Flow rate defective boundary conditions in haemodynamics simulations
Thursday 8th April 2004
Veneziani, Alessandro; Vergara, Christian
In the numerical simulation of blood flow problems it might happen that the only available boundary conditions prescribe the flow rate incoming/outcoming the vascular district at hand. In order to have a well posed Navier-Stokes problem, these conditions need to be completed. In the bioengineering community, this problem is usually faced by choosing a priori a velocity profile on the inflow/outflow sections, to be fitted with the assigned flow rates. This approach strongly influences the accuracy of the numerical solutions. A less perturbative strategy is based on the so-called do-nothing approach, advocated in Heywood, Rannacher, turek, Int. J. Num. Fl, 1996. An equivalent approach, but easier from the numerical discretization viewpoint, has been proposed in Formaggia, Gerbeau, Nobile, Quarteroni, SIAM J Num An, 2002. It is based on an augmented formulation of the problem, in which the conditions on the flow rates are prescribed in a weak sense by means of Lagrangian multipliers. In this paper we analyze the unsteady augmented Navier-Stokes problem, proving a well posedness result. Moreover, we present some numerical methods for solving the augmented problem, based on a splitting of the computation of velocity and pressure on one side and the Lagrangian multiplier on the other one. In this way, we show how it is possible to solve the augmented problem resorting to available Navier-Stokes solvers.