Discrete Geometric Mechanics for Variational Time Integrators
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Transcript of Discrete Geometric Mechanics for Variational Time Integrators
Discrete Geometric Mechanics for
Variational Time Integrators
Ari Stern
Mathieu Desbrun
Geometric, Variational
Integrators for Computer Animation
L. KharevychWeiweiY. Tong
E. KansoJ. E. MarsdenP. SchröderM. Desbrun
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 care more for qualitative behavior
• Global behavior > Local behavior for our purposes
• A geometric approach can guarantee both
Simple Example: Swinging Pendulum
• Equation of motion:
• Rewrite as first-order equations:
𝑞 (𝑡)
𝑙
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
Taylor Approximation
• First order approximation using tangent to curve:
v
• As , approximations approach continuous values
(𝑞𝑘 ,𝑣𝑘)
(𝑞𝑘+1 ,𝑣𝑘+1)
Explicit Euler Method
• Direct first order approximations:
• Pros:• Fast
• Cons:• Energy “blows up”• Numerically unstable• Bad global accuracy
Implicit Euler Method
• Evaluate RHS using next time step:
• Pros:• Numerically stable
• Cons:• Energy dissipation• Needs non-linear solver• Bad global accuracy
Symplectic Euler Method
• Evaluate explicitly, then :
• Energy is conserved!• Numerically stable• Fast• Good global accuracy
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
Geometric View: Lagrangian Mechanics
• Lagrangian: • Action Functional:• Least Action Principle:
• Action Functional “Measure of Curvature”• Least Action “Curvature” is extremized
𝑡 0
𝑇
Euler-Lagrange Equation
=
= 0
Lagrangian Example: Falling Mass
The Discrete Lagrangian
• Derive discrete equations of motion from a Discrete Lagrangian to recover symplecticity:
• RHS can be approximated using one-point quadrature:
The Discrete Action Functional
• Continuous version:
• Discrete version:
Discrete Euler-Lagrange Equation
Discrete Lagrangian Example: Falling Mass
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:
Discrete Hamilton-Pontryagin Principle
Faster Update via Minimization
• Minimization > Root-Finding
• Variational Integrability Assumption:
• Above satisfied by most current models in computer animation
Minimization: The Lilyan