Space--time formulation of geometrically exact beams

Abstract

We formulate the dynamics of a straight-reference geometrically exact Cosserat beam as a directed world sheet in non-relativistic space--time. The centerline history is embedded in \(\mathbb R^4\), while absolute time remains prescribed and is not an additional mechanical degree of freedom. Spatial force and moment resultants and temporal linear and intrinsic angular momenta are combined into common space--time fluxes, so that spatial Neumann data and temporal inflow and outflow are represented by one co-normal boundary operator. A mixed configuration--momentum system is discretized by continuous tensor-product finite elements and stabilized by a future-directed Petrov--Galerkin perturbation. The resulting time-aligned \(Q_2/Q_2\) method requires one scalar stabilization parameter and retains the interior momentum trace as the free terminal outflow. The numerical study verifies the terminal flux treatment by an exact rigid-motion state, establishes monotone convergence for a smooth manufactured shear--bending solution, quantifies the pre-response--accuracy trade-off under delayed loading, and verifies covariance under constant superposed spatial rotations. The stabilization is not a proof of strict domain-of-dependence causality; it biases the global space-time approximation in the future direction and reduces the measured pre-activation response. A conservative extension to non-aligned simplex facets is also analyzed. Although this extension is consistent and locally conservative, continuous equal-order \(P_2/P_2\) simplex spaces exhibit strongly mesh- and orientation-dependent inverse amplification and do not show mesh-uniform stability. The aligned \(Q_2/Q_2\) discretization is therefore recommended as the practical realization of the formulation.