We consider the coupling between three-dimensional (D) and one-dimensional (D) fluid-structure interaction (FSI) models describing blood flow inside compliant vessels. The D model is a hyperbolic system of partial differential equations. The 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 D-D FSI coupling. We consider several possible models for the mechanics of the vessel wall in the D problem and show how the D-D coupling depends on them. Several comparative numerical tests illustrating the coupling are presented.
@article{M2AN_2007__41_4_743_0,
author = {Formaggia, Luca and Moura, Alexandra and Nobile, Fabio},
title = {On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {41},
year = {2007},
pages = {743-769},
doi = {10.1051/m2an:2007039},
mrnumber = {2362913},
zbl = {1139.92009},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2007__41_4_743_0}
}
Formaggia, Luca; Moura, Alexandra; Nobile, Fabio. On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) pp. 743-769. doi : 10.1051/m2an:2007039. http://gdmltest.u-ga.fr/item/M2AN_2007__41_4_743_0/
[1] ,, and, 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 0687.35069
[2] , On the existence of strong solutions to a coupled fluid-structure evolution problem. J. Math. Fluid Mechanics 6 (2004) 21-52. | Zbl 1068.35087
[3] and, 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] , and, Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput. Methods Appl. Mech. Engrg. 194 (2005) 4506-4527. | Zbl 1101.74027
[5] ,, and, 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 1080.74024
[6] , Mathematical Elasticity. Volume 1: Three Dimensional Elasticity. Elsevier, second edition (2004). | Zbl 0888.73001
[7] , and, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure. Japan J. Math. 20 (1994) 279-318. | Zbl 0826.35093
[8] and, The interaction between quasilinear elastodynamics and the Navier-Stokes equations. Arch. Rational Mech. Anal. 179 (2006) 303-352. | Zbl 1138.74325
[9] ,, and, 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 1007.74035
[10] , Principia pro motu sanguinis per arterias determinando. Opera posthima mathematica et physica anno 1844 detecta 2 (1775) 814-823.
[11] and, A Newton method using exact Jacobian for solving fluid-structure coupling. Comput. Struct. 83 (2005) 127-142.
[12] , and, 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] and, Reduced and multiscale models for the human cardiovascular system. Lecture notes VKI Lecture Series 2003-07, Brussels (2003).
[14] ,, and, Multiscale modeling of the circulatory system: a preliminary analysis. Comput. Visual. Sci. 2 (1999) 75-83. | Zbl 1067.76624
[15] ,, and, Numerical treatment of defective boundary conditions for the Navier-Stokes equations. SIAM J. Num. Anal. 40 (2002) 376-401. | Zbl 1020.35070
[16] ,, and, 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] , and, 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] ,,, and, Time domain computational modelling of 1D arterial networks in monochorionic placentas. ESAIM: M2AN 37 (2003) 557-580. | Numdam | Zbl 1065.92017
[19] and, 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 1070.74047
[20] , and, Fluid-structure interaction in blood flows on geometries coming from medical imaging. Comput. Struct. 83 (2005) 155-165.
[21] ,, and, The influence of the non-Newtonian properies of blood on the flow in large arteries: unsteady flow in a curved tube. J. Biomechanics 32 (1999) 705-713.
[22] and, Finite element method fo the Navier-Stokes equations, in Computer Series in Computational Mathematics 5, Springer-Verlag (1986). | Zbl 0585.65077
[23] , Une inégalité fondamentale de la théorie de l'élasticité3-4) (1962) 182-191. | Zbl 0112.38902
[24] , and, Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int. J. Num. Meth. Fluids 22 (1996) 325-352. | Zbl 0863.76016
[25] ,,,,, and modelling as a tool to prescribe realistic boundary conditions for the study of surgical procedures. Biorheology 39 (2002) 359-364.
[26] , Blood flow in arteries. Edward Arnold Ltd (1990).
[27] , The Geometrical Multiscale Modelling of the Cardiovascular System: Coupling D and D FSI models. Ph.D. thesis, Politecnico di Milano (2007).
[28] and, The role of fluid mechanics in artherogenesis. J. Biomech. Eng. 102 (1980) 181-189.
[29] and, An effective fluid-structure interaction formulation for vascular dynamics by generalized Robin conditions. Technical Report 97, MOX (2007). | MR 2385883
[30] ,,,, and, Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann. Biomed. Eng. 28 (2000) 1281-1299.
[31] , The fluid mechanics of large blood vessels. Cambridge University Press (1980). | Zbl 0449.76100
[32] , Mathematical modelling of arterial fluid dynamics. J. Eng. Math. 47 (2003) 419-444. | Zbl 1065.76212
[33] and, 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 0809.76098
[34] , and, Pulsatile non-Newtonian flow characteristics in a three-dimensional human carotid bifurcation model. J. Biomech. Eng. 113 (1991) 464-475.
[35] and, A semi-implicit approach for fluid-structure interaction based on an algebraic fractional step method. Technical Report 90, MOX (2006). | Zbl pre05176130
[36] , 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 1121.92022
[37] , and, Computational vascular fluid dynamics: problems, models and methods. Comput. Visual. Sci. 2 (2000) 163-197. | Zbl 1096.76042
[38] , and, Coupling between lumped and distributed models for blood flow problems. Comput. Visual. Sci. 4 (2001) 111-124. | Zbl 1097.76615
[39] ,, and, 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 1008.92011
[40] and, Flow rate defective boundary conditions in haemodinamics simulations. Int. J. Num. Meth. Fluids 47 (2005) 801-183. | Zbl 1134.76748
[41] ,, and, 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. | Zbl pre05194200