Momentum and near-energy conserving/decaying time integrator for beams with higher-order interpolation on SE(3)

Abstract

This study aims to develop a simple yet robust time-integration scheme for configuration-interpolated beam finite elements. Geometrically exact theory is employed to model the beams. It utilises the rotations which belong to a non-commutative Lie group and thus require special attention. The configuration is interpolated using a two-node SE(3) interpolation or its generalised implicit variant which enables higher orders. Such interpolation treats position and orientation as a unit and a member of the SE(3) Lie group. This unified approach allows for elegant mathematical manipulation. By adapting the Lie midpoint rule appropriately, it becomes possible to express energy changes in terms of inertial and internal forces, thus enabling the derivation of a momentum conserving and almost energy conserving time integration algorithm. The precision of energy conservation depends solely on the length of the finite elements. With further modification, this algorithm can also become an energy-decaying algorithm. Despite the configuration-dependent nature of the interpolation, the need to deal with its derivatives is avoided, which simplifies the implementation. The method is tested using 3D numerical example with finite rotations which confirms, that the method indeed has the conservation properties and is stable and robust.