Section: New Results

Mathematical and numerical analysis of fluid-structure interaction problems

Participants : Benoit Fabrèges, Miguel Ángel Fernández Varela, Mikel Landajuela Larma, Jimmy Mullaert, Marina Vidrascu.

• In [54] we introduce two new classes of numerical methods for the solution of incompressible fluid/thin-walled structure interaction problems with unfitted meshes. The semi-implicit or explicit nature of the splitting in time is dictated by the order in which the spatial and time discretizations are performed. Stability and optimal accuracy are achieved without restrictive CFL conditions or correction iterations. Results presented by M. Landajuela at the 11th World Congress on Computational Mechanics (WCCM XI), July 20-25, 2014, Barcelona (Spain).

• In [47] we introduce a class of fully decoupled time-marching schemes (velocity-pressure-displacement splitting) for the coupling of an incompressible fluid with a thin-walled viscoelastic structure. The time splitting combines a projection method in the fluid with a specific Robin-Neumann treatment of the interface coupling. A priori energy estimates guaranteeing unconditional stability are established for some of the schemes. The accuracy and performance of the methods proposed is illustrated by a thorough numerical study.

• We have performed an a priori error analysis for the generalized Robin-Neumann explicit coupling schemes introduced in [30] . The analysis confirms the $𝒪({\tau }^{2^{r-1}}/h^{\frac{1}{2}})$ error perturbation anticipated by the numerical evidence of [30] . Another fundamental result of this work is that the $h$-non-uniformity of the splitting error is not a consequence of the mass-lumping approximation (which simply dictates the explicit or semi-implicit nature of the coupling scheme). The analysis indicates that the genesis of the $𝒪\left(h^{-\frac{1}{2}}\right)$ is the non-uniformity of discrete viscoelastic operator, which is a consequence of thick-walled character of the solid. These results have been reported in [48] and presented by M.A. Fernández at the 11th World Congress on Computational Mechanics (WCCM XI), July 20-25, 2014, Barcelona (Spain).

• We consider the extension of the Nitsche-XFEM method to fluid-structure interaction problems involving a thin-walled elastic structure (Lagrangian formalism) immersed in an incompressible 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 kinematic/kinetic fluid-solid coupling is enforced consistently using a variant of Nitsche’s method involving cut elements. Robustness with respect to arbitrary interface/element intersections is guaranteed through a ghost penalty stabilization. Different coupling schemes, either fully implicit or loosely coupled, are proposed. Several numerical examples, involving static and moving interfaces, illustrate the performance of the methods. A paper in collaboration with F. Alauzet (project-team Gamma3) is under preparation. Results presented by B. Fabrèges at the 11th World Congress on Computational Mechanics (WCCM XI), July 20-25, 2014, Barcelona (Spain).