Analysis and optimal control of some solidification processes

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October 2014, 34(10): 3985-4017. doi: 10.3934/dcds.2014.34.3985

Analysis and optimal control of some solidification processes

Roberto C. Cabrales ^1, , Gema Camacho ^2, and Enrique Fernández-Cara ^2,

1.	Departamento de Ciencias Básicas, Universidad del Bío-Bío, Campus Fernando May, Casilla 447, Chillán, Chile
2.	Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain

Received October 2012 Revised January 2013 Published April 2014

Abstract
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In this paper we consider a mathematical model that describes the solidification of a binary alloy. We prove some existence and uniqueness results for a regularized problem, depending on a small parameter $\epsilon$. We also analyze the behavior of the regularized solutions as $\epsilon \to 0$. Then, we consider some associated optimal control problems. We prove existence and optimality results and we present and discuss some iterative methods.

Keywords: weak solutions, Nonlinear PDEs, regularization, optimal control., solidification models.

Mathematics Subject Classification: Primary: 93C20, 35Q93, 49K20, 35A01, 35A02; Secondary: 76D0.

Citation: Roberto C. Cabrales, Gema Camacho, Enrique Fernández-Cara. Analysis and optimal control of some solidification processes. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 3985-4017. doi: 10.3934/dcds.2014.34.3985

References:

[1]	R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975.
[2]	W. J. Boettinger, J. A. Warren, C. Beckermann and A. Karma, Phase-field simulation of solidification, Annual Review of Materials Research, 32 (2002), 163-194. doi: 10.1007/BF02648953.
[3]	J. Ni and C. Beckermann, A volume-averaged two-phase model for solidication transportphenomena, Metallurgical Transactions B, 22 (1991), 349-361.
[4]	W. D. Bennon and F. P. Incropera, A Continuum Model for Momentum, Heat and Species Transport in Binary Solid-Liquid Phase Change Systems: I. Model Formulation, Int. J. Heat Mass Transfer, 30 (1987), 2161-2170. doi: 10.1016/0017-9310(87)90094-9.
[5]	W. D. Bennon and F. P. Incropera, A Continuum Model for Momentum, Heat and Species Transport in Binary Solid Liquid Phase Change Systems: II. Application to Solidification in a Rectangular Cavity, Int. J. Heat Mass Transfer, 30 (1987), 2171-2187. doi: 10.1016/0017-9310(87)90095-0.
[6]	Ph. Blanc, L. Gasser and J. Rappaz, Existence for a stationary model of binary alloy solidification, RAIRO Modél. Math. Anal. Numér., 29 (1995), 687-699.
[7]	J. L. Boldrini and G. Planas, A tridimensional phase-field model with convection for phase change of an alloy, Discrete Contin. Dyn. Syst., 13 (2005), 429-450. doi: 10.3934/dcds.2005.13.429.
[8]	P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.
[9]	M. Durán, E. Ortega-Torres and J. Rappaz, Weak solution of a stationary convection-diffusion model describing binary alloy solidification processes, Math. Comput. Modelling, 42 (2005), 1269-1286. doi: 10.1016/j.mcm.2005.01.035.
[10]	M. Durán and E. Ortega-Torres, Existence of weak solutions for a non-homogeneous solidification model, Electron. J. Differential Equations, 2006 17 pp.
[11]	I. Ekeland and R. Temam, Convex analysis and Variational Problems, Society for Industrial and Applied Mathematics, Philadelphia, 1999. doi: 10.1137/1.9781611971088.
[12]	J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach, Encyclopedia of Mathematics and its Applications, 117, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511721595.
[13]	O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltzeva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R.I., 1967.
[14]	G. Planas and J. L. Boldrini, A bidimensional phase-field model with convection for change phase of an alloy, J. Math. Anal. Appl., 303 (2005), 669-687. doi: 10.1016/j.jmaa.2004.08.068.
[15]	J. C. Ramirez and C. Beckermann, Examination of binary alloy free dendritic growth theories with a phase-field model, Acta Materialia, 53 (2005), 1721-1736. doi: 10.1016/j.actamat.2004.12.021.
[16]	J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differ. Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X.
[17]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, 146 (1986), 64-96. doi: 10.1007/BF01762360.
[18]	J. Simon, A constructive proof of a theorem of G. de Rham, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 1167-1172.
[19]	R. Temam, Navier Stokes Equations. Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001.

show all references

References:

[1]	R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975.
[2]	W. J. Boettinger, J. A. Warren, C. Beckermann and A. Karma, Phase-field simulation of solidification, Annual Review of Materials Research, 32 (2002), 163-194. doi: 10.1007/BF02648953.
[3]	J. Ni and C. Beckermann, A volume-averaged two-phase model for solidication transportphenomena, Metallurgical Transactions B, 22 (1991), 349-361.
[4]	W. D. Bennon and F. P. Incropera, A Continuum Model for Momentum, Heat and Species Transport in Binary Solid-Liquid Phase Change Systems: I. Model Formulation, Int. J. Heat Mass Transfer, 30 (1987), 2161-2170. doi: 10.1016/0017-9310(87)90094-9.
[5]	W. D. Bennon and F. P. Incropera, A Continuum Model for Momentum, Heat and Species Transport in Binary Solid Liquid Phase Change Systems: II. Application to Solidification in a Rectangular Cavity, Int. J. Heat Mass Transfer, 30 (1987), 2171-2187. doi: 10.1016/0017-9310(87)90095-0.
[6]	Ph. Blanc, L. Gasser and J. Rappaz, Existence for a stationary model of binary alloy solidification, RAIRO Modél. Math. Anal. Numér., 29 (1995), 687-699.
[7]	J. L. Boldrini and G. Planas, A tridimensional phase-field model with convection for phase change of an alloy, Discrete Contin. Dyn. Syst., 13 (2005), 429-450. doi: 10.3934/dcds.2005.13.429.
[8]	P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.
[9]	M. Durán, E. Ortega-Torres and J. Rappaz, Weak solution of a stationary convection-diffusion model describing binary alloy solidification processes, Math. Comput. Modelling, 42 (2005), 1269-1286. doi: 10.1016/j.mcm.2005.01.035.
[10]	M. Durán and E. Ortega-Torres, Existence of weak solutions for a non-homogeneous solidification model, Electron. J. Differential Equations, 2006 17 pp.
[11]	I. Ekeland and R. Temam, Convex analysis and Variational Problems, Society for Industrial and Applied Mathematics, Philadelphia, 1999. doi: 10.1137/1.9781611971088.
[12]	J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach, Encyclopedia of Mathematics and its Applications, 117, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511721595.
[13]	O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltzeva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R.I., 1967.
[14]	G. Planas and J. L. Boldrini, A bidimensional phase-field model with convection for change phase of an alloy, J. Math. Anal. Appl., 303 (2005), 669-687. doi: 10.1016/j.jmaa.2004.08.068.
[15]	J. C. Ramirez and C. Beckermann, Examination of binary alloy free dendritic growth theories with a phase-field model, Acta Materialia, 53 (2005), 1721-1736. doi: 10.1016/j.actamat.2004.12.021.
[16]	J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differ. Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X.
[17]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, 146 (1986), 64-96. doi: 10.1007/BF01762360.
[18]	J. Simon, A constructive proof of a theorem of G. de Rham, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 1167-1172.
[19]	R. Temam, Navier Stokes Equations. Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001.

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