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

  • John W. Barrett (a1), Harald Garcke (a2) and Robert Nürnberg (a1)


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.


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[1]Al-Rawahi, N. and Tryggvason, G., Numerical simulation of dendritic solidification with convection: Three-dimensional flow, J. Comput. Phys., 194 (2004), 677–696.
[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.
[3]Almgren, R. F., Variational algorithms and pattern formation in dendritic solidification, J. Comput. Phys., 106 (1993), 337–354.
[4]Almgren, R. F., Second-order phase field asymptotics for unequal conductivities, SIAM J. Appl. Math., 59 (1999), 2086–2107.
[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.
[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).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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).
[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).
[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.
[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.
[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.
[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.
[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.
[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.
[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).
[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.
[26]Brochu, T. and Bridson, R., Robust topological operations for dynamic explicit surfaces, SIAM J. Sci. Comput., 31 (2009), 2472–2493.
[27]Caginalp, G., An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205–245.
[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.
[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.
[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.
[31]Caginalp, G. and Lin, J.-T., A numerical analysis of an anisotropic phase field model, IMA J. Appl. Math., 39 (1987), 51–66.
[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.
[33]Chen, L.-Q., Phase-field models for microstructure evolution, Annu. Rev. Mater. Res., 32 (2002), 113–140.
[34]Chen, X., Caginalp, G. and Eck, C., A rapidly converging phase field model, Discrete Contin. Dynam. Systems, 15 (2006), 1017–1034.
[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.
[36]Collins, J. B. and Levine, H., Diffuse interface model of diffusion-limited crystal growth, Phys. Rev. B, 31 (1985), 6119–6122.
[37]Davis, S. H., Theory of Solidification, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge (2001).
[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.
[39]Dziuk, G., An algorithm for evolutionary surfaces, Numer. Math., 58 (1991), 603–611.
[40]Dziuk, G. and Elliott, C. M., Finite elements on evolving surfaces, IMA J. Numer. Anal., 27 (2007), 262–292.
[41]Dziuk, G. and Elliott, C. M., Surface finite elements for parabolic equations, J. Comput. Math., 25 (2007), 385–407.
[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).
[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.
[44]Elliott, C. M. and Stuart, A. M., The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal., 30 (1993), 1622–1663.
[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.
[46]Fix, G. J. and Lin, J. T., Numerical simulations of nonlinear phase transitions. I. The isotropic case, Nonlinear Anal., 12 (1988), 811–823.
[47]Garcke, H. and Stinner, B., Second order phase field asymptotics for multi-component systems, Interfaces Free Bound., 8 (2006), 131–157.
[48]Garcke, H., Stoth, B. and Nestler, B., Anisotropy in multi-phase systems: a phase field approach, Interfaces Free Bound., 1 (1999), 175–198.
[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).
[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.
[51]Gurtin, M. E., Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York (1993).
[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.
[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.
[54]Juric, D. and Tryggvason, G., A front-tracking method for dendritic solidification, J. Comput. Phys., 123 (1996), 127–148.
[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.
[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.
[57]Kessler, D. A., Koplik, J. and Levine, H., Numerical simulation of two-dimensional snowflake growth, Phys. Rev. A, 30 (1984), 2820–2823.
[58]Kobayashi, R., Modeling and numerical simulations of dendritic crystal growth, Phys. D, 63 (1993), 410–423.
[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).
[60]Lin, J. T., The numerical analysis of a phase field model in moving boundary problems, SIAM J. Numer. Anal., 25 (1988), 1015–1031.
[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.
[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).
[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.
[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.
[65]Nestler, B., A 3D parallel simulator for crystal growth and solidification in complex alloy systems, J. Cryst. Growth, 275 (2005), e273–e278.
[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.
[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.
[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.
[69]Osher, S. and Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces, vol. 153 of Applied Mathematical Sciences, Springer-Verlag, New York (2003).
[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.
[71]Provatas, N., Goldenfeld, N. and Dantzig, J., Efficient computation of dendritic microstructures using adaptive mesh refinement, Phys. Rev. Lett., 80 (1998), 3308–3311.
[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.
[73]Schmidt, A., Die Berechnung dreidimensionaler Dendriten mit Finiten Elementen, Ph.D. thesis, University Freiburg, Freiburg (1993).
[74]Schmidt, A., Computation of three dimensional dendrites with finite elements, J. Comput. Phys., 195 (1996), 293–312.
[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.
[76]Sethian, J. A., Level Set Methods and Fast Marching Methods, Cambridge University Press, Cambridge (1999).
[77]Singer-Loginova, I. and Singer, H. M., The phase field technique for modeling multiphase materials, Rep. Progr. Phys., 71 (2008), 106501.
[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.
[79]Steinbach, I., Phase-field models in materials science, Modelling Simul. Mater. Sci. Eng., 17 (2009), 073001.
[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.
[81]Strain, J., A boundary integral approach to unstable solidification, J. Comput. Phys., 85 (1989), 342–389.
[82]Veeser, A., Error estimates for semi-discrete dendritic growth, Interfaces Free Bound., 1 (1999), 227–255.
[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.
[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.
[85]Yue, X. Y., Finite element analysis of the phase field model with nonsmooth initial data, Acta Math. Appl. Sinica, 19 (1996), 15–24.
[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.


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

  • John W. Barrett (a1), Harald Garcke (a2) and Robert Nürnberg (a1)


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