A simple level set method for solving Stefan problems. (English) Zbl 0889.65133
A numerical method is presented for solving Stefan problems and for simulating the behaviour that arises from the unstable solidification of pure substances. This method can be applied to problems involving dendritic solidification. The method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance. The method accurately computes the boundary between the solid and liquid phases as it undergoes the process of solidification, as well as the temperature of the material as it evolves over time.
Many interesting numerical results are presented which are found after applying that method to solving Stefan problems and to modelling unstable crystal growth.
Many interesting numerical results are presented which are found after applying that method to solving Stefan problems and to modelling unstable crystal growth.
Reviewer: Z.Dżygadło (Warszawa)
MSC:
| 65Z05 | Applications to the sciences |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35R35 | Free boundary problems for PDEs |
| 35K05 | Heat equation |
| 80A22 | Stefan problems, phase changes, etc. |
Keywords:
level set method; Stefan problems; finite difference scheme; heat equation; numerical resultsReferences:
| [1] | Almgren, R., Variational algorithms and pattern formation in dendritic solidification, J. Comput. Phys., 106, 337 (1993) · Zbl 0787.65095 |
| [2] | Brattkus, K.; Meiron, D. I., Numerical simulations of unsteady crystal growth, SIAM J. Appl. Math., 52, 1303 (1992) · Zbl 0753.35124 |
| [3] | Caginalp, G.; Socolovsky, E. A., Computation of sharp phase boundaries by spreading: the planar and spherically symmetric cases, J. Comput. Phys., 95, 85 (1991) · Zbl 0732.65116 |
| [4] | Fix, G., Phase field models for free boundary problems, Free Boundary Problems: Theory and Applications, Vol. II (1983), Piman: Piman Boston, p. 580-600 · Zbl 0518.35086 |
| [5] | Harabetian, E.; Osher, S., Regularization of Ill-Posed Problems Via the Level Set Approach, UCLA CAM Report, 95-41 (1995) |
| [6] | Ivantsov, G. P., Temperature field around spherical, cylindrical and acircular crystal growing in a supercooled melt, Dokl. Akad. Nauk SSSR, 58, 567 (1947) |
| [7] | Juric, D.; Tryggvason, G., A front tracking method for dendritic solidification, J. Comput. Phys., 123, 127 (1996) · Zbl 0843.65093 |
| [8] | A. Karma, W. J. Rappel, Numerical simulation of three-dimensional dendritic growth, preprint; A. Karma, W. J. Rappel, Numerical simulation of three-dimensional dendritic growth, preprint |
| [9] | Karma, A.; Rappel, W. J., Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics, Phys. Rev. E, 53, 3017 (1996) |
| [10] | Kobayashi, R., Modeling and numerical simulations of dendritic crystal growth, Physica D, 63, 410 (1993) · Zbl 0797.35175 |
| [11] | Langer, J. S., Instabilities and pattern formation in crystal growth, Rev. Mod. Phys., 52, 1 (1980) |
| [12] | Langer, J. S., Lectures in the theory of pattern formation, (Souletie, J.; Vannimenus, J.; Stora, R., Chance and Matter (1987), North Holland: North Holland Amsterdam), 629-711 |
| [13] | Langer, J. S.; Muller-Krumbhaar, H., Theory of dendritic growth, Acta. Metall., 26, 1681 (1978) |
| [14] | Meirmanov, A., The Stefan Problem (1992), DeGruyter: DeGruyter Berlin, p. 10-30 |
| [15] | Merriman, B.; Bence, J. K.; Osher, S. J., Motion of multiple junctions: A level set approach, J. Comput. Phys., 112, 334 (1994) · Zbl 0805.65090 |
| [16] | Mullins, W. W.; Sekerka, R. F., Stability of a planar interface during solidification of a dilute binary alloy, J. Appl. Phys., 35, 444 (1964) |
| [17] | Nochetto, R. H.; Paolini, M.; Verdi, C., An adaptive finite element method for two-phase Stefan problems in two space dimensions. Part II: Implementation and numerical experiments, SIAM J. Sci. Stat. Comput., 12, 1207 (1991) · Zbl 0733.65088 |
| [18] | Osher, S.; Sethian, J. A., Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79, 12 (1988) · Zbl 0659.65132 |
| [19] | Pelce, P., Dynamics of Curved Fronts (1988), Academic Press: Academic Press San Diego, p. 1-102 · Zbl 0712.76009 |
| [20] | Penrose, O.; Fife, P. C., Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43, 44 (1990) · Zbl 0709.76001 |
| [21] | Roosen, A. R.; Taylor, J. E., Modeling crystal growth in a diffusion field using fully faceted interfaces, J. Comput. Phys., 114, 113 (1994) · Zbl 0805.65128 |
| [22] | Rose, M. E., An enthalpy scheme for Stefan problems in several dimensions, App. Numerical Math., 12, 229 (1993) · Zbl 0787.65094 |
| [23] | Schmidt, A., Computation of three dimensional dendrites with finite elements, J. Comput. Phys., 125, 293 (1996) · Zbl 0844.65096 |
| [24] | Sethian, J.; Strain, J., Crystal growth and dendritic solidification, J. Comput. Phys., 98, 231 (1992) · Zbl 0752.65088 |
| [25] | Shu, C. W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. Comput. Phys., 77, 439 (1988) · Zbl 0653.65072 |
| [26] | Strain, J., Linear stability of planar solidification fronts, Physica D, 30, 297 (1988) · Zbl 0696.35197 |
| [27] | Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114, 146 (1994) · Zbl 0808.76077 |
| [28] | Wang, S-L.; Sekerka, R. F., Algorithms for phase field computation of the dendritic operating state at large supercoolings, L. Comput. Phys., 127, 110 (1996) · Zbl 0859.65131 |
| [29] | Warren, J., How does a metal freeze? A phase-field model of alloy solidification, Comput. Sci. Eng., 2, 38 (1995) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
