72,197 research outputs found
An alternative to the Allen-Cahn phase field model for interfaces in solids - numerical efficiency
The derivation of the Allen-Cahn and Cahn-Hilliard equations is based on the
Clausius-Duhem inequality. This is not a derivation in the strict sense of the
word, since other phase field equations can be fomulated satisfying this
inequality. Motivated by the form of sharp interface problems, we formulate
such an alternative equation and compare the properties of the models for the
evolution of phase interfaces in solids, which consist of the elasticity
equations and the Allen-Cahn equation or the alternative equation. We find that
numerical simulations of phase interfaces with small interface energy based on
the alternative model are more effective then simulations based on the
Allen-Cahn model.Comment: arXiv admin note: text overlap with arXiv:1505.0544
Symmetry of Traveling Wave Solutions to the Allen-Cahn Equation in \Er^2
In this paper, we prove even symmetry of monotone traveling wave solutions to
the balanced Allen-Cahn equation in the entire plane. Related results for the
unbalanced Allen-Cahn equation are also discussed
Nonlinear diffusion equations as asymptotic limits of Cahn-Hilliard systems
An asymptotic limit of a class of Cahn-Hilliard systems is investigated to
obtain a general nonlinear diffusion equation. The target diffusion equation
may reproduce a number of well-known model equations: Stefan problem, porous
media equation, Hele-Shaw profile, nonlinear diffusion of singular logarithmic
type, nonlinear diffusion of Penrose-Fife type, fast diffusion equation and so
on. Namely, by setting the suitable potential of the Cahn-Hilliard systems, all
of these problems can be obtained as limits of the Cahn-Hilliard related
problems. Convergence results and error estimates are proved
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