Simple deformation measures for Discrete elastic rods and ribbons

TL;DR

This paper introduces a simplified formulation for discrete elastic rods and ribbons, making the computation of deformation measures and their gradients more straightforward while maintaining consistency with continuous models.

Contribution

It proposes a simpler, consistent definition of discrete deformation measures and derives formulas for their gradients, extending the approach to inextensible ribbons.

Findings

Simpler formulas for deformation measures and their gradients.

Extension of the method to inextensible ribbons with developability constraints.

Numerical illustrations demonstrating the approach's effectiveness.

Abstract

The Discrete elastic rod method (Bergou et al., 2008) is a numerical method for simulating slender elastic bodies. It works by representing the center-line as a polygonal chain, attaching two perpendicular directors to each segment, and defining discrete stretching, bending and twisting deformation measures and a discrete strain energy. Here, we investigate an alternative formulation of this model based on a simpler definition of the discrete deformation measures. Both formulations are equally consistent with the continuous rod model. Simple formulas for the first and second gradients of the discrete deformation measures are derived, making it easy to calculate the Hessian of the discrete strain energy. A few numerical illustrations are given. The approach is also extended to inextensible ribbons described by the Wunderlich model, and both the developability constraint and the dependence of the energy of the strain gradients are handled naturally.