Lie Group Variational Integrator for the Geometrically Exact Rod with Circular Cross-Section Incorporating Cross-Sectional Deformation

TL;DR

This paper develops a structure-preserving numerical integrator for a 3D Cosserat rod model that includes cross-sectional deformation, ensuring volume conservation, energy stability, and accurate physical behavior in simulations.

Contribution

It introduces a Lie group variational integrator for a geometrically exact rod model with cross-sectional deformation, combining rotational and deformation effects in a structure-preserving manner.

Findings

The integrator conserves volume and energy with bounded error.

It accurately replicates the physics of the Cosserat rod system.

Numerical results validate the model's effectiveness under various conditions.

Abstract

In this paper, we derive the continuous space-time equations of motion of a three-dimensional geometrically exact rod, or the Cosserat rod, incorporating planar cross-sectional deformation. We then adopt the Lie group variational integrator technique to obtain a discrete model of the rod incorporating both rotational motion and cross-sectional deformation as well. The resulting discrete model possesses several desirable features: it ensures volume conservation of the discrete elements by considering cross-sectional deformation through a local dilatation factor, it demonstrates the beneficial properties associated with the variational integrator technique, such as the preservation of the rotational configuration, and energy conservation with a bounded error. An exhaustive set of numerical results under various initial conditions of the rod demonstrates the efficacy of the model in replicating the physics of the system.