Discrete Elastic Rods Mikl´ os Bergou Columbia University Max Wardetzky Freie Universit¨ at 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 simple, robust, and efficient tools for geometry processing and physical simulation. The DDG approach to simulation begins with the laying out of a physical model that is discrete from the ground up; the primary directive in designing this model is a focus on the preservation of key geometric structures that characterize the actual (smooth) physical system [Grinspun 2006]. Notably lacking is the application of DDG to physical modeling of elastic rods—curve-like elastic bodies that have one dimension (“length”) much larger than the others (“cross-section”). Rods have many interesting potential applications in animating knots, sutures, plants, and even kinematic skeletons. They are ideal for model- ing deformations characterized by stretching, bending, and twist- ing. Stretching and bending are captured by the deformation of a curve called the centerline, while twisting is captured by the rota- tion of a material frame associated to each point on the centerline. 1.1 Goals and contributions Our goal is to develop a principled model that is (a) simple to im- plement and efficient to execute and (b) easy to validate and test for convergence, in the sense that solutions to static problems and trajectories of dynamic problems in the discrete setup approach the solutions of the corresponding smooth problem. In pursuing this goal, this paper advances our understanding of discrete differential geometry, physical modeling, and physical simulation. Elegant model of elastic rods We build on a representation of elastic rods introduced for purposes of analysis by Langer and Singer [1996], arriving at a reduced coordinate formulation with a minimal number of degrees of freedom for extensible rods that rep- resents the centerline of the rod explicitly and represents the mate- rial frame using only a scalar variable (§4.2). Like other reduced coordinate models, this avoids the need for stiff constraints that couple the material frame to the centerline, yet unlike other (e.g., curvature-based) reduced coordinate models, the explicit centerline representation facilitates collision handling and rendering. Efficient quasistatic treatment of material frame We addition- ally emphasize that the speed of sound in elastic rods is much faster for twisting waves than for bending waves. While this has long been established, to the best of our knowledge it has not been used to simulate general elastic rods. Since in most applications the slower waves are of interest, we treat the material frame quasistat- ically (§5). When we combine this assumption with our reduced coordinate representation, the resulting equations of motion (§7) become very straightforward to implement and efficient to execute. Geometry of discrete framed curves and their connections Because our derivation is based on the concepts of DDG, our dis- crete model retains very distinctly the geometric structure of the smooth setting—in particular, that of parallel transport and the forces induced by the variation of holonomy (§6). We introduce