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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 and 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), 797-823. doi: 10.1137/0732037.

[2]

V. Berti and M. Fabrizio, A non-isothermal Ginzburg-Landau model in superconductivity: Existence, uniqueness and asymptotic behaviour, Nonlin. Anal., 66 (2007), 2565-2578. doi: 10.1016/j.na.2006.03.039.

[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), 035002. doi: 10.1088/0965-0393/19/3/035002.

[4]

C.-G. Canuto, M.-Y. Hussaini, A. Quarteroni and T.-A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, 2006.

[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), 15752-15756. doi: 10.1103/PhysRevB.50.15752.

[6]

L.-Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations, Comp. Phys. Comm., 108 (1998), 147-158. doi: 10.1016/S0010-4655(97)00115-X.

[7]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics, 37, John-Wiley, New York, 1994.

[8]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7.

[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), 703-730. doi: 10.1017/S030821050000408X.

[10]

C.-M. Elliott and S.-M. Zheng, On the Cahn-Hilliard equation, Arch. Rat. Mech. Anal., 96 (1986), 339-357. doi: 10.1007/BF00251803.

[11]

G.-S. Ganot, Laser Crystallization of Silicon Thin Films for Three-Dimensional Integrated Circuits, Ph.D. thesis, Colmbia University, 2012.

[12]

D. Gilbarg and N.-S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 2001.

[13]

J.-K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, 1988.

[14]

A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, Masson, Paris, 1991.

[15]

A. Haraux and M.-A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asymptot. Anal., 26 (2001), 21-36.

[16]

S.-Z. Huang, Gradient Inequalities, with Applications to Asymptotic Behavior and Stability of Gradient-Like Systems, Mathematical Surveys and Monographs 126, AMS, 2006. doi: 10.1090/surv/126.

[17]

M. A. Jendoubi, A simple unified approach to some convergence theorem of L. Simon, J. Func. Anal., 153 (1998), 187-202. doi: 10.1006/jfan.1997.3174.

[18]

A.-K. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput., 26 (2005), 1214-1233. doi: 10.1137/S1064827502410633.

[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), 2491-2502. doi: 10.1016/S1359-6454(02)00078-2.

[20]

C.-E. Krill and L.-Q. Chen, Computer simulation of 3-D grain growth using a phasefield model, Acta Mat., 50 (2002), 3057-3073.

[21]

D. Kinderlehrer and C. Liu, Evolution of grain boundaries, Math. Models Methods Appl. Sci., 11 (2001), 713-729. doi: 10.1142/S0218202501001069.

[22]

R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth, Physica D, 63 (1993), 410-423. doi: 10.1016/0167-2789(93)90120-P.

[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), 396-413. doi: 10.1016/j.jcp.2013.05.040.

[24]

N. Moelans, B. Blanpain and P. Wollants, An introduction to phase-field modeling of microstructure evolution, Comput. Coupling Phase Diagr. Thermochem., 32 (2008), 268-294. doi: 10.1016/j.calphad.2007.11.003.

[25]

J. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, 2001. doi: 10.1007/978-94-010-0732-0.

[26]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Science, 68, Springer-Verlag, 1988. doi: 10.1007/978-1-4684-0313-8.

[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), 6035-6058. doi: 10.1016/S1359-6454(03)00388-4.

[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), 125-153. doi: 10.1142/S0218202507001851.

[29]

X. Ye, The Fourier collocation method for the Cahn-Hilliard equation, Comp. Math. Appl., 44 (2002), 213-229. doi: 10.1016/S0898-1221(02)00142-6.

[30]

S.-M. Zheng, Nonlinear Evolution Equations, Pitman series Monographs and Survey in Pure and Applied Mathematics, 133, Chapman Hall/CRC, Boca Raton, Florida, 2004. doi: 10.1201/9780203492222.

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), 797-823. doi: 10.1137/0732037.

[2]

V. Berti and M. Fabrizio, A non-isothermal Ginzburg-Landau model in superconductivity: Existence, uniqueness and asymptotic behaviour, Nonlin. Anal., 66 (2007), 2565-2578. doi: 10.1016/j.na.2006.03.039.

[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), 035002. doi: 10.1088/0965-0393/19/3/035002.

[4]

C.-G. Canuto, M.-Y. Hussaini, A. Quarteroni and T.-A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, 2006.

[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), 15752-15756. doi: 10.1103/PhysRevB.50.15752.

[6]

L.-Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations, Comp. Phys. Comm., 108 (1998), 147-158. doi: 10.1016/S0010-4655(97)00115-X.

[7]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics, 37, John-Wiley, New York, 1994.

[8]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7.

[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), 703-730. doi: 10.1017/S030821050000408X.

[10]

C.-M. Elliott and S.-M. Zheng, On the Cahn-Hilliard equation, Arch. Rat. Mech. Anal., 96 (1986), 339-357. doi: 10.1007/BF00251803.

[11]

G.-S. Ganot, Laser Crystallization of Silicon Thin Films for Three-Dimensional Integrated Circuits, Ph.D. thesis, Colmbia University, 2012.

[12]

D. Gilbarg and N.-S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 2001.

[13]

J.-K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, 1988.

[14]

A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, Masson, Paris, 1991.

[15]

A. Haraux and M.-A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asymptot. Anal., 26 (2001), 21-36.

[16]

S.-Z. Huang, Gradient Inequalities, with Applications to Asymptotic Behavior and Stability of Gradient-Like Systems, Mathematical Surveys and Monographs 126, AMS, 2006. doi: 10.1090/surv/126.

[17]

M. A. Jendoubi, A simple unified approach to some convergence theorem of L. Simon, J. Func. Anal., 153 (1998), 187-202. doi: 10.1006/jfan.1997.3174.

[18]

A.-K. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput., 26 (2005), 1214-1233. doi: 10.1137/S1064827502410633.

[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), 2491-2502. doi: 10.1016/S1359-6454(02)00078-2.

[20]

C.-E. Krill and L.-Q. Chen, Computer simulation of 3-D grain growth using a phasefield model, Acta Mat., 50 (2002), 3057-3073.

[21]

D. Kinderlehrer and C. Liu, Evolution of grain boundaries, Math. Models Methods Appl. Sci., 11 (2001), 713-729. doi: 10.1142/S0218202501001069.

[22]

R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth, Physica D, 63 (1993), 410-423. doi: 10.1016/0167-2789(93)90120-P.

[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), 396-413. doi: 10.1016/j.jcp.2013.05.040.

[24]

N. Moelans, B. Blanpain and P. Wollants, An introduction to phase-field modeling of microstructure evolution, Comput. Coupling Phase Diagr. Thermochem., 32 (2008), 268-294. doi: 10.1016/j.calphad.2007.11.003.

[25]

J. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, 2001. doi: 10.1007/978-94-010-0732-0.

[26]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Science, 68, Springer-Verlag, 1988. doi: 10.1007/978-1-4684-0313-8.

[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), 6035-6058. doi: 10.1016/S1359-6454(03)00388-4.

[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), 125-153. doi: 10.1142/S0218202507001851.

[29]

X. Ye, The Fourier collocation method for the Cahn-Hilliard equation, Comp. Math. Appl., 44 (2002), 213-229. doi: 10.1016/S0898-1221(02)00142-6.

[30]

S.-M. Zheng, Nonlinear Evolution Equations, Pitman series Monographs and Survey in Pure and Applied Mathematics, 133, Chapman Hall/CRC, Boca Raton, Florida, 2004. doi: 10.1201/9780203492222.

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