Geodesic
Approximation of a Solidification Problem
International Journal of Applied Mathematics and Computer Science, Tome 11 (2001) no. 4, pp. 921-955

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A two-dimensional Stefan problem is usually introduced as a model of solidification, melting or sublimation phenomena. The two-phase Stefan problem has been studied as a direct problem, where the free boundary separating the two regions is eliminated using a variational inequality (Baiocci, 1977; Baiocchi et al., 1973; Rodrigues, 1980; Saguez, 1980; Srunk and Friedman, 1994), the enthalpy function (Ciavaldini, 1972; Lions, 1969; Nochetto et al.., 1991; Saguez, 1980), or a control problem (El Bagdouri, 1987; Peneau, 1995; Saguez, 1980). In the present work, we provide a new formulation leading to a shape optimization problem. For a semidiscretization in time, we consider an Euler scheme. Under some restrictions related to stability conditions, we prove an L^2-rate of convergence of order 1 for the temperature. In the last part, we study the existence of an optimal shape, compute the shape gradient, and suggest a numerical algorithm to approximate the free boundary. The numerical results obtained show that this method is more efficient compared with the others.
Keywords: Stefan problem, free boundary, shape optimization, Euler method, finite element method
Mots-clés : optymalizacja, metoda elementów skończonych 
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     title = {Approximation of a {Solidification} {Problem}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IJAMCS_2001_11_4_a8/}
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Aboulaich, R.; Haggouch, I.; Souissi, A. Approximation of a Solidification Problem. International Journal of Applied Mathematics and Computer Science, Tome 11 (2001) no. 4, pp. 921-955. http://geodesic.mathdoc.fr/item/IJAMCS_2001_11_4_a8/