Abstract
We consider the numerical solution of a phase field model for polycrystallization in the solidification of binary mixtures in a domain \( \varOmega \subset \mathbb {R}^2\). The model is based on a free energy in terms of three order parameters: the local orientation \(\varTheta \) of the crystals, the local crystallinity \(\phi \), and the concentration c of one of the components of the binary mixture. The equations of motion are given by an initial-boundary value problem for a coupled system of partial differential equations consisting of a regularized second order total variation flow in \( \varTheta \), an \(L^2\) gradient flow in \(\phi \), and a \(W^{1,2}(\varOmega )^*\) gradient flow in c. Based on an implicit discretization in time by the backward Euler scheme, we suggest a splitting method such that the three semidiscretized equations can be solved separately and prove existence of a solution. As far as the discretization in space is concerned, the fourth order Cahn–Hilliard type equation in c is taken care of by a \(\hbox {C}^0\) Interior Penalty Discontinuous Galerkin approximation which has the advantage that the same finite element space can be used as well for the spatial discretization of the equations in \( \varTheta \) and \( \phi \). The fully discretized equations represent parameter dependent nonlinear algebraic systems with the discrete time as a parameter. They are solved by a predictor corrector continuation strategy featuring an adaptive choice of the time-step. Numerical results illustrate the performance of the suggested numerical method.
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References
Andreu, F., Caselles, V., Mazón, J.M.: Existence and uniqueness of solutions for a parabolic quasilinear problem for linear growth functionals with L1 data. Math. Ann. 322, 139–206 (2002)
Andreu-Vaillo, F., Caselles, V., Mazón, J.M.: Parabolic Quasilinear Equations Minimizing Linear Growth Functionals. Birkhäuser, Basel (2004)
Andreu, F., Mazòn, J.M., Segura, S., Toledo, J.: Existence and uniqueness for a degenerate parabolic equation with L1-data. Trans. Am. Math. Soc. 315, 285–306 (1999)
Baldi, A.: Weighted BV functions. Houst. J. Math. 27, 1–23 (2001)
Bartels, S.: Methods for Nonlinear Partial Differential Equations. Springer, Berlin (2015)
Bartkowiak, L., Pawlow, I.: The Cahn–Hilliard–Gurtin system coupled with elasticity. Control Cybern. 34, 1005–1043 (2005)
Bebendorf, M.: A note on the Poincaré inequality for convex domains. Z. Anal. Anwend. 22, 751–756 (2003)
Bellettini, G., Novaga, M., Paolini, M.: On a crystalline variational problem, part I: first variation and global L1-regularity. Arch. Ration. Mech. Anal. 157, 165–191 (2001)
Bellettini, G., Novaga, M., Paolini, M.: On a crystalline variational problem, part II: BV regularity and structure of minimizers on facets. Arch. Ration. Mech. Anal. 157, 193–217 (2001)
Bonetti, E., Colli, P., Dreyer, W., Giliardi, G., Schimperna, G., Sprekels, J.: On a model for phase separation in binary alloys driven by mechanical effects. Physica D 165, 48–65 (2002)
Braess, D., Hoppe, R.H.W., Linsenmann, C.: A two-energies principle for the biharmonic equation and an a posteriori error estimator for an interior penalty discontinuous Galerkin approximation. ESAIM: M2AN (2016). https://doi.org/10.1051/m2an/2016074
Brenner, S.C., Sung, L.-Y.: \(C^0\) interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22(23), 83–118 (2005)
Burger, M., Frick, K., Osher, S., Scherzer, O.: Inverse total variation flow. Multiscale Model. Simul. 6, 365–395 (2007)
Carrive, M., Miranville und, A., Piétrus, A.: The Cahn–Hilliard equation for deformable elastic continua. Adv. Math. Sci. Appl. 10, 539–569 (2000)
Carrive, M., Miranville, A., Piétrus, A., Rakotoson, J.: The Cahn-Hilliard equation for anisotropic deformable elastic continuum. Appl. Math. Lett. 12, 23–28 (1999)
Deuflhard, P.: Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms. Springer, Berlin (2004)
Engel, G., Garikipati, K., Hughes, T.J.R., Larson, M.G., Mazzei, L., Taylor, R.L.: Continuous/discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Eng. 191, 3669–3750 (2002)
Feng, X., von Oehsen, M., Prohl, A.: Rate of convergence of regularization procedures and finite element approximations for the total variation flow. Numer. Math. 100, 441–456 (2005)
Garcke, H.: On Cahn–Hilliard systems with elasticity. Proc. R. Soc. Edinb. Sect. A Math. 133, 307–331 (2003)
Garcke, H.: On a Cahn–Hilliard system for phase separation with elastic misfit. Ann. Inst. Henri Poincaré (C) Nonlinear Anal. 22, 165–185 (2005)
Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Basel (1984)
Gránásy, L., Börzsönyi, L., Pusztai, T.: Nucleation and bulk crystallization in binary phase field theory. Phys. Rev. Lett. 88, 206105 (2002)
Gránásy, L., Börzsönyi, L., Pusztai, T.: Crystal nucleation and growth in binary phase-field theory. J. Cryst. Growth 237, 1813–1817 (2002)
Gránásy, L., Pusztai, T., Börzsönyi, L., Warren, J.A., Douglas, J.F.: A general mechanism of polycrystalline growth. Nat. Mater. 3, 645–650 (2004)
Gránásy, L., Pusztai, T., Warren, J.A.: Modeling polycrystalline solidification using phase field theory. J. Phys. Condens. Matter 16, R1205–R1235 (2004)
Gránásy, L., Pusztai, T., Saylor, D., Warren, J.A.: Phase field theory of heterogeneous crystal nucleation. Phys. Rev. Lett. 98, 035703 (2007)
Gránásy, L., Pusztai, T., Tegze, G., Warren, J.A., Douglas, J.F.: Growth and form of spherulites. Phys. Rev. E 72, 011605 (2004)
Gránásy, L., Ratkai, L., Szallas, A., Korbuly, B., Toth, G., Környei, L., Pusztai, T.: Phase-field modeling of polycrystalline solidification: from needle crystals to spherulites: a review. Metall. Mater. Trans. A 45A, 1694–1719 (2014)
Gurtin, M.E.: Generalised Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance. Physica D 92, 178–192 (1996)
Hoppe, R.H.W., Linsenmann, C.: An adaptive Newton continuation strategy for the fully implicit finite element immersed boundary method. J. Comput. Phys. 231, 4676–4693 (2012)
Kobayashi, R., Warren, J.A., Carter, W.C.: A continuum model of grain boundaries. Phys. D Nonlinear Phenom. 140, 141–150 (2000)
Larché, F.C., Cahn, J.W.: The effect of self-stress on diffusion in solids. Acta Metall. 30, 1835–1845 (1982)
Larché, F.C., Cahn, J.W.: The interactions of composition and stress in crystalline solids. Acta Metall. 33, 331–357 (1985)
Larché, F.C., Cahn, J.W.: Phase changes in a thin plate with non-local self-stress effects. Acta Metall. 40, 947–955 (1992)
Leo, P.H., Lowengrub, J.S., Jou, H.J.: A diffuse interface model for microstructural evolution in elastically stressed solids. Acta Mater. 46, 2113–2130 (1998)
Miranville, A.: Some generalizations of Cahn–Hilliard equation. Asymptot. Anal. 22, 235–259 (2000)
Miranville, A.: Long-time behavior of some models of Cahn–Hilliard equations in deformable continua. Nonlinear Anal. Real World Appl. 2, 273–304 (2001)
Miranville, A.: Consistent models of Cahn–Hilliard–Gurtin equations with Neumann boundary conditions. Physica D 158, 233–257 (2001)
Miranville, A.: Generalized Cahn–Hilliard equations based on a microforce balance. J. Appl. Math. 4, 165–185 (2003)
Moll, S., Shirakawa, K.: Existence of solutions to the Kobayashi–Warren–Carter system. Calc. Var. Partial Differ. Equ. 51, 621–656 (2014)
Moll, S., Shirakawa, K., Watanabe, H.: Energy dissipative solutions to the Kobayashi–Warren–Carter system. Nonlinearity 30, 2752–2784 (2017)
Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960)
Provatas, N., Elder, K.: Phase-Field Methods in Materials Science. Wiley, Weinheim (2010)
Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1986)
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D Nonlinear Phenom. 60, 259–268 (1992)
Tartar, L.: Introduction to Sobolev Spaces and Interpolation Theory. Springer, Berlin (2007)
Warren, J.A., Kobayashi, R., Carter, W.C.: Modeling grain boundaries using a phase field technique. J. Cryst. Growth 211, 18–20 (2000)
Wells, G.N., Kuhl, E., Garikipati, K.: A discontinuous Galerkin method for the Cahn–Hilliard equation. J. Comput. Phys. 218, 860–877 (2006)
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Communicated by Gabriel Wittum.
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J. J. Winkle: The authors acknowledge support by the NSF Grant DMS-1520886.
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Hoppe, R.H.W., Winkle, J.J. Numerical solution of a phase field model for polycrystallization processes in binary mixtures. Comput. Visual Sci. 20, 13–27 (2019). https://doi.org/10.1007/s00791-018-00307-5
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DOI: https://doi.org/10.1007/s00791-018-00307-5