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Published 2006 | Published
Book Section - Chapter Open

Geometric, Variational Integrators for Computer Animation

Abstract

We present a general-purpose numerical scheme for time integration of Lagrangian dynamical systems—an important computational tool at the core of most physics-based animation techniques. Several features make this particular time integrator highly desirable for computer animation: it numerically preserves important invariants, such as linear and angular momenta; the symplectic nature of the integrator also guarantees a correct energy behavior, even when dissipation and external forces are added; holonomic constraints can also be enforced quite simply; finally, our simple methodology allows for the design of high-order accurate schemes if needed. Two key properties set the method apart from earlier approaches. First, the nonlinear equations that must be solved during an update step are replaced by a minimization of a novel functional, speeding up time stepping by more than a factor of two in practice. Second, the formulation introduces additional variables that provide key flexibility in the implementation of the method. These properties are achieved using a discrete form of a general variational principle called the Pontryagin-Hamilton principle, expressing time integration in a geometric manner. We demonstrate the applicability of our integrators to the simulation of non-linear elasticity with implementation details.

Additional Information

© 2006 The Eurographics Association. We thank Rasmus Tamstorf, Eitan Grinspun, Matt West, Hiroaki Yoshimura, and Michael Ortiz for helpful comments. This research was partially supported by NSF (ACI-0204932, DMS-0453145, CCF-0503786 & 0528101, CCR-0133983), DOE (W-7405-ENG-48/B341492 & DE-FG02- 04ER25657), Caltech Center for Mathematics of Information, nVidia, Autodesk, and Pixar.

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