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Force‐stepping integrators in Lagrangian mechanics
Author(s) -
Gonzalez M.,
Schmidt B.,
Ortiz M.
Publication year - 2010
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2942
Subject(s) - lagrangian mechanics , mathematics , piecewise , scheme (mathematics) , momentum (technical analysis) , conservation law , hamiltonian mechanics , symplectic geometry , classical mechanics , mathematical analysis , analytical mechanics , physics , finance , quantum mechanics , phase space , quantum dynamics , economics , quantum , thermodynamics
We formulate an integration scheme for Lagrangian mechanics, referred to as the force‐stepping scheme , which is symplectic, energy conserving, time‐reversible, and convergent with automatic selection of the time‐step size. The scheme also conserves approximately all the momentum maps associated with the symmetries of the system. The exact conservation of momentum maps may additionally be achieved by recourse to the Lagrangian reduction. The force‐stepping scheme is obtained by replacing the potential energy by a piecewise affine approximation over a simplicial grid or regular triangulation. By taking triangulations of diminishing size, an approximating sequence of energies is generated. The trajectories of the resulting approximate Lagrangians can be characterized explicitly and consist of piecewise parabolic motion, or free fall . Selected numerical tests demonstrate the excellent long‐term behavior of force‐stepping, its automatic time‐step selection property, and the ease with which it deals with constraints, including contact problems. Copyright © 2010 John Wiley & Sons, Ltd.