Efficient computation of a sharp interface by spreading via phase field methods

Abstract.We study in this paper the exponential time differencing (ETD) schemes and their modifications via complex contour integrations for the numerical solutions of parabolic type equations. We illustrate that the contour integration shares an added advantage of improving the stability of the time integration. In addition, we demonstrate the effectiveness of the ETD type schemes through the numerical solution of a typical problem in phase field modeling and through the comparisons with other existing methods.AMS subject classification (2000): 65M15, 65M70, 82D99.

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“…Numerical experiments on the similar problems have been conducted in, for example, [3,5] and the references cited therein.…”

Section: Numerical Resultsmentioning

confidence: 99%

Analysis and Applications of the Exponential Time Differencing Schemes and Their Contour Integration Modifications

Zhu

2005

Bit Numer Math

110

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“…On the other hand restriction (a) is fundamental and is intrinsic to the physical problem. In particular, independent of the specific method, one does not expect to resolve an interface with a radius of curvature, R, by means of a mesh spacing, Ax, which is greater than R. Within this natural limitation, [CS1,2] showed that quantitatively reliable calculations could be performed using the equations [(1.6)-(1.7)].…”

Section: Z Oclqmentioning

confidence: 99%

“…Computations performed with 400 mesh point were very similar though much more costly. We use the same type of explicit schemes as in [CS1,2] and similar time scales. The time step in all cases is the largest allowed by numerical stability considerations.…”

Section: X(y) = a + B Cos Mymentioning

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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“…Here, the 3J Â 3J matrix A n corresponds to the time derivative and interfacial energy terms, the J Â 3J matrix B corresponds to the constraint (1), and the J Â J matrix D n to the time derivative and diffusion terms in (5). In Appendix A, we prove that the linear system (11) is uniquely solvable without any restrictions concerning the space and time steps h and s, using the classical theory of saddle-point problems (see, for instance [28]), provided anisotropy of interfacial energy is neglected.…”

Section: Time and Space Discretizationmentioning

confidence: 99%

“…Although sharp interface models are still in development for the description of dendritic growth (see, for instance [2,3] for recent works), the phase-field has emerged as a method of choice, namely due to the advantage of avoiding the difficult problem of tracking sharp interfaces in two or three space dimensions. The principles of the phase-field model have been described in detail in numerous publications [4][5][6][7][8][9][10][11][12][13][14][15][16][17] (see [14,15] for recent reviews and [4][5][6][7]16,17] for some of the mathematical aspects). The idea is to describe the location of the solid and liquid phases in the computational www.elsevier.com/locate/jcp Journal of Computational Physics 195 (2004) domain by introducing an order parameter -the phase field -which varies smoothly from one to zero (sometimes from )1 to +1) through a diffuse interface.…”

Section: Introductionmentioning

confidence: 99%

Adaptive finite elements with high aspect ratio for the computation of coalescence using a phase-field model

Burman

Jacot

Picasso

2004

Journal of Computational Physics

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Efficient computation of a sharp interface by spreading via phase field methods

Cited by 36 publications

References 2 publications

Analysis and Applications of the Exponential Time Differencing Schemes and Their Contour Integration Modifications

Analysis and Applications of the Exponential Time Differencing Schemes and Their Contour Integration Modifications

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

Adaptive finite elements with high aspect ratio for the computation of coalescence using a phase-field model

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