Symplectic Hamiltonian finite element methods for linear elastodynamics

Manuel A. Sánchez, Bernardo Cockburn, Ngoc Cuong Nguyen, Jaime Peraire

Research output: Contribution to journalArticlepeer-review

Abstract

We present a class of high-order finite element methods that can conserve the linear and angular momenta as well as the energy for the equations of linear elastodynamics. These methods are devised by exploiting and preserving the Hamiltonian structure of the equations of linear elastodynamics. We show that several mixed finite element, discontinuous Galerkin, and hybridizable discontinuous Galerkin (HDG) methods belong to this class. We discretize the semidiscrete Hamiltonian system in time by using a symplectic integrator in order to ensure the symplectic properties of the resulting methods, which are called symplectic Hamiltonian finite element methods. For a particular semidiscrete HDG method, we obtain optimal error estimates and present, for the symplectic Hamiltonian HDG method, numerical experiments that confirm its optimal orders of convergence for all variables as well as its conservation properties.

Original languageEnglish (US)
Article number113843
JournalComputer Methods in Applied Mechanics and Engineering
Volume381
DOIs
StatePublished - Aug 1 2021

Bibliographical note

Funding Information:
M.A. S?nchez was partially supported by FONDECYT, Chile Iniciaci?n n.11180284 grant.B. Cockburn was partially supported by NSF, USA via DMS-1912646 grant.N.-C. Nguyen and J. Peraire were partially supported by NASA, USA (under grant number NNX16AP15A) and the AFOSR, USA (under grant number FA9550-16-1-0214).

Publisher Copyright:
© 2021 Elsevier B.V.

Keywords

  • Discontinuous Galerkin methods
  • Elastodynamics
  • Hybridizable discontinuous Galerkin methods
  • Mixed methods
  • Symplectic Hamiltonian finite element methods

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