Algorithms for the partitioned solution of weakly coupled fluid models for cardiovascular flows

32/2010

Algorithms for the partitioned solution of weakly coupled fluid models for cardiovascular flows

Sunday 7th November 2010

Malossi, A. Cristiano I.; Blanco, Pablo J.; Deparis, Simome; Quarteroni, Alfio

The main goal of the present work is to devise robust iterative strategies to partition the solution of the Navier–Stokes equations in a three-dimensional(3D) computational domain, into non overlapping 3D subdomains,which communicate through the exchange of integrated quantities across the interfaces. The novel aspect of the present approach is that at coupling boundaries the conservation of flow rates and of the associated dual variables is imposed, entailing a weak physical coupling. For the solution of the non-linear problem, written in terms of interfaces variables, two strategies are compared: relaxed fixed point iterations and Newton iterations. The algorithm is tested in several configurations for problems which involve more than two components at each coupling interface. In such cases it is shown that relaxed fixed point methods are not convergent, whereas the Newton method leads in all the tested cases to convergent schemes. One of the appealing aspects of the strategy proposed here is the flexibility in the setting of boundary conditions at branching points, where no hierarchy is established a priori, unlike classical Gauss–Seidel methods. Such an approach can be applied in two other different contexts: (i) when coupling dimensionally-heterogeneous models, just by replacing some of the 3D models by one-dimensional (or zero-dimensional) condensed ones, and (ii) as a preconditioner method for domain decomposition methods for the Navier–Stokes equations. These two issues are also addressed in the present work. Finally, several examples of application are presented, ranging from academic examples to some related to the computational hemodynamics field.