Lie group and Lie algebra variational integrators for flexible beam and plate in R

The purpose of this research project is to develop variational integrators synchronous or asynchronous, which can be used as tools to study complex structures composed of plates and beams subjected to large deformations and stress. We consider the geometrically exact models of beam and plate, whose configuration spaces are Lie groups. These models are suitable for modeling objects subjected to large deformations, where the stored energy chosen is adapted for the types of materials used in our field (isotropic or composite). The work of J. E. Marsden, and of his doctoral and post-doctoral students, were the basis for the development of variational integrators which are symplectic and perfectly preserve symmetries. Furthermore, discrete mechanical systems with symmetry can be reduced. In addition, by a ”good discretization”, the strain measures are unchanged by superposed rigid motion. The idea behind this work is to take advantage of the properties of these integrators to define the equilibrium position of structures, which are generally unknown, as well as to determine the constraints, while preserving the invariants of the structure. Along with solving this problem, we continue the approach of J.E. Marsden which consists to lay the foundations of discrete mechanics, with its theorems, its axioms, its definitions, which have the same value as the laws of continuous mechanics but for a discrete domain. That is, the discrete trajectories of a motion obtained by variational integrators satisfy these discrete laws.