Publications

Asynchronous Contact Mechanics. David Harmon, Etienne Vouga, Breannan Smith, Rasmus Tamstorf, and Eitan Grinspun, SIGGRAPH ( ACM Transactions on Graphics ), 2009.

We develop a method for reliable simulation of elastica in complex contact scenarios. Our focus is on firmly establishing three parameter-independent guarantees: that simulations of well-posed problems (a) have no interpenetrations, (b) obey causality, momentum- and energy-conservation laws, and (c) complete in finite time. We achieve these guarantees through a novel synthesis of asynchronous variational integrators, kinetic data structures, and a discretization of the contact barrier potential by an infinite sum of nested quadratic potentials. In a series of two- and three-dimensional examples, we illustrate that this method more easily handles challenging problems involving complex contact geometries, sharp features, and sliding during extremely tight contact.

Robust Treatment of Simultaneous Collisions. David Harmon, Etienne Vouga, Rasmus Tamstorf, and Eitan Grinspun, SIGGRAPH ( ACM Transactions on Graphics ), 2008.

Robust treatment of complex collisions is a challenging problem in cloth simulation. Some state of the art methods resolve collisions iteratively, invoking a fail-safe when a bound on iteration count is exceeded. The best-known fail-safe rigidifies the contact region, causing simulation artifacts. We present a fail-safe that cancels impact but not sliding motion, considerably reducing artificial dissipation. We equip the proposed fail-safe with an approximation of Coulomb friction, allowing finer control of sliding dissipation.

Two Blossoming Proofs of the Lane-Riesenfeld Algorithm. Vouga, E. and Goldman, R. 2007. Computing 79, 2 (Apr. 2007), 153-162.

The standard proof of the Lane-Riesenfeld algorithm for inserting knots into uniform B-spline curves is based on the continuous convolution formula for the uniform B-spline basis functions. Here we provide two new, elementary, blossoming proofs of the Lane-Riesenfeld algorithm for uniform B-spline curves of arbitrary degree.

Nonlinear Subdivision Through Nonlinear Averaging. Schaefer S., Vouga E. and Goldman R. Computer Aided Geometric Design, Vol. 25, No. 3 (2008), pages 162-180.

We investigate a general class of nonlinear subdivision algorithms for functions of a real or complex variable built from linear subdivision algorithms by replacing binary linear averages such as the arithmetic mean by binary nonlinear averages such as the geometric mean. Using our method, we can easily create stationary subdivision schemes for Gaussian functions, spiral curves, and circles with uniform parametrizations. More generally, we show that stationary subdivision schemes for e^p(x), cos(p(x)) and sin(p(x)) for any polynomial or piecewise polynomial p(x) can be generated using only addition, subtraction, multiplication, and square roots. The smoothness of our nonlinear subdivision schemes is inherited from the smoothness of the original linear subdivision schemes and the differentiability of the corresponding nonlinear averaging rules. While our results are quite general, our proofs are elementary, based mainly on the observation that generic nonlinear averaging rules on a pair of real or complex numbers can be constructed by conjugating linear averaging rules with locally invertible nonlinear maps. In a forthcoming paper we show that every continuous nonlinear averaging rule on a pair of real or complex numbers can be constructed by conjugating a linear averaging rule with an associated continuous locally invertible nonlinear map. Thus the averaging rules considered in this paper are actually the general case. As an application we show how to apply our nonlinear subdivision algorithms to intersect some common transcendental functions.