# Estimates and computations for melting and solidification problems

• Volume: 35, Issue: 4, page 607-630
• ISSN: 0764-583X

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## Abstract

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In this paper we focus on melting and solidification processes described by phase-field models and obtain rigorous estimates for such processes. These estimates are derived in Section 2 and guarantee the convergence of solutions to non-constant equilibrium patterns. The most basic results conclude with the inequality (E2.31). The estimates in the remainder of Section 2 illustrate what obtains if the initial data is progressively more regular and may be omitted on first reading. We also present some interesting numerical simulations which demonstrate the equilibrium structures and the approach of the system to non-constant equilibrium patterns. The novel feature of these calculations is the linking of the small parameter in the system, $\delta$, to the grid spacing, thereby producing solutions with approximate sharp interfaces. Similar ideas have been used by Caginalp and Sokolovsky [5]. A movie of these simulations may be found at http:www.math.cmu.edu/math/people/greenberg.html

## How to cite

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Greenberg, James M.. "Estimates and computations for melting and solidification problems." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.4 (2001): 607-630. <http://eudml.org/doc/194065>.

@article{Greenberg2001,
abstract = {In this paper we focus on melting and solidification processes described by phase-field models and obtain rigorous estimates for such processes. These estimates are derived in Section 2 and guarantee the convergence of solutions to non-constant equilibrium patterns. The most basic results conclude with the inequality (E2.31). The estimates in the remainder of Section 2 illustrate what obtains if the initial data is progressively more regular and may be omitted on first reading. We also present some interesting numerical simulations which demonstrate the equilibrium structures and the approach of the system to non-constant equilibrium patterns. The novel feature of these calculations is the linking of the small parameter in the system, $\delta$, to the grid spacing, thereby producing solutions with approximate sharp interfaces. Similar ideas have been used by Caginalp and Sokolovsky [5]. A movie of these simulations may be found at http:www.math.cmu.edu/math/people/greenberg.html},
author = {Greenberg, James M.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {phase-field models; melting and solidification},
language = {eng},
number = {4},
pages = {607-630},
publisher = {EDP-Sciences},
title = {Estimates and computations for melting and solidification problems},
url = {http://eudml.org/doc/194065},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Greenberg, James M.
TI - Estimates and computations for melting and solidification problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 4
SP - 607
EP - 630
AB - In this paper we focus on melting and solidification processes described by phase-field models and obtain rigorous estimates for such processes. These estimates are derived in Section 2 and guarantee the convergence of solutions to non-constant equilibrium patterns. The most basic results conclude with the inequality (E2.31). The estimates in the remainder of Section 2 illustrate what obtains if the initial data is progressively more regular and may be omitted on first reading. We also present some interesting numerical simulations which demonstrate the equilibrium structures and the approach of the system to non-constant equilibrium patterns. The novel feature of these calculations is the linking of the small parameter in the system, $\delta$, to the grid spacing, thereby producing solutions with approximate sharp interfaces. Similar ideas have been used by Caginalp and Sokolovsky [5]. A movie of these simulations may be found at http:www.math.cmu.edu/math/people/greenberg.html
LA - eng
KW - phase-field models; melting and solidification
UR - http://eudml.org/doc/194065
ER -

## References

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1. [1] G. Caginalp, An analysis of a phase-field model of a free boundary. Arch. Rat. Mech. Anal. 92 (1986) 205–245. Zbl0608.35080
2. [2] G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of the phase field equation. Phys. Rev. A 39 (1989) 5887–5896. Zbl1027.80505
3. [3] G. Caginalp, Phase field models and sharp interface limits: some differences in subtle situations. Rocky Mountain J. Math. 21 (1996) 603–616. Zbl0753.35125
4. [4] G. Caginalp and X. Chen, Phase field equations in the singular limit of sharp interface problems, in On the evolution of phase boundaries, IMA 43 (1990-1991) 1–28. Zbl0760.76094
5. [5] G. Caginalp and E. Sokolovsky, Phase field computations of single-needle crystals, crystal growth, and motion by mean curvature. SIAM J. Sci. Comput. 15 (1994) 106–126. Zbl0793.65099
6. [6] M. Fabbri and V.R. Vollmer, The phase-field method in the sharp-interface limit: A comparison between model potentials. J. Comp. Phys. 130 (1997) 256–265. Zbl0868.65094
7. [7] G.B. McFadden, A.A. Wheeler, R.J. Brown, S.R. Coriell and R.F. Sekerka, Phase-field models for anisotropic interfaces. Phys. Rev. E 48 (1993) 2016–2024.
8. [8] O. Penrose and P. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Physica D 43 (1990) 44–62. Zbl0709.76001
9. [9] O. Penrose and P. Fife, On the relation between the standard phase-field model and a “thermodynamically consistent” phase-field model. Physica D 69 (1993) 107–113. Zbl0799.76084
10. [10] S.L. Wang, R.F. Sekerka, A.A. Wheeler, B.T. Murray, S.R. Coriell, R.J. Braun and G.B. McFadden, Thermodynamically-consistent phase-field models. Physica D 69 (1993) 189–200. Zbl0791.35159
11. [11] S.L. Wang and R.F. Sekerka, Algorithms for phase field computations of the dendritic operating state at large supercoolings. J. Comp. Phys. 127 (1996) 110–117. Zbl0859.65131

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