ESAIM: PROCEEDINGS, September 2005, Vol.14, 89-99 Eric Canc` es & Jean-Fr´ ed´ eric Gerbeau, Editors DOI: 10.1051/proc:2005008 DEFECTIVE BOUNDARY CONDITIONS APPLIED TO MULTISCALE ANALYSIS OF BLOOD FLOW ∗ Fern´ andez M. 1 , Moura A. 2 and Vergara C. 2 Abstract. In hemodynamics, the prescription of suitable boundary conditions for the Navier-Stokes equations (3D model) on the artificial sections (i.e. the parts of the boundary not corresponding to the physical artery wall) is critical. A first solution is to prescribe experimental data, whenever available from specific measurements, or we can use reduced models, i.e. one-dimensional (1D) or zero-dimensional (0D) models, to get the proper interface conditions accounting for global behavior (see [4, 5]). In this work we couple a 0D model with a 3D local model of a non-compliant cylindric vessel. We propose two techniques. In the first approach the reduced model provides the mean pressure to be imposed as defective boundary condition to the 3D model, which conversely will make ready the flow rate to the reduced model. In the second strategy the type of data to be exchanged is reversed. Mean value conditions are not natural to Navier-Stokes equations, which would need a vector con- dition at each point of Γj . Special techniques have to be implemented. For what concerns the mean pressure problem, we follow the approach proposed in [9], that suggests to impose on the artificial section some natural (Neumann) conditions obtained from a suitable variational formulation. For the flow rate problem, we use the augmented formulation proposed in [2] and [16]. In these works, the flux conditions are regarded as constraints to be fulfilled by the solution by introducing a Lagrange multiplier for each defective condition. Introduction Over the last years, the advances on fluid-dynamics numerical simulations, together with the development of new technologies in experimental data acquisition have allowed to obtain quantitative information of the blood flow behavior. For instance, measures of shear stresses in the vessel wall, identification of recirculation zones in vascular districts, the blood-vessel mechanical interaction or the adaptation of the vascular walls after surgical intervention (see for example [12]). Specifically, research focused on the determination of the velocity and pressure field, based on the knowledge of some global quantities as the flux or the mean pressure on specific boundaries of the vascular district under study. In medium and large vessels, blood behaves as an incompressible Newtonian fluid. The incompressible Navier-Stokes equations are then a mathematical model capable to provide the velocity and the pressure of the blood. Nowadays, three-dimensional simulations of blood flow based on these equations are a common practice [11]. Obviously, these simulations require the enforcement of specific data on the artificial boundary * This work has been supported by the European Network HAEMODEL - Mathematical Modelling of the Cardiovascular System under contract HPRN - CT - 2002 - 002670. 1 Institut National de Recherche en Informatique et en Automatique; e-mail: miguel.fernandez@inria.fr 2 MOX (Modeling and Scientific Computing) Department of Mathematics, Politecnico di Milano; e-mail: alexandra.moura@mate.polimi.it & christian.vergara@mate.polimi.it c  EDP Sciences, SMAI 2005 Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx.doi.org/10.1051/proc:2005008