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Discrete Geometric Mechanics for Variational Time Integrators

Geometric, Variational Integrators for Computer Animation. Discrete Geometric Mechanics for Variational Time Integrators. L. Kharevych Weiwei Y. Tong E. Kanso J. E. Marsden P. Schr ö der M. Desbrun. Ari Stern Mathieu Desbrun. Time Integration. Interested in D ynamic Systems

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Discrete Geometric Mechanics for Variational Time Integrators

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

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

  3. 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

  4. Simple Example: Swinging Pendulum • Equation of motion: • Rewrite as first-order equations:

  5. Discretizing the Problem • Break time into equal steps of length : • Replace continuous functions and with discrete functions and • Approximate the differential equation by finding values for • Various methods to compute

  6. Taylor Approximation • First order approximation using tangent to curve: v • As , approximations approach continuous values

  7. Explicit Euler Method • Direct first order approximations: • Pros: • Fast • Cons: • Energy “blows up” • Numerically unstable • Bad global accuracy

  8. Implicit Euler Method • Evaluate RHS usingnext time step: • Pros: • Numerically stable • Cons: • Energy dissipation • Needs non-linear solver • Bad global accuracy

  9. Symplectic Euler Method • Evaluate explicitly, then: • Energy is conserved! • Numerically stable • Fast • Good global accuracy

  10. Symplecticity • Sympletic motions preserve thetwo-form: • For a trajectory of points inphase space: • Area of 2D-phase-space region is preserved in time • Liouville’s Theorem

  11. Geometric View: Lagrangian Mechanics • Lagrangian: • Action Functional: • Least Action Principle: • Action Functional “Measure of Curvature” • Least Action “Curvature” is extremized

  12. Euler-Lagrange Equation = = 0

  13. Lagrangian Example: Falling Mass

  14. The Discrete Lagrangian • Derive discrete equations of motion from a Discrete Lagrangian to recover symplecticity: • RHS can be approximated using one-point quadrature:

  15. The Discrete Action Functional • Continuous version: • Discrete version:

  16. Discrete Euler-Lagrange Equation

  17. Discrete Lagrangian Example: Falling Mass

  18. More General: Hamilton-Pontryagin Principle • Equations of motion given by critical points of Hamilton-Pontryagin action • 3 variations now: • is a Lagrange Multiplier to equate and • Analog to Euler-Lagrange equation:

  19. Discrete Hamilton-Pontryagin Principle

  20. Faster Update via Minimization • Minimization > Root-Finding • VariationalIntegrability Assumption: • Above satisfied by most current models in computer animation

  21. Minimization: The Lilyan

  22. Results http://tinyurl.com/n5sn3xq

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