Numerical solution of a bending-torsion model for elastic rods

TL;DR

This paper develops a numerical method for simulating elastic rods based on hyperelasticity, proving convergence and stability, and exploring complex behaviors like instability thresholds and energy landscapes.

Contribution

It introduces a discretization scheme for a hyperelastic rod model, with proven convergence and stability, and provides numerical insights into instability and energy landscape complexities.

Findings

Numerical confirmation of Michell's instability thresholds

Identification of complex energy landscapes in elastic rods

Stability of the discretized model

Abstract

Aiming at simulating elastic rods, we discretize a rod model based on a general theory of hyperelasticity for inextensible and unshearable rods. After reviewing this model and discussing topological effects of periodic rods, we prove convergence of the discretized functionals and stability of a corresponding discrete flow. Our experiments numerically confirm thresholds e.g. for Michell's instability and indicate a complex energy landscape, in particular in the presence of impermeability.