Γ-convergence of a discrete Kirchhoff rod energy

¹ Abteilung für Angewandte Mathema tik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 10, 79104 Freiburg i. Br., Germany
² Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 10, 79104 Freiburg i. Br., Germany
³ Univerza v Ljubljani Fakulteta za gradbeništvo in geodezijo, Jamova cesta 2, 1001 Ljubljana, Slovenia and Inštitut za matematiko, fiziko in mehaniko, Jadranska ulica 19, 1000 Ljubljana, Slovenia

^* Corresponding author: mjesenko@fgg.uni-lj.si

Received: 20 June 2023
Accepted: 8 May 2024

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 Γ-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 on each discrete segment only on angle and adjacent edge lengths. For the lim inf-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.