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September  2014, 19(7): 2227-2246. doi: 10.3934/dcdsb.2014.19.2227

## Analysis and simulation for an isotropic phase-field model describing grain growth

 1 Institute of Mathematics, Technical University Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany 2 School of Mathematical Sciences, Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Han Dan Road 220, 200433 Shanghai, China

Received  March 2013 Revised  June 2013 Published  August 2014

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.
Citation: Maciek D. Korzec, Hao Wu. Analysis and simulation for an isotropic phase-field model describing grain growth. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2227-2246. doi: 10.3934/dcdsb.2014.19.2227
##### References:
 [1] U.-M. Ascher, S.-J. Ruuth and B. Wetton, Implicit-explicit methods for time-dependent partial differential equations,, SIAM J. Numer. Anal., 32 (1995), 797. doi: 10.1137/0732037. Google Scholar [2] V. Berti and M. Fabrizio, A non-isothermal Ginzburg-Landau model in superconductivity: Existence, uniqueness and asymptotic behaviour,, Nonlin. Anal., 66 (2007), 2565. doi: 10.1016/j.na.2006.03.039. Google Scholar [3] S. Bhattacharyya, T.-W. Heo, K. Chang and L.-Q. Chen, A phase-field model of stress effect on grain boundary migration,, Modelling Simul. Mater. Sci. Eng., 19 (2011). doi: 10.1088/0965-0393/19/3/035002. Google Scholar [4] C.-G. Canuto, M.-Y. Hussaini, A. Quarteroni and T.-A. Zang, Spectral Methods: Fundamentals in Single Domains,, Springer-Verlag, (2006). Google Scholar [5] L.-Q. Chen and W. Yang, 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 (1994), 15752. doi: 10.1103/PhysRevB.50.15752. Google Scholar [6] L.-Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations,, Comp. Phys. Comm., 108 (1998), 147. doi: 10.1016/S0010-4655(97)00115-X. Google Scholar [7] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations,, Research in Applied Mathematics, 37 (1994). Google Scholar [8] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbbR^3$,, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713. doi: 10.1016/S0764-4442(00)00259-7. Google Scholar [9] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703. doi: 10.1017/S030821050000408X. Google Scholar [10] C.-M. Elliott and S.-M. Zheng, On the Cahn-Hilliard equation,, Arch. Rat. Mech. Anal., 96 (1986), 339. doi: 10.1007/BF00251803. Google Scholar [11] G.-S. Ganot, Laser Crystallization of Silicon Thin Films for Three-Dimensional Integrated Circuits,, Ph.D. thesis, (2012). Google Scholar [12] D. Gilbarg and N.-S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (2001). Google Scholar [13] J.-K. Hale, Asymptotic Behavior of Dissipative Systems,, AMS, (1988). Google Scholar [14] A. Haraux, Systèmes Dynamiques Dissipatifs et Applications,, Masson, (1991). Google Scholar [15] A. Haraux and M.-A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity,, Asymptot. Anal., 26 (2001), 21. Google Scholar [16] S.-Z. Huang, Gradient Inequalities, with Applications to Asymptotic Behavior and Stability of Gradient-Like Systems,, Mathematical Surveys and Monographs 126, 126 (2006). doi: 10.1090/surv/126. Google Scholar [17] M. A. Jendoubi, A simple unified approach to some convergence theorem of L. Simon,, J. Func. Anal., 153 (1998), 187. doi: 10.1006/jfan.1997.3174. Google Scholar [18] A.-K. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff PDEs,, SIAM J. Sci. Comput., 26 (2005), 1214. doi: 10.1137/S1064827502410633. Google Scholar [19] A. Kazaryan, Y. Wang, S.-A. Dregia and B. R. Patton, Grain growth in anisotropic systems: Comparison of effects of energy and mobility,, Acta Mat., 50 (2002), 2491. doi: 10.1016/S1359-6454(02)00078-2. Google Scholar [20] C.-E. Krill and L.-Q. Chen, Computer simulation of 3-D grain growth using a phasefield model,, Acta Mat., 50 (2002), 3057. Google Scholar [21] D. Kinderlehrer and C. Liu, Evolution of grain boundaries,, Math. Models Methods Appl. Sci., 11 (2001), 713. doi: 10.1142/S0218202501001069. Google Scholar [22] R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth,, Physica D, 63 (1993), 410. doi: 10.1016/0167-2789(93)90120-P. Google Scholar [23] M.-D. Korzec and T. Ahnert, Time-stepping methods for the simulation of the self-assembly of nano-crystals in MATLAB on a GPU,, J. Comp. Phys., 251 (2013), 396. doi: 10.1016/j.jcp.2013.05.040. Google Scholar [24] N. Moelans, B. Blanpain and P. Wollants, An introduction to phase-field modeling of microstructure evolution,, Comput. Coupling Phase Diagr. Thermochem., 32 (2008), 268. doi: 10.1016/j.calphad.2007.11.003. Google Scholar [25] J. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge University Press, (2001). doi: 10.1007/978-94-010-0732-0. Google Scholar [26] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematical Science, 68 (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar [27] J.-A. Warren, R. Kobayashi, A.-E. Lobkovsky and W.-C. Carter, Extending phase field models of solidification to polycrystalline materials,, Acta Mat., 51 (2003), 6035. doi: 10.1016/S1359-6454(03)00388-4. Google Scholar [28] H. Wu, M. Grasselli and S. Zheng, Convergence to equilibrium for a parabolic-hyperbolic phase-field system with Neumann boundary conditions,, Math. Models Methods Appl. Sci., 17 (2007), 125. doi: 10.1142/S0218202507001851. Google Scholar [29] X. Ye, The Fourier collocation method for the Cahn-Hilliard equation,, Comp. Math. Appl., 44 (2002), 213. doi: 10.1016/S0898-1221(02)00142-6. Google Scholar [30] S.-M. Zheng, Nonlinear Evolution Equations,, Pitman series Monographs and Survey in Pure and Applied Mathematics, 133 (2004). doi: 10.1201/9780203492222. Google Scholar

show all references

##### References:
 [1] U.-M. Ascher, S.-J. Ruuth and B. Wetton, Implicit-explicit methods for time-dependent partial differential equations,, SIAM J. Numer. Anal., 32 (1995), 797. doi: 10.1137/0732037. Google Scholar [2] V. Berti and M. Fabrizio, A non-isothermal Ginzburg-Landau model in superconductivity: Existence, uniqueness and asymptotic behaviour,, Nonlin. Anal., 66 (2007), 2565. doi: 10.1016/j.na.2006.03.039. Google Scholar [3] S. Bhattacharyya, T.-W. Heo, K. Chang and L.-Q. Chen, A phase-field model of stress effect on grain boundary migration,, Modelling Simul. Mater. Sci. Eng., 19 (2011). doi: 10.1088/0965-0393/19/3/035002. Google Scholar [4] C.-G. Canuto, M.-Y. Hussaini, A. Quarteroni and T.-A. Zang, Spectral Methods: Fundamentals in Single Domains,, Springer-Verlag, (2006). Google Scholar [5] L.-Q. Chen and W. Yang, 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 (1994), 15752. doi: 10.1103/PhysRevB.50.15752. Google Scholar [6] L.-Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations,, Comp. Phys. Comm., 108 (1998), 147. doi: 10.1016/S0010-4655(97)00115-X. Google Scholar [7] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations,, Research in Applied Mathematics, 37 (1994). Google Scholar [8] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbbR^3$,, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713. doi: 10.1016/S0764-4442(00)00259-7. Google Scholar [9] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703. doi: 10.1017/S030821050000408X. Google Scholar [10] C.-M. Elliott and S.-M. Zheng, On the Cahn-Hilliard equation,, Arch. Rat. Mech. Anal., 96 (1986), 339. doi: 10.1007/BF00251803. Google Scholar [11] G.-S. Ganot, Laser Crystallization of Silicon Thin Films for Three-Dimensional Integrated Circuits,, Ph.D. thesis, (2012). Google Scholar [12] D. Gilbarg and N.-S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (2001). Google Scholar [13] J.-K. Hale, Asymptotic Behavior of Dissipative Systems,, AMS, (1988). Google Scholar [14] A. Haraux, Systèmes Dynamiques Dissipatifs et Applications,, Masson, (1991). Google Scholar [15] A. Haraux and M.-A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity,, Asymptot. Anal., 26 (2001), 21. Google Scholar [16] S.-Z. Huang, Gradient Inequalities, with Applications to Asymptotic Behavior and Stability of Gradient-Like Systems,, Mathematical Surveys and Monographs 126, 126 (2006). doi: 10.1090/surv/126. Google Scholar [17] M. A. Jendoubi, A simple unified approach to some convergence theorem of L. Simon,, J. Func. Anal., 153 (1998), 187. doi: 10.1006/jfan.1997.3174. Google Scholar [18] A.-K. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff PDEs,, SIAM J. Sci. Comput., 26 (2005), 1214. doi: 10.1137/S1064827502410633. Google Scholar [19] A. Kazaryan, Y. Wang, S.-A. Dregia and B. R. Patton, Grain growth in anisotropic systems: Comparison of effects of energy and mobility,, Acta Mat., 50 (2002), 2491. doi: 10.1016/S1359-6454(02)00078-2. Google Scholar [20] C.-E. Krill and L.-Q. Chen, Computer simulation of 3-D grain growth using a phasefield model,, Acta Mat., 50 (2002), 3057. Google Scholar [21] D. Kinderlehrer and C. Liu, Evolution of grain boundaries,, Math. Models Methods Appl. Sci., 11 (2001), 713. doi: 10.1142/S0218202501001069. Google Scholar [22] R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth,, Physica D, 63 (1993), 410. doi: 10.1016/0167-2789(93)90120-P. Google Scholar [23] M.-D. Korzec and T. Ahnert, Time-stepping methods for the simulation of the self-assembly of nano-crystals in MATLAB on a GPU,, J. Comp. Phys., 251 (2013), 396. doi: 10.1016/j.jcp.2013.05.040. Google Scholar [24] N. Moelans, B. Blanpain and P. Wollants, An introduction to phase-field modeling of microstructure evolution,, Comput. Coupling Phase Diagr. Thermochem., 32 (2008), 268. doi: 10.1016/j.calphad.2007.11.003. Google Scholar [25] J. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge University Press, (2001). doi: 10.1007/978-94-010-0732-0. Google Scholar [26] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematical Science, 68 (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar [27] J.-A. Warren, R. Kobayashi, A.-E. Lobkovsky and W.-C. Carter, Extending phase field models of solidification to polycrystalline materials,, Acta Mat., 51 (2003), 6035. doi: 10.1016/S1359-6454(03)00388-4. Google Scholar [28] H. Wu, M. Grasselli and S. Zheng, Convergence to equilibrium for a parabolic-hyperbolic phase-field system with Neumann boundary conditions,, Math. Models Methods Appl. Sci., 17 (2007), 125. doi: 10.1142/S0218202507001851. Google Scholar [29] X. Ye, The Fourier collocation method for the Cahn-Hilliard equation,, Comp. Math. Appl., 44 (2002), 213. doi: 10.1016/S0898-1221(02)00142-6. Google Scholar [30] S.-M. Zheng, Nonlinear Evolution Equations,, Pitman series Monographs and Survey in Pure and Applied Mathematics, 133 (2004). doi: 10.1201/9780203492222. Google Scholar
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