An enthalpy-based finite element method for solving two-phase Stefan problem

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Stefan problem is a problem involving phase transition from solid to liquid or vice versa where boundary between solid and liquid regions moves as function of time. This paper presents numerical solution of one-dimensional two-phase Stefan problem by using finite element method. The governing equations involved in Stefan problem consist of heat conduction equation for solid and liquid regions, and also transition equation in interface position (moving boundary). The equations are difficult to solve by ordinary numerical method because of the presence of moving boundary. As consequence, the equations is reformulated into the form of internal energy (enthalpy). By the enthalpy formulation, solution of the heat conduction equations is no longer concerning the phase state of material. The advantage of the enthalpy formulation is that, finite element method can be easily implemented to solve Stefan problem. Numerical simulation of interface position, temperature profile, and temperature history has good agreement with the exact solution. The approximation of interface position using finite element method was found that it is more accurate than the approximation by using Godunov method. The simulation results also reveal that the finite element method for solving Stefan problem have smaller mean absolute error than the Godunov method.


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Vasilios Alexiades. Mathematical modeling of melting and freezing processes. Hemisphere Publishing Corporation, 1981.

NS Asaithambi. A galerkin method for stefan problems. Applied Mathematics and Computation, 52(2-3):239–250, 1992.

G Beckett, John A Mackenzie, and ML Robertson. A moving mesh finite element method for the solution of two-dimensional stefan problems. Journal of Computational Physics, 168(2):500–518, 2001.

Alaattin Esen and Selçuk Kutluay. A numerical solution of the stefan problem with a neumann-type boundary condition by enthalpy method. Applied Mathematics and Computation, 148(2):321–329, 2004.

Chiawen W Lan, Chunchang C Liu, and Chiaming M Hsu. An adaptive finite volume method for incompressible heat flow problems in solidification. Journal of Computational Physics, 178(2):464–497, 2002.

Sarah L Mitchell and M Vynnycky. Finite-difference methods with increased accuracy and correct initialization for one- dimensional stefan problems. Applied Mathematics and Computation, 215(4):1609–1621, 2009.

Joseph J Monaghan, Herbert E Huppert, and M Grae Worster. Solidification using smoothed particle hydrodynamics. Journal of Computational Physics, 206(2):684–705, 2005.

SR Pudjaprasetya. Transport phenomena equations and numerical methods. ITB Press, 2018.

Svetislav Savovi´ c and James Caldwell. Finite difference solution of one-dimensional stefan problem with periodic boundary conditions. International journal of heat and mass transfer, 46(15):2911–2916, 2003.

Larry J Segerlind. Applied finite element analysis, volume 316. Wiley New York, 1976.

D Tarwidi. Godunov method for computerized lung cancer cryosurgery planning with efficient freezing time. In Information and Communication Technology (ICoICT), 2015 3rd International Conference on, pages 494–499. IEEE, 2015.

D Tarwidi. Modeling and numerical simulation of solar cooker with pcm as thermal energy storage. In Information and Communication Technology (ICoICT), 2015 3rd International Conference on, pages 584–589. IEEE, 2015.

D Tarwidi and SR Pudjaprasetya. Godunov method for stefan problems with enthalpy formulations. East Asian Journal on Applied Mathematics, 3(2):107–119, 2013.

Fedor Pavlovich Vasil’ev and Aleksandr Borisovich Uspenskii. A difference method for the solution of the two-phase stefan problem. USSR Computational Mathematics and Mathematical Physics, 3(5):1192–1208, 1963.

V Voller and M Cross. Accurate solutions of moving boundary problems using the enthalpy method. International journal of heat and mass transfer, 24(3):545–556, 1981.

VR Voller and L Shadabi. Enthalpy methods for tracking a phase change boundary in two dimensions. International communications in heat and mass transfer, 11(3):239–249, 1984.

VR Voller, CR Swaminathan, and Brian G Thomas. Fixed grid techniques for phase change problems: a review. International Journal for Numerical Methods in Engineering, 30(4):875–898, 1990.

Bengt Winzell. Finite element galerkin methods for multi-phase stefan problems. Applied Mathematical Modelling, 7(5):329– 344, 1983.

How to Cite
Tarwidi, D. (2019). An enthalpy-based finite element method for solving two-phase Stefan problem. Indonesian Journal on Computing (Indo-JC), 4(1), 43-56.
Computational and Simulation