On a generalized energy conservation/dissipation time finite element method for Hamiltonian mechanics

Tao Xue, Yazhou Wang, Mridul Aanjaneya, Kumar K. Tamma, Guoliang Qin

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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 (ρqp)=(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.

Original languageEnglish (US)
Article number113509
JournalComputer Methods in Applied Mechanics and Engineering
Volume373
DOIs
StatePublished - Jan 1 2021
Externally publishedYes

Bibliographical note

Funding Information:
The authors thank the anonymous reviewers for their constructive feedback. This research was partly supported by the Rutgers University start-up grant, and the Ralph E. Powe Junior Faculty Enhancement Award . The author Yazhou Wang would like to thank China Scholarship Council (No. 201906280340 ) for the financial support.

Publisher Copyright:
© 2020 Elsevier B.V.

Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

Keywords

  • Controllable numerical dissipation
  • High-order time accuracy
  • Petrov–Galerkin
  • Stabilized time-weighted residual
  • Unconditionally stable

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