Space--time formulation of geometrically exact beams
2026-08-03 • Computational Engineering, Finance, and Science
Computational Engineering, Finance, and Science
AI summaryⓘ
The authors describe a way to model the motion of a special type of beam using a space-time surface in four dimensions, keeping time separate from other movements. They combine forces and momenta into common quantities and use a mathematical method involving finite elements to simulate the beam's behavior accurately over time. Their numerical tests show the method works well under different conditions and preserves important physical properties, though some extensions to other mesh shapes are less stable. They recommend using their specific aligned element approach for practical simulations.
Cosserat beamgeometrically exact beam theoryspace-time formulationfinite element methodPetrov-Galerkin stabilizationtensor-product elementsNeumann boundary conditionsmomentum conservationrigid body motionnumerical stability
Authors
C. Hesch, E. Sahin
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.