Hermite-based, One-step, Variational and Galerkin Time Integrators for Mechanical Systems

TL;DR

This paper introduces Hermite polynomial-based one-step variational and Galerkin integrators for mechanical systems, demonstrating superior energy conservation and trajectory accuracy through numerical examples including particles, oscillators, and aeroelastic systems.

Contribution

The paper develops novel Hermite-based one-step variational and Galerkin integrators, enhancing numerical accuracy and energy behavior in mechanical system simulations.

Findings

Galerkin methods outperform variational methods in energy conservation.

Both methods accurately capture conservative and dissipative dynamics.

Numerical examples validate improved trajectory and energy performance.

Abstract

In this paper, we present two Hermite polynomial based approaches to derive one-step numerical integrators for mechanical systems. These methods are based on discretizing the configuration using Hermite polynomials which leads to numerical trajectories continuous in both configuration and velocity. First, we incorporate Hermite polynomials for time-discretization and derive one-step variational methods by discretizing the Lagrange-d'Alembert principle over a single time step. Second, we present the Galerkin approach to derive one-step numerical integrators by setting the weighted average of the residual of the equations of motion over a time step to zero. We consider three numerical examples to understand the numerical performance of the one-step variational and Galerkin methods. We first study a particle in a double-well potential and compare the variational approach results with the corresponding results for the Galerkin approach. We then study the Duffing oscillator to understand the numerical behavior in presence of dissipative forces. Finally, we apply the proposed methods to a nonlinear aeroelastic system with two degrees of freedom. Both variational and Galerkin one-step methods capture conservative and nonconservative dynamics accurately with excellent energy behavior. The one-step Galerkin methods exhibit better trajectory and energy performance than the one-step variational methods and the variational integrators.