Abstract
As an alternative to the sharp interface formulation, the phase field approach is a widely used technique for modeling phase transitions. The governing system of reaction-diffusion equations captures the instability of the underlying physical problem and is capable of modeling the evolution of complicated crystal shapes during solidification of an undercooled melt. For its numerical solution, we propose our novel anti-diffusive multipoint flux approximation (MPFA) finite volume scheme on a Cartesian mesh. The scheme is verified against the analytical solution of the modified sharp interface model. Experimental order of convergence (EOC) is measured for the temperature field in the usual norms. In addition, EOC is also obtained for the phase interface through approximating the volume of the symmetric difference of the solid phase subdomains. In the anisotropic cases including unusual higher order symmetries, computational studies with various settings also confirm convergence of our MPFA scheme which is faster than in the case of the reference finite volume scheme with 2nd order flux approximation.
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References
Allen, S., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1084–1095 (1979)
Barrett, J.W., Garcke, H., 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, 6270–6299 (2010)
Bellettini, G., Paolini, M.: Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Math. J. 25(3), 537–566 (1996)
Beneš, M.: Anisotropic phase-field model with focused latent-heat release. In: FREE BOUNDARY PROBLEMS: Theory and Applications II, GAKUTO International Series in Mathematical Sciences and Applications, vol. 14, pp. 18–30 (2000)
Beneš, M.: Mathematical analysis of phase-field equations with numerically efficient coupling terms. Interface. Free. Bound. 3, 201–221 (2001)
Beneš, M.: Mathematical and computational aspects of solidification of pure substances. Acta Math. Univ. Comenianae 70(1), 123–151 (2001)
Beneš, M.: Diffuse-interface treatment of the anisotropic mean-curvature flow. Appl. Math-Czech. 48(6), 437–453 (2003)
Beneš, M.: Computational studies of anisotropic diffuse interface model of microstructure formation in solidification. Acta Math. Univ. Comenianae 76, 39–59 (2007)
Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley (2003)
Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: P.G. Ciarlet, J.L. Lions (eds.) Handbook of Numerical Analysis, vol. 7, pp. 715–1022. Elsevier (2000)
Green, J.R., Jimack, P.K., Mullis, A.M., Rosam, J.: An adaptive, multilevel scheme for the implicit solution of three-dimensional phase-field equations. Numer. Meth. Part. D. E. 27, 106–120 (2010)
Gurtin, M.E.: Thermomechanics of Evolving Phase Boundaries in the Plane. Oxford Mathematical Monographs. Oxford University Press (1993)
Karma, A., Rappel, W.J.: Numerical simulation of three-dimensional dendritic growth. Phys. Rev. Lett. 77(19), 4050–4053 (1996)
Karma, A., Rappel, W.J.: Quantitative phase-field modeling of dendritic growth in two and three dimensions. Phys. Rev. E 57(4), 4 (1998)
Kupferman, R., Shochet, O., Ben-Jacob, E.: Numerical study of a morphology diagram in the large undercooling limit using a phase-field model. Phys. Rev. E. 50(2), 1005–1008 (1993)
McFadden, G.B., Wheeler, A.A., Braun, R.J., Coriell, S.R.: Phase-field models for anisotropic interfaces. Phys. Rev. E 48(3), 2016–2024 (1993)
Meirmanov, A.M.: The Stefan Problem. De Gruyter Expositions in Mathematics. Walter de Gruyter (1992)
Mullis, A.M., Cochrane, R.F.: A phase field model for spontaneous grain refinement in deeply undercooled metallic melts. Acta Mater. 49, 2205–2214 (2001)
PunKay, M.: Modeling of anisotropic surface energies for quantum dot formation and morphological evolution. In: NNIN REU Research Accomplishments, pp. 116–117. University of Michigan (2005)
R. E. Napolitano, S.L.: Three-dimensional crystal-melt Wulff-shape and interfacial stiffness in the Al-Sn binary system. Phys. Rev. B 70(21), 214,103 (2004)
Rice, J.R., Mu, M.: An experimental performance analysis for the rate of convergence of 5-point star on general domains. Tech. rep., Department of Computer Sciences, Purdue University (1988)
Schiesser, W.E.: The Numerical Method of Lines: Integration of Partial Differential Equations. Academic Press, San Diego (1991)
Schmidt, A.: Computation of three dimensional dendrites with finite elements. J. Comput. Phys. 125, 293–3112 (1996)
Singer, H.M., Singer-Loginova, I., Bilgam, J.H., Amberg, G.: Morphology diagram of thermal dendritic solidification by means of phase-field models in two and three dimensions. J. Cryst. Growth. 296, 58–68 (2006)
Strachota, P., Beneš, M.: A multipoint flux approximation finite volume scheme for solving anisotropic reaction-diffusion systems in 3D. In: J. Fořt, J. Fürst, J. Halama, R. Herbin, F. Hubert (eds.) Finite Volumes for Complex Applications VI - Problems & Perspectives, pp. 741–749. Springer (2011). DOI 10.1007/978-3-642-20671-9_78
Strachota, P., Beneš, M., Tintěra, J.: Towards clinical applicability of the diffusion-based DT-MRI visualization algorithm. J. Vis. Commun. Image R. 23(2), 387–396 (2012). DOI 10.1016/j.jvcir.2011.11.009
Acknowledgements
This work was partially supported by the following projects: The project of the Ministry of Education of the Czech Republic MSM6840770010 Applied Mathematics in Technical and Physical Sciences. The Grant Agency of the Czech Technical University in Prague, grant No. SGS11/161/OHK4/3T/14.
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Strachota, P., Beneš, M. (2013). Design and Verification of the MPFA Scheme for Three-Dimensional Phase Field Model of Dendritic Crystal Growth. In: Cangiani, A., Davidchack, R., Georgoulis, E., Gorban, A., Levesley, J., Tretyakov, M. (eds) Numerical Mathematics and Advanced Applications 2011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33134-3_49
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