# Team : reo

## Section: New Results

### Mathematical modelling and numerical methods in fluid dynamics

Participants : Paola Causin, Nuno Diniz dos Santos, Miguel Ángel Fernández, Jean-Frédéric Gerbeau, Céline Grandmont, Yvon Maday.

#### Existence results in fluid-structure interaction

In [13] and [31] we study 2D or 3D viscous incompressible fluid governed by the Navier–Stokes equations, interacting with elastic plate or beam located on one part of the fluid boundary. These models can be viewed as a first attempt to described the evolution of the blood flow past arteries. In [13] existence of weak solutions is proven for a three dimensional fluid interacting with a ``viscous'' plate in flexion. From the mechanical point of view, adding a viscous term is a way to introduce dissipation in the beam model and from the mathematical point of view, this is a way to regularize the structure velocity. In [31] we show that for a 2D–fluid coupled with a beam one can pass to the limit as the coefficient modeling the viscoelasticity (resp. the rotatory inertia) of the beam tends to zero. In this case the dissipation coming from the fluid enables us to control the high frequencies of the structure velocity. As a consequence, we obtain the existence of at least one weak solution for the limit problem (Navier–Stokes equation coupled with Euler–Bernoulli equation) as long as the beam does not touch the bottom of the fluid cavity. In [32], we present and analyse a dynamical geometrically nonlinear formulation that models the motion of two–dimensional and three–dimensional elastic structures in large displacements–small strains. In a first part we derive the equations describing the motion of the body. In a second part, existence of a weak solution is proven using a Galerkin method. We also prove that the solution is unique. Those type of models can be used for instance to described the motion of red blood cells.

#### Numerical methods in fluid-structure interaction

This activity is done in close collaboration with the MACS project, in particular with Marina Vidrascu.

We have proposed in [6] a numerical method to solve the coupling between the Navier-Stokes equations on moving domains (ALE formulation) and a shell model in large displacements. The core of the algorithm is a Newton-Krylov method based on a reduced model which offers a significative gain in robustness and efficiency compared to standard methods commonly used in this field. In [29], we propose a nested preconditionner of GMRES and acceleration techniques which improve the efficiency of Newton-Krylov methods. The basic idea is to use previously computed Krylov basis in order to build a cheap and efficient preconditionner for subsequent problems.

In [11], we give, on a simplified model, a theoretical explanation of several empirical facts observed in the simulation of blood flows in compliant vessels. In particular, we show that under certain choices of the physical parameters, typically when the densities of the fluid and of the structure are close or when the domain is a slender geometry, loosely coupled schemes are unstable irrespectively of the time step.

##### Work in progress

We are currently investigating numerical methods, in particular based on ALE formulation and fictitious domain method, to simulate the behaviour of biological valves.

#### Stabilized finite element methods in fluid mechanics

We first addressed the problem from the classical point of view, namely, by stabilizing both the velocities and the pressure using a residual based stabilization (SUPG/SDEFM). Hence our method consists in subtracting a mesh dependent term (including the equation residual) from the formulation without compromising consistency. Our contribution relies on the fact that the design of mesh dependent term, as well as the stabilization parameter involved, are suggested by bubble condensation. As a result, no free constants have to be set. Stability was proved for any combination of velocity and pressure spaces, under the hypotheses of continuity for the pressure space. Optimal order error estimates were derived for the velocity and the pressure. Numerical experiments in 2D confirmed these theoretical results [25]. This work was carried out in collaboration with G. Barrenechea and C. Vidal.

Although the previous approach gives good results in practice, it has several undesirable features. Among others, artificial boundary conditions and artificial pressure-velocity couplings are introduced, which (in particular) makes time stepping awkward. To overcome these disadvantages we have recently introduced a new method based on the addition of gradient jumps to the discrete formulation. Stability is obtained in a unified fashion without introducing pressure velocity couplings or additional unknowns. The method has been analyzed and tested with very promising results in 3D [27]. This work was carried out in collaboration with E. Burman and P. Hansbo.