We propose a model for a medical device, called a stent, designed for the treatment of cerebral aneurysms. The stent consists of a grid, immersed in the blood flow and located at the inlet of the aneurysm. It aims at promoting a clot within the aneurysm. The blood flow is modelled by the incompressible Navier-Stokes equations and the stent by a dissipative surface term. We propose a stabilized finite element method for this model and we analyse its convergence in the case of the Stokes equations. We present numerical results for academical test cases, and on a realistic aneurysm obtained from medical imaging.
Mots-clés : stabilized finite element, sieve problem, blood flow, terminal aneurysm, stent, fluid-structure interaction
@article{M2AN_2008__42_6_961_0,
author = {Fern\'andez, Miguel A. and Gerbeau, Jean-Fr\'ed\'eric and Martin, Vincent},
title = {Numerical simulation of blood flows through a porous interface},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {961--990},
publisher = {EDP-Sciences},
volume = {42},
number = {6},
year = {2008},
doi = {10.1051/m2an:2008031},
mrnumber = {2473316},
zbl = {1148.92017},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an:2008031/}
}
TY - JOUR AU - Fernández, Miguel A. AU - Gerbeau, Jean-Frédéric AU - Martin, Vincent TI - Numerical simulation of blood flows through a porous interface JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2008 SP - 961 EP - 990 VL - 42 IS - 6 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2008031/ DO - 10.1051/m2an:2008031 LA - en ID - M2AN_2008__42_6_961_0 ER -
%0 Journal Article %A Fernández, Miguel A. %A Gerbeau, Jean-Frédéric %A Martin, Vincent %T Numerical simulation of blood flows through a porous interface %J ESAIM: Modélisation mathématique et analyse numérique %D 2008 %P 961-990 %V 42 %N 6 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an:2008031/ %R 10.1051/m2an:2008031 %G en %F M2AN_2008__42_6_961_0
Fernández, Miguel A.; Gerbeau, Jean-Frédéric; Martin, Vincent. Numerical simulation of blood flows through a porous interface. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 6, pp. 961-990. doi : 10.1051/m2an:2008031. http://archive.numdam.org/articles/10.1051/m2an:2008031/
[1] , Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. II. Noncritical sizes of the holes for a volume distribution and a surface distribution of holes. Arch. Rational Mech. Anal. 113 (1991) 261-298. | MR | Zbl
[2] , , , and , Experimental and computational flow evaluation of coronary stents. Ann. Biomed. Eng. 28 (2000) 386-398.
[3] and , Mixed and Hybrid Finite Element Methods. Springer-Verlag, Berlin (1991). | MR | Zbl
[4] , Asymptotic flow of a viscous and incompressible fluid through a plane sieve, in Progress in partial differential equations: calculus of variations, applications (Pont-à-Mousson, 1991), Pitman Res. Notes Math. Ser. 267, Longman Sci. Tech., Harlow (1992) 158-172. | MR | Zbl
[5] and , Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Comput. Methods Appl. Mech. Engrg. 195 (2006) 2393-2410. | MR | Zbl
[6] and , The finite element analysis of shell - Fundamentals. Springer-Verlag (2004). | MR | Zbl
[7] , The finite element method for elliptic problems, Classics in Applied Mathematics 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. | MR | Zbl
[8] , Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. 9(R-2) (1975) 77-84. | Numdam | MR | Zbl
[9] and , A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation. Comput. Methods Appl. Mech. Engrg. 143 (1997) 373-391. | MR | Zbl
[10] , Étude d'un fluide traversant une paroi perforée, I. Comportement limite près de la paroi. J. Math. Pures Appl. 66 (1987) 1-43. | MR | Zbl
[11] , Étude d'un fluide traversant une paroi perforée, II. Comportement limite loin de la paroi. J. Math. Pures Appl. 66 (1987) 45-70. | MR | Zbl
[12] and , Numerical results in the Stokes sieve problem. Rev. Internac. Métod. Numér. Cálc. Diseñ. Ingr. 5 (1989) 435-452. | MR
[13] and , Theory and practice of finite elements, Applied Mathematical Sciences 159. Springer-Verlag, New York (2004). | MR | Zbl
[14] , , 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. | MR | Zbl
[15] , Yams: A fully automatic adaptive isotropic surface remeshing procedure. Technical report 0252, INRIA, Rocquencourt, France, Nov. (2001).
[16] , Medit: An interactive mesh visualisation software. Technical report 0253, INRIA, Rocquencourt, France, Dec. (2001).
[17] and , A quasi-Newton algorithm based on a reduced model for fluid structure problems in blood flows. ESAIM: M2AN 37 (2003) 631-647. | Numdam | MR | Zbl
[18] , and , Fluid-structure interaction in blood flows on geometries coming from medical imaging. Comput. Struct. 83 (2005) 155-165.
[19] and , Finite element methods for Navier-Stokes equations - Theory and algorithms, Springer Series in Computational Mathematics 5. Springer-Verlag, Berlin (1986). | MR | Zbl
[20] , and , A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comp. Meth. App. Mech. Eng. 59 (1986) 85-99. | MR | Zbl
[21] and , Mathematical modelling and numerical simulation of the cardiovascular system, in Handbook of Numerical Analysis XII, North-Holland, Amsterdam (2004) 3-127. | MR
[22] , and , Medical image-based computational model of pulsatile flow in saccular aneurisms. ESAIM: M2AN 37 (2003) 663-679. | Numdam | MR | Zbl
[23] , Problèmes mathématiques liés à l'écoulement d'un fluide visqueux à travers une grille, in Ennio De Giorgi colloquium (Paris, 1983), Res. Notes in Math. 125, Pitman, Boston, USA (1985) 126-138. | Zbl
[24] and , Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190) (1990) 483-493. | MR | Zbl
[25] , , , and , Image-based computational simulation of flow dynamics int a giant intracranial aneurysms. Am. J. Neuroradiol. 24 (2003) 559-566.
[26] and , Finite-element modeling of the hemodynamics of stented aneurysms. J. Biomech. Eng. 126 (2004) 382-387.
[27] , Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics 25. Springer-Verlag, Berlin, second edition (2006). | MR | Zbl
[28] and , Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations. SIAM J. Numer. Anal. 33 (1996) 107-127. | MR | Zbl
[29] , , 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. | MR
[30] and , Computational methods for continuum models of platelet aggregation. J. Comput. Phys. 151 (1999) 649-675. | MR | Zbl
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