Discrete viscous threads

Abstract

We present a continuum-based discrete model for thin threads of viscous fluid by drawing upon the Rayleigh analogy to elastic rods, demonstrating canonical coiling, folding, and breakup in dynamic simulations. Our derivation emphasizes space-time symmetry, which sheds light on the role of time-parallel transport in eliminating-without approximation-all but an O(n) band of entries of the physical system's energy Hessian. The result is a fast, unified, implicit treatment of viscous threads and elastic rods that closely reproduces a variety of fascinating physical phenomena, including hysteretic transitions between coiling regimes, competition between surface tension and gravity, and the first numerical fluidmechanical sewing machine. The novel implicit treatment also yields an order of magnitude speedup in our elastic rod dynamics.

A Geometric Model for the Coiling of an Elastic Rod Deployed Onto a Moving Substrate

“…The simulations were based on the discrete elastic rod method (DER) [17,21] (a robust and predictive simulation tool that was originally developed by the computer graphics community) and were found to be in excellent agreement with our own experiments. Subsequently [2], we systematically applied DER to further explore the parameter space of this problem and pin-point the paramount role played by geometry.…”

Boundary handling and porous flow for fluid–hair interactions

“…Although our method can be applied to various fluid and hair solvers, in our implementation, the time step of the simulation is determined by choosing the smaller one of both systems. In this case, IISPH [28] together with implicit DER [26] allows us to use a much larger time step which leads to a better performance of the simulation.…”

“…The techniques described in Sections 3 and 4 can be added to various SPH fluid solvers easily by following the steps described in Algorithm 1. In this paper, we have implemented both explicit and implicit integration of discrete elastic rods (DER) [25,26] for the simulation of hair dynamics. We use a fast projection method [36] to enforce inextensibility of hair strands, achieving a stable hair simulation.…”

“…Although our algorithm is generic and should be able to be integrated into any fluid and hair solvers with slight modifications, we choose to use discrete elastic rods (DER) [25,26] for the simulation of hair dynamics because of its stability and the ability to simulate various behaviors of rods. For fluid solvers, we choose PCISPH [27] and IISPH [28] as our pressure solvers to validate our algorithm.…”

Mechanical growth and morphogenesis of seashells

“…However, such a method can be computationally expensive and does not utilise the previous data or the mathematical structure inherent in the continuous equations. Instead, the method we propose uses the concept of parallel transport, a familiar concept in differential geometry and used in discrete differential geometry (Bergou, 2010). As stated above, u 2 is the rotation of the frames about the binormal along the curve for fixed time.…”