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Numerical Algorithms

, Volume 80, Issue 2, pp 447–467 | Cite as

Parallel computing investigations for the projection method applied to the interface transport scheme of a two-phase flow by the method of characteristics

  • Mireille HaddadEmail author
  • Frédéric Hecht
  • Toni Sayah
  • Pierre Henri Tournier
Original Paper
First Online:
  • 28 Downloads

Abstract

This paper deals with the discretization of the problem of mould filling in iron foundry and its numerical solution using a Schwarz domain decomposition method. An adapted technique for domain decomposition methods that suits the discretization in time by the method of characteristics is introduced. Furthermore, the projection method is used to reduce the computation time. Finally, numerical experiments show and validate the effectiveness of the proposed scheme.

Keywords

Two-phase flow Level-set function Finite element method Method of characteristics Domain decomposition methods Overlapping decomposition Parallel computing 
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References

  1. 1.
    Amestoy, P.R., Duff, I.S., L’Excellent, J.Y.: Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods in Appl. Mech. Eng. 184, 501–520 (2000)CrossRefzbMATHGoogle Scholar
  2. 2.
    Amestoy, P.R., Duff, I.S., Koster, J., L’Excellent, J.Y.: A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23, 15–41 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Amestoy, P.R., Guermouche, A., L’Excellent, J.Y., Pralet, S.: Hybrid scheduling for the parallel solution of linear systems. Parallel Comput. 32, 136–156 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Babuska, I.: The finite element method with penalty. Math. Comput. 27, 221–228 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press (1967)Google Scholar
  6. 6.
    Benqué, J.P., Ibler, B., Keramsi, A., Labadie, G.: A finite element method for the Navier-Stokes equations. In: Proceedings of the Third International Conference on Finite Elements in Flow Problems. Banff. Alberta (1980)Google Scholar
  7. 7.
    Cai, X.C., Farhat, C., Sarkis, M.: A minimum overlap restricted additive Schwarz preconditioner and applications to 3D flow simulations. Contemp. Math. 218, 479–485 (1998)CrossRefzbMATHGoogle Scholar
  8. 8.
    Cai, X.C., Sarkis, M.: A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM J. Sci. Comput. 21, 239–247 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chevalier, C., Pellegrini, F.: PT-SCOTCH: A tool for efficient parallel graph ordering. Parallel Comput. 34, 318–331 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chorin, A.J.: Numerical solution of the Navier-Stokes equations. Math. Comp. 22, 745–762 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chorin, A.J.: On the convergence of discrete approximations to the Navier- Stokes equations. Math. Comp. 23, 341–353 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Davis, T.A., Duff, I.S.: A combined unifrontal/multifrontal method for unsymmetric sparse matrices. ACM Trans. Math. Softw. 25, 1–19 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dervieux, A., Thomasset, F.: A finite element method for the simulation of a Rayleigh-Taylor instablity. In: Approximation Methods for Navier-Stokes Problems, Lecture Notes in Mathematics, vol. 771, pp. 145–158. Springer, Berlin (1980)Google Scholar
  14. 14.
    Dolean, V., Jolivet, P., Nataf, F.: An Introduction to Domain Decomposition Methods: Algorithms, Theory and Parallel Implementation. SIAM, PA (2015)CrossRefzbMATHGoogle Scholar
  15. 15.
    Gander, M.J.: Schwarz methods over the course of time. Electron. Trans. Numer. Anal. 31, 228–255 (2008)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gupta, P.K., Pagalthivarthi, K.V.: Application of multifrontal and GMRES solvers for multisize particulate flow in rotating channels. Prog. Comput. Fluid Dynam. 7, 323–336 (2007)CrossRefGoogle Scholar
  17. 17.
    Haddad, M., Hecht, H., Sayah, T: Interface transport scheme of a two-phase flow by the method of characteristics. Int. J. Numer. Methods Fluids 68, 513–543 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hecht, F.: New development in FreeFem++. J. Numer. Math. 20, 251–256 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Jolivet, P.: Domain decomposition methods. Application to high-performance computing, PhD thesis University of Grenoble, France (2014)Google Scholar
  20. 20.
    Jolivet, P., Hecht, F., Nataf, F., Prud’homme, C.: Scalable domain decomposition preconditioners for heterogeneous elliptic problems. Sci. Program. 22, 157–171 (2014)Google Scholar
  21. 21.
    Karypis, G., Kumar. V.: Unstructured graph partitioning and sparse matrix ordering system (version 2.0. Technical report), University of Minnesota, Department of Computer Science. Minneapolis, MN 55455 (1995)Google Scholar
  22. 22.
    Lamb, H.: Hydrodynamics. Cambridge University Press (1932)Google Scholar
  23. 23.
    Navier, C.L.M.H.: Sur les lois de l’équilibre et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst., vol. 6. France, p. 369 (1827)Google Scholar
  24. 24.
    Peng, D., Merriman, B., Osher, S., Zhao, H., Kang, M.: A PDE-based fast local level-set method. J. Comput. Phys. 155, 410–438 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Raju, M.P.: Parallel computation of finite element Navier-Stokes codes using MUMPS solver. Int. J. Comput. Sci. 4, 20–24 (2009)Google Scholar
  26. 26.
    Saad, Y.: Iterative Methods For Sparse Linear Systems, 2nd edn. SIAM Publishers (2000)Google Scholar
  27. 27.
    Saad, Y., Schultz, M.: A generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)CrossRefzbMATHGoogle Scholar
  28. 28.
    Schwarz, H.A.: Uber einen Grenzubergang durch alternierendes Verfahren. Vierteljahresschrift der Naturforschenden Gesellschaft in Zurich 15, 272–286 (1870)Google Scholar
  29. 29.
    Smalonski, A.: Numerical Modeling of Two-Fluid Interfacial Flows, PhD thesis University of Jyvaskyla Finland (2001)Google Scholar
  30. 30.
    Sussmann, M., Smereka, P., Osher, S.: A level-set approach for computing solutions to incompressible two-phase flows. J. Comput. Phys. 114, 146–159 (1994)CrossRefzbMATHGoogle Scholar
  31. 31.
    Toselli, A., Widlund, O.: Domain Decomposition Methods - Algorithms and Theory. Springer Series in Computational Mathematics, p. 34 (2005)Google Scholar
  32. 32.
    Toselli, A., Widlund, O.B.: Domain Decomposition Methods: Algorithms and Theory. Springer (2005)Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Mireille Haddad
    • 1
    • 2
    • 3
    Email author
  • Frédéric Hecht
    • 1
    • 2
  • Toni Sayah
    • 3
  • Pierre Henri Tournier
    • 1
    • 2
  1. 1.Laboratoire Jacques-Louis LionsParisFrance
  2. 2.Sorbonne Université, UPMC Univ Paris 06, UMR 7598Laboratoire Jacques-Louis LionsParisFrance
  3. 3.Unité de recherche EGFEM, Faculté des SciencesUniversité Saint-JosephBeyrouthLebanon

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