Fluid Structure Interaction

The interaction between the blood and the deformable artery leads to a very complex coupled problem

The solution of this fluid-structure interaction problem has been addressed by means of partitioned algorithms, whose convergence properties are in general very poor due to the high added mass effect characterizing the haemodynamic regime.

A theoretical analysis of the classical Dirichlet-Neumann scheme highlighted the instability of an explicit algorithm and the slowness of an implicit one. To improve the convergence rate Robin-type interface conditions have been studied. With a suitable choice of the interface parameters (based on the Optimized Schwarz Method), the convergence is quite independent of the added-mass effect

A different class of partitioned algorithms is derived by an algebraic fractional step method, leading to semi-implicit algorithms, where the geometrical problem is solved not exactly. In view of the applications to real clinical problems, semi-implicit algorithms have been extended to the case of non-linear models of elasticity for the vessel, obtaining efficient and accurate partitioned algorithms

A completely different approach consists in the monolithic schemes where suitable global preconditioners have been studied in view of an efficient parallel solution

We mention also the research on reduced models, where the structure is modelled as a membrane leading to simplified FSI problems involving only the fluid problem with suitable boundary conditions, and that on parameter estimation in FSI problems



  1. S. Deparis, M. Fernandez, L. Formaggia, and F. Nobile, Modified fixed point algorithm in fluid-structure interaction, C.R. Mecanique, 331, pp. 525–530, 2003.
  2. Causin P., Gerbeau J.F., Nobile F., Added-mass effect in the design of partitioned algorithms for fluid–structure problems, Comp. Meth. Appl. Meth. Eng.194(42-44), pp. 4506-4527, 2005
  3. S. Deparis, M. Discacciati, G. Fourestey, A. Quarteroni, Fluid-structure algorithms based on Steklov-Poincaré operatorsComputer Methods in Applied Mechanics and Engineering, 195(41-43), pp. 5797-812, 2006
  4. A. Quaini, A. Quarteroni, A semi-implicit approach for fluid-structure interaction based on an algebraic fractional step methodMathematical Models and Methods in Applied Sciences, 17(6), pp. 957-983, 2007
  5. Nobile F., Vergara C., An effective fluid-structure interaction formulation for vascular dynamics by generalized Robin conditionsSIAM J. Sc. Comp., 30(2), pp. 731-763, 2008
  6. Badia S., Nobile F., Vergara C., Fluid-structure partitioned procedures based on Robin transmission conditionsJ. Comp. Phys., 227, pp. 7027-7051, 2008
  7. S. Badia, A. Quaini, A. Quarteroni, Splitting methods based on algebraic factorization for fluid-structure interaction, SIAM Journal on Scientific Computing, 30(4), pp. 1778-1805, 2008
  8. S. Badia, A. Quaini, A. Quarteroni, Modular vs. non-modular preconditioners for fluid-structure systems with large added-mass effectComputer Methods in Applied Mechanics and Engineering, 197(49-50), pp. 4216-4232, 2008
  9. Badia S., Nobile F., Vergara C., Robin-Robin preconditioned Krylov methods for fluid-structure interaction problems Comp. Meth. Appl. Mech. Eng., 198 (33-36), pp. 2768-2784, 2009
  10. S. Badia, A. Quaini, A. Quarteroni, Coupling Biot and Navier-Stokes equations for modelling fluid-poroelastic media interactionJournal Of Computational Physics, 228, pp. 7986-8014, 2009
  11. F. Nobile, Coupling strategies for the numerical simulation of blood
    ow in deformable arteries by 3D and 1D models,
     Mathematical and Computer Modelling, 49(11-12), pp. 2152–2160, 2009.
  12. Formaggia L., Veneziani A., Vergara C., Flow rate boundary problems for an incompressible fluid in deformable domains: formulations and solution methods Comp. Meth. Appl. Mech. Eng., 199 (9-12), pp. 677-688, 2010
  13. Gerardo-Giorda L., Nobile F., Vergara C., Analysis and optimization of Robin-Robin partitioned procedures in fluid-structure interaction problems. SIAM J. Num. Anal. , 48(6), pp. 2091-2116, 2010
  14. P. Crosetto, S. Deparis, G. Fourestey, A. Quarteroni, Parallel Algorithms for Fluid-Structure Interaction Problems in HaemodynamicsSiam Journal on Scientific Computing, 33(4), pp. 1598-1622, 2011
  15. Perego M., Veneziani A., Vergara C., A variational approach for estimating the compliance of the cardiovascular tissue: An Inverse fluid-structure interaction problemSIAM J. Sc. Comp, 33(3), pp. 1181-1211, 2011
  16. P. Crosetto, P. Reymond, S. Deparis, D. Kontaxakis, N. Stergiopulos, A. Quarteroni, Fluid Structure Interaction Simulations of Physiological Blood Flow in the AortaComputers and Fluids, 43(1), pp. 46-57, 2011
  17. F. Nobile, C. Vergara, Partitioned algorithms for fluid-structure interaction problems in haemodynamicsMilan Journal of Mathematics, 80(2), pp. 443-467, 2012
  18. Pozzoli M., Vergara C., Nobile F. Efficient algorithms for the solution of fluid-structure interaction problems inhaemodynamic applications, Proceedings of the Conference “Numerical Methods for Hyperbolic EquationsTheory and Applications”, Santiago de Compostela, pp. 355-364, 2012
  19. T. M. Lassila, A. Quarteroni and G. Rozza. A reduced basis model with parametric coupling for fluid-structure interaction problemsSiam Journal on Scientific Computing, 34, pp. A1187 – A1213, 2012
  20. F. Nobile, M. Pozzoli, C. Vergara, Time accurate partitioned algorithms for the solution of fluid-structure interaction problems in haemodynamicsComputer and Fluids, 86, pp. 470-482, 2013
  21. F. Nobile, M. Pozzoli, C. Vergara, Inexact accurate partitioned algorithms for fluid-structure interaction problems with finite elasticity in haemodynamicsJournal of Computational Physics. 273, pp. 598-617, 2014
  22. C. M. Colciago, S. Deparis, A. Quarteroni, Comparisons Between Reduced Order Models and Full 3D Models for Fluid-Structure Interaction Problems in HaemodynamicsJournal of Computational and Applied Mathematics, 265, pp. 120-138, 2014
  23. Gigante G., Vergara C., Analysis and optimization of the generalized Schwarz method for elliptic problems with application to fluid-structure interactionNumer. Math., 131(2), pp. 369–404, 2015
  24. G. Gigante, C. Vergara, Optimized Schwarz method for the fluid-structure interaction with cylindrical interfaces. Domain Decomposition Methods in Science and Engineering XXII – Lecture Notes in Computational Science and Engineering – Proceedings of the 22nd International Conference on Domain Decomposition Methods, 104, pp. 521-529, 2015
  25. S. DeparisD. FortiA. HeinleinA. Klawonn and A. Quarteroni et alA Comparison of Preconditioners for the Steklov–Poincaré Formulation of the Fluid-Structure Coupling in Hemodynamicsin Proceedings in Applied Mathematics and Mechanics, vol. 15, p. 93-94, 2015.
  26. P. TricerriL. Dede’S. DeparisA. Quarteroni and A. M. Robertson et alFluid-structure interaction simulations of cerebral arteries modeled by isotropic and anisotropic constitutive lawsin Computational Mechanics -International Journal then Research Journal-, vol. 55, num. 3, p. 479-498, 2015.
  27. D.Balzani, S.Deparis, S.Fausten, D.Forti, A.Heinlein, A.Klawonn, A.Quarteroni, O.Rheinbach and J.Schroeder, Numerical Modeling of Fluid Structure Interaction in Arteries with Anisotropic Polyconvex Hyperelastic and Anisotropic Viscoelastic Material Models at Finite Strains,
    IJNMBE, to appear, 2015
  28. S. Deparis, D. Forti, G. Grandperrin, and A. Quarteroni. Facsi: A block parallel preconditioner for fluid-structure interaction in hemodynamics. J. Comput. Physics, (327):700–718, 2016.
  29. Quarteroni A., Manzoni A., Vergara C., The Cardiovascular System: Mathematical Modeling, Numerical Algorithms, Clinical Applications. MOX Report n. 38/2016.



  1. C. Corrado – Studio di un problema di interazione fluido struttura con un metodo ad elementi finiti immersi – MSc in Aerospace Engineering – Advisor: F. Nobile – A.A ’05-’06
  2. S. Carcano, M. Cogliati – Modelli ed algoritmi per lo studio dell’interazione fluido-struttura in emodinamica – BSc in Mathematical Engineering – Politecnico di Milano – Advisor: F. Nobile – Accademic year ’06-’07
  3. M. Pozzoli – Simulazioni numeriche per l’interazione fluido-struttura in emodinamica computazionale – MSc in Mathematical Engineering – Politecnico di Milano – Accademic year ’07-’08. Advisors: F. Nobile and C. Vergara
  4. F. Premoli – Emodinamica e aterosclerosi: simulazioni numeriche per problemi di interazione fluido-struttura -MSc in Aerospace Engineering – Advisor: F. Nobile – A.A ’08-’09
  5. L. Cassani – Metodi di domini fittizi per problemi di interazione fluido struttura -Advisors: L. Formaggia, F. Nobile – A.A ’08-’09
  6. A. Mencarelli, A. Meschini – Modelli di interazione fluido-membrana e applicazioni in ambito aerodinamico – BSc in Aerospace Engineering – Politecnico di Milano – Accademic year ’08-’09. Advisors: M. Verani and C. Vergara
  7. G. Mengaldo – Nonlinear fluid-structure interaction with application in computational haemodynamics – MSc in Aerospace Engineering – Advisors: L. Formaggia, F. Nobile – A.A ’10-’11
  8. M. Lancellotti – Numerical Computations of Deflated Vascular Geometries for Fluid-Structure Interaction in Haemodynamics – MSc in Aerospacial Engineering – Università degli Studi di Napoli – Accademic year ’11-’12. Advisors: S. De Rosa and C. Vergara
  9. Davide Forti, Comparison of shape parametrization techniques for fluid structure interaction problems, – MSc in Aerospace Engineering – Advisors: A. Quarteroni, A.A. ’11-’12
  10. M. Pozzoli – Efficient partitioned algorithms for the solution of fluid-structure interaction problems in haemodynamics – Ph.D in Mathematical
    Models and Methods in Engineering, Dipartimento di Matematica, Politecnico di Milano – 2012. Advisors: F. Nobile and C. Vergara
  11. E. Orso – Analisi computazionale dell’interazione fluido-struttura in aorta ascendente con valvola aortica stentless, MSc in Biomedial Engineering, Politecnico di Milano, Accademic year ’12-’13. Advisors: P. Antonietti and C: Vergara