Reduced-Order Semi-Implicit Schemes for Fluid-Structure Interaction Problems

  • Francesco BallarinEmail author
  • Gianluigi Rozza
  • Yvon Maday
Part of the MS&A book series (MS&A, volume 17)


POD–Galerkin reduced-order models (ROMs) for fluid-structure interaction problems (incompressible fluid and thin structure) are proposed in this paper. Both the high-fidelity and reduced-order methods are based on a Chorin-Temam operator-splitting approach. Two different reduced-order methods are proposed, which differ on velocity continuity condition, imposed weakly or strongly, respectively. The resulting ROMs are tested and compared on a representative haemodynamics test case characterized by wave propagation, in order to assess the capabilities of the proposed strategies.



We acknowledge the support by European Union Funding for Research and Innovation—Horizon 2020 Program—in the framework of European Research Council Executive Agency: H2020 ERC CoG 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics” and the PRIN project “Mathematical and numerical modelling of the cardiovascular system, and their clinical applications”. We also acknowledge the INDAM-GNCS project “Tecniche di riduzione della complessità computazionale per le scienze applicate” and INDAM-GNCS young researchers project “Numerical methods for model order reduction of PDEs”.


  1. 1.
    Amsallem, D., Cortial, J., Farhat, C.: Towards real-time computational-fluid-dynamics-based aeroelastic computations using a database of reduced-order information. AIAA J. 48(9), 2029–2037 (2010)CrossRefGoogle Scholar
  2. 2.
    Astorino, M., Chouly, F., Fernández, M.A.: Robin based semi-implicit coupling in fluid-structure interaction: Stability analysis and numerics. SIAM J. Sci. Comput. 31(6), 4041–4065 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Badia, S., Nobile, F., Vergara, C.: Fluid–structure partitioned procedures based on Robin transmission conditions. J. Comput. Phys. 227(14), 7027–7051 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Badia, S., Quaini, A., Quarteroni, A.: Splitting methods based on algebraic factorization for fluid-structure interaction. SIAM J. Sci. Comput. 30(4), 1778–1805 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ballarin, F., Manzoni, A., Quarteroni, A., Rozza, G.: Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations. Int. J. Numer. Methods Eng. 102(5), 1136–1161 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ballarin, F., Rozza, G.: POD–Galerkin monolithic reduced order models for parametrized fluid-structure interaction problems. Int. J. Numer. Methods Fluids 82(12), 1010–1034 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ballarin, F., Sartori, A., Rozza, G.: RBniCS – reduced order modelling in fenics. (2016)
  8. 8.
    Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. C.R. Math. 339(9), 667–672 (2004)Google Scholar
  9. 9.
    Colciago, C.M.: Reduced order fluid-structure interaction models for haemodynamics applications. Ph.D. Thesis, École Polytechnique Fédérale de Lausanne, N. 6285 (2014)Google Scholar
  10. 10.
    Fernández, M.A.: Incremental displacement-correction schemes for incompressible fluid-structure interaction. Numer. Math. 123(1), 21–65 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fernández, M.A., Gerbeau, J.F., Grandmont, C.: A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid. Int. J. Numer. Methods Eng. 69(4), 794–821 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fernández, M.A., Landajuela, M., Mullaert, J., Vidrascu, M.: Robin-Neumann schemes for incompressible fluid-structure interaction. Domain Decomposition Methods in Science and Engineering, vol. XXII. pp. 65–76. Springer, Cham (2016)Google Scholar
  13. 13.
    Fernández, M.A., Mullaert, J., Vidrascu, M.: 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
  14. 14.
    Formaggia, L., Gerbeau, J., Nobile, F., Quarteroni, A.: On the coupling of 3D and 1D Navier–Stokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Eng. 191(6–7), 561–582 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Guermond, J.L., Quartapelle, L.: On stability and convergence of projection methods based on pressure poisson equation. Int. J. Numer. Methods Fluids 26(9), 1039–1053 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Guidoboni, G., Glowinski, R., Cavallini, N., Canic, S.: Stable loosely-coupled-type algorithm for fluid–structure interaction in blood flow. J. Comput. Phys. 228(18), 6916–6937 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hesthaven, J.S., Rozza, G., Stamm, B.: Certified reduced basis methods for parametrized partial differential equations. SpringerBriefs in Mathematics. Springer, New York (2015)zbMATHGoogle Scholar
  18. 18.
    Lassila, T., Manzoni, A., Quarteroni, A., Rozza, G.: A reduced computational and geometrical framework for inverse problems in hemodynamics. Int. J. Numer. Methods Biomed. Eng. 29(7), 741–776 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lassila, T., Quarteroni, A., Rozza, G.: A reduced basis model with parametric coupling for fluid-structure interaction problems. SIAM J. Sci. Comput. 34(2), A1187–A1213 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Logg, A., Mardal, K.A., Wells, G.N.: Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  21. 21.
    Quarteroni, A., Formaggia, L.: Mathematical modelling and numerical simulation of the cardiovascular system. In: Computational Models for the Human Body. Handbook of Numerical Analysis, vol. 12, pp. 3–127. Elsevier, Amsterdam (2004)Google Scholar
  22. 22.
    Quarteroni, A., Tuveri, M., Veneziani, A.: Computational vascular fluid dynamics: problems, models and methods. Comput. Vis. Sci. 2(4), 163–197 (2000). doi:10.1007/s007910050039. CrossRefzbMATHGoogle Scholar
  23. 23.
    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations, vol. 23. Springer, Berlin (2008)zbMATHGoogle Scholar
  24. 24.
    Rozza, G.: Reduced basis methods for Stokes equations in domains with non-affine parameter dependence. Comput. Vis. Sci. 12(1), 23–35 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rozza, G., Huynh, D.B.P., Manzoni, A.: Reduced basis approximation and a posteriori error estimation for stokes flows in parametrized geometries: roles of the inf-sup stability constants. Numer. Math. 125(1), 115–152 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Rozza, G., Huynh, D.B.P., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Meth. Eng. 15, 1–47 (2007)CrossRefzbMATHGoogle Scholar
  27. 27.
    Rozza, G., Veroy, K.: On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Eng. 196(7), 1244–1260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Francesco Ballarin
    • 1
    Email author
  • Gianluigi Rozza
    • 1
  • Yvon Maday
    • 2
    • 3
  1. 1.Mathematics Area, mathLab, SISSATriesteItaly
  2. 2.Sorbonne Universités, UPMC Université Paris 06 and CNRS UMR 7598, Laboratoire Jacques-Louis LionsParisFrance
  3. 3.Division of Applied MathematicsBrown UniversityProvidenceUSA

Personalised recommendations