## Section: New Results

### Mathematical and numerical analysis of fluid-structure interaction problems

Participants : Matteo Aletti, Faisal Amlani, Benoit Fabrèges, Miguel Ángel Fernández Varela, Jean-Frédéric Gerbeau, Mikel Landajuela Larma, Damiano Lombardi, Marina Vidrascu.

In [55] we present a numerical study in which several partitioned solution procedures for incompressible fluid-structure interaction are compared and validated against the results of an experimental FSI benchmark. The numerical methods discussed cover the three main families of coupling schemes: strongly coupled, semi-implicit and loosely coupled. Very good agreement is observed between the numerical and experimental results. The comparisons confirm that strong coupling can be efficiently avoided, via semi-implicit and loosely coupled schemes, without compromising stability and accuracy.

In [14] we introduce a Nitsche-XFEM method for fluid-structure interaction problems involving a thin-walled elastic structure (Lagrangian formalism) immersed in an incompressible viscous fluid (Eulerian formalism). The fluid domain is discretized with an unstructured mesh not fitted to the solid mid-surface mesh. Weak and strong discontinuities across the interface are allowed for the velocity and pressure, respectively. The fluid-solid coupling is enforced consistently using a variant of Nitsche's method with cut-elements. Robustness with respect to arbitrary interface intersections is guaranteed through suitable stabilization. Several coupling schemes with different degrees of fluid-solid time splitting (implicit, semi-implicit and explicit) are investigated. A series of numerical tests in 2D, involving static and moving interfaces, illustrates the performance of the different methods proposed.

In [15] we investigated the autoregulation in the retinal haemodynamics by means of three-dimensional simulations. The autoregulation is a key phenomenon from a physiological standpoint, consisting in the ability of the vasculature to control the flow in different pressure conditions. A simplified fluid-structure interaction method was devised in order to render the vessels wall contraction in a large network, with an affordable computational cost. Several test cases were performed on a patient-specific arteriolar network, whose geometry was reconstructed by using fundus camera images. The tests were in agreement with experimental trends and confirm the ability of the approach to reproduce the phenomena involved.

In [33] we study an unsteady nonlinear fluid-structure interaction problem which is a simplified model to describe blood flow through viscoelastic arteries. We consider a Newtonian incompressible two-dimensional flow described by the Navier-Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear viscoelastic beam equation. The fluid and the structure are fully coupled via interface conditions prescribing the continuity of the velocities at the fluid-structure interface and the action-reaction principle. We prove that strong solutions to this problem are global-in-time. We obtain in particular that contact between the viscoleastic wall and the bottom of the fluid cavity does not occur in finite time. To our knowledge, this is the first occurrence of a no-contact result, but also of existence of strong solutions globally in time, in the frame of interactions between a viscous fluid and a deformable structure.

In [27] and [45] we study the effect of wall bending resistance on the motion of an initially spherical capsule freely suspended in shear flow or in a a planar hyperbolic flow. We consider a capsule with a given thickness made of a three–dimensional homogeneous elastic material. A numerical method is used to model the coupling of a boundary integral method for the fluids with a shell finite element method for the capsule envelope. For a given wall material, the capsule deformability strongly decreases when the wall bending resistance increases. In addition, if one expresses the same results as a function of the two–dimensional mechanical properties of the mid–surface, which is how the capsule wall is modeled in the thin–shell model, the capsule deformed shape is identical to the one predicted for a capsule devoid of bending resistance. The bending rigidity is found to have a negligible influence on the overall deformation of an initially spherical capsule, which therefore depends only on the elastic stretching of the mid–surface. Still, the bending resistance of the wall must be accounted for to model the buckling phenomenon, which is observed locally at low flow strength and persist at steady state. We show that the wrinkle wavelength only depends on the bending number, which compares the relative importance of bending and shearing phenomena, and provide the correlation law. Such results can then be used to infer values of the bending modulus and wall thickness from experiments on spherical capsules in simple shear flow.

In [57] we consider the motion of an elastic structure represented by the nonlinear Saint-Venant Kirchhoff model immersed in a compressible fluid modeled by the compressible Navier-Stokes equations. Existence and uniqueness of a regular solution defined locally in time is proved.