In this paper, we give a new characterization of the cut locus of a
point on a compact Riemannian manifold as the zero set of the optimal
transport density solution of the Monge-Kantorovich equations, a PDE
formulation of the optimal transport problem with cost equal to the
geodesic distance. Combining this result with an optimal transport
numerical solver, based on the so-called dynamical Monge-Kantorovich
approach, we propose a novel framework for the numerical approximation
of the cut locus of a point in a manifold. We show the applicability
of the proposed method on a few examples settled on 2d-surfaces
embedded in $R^{3}$, and discuss advantages and limitations.
We address the swimming problem at low Reynolds number. This regime, which is typically used for micro-swimmers, is described by Stokes equations. We couple a PDE solver of Stokes equations, derived from the Feel++ finite elements library, to a quaternion-based rigid-body solver. We validate our numerical results both on a 2D exact solution and on an exact solution for a rotating rigid body respectively. Finally, we apply them to simulate the motion of a one-hinged swimmer, which obeys to the scallop theorem.Resumé. Nous considérons le problème de la nage à bas nombre de Reynolds. Ce régime, typique des micro-organismes nageurs, est décrit par les équations de Stokes. Nous couplons un solveur d'EDP des équations de Stokes, construit à l'aide de la librairie aux éléments finis Feel++, avec un solveur de corps rigide basé sur les quaternions. Nous validons les deux solveurs à l'aide d'une solution exacte en 2D pour le fluide et d'une solution exacte pour un corps rigide qui tourne. Nous les appliquons pour simuler la nage d'un micro-organisme qui obéit au théorème de la coquille de Saint-Jacques.
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