Section: New Results

Mathematical and numerical analysis of fluid-structure interaction problems

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

• In [26] , we study a three-dimensional fluid-structure interaction problem. The motion of the fluid is modeled by the Navier-Stokes equations and we consider for the elastic structure a finite-dimensional approximation of the equation of linear elasticity. The time variation of the fluid domain is not known a priori, so we deal with a free boundary value problem. Our main result yields the local in time existence and uniqueness of strong solutions for this system.

• In [31] , a robust finite volume method for the solution of high-speed compressible flows in multi-material domains involving arbitrary equations of state and large density jumps is presented. One of the main contributions of this paper is a tabulation method based on a sparsegrid approximation to solve very efficiently two-phase Riemann problems for arbitrary equations of state. The proposed computational method is illustrated with the three-dimensional simulation of the dynamics of an underwater explosion bubble.

• In [52] we analyze the performances of several Luenberger observers to estimate the state of a fluid-structure interaction model for hemodynamics, when the measurements are assumed to be restricted to displacements or velocities in the solid. The present framework establishes that these methods are very attractive strategies (compared, e.g., to classical variational techniques) to perform state estimation.

• In [51] we analyze two 3D-0D coupling approaches in which a fractional-step projection scheme is used in the fluid. We introduce and analyze an implicitly 3D-0D coupled formulation with enhanced stability properties and which requires a negligible additional computational cost. The theoretical stability results are confirmed by meaningful numerical experiments in patient specific geometries coming from medical imaging.

• In [55] we introduce a class of explicit Robin-Neumann schemes for the explicit coupling of a general thin-structure (e.g., viscoelastic and non-linear) with an incompressible fluid. These methods generalize the displacement correction schemes introduced in [32] . A priori stability and convergence error estimates show that optimal first-order accuracy can be achieved with appropriate extrapolation and without compromising stability. A deep numerical study confirms the theoretical findings.

• In [64] we present two-dimensional simulations of chemotactic self-propelled bacteria swimming in a viscous fluid. Self-propulsion is modelled by a couple of forces of same intensity and opposite direction applied on the rigid bacterial body and on an associated region in the fluid representing the flagellar bundle. The orientations of the individual bacteria are subjected to random changes, with a frequency that depends on the surrounding oxygen concentration, in order to favor the direction of the concentration gradient.

• In [40] we propose a method of modeling sail structures which captures the wrinkling behavior of such structures. The method is validated through experimental and analytical test cases, particularly in terms of wrinkling prediction. An enhanced wrinkling index is proposed as a valuable measure characterizing the global wrinkling development on the deformed structure. The method is based on a pseudo-dynamic finite element procedure involving non-linear MITC shell elements. The major advantage compared to membrane models generally used for this type of analysis is that no ad hoc wrinkling model is required to control the stability of the structure. We demonstrate our approach to analyse the behavior of various structures with spherical and cylindrical shapes, characteristic of downwind sails over a rather wide range of shape constitutive parameters. In all cases convergence is reached and the overall flying shape is most adequately represented, which shows that our approach is a most valuable alternative to standard techniques to provide deeper insight into the physical behaviour. Limitations appear only in some very special instances in which local wrinkling-related instabilities are extremely high and would require specific additional treatments.