Discrete Geometric Mechanics for Variational Time Integrators Geometric

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Discrete Geometric Mechanics for Variational Time Integrators Geometric, Variational Integrators for Computer Animation Ari

Discrete Geometric Mechanics for Variational Time Integrators Geometric, Variational Integrators for Computer Animation Ari Stern L. Kharevych Weiwei Y. Tong E. Kanso J. E. Marsden P. Schröder M. Desbrun Mathieu Desbrun

Time Integration • Interested in Dynamic Systems • Analytical solutions usually difficult or impossible

Time Integration • Interested in Dynamic Systems • Analytical solutions usually difficult or impossible • Need numerical methods to compute time progression

Local vs. Global Accuracy • Local accuracy (in scientific applications) • In CG, we

Local vs. Global Accuracy • Local accuracy (in scientific applications) • In CG, we care more for qualitative behavior • Global behavior > Local behavior for our purposes • A geometric approach can guarantee both

Simple Example: Swinging Pendulum •

Simple Example: Swinging Pendulum •

Discretizing the Problem •

Discretizing the Problem •

Taylor Approximation •

Taylor Approximation •

Explicit Euler Method •

Explicit Euler Method •

Implicit Euler Method •

Implicit Euler Method •

Symplectic Euler Method •

Symplectic Euler Method •

Symplecticity •

Symplecticity •

Geometric View: Lagrangian Mechanics •

Geometric View: Lagrangian Mechanics •

Euler-Lagrange Equation •

Euler-Lagrange Equation •

Lagrangian Example: Falling Mass •

Lagrangian Example: Falling Mass •

The Discrete Lagrangian •

The Discrete Lagrangian •

The Discrete Action Functional •

The Discrete Action Functional •

Discrete Euler-Lagrange Equation •

Discrete Euler-Lagrange Equation •

Discrete Lagrangian Example: Falling Mass •

Discrete Lagrangian Example: Falling Mass •

More General: Hamilton-Pontryagin Principle •

More General: Hamilton-Pontryagin Principle •

Discrete Hamilton-Pontryagin Principle •

Discrete Hamilton-Pontryagin Principle •

Faster Update via Minimization •

Faster Update via Minimization •

Minimization: The Lilyan •

Minimization: The Lilyan •

Results http: //tinyurl. com/n 5 sn 3 xq

Results http: //tinyurl. com/n 5 sn 3 xq