Super-Clothoids

Florence Bertails-Descoubes

Piecewise clothoids are 2D curves with continuous, piecewise linear curvature. Due to their smoothness properties, they have been extensively used in road design and robot path planning, as well as for the compact representation of hand-drawn curves. In this paper we present the Super-Clothoid model, a new mechanical model that for the first time allows for the computing of the dynamics of an elastic, inextensible piecewise clothoid. We first show that the kinematics of this model can be computed analytically depending on the Fresnel integrals, and precisely evaluated when required. Secondly, the discrete dynamics, naturally emerging from the Lagrange equations of motion, can be robustly and efficiently computed by performing and storing formal computations as far as possible, recoursing to numerical evaluation only when assembling the linear system to be solved at each time step. As a result, simulations turn out to be both interactive and stable, even for large displacements of the rod. Finally, we demonstrate the versatility of our model by handling various boundary conditions for the rod as well as complex external constraints such as frictional contact, and show that our model is perfectly adapted to inverse statics. Compared to lower-order models, the super-clothoid appears as a more natural and aesthetic primitive for bridging the gap between 2D geometric design and physics-based deformation.

Super-Clothoids

Large-Scale Dynamic Simulation of Highly Constrained Strands

Shinjiro Sueda, Garrett L. Jones, David I. W. Levin, Dinesh K. Pai

A significant challenge in applications of computer animation is the simulation of ropes, cables, and other highly constrained strand-like physical curves. Such scenarios occur frequently, for instance, when a strand wraps around rigid bodies or passes through narrow sheaths. Purely Lagrangian methods designed for less constrained applications such as hair simulation suffer from difficulties in these important cases. To overcome this, we introduce a new framework that combines Lagrangian and Eulerian approaches. The two key contributions are the reduced node, whose degrees of freedom precisely match the constraint, and the Eulerian node, which allows constraint handling that is independent of the initial discretization of the strand. The resulting system generates robust, efficient, and accurate simulations of massively constrained systems of rigid bodies and strands.

Large-Scale Dynamic Simulation of Highly Constrained Strands

Solid Simulation with Oriented Particles

We propose a new fast and robust method to simulate various types of solid including rigid, plastic and soft bodies as well as one, two and three dimensional structures such as ropes, cloth and volumetric objects. The underlying idea is to use oriented particles that store rotation and spin, along with the usual linear attributes, i.e. position and velocity. This additional information adds substantially to traditional particle methods. First, particles can be represented by anisotropic shapes such as ellipsoids, which approximate surfaces more accurately than spheres. Second, shape matching becomes robust for sparse structures such as chains of particles or even single particles because the undefined degrees of freedom are captured in the rotational states of the particles. Third, the full transformation stored in the particles, including translation and rotation, can be used for robust skinning of graphical meshes and for transforming plastic deformations back into the rest state.

Solid Simulation with Oriented Particles

A Nonsmooth Newton Solver for Capturing Exact Coulomb Friction in Fiber Assemblies

We focus on the challenging problem of simulating thin elastic rods in contact, in the presence of friction. Most previous approaches in computer graphics rely on a linear complementarity formulation for handling contact in a stable way, and approximate Coulombs’s friction law for making the problem tractable. In contrast, following the seminal work by Alart and Curnier in contact mechanics, we simultaneously model contact and exact Coulomb friction as a zero finding problem of a nonsmooth function. A semi-implicit time-stepping scheme is then employed to discretizethe dynamics of rods constrained by frictional contact: this leads to a set of linear equations subject to an equality constraint involving a non-differentiable function. To solve this one-step problem we introduce a simple and practical nonsmooth Newton algorithm, which proves to be reasonably efficient and robust for systems that are not over-constrained. We show that our method is able to finely capture the subtle effects that occur when thin elastic rods with various geometries enter into contact, such as stick-slip instabilities in free configurations, entangling curls, resting contacts in braid-like structures, or the formation of tight knots under large constraints. Our method can be viewed as a first step towards the accurate modeling of dynamic fibrous materials.

A Nonsmooth Newton Solver for Capturing Exact Coulomb Friction in Fiber Assemblies

Stable Inverse Dynamic Curves

2d animation is a traditional but fascinating domain that has recently regained popularity both in animated movies and video games. This paper introduces a method for automatically converting a smooth sketched curve into a 2d dynamic curve at stable equilibrium under gravity. The curve can then be physically animated to produce secondary motions in 2d animations or simple video games. Our approach proceeds in two steps. We first present a new technique to fit a smooth piecewise circular arcs curve to a sketched curve. Then we show how to compute the physical parameters of a dynamic rod model (super-circle) so that its stable rest shape under gravity exactly matches the fitted circular arcs curve. We demonstrate the interactivity and controllability of our approach on various examples where a user can intuitively setup efficient and precise 2d animations by specifying the input geometry.

Stable Inverse Dynamic Curves

Discrete Viscous Threads

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 fluid-mechanical sewing machine. The novel implicit treatment also yields an order of magnitude speedup in our elastic rod dynamics.

Discrete Viscous Threads

Unified Simulation of Elastic Rods, Shells, and Solids

We develop an accurate, unified treatment of elastica. Following the method of resultant-based formulation to its logical extreme, we derive a higher-order integration rule, or elaston, measuring stretching, shearing, bending, and twisting along any axis. The theory and accompanying implementation do not distinguish between forms of different dimension (solids, shells, rods), nor between manifold regions and non-manifold junctions. Consequently, a single code accurately models a diverse range of elastoplastic behaviors, including buckling, writhing, cutting and merging. Emphasis on convergence to the continuum sets us apart from early unification efforts.

Unified Simulation of Elastic Rods, Shells, and Solids

Linear Time Super-Helices

Thin elastic rods such as cables, phone coils, tree branches, or hair, are common objects in the real world but computing their dynamics accurately remains challenging. The recent Super-Helix model, based on the discrete equations of Kirchhoff for a piecewise helical rod, is one of the most promising models for simulating non-stretchable rods that can bend and twist. However, this model suffers from a quadratic complexity in the number of discrete elements, which, in the context of interactive applications, makes it limited to a few number of degrees of freedom – or equivalently to a low number of variations in curvature along the mean curve. This paper proposes a new, recursive scheme for the dynamics of a Super-Helix, inspired by the popular algorithm of Featherstone for serial multibody chains. Similarly to Featherstone’s algorithm, we exploit the recursive kinematics of a Super-Helix to propagate elements inertias from the free end to the clamped end of the rod, while the dynamics is solved within a second pass traversing the rod in the reverse way. Besides the gain in linear complexity, which allows us to simulate a rod of complex shape much faster than the original approach, our algorithm makes it straightforward to simulate tree-like structures of Super-Helices, which turns out to be particularly useful for animating trees and plants realistically, under large displacements.

Linear Time Super-Helices