Discrete Geometric Simulation of Elastic Ribbons*

We report a discrete differential geometry based numerical simulation for elastic ribbons. Ribbon is a mechanical structure whose three dimensions are very different: length >> width >> thickness. In our framework, we use the elastic energy form, with two essential geometric constraints, of a one-dimensional model of the ribbon [Dias and Audoly, J. Elast. 2015]. As non-zero natural curvature in both in-plane (geodesic curvature) and out-of-plane directions have been considered, this model allows a unified treatment of various types of ribbons, e.g. annular and rectangular. This continuous model is discretized into a mass-spring system in a manner similar to well established Discrete Elastic Rods algorithm for simulation of elastic rods. A system of discrete equations of motions are developed that can be solved implicitly using Newton's method. In parallel with simulations, we perform experiments with several test cases, e.g. large deformation of elastic ribbons under gravity, coupling of twisting and bending in rectangular ribbons, shape of Mobius strips, and buckling in annular ribbons. Quantitative comparison between experiments and simulations validates the correctness of our numerical method.