@article{Xue:2021:GECDHM,
title = "On a generalized energy conservation/dissipation time finite element method for Hamiltonian mechanics",
journal = "Computer Methods in Applied Mechanics and Engineering",
volume = "373",
pages = "113509",
year = "2021",
issn = "0045-7825",
doi = "https://doi.org/10.1016/j.cma.2020.113509",
url = "http://www.sciencedirect.com/science/article/pii/S0045782520306940",
author = "Tao Xue and Yazhou Wang and Mridul Aanjaneya and Kumar K. Tamma and Guoliang Qin",
keywords = "Petrov–Galerkin, Stabilized time-weighted residual, Unconditionally stable, Controllable numerical dissipation, High-order time accuracy",
abstract = "Energy conserving and dissipative algorithm designs in Hamilton’s canonical equations via Petrov–Galerkin time finite element methodology are proposed in this paper that provide new avenues with high-order convergence rate, improved solution accuracy, and controllable numerical dissipation. Lagrange quadratic shape function in time with flexible interpolation points are considered to approximate the solution over a time interval. Instead of specifying the weight functions, two algorithmic parameters, namely, a principal root (ρq∞) and a spurious root (ρp∞), are introduced to formulate a generalized weight function, which enables us to introduce controllable numerical dissipation with respect to displacement and momenta while preserving high-order convergence, and features with improved solution accuracy. A family of third-order accurate time finite element algorithms with controllable dissipation and improved solution accuracy is presented in both the homogeneous and non-homogeneous dynamic problems; and via setting (ρq∞,ρp∞)=(1,1), this third-order family of algorithms directly leads to a new family of fourth-order accurate non-dissipative algorithms in general homogeneous problems; and the fourth-order accuracy is also preserved in non-homogeneous problems when the third-order time derivative of the external excitation has the order of O(Δtn)(n≤1). Numerical examples are performed to demonstrate the pros/cons for the conserving properties of various schemes in the proposed Petrov–Galerkin time finite element of algorithms."
}