Numerical simulation of blood flows through a porous interface
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) no. 6, p. 961-990
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.
DOI : https://doi.org/10.1051/m2an:2008031
Classification:  65M60,  74K25,  76D05,  76Z05
@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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {42},
number = {6},
year = {2008},
pages = {961-990},
doi = {10.1051/m2an:2008031},
zbl = {1148.92017},
mrnumber = {2473316},
language = {en},
url = {http://www.numdam.org/item/M2AN_2008__42_6_961_0}
}

Fernández, Miguel A.; Gerbeau, Jean-Frédéric; Martin, Vincent. Numerical simulation of blood flows through a porous interface. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) no. 6, pp. 961-990. doi : 10.1051/m2an:2008031. http://www.numdam.org/item/M2AN_2008__42_6_961_0/

[1] G. Allaire, 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 1079190 | Zbl 0724.76021

[2] J.L. Berry, A. Santamarina, J.E. Jr. Moore, S. Roychowdhury and W.D. Routh, Experimental and computational flow evaluation of coronary stents. Ann. Biomed. Eng. 28 (2000) 386-398.

[3] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, Berlin (1991). | MR 1115205 | Zbl 0788.73002

[4] A. Brillard, 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 1194196 | Zbl 0795.76079

[5] E. Burman and P. Hansbo, Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Comput. Methods Appl. Mech. Engrg. 195 (2006) 2393-2410. | MR 2207476 | Zbl 1125.76038

[6] D. Chapelle and K.J. Bathe, The finite element analysis of shell - Fundamentals. Springer-Verlag (2004). | MR 2143259 | Zbl 1103.74003

[7] P.G. Ciarlet, 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 1930132 | Zbl 0999.65129

[8] P. Clément, 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 400739 | Zbl 0368.65008

[9] R. Codina and J. Blasco, A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation. Comput. Methods Appl. Mech. Engrg. 143 (1997) 373-391. | MR 1445157 | Zbl 0893.76040

[10] C. Conca, Étude d'un fluide traversant une paroi perforée, I. Comportement limite près de la paroi. J. Math. Pures Appl. 66 (1987) 1-43. | MR 884812 | Zbl 0622.35061

[11] C. Conca, Étude d'un fluide traversant une paroi perforée, II. Comportement limite loin de la paroi. J. Math. Pures Appl. 66 (1987) 45-70. | MR 884813 | Zbl 0622.35062

[12] C. Conca and M. Sepúlveda, Numerical results in the Stokes sieve problem. Rev. Internac. Métod. Numér. Cálc. Diseñ. Ingr. 5 (1989) 435-452. | MR 1036224

[13] A. Ern and J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences 159. Springer-Verlag, New York (2004). | MR 2050138 | Zbl 1059.65103

[14] 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. | MR 1873203 | Zbl 1007.74035

[15] P. Frey, Yams: A fully automatic adaptive isotropic surface remeshing procedure. Technical report 0252, INRIA, Rocquencourt, France, Nov. (2001).

[16] P. Frey, Medit: An interactive mesh visualisation software. Technical report 0253, INRIA, Rocquencourt, France, Dec. (2001).

[17] J.-F. Gerbeau and M. Vidrascu, A quasi-Newton algorithm based on a reduced model for fluid structure problems in blood flows. ESAIM: M2AN 37 (2003) 631-647. | Numdam | MR 2018434 | Zbl 1070.74047

[18] 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.

[19] V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations - Theory and algorithms, Springer Series in Computational Mathematics 5. Springer-Verlag, Berlin (1986). | MR 851383 | Zbl 0585.65077

[20] T.J.R. Hughes, L.P. Franca and M. Balestra, 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 868143 | Zbl 0622.76077

[21] A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system, in Handbook of Numerical Analysis XII, North-Holland, Amsterdam (2004) 3-127. | MR 2087609

[22] S. Salmon, M. Thiriet and J.-F. Gerbeau, Medical image-based computational model of pulsatile flow in saccular aneurisms. ESAIM: M2AN 37 (2003) 663-679. | Numdam | MR 2018436 | Zbl 1065.92029

[23] E. Sánchez-Palencia, 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 0602.76036

[24] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190) (1990) 483-493. | MR 1011446 | Zbl 0696.65007

[25] D.A. Steinman, J.S. Milner, C.J. Norley, S.P. Lownie and D.W. Holdsworth, Image-based computational simulation of flow dynamics int a giant intracranial aneurysms. Am. J. Neuroradiol. 24 (2003) 559-566.

[26] G.R. Stuhne and D.A. Steinman, Finite-element modeling of the hemodynamics of stented aneurysms. J. Biomech. Eng. 126 (2004) 382-387.

[27] V. Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics 25. Springer-Verlag, Berlin, second edition (2006). | MR 2249024 | Zbl 1105.65102

[28] L. Tobiska and V. Verfurth, Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations. SIAM J. Numer. Anal. 33 (1996) 107-127. | MR 1377246 | Zbl 0843.76052

[29] 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. | MR 2221774 | Zbl pre05194200

[30] N.T. Wang and A.L. Fogelson, Computational methods for continuum models of platelet aggregation. J. Comput. Phys. 151 (1999) 649-675. | MR 1686379 | Zbl 0981.92005