zbMATH — the first resource for mathematics

Analysis and simulation for an isotropic phase-field model describing grain growth. (English) Zbl 1304.35094
Summary: A phase-field system of coupled Allen-Cahn type PDEs describing grain growth is analyzed and simulated. In the periodic setting, we prove the existence and uniqueness of global weak solutions to the problem. Then we investigate the long-time behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. Namely, the problem possesses a global attractor as well as an exponential attractor, which entails that the global attractor has finite fractal dimension. Moreover, we show that each trajectory converges to a single equilibrium. A time-adaptive numerical scheme based on trigonometric interpolation is presented. It allows to track the approximated long-time behavior accurately and leads to a convergence rate. The scheme exhibits a physically consistent discrete free energy dissipation.
MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 35K57 Reaction-diffusion equations 74H40 Long-time behavior of solutions for dynamical problems in solid mechanics 74N20 Dynamics of phase boundaries in solids 35B41 Attractors
Full Text:
References:
 [1] U.-M. Ascher, Implicit-explicit methods for time-dependent partial differential equations,, SIAM J. Numer. Anal., 32, 797, (1995) · Zbl 0841.65081 [2] V. Berti, A non-isothermal Ginzburg-Landau model in superconductivity: Existence, uniqueness and asymptotic behaviour,, Nonlin. Anal., 66, 2565, (2007) · Zbl 1119.82045 [3] S. Bhattacharyya, A phase-field model of stress effect on grain boundary migration,, Modelling Simul. Mater. Sci. Eng., 19, (2011) [4] C.-G. Canuto, Spectral Methods: Fundamentals in Single Domains,, Springer-Verlag, (2006) · Zbl 1093.76002 [5] L.-Q. Chen, Computer simulation of the domain dynamics of a quenched system with a large number of nonconserved order parameters: The grain-growth kinetics,, Phys. Rev. B, 50, 15752, (1994) [6] L.-Q. Chen, Applications of semi-implicit Fourier-spectral method to phase field equations,, Comp. Phys. Comm., 108, 147, (1998) · Zbl 1017.65533 [7] A. Eden, Exponential Attractors for Dissipative Evolution Equations,, Research in Applied Mathematics, 37, (1994) · Zbl 0842.58056 [8] M. Efendiev, Exponential attractors for a nonlinear reaction-diffusion system in $$\mathbbR^3$$,, C. R. Acad. Sci. Paris Sér. I Math., 330, 713, (2000) · Zbl 1151.35315 [9] M. Efendiev, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems,, Proc. Roy. Soc. Edinburgh Sect. A, 135, 703, (2005) · Zbl 1088.37005 [10] C.-M. Elliott, On the Cahn-Hilliard equation,, Arch. Rat. Mech. Anal., 96, 339, (1986) · Zbl 0624.35048 [11] G.-S. Ganot, Laser Crystallization of Silicon Thin Films for Three-Dimensional Integrated Circuits,, Ph.D. thesis, (2012) [12] D. Gilbarg, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (2001) · Zbl 1042.35002 [13] J.-K. Hale, Asymptotic Behavior of Dissipative Systems,, AMS, (1988) · Zbl 0642.58013 [14] A. Haraux, Systèmes Dynamiques Dissipatifs et Applications,, Masson, (1991) · Zbl 0726.58001 [15] A. Haraux, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity,, Asymptot. Anal., 26, 21, (2001) · Zbl 0993.35017 [16] S.-Z. Huang, Gradient Inequalities, with Applications to Asymptotic Behavior and Stability of Gradient-Like Systems,, Mathematical Surveys and Monographs 126, 126, (2006) · Zbl 1132.35002 [17] M. A. Jendoubi, A simple unified approach to some convergence theorem of L. Simon,, J. Func. Anal., 153, 187, (1998) · Zbl 0895.35012 [18] A.-K. Kassam, Fourth-order time-stepping for stiff PDEs,, SIAM J. Sci. Comput., 26, 1214, (2005) · Zbl 1077.65105 [19] A. Kazaryan, Grain growth in anisotropic systems: Comparison of effects of energy and mobility,, Acta Mat., 50, 2491, (2002) [20] C.-E. Krill, Computer simulation of 3-D grain growth using a phasefield model,, Acta Mat., 50, 3057, (2002) [21] D. Kinderlehrer, Evolution of grain boundaries,, Math. Models Methods Appl. Sci., 11, 713, (2001) · Zbl 1036.74041 [22] R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth,, Physica D, 63, 410, (1993) · Zbl 0797.35175 [23] M.-D. Korzec, Time-stepping methods for the simulation of the self-assembly of nano-crystals in MATLAB on a GPU,, J. Comp. Phys., 251, 396, (2013) · Zbl 1349.82160 [24] N. Moelans, An introduction to phase-field modeling of microstructure evolution,, Comput. Coupling Phase Diagr. Thermochem., 32, 268, (2008) [25] J. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge University Press, (2001) · Zbl 1026.37500 [26] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematical Science, 68, (1988) · Zbl 0662.35001 [27] J.-A. Warren, Extending phase field models of solidification to polycrystalline materials,, Acta Mat., 51, 6035, (2003) [28] H. Wu, Convergence to equilibrium for a parabolic-hyperbolic phase-field system with Neumann boundary conditions,, Math. Models Methods Appl. Sci., 17, 125, (2007) · Zbl 1120.35024 [29] X. Ye, The Fourier collocation method for the Cahn-Hilliard equation,, Comp. Math. Appl., 44, 213, (2002) · Zbl 1034.65085 [30] S.-M. Zheng, Nonlinear Evolution Equations,, Pitman series Monographs and Survey in Pure and Applied Mathematics, 133, (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.