Coupling between 3D and 1D fluid-structure interaction models for blood flow

Three-dimensional ($3$D) simulations of blood flow provide detailed information important for the comprehension of the cardiovascular system. These consist in coupling the $3$D Navier-Stokes equations with a model for the vessel wall structure. Due to the computational cost of fully $3$D fluid-structure interaction (FSI) problems and to the complexity of the cardiovascular system, these models can be applied only to truncated regions of interest. Following the geometrical multiscale modelling of the cardiovascular system, the remaining parts can be accounting resorting to reduced models, one-dimensional ($1$D) or zero-dimensional ($0$D). In particular, the $1$D models describe very well the wave propagation nature of blood flow and coupled with the $3$D model act as proper absorbing boundary conditions. We address the coupling of $3$D and $1$D FSI models. The difficulty of this coupling lays in putting together such different models, namely it is not evident which conditions to impose at the coupling interface. We study the stability of such coupling, bringing forth proper matching conditions between the models. Different coupling strategies are discussed and several numerical results presented.