We consider self-propelled rigid bodies interacting through local body-attitude alignment modelled by stochastic differential equations. We derive a hydrodynamic model of this system at large spatio-temporal scales and particle numbers in any dimension $n \geq 3$. This goal was already achieved in dimension $n=3$ or in any dimension $n \geq 3$ for a different system involving jump processes. However, the present work corresponds to huge conceptual and technical gaps compared with earlier ones. The key difficulty is to determine an auxiliary but essential object, the generalised collision invariant. We achieve this aim by using the geometrical structure of the rotation group, namely its maximal torus, Cartan subalgebra and Weyl group as well as other concepts of representation theory and Weyl’s integration formula. The resulting hydrodynamic model appears as a hyperbolic system whose coefficients depend on the generalised collision invariant.

A mathematic model is established to describe a swarm with multi-behavior. Regarding a swarm designed for cooperative task, we propose a model which includes a macroscopic model and a individual-based model. The macroscopic framework model describes global dynamics of swarms, which is normally expressed by dynamical populations' densities with different behaviors, while the individual-based framework model describes a individual agent's trajectory. Based on these models, we prove that all objects can be collected to the “home” area under conditions of individual agents subject to sensor constraints.

La modélisation de la mise en forme des structures souples présente un outil précieux d'aide à la conception. Elle permet d'étudier la faisabilité du procédé de fabrication ainsi que la prédiction des défauts qui puissent être générés. Cependant, il est nécessaire d'avoir une connaissance du comportement mécanique de la structure. Une modélisation du comportement mécanique des étoffes souples est présentée. L'identification des paramètres du modèle s'appuie sur des essais de traction uni-axiale. Une méthode d'identification par approche inverse basée sur des algorithmes d'optimisation est proposée. Des simulations de mise en forme par emboutissage avec remaillage adaptatif sont étudiées dans cet article.