Numerical methods for fluid-structure interaction with immersed thin-walled bodies

The numerical simulation of elastic thin-walled bodies immersed in an incompressible viscous fluid is an essential ingredient in the mathematical modeling of many living systems: From the opening and closing dynamics of heart valves to the wings of a bird interacting with the air or the fins of a fish moving in water.
The numerical methods for the simulation of these systems generally fall into one of the following two categories: fitted and unfitted mesh methods. Fitted mesh methods are known to deliver optimal accuracy for moderate interface displacements, but they become cumbersome or lose efficiency in presence of topological changes (e.g., due to contacting solids). Unfitted mesh methods, such as the Immersed Boundary/Fictitious Domain methods or the recently developed Nitsche-XFEM method, allow for arbitrary interface displacements but this flexibility comes at a price: the mismatch between the fluid and solid meshes complicates the interface coupling. In this talk, we will review some of these approaches by comparing them on some known FSI benchmarks involving moving interfaces and topology changes. We will also introduce a new time splitting scheme for a particular class of fictitious domain approximations, which invokes the fluid and solid solvers only once per time-step without compromising stability and accuracy.