Computation of sharp phase boundaries by spreading: The planar and spherically symmetric cases

Caginalp, Gunduz; Socolovsky, E. A.

doi:10.1016/0021-9991(91)90254-i

Cited by 68 publications

(63 citation statements)

References 8 publications

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“…Throughout this study we use finite difference methods ([CL], [CS2]) with a fixed grid in order to isolate the issues that are of specific interest. In principle, a combination of phase field methods and adaptive grids with finite elements should provide similar accuracy at a lower cost.…”

Section: Z Oclqmentioning

confidence: 99%

Phase Field Computations of Single-Needle Crystals, Crystal Growth, and Motion by Mean Curvature

Caginalp¹,

Socolovsky²

1994

SIAM J. Sci. Comput.

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ABSTRACT. The phase field model for free boundaries consists of a system of parabolic differential equations in which the variables represent a phase (or "order") parameter and temperature respectively. The parameters in the equations are related directly to the physical observables including the interfacial width c which we can regard as a free parameter in computation. The phase field equations can be used to compute a wide range of sharp interface problems including the classical Stefan model, its modification to incorporate surface-tension and/or surface kinetic terms, the Cahn-Allen motion by mean curvature, the Hele-Shaw model, etc. Also included is anisotropy in the equilibrium and dynamical forms generally considered by materials scientists. By adjusting the parameters, the computations can be varied continuously from single needle dendritic to faceted crystals.The computational method consists of smoothing a sharp interface problem within the scaling of distinguished limits of the phase field equations which preserve the physically important parameters. The two-dimensional calculations indicate that this efficient method for treating these stiff problems results in very accurate interface determination without interface tracking. We test these methods against exact and analytical results available in planar waves, faceted growth and motion by mean curvature up to extinction time.The results obtained for the single needle crystal show a constant velocity growth as expected from laboratory experiments. + Supported by NSF Grant DMS-9002242 .

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Section: Z Oclqmentioning

confidence: 99%

Phase Field Computations of Single-Needle Crystals, Crystal Growth, and Motion by Mean Curvature

Caginalp¹,

Socolovsky²

1994

SIAM J. Sci. Comput.

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“…In other words, if one is to use the phase field equations for realistic computations the interface motion for the system with ε = 10 −4 and ε = 10 −8 must be nearly identical. This physical ansatz that the interface thickness could be stretched out by a factor of ten thousand without significantly altering the motion of the interface was suggested and confirmed numerically in [13]. Since then, phase field equations have been used for a number of applications, and appear to be one of the most viable methods for computing interface motion.…”

mentioning

confidence: 77%

A rapidly converging phase field model

Chen¹,

Caginalp²,

Eck³

2006

Discrete &Amp; Continuous Dynamical Systems - A

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Abstract. We propose a phase field model that approximates its limiting sharp interface model (free boundary problem) up to second order in interface thickness. A broad range of double-well potentials can be utilized so long as the dynamical coefficient in the phase equation is adjusted appropriately. This model thereby assures that computation with particular value of interface thickness ε, will differ at most by O(ε 2 ) from the limiting sharp interface problem. As an illustration, the speed of a traveling wave of the phase field model is asymptotically expanded to demonstrate that it differs from the speed of the traveling wave of the limit problem by O(ε 2 ). 1.Introduction. Interface problems arising from solidification have been studied extensively in mathematics and physics for more than a century [29]. The mathematical study began with Lamè and Clapeyron [23] in 1831 who modeled the freezing of the ground using the heat equation, latent heat across the interface, and the condition that the temperature at the interface remains at the equilibrium freezing temperature. Reformulated in 1889, this problem became known as the classical Stefan model [31], and can be stated as follows. Determine the temperature, T (x, t), and the interface, Γ(t), satisfying the system of equations,

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“…We refer the readers to [6,10,31,32] and references therein for more details. The boundary conditions for the phase-field equations are the same as the sharp interface model for , with compatible conditions for p. For example, if Dirichlet conditions are imposed on = ± , where ± denotes the liquid and solid boundaries, respectively, then the corresponding values of p are the largest ( p + ) and smallest ( p − ) roots of…”

Section: Two-dimensional Phase-field Problemsmentioning

confidence: 99%

“…[6,7,10]). Then the two phases are characterized by p taking values close to p + and p − in each phase.…”

Section: Two-dimensional Phase-field Problemsmentioning

confidence: 99%

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An alternating Crank–Nicolson method for the numerical solution of the phase‐field equations using adaptive moving meshes

Tan

Huang

2007

Numerical Methods in Fluids

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SUMMARYAn alternating Crank-Nicolson method is proposed for the numerical solution of the phase-field equations on a dynamically adaptive grid, which automatically leads to two decoupled algebraic subsystems, one is linear and the other is semilinear. The moving mesh strategy is based on the approach proposed by Li et al. (J. Comput. Phys. 2001; 170:562-588) to separate the mesh-moving and partial differential equation evolution. The phase-field equations are discretized by a finite volume method in space, and the mesh-moving part is realized by solving the conventional Euler-Lagrange equations with the standard gradient-based monitors. The algorithm is computationally efficient and has been successfully used in numerical simulations.

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Computation of sharp phase boundaries by spreading: The planar and spherically symmetric cases

Cited by 68 publications

References 8 publications

Phase Field Computations of Single-Needle Crystals, Crystal Growth, and Motion by Mean Curvature

Phase Field Computations of Single-Needle Crystals, Crystal Growth, and Motion by Mean Curvature

A rapidly converging phase field model

An alternating Crank–Nicolson method for the numerical solution of the phase‐field equations using adaptive moving meshes

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