Afrika Matematika

, Volume 28, Issue 3–4, pp 417–441 | Cite as

Numerical simulation of steel solidification in continuous casting

  • Ghania Khenniche
  • Pierre SpiteriEmail author
  • Salah Bouhouche
  • Hocine Sissaoui


The present study deals with the numerical simulation of steel solidification in continuous casting. We consider a semi-discretization with respect to the time of the studied evolution problem; then we have to solve a sequence of stationary coupled problems. So, due to the fact that the temperature is assumed to be positive, after reformulation of the problem into a variational inequality, we study under appropriate assumptions the existence and uniqueness of the solution of the stationary coupled problems. We also consider a multivalued formulation of the same problem which allows to analyze the behavior of the iterative relaxation algorithms used for the solution of the discretized problems. Finally the numerical experiments are presented.


Continuous casting Solidification of steel Sparse nonlinear systems Relaxation method 

Mathematics Subject Classification

58J35 35B09 58J26 35R70 



This study was possible thanks to a grant conceded to Miss Ghania Khenniche by the Algerian govermment. Also Miss Ghania Khenniche gratefully acknowledges support provided by the National Polytechnic Institute of Toulouse and the Institute of Research and computer science of Toulouse during her stage in France.


  1. 1.
    Axelson, O., Barker, V.A.: Finite element solution of boundary value problems, theory and computation. Academic Press, USA (1984)Google Scholar
  2. 2.
    Barbu, V. : Nonlinear semigroups and differential equations in Banach spaces, Noordhoff International Publishing (1976)Google Scholar
  3. 3.
    Briozzo, A.C., Natale, M.F., Tarzia, D.A.: Determination of unknown thermal coeffcients for Storm’s-type materials through a phase-change process. Int. J. Non Linear Mech. 34, 324–340 (1999)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chau, M., Couturier, R., Bahi, J., Spiteri, P.: Parallel solution of the obstacle problem in Grid environments. Int. J. High Perform. Comput. Appl. 25(4), 488–495 (2011)CrossRefGoogle Scholar
  5. 5.
    Chau, M., Tauber, C., Spiteri, P.: Parallel Schawarz alternating methods for anisotropic diffusion of speckled medical images. Numer. Algorithms 51, 85–114 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Comini, G., Del Guiduce, S., Lewwis, R.W., Zienkiewicz, O.C.: Finite element solution of non-linear heat conduction problems with special reference to phase change. Int. J. Numer. Methods Eng 8, 613–624 (1974)CrossRefGoogle Scholar
  7. 7.
    Costes, F.: Modélisation thermomécanique tridimensionnelle par éléments finis de la coulée continue d’acier. Thèse de doctorat, ENSM Paris (2004)Google Scholar
  8. 8.
    Duvaut, G., Lions, J.L. : Les inéquations en mécanique et physique, Dunod (1972)Google Scholar
  9. 9.
    El Tarazi, M.: Some convergence results for asynchronous algorithms. Numer. Math. 39, 325–340 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fortin, A., Garon, A.: Les éléments finis de la théorie à la pratique. Ecole Polytechnique de Montréal (1999)Google Scholar
  11. 11.
    George, P.L.: Simulation Numérique d’un problème Parabolique non linéaire Trempe d’un Bareau métallique, INRIA (1982)Google Scholar
  12. 12.
    Glowinski. R., Lions, J.L., Tremolieres, R.: Analyse numérique des inéquations variationnelles, Dunod, tome 1 and 2 (1976)Google Scholar
  13. 13.
    Kandeil, A.Y., Tag, I.A., Hassab, M.A.: Solidification of Steel billets in continuous casting. Eng. J. Qatar Univ. 4, 103–120 (1991)Google Scholar
  14. 14.
    Lewis, R.W., Ravidrans, K.: Finite element simulation of metal casting. Int. J. Numer. Methods Eng. 47(1–3), 29–59 (2000)CrossRefzbMATHGoogle Scholar
  15. 15.
    Miellou, J.C.: Algorithmes de relaxation chaotique à retards. RAIRO Anal. Numérique R1, 55–82 (1975)zbMATHGoogle Scholar
  16. 16.
    Miellou, J.C., Spiteri, P. : Un critère de convergence pour des méthodes générales de point fixe. M2AN 19, 645–669 (1985)Google Scholar
  17. 17.
    Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. Academic Press, New York (1970)zbMATHGoogle Scholar
  18. 18.
    Tauber, C., Spiteri, P., Batatia, H.: Iterative methods for anisotropic diffusion of speckled medical images. Appl. Num. Math. 60, 1115–1130 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ziane-Khodja, L., Chau, M., Couturier, R., Bahi, J., Spiteri, P.: Parallel solution of American option derivatives on GPU clusters. Comp. Math. Appl. 65, 1830–1848 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Ghania Khenniche
    • 1
    • 2
  • Pierre Spiteri
    • 2
    Email author
  • Salah Bouhouche
    • 3
  • Hocine Sissaoui
    • 4
  1. 1.Laboratory LAMAHIS, Department of Mathematics, Faculty of sciencesUniversity 20 August 1955SkikdaAlgeria
  2. 2.Laboratory IRIT-INP-ENSEEIHTToulouse CEDEXFrance
  3. 3.Welding and NDT Research CentreIron and Steel Applied Research Unit-CSCAnnabaAlgeria
  4. 4.Laboratory LANOSBadji Mokhtar UniversityAnnabaAlgeria

Personalised recommendations