On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations

Formaggia, Luca; Moura, Alexandra; Nobile, Fabio

ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 4, pp. 743-769.

We consider the coupling between three-dimensional ($3$D) and one-dimensional ($1$D) fluid-structure interaction (FSI) models describing blood flow inside compliant vessels. The $1$D model is a hyperbolic system of partial differential equations. The $3$D model consists of the Navier-Stokes equations for incompressible newtonian fluids coupled with a model for the vessel wall dynamics. A non standard formulation for the Navier-Stokes equations is adopted to have suitable boundary conditions for the coupling of the models. With this we derive an energy estimate for the fully $3$D-$1$D FSI coupling. We consider several possible models for the mechanics of the vessel wall in the $3$D problem and show how the $3$D-$1$D coupling depends on them. Several comparative numerical tests illustrating the coupling are presented.

Classification : 65M12, 65M60, 92C50, 74F10, 76Z05
Mots clés : fluid-structure interaction, 3D-1D FSI coupling, energy estimate, multiscale models

[1] C. Begue, C. Conca, F. Murat and O. Pironneau, Les équations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression, in Nonlinear partial differential equations and their applications, Collège de France Seminar, in Pitman Res. Notes Math. Ser. 181, Longman Sci. Tech., Harlow (1986) 179-264. | Zbl

[2] H. Beirão Da Veiga, On the existence of strong solutions to a coupled fluid-structure evolution problem. J. Math. Fluid Mechanics 6 (2004) 21-52. | Zbl

[3] C.G. Caro and K.H. Parker, The effect of haemodynamic factors on the arterial wall, in Atherosclerosis - Biology and Clinical Science, A.G. Olsson Ed., Churchill Livingstone, Edinburgh (1987) 183-195.

[4] P. Causin, J.-F. Gerbeau and F. Nobile, Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput. Methods Appl. Mech. Engrg. 194 (2005) 4506-4527. | Zbl

[5] A. Chambolle, B. Desjardins, M. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. J. Math. Fluid Mech. 7 (2005) 368-404. | Zbl

[6] P.G. Ciarlet, Mathematical Elasticity. Volume 1: Three Dimensional Elasticity. Elsevier, second edition (2004). | Zbl

[7] C. Conca, F. Murat and O. Pironneau, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure. Japan J. Math. 20 (1994) 279-318. | Zbl

[8] D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations. Arch. Rational Mech. Anal. 179 (2006) 303-352. | Zbl

[9] L. Formaggia, J.F. Gerbeau, F. Nobile and A. Quarteroni, On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Engrg. 191 (2001) 561-582. | Zbl

[10] L. Euler, Principia pro motu sanguinis per arterias determinando. Opera posthima mathematica et physica anno 1844 detecta 2 (1775) 814-823.

[11] M.A. Fernández and M. Moubachir, A Newton method using exact Jacobian for solving fluid-structure coupling. Comput. Struct. 83 (2005) 127-142.

[12] M.A. Fernández, J.-F. Gerbeau and C. Grandmont, A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid. Inter. J. Num. Meth. Eng. 69 (2007) 794-821.

[13] L. Formaggia and A. Veneziani, Reduced and multiscale models for the human cardiovascular system. Lecture notes VKI Lecture Series 2003-07, Brussels (2003).

[14] L. Formaggia, F. Nobile, A. Quarteroni and A. Veneziani, Multiscale modeling of the circulatory system: a preliminary analysis. Comput. Visual. Sci. 2 (1999) 75-83. | Zbl

[15] L. Formaggia, J.F. Gerbeau, F. Nobile and A. Quarteroni, Numerical treatment of defective boundary conditions for the Navier-Stokes equations. SIAM J. Num. Anal. 40 (2002) 376-401. | Zbl

[16] L. Formaggia, D. Lamponi, M. Tuveri and A. Veneziani, Numerical modeling of 1D arterial networks coupled with a lumped parameters description of the heart. Comput. Methods Biomech. Biomed. Eng. 9 (2006) 273-288.

[17] L. Formaggia, A. Quarteroni and A. Veneziani, The circulatory system: from case studies to mathematical modelling, in Complex Systems in Biomedicine, A. Quarteroni, L. Formaggia and A. Veneziani Eds., Springer, Milan (2006) 243-287.

[18] V.E. Franke, K.H. Parker, L.Y. Wee, N.M. Fisk and S.J. Sherwin, Time domain computational modelling of 1D arterial networks in monochorionic placentas. ESAIM: M2AN 37 (2003) 557-580. | Numdam | Zbl

[19] J.-F. Gerbeau and M. Vidrascu, A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows. ESAIM: M2AN 37 (2003) 631-647. | Numdam | Zbl

[20] J.-F. Gerbeau, M. Vidrascu and P. Frey, Fluid-structure interaction in blood flows on geometries coming from medical imaging. Comput. Struct. 83 (2005) 155-165.

[21] F.J.H. Gijsen, E. Allanic, F.N. Van De Vosse and J.D. Janssen, The influence of the non-Newtonian properies of blood on the flow in large arteries: unsteady flow in a ${90}^{\circ }$ curved tube. J. Biomechanics 32 (1999) 705-713.

[22] V. Giraut and P.-A. Raviart, Finite element method fo the Navier-Stokes equations, in Computer Series in Computational Mathematics 5, Springer-Verlag (1986). | Zbl

[23] J. Gobert, Une inégalité fondamentale de la théorie de l'élasticité3-4) (1962) 182-191. | Zbl

[24] J. Heywood, R. Rannacher and S. Turek, Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int. J. Num. Meth. Fluids 22 (1996) 325-352. | Zbl

[25] K. Laganà, G. Dubini, F. Migliavaca, R. Pietrabissa, G. Pennati, A. Veneziani and A. Quarteroni Multiscale modelling as a tool to prescribe realistic boundary conditions for the study of surgical procedures. Biorheology 39 (2002) 359-364.

[26] D.A. Mcdonald, Blood flow in arteries. Edward Arnold Ltd (1990).

[27] A. Moura, The Geometrical Multiscale Modelling of the Cardiovascular System: Coupling $3$D and $1$D FSI models. Ph.D. thesis, Politecnico di Milano (2007).

[28] R.M. Nerem and J.F. Cornhill, The role of fluid mechanics in artherogenesis. J. Biomech. Eng. 102 (1980) 181-189.

[29] F. Nobile and C. Vergara, An effective fluid-structure interaction formulation for vascular dynamics by generalized Robin conditions. Technical Report 97, MOX (2007). | MR

[30] M.S. Olufsen, C.S. Peskin, W.Y. Kim, E.M. Pedersen, A. Nadim and J. Larsen, Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann. Biomed. Eng. 28 (2000) 1281-1299.

[31] T.J. Pedley, The fluid mechanics of large blood vessels. Cambridge University Press (1980). | Zbl

[32] T.J. Pedley, Mathematical modelling of arterial fluid dynamics. J. Eng. Math. 47 (2003) 419-444. | Zbl

[33] K. Perktold and G. Rappitsch, Mathematical modeling of local arterial flow and vessel mechanics, in Computational Methods for Fluid Structure Interaction, Pitman Research Notes in Mathematics 306, J. Crolet and R. Ohayon Eds., Harlow, Longman (1994) 230-245. | Zbl

[34] K. Perktold, M. Resch and H. Florian, Pulsatile non-Newtonian flow characteristics in a three-dimensional human carotid bifurcation model. J. Biomech. Eng. 113 (1991) 464-475.

[35] A. Quaini and A. Quarteroni, A semi-implicit approach for fluid-structure interaction based on an algebraic fractional step method. Technical Report 90, MOX (2006).

[36] A. Quarteroni, Cardiovascular mathematics, in Proceedings of the International Congress of Mathematicians, Vol. 1, M. Sanz-Solé, J. Soria, J.L. Varona and J. Vezdeza Eds., European Mathematical Society (2007) 479-512. | Zbl

[37] A. Quarteroni, M. Tuveri and A. Veneziani, Computational vascular fluid dynamics: problems, models and methods. Comput. Visual. Sci. 2 (2000) 163-197. | Zbl

[38] A. Quarteroni, S. Ragni and A. Veneziani, Coupling between lumped and distributed models for blood flow problems. Comput. Visual. Sci. 4 (2001) 111-124. | Zbl

[39] S. Sherwin, L. Formaggia, J. Peiró and V. Franke, Computational modelling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system. Int. J. Num. Meth. Fluids 12 (2002) 48-54. | Zbl

[40] A. Veneziani and C. Vergara, Flow rate defective boundary conditions in haemodinamics simulations. Int. J. Num. Meth. Fluids 47 (2005) 801-183. | Zbl

[41] I.E. Vignon-Clementel, C.A. Figueroa, K.E. Jansen and C.A. Taylor, Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3776-3796.