# Fluid-structure interaction

We have developed and analysed new schemes for the numerical resolution of fluid-structure interaction, involving an incompressible viscous fluid and a structure in large displacements and/or deformations. This issue is particularly difficult to face when added-mass effect, of the fluid acting on the structure, is strong, since the standard fully explicit schemes are numerically unstable. On the other hand, standard partitioned methods for the solution of fully coupled implicit schemes require expensive iterative procedures to solve the resulting non-linear systems. Within this framework, we have addressed the following two major issues:

- how to improve efficiency in implicit coupling;
- how to avoid implicit coupling without compromising stability and accuracy.

A number of iterative solution procedures and preconditioners (e.g., exact and inexact Dirichlet-Neumann Newton-GMRES solver) have been proposed and investigated in the framework of the first issue. The second problem has been addressed through the following three different coupling paradigms:

- Projection based semi-implicit coupling. This scheme avoids strong coupling without compromising stability;
- Stabilized explicit coupling. This explicit coupling scheme is stable irrespectively of the added-mass effect;
- Robin-Neumann schemes. These explicit coupling schemes are stable, irrespectively of the added-mass effect, and in the case of the coupling with thin-walled structures yield optimal first-order accuracy.

Incompressible Navier-Stokes equations in ALE formulation coupled with a linear shell (Reissner-Mindlin) model. Simulation performed with a Robin-Neumann explicit coupling scheme. | Incompressible Navier-Stokes equations in ALE formulation coupled with a non-linear shell (Reissner-Mindlin) model. Simulation performed with a Robin-Neumann explicit coupling scheme. |

Incompressible Navier-Stokes equations in ALE formulation coupled with non-linear elastodynamics (Saint-Venant-Kirchhoff material). Simulation performed with a generalized Robin-Neumann explicit coupling scheme. | Incompressible Navier-Stokes equations in ALE formulation coupled with non-linear elastodynamics (Saint-Venant-Kirchhoff material). Simulation performed with a generalized Robin-Neumann explicit coupling scheme. |

# Stabilized finite element methods in fluid mechanics

This research activity is essentially devoted to the development and analysis of stabilized finite element methods for transient problems in fluid mechanics. Gold-standard *residual based* finite element stabilizations overcome the instabilities related to the discrete inf-sup incompatibility or to the presence of dominant convection. In the steady case, the numerical analysis of these methods is well established. However, their formulation and analysis in the transient case raises many issues. For instance, these methods do not seem to be compatible with explicit Runge-Kutta schemes and their combination with standard *A*-stable time-marching can lead to numerical instabilities. We have worked on the formulation and the analysis symmetric stabilization methods (e.g., CIP) and shown that these methods are particularly well adapted to the spatial approximation of transient problems (e.g., reaction-advection-diffusion, Stokes and Navier-Stokes equations). We have combined these approaches with *A*-stable monolithic time-marching, explicit Runge-Kutta schemes and fractional-step methods.

Incompressible Navier-Sokes equations (Re=100). Simulation performed with a BDF1 fractional-step scheme with CIP spatial stabilization | Incompressible Navier-Sokes equations (Re=100000). Simulation performed with a BDF2 monolithic scheme with CIP spatial stabilization |

# Reduced-Order Modeling

Inverse Problems as well as optimization problems requires hundreds or thousands direct numerical simulations to be performed. The computational cost often makes them unfeasible in realistic applications. Reduced-order modeling aims at defining a system of equations that emulate the original one but whose integration is significantly cheaper from a computational standpoint.

Several numerical methods have been proposed to this end. Among them POD (Proper Orthogonal Decomposition) has been used in many contexts and applications. POD is well performing for problems featured by “global” solutions, but its performances are poor when transport phenomena are involved. Moreover, it is based on a database of precomputed solutions and its performances “out of database” are often not robust.

To overcome these drawbacks, a method (ALP) is currently under development, that consists in defining a time varying basis expansion. The basis is moved by asking that the representation of a given operator on the basis is diagonal in time. This operator has been linearly perturbed with the solution of the system itself. Hence:

- the basis try to “follow” the system dynamics
- there is no need to rely totally on precomputed solutions, so that the robustness is increased

An example that has been extensively investigated concerns the Schroedinger operator in which the potential is the solution of the system itself.

The equations in the reduced-order space have been obtained by Galerkin projection.

Two realistic applications are currently under scrutiny:

- the ultra-fast integration of 1D fluid-structure interaction on systemic networks
- the electrophysiology