Stabilized Pressure Segregation methods and their application to Fluid-Structure Interaction problems

The numerical approximation of
the Navier-Stokes equations is a difficult task that has focused an
active research in the last decades. The velocity and pressure
interpolation spaces must satisfy a undesired compatibility
condition when using the Galerkin method. Another unrelated
complication is the numerical instability that arises for convective
dominated flows. About the computational cost, the incompressibility
constraint couples velocity and pressure computation, being its
numerical solution expensive.

In this seminar we consider some topics in order to facilitate the
finite element discretization of the Navier-Stokes equations. We
suggest the use of a stabilization technique in order to circumvent
the compatibility condition over the velocity and pressure discrete
spaces and be able to solve convective dominant flows. We introduce
some novel concepts for the stabilization of transient equations. In
order to reduce the computational cost, we explore some pressure
segregation methods motivated at the discrete level. We state and
analyze classical pressure correction methods and new velocity
correction methods motivated by the use of a discrete pressure
Poisson equation (where is the velocity instead of the pressure the
extrapolated variable).

The second part of the seminar tackles the fluid-structure
interaction problem. We apply the ALE framework in order for the
fluid governing equations to be formulated on moving domains. We
analyze the blend of the ALE framework and a stabilized finite
element method.For the interaction, we consider a coupling procedure
that profits from the ingredients previously introduced. The final
algorithm, using one loop, tends to the monolithic (fluid-structure)
system.

This method has been applied to the analysis of bridge aerodynamics,
obtaining a good convergence behavior. We end with the simulation of
wind turbines. The fact that we have a rotary body surrounded by the
fluid (air) has motivated the introduction of a selective remeshing
procedure.