Fluid dynamics of an idealized left ventricle: the extended Nitsche's method for the treatment of heart valves as mixed time varying boundary conditions

In this work, we study the blood flow dynamics in idealized left ventricles (LV) of the human heart modelled by the Navier–Stokes equations with mixed time varying boundary conditions. The latter are introduced for simulating the functioning of the aortic and mitral valves. On the basis of the extended Nitsche's method firstly presented in [Juntunen and Stenberg, Mathematics of Computation, 2009], we propose a formulation allowing an efficient and straightforward numerical treatment of the opening and closing phases of the heart valves that are associated with different kind of boundary conditions, namely, natural and essential, switching during each heartbeat. Moreover, our formulation already includes terms preventing the numerical instabilities associated to backflow divergence, that is, nonphysical reinflow at the valves. We present and discuss numerical results for the LV obtained by means of isogeometric analysis for the spatial approximation with the aim of both analysing the formulation and showing the effectiveness of the approach. In particular, we show that the formulation allows to reproduce meaningful results even in idealized LV