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

TL;DR

The paper tackles the challenge of efficiently simulating the dynamics of geometrically exact shear-deformable beams with high spatial order. It introduces a fully explicit IGA-C method by decoupling Neumann boundary conditions, reorganizing and scaling the mass matrix, and linearizing the rotational balance to avoid Newton iterations, yielding a diagonalizable system while preserving accuracy. The approach, termed LU L, is benchmarked against the existing CN NL formulation and a lumped nonlinear variant across cantilever, pendulum, 3D flying, and spinning beam tests, showing comparable accuracy with substantial runtime reductions, particularly as the discretization grows. This work enhances the practicality of explicit beam dynamics in complex, high-fidelity simulations and opens avenues for multi-patch and SO(3)-consistent extensions in advanced engineering applications.

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.

Figures (19)

• Figure 1: Spectral radius versus number of collocation points for different basis function degree $p$ and different combinations of boundary conditions. Results are the same for both values $k$ can take.
• Figure 2: Cantilever beam under impulsive load: geometry and applied forces.
• Figure 3: Tip displacement of a cantilever beam subjected to a tip force $F_3 = -100N$. $p=4$, $\rm n = 20$, $h = 1E-6s$: results for the entire simulation time (left) and close-up for $t\in[0.44,0.48]\,s$ (right).
• Figure 4: Cantilever beam under tip load: convergence plots for $p=2,4,6$ vs. number of collocation points of the three solution procedures.
• Figure 5: Swinging flexible pendulum subjected to a distributed vertical load.