Energy-preserving Variational Integrators for Forced Lagrangian Systems
For researchers in computational mechanics, this work provides a theoretical framework for adaptive variational integrators but is incremental, as it extends existing methods to forced systems with limited practical validation.
This paper develops energy-preserving variational integrators for forced Lagrangian systems by extending the Lagrange-d'Alembert principle to adaptive time-stepping. Numerical tests show a trade-off between energy preservation and trajectory accuracy, and reveal ill-conditioning and unexpected energy behavior in dissipative systems.
The goal of this paper is to develop energy-preserving variational integrators for time-dependent mechanical systems with forcing. We first present the Lagrange-d'Alembert principle in the extended Lagrangian mechanics framework and derive the extended forced Euler-Lagrange equations in continuous-time. We then obtain the extended forced discrete Euler-Lagrange equations using the extended discrete mechanics framework and derive adaptive time step variational integrators for time-dependent Lagrangian systems with forcing. We consider two numerical examples to study the numerical performance of energy-preserving variational integrators. First, we consider the example of a nonlinear conservative system to illustrate the advantages of using adaptive time-stepping in variational integrators. We show a trade-off between energy-preserving performance and accurate discrete trajectories while choosing an initial time step. In addition, we demonstrate how the implicit equations become more ill-conditioned as the adaptive time step increases through a condition number analysis. As a second example, we numerically simulate a damped harmonic oscillator using the adaptive time step variational integrator framework. The adaptive time step increases monotonically for the dissipative system leading to unexpected energy behavior.