Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T12:31:26.688Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  04 October 2007

Luca Formaggia ,
Alexandra Moura  and
Fabio Nobile
Luca Formaggia
Affiliation:
MOX - Modelling and Scientific Computing, Department of Mathematics, Politecnico di Milano, Italy. [email protected]; [email protected]; [email protected]
Alexandra Moura
Affiliation:
MOX - Modelling and Scientific Computing, Department of Mathematics, Politecnico di Milano, Italy. [email protected]; [email protected]; [email protected]
Fabio Nobile
Affiliation:
MOX - Modelling and Scientific Computing, Department of Mathematics, Politecnico di Milano, Italy. [email protected]; [email protected]; [email protected]
Get access

Abstract

We consider the coupling between three-dimensional(3D) and one-dimensional (1D) fluid-structure interaction (FSI) models describing blood flow inside compliant vessels. The 1D model is a hyperbolicsystem of partial differential equations.The 3D model consists of the Navier-Stokes equationsfor incompressible Newtonian fluids coupled with a model for the vessel wall dynamics. A non standard formulationfor the Navier-Stokes equations is adopted tohave suitable boundary conditions for the couplingof the models. With this we derive an energy estimatefor the fully 3D-1D FSI coupling. We consider several possiblemodels for the mechanics of the vessel wall in the 3D problemand show how the 3D-1D coupling depends on them.Several comparative numerical tests illustrating the coupling are presented.

Keywords

Fluid-structure interaction3D-1D FSI couplingenergy estimatemultiscale models.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 4 , July 2007 , pp. 743 - 769
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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.
Beirão da, H. Veiga, On the existence of strong solutions to a coupled fluid-structure evolution problem. J. Math. Fluid Mechanics 6 (2004) 21–52. CrossRef
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.
Causin, P., Gerbeau, J.-F. and Nobile, F., Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput. Methods Appl. Mech. Engrg. 194 (2005) 4506–4527. CrossRef
Chambolle, A., Desjardins, B., Esteban, M. and Grandmont, C., Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. J. Math. Fluid Mech. 7 (2005) 368–404. CrossRef
P.G. Ciarlet, Mathematical Elasticity. Volume 1: Three Dimensional Elasticity. Elsevier, second edition (2004).
Conca, C., Murat, F. and Pironneau, O., The Stokes and Navier-Stokes equations with boundary conditions involving the pressure. Japan J. Math. 20 (1994) 279–318.
Coutand, D. and Shkoller, S., The interaction between quasilinear elastodynamics and the Navier-Stokes equations. Arch. Rational Mech. Anal. 179 (2006) 303–352. CrossRef
Formaggia, L., Gerbeau, J.F., Nobile, F. and Quarteroni, A., 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. CrossRef
Euler, L., Principia pro motu sanguinis per arterias determinando. Opera posthima mathematica et physica anno 1844 detecta 2 (1775) 814–823.
Fernández, M.A. and Moubachir, M., Newton, A method using exact Jacobian for solving fluid-structure coupling. Comput. Struct. 83 (2005) 127–142. CrossRef
Fernández, M.A., Gerbeau, J.-F. and Grandmont, C., 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. CrossRef
L. Formaggia and A. Veneziani, Reduced and multiscale models for the human cardiovascular system. Lecture notes VKI Lecture Series 2003-07, Brussels (2003).
Formaggia, L., Nobile, F., Quarteroni, A. and Veneziani, A., Multiscale modeling of the circulatory system: a preliminary analysis. Comput. Visual. Sci. 2 (1999) 75–83. CrossRef
Formaggia, L., Gerbeau, J.F., Nobile, F. and Quarteroni, A., Numerical treatment of defective boundary conditions for the Navier-Stokes equations. SIAM J. Num. Anal. 40 (2002) 376–401. CrossRef
Formaggia, L., Lamponi, D., Tuveri, M. and Veneziani, A., Numerical modeling of 1D arterial networks coupled with a lumped parameters description of the heart. Comput. Methods Biomech. Biomed. Eng. 9 (2006) 273–288. CrossRef
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.
Franke, V.E., Parker, K.H., Wee, L.Y., Fisk, N.M. and Sherwin, S.J., Time domain computational modelling of 1D arterial networks in monochorionic placentas. ESAIM: M2AN 37 (2003) 557–580. CrossRef
Gerbeau, J.-F. and Vidrascu, M., A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows. ESAIM: M2AN 37 (2003) 631–647. CrossRef
Gerbeau, J.-F., Vidrascu, M. and Frey, P., Fluid-structure interaction in blood flows on geometries coming from medical imaging. Comput. Struct. 83 (2005) 155–165. CrossRef
Gijsen, F.J.H., Allanic, E., van de Vosse, F.N. and Janssen, J.D., 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. CrossRef
V. Giraut and P.-A. Raviart, Finite element method fo the Navier-Stokes equations, in Computer Series in Computational Mathematics 5, Springer-Verlag (1986).
J. Gobert, Une inégalité fondamentale de la théorie de l'élasticité. Bull. Soc. Royale Sciences Liège, 31 e année (3-4) (1962) 182–191.
Heywood, J., Rannacher, R. and Turek, S., Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int. J. Num. Meth. Fluids 22 (1996) 325–352. 3.0.CO;2-Y>CrossRef
Laganà, K., Dubini, G., Migliavaca, F., Pietrabissa, R., Pennati, G., Veneziani, A. and Quarteroni Multiscale, A. modelling as a tool to prescribe realistic boundary conditions for the study of surgical procedures. Biorheology 39 (2002) 359–364.
D.A. McDonald, Blood flow in arteries. Edward Arnold Ltd (1990).
A. Moura, The Geometrical Multiscale Modelling of the Cardiovascular System: Coupling 3D and 1D FSI models. Ph.D. thesis, Politecnico di Milano (2007).
Nerem, R.M. and Cornhill, J.F., The role of fluid mechanics in artherogenesis. J. Biomech. Eng. 102 (1980) 181–189. CrossRef
F. Nobile and C. Vergara, An effective fluid-structure interaction formulation for vascular dynamics by generalized Robin conditions. Technical Report 97, MOX (2007).
Olufsen, M.S., Peskin, C.S., Kim, W.Y., Pedersen, E.M., Nadim, A. and Larsen, J., Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann. Biomed. Eng. 28 (2000) 1281–1299. CrossRef
T.J. Pedley, The fluid mechanics of large blood vessels. Cambridge University Press (1980).
Pedley, T.J., Mathematical modelling of arterial fluid dynamics. J. Eng. Math. 47 (2003) 419–444. CrossRef
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.
Perktold, K., Resch, M. and Florian, H., Pulsatile non-Newtonian flow characteristics in a three-dimensional human carotid bifurcation model. J. Biomech. Eng. 113 (1991) 464–475. CrossRef
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).
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.
Quarteroni, A., Tuveri, M. and Veneziani, A., Computational vascular fluid dynamics: problems, models and methods. Comput. Visual. Sci. 2 (2000) 163–197. CrossRef
Quarteroni, A., Ragni, S. and Veneziani, A., Coupling between lumped and distributed models for blood flow problems. Comput. Visual. Sci. 4 (2001) 111–124. CrossRef
Sherwin, S., Formaggia, L., Peiró, J. and Franke, V., 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.
Veneziani, A. and Vergara, C., Flow rate defective boundary conditions in haemodinamics simulations. Int. J. Num. Meth. Fluids 47 (2005) 801–183. CrossRef
Vignon-Clementel, I.E., Figueroa, C.A., Jansen, K.E. and Taylor, C.A., 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. CrossRef