Fluid-structure interaction for vascular flows: from supercomputers to laptops

There exists several models for the simulation of vascular flows; they span from simple circuit models, to full three dimensional ones which take into account detailed features of the blood and of the arterial wall. Each model comes with benefits and drawbacks, the main denominator being a compromise between detailed resolution requirements versus computational time. We first present a fluid-structure interaction computational model where both the fluid and the structure are three dimensional; in particular, the fluid includes modeling of large eddies by the variational multiscale method. After time and space discretisations carried out by finite differences and finite elements, respectively, we set up a parallel solver based on domain decomposition and a FaCSI preconditioner. These simulations allows to capture details of the flow dynamics and of the structure deformation even in the transitional regime characterizing the hemodynamics in the aorta. To complete a simulation of one heartbeat with 35 millions of degrees of freedom on 2048 cores it takes roughly 10 hours. We then reduce both the model and its numerical complexity. The structural model is simplified to a two dimensional membrane located at the fluid-structure interface and the fluid computational domain is fixed. For a fixed geometry and mesh, these assumptions allow to apply proper orthogonal decomposition and generate a space discretisation which has only few dozen degrees of freedom. It is then possible to perform the simulation of one hearbeat on a laptop in less than one second. The modeling and numerical reductions allows therefore a dramatic reduction of the computational time. However, the price to pay comes, on the one hand, in terms of the preparation of a reduced basis specific to the patient and the geometry of the vessel and, on the other hand, with a detriment of certain quantities of interest. For example, when using a finite element discretization with 9 millions of degrees of freedom, the offline part takes about 12 hours on 720 cores for the example provided in this work; in this case, the flow profiles in the aorta are pretty close to the full three dimensional model, but the wall shear stress is overestimated (although it follows the same temporal patterns).