Fluid-Structure Interactions in Discrete Mechanics

The primary objective of discrete mechanics is to unify various laws from different areas of physics, such as fluid mechanics and solid mechanics. The same objective was also pursued by continuum mechanics, but the latter has not been entirely successful in accomplishing it. The Galilean invariance and the principle of equivalence make it possible to rewrite the law of dynamics as an equality between accelerations, the one undergone by the medium and the external accelerations applied to it. The derivation of the equation of discrete motion leads to writing the acceleration as a Hodge-Helmholtz decomposition, i.e. the sum of a gradient of a scalar potential and the rotational of a vector potential. By choosing the acceleration as being a primary variable, we can express the velocity and the displacement simply as quantities that accumulate over time. Potentials represent energies per unit mass and are also stored over time. The resulting formulation is able to describe the motion and dynamics of complex media, that can be both fluid and solid, under large deformations and large displacements. Two examples of fluid-structure coupling, an analytical solution and a numerical solution used for a benchmark, are presented here. They show the ability of the model to reproduce the behavior of interacting fluid and solid media.