Convergence of the Phase-Field Equations to the Mullins-Sekerka Problem with Kinetic Undercooling

1999 ◽  
pp. 413-471 ◽  
Author(s):  
H. M. Soner
Keyword(s):  
Phase Field ◽  
Field Equations ◽  
Kinetic Undercooling ◽  
Phase Field Equations

A sharp interface limit of the phase field equations: one-dimensional and axisymmetric

1996 ◽  
Vol 7 (6) ◽  
pp. 603-633 ◽  
Author(s):  
Barbara E. E. Stoth
Keyword(s):  
Stefan Problem ◽  
Phase Field ◽  
Space Dimension ◽  
Main Tool ◽  
Field Equations ◽  
First Case ◽  
Kinetic Undercooling ◽  
Phase Field Equations ◽  
Image Position ◽  
Energy Type

We study the singular limit of the dimensionless phase-field equationsWe consider two cases: either the space dimension is 1 and then ɛ tends to zero; or the solutions are radially symmetric and then both ɛ and α tend to zero. It turns out that, in the first case, the limiting functions solve the Stefan problem with kinetic undercooling, provided the initial temperature is small compared to the surface tension and the latent heat. In the second case, the limiting functions satisfy the Stefan problem coupled with the Gibbs–Thomson law for the melting temperature. We show, in addition, that the multiplicity of the interface is always one, in a sense to be explained at the end of § 1. As main tool we use energy type estimates, and prove that the formal first-order asymptotic expansion with respect to ɛ in fact gives an approximation of the exact solution. Our results hold without smoothness assumptions on the limiting Stefan problem.

scholarly journals Distributed optimal control of a nonstandard system of phase field equations

2011 ◽  
Vol 24 (4-6) ◽  
pp. 437-459 ◽  
Author(s):  
Pierluigi Colli ◽  
Gianni Gilardi ◽  
Paolo Podio-Guidugli ◽  
Jürgen Sprekels
Keyword(s):  
Optimal Control ◽  
Phase Field ◽  
Field Equations ◽  
Distributed Optimal Control ◽  
Phase Field Equations

A Hyperbolic Phase-Field Approach for Solidification With Supercooling

1999 ◽  
Author(s):  
Yuqi Chen ◽  
James M. McDonough ◽  
Kaveh A. Tagavi
Keyword(s):  
Phase Field ◽  
Supercooled Liquid ◽  
Field Method ◽  
Phase Field Method ◽  
Uniform Mesh ◽  
Phase Variable ◽  
Field Equations ◽  
Pure Material ◽  
Melt Temperature ◽  
Phase Field Equations

Abstract This report concerns the solidification of a “supercooled” liquid, whose temperature is initially below the equilibrium melt temperature, Tm of the solid. A new approach, the phase-field method, will be applied for this Stefan problem with supercooling, which simulates the solidification process of a pure material into a supercooled liquid in a spherical region. The advantage of the phase-field method is that it bypasses explicitly tracking the freezing front. In this approach the solid-liquid interface is treated as diffuse, and a dynamic equation for the phase variable is introduced in addition to the equation for heat flow. Thus, there are two coupled partial differential equations for temperature and phase field. In the reported study, an implicit numerical scheme using finite-difference techniques on a uniform mesh is employed to solve both Fourier phase-field equations and non-Fourier (known as damped wave or telegraph) phase-field equations. The latter gurantees a finite speed of propagation for the solidification front. Both Fourier (parabolic) and non-Fourier (hyperbolic) Stefan problems with supercooling are satisfactorily simulated and their solutions compared in the present work.

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