Defective boundary conditions applied to multiscale analysis of blood flow

¹ Institut National de Recherche en Informatique et en Automatique;
² MOX (Modeling and Scientific Computing) Department of Mathematics, Politecnico di Milano;

Corresponding authors: miguel.fernandez@inria.fr alexandra.moura@mate.polimi.it christian.vergara@mate.polimi.it

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 condition at each point of . 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.