The Stefan model is a classical phase-change model in which motion is neglected: fluid and solid are at rest (and expansion due to freezing is neglected). The phase-change takes place at a given temperature (0 degrees Celsius for water-ice) and the free energy is continuous across the front. The internal energy on the contrary is discontinuous and the jump gives the amount of energy (latent heat) released by the fluid as it becomes solid. The heat flux is also discontinuous across the front, it is equal to the product of the latent heat times the front velocity.

The Stefan model leads to a stable solidification process because surface tension and kinetic mobility are neglected [1]. To model unstable processes as dendritic growth, the fixed front temperature must be replaced by the so-called Gibbs-Thomson condition. This relation links the front temperature to a reference melting temperature, the front curvature and the front speed. There also exist anisotropic versions of the Gibbs-Thomson which take into account the local orientation of the interface surface. It influences the both the surface tension and kinetic mobility. For a flat front with very slow motion, the classical front temperature (273.15 Kelvin) is recovered.

Numerical schemes to address the Stefan model may be classified in tracking approaches or capturing approaches. In the first category, the mesh moves with the interface. This is the case for the Arbitrary Eulerian Lagrangian approach [2, 3, 4]. Unfortunately ALE is not able to handle topological changes of the front and requires remeshing when the

mesh is too distorted. Remeshing requires projection of the solution between successive meshes which is highly detrimental to the continuity of the solution in time. Capturing approaches use a fixed mesh. This is the case for the level set approach [5, 6] and the extended finite element method [7, 8, 9].

We consider in this paper a new paradigm for mesh movements allowing elements to reach zero measure at some instant in their evolution. This extreme mesh deformation (X-MESH) allows an interface to be relayed from one node to another in a continuous fashion. It also allows interface annihilation or seeding. Moreover, no remeshig is needed and the mesh topology can be kept fixed. Only mesh movements are needed.

REFERENCES

[1] M.A. Jaafar, D.R. Rousse, S. Gibout, and J.-P. B´ed´ecarrats. A review of dendritic growth during solidification: Mathematical modeling and numerical simulations. 2017.

[2] M. J. Baines, M. E. Hubbard, P. K. Jimack, and R. Mahmood. A moving-mesh finite element method and its application to the numerical solution of phase-change problems. Communications in Computational Physics, 6(3):595–624, 2009.

[3] E. Gros, G. Anjos, and J. Thome. Moving mesh method for direct numerical simulation of two-phase flow with phase change. Applied Mathematics and Computation, 339:636–650, 2018.

[4] Yu Zhang, Anirban Chandra, Fan Yang, Ehsan Shams, Onkar Sahni, Mark Shephard, and Assad A. Oberai. A locally discontinuous ALE finite element formulation for compressible phase change problems. Journal of Computational Physics, 393:438–464, 2019.

[5] S. Chen, B. Merriman, S. Osher, and P. Smereka. A Simple Level Set Method for Solving Stefan Problems. Journal of Computational Physics, 135(1):8–29, 1997.

[6] Javed Shaikh, Atul Sharma, and Rajneesh Bhardwaj. On sharp-interface level-set method for heat and/or mass transfer induced Stefan problem. International Journal of Heat and Mass Transfer, 96:458–473, 2016.

[7] R. Merle and J. Dolbow. Solving thermal and phase change problems with the eXtended finite element method. Computational Mechanics, 28(5):339–350, 2002.

[8] H. Ji, D.L. Chopp, and J. Dolbow. A hybrid extended finite element/level set method for modeling phase transformations. International Journal For Numerical Methods in Engineering, 54:1209–1233, 2002.

[9] Min He, Qing Yang, Ning Li, Xiaopeng Feng, and Naifei Liu. An Extended Finite Element Method for Heat Transfer with Phase Change in Frozen Soil. Soil Mechanics and Foundation Engineering, 57(6):497–505, 2021