Equivalence between the co-rotational finite element method and the absolute coordinate formulation in multibody dynamics

Abstract

Geometric nonlinearity arises in problems involving large displacements and rotations, where traditional linear assumptions fail. In the finite element community, several formulations have been developed to address these complexities; the corotational finite element formulation (CRF) has proven to be an efficient alternative to generic total Lagrangian (TL) and updated Lagrangian (UL) approaches for small-strain problems. In the multibody dynamics community, the floating frame of reference formulation (FFRF) is commonly used, employing a gross-motion-following local reference frame per body (also referred to as co-rotational or floating frame), in contrast to CRF's element-based approach. A less known but fully equivalent multibody formulation to FFRF is the absolute coordinate formulation (ACF), which uses absolute coordinates in contrast to rigid body coordinates plus local deformation as in FFRF. This paper demonstrates the equivalence of ACF and CRF, with the only difference being the number of reference frames-body-based versus element-based. Moreover, since CRF, while efficient, faces challenges in real-world multibody simulations due to the computational burden of assigning a reference frame to each finite element, this paper also discusses how CR/ACF can be applied when partitioning bodies into substructures, each equipped with its own reference frame.