Energy-Momentum Integrators for Elastic Cosserat Points, Rigid Bodies, and Multibody Systems

The goal of this chapter is to present the development of energy-momentum (EM) schemes in the framework of discrete (or finite-dimensional) mechanical systems. EM integrators belong to the class of structure-preserving numerical methods and have been originally developed in the field of nonlinear solid and structural mechanics. EM schemes and energy dissipating variants thereof typically exhibit improved numerical stability and robustness when compared to standard integrators. Due to their superior numerical properties, EM schemes have soon been extended to more involved applications such as flexible multibody dynamics and coupled thermomechanical problems. In this chapter, we start the development of second-order EM schemes in the context of the Cosserat point (or pseudo-rigid body). The theory of a Cosserat point shares main structural properties with semi-discrete formulations of elastodynamics. Indeed, the Cosserat point can be directly linked to the 4-node tetrahedral finite element. Besides its usefulness in explaining main ingredients of EM schemes such as the algorithmic stress formula, the Cosserat point is ideally suited to perform the transition to rigid body dynamics. In particular, in the present work, the rigid body formulation is obtained by imposing the zero strain condition on the Cosserat point. This way the rigid body is treated as constrained mechanical system. Moreover, we show that the EM discretization of constrained mechanical systems can be derived in a straightforward way from the EM scheme for the Cosserat point. The resulting rigid body formulation is closely connected to natural coordinates. Eventually, we deal with the extension to multibody systems which can be done in a straightforward way due to the presence of holonomic constraints in the present rigid body formulation.