Γ-convergence of a discrete Kirchhoff rod energy

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.