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Numerical analysis of an isotropic phase-field model with magnetic-field effect. (Analyse numérique d’un modéle isotrope de champ de phase sous l’effet d’un champ magnétique.) (English. French summary) Zbl 1319.78001
The aim of this paper is to provide a numerical scheme based on FEM, and a numerical error and stability analysis for the model proposed by the first author and A. Belmiloudi [J. Math. Anal. Appl. 390, No. 1, 244–273 (2012; Zbl 1387.80004)] and by A. Rasheed et al. [Discrete Contin. Dyn. Syst. 2011, Suppl., 1224–1233 (2011; Zbl 1306.82018)] in an isotropic and isothermal regime. The semidiscrete weak form yields the differential-algebraic system which is fully discretized first by invoking Euler’s backward difference method and then resolved by using the Newton iteration technique.
78A25 Electromagnetic theory, general
80A22 Stefan problems, phase changes, etc.
80M10 Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
65L80 Numerical methods for differential-algebraic equations
Full Text: DOI
[1] Anderson, M. D.; McFadden, G. B.; Wheeler, A. A., A phase-field model of solidification with convection, Physica D, 135, 175-194, (2000) · Zbl 0951.35112
[2] Belmiloudi, A., Method of characteristics and error estimates of the perturbation of given mean flow, (Application of Mathematics in Engineering and Business Sozopol, Proc. XXIInd Summer School, (1996)), 25-38
[3] Grujicic, M.; Cao, G.; Millar, R. S., Computer modelling of the evolution of dendrite microstructure in binary alloys during non-isothermal solidification, J. Mater. Synth. Process., 10, 191-203, (2002)
[4] Hadid, H. B.; Henry, D.; Kaddeche, S., Numerical study of convection in the horizontal bridgman configuration under the action of a constant magnetic field. part 1. two-dimensional flow, J. Fluid Mech., 333, 23-56, (1997) · Zbl 0897.76033
[5] Li, M.; Takuya, T.; Omura, N.; Miwa, K., Effects of magnetic field and electric current on the solidification of AZ91D magnesium alloys using an electromagnetic vibration technique, J. Alloys Compd., 487, 187-193, (2009)
[6] Petzold, L. R., A description of DASSL: a differential/algebraic system solver, (Scientific Computing, IMACS Trans. Sci. Comput., (1983)), 65-68
[7] Prescott, P.; Incropera, F., Magnetically damped convection during solidification of a binary metal alloy, Trans. Amer. Soc. Mech. Eng., 115, 302-310, (1993)
[8] Ramizer, J. C.; Beckermann, C., Examination of binary alloy free dendritic growth theories with a phase-field model, Acta Mater., 53, 1721-1736, (2005)
[9] Rasheed, A., Dendritic solidification of binary mixtures of metals under the action of magnetic field: modeling, mathematical and numerical analysis, (2010), INSA de Rennes France, Ph.D. dissertation
[10] Rasheed, A.; Belmiloudi, A., An analysis of the phase-field model for isothermal binary alloy solidification with convection under the influence of magnetic field, J. Math. Anal. Appl., 390, 244-273, (2012) · Zbl 1387.80004
[11] Rasheed, A.; Belmiloudi, A., Mathematical modelling and numerical simulation of dendrite growth using phase-field method with a magnetic field effect, Commun. Comput. Phys., 14, 477-508, (2013) · Zbl 1373.80005
[12] Rasheed, A.; Belmiloudi, A.; Mahé, F., Dynamics of dendrite growth in a binary alloy with magnetic field affect, Discrete Contin. Dyn. Syst., 1224-1233, (2011), (special issue) · Zbl 1306.82018
[13] Sampath, R., The adjoint method for the design of directional binary alloy solidification processes in the presence of a strong magnetic field, (2001), Cornell University Ithaca, NY, USA, Ph.D. dissertation
[14] Süli, E., Convergence and non-linear stability of Lagrange-Galerkin method for the Navier-Stokes equations, Numer. Math., 53, 459-483, (1988) · Zbl 0637.76024
[15] Tong, X.; Beckermann, C.; Kerma, A.; Li, Q., Phase-field simulations of dendritic crystal growth in a forced flow, Phys. Rev. E, 63, 061601, (2001)
[16] Warren, J. A.; Boettinger, W. J., Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method, Acta Metall. Mater., 43, 689-703, (1995)
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