Discrete Elastic Rods(Kirchhoff模型-能量法-一维弹性杆缠绕扭转).pdf

Discrete Elastic Rods Miklo?s Bergou Columbia University Max Wardetzky Freie Universita?t Berlin Stephen Robinson Columbia University Basile Audoly CNRS / UPMC Univ Paris 06 Eitan Grinspun Columbia University Figure 1: Experiment and simulation: A simple (trefoil) knot tied on an elastic rope can be turned into a number of fascinating shapes when twisted. Starting with a twist-free knot (left), we observe both continuous and discontinuous changes in the shape, for both directions of twist. Using our model of Discrete Elastic Rods, we are able to reproduce experiments with high accuracy. Abstract We present a discrete treatment of adapted framed curves, paral- lel transport, and holonomy, thus establishing the language for a discrete geometric model of thin flexible rods with arbitrary cross section and undeformed configuration. Our approach differs from existing simulation techniques in the graphics and mechanics lit- erature both in the kinematic description—we represent the mate- rial frame by its angular deviation from the natural Bishop frame— as well as in the dynamical treatment—we treat the centerline as dynamic and the material frame as quasistatic. Additionally, we describe a manifold projection method for coupling rods to rigid- bodies and simultaneously enforcing rod inextensibility. The use of quasistatics and constraints provides an efficient treatment for stiff twisting and stretching modes; at the same time, we retain the dy- namic bending of the centerline and accurately reproduce the cou- pling between bending and twisting modes. We validate the discrete rod model via quantitative buckling, stability, and coupled-mode experiments, and via qualitative knot-tying comparisons. CR Categories: I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—Animation Keywords: rods, strands, discrete holonomy, discrete differential geometry 1 Introduction Recent activity in the field of discrete differential geometry (DDG) has fueled the development of