Robin-Neumann Schemes for Incompressible Fluid-Structure Interaction

  • Miguel A. Fernández
  • Mikel Landajuela
  • Jimmy Mullaert
  • Marina VidrascuEmail author
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)


Mathematical problems involving the coupling of an incompressible viscous flow with an elastic structure appear in a large variety of engineering fields (see, e.g., [14, 17, 19–21]). This problem is considered here within a heterogenous domain decomposition framework, with the aim of using independent well-suited solvers for the fluid and the solid. One of the main difficulties that have to be faced under this approach is that the coupling can be very stiff. In particular, traditional Dirichlet-Neumann explicit coupling methods, which solve for the fluid (Dirichlet) and for the solid (Neumann) only once per time-step, are unconditionally unstable whenever the amount of added-mass effect in the system is large (see, e.g., [5, 12]). Typically this happens when the fluid and solid densities are close and the fluid domain is slender, as in hemodynamical applications. This explains, in part, the tremendous amount of work devoted over the last decade to the development of alternative coupling paradigms (see, e.g., [7] for a review).


Implicit Scheme Incompressible Viscous Flow Fundamental Ingredient Explicit Coupling Kinematic Perturbation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was supported by the French National Research Agency (ANR) through the EXIFSI project (ANR-12-JS01-0004)


  1. 1.
    S. Badia, F. Nobile, C. Vergara, Fluid-structure partitioned procedures based on Robin transmission conditions. J. Comput. Phys. 227(14), 7027–7051 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    M. Bukač, S. Čanić, R. Glowinski, J. Tambača, A. Quaini, Fluid-structure interaction in blood flow capturing non-zero longitudinal structure displacement. J. Comput. Phys. 235, 515–541 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    E. Burman, M.A. Fernández, Stabilization of explicit coupling in fluid-structure interaction involving fluid incompressibility. Comput. Methods Appl. Mech. Eng. 198(5–8), 766–784 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    E. Burman, M.A. Fernández, Explicit strategies for incompressible fluid-structure interaction problems: Nitsche type mortaring versus Robin-Robin coupling. Int. J. Numer. Methods Eng. 97(10), 739–758 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    P. Causin, J.F. Gerbeau, F. Nobile, Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput. Methods Appl. Mech. Eng. 194(42–44), 4506–4527 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    D. Chapelle, K.J. Bathe, The Finite Element Analysis of Shells—Fundamentals. Computational Fluid and Solid Mechanics (Springer, Berlin, 2011)CrossRefzbMATHGoogle Scholar
  7. 7.
    M.A. Fernández, Coupling schemes for incompressible fluid-structure interaction: implicit, semi-implicit and explicit. S \({\boldsymbol {\mathrm{e}}}\) MA J. 55(55), 59–108 (2011)Google Scholar
  8. 8.
    M.A. Fernández, Incremental displacement-correction schemes for incompressible fluid-structure interaction: stability and convergence analysis. Numer. Math. 123(1), 21–65 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M.A. Fernández, J. Mullaert, Convergence analysis for a class of explicit coupling schemes in incompressible fluid-structure interaction (2014). Research report RR-8670, Inria, 2015Google Scholar
  10. 10.
    M.A. Fernández, J. Mullaert, M. Vidrascu, Explicit Robin-Neumann schemes for the coupling of incompressible fluids with thin-walled structures. Comput. Methods Appl. Mech. Eng. 267, 566–593 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    M.A. Fernández, J. Mullaert, M. Vidrascu, Generalized Robin-Neumann explicit coupling schemes for incompressible fluid-structure interaction: stability analysis and numerics. Int. J. Numer. Methods Eng. 101(3), 199–229 (2015). doi:10.1002/nme.4785. MathSciNetCrossRefGoogle Scholar
  12. 12.
    C. Förster, W.A. Wall, E. Ramm, Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows. Comput. Methods Appl. Mech. Eng. 196(7), 1278–1293 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    G. Guidoboni, R. Glowinski, N. Cavallini, S. Čanić, Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow. J. Comput. Phys. 228(18), 6916–6937 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    M. Heil, A.L. Hazel, Fluid-structure interaction in internal physiological flows. Annu. Rev. Fluid Mech. 43, 141–162 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    P. Kalita, R. Schaefer. Mechanical models of artery walls. Arch. Comput. Methods Eng.15(1), 1–36 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    U. Küttler, C. Förster, W.A. Wall, A solution for the incompressibility dilemma in partitioned fluid-structure interaction with pure Dirichlet fluid domains. Comput. Mech. 38, 417–429 (2006)CrossRefzbMATHGoogle Scholar
  17. 17.
    M. Lombardi, N. Parolini, A. Quarteroni, G. Rozza, Numerical simulation of sailing boats: dynamics, FSI, and shape optimization, in Variational Analysis and Aerospace Engineering: Mathematical Challenges for Aerospace Design, ed. by G. Buttazzo, A. Frediani. Springer Optimization and Its Applications (Springer, Berlin, 2012), pp. 339–377Google Scholar
  18. 18.
    M. Lukáčová-Medvid’ová, G. Rusnáková, A. Hundertmark-Zaušková, Kinematic splitting algorithm for fluid-structure interaction in hemodynamics. Comput. Methods Appl. Mech. Eng. 265(1), 83–106 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    P. Moireau, N. Xiao, M. Astorino, C.A. Figueroa, D. Chapelle, C.A. Taylor, J.-F. Gerbeau, External tissue support and fluid-structure simulation in blood flows. Biomech. Model. Mechanobiol. 11, 1–18 (2012)CrossRefGoogle Scholar
  20. 20.
    M.P. Païdoussis, S.J. Price, E. de Langre, Fluid-Structure Interactions: Cross-Flow-Induced Instabilities (Cambridge University Press, Cambridge, 2011)zbMATHGoogle Scholar
  21. 21.
    K. Takizawa, T.E. Tezduyar, Computational methods for parachute fluid-structure interactions. Arch. Comput. Methods Eng. 19, 125–169 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    D. Valdez-Jasso, H.T. Banks, M.A. Haider, D. Bia, Y. Zocalo, R.L. Armentano, M.S. Olufsen, Viscoelastic models for passive arterial wall dynamics. Adv. Appl. Math. Mech. 1(2), 151–165 (2009)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Miguel A. Fernández
    • 1
    • 2
  • Mikel Landajuela
    • 1
    • 2
  • Jimmy Mullaert
    • 1
    • 2
  • Marina Vidrascu
    • 1
    • 2
    Email author
  1. 1.Inria Paris-RocquencourtLe Chesnay CedexFrance
  2. 2.Sorbonne Universités, UPMC University Paris 6, Laboratoire Jacques-Louis LionsParis Cedex 05France

Personalised recommendations