The paper is focused on the study of the solidification process of pure metals, in which the solidification front is smooth. It has the shape of a surface separating liquid from solid in three dimensional space or a curve in 2D. The location and topology of moving interface change over time and its velocity depends on the values of heat fluxes on the solid and liquid side of it.
Such a formulation belongs to a group called Stefan problems. A mathematical model of the Stefan problem is based on differential equations of heat conduction and interface motion. This system of equations is supplemented by appropriate initial and boundary conditions as well as the continuity conditions at the solidification interface. The solution involves the determination of temporary temperature field and interface position. Typically, it is impossible to obtain the exact solution of such problem.
This paper presents a mathematical model for the two-dimensional problem. The equation of heat conduction is supplemented with Dirichlet and Neumann boundary conditions. Interface motion is described by the level set equation which solution is sought in the form of temporary distribution of the signed distance function. Zero level of the distance field coincides with the position of the front. Values of the signed distance function obtained from the level set equation require systematic reinitialization.
Numerical model of the process based on the finite element method (FEM) is also presented. FEM equations are derived and discussed. The explicit time integration scheme is proposed. It helps to avoid solving the system of equations during each time step. The reinitialization procedure of the signed distance function is described in detail. Examples of numerical analysis of the solidification process of pure copper within the complex geometry are presented. Results obtained from the use of constant material properties are compared with those obtained from the use of temperature dependent properties.
 B. Chalmers, Principles of solidification, Wiley, New York, 1964.
 K. Fisher, Fundamentals of solidification, fourth ed, Trans Tech Publications Ltd, Switzerland, 45-55 (1998).
 S. Chakraborty, P. Dutta, The effect of solutal undercooling on double-diffusive convection and macrosegregation during binary alloy solidification: a numerical investigation, International Journal for Numerical Methods in Fluids 38, 895-917 (2002).
 S. Chakraborty, P. Dutta, Three-dimensional double-diffusive convection and macrosegregation during non-equilibrium solidification of binary mixtures, International Journal of Heat and Mass Transfer 46, 2115-2134 (2003).
 T. Telejko, Z. Malinowski, M. Rywotycki, Analysis of heat transfer and fluid flow in continuous steel casting, Archives of Metallurgy and Materials 54, 3, 837-844 (2009).
 L. Sowa, Numerical analysis of solid phase growing in a continuous steel caster, ZAMM 76, 491-492 (1996).
 L. Sowa, A. Bokota, Numerical model of thermal and flow phenomena the process growing of the CC slab, Archives of Metallurgy and Materials 56, 2, 359-366 (2011).
 W. Piekarska, M. Kubiak, A. Bokota, Numerical simulation of thermal phenomena and phase transformations in laser-arc hybrid welded joints, Archives of Metallurgy and Materials 56, 2, 409-421 (2011).
 W. Piekarska, M. Kubiak, Three-dimensional model for numerical analysis of thermal phenomena in laser-arc hybrid welding process, International Journal of Heat and Mass Transfer 54, 23-24, 4966-4974 (2011).
 A. Burbelko, J. Falkus, W. Kapturkiewicz, K. Sołek, P. Drożdż, M. Wróbel, Modeling of the grain structure formation in the steel continuous ingot by CAFE method, Archives of Metallurgy and Materials 57, 1, 379-384 (2012).
 D. Gurgul, A. Burbelko, Simulation of austenite and graphite growth in ductile iron by means of cellular automata, Archives of Metallurgy and Materials 55, 1, 53-60 (2010).
 W.J. Boettinger, J.A. Warren, C. Beckermann, A. Karma, Phase-field simulation of solidification, Annual Review of Materials Research 32, 163-194 (2002).
 A. Karma, W.J. Rappel, Quantitative phase-field modelling of dendritic growth in two and three dimensions, Physical Review E 57, 4323-4349 (1998).
 D.M. Anderson, G.B. McFadden, A.A. Wheeler, A phase-field model of solidification with convection, Physica D 135, 175-194 (2000).
 S. Chen, B. Merriman, S. Osher, P. Smereka, A simple level set method for solving Stefan problems, Journal of Computational Physics 135, 8-29 (1997).
 J. Chessa, P. Smolinski, T. Belytschko, The extended finite element method (XFEM) for solidification problems, International Journal for Numerical Methods in Engineering 53, 1959-1977 (2002).
 G. Bell, A refinement of the heat balance integral method applied to a melting problem, International Journal of Heat and Mass Transfer 21, 1357-1362 (1986).
 G. Comini, S.D. Guidice, R.W. Lewis, O.C. Zienkiewicz, Finite element solution on non-linear heat conduction problems with special reference to phase change, International Journal of Numerical Methods in Engineering 8, 613-624 (1974).
 D. Rolph III, K.J. Bathe, An efficient algorithm for analysis of nonlinear heat transfer with phase change, International Journal of Numerical Methods in Engineering 18, 119-134 (1982).
 D. Adalsteinsson, J.A. Sethian, A fast level set method for propagating interfaces, Journal of Computational Physics 118, 269-277 (1995).
 T. Barth, J. Sethian, Numerical schemes for the hamilton-jacobi and level set equations on triangulated domains, Journal of Computational Physics 145, 1-40 (1998).
 S. Osher, J.A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on the Hamilton-Jacobi formulation, Journal of Computational Physics 79, 12-49 (1988).
 T. Skrzypczak, E. Węgrzyn-Skrzypczak, Mathematical and numerical model of solidification process of pure metals, International Journal of Heat and Mass Transfer 55, 4276-4284 (2012).
 D. Peng, B. Merriman, S. Osher, H. Zhao, A PDE-based fast local level set method, Journal of Computational Physics 155, 410-438 (1999).
 M. Sussman, E. Fatemi, An efficient interface preserving level set re-distancing algorithm and its applications to interfacial incompressible fluid flow, Journal of Scientific Computing 20, 1165-1191 (1999).
 B. Mochnacki, J.S. Suchy, Modelowanie i symulacja krzepnięcia odlewów, Wydawnictwo Naukowe PWN, Warszawa, 208, 1993.
 M.J. Assael, A.E. Kalyva, K.D. Antoniadis, Reference data for the density and viscosity of liquid copper and liquid tin, Journal of Physical and Chemical Reference Data 39, 3, 033105-033105-8 (2010).
 G.K. White, S.J. Collocott, Heat capacity of reference materials: Cu and W, Journal of Physical and Chemical Reference Data 13, 4, 1251-1257 (1984).
 T.J. Miller, S.J. Zinkle, B.A. Chin, Strength and fatigue of dispersion-strengthened Copper, Journal of Nuclear Materials 179-181, 263-266 (1991).
 F.P. Incropera, D.P. Dewitt, Fundamentals of heat and mass transfer, 2nd ed., Wiley, New York, 1995.