# 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

**Publications:**

- 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. - 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 - S. Deparis, M. Discacciati, G. Fourestey, A. Quarteroni, Fluid-structure algorithms based on Steklov-Poincaré operators,
**Computer Methods in Applied Mechanics and Engineering**, 195(41-43), pp. 5797-812, 2006 - A. Quaini, A. Quarteroni, A semi-implicit approach for fluid-structure interaction based on an algebraic fractional step method,
**Mathematical Models and Methods in Applied Sciences**, 17(6), pp. 957-983, 2007 - Nobile F., Vergara C., An effective fluid-structure interaction formulation for vascular dynamics by generalized Robin conditions,
**SIAM J. Sc. Comp.**, 30(2), pp. 731-763, 2008 - Badia S., Nobile F., Vergara C., Fluid-structure partitioned procedures based on Robin transmission conditions,
**J. Comp. Phys.**, 227, pp. 7027-7051, 2008 - 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 - S. Badia, A. Quaini, A. Quarteroni, Modular vs. non-modular preconditioners for fluid-structure systems with large added-mass effect,
**Computer Methods in Applied Mechanics and Engineering**, 197(49-50), pp. 4216-4232, 2008 - 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 - S. Badia, A. Quaini, A. Quarteroni, Coupling Biot and Navier-Stokes equations for modelling fluid-poroelastic media interaction,
**Journal Of Computational Physics**, 228, pp. 7986-8014, 2009 - 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. - 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 - 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 - P. Crosetto, S. Deparis, G. Fourestey, A. Quarteroni, Parallel Algorithms for Fluid-Structure Interaction Problems in Haemodynamics,
**Siam Journal on Scientific Computing**, 33(4), pp. 1598-1622, 2011 - Perego M., Veneziani A., Vergara C., A variational approach for estimating the compliance of the cardiovascular tissue: An Inverse fluid-structure interaction problem.
**SIAM J. Sc. Comp**, 33(3), pp. 1181-1211, 2011 - P. Crosetto, P. Reymond, S. Deparis, D. Kontaxakis, N. Stergiopulos, A. Quarteroni, Fluid Structure Interaction Simulations of Physiological Blood Flow in the Aorta,
**Computers and Fluids**, 43(1), pp. 46-57, 2011 - F. Nobile, C. Vergara, Partitioned algorithms for fluid-structure interaction problems in haemodynamics.
**Milan Journal of Mathematics**, 80(2), pp. 443-467, 2012 - 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
- T. M. Lassila, A. Quarteroni and G. Rozza. A reduced basis model with parametric coupling for fluid-structure interaction problems,
**Siam Journal on Scientific Computing**, 34, pp. A1187 – A1213, 2012 - F. Nobile, M. Pozzoli, C. Vergara, Time accurate partitioned algorithms for the solution of fluid-structure interaction problems in haemodynamics.
**Computer and Fluids**, 86, pp. 470-482, 2013 - F. Nobile, M. Pozzoli, C. Vergara, Inexact accurate partitioned algorithms for fluid-structure interaction problems with finite elasticity in haemodynamics.
**Journal of Computational Physics**. 273, pp. 598-617, 2014 - C. M. Colciago, S. Deparis, A. Quarteroni, Comparisons Between Reduced Order Models and Full 3D Models for Fluid-Structure Interaction Problems in Haemodynamics,
**Journal of Computational and Applied Mathematics**, 265, pp. 120-138, 2014 - Gigante G., Vergara C., Analysis and optimization of the generalized Schwarz method for elliptic problems with application to fluid-structure interaction.
**Numer. Math.**, 131(2), pp. 369–404, 2015 - 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
- A Comparison of Preconditioners for the Steklov–Poincaré Formulation of the Fluid-Structure Coupling in Hemodynamics, in
**Proceedings in Applied Mathematics and Mechanics**, vol. 15, p. 93-94, 2015.
. - Fluid-structure interaction simulations of cerebral arteries modeled by isotropic and anisotropic constitutive laws, in
**Computational Mechanics**-International Journal then Research Journal-, vol. 55, num. 3, p. 479-498, 2015.
. -
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 - 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. - Quarteroni A., Manzoni A., Vergara C., The Cardiovascular System: Mathematical Modeling, Numerical Algorithms, Clinical Applications. MOX Report n. 38/2016.

**Thesis**

- 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

- 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
- 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
- F. Premoli – Emodinamica e aterosclerosi: simulazioni numeriche per problemi di interazione fluido-struttura -MSc in Aerospace Engineering – Advisor: F. Nobile – A.A ’08-’09
- L. Cassani – Metodi di domini fittizi per problemi di interazione fluido struttura -Advisors: L. Formaggia, F. Nobile – A.A ’08-’09
- 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
- G. Mengaldo –
*Nonlinear fluid-structure interaction with application in computational haemodynamics –*MSc in Aerospace Engineering – Advisors: L. Formaggia, F. Nobile – A.A ’10-’11 - 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
- Davide Forti, Comparison of shape parametrization techniques for fluid structure interaction problems, – MSc in Aerospace Engineering – Advisors: A. Quarteroni, A.A. ’11-’12
- 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 - 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

**Projects**

**Links**

– Application to stentless aortic valve prosthesis

– The geometrical multiscale approach

– FSI for the heart