Explicit high-order time and space accurate isogeometric collocation method for the dynamics of geometrically exact beams

The efficient simulation of highly deformable mechanical systems under fast and complex dynamic loads necessitates the advancement of explicit computational methods. Traditional explicit schemes, which leverage diagonal mass matrices and mass-scaling techniques, have improved efficiency. However, for problems evolving on nonlinear manifolds, such as IR3×SO(3), their accuracy remains limited to second-order in time. This limitation becomes even more critical when combined with high-order spatial techniques, such as Isogeometric Collocation (IGA-C), since it prevents the full exploitation of the spatial discretization potentialities. To address this gap, we propose a novel, fully explicit, high-order method for the dynamics of geometrically exact beams. The approach extends the Runge-Kutta-Munthe-Kaas (RKMK) scheme, originally developed for rigid body systems, to nonlinear partial differential equations governing deformable structures. The strong form of the problem is efficiently discretized in space using IGA-C, eliminating the need for element integration. Through various numerical tests, we demonstrate the ability of the proposed formulation to achieve high-order accuracy in time and space. Although the paper is focused on geometrically exact beams, the method is general and lays the ground for a promising extension to geometrically exact shells.