Finite Element Approximation of Fluid-Structure Interaction Problems

Overview

Many phenomena, from blood flowing through a vessel to a flag waving in the wind, involve a fluid and a deformable solid that influence each other: the fluid pushes the structure, the structure moves, and by moving it changes the region occupied by the fluid. Simulating such fluid-structure interaction problems is challenging precisely because the two subproblems are of different nature and are posed on domains that are not known in advance. After an elementary introduction to the equations involved and to the main difficulties one faces when discretizing them, we review the classical strategies proposed in the literature, and in particular the distinction between methods that follow the moving domain with a deforming mesh and methods that keep the mesh fixed. We then concentrate on one possible strategy, the fictitious domain approach: the fluid problem is extended to a fixed background domain, the structure is described on its own mesh, and the two are coupled through a Lagrange multiplier enforcing the matching conditions. We illustrate the main features of this formulation and the questions it raises from both the theoretical and the numerical point of view, together with some representative examples.