An Implicit Contact Method for Tying Discrete Elastic Knots
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An Implicit Contact Method for Tying Discrete Elastic Knots


Rod-rod contact is critical in simulating knots and tangles. To simulate contact, typicallya contact force is applied to enforce the non-penetration condition. This force is often applied explicitly (forward Euler) such that at every time step of the dynamic simulation, the equations of motions are solved repeatedly until the right amount of contact force successfully imposes the condition. There are two drawbacks to this method: (1) the explicit implementation suffers from numerical convergence issues and (2) solving the equations of motion iteratively to find the right contact force slows down the simulation. In this thesis, we propose a simple, efficient, and fully-implicit contact model with good convergence properties. Compared to previous methods, ours is shown to be capable of taking large time steps without forfeiting accuracy during knot tying simulations. It involves 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 our contact potential is differentiable, we can readily incorporate its force (negative gradient of the energy) and Jacobian (negative Hessian of the energy) into the elastic rod simulation.

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