Fluidstructure interaction modelization using the enriched spacetime finite element
Fluidstructure interaction problems can be stronly nonlinear and thus difficult to model specially when thin flexible structures are immersed in a fluid flow as for instance for an artificial cardiac valve. The classical ALE (Arbitrary Lagrangian Eulerian) approach is largely use to treat these problems. However, it has the disadvantage of needed a conforming mesh on the fluidstructure interface: the fluid mesh needs to be updated during the time integration process.
The developed method does not need any remeshing strategy. It is based on a spacetime discretization of the coupled problem. The fluid is incompressible, Eulerian and the NavierStokes equations are used. One or several thin structures are emmbeded in the fluid using a total Lagrangian formulation. The structures can moved through the fluid fixed spacetime mesh, they are catched by the isocontour of a level set function choosen as the signed distance function to the interface. Embedding a thin structure into the flow field results in nonsmooth fields of the physical state variables of the fluid. Based on the concept of the partition of unity and the extended finite element method, the spacetime approximations of the fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. The enrichment is a Heaviside function based on the sign of the level set function.
Spacetime meshing

Enriched elements

Finally, the method leads to a monolithic approach where a nonlinear system has to be solved at each time step. The unknowns are fluid velocity and pressure, Lagrange multipliers to ensure velocity continuity at the interface and also structure velocity.
Reference: A. Zilian, A. Legay, The enriched spacetime finite element method (EST) for simultaneous solution of fluidstructure interaction, International Journal for Numerical Methods in Engineering, 75 (3), 305334, 2008. doi