One-Dimensional Elastic Objects
Jonas Spillmann
Computer Graphics
University of Freiburg

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Cosserat Nets


Cosserat nets are networks of elastic rods that are linked by elastic joints. They allow to represent a large variety of objects such as elastic rings, coarse nets, or truss structures. In this paper, we propose a novel approach to model and dynamically simulate such Cosserat nets. We first derive the static equilibrium of the elastic rod model that supports both bending and twisting deformation modes. We further propose a dynamic model that allows
for the efficient simulation of elastic rods. We then focus on the simulation of the Cosserat nets by extending the elastic rod deformation model to branched and looped topologies.

To round out the discussion, we evaluate our deformation model. By comparing our deformation model to a reference model, we illustrate both the physical plausibility and the conceptual advantages of the proposed approach.


Jonas Spillmann, Matthias Teschner, "Cosserat Nets",
to appear in IEEE Transactions on Visualization and Computer Graphics. [pdf]

[divx] Movie (23MB)


Figure 1: Cosserat nets are non-manifold branched structures whose struts are modelled as CORDE rods, with elastic joints. They can be used to represent a large variety of different objects, such as rings, nets and trusses.


Figure 2: Comparison of CORDE (blue rod) to an accurate mechanical reference deformation model (red rod). In this experiment, both rods are subject to torsional deformation. As long as the centerlines stay straight, the rods are in an unstable equilibrium (left). When the centerlines are perturbed, then the rods start to bend in order to balance stretching, bending and torsional deformation, until a stable equilibrium is found (right).






An Adaptive Contact Model for the Robust Simulation of Knots


In this paper, we present an adaptive model for dynamically deforming hyper-elastic rods. In contrast to existing approaches, adaptively introduced control points are not governed by geometric subdivision rules. Instead, their states are determined by employing a non-linear energy-minimization approach. Since valid control points are computed instantaneously, post-stabilization schemes are avoided and the stability of the dynamic simulation is improved.
Due to inherently complex contact configurations, the simulation of knot tying using rods is a challenging task. In order to address this problem, we combine our adaptive model with a robust and accurate collision handling method for elastic rods. By employing our scheme, complex knot configurations can be simulated in a physically plausible way.


Jonas Spillmann, Matthias Teschner, "An Adaptive Contact Model for the Robust Simulation of Knots",
Computer Graphics Forum, vol. 27, no. 2 (Proc. Eurographics),pp. 497-506, Crete, Greece, Apr. 14-18, 2008. [pdf]

[divx] Movie (23MB)



Figure 1: In knot-tying simulations, large parts of the elastic rod are undeformed while a high mechanical accuracy is required to simulate the knot, which calls for adaptive methods. A network linked by Prusik knots carries a bar element. The delicate equilibrium requires a robust collision handling method.
[DivX movie sequence (3 MB)]
  Figure 2: Interactive simulation of adaptive knot-tying. A thread is
tied around four poles, and a shoelace knot prevents it from slipping down. As the user unties the knot, the pressure is reduced and the thread slips to the ground. This simulation illustrates the effect of Coulomb friction. [DivX movie sequence (2 MB)]



  High res. Adaptive Low res.
Nodes 252 116 63
Avg. collisions 70 69 22
Simulation step [ms] 13.3 7.3 2.6

Figure 3: Simulation of two ropes tied together with the double Fisherman's knot. Left: A high-resolution rope with 252 nodes. Middle: An adaptive rope with 116 nodes and the same maximum resolution as the high-resolution rope. Right: A low-resolution rope with 63 nodes.
As the measurements indicate, simulating the knot with adaptive ropes is almost twice as efficient and requires less than half the number of nodes than uniformly sampled ropes at the same maximum resolution. Still, the number of collisions does not vary significantly, illustrating that the knot is simulated with similar accuracy. In contrast, the low-resolution ropes cannot reproduce the knot configuration accurately enough.
[DivX movie sequence (4 MB)]


[Another DivX movie sequence (8MB) illustrating a massive stacking of rods]




CORDE - Cosserat Rod Elements for the Dynamic Simulation of One-Dimensional Elastic Objects


Simulating one-dimensional elastic objects such as threads, ropes or hair strands is a difficult problem, especially if material torsion is considered. In this paper, we present CORDE (french 'rope'), a novel deformation model for the dynamic interactive simulation of elastic rods with torsion. We derive continuous energies for a dynamically deforming rod based on the Cosserat theory of elastic rods. We then discretize the rod and compute energies per element by employing finite element methods. Thus, the global dynamic behavior is independent of the discretization. The dynamic evolution of the
rod is obtained by numerical integration of the resulting Lagrange equations of motion. We further show how this system of equations can be decoupled and efficiently solved.
Since the centerline of the rod is explicitly represented, the deformation model allows for accurate contact and self-contact handling. Thus, we can reproduce many important looping phenomena. Further, a broad variety of different materials can be simulated at interactive rates. Experiments underline the physical plausibility of our deformation model.


J. Spillmann, M. Teschner, "CORDE: Cosserat Rod Elements for the Dynamic Simulation of One-Dimensional Elastic Objects,"
ACM SIGGRAPH / Eurographics Symposium on Computer Animation, San Diego, USA, Aug. 3-4, 2007. [pdf]

[divx] Movie (25MB)



Figure 1: Dynamic looping phenomenon of a rod under torsional strain: A rod is spanned between two anchors, and its ends are clamped. A torque transducer on the left varies the end-to-end rotation of the rod. We observe a bifurcation sequence that results in a looping with an increasing number of self-contacts. Bottom: Comparison to a real rope under torsional strain. [DivX movie sequence (2 MB)]



Figure 2: Interactive simulation of threads. The user can freely interact with the thread. Oscillations are minimized by modeling internal friction. A robust collision handling enables the simulation of difficult collision and self-collision configurations. Threads with intrinsic twist and bend are also supported. [Torsion: DivX movie (4.5 MB)] [Intrinsic twist: DivX movie (4.3 MB)] [Knot-tying: DivX movie (8 MB)]