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On the coupling of 3D and 1D Navier±Stokes equations for ¯ow problems in compliant vessels q L. Formaggia a , J.F. Gerbeau b , F. Nobile a , A. Quarteroni a,c, * a D epartement de Math ematiques, Ecole Polytechnique F ed erale de Lausanne, CH-1015 Lausanne, Switzerland b INRIA ± Projet M3N ± Rocquencourt, B.P.105, F-78153 Le Chesnay Cedex, France c Dipartimento di Matematica ``F. Brioschi'', Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy Received 3 January 2000; received in revised form 27 October 2000 Abstract For the analysis of ¯ows in compliant vessels, we propose an approach to couple the original 3D equations with a convenient 1D model. This multi-scale strategy allows for a dramatic reduction of the computational complexity and is suitable for ``absorbing'' outgoing pressure waves. In particular, it is of utmost interest for the description of blood motion in the arterial system. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Fluid±structure interaction; Computational ¯uid dynamics; Absorbing boundary conditions; Multi-scale models; Haemodynamics 1. Introduction The study of a transient incompressible ¯ow inside a compliant vessel, that is a vessel whose walls de- form under the action of the ¯uid with possibly large displacements, is of considerable interest, for instance in the simulation of blood ¯ow in large arteries [17]. The mathematical problem consists of the 2D or 3D Navier±Stokes equations describing the ¯uid motion, coupled with a suitable model that describes the displacement of the vessel wall hereafter called ``the structure''). We will refer to this coupled system as the 2D/3D ¯uid±structure model. This problem is interesting from the mathematical and numerical modelling viewpoints, as it exhibits some peculiar features like the appearance of travelling pressure waves along the vessel. Indeed, even if the ¯ow is governed by parabolic equations such as the incompressible Navier±Stokes, the behaviour of the coupled ¯uid±structure system is in many aspects more akin to that of a hyperbolic problem. As a con- sequence, an additional complexity arises in the treatment of the ``in¯ow'' and ``out¯ow'' boundaries, where one would like to have a correct representation of the travelling waves, without spurious re¯ections. The case of absorbing boundary conditions for compressible ¯ows and more general hyperbolic equations has been widely studied, see for instance [3,6,7]. However, in our case, where the propagation phenomena are www.elsevier.com/locate/cma Comput. Methods Appl. Mech. Engrg. 191 2001) 561±582 q Supported by the Swiss N.S.F. Project 21-54139.98) and by MURST Co®n. 1998, ``Advanced Numerical Methods for Scienti®c Computing''. * Corresponding author. Tel.: 21-693-5546; fax: 41-21-693-4303. E-mail addresses: luca.formaggia@ep¯.ch L. Formaggia), [email protected] J.F. Gerbeau), fabio.nobile@ep¯.ch F. Nobile), al®o.quarteroni@ep¯.ch A. Quarteroni). 0045-7825/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII:S0045-782501)00302-4

On the Coupling FINAL

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