Geometric, Variational Integrators for Computer Animation

We present a general-purpose numerical scheme for time integration of Lagrangian dynamical systems---an important computational tool at the core of physics-based animation. Several features make this particular time integrator highly desirable for computer animation. First, it guarantees preservation of 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. Constraints can also be enforced quite elegantly. Finally, higher-order accurate schemes can easily be derived if needed. Two key properties set the method apart from earlier approaches. First, the nonlinear equations that must be solved for updates are replaced by a minimization of a novel energy-like function. 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. We show its application to a simulation in non-linear elasticity with implementation details.

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Geometric, Variational Integrators for Computer Animation

L. Kharevych
Weiwei
Y. Tong
E. Kanso
J. E. Marsden
P. Schroeder
M. Desbrun

Elasticity.avi (30MB)