An analysis of a phase field model of a free boundary

Caginalp, Gunduz

doi:10.1007/bf00254827

Cited by 728 publications

(563 citation statements)

References 28 publications

Supporting

Mentioning

544

Contrasting

Unclassified

Order By: Relevance

“…In the limit when the interfacial thickness vanishes ('If becomes a step function), asymptotic analysis produces several modified Stefan problems including boundary conditions like (1) and (2); see [15] and references therein for the details. Some equilibrium existence results are also known for these phase field models; see [81,14,63].…”

Section: Fmentioning

confidence: 99%

A boundary integral approach to unstable solidification

Strain¹

1989

Journal of Computational Physics

View full text Add to dashboard Cite

We consider the supercooled Stefan problem with a general anisotropic curvature-and velocity-dependent boundary c<;mdition on the moving interface. This is a well-known model for pattern formation in unstable solidification.We reformulate the problem in terms of a quasilinear history-dependent singular integral equation for the velocity of the boundary. Using this equation, we carry out a new linear stability analysis of a planar solidification front with a general boundary condition. This analysis disagrees with the classical linear stability theory, because our approach includes transient effects due to initial conditions and linearizes the boundary conditions more accurately. Our analysis exhibits the smoothing role of velocity-dependence and the destabilizing effect of anisotropy.We then present numerical methods, also based on the integral equation formulation. These methods are able to follow the evolution of a periodic solidification front far into the nonlinear regime, with 0 (M) accuracy, where M is the time step. Previous work has been limited to short times and achieved slightly less than 0 (M 112 ) accuracy. Our methods also include a new algorithm for moving curves with curvature-dependent velocity.We present numerical results obtained with these new methods. After demonstrating first-order convergence for short time spans, we compare accurate numerical results with the predictions of the new and classical linear stability theories. Our results agree very well with the new theory, and disagree with the classical theory by as much as 25%. Then we study the long-time evolution of a dendritic front. We confirm the smoothing effect of velocity-dependence numerically. Our computations exhibit the beginnings of a sidebranching instability, followed by formation of a periodic cellular front, with an anisotropic boundary condition. Tip-splitting occurs instead, in the isotropic case .

show abstract

Section: Fmentioning

confidence: 99%

A boundary integral approach to unstable solidification

Strain¹

1989

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…The dynamics of φ(r, t) are designed to follow the evolving solidification front [13][14][15][16][17][18][19]. The phasefield interpolates between the solid and liquid phases, attaining a different constant value in each phase (typically ±1), with a rapid transition region in the vicinity of the solidification front.…”

mentioning

confidence: 99%

Universal dynamics of phase-field models for dendritic growth

et al. 1999

View full text Add to dashboard Cite

We compare time-dependent solutions of different phase-field models for dendritic solidification in two dimensions, including a thermodynamically consistent model and several ad hoc models. The results are identical when the phase-field equations are operating in their appropriate sharp interface limit. The long time steady state results are all in agreement with solvability theory. No computational advantage accrues from using a thermodynamically consistent phase-field model. PACS Numbers: 81.10. Aj, 05.70.Ln, 64.70.Dv, 81.30.Fb Dendrites are the most commonly observed solidification microstructures in metals. The free growth of a single dendrite is a prototype for problems of pattern selection in materials science [1][2][3][4][5] and has been extensively studied experimentally and theoretically. It is still not possible to compare theory with experiment because of the difficulties in computing three dimensional microstructures in the temperature and material's parameter range of the experiments.Recently, a significant step forward was taken by Karma and Rappel [6,7], who not only showed how to compute accurately two dimensional dendritic growth but also were able to compare their results with theoretical predictions. Their calculations used the so-called phase-field formulation of solidification, in which a mathematically sharp solid-liquid interface is smeared out or regularized and treated as a boundary layer, with its own equation of motion. The resulting formulation, described in more detail below, no longer requires front tracking and the imposition of boundary conditions, but must be related to the sharp interface model by an asymptotic analysis. In fact, there are many ways to prescribe a smoothing and dynamics of a sharp interface consistent with the original sharp interface model, so that there is no unique phase-field model, but rather a family of related models. An important part of Karma and Rappel's work was an improved asymptotic analysis which allows coarser spatial grids to be used in the numerical computations than was previously possible.Although the phase-field method has gained acceptance as a useful way to study solidification problems, a debate still exists over the interpretation and validity of the phase-field models themselves. Each model includes a double-well potential field which enforces the above properties of the phase-field. Some models can be shown rigorously to satisfy entropy inequality [8,9]. These are sometimes called "thermodynamically consistent" models. On the other hand, it has been argued that the precise form of the phase-field equations should be irrelevant so long as the computations are performed at the asymptotic limit where the phase-field model converges to the sharp interface limit [10].The purpose of this Letter is to compare the dynamics of the different phase-field models proposed. To this end, we have performed accurate and extensive computations using a specially developed adaptive mesh refinement algorithm [11,12]. We find that when properly used, all ...

show abstract

“…where χ [1,2] is the characteristic function of the interval [1,2]. Since from Lemma 4.9(ii) we know that α(γ)[h k (γ)](r) satisfies (4.9) and χ(r) = 1 for all r ≥ 2 and χ(r) = 0 for all r ∈ [0, 1], we obtain that…”

Section: The Eigenvalue Problem Nearmentioning

confidence: 95%