Energy-stepping integrators in

Transcription

1 Energy-stepping integrators in Lagrangian mechanics M. Gonzalez, B. Schmidt, M. Ortiz California Institute of Technology Technische Universität München Structured Integrators Workshop Caltech, May 8, 2009

2 Classical Lagrangian mechanics

3 Hamiltonian picture

4 Lagrangian mechanics Noether s thm

5 Lagrangian mechanics Noether s thm

6 Conservation of energy - Spacetime

7 Time integration ODE approach

8 Discrete Lagrangian mechanics

9 Discrete Noether s theorem

10 Discrete Noether s theorem

11 Discrete Noether s theorem

12 Discrete Symplecticity

13 Variational integrators Appraisal Variational integrators are symplectic Discrete trajectories conserve discrete momentum maps exactly Discrete momentum maps approximate (but differ from) time-continuous momentum maps Ambiguity in choice of discrete Lagrangian No guarantee of solvability of the discrete Euler- Lagrange equations (e.g., energy equation) Variational structure is no guarantee of convergence of the discrete trajectory Some schemes of interest may not be variational (e.g., absorbing/radiating boundaries)

14 Energy Stepping, Force Stepping q q q q q q2 1 Exact potential Piecewise constant Piecewise linear

15 Energy Stepping, Force Stepping Piecewise constant approximation Piecewise linear approximation Diffraction by Reflection by downhill/uphill uphill energy step energy step Energy stepping System in free-fall within simplices Force stepping

16 Energy Stepping Exact trajectory

17 Energy Stepping Conservation props.

18 Energy Stepping Convergence

19 Energy Stepping Convergence

20 Energy Stepping Convergence

21 Energy stepping Appraisal Energy-stepping scheme is symplectic, energymomentum time-reversible integrator with automatic selection of the time step size Energy-stepping automatically conserves all the invariants of the system, whether explicitly known or not (hidden symmetries) The exact invariants of the system, as opposed to discrete approximations thereof, are exactly conserved by energy stepping Convergence subject to transversality condition

22 Energy stepping Examples Dynamics of a frozen argon cluster Spinning neo-hookean cube Dynamic contact of deformable bodies

23 Energy stepping Argon cluster Kinetic energy Potential energy Lennard-Jones pair potential ti ES NM Explicit Newmark (velocity Verlet) NM NM Total time simulated....

24 Energy stepping Argon cluster ES NM Ex ES NM NM ES ES

25 Energy stepping Spinning cube Finite element mesh: node tetrahedral isoparametric elements and 2969 nodes.- Strain-energy density: ES ES NM NM Automatic selection of time step size!!

26 Energy stepping Spinning cube The ball on the left is initially in rest. The ball on the right is set into motion with initial linear and angular velocities. ES NM Energy-stepping Explicit Newmark Complex dynamics of non-smooth contact Hyperelastic material properties Explicit treatment of vibrational energy Contact duration Coefficient of restitution Static Hertz law // Perfectly smooth rigid bodies

27 Concluding remarks Main idea: Replace the original Lagrangian by an approximate one that can be solved exactly Energy stepping scheme: Replace potential energy by terraced approximation Energy stepping trajectories consist of piecewise rectilinear motion, automatic time-step selection Energy stepping is symplectic, energymomentum conserving, time-reversible Convergence subject to transversality Transversality does not appear to be a concern in practice for large problems, general conditions