M. Borri, C.L. Bottasso, L. Trainelli

An Invariant-Preserving Approach to Robust Finite-Element Multibody Simulation

Zeitschrift für Angewandte Mathematik und Mechanik, 83 (10): 663-676 (2003).

Abstract

We briefly describe the main concepts that have inspired a novel approach for the integration of general non-linear structural and multibody dynamics problems within a finite-element framework that includes geometrically exact beams. In this approach, a numerically robust family of second to fourth order accurate algorithms are devised, which retain several important qualitative features of the exact solution. Among these, we address frame indifference, invariance with respect to reference entities (such as beam axes and shell mid-surfaces), and, for load-free motions, preservation of the system total linear and fixed-pole angular momenta and a rigorous bound on the system total mechanical energy. When constrained systems are analyzed, these results are obtained by additionally requiring at the discrete level the preservation of Newton's Third Law of Action and Reaction and the vanishing of the work of the reaction forces for each constrained body pair. As a result, the schemes exhibit full non-linear unconditional stability according to the Energy Method. The underlying single-step, two-stage, implicit difference schemes provide tunable high-frequency damping ranging from null ($\rho_{\infty}=1$) to asymptotic annihilation ($\rho_{\infty}=0$). The scheme performances are shown to improve on those provided by widely adopted algorithms stemmed from the Newmark family such as the generalized-$\alpha$ methods. Representative numerical exercises are included to illustrate the main features of the methodology.