$\Gamma$-convergence of a discrete Kirchhoff rod energy

TL;DR

This paper establishes a rigorous mathematical link between a discrete elastic rod model and the classical continuous Kirchhoff elastic energy through $ ext{Gamma}$-convergence, providing a new discrete energy formulation based on polynomial interpolation.

Contribution

It introduces a discrete Kirchhoff energy model that converges to the continuous model, with a novel formulation involving polynomial interpolation and careful treatment of bending and torsion interactions.

Findings

Discrete energy converges to continuous Kirchhoff energy via $ ext{Gamma}$-convergence.

A simple formula for discrete bending energy depending on angles and edge lengths.

Construction of a recovery sequence with equal Euclidean distances between points.

Abstract

This work is motivated by the classical discrete elastic rod model by Audoly et al. We derive a discrete version of the Kirchhoff elastic energy for rods undergoing bending and torsion and prove $\Gamma$-convergence to the continuous model. This discrete energy is given by the bending and torsion energy of an interpolating conforming polynomial curve and provides a simple formula for the bending energy depending in each discrete segment only on angle and adjacent edge lengths. For the $\liminf$-inequality, we need to introduce penalty terms to ensure arc-length parametrization in the limit. For the recovery sequence a discretization with equal Euclidean distance between consecutive points is constructed. Particular care is taken to treat the interaction between bending and torsion by employing a discrete version of the Bishop frame.