Section: New Results

Mathematical and numerical analysis of fluid-structure interaction problems

Participants : Cristóbal Bertoglio Beltran, Muriel Boulakia, Miguel Ángel Fernández Varela, Jean-Frédéric Gerbeau, Jimmy Mullaert.

• Over the last decade, the numerical simulation of incompressible fluid-structure interaction has been a very active research field and the subject of numerous works. In [19] , we review some of the coupling schemes recently proposed in the literature. Some numerical results that show the effectiveness of the novel approaches are also presented.

• In [21] , we propose a new class of time-marching schemes for the explicit coupling of an incompressible fluid and a general elastic solid (i.e., not necessarily thin [46] and possibly damped). We state a general energy-based stability result and illustrate the accuracy of the different variants in a numerical benchmark.

• [30] : This paper focuses on Eulerian-based algorithms for fluid-structure applications featuring large structural motions and/or deformations in the context of compressible flows. First, it presents a numerical method for treating simultaneously the fluid pressure and velocity conditions on static and dynamic embedded interfaces. This method is based on the exact solution of local, one-dimensional, fluid-structure Riemann problems. Next, it describes two consistent and conservative approaches for computing the flow-induced loads on rigid and flexible embedded structures.

• In [39] , we present some issues encountered in fluid-structure interaction simulation (this text is targeted to a non-expert audience).

• In [42] , we present a robust and efficient parameter estimation strategy for fluid-structure interaction problems. The method is based on a filtering algorithm restricted to the parameter space, known as the reduced order Unscented Kalman Filter. We illustrate our methodology with the estimation of the artery wall stiffness and the proximal Windkessel resistance.

• [44] : In this paper, we are interested in the three-dimensional coupling between an incompressible fluid and a rigid body. The fluid is modeled by the Navier-Stokes equations, while the solid satisfies the Newton's laws. In the main result of the paper we prove that, with the help of a distributed control, we can drive the fluid and structure velocities to zero and the solid to a reference position provided that the initial velocities are small enough and the initial position of the structure is close to the reference position.

• In [46] , we introduce a class of incremental displacement-correction schemes for the explicit coupling of a thin-structure with an incompressible fluid. We provide a general stability and convergence analysis that covers both the incremental and the non-incremental variants, and also the fully implicit case. The incremental variant with first-order extrapolation is unconditionally stable and yields optimal first-order accuracy in time.