We 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 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 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.

We consider resonance problems at an arbitrary eigenvalue of the p-Laplacian, and prove the existence of weak solutions assuming a standard Landesman Lazer condition. We use variational arguments to characterize certain eigenvalues and then to establish the solvability of the given boundary value problem.

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

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In this paper we consider the analogue of the Courant nodal domain theorem for the nonlinear eigenvalue problem for the p-Laplacian. In particular we prove that if u ln is an eigenfunction associated with the nth variational eigenvalue, l n , then u ln has at most 2n − 2 nodal domains. Also, if u ln has n+k nodal domains, then there is another eigenfunction with at most n − k nodal domains. © 2002 Elsevier Science (USA)