A fully explicit isogeometric collocation formulation for the dynamics of geometrically exact beams

TL;DR

This paper introduces a fully explicit isogeometric collocation method for geometrically exact beam dynamics, improving computational efficiency and high-order accuracy by decoupling boundary conditions and simplifying the mass matrix.

Contribution

It extends a predictor--multicorrector approach to rotational beam dynamics, removing nonlinear terms and system matrix inversion for enhanced efficiency and accuracy.

Findings

Achieves high-order spatial accuracy in beam simulations.

Reduces computational cost by removing nonlinear terms.

Demonstrates improved efficiency over previous formulations.

Abstract

We present a fully explicit dynamic formulation for geometrically exact shear-deformable beams. The starting point of this work is an existing isogeometric collocation (IGA-C) formulation which is explicit in the strict sense of the time integration algorithm, but still requires a system matrix inversion due to the use of a consistent mass matrix. Moreover, in that work, the efficiency was also limited by an iterative solution scheme needed due to the presence of a nonlinear term in the time-discretized rotational balance equation. In the present paper, we address these limitations and propose a novel fully explicit formulation able to preserve high-order accuracy in space. This is done by extending a predictor--multicorrector approach, originally proposed for standard elastodynamics, to the case of the rotational dynamics of geometrically exact beams. The procedure relies on decoupling the Neumann boundary conditions and on a rearrangement and rescaling of the mass matrix. We demonstrate that an additional gain in terms of computational cost is obtained by properly removing the angular velocity-dependent nonlinear term in the rotational balance equation without any significant loss in terms of accuracy. The high-order spatial accuracy and the improved efficiency of the proposed formulation compared to the existing one are demonstrated through some numerical experiments covering different combinations of boundary conditions.