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Approximation of a solidification problem

Rajae Aboulaïch; Ilham Haggouch; Ali Souissi

International Journal of Applied Mathematics and Computer Science (2001)

  • Volume: 11, Issue: 4, page 921-955
  • ISSN: 1641-876X

Abstract

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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 (Baiocchi, 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.

How to cite

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Aboulaïch, Rajae, Haggouch, Ilham, and Souissi, Ali. "Approximation of a solidification problem." International Journal of Applied Mathematics and Computer Science 11.4 (2001): 921-955. <http://eudml.org/doc/207538>.

@article{Aboulaïch2001,
abstract = {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 (Baiocchi, 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.},
author = {Aboulaïch, Rajae, Haggouch, Ilham, Souissi, Ali},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {free boundary; shape optimization; finite-element method; Euler method; Stefan problem; finite element method},
language = {eng},
number = {4},
pages = {921-955},
title = {Approximation of a solidification problem},
url = {http://eudml.org/doc/207538},
volume = {11},
year = {2001},
}

TY - JOUR
AU - Aboulaïch, Rajae
AU - Haggouch, Ilham
AU - Souissi, Ali
TI - Approximation of a solidification problem
JO - International Journal of Applied Mathematics and Computer Science
PY - 2001
VL - 11
IS - 4
SP - 921
EP - 955
AB - 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 (Baiocchi, 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.
LA - eng
KW - free boundary; shape optimization; finite-element method; Euler method; Stefan problem; finite element method
UR - http://eudml.org/doc/207538
ER -

References

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  2. Baiocchi C. (1977): Problèmes à frontière libre en hydraulique: milieu non homogène. - Annali della Scuola Norm. Sup. di Pisa, Vol.28, pp.429-453. Zbl0386.35044
  3. Ciavaldini J.F. (1972): Resolution numerique d'un problème de Stefan à deux phases. - Ph.D. Thesis, Rennes, France. 
  4. El Bagdouri M. (1987): Commande optimale d'un système thermique non-lineaire. - Thèse de Doctorat d'Etates-Sciences, Ecole Nationale Superieure de Mecanique, Universite de Nantes. 
  5. Haggouch I. (1997): Resolution d'un problème de Stefan à deux phases par la methode d'optimisation de forme. - Ph.D. Thesis, Rabat, Morocco. 
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  8. Lions J.L. (1968): Contrôle Optimal d'un Système Gouverne par des Equations aux Derivees Partielles. - Paris: Dunod. 
  9. Lions J.L. (1969): Quelques Methodes de Resolution des Problèmes aux Limites Non Lineaires. - Paris: Dunod. Zbl0189.40603
  10. Lyaghfouri A. (1996): The inhomogeneous dam with linear Darcy's law and Dirichlet boundary conditions. - Math. Models Meth. Appl. Sci., Vol.6, No.8, pp.1051-1077. Zbl0869.76088
  11. Nochetto R.H., Paolin M. and Verdi C. (1991): Anadaptive finite element method for two phase Stefan problems in two space dimension, Part 1: Stability and error estimates. - Math. Comp., Vol.57, No.57, pp.73-108, S1-S11 (supplement); Part 2: Implementation and Numerical Experiments. - SIAM J. Sci.Stat. Comput., Vol.12, No.5, pp.1207-1244. 
  12. Peneau S. (1995): Contrôle optimal et optimisation de forme dans des problèmes à frontière libre. Application à un système thermique avec changement de phase. - Ph.D., Ecole Central de Nantes, France. 
  13. Pironneau O. (1983): Optimal Shape Design for Elliptic Systems. - Berlin: Springer. Zbl0496.93029
  14. Raviart P.A. and Girault V. (1981): Finite Element Approximation of the Navier-Stokes Equations. - Berlin: Springer. Zbl0441.65081
  15. Rodrigues J.F. (1980): Sur la cristallisation d'un métal en coulée continue par des méthodes variationnelles. - Ph.D. Thesis, Universite Paris 6. 
  16. Saguez C. (1980): Contrôle optimal de systèmes à frontière libre. - Ph.D. Thesis, Université de Technologie de Compiègne, France. Zbl0439.73076
  17. Srunk and Friedman A. (1994): Variational and Free Boundary Problems. - Berlin: Springer. 
  18. Zolésio J.P. (1981): The material derivative (or speed method) for shape optimisation, In: Optimisation of Parameter Structures, Vol.II (E.J. Haug and J. Cea, Eds.). - Alphen aan den Rijn, the Netherlands: Sijthoff, pp.1098-1151. 
  19. Zolésio J.P. (1979): Identification de domaines par déformations. - Thèse d'Etat, Université de Nice, France. 

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