COUPLED 2023

Mechanically Consistent Modeling of Fluid-Structure-Contact Interaction without Collision Paradox

  • Champion, Marguerite (Inria Paris)
  • Fernandez, Miguel (Inria Paris)
  • Grandmont, Céline (Inria Paris & Université Libre de Bruxelles)
  • Vergnet, Fabien (Inria Paris)
  • Vidrascu, Marina (Inria Paris)

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The numerical simulation of systems involving fluid-structure-contact interaction is an oustanding problem that raises many modeling, mathematical and numerical issues. It is also crucial for numerous biomedical applications. For instance, contact modeling is of fundamental importance for the simulation of native or artificial cardiac valve dynamics. If fluid-structure interaction (FSI) is already highly complex in itself, modeling contact between solids adds challenging difficulties. First, in some configurations and with no-slip boundary conditions, FSI models are unable to predict contact (see, e.g., [1]); this is the so called no collision paradox. A second major issue is to obtain mechanically consistant models. Even in FSI models which allows contact, the simple addition of a contact constraint (variational inequality) leads to mechanical inconsistancies like unphysical void creation at releases from contact or unbalanced stress at contact. A favored approach is to consider a poroelastic modeling of the fluid seepage induced by the roughness of the contacting solid (see [2]). Yet, very little is known on the mathematical foundations of this approach. In this work, we analyze the ability of the reduced porous model reported in [2] to encompass contact. To this purpose, we build on an approach proposed in [3] to evaluate the behaviour of the drag force with respect to the gap in a simplified 2D setting of a sphere falling on a porous layer in a Stokes flow. As the porous layer is more permissive than Navier or Dirichlet boundary conditions, we show that the incompressibility constraint does not create a singularity at contact. The asymptotics of the model with respect to the porous layer parameters are also investigated. In particular, when the porous conductivity vanishes, we recover a Navier boundary condition. Finally, numerical evidence of these theoretical results is provided. REFERENCES [1] M. Hillairet, “Lack of collision between solid bodies in a 2D incompressible viscous flow,” Commun. Part. Diff. Eq., vol. 32, no. 9, pp. 1345-1371, 2007. [2] S. Frei, F. Gerosa, E. Burman and M. Fernandez, “A mechanically consistent model for fluid-structure interactions with contact including seepage”, Computer Methods in Applied Mechanics and Engineering, vol. 392, 202 [3] D. Gérard-Varet, D., M Hillairet, “Computation of the Drag Force on a Sphere Close to a Wall: The Roughness Issue.” ESAIM: Mathematical Modelling and Numerical Analysis 4