Flow rate defective boundary conditions in haemodynamics simulations

In the numerical simulation of blood flow problems it might happen that the only available physical boundary conditions prescribe the flow rate incoming/outgoing the vascular district at hand. In order to have a well-posed Navier–Stokes (NS) 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, that should fit 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 et al. (Int. J. Num. Meth. Fluids 1996; 22:325–352). An equivalent approach, however, easier from the numerical discretization viewpoint, has been proposed in Formaggia et al. (SIAM J. Numer. Anal. 2002; 40(1):376–401). 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 extend this analysis to the unsteady augmented NS 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 NS solvers.