Robin-Neumann schemes for incompressible fluid-structure interaction

International audienceMathematical problems involving the coupling of an incompressible viscous flow with an elastic structure appear in a large variety of engineering fields. This problem is considered here within a heterogenous domain decomposition framework, with the aim of using independent well-suited solvers for the fluid and the solid. One of the main difficulties that have to be faced under this approach is that the coupling can be very stiff. In particular, traditional Dirichlet-Neumann explicit coupling methods, which solve for the fluid (Dirichlet) and for the solid (Neumann) only once per time-step, are unconditionally unstable whenever the amount of added-mass effect in the system is large. Typically this happens when the fluid and solid densities are close and the fluid domain is slender, as in hemodynamic applications. This explains, in part, the tremendous amount of work devoted over the last decade to the development of alternative coupling paradigms. In this paper we will review several explicit coupling procedures recently reported in the literature and present some new developments. The common feature of these methods is that they are based on Robin-Neumann transmission conditions, whose nature depends on the thin- or thick-walled character of the structure