Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations

Caginalp, Gunduz

doi:10.1103/physreva.39.5887

Cited by 432 publications

(346 citation statements)

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“…In recent years further progress has been made towards understanding solidification phenomena [10] by the advent of by phase-field modelling [11,12,13], particularly through the formulation of the 'thin interface model', due to Karma & Rappel [14]. In the thininterface model, asymptotic expansions of the solution on the solid and liquid sides of the boundary are matched such that a solution is obtained which is independent of the length scale chosen for the mesoscopic diffuse interface width.…”

Section: Introductionmentioning

confidence: 99%

Prediction of the operating point of dendrites growing under coupled thermosolutal control at high growth velocity

Mullis

2011

Phys. Rev. E

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We use a phase-field model for the growth of dendrites in dilute binary alloys under coupled thermo-solutal control to explore the dependence of the dendrite tip velocity and radius of curvature upon undercooling, Lewis number (ratio of thermal to solutal diffusivity), alloy concentration and equilibrium partition coefficient. Constructed in the quantitatively valid thin-interface limit the model uses advanced numerical techniques such as mesh adaptivity, multigrid and implicit time-stepping to solve the non-isothermal alloy solidification problem for materials parameters that are realistic for metals. From the velocity and curvature data we estimate the dendrite operating point parameter, σ*. We find that σ* is non-constant and, over a wide parameter space, displays first a local minimum followed by a local maximum as the undercooling is increased. This behaviour is contrasted with a similar type of behaviour to that predicted by simple marginal stability models to occur in the radius of curvature, on the assumption of constant σ*.

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Section: Introductionmentioning

confidence: 99%

Prediction of the operating point of dendrites growing under coupled thermosolutal control at high growth velocity

Mullis

2011

Phys. Rev. E

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“…First proposed by Langer [5] and subsequently developed by, among others, Caginalp [6] and Penrose & Fife [7], the basis of the phase-field method is the definition of a phase variable (say ) the value of which describes the phase state of the material. At it simplest, for a single-phase solid, this might for instance equate   1 with the solid and   -1 with the liquid.…”

Section: Introductionmentioning

confidence: 99%

Quantification of mesh induced anisotropy effects in the phase-field method

Mullis

2006

Computational Materials Science

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Phase-field modelling is one of the most powerful techniques currently available for the simulation from first principles the time dependant evolution of complex solidification microstructures. However, unless care is taken the computational mesh used to solve the set of partial differential equations that result from the phase-field formulation of the solidification problem may introduce a stray, or implicit, anisotropy, which would be highly undesirable in quantitative calculations. In this paper we quantify this effect as a function of various computational parameters and subsequently suggest techniques for mitigating the effect of this stray anisotropy. 81.30.Fb, PACS:

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“…(19) and (28), let us verify that it indeed gives rise to diffuse interfaces with the hyperbolic tangent profile (7) as → 0. We give a heuristic outline, but this procedure can be formalized using the theory of matched asymptotic expansions (Caginalp, 1989;Fife, 1988). Let us first verify separation into phases on short time-scales.…”

Section: The Diffuse-interface Transition Profile: Asymptotic Analysismentioning

confidence: 99%

Computational Phase‐Field Modeling

Gómez

Zee

2017

Encyclopedia of Computational Mechanics Second Edition

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Phase-field modeling is emerging as a promising tool for the treatment of problems with interfaces. The classical description of interface problems requires the numerical solution of partial differential equations on moving domains in which the domain motions are also unknowns. The computational treatment of these problems requires moving meshes and is very difficult when the moving domains undergo topological changes. Phase-field modeling may be understood as a methodology to reformulate interface problems as equations posed on fixed domains. In some cases, the phase-field model may be shown to converge to the moving-boundary problem as a regularization parameter tends to zero, which shows the mathematical soundness of the approach. However, this is only part of the story because phase-field models do not need to have a moving-boundary problem associated and can be rigorously derived from classical thermomechanics. In this context, the distinguishing feature is that constitutive models depend on the variational derivative of the free energy. In all, phase-field models open the opportunity for the efficient treatment of outstanding problems in computational mechanics, such as, the interaction of a large number of cracks in three dimensions, cavitation, film and nucleate boiling, tumor growth or fully three-dimensional air-water flows with surface tension. In addition, phase-field models bring a new set of challenges for numerical discretization that will excite the computational mechanics community.

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Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations

Cited by 432 publications

References 9 publications

Prediction of the operating point of dendrites growing under coupled thermosolutal control at high growth velocity

Prediction of the operating point of dendrites growing under coupled thermosolutal control at high growth velocity

Quantification of mesh induced anisotropy effects in the phase-field method

Computational Phase‐Field Modeling

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