, M. and Garcke, H., Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method, ESAIM Control Optim. Calc. Var., 17 (2011), 931–954.Google Scholar
[22]Blank, L., Garcke, H., Sarbu, L. and Styles, V., Primal-dual active set methods for Allen-Cahn variational inequalities with non-local constraints, Numer. Methods Partial Differential Equations, 29 (2013), 999–1030.Google Scholar
[23]Blowey, J. F. and Elliott, C. M., The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part II: Numerical analysis, European J. Appl. Math., 3 (1992), 147–179.Google Scholar
[24]Blowey, J. F. andElliott, C. M., A phase-field model with a double obstacle potential, in But-tazzo, G. and Visintin, A., editors, Motion by mean curvature and related topics (Trento, 1992), pages 1–22, de Gruyter, Berlin (1994).Google Scholar
[25]Boettinger, W. J., Warren, J. A., Beckermann, C. and Karma, A., Phase-field simulation of solidification, Annu. Rev. Mater. Res., 32 (2002), 163–194.Google Scholar
[26]Brochu, T. and Bridson, R., Robust topological operations for dynamic explicit surfaces, SIAM J. Sci. Comput., 31 (2009), 2472–2493.Google Scholar
[27]Caginalp, G., An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205–245.Google Scholar
[28]Caginalp, G., Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations, Phys. Rev. A (3), 39 (1989), 5887–5896.Google Scholar
[29]Caginalp, G. and Chen, X., Convergence of the phase field model to its sharp interface limits, European J. Appl. Math., 9 (1998), 417–445.Google Scholar
[30]Caginalp, G., Chen, X. and Eck, C., Numerical tests of a phase field model with second order accuracy, SIAM J. Appl. Math., 68 (2008), 1518–1534.Google Scholar
[31]Caginalp, G. and Lin, J.-T., A numerical analysis of an anisotropic phase field model, IMA J. Appl. Math., 39 (1987), 51–66.Google Scholar
[32]Cahn, J. W. and Hoffman, D. W., A vector thermodynamics for anisotropic surfaces – II. Curved and faceted surfaces, Acta Metall., 22 (1974), 1205–1214.CrossRefGoogle Scholar
[33]Chen, L.-Q., Phase-field models for microstructure evolution, Annu. Rev. Mater. Res., 32 (2002), 113–140.CrossRefGoogle Scholar
[34]Chen, X., Caginalp, G. and Eck, C., A rapidly converging phase field model, Discrete Contin. Dynam. Systems, 15 (2006), 1017–1034.CrossRefGoogle Scholar
[35]Chen, Z. M. and Hoffmann, K.-H., An error estimate for a finite-element scheme for a phase field model, IMA J. Numer. Anal., 14 (1994), 243–255.Google Scholar
[36]Collins, J. B. and Levine, H., Diffuse interface model of diffusion-limited crystal growth, Phys. Rev. B, 31 (1985), 6119–6122.Google Scholar
[37]Davis, S. H., Theory of Solidification, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge (2001).Google Scholar
[38]Deckelnick, K., Dziuk, G. and Elliott, C. M., Computation of geometric partial differential equations and mean curvature flow, Acta Numer., 14 (2005), 139–232.Google Scholar
[39]Dziuk, G., An algorithm for evolutionary surfaces, Numer. Math., 58 (1991), 603–611.Google Scholar
[40]Dziuk, G. and Elliott, C. M., Finite elements on evolving surfaces, IMA J. Numer. Anal., 27 (2007), 262–292.CrossRefGoogle Scholar
[41]Dziuk, G. and Elliott, C. M., Surface finite elements for parabolic equations, J. Comput. Math., 25 (2007), 385–407.Google Scholar
[42]Elliott, C. M., Approximation of curvature dependent interface motion, in Duff, I. S. and Watson, G. A., editors, The state of the art in numerical analysis (York, 1996), vol. 63 of Inst. Math. Appl. Conf. Ser. New Ser., pages 407–440, Oxford Univ. Press, New York (1997).Google Scholar
[43]Elliott, C. M. and Gardiner, A. R., Double obstacle phase field computations of dendritic growth (1996), university of Sussex CMAIA Research report 96-19, http://homepages. warwick.ac.uk/staff/C.M.Elliott/PAPERS/DoubleObstaclePhaseField/EllGar96.pdf.Google Scholar
[44]Elliott, C. M. and Stuart, A. M., The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal., 30 (1993), 1622–1663.Google Scholar
[45]Feng, X. and Prohl, A., Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits, Math. Comp., 73 (2004), 541–567.Google Scholar
[46]Fix, G. J. and Lin, J. T., Numerical simulations of nonlinear phase transitions. I. The isotropic case, Nonlinear Anal., 12 (1988), 811–823.Google Scholar
[47]Garcke, H. and Stinner, B., Second order phase field asymptotics for multi-component systems, Interfaces Free Bound., 8 (2006), 131–157.Google Scholar
[48]Garcke, H., Stoth, B. and Nestler, B., Anisotropy in multi-phase systems: a phase field approach, Interfaces Free Bound., 1 (1999), 175–198.Google Scholar
[49]Gräser, C., Kornhuber, R. and Sack, U., Time discretizations of anisotropic Allen-Cahn equations, IMA J. Numer. Anal., (2013), (DOI: 10.1093/imanum/drs043, to appear).Google Scholar
[50]Gurtin, M. E., Multiphase thermomechanics with interfacial structure. 1. Heat conduction and the capillary balance law, Arch. Rational Mech. Anal., 104 (1988), 195–221.Google Scholar
[51]Gurtin, M. E., Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York (1993).Google Scholar
[52]Hou, T. Y., Lowengrub, J. S. and Shelley, M. J., Removing the stiffness from interfacial flows with surface tension, J. Comput. Phys., 114 (1994), 312–338.CrossRefGoogle Scholar
[53]Ihle, T. and Müller-Krumbhaar, H., Diffusion-limited fractal growth morphology in thermo-dynamical two-phase systems, Phys. Rev. Lett., 70 (1993), 3083–3086.CrossRefGoogle Scholar
[54]Juric, D. and Tryggvason, G., A front-tracking method for dendritic solidification, J. Comput. Phys., 123 (1996), 127–148.Google Scholar
[55]Karma, A. and Rappel, W.-J., Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics, Phys. Rev. E, 53 (1996), R3017–R3020.Google Scholar
[56]Karma, A. and Rappel, W.-J., Quantitative phase-field modeling of dendritic growth in two and three dimensions, Phys. Rev. E, 57 (1998), 4323–4349.CrossRefGoogle Scholar
[57]Kessler, D. A., Koplik, J. and Levine, H., Numerical simulation of two-dimensional snowflake growth, Phys. Rev. A, 30 (1984), 2820–2823.Google Scholar
[58]Kobayashi, R., Modeling and numerical simulations of dendritic crystal growth, Phys. D, 63 (1993), 410–423.CrossRefGoogle Scholar
[59]Langer, J. S., Models of pattern formation in first-order phase transitions, in Directions in condensed matter physics, vol. 1 of World Sci. Ser. Dir. Condensed Matter Phys., pages 165– 186, World Sci. Publishing, Singapore (1986).Google Scholar
[60]Lin, J. T., The numerical analysis of a phase field model in moving boundary problems, SIAM J. Numer. Anal., 25 (1988), 1015–1031.Google Scholar
[61]Luckhaus, S., Solutions for the two-phase Stefan problem with the Gibbs–Thomson law for the melting temperature, European J. Appl. Math., 1 (1990), 101–111.Google Scholar
[62]McFadden, G. B., Phase-field models of solidification, in Recent advances in numerical methods for partial differential equations and applications (Knoxville, TN, 2001), vol. 306 of Con-temp. Math., pages 107–145, Amer. Math. Soc., Providence, RI (2002).Google Scholar
[63]Mikula, K. and Ševčovič, D., Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM J. Appl. Math., 61 (2001), 1473–1501.Google Scholar
[64]Mullins, W. W. and Sekerka, R. F., Morphological stability of a particle growing by diffusion or heat flow, J. Appl. Phys., 34 (1963), 323–329.Google Scholar
[65]Nestler, B., A 3D parallel simulator for crystal growth and solidification in complex alloy systems, J. Cryst. Growth, 275 (2005), e273–e278.Google Scholar
[66]Nochetto, R. H. and Verdi, C., Combined effect of explicit time-stepping and quadrature for curvature driven flows, Numer. Math., 74 (1996), 105–136.Google Scholar
[67]Nochetto, R. H. and Verdi, C., Convergence past singularities for a fully discrete approximation of curvature-driven interfaces, SIAM J. Numer. Anal., 34 (1997), 490–512.CrossRefGoogle Scholar
[68]Nochetto, R. H. and Walker, S. W., A hybrid variational front tracking-level set mesh generator for problems exhibiting large deformations and topological changes, J. Comput. Phys., 229 (2010), 6243–6269.Google Scholar
[69]Osher, S. and Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces, vol. 153 of Applied Mathematical Sciences, Springer-Verlag, New York (2003).Google Scholar
[70]Penrose, O. and Fife, P. C., Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Phys. D, 43 (1990), 44–62.Google Scholar
[71]Provatas, N., Goldenfeld, N. and Dantzig, J., Efficient computation of dendritic microstructures using adaptive mesh refinement, Phys. Rev. Lett., 80 (1998), 3308–3311.Google Scholar
[72]Roosen, A. R. and Taylor, J. E., Modeling crystal growth in a diffusion field using fully faceted interfaces, J. Comput. Phys., 114 (1994), 113–128.CrossRefGoogle Scholar
[73]Schmidt, A., Die Berechnung dreidimensionaler Dendriten mit Finiten Elementen, Ph.D. thesis, University Freiburg, Freiburg (1993).Google Scholar
[74]Schmidt, A., Computation of three dimensional dendrites with finite elements, J. Comput. Phys., 195 (1996), 293–312.Google Scholar
[75]Schmidt, A., Approximation of crystalline dendrite growth in two space dimensions, in Kaçur, J. and Mikula, K., editors, Proceedings of the Algoritmy’97 Conference on Scientific Computing (Zuberec), vol. 67, Slovak University of Technology, Bratislava (1998) pages 57–68.Google Scholar
[76]Sethian, J. A., Level Set Methods and Fast Marching Methods, Cambridge University Press, Cambridge (1999).Google Scholar
[77]Singer-Loginova, I. and Singer, H. M., The phase field technique for modeling multiphase materials, Rep. Progr. Phys., 71 (2008), 106501.CrossRefGoogle Scholar
[78]Soner, H. M., Convergence of the phase-field equations to the Mullins–Sekerka problem with kinetic undercooling, Arch. Rational Mech. Anal., 131 (1995), 139–197.Google Scholar
[79]Steinbach, I., Phase-field models in materials science, Modelling Simul. Mater. Sci. Eng., 17 (2009), 073001.Google Scholar
[80]Stoth, B. E. E., Convergence of the Cahn–Hilliard equation to the Mullins–Sekerka problem in spherical symmetry, J. Differential Equations, 125 (1996), 154–183.Google Scholar
[81]Strain, J., A boundary integral approach to unstable solidification, J. Comput. Phys., 85 (1989), 342–389.Google Scholar
[82]Veeser, A., Error estimates for semi-discrete dendritic growth, Interfaces Free Bound., 1 (1999), 227–255.Google Scholar
[83]Wang, S.-L., Sekerka, R. F., Wheeler, A. A., Murray, B. T., Coriell, S. R., Braun, R. J. and McFadden, G. B., Thermodynamically-consistent phase-field models for solidification, Phys. D, 69 (1993), 189–200.Google Scholar
[84]Wheeler, A. A., Murray, B. T. and Schaefer, R. J., Computation of dendrites using a phase field model, Phys. D, 66 (1993), 243–262.Google Scholar
[85]Yue, X. Y., Finite element analysis of the phase field model with nonsmooth initial data, Acta Math. Appl. Sinica, 19 (1996), 15–24.Google Scholar
[86]Zhao, P., Heinrich, J. C. and Poirier, D. R., Numerical simulation of crystal growth in three dimensions using a sharp-interface finite element method, Internat. J. Numer. Methods En-grg., 71 (2007), 25–46.Google Scholar