Applied Mathematics Seminar

An Eulerian Monolithic Fluid-Structure Formulation
Monday, 13 February 2017 - 10:00 am to 10:30 am
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SPEAKER: Olivier Pironneau (Sorbonne Universités, UPMC (Paris VI) Laboratoire Jacques-Louis Lions) DATE: Monday, February 13, 2017 TIME: 10:00 am ROOM: B015 ABSTRACT: The conservation laws of continuum mechanic are naturally written in an Eulerian frame where the difference between a fluid and a solid is only in the expression of the stress tensors, usually with Newton?s hypothesis for the fluids and Helmholtz potentials of energy for hyperelastic solids. There are currently two favoured approaches to Fluid Structured Interactions (FSI) both working with the equations for the solid in the initial domain; one uses an ALE formulation for the fluid and the other matches the fluid-structure interfaces using Lagrange multipliers and the immersed boundary method. By contrast the proposed formulation works in the frame of physically deformed solids and proposes a discretization where the structures have large displacements computed in the deformed domain together with the fluid in the same; in such a monolithic formulation velocities of solids and fluids are computed all at once in a single variational formulation by a semi-implicit in time and the finite element method. The idea is not new but the progress of mesh generators renders this approach feasible and even robust. In this lecture the method and its discretization are presented, stability is shown by an energy estimate even in the discrete case. Extension to problems with contact will also be shown using the semi-smooth Newton algorithm on the variational inequality of the problem. A numerical section discusses implementation and presents benchmark cases in 2D.