Discrete elastic rods

Abstract

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. AbstractWe present a discrete treatment of adapted framed curves, parallel 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 literature both in the kinematic description-we represent the material frame by its angular deviation from the natural Bishop frameas 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 rigidbodies 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 dynamic bending of the centerline and accurately reproduce the coupling 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. Elegant model of elastic rodsWe 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 represents the centerline of the rod explicitly and represents the material 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 frameWe additionally 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 quasistatically ( §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 connectionsBecause our derivation is based on the concepts of DDG, our discrete 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

Inverse dynamic hair modeling with frictional contact

“…For instance, the Discrete Elastic Rods model [Bergou et al 2008] would naturally fit into our framework, the sole difference being the structure of the stiffness matrix K which would be banded instead of diagonal.…”

“…In practice we use up to a few thousands thin elastic rods, modeled as super-helices [Bertails et al 2006], to represent a collection of guide strands. Our inversion method is however generic and could accommodate any other rod model accounting for arbitrary hair curliness [Spillmann and Teschner 2007;Bergou et al 2008].…”

Numerical aspects in the dynamic simulation of geometrically exact rods

“…Further, a staggered grid discretisation approach is much closer to the geometry of the framed curve: From a discrete differential geometric point of view, discrete curvature K n must be situated on the vertices. These rod models are 'discrete framed curves' [7,10,27]. Where can this work be situated within the state of the art rod models?…”

“…The choice of a discrete curvature measure is by no means unique. We propose several choices from literature [6,7,10,24,27,32,40], which can be expressed in terms of the material unit axis u n and the material angle ϕ n of difference rotation. Typical examples are…”

“…For the method of lines, in section 3, we summarise a recently introduced new spatial discretisation approach, which is based on geometric finite differences on a staggered grid [26,27]. Our ansatz generalises a discrete differential geometric approach in [7,10], where inextensible Kirchhoff rods have been examined. The formulation is appropriate for multibody dynamics, i. e. classical Lagrangian mechanics on manifolds.…”