Abstract When an overhand knot tied in an elastic rod is tightened, it can undergo a sudden change in shape through snap buckling. In this article, we use a combination of discrete differential geometry (DDG)-based simulations and tabletop experiments to explore the onset of buckling as a function of knot topology, rod geometry, and friction. In our setup, two open ends of an overhand knot are slowly pulled apart, which leads to snap buckling in the knot loop. We call this phenomenon “inversion” since the loop appears to move dramatically from one side of the knot to the other. This inversion occurs due to the coupling of elastic energy between the braid (the portion of the knot in self-contact) and the loop (the portion of the knot with two ends connected to the braid). A numerical framework is implemented that combines discrete elastic rods with a constraint-based method for frictional contact to explore inversion in overhand knots. The numerical simulation robustly captures inversion in the knot and is found to be in good agreement with experimental results. In order to gain physical insight into the inversion process, we also develop a simplified model of the knot that does not require simulation of self-contact, which allows us to visualize the bifurcation that results in snap buckling.
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Implicit Contact Model for Discrete Elastic Rods in Knot Tying
Abstract Rod–rod contact is critical in simulating knots and tangles. To simulate contact, typically a contact force is applied to enforce nonpenetration condition. This force is often applied explicitly (Euler forward). At every time-step in a dynamic simulation, the equations of motions are solved over and over again until the right amount of contact force successfully imposes the nonpenetration condition. There are two drawbacks: (1) Explicit implementation brings numerical convergence issues. (2) Solving equations of motion iteratively to find this right contact force slows down the simulation. In this article, we propose a simple, efficient, and fully implicit contact model with high convergence properties. This model is shown to be capable of taking large time-steps without forfeiting accuracy during knot tying simulations when compared to previous methods. We introduce a new contact potential, based on a smoothed segment–segment distance function, that is an analytic function of the four endpoints of the two contacting edges. Since this contact potential is differentiable, we can incorporate its force (negative gradient of the energy) and Jacobian (negative Hessian of the energy) into the elastic rod simulation.
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- Award ID(s):
- 1925360
- NSF-PAR ID:
- 10292349
- Date Published:
- Journal Name:
- Journal of Applied Mechanics
- Volume:
- 88
- Issue:
- 5
- ISSN:
- 0021-8936
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1115/1.4050238
