Optimized Schwarz Method for the Fluid-Structure Interaction with Cylindrical Interfaces

Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)

Abstract

The Optimized Schwarz Method (OSM) is a domain decomposition method based on the introduction of generalized Robin interface conditions obtained by linearly combining the two physical interface conditions through the introduction of suitable symbols, and then on the optimization of such symbols within a proper subset, see [10, 13]. This method has been considered so far for many problems in the case of flat interfaces, see, e.g., [3, 5–7, 11, 16, 17]. Recently, OSM has been considered and analyzed for the case of cylindrical interfaces in [8, 9], and for the case of circular interfaces in [2]. In particular, in [8] we developed a general convergence analysis of the Schwarz method for elliptic problems and an optimization procedure within the constants, with application to the fluid-structure interaction (FSI) problem.

References

  1. 1.
    S. Badia, F. Nobile, C. Vergara, Fluid-structure partitioned procedures based on Robin transmission conditions. J. Comput. Phys. 227, 7027–7051 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    H. Barucq, M.J. Gander, Y. Xu, On the Influence of Curvature on Transmission Conditions, in Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering (Springer, Berlin, 2013)Google Scholar
  3. 3.
    V. Dolean, M.J. Gander, L.G. Giorda, Optimized Schwarz Methods for Maxwell’s equations. SIAM J. Sci. Comput. 31(3), 2193–2213 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    J. Donea, An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interaction. Comput. Meth. Appl. Mech. Eng. 33, 689–723 (1982)CrossRefMATHGoogle Scholar
  5. 5.
    M.J. Gander, Optimized Schwarz Methods. SIAM J. Numer. Anal. 44(2), 699–731 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    M.J. Gander, F. Magoulès, F. Nataf, Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24, 38–60 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    L. Gerardo Giorda, F. Nobile, C. Vergara, Analysis and optimization of Robin-Robin partitioned procedures in fluid-structure interaction problems. SIAM J. Numer. Anal. 48(6), 2091–2116 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    G. Gigante, C. Vergara, Analysis and optimization of the generalized Schwarz method for elliptic problems with application to fluid-structure interaction. Numer. Math., doi:10.1007/s00211-014-0693-2Google Scholar
  9. 9.
    G. Gigante, M. Pozzoli, C. Vergara, Optimized Schwarz Methods for the diffusion-reaction problem with cylindrical interfaces. SIAM J. Numer. Anal. 51(6), 3402–3420 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    C. Japhet, Optimized Krylov-Ventcell method. Application to convection-diffusion problems, in Proceedings of the Ninth International Conference on Domain Decomposition Methods, ed. by P.E. Bjorstad, M.S. Espedal, D.E. Keyes (1998) pp. 382–389Google Scholar
  11. 11.
    C. Japhet, N. Nataf, F. Rogier, The optimized order 2 method. Application to convection-diffusion problems. Futur. Gener. Comput. Syst. 18, 17–30 (2001)CrossRefMATHGoogle Scholar
  12. 12.
    N. Lebedev, Special Functions and Their Applications (Courier Dover Publications, New York, 1972)MATHGoogle Scholar
  13. 13.
    P.L. Lions, On the Schwartz alternating method III, in Proceedings of the Third International Symposium on Domain Decomposition Methods for PDE’s, ed. by T. Chan, R. Glowinki, J. Periaux, O.B. Widlund (SIAM, Philadelphia, 1990), pp. 202–223Google Scholar
  14. 14.
    F. Nobile, C. Vergara, Partitioned algorithms for fluid-structure interaction problems in haemodynamics. Milan J. Math. 80(2), 443–467 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    F. Nobile, M. Pozzoli, C. Vergara, Time accurate partitioned algorithms for the solution of fluid-structure interaction problems in haemodynamics. Comput. Fluids 86, 470–482 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    A. Qaddouria, L. Laayounib, S. Loiselc, J. Cotea, M.J. Gander, Optimized Schwarz methods with an overset grid for the shallow-water equations: preliminary results. Appl. Numer. Math. 58, 459–471 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    B. Stupfel, Improved transmission conditions for a one-dimensional domain decomposition method applied to the solution of the Helmhotz equation. J. Comput. Phys. 229, 851–874 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria Gestionale, dell’Informazione e della ProduzioneUniversità di BergamoDalmine (BG)Italy
  2. 2.MOX, Dipartimento di MatematicaPolitecnico di MilanoMilanItaly

Personalised recommendations