Force-stepping integrators in Lagrangian mechanics
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.
© 2010 John Wiley & Sons, Ltd. Received 6 January 2010; Revised 1 April 2010; Accepted 17 April 2010, Article first published online: 4 Aug 2010. The authors gratefully acknowledge the support of the US Department of Energy through Caltech's PSAAP Center for the Predictive Modeling and Simulation of High-Energy Density Dynamic Response of Materials.