A hybridizable discontinuous Galerkin formulation for non-linear elasticity

Hardik Kabaria, Adrian J. Lew, Bernardo Cockburn

Research output: Contribution to journalArticlepeer-review

41 Scopus citations

Abstract

We revisit the hybridizable discontinuous Galerkin method for non-linear elasticity introduced by S.-C. Soon (2008). We show that it can be recast as a minimization problem of a non-linear functional over a space of discontinuous approximations to the displacement. The functional can be written as the sum over the elements of the classic potential energy plus a new energy associated to the inter-element jumps of the displacement. We then show that if this new energy is not properly weighted, the minimizers might not converge to the exact solution. We construct an example illustrating this phenomenon and show how to overcome it by suitably increasing the weight of the energy of the inter-element jumps. Finally, we explore the performance of the method for the case of piecewise-linear approximations in rather demanding situations in both two-dimensional and, for the first time, three-dimensional situations. They include almost incompressible materials, large deformations with large-shear layers, and cavitation. We also compare the method with the continuous Galerkin method and a previously explored discontinuous Galerkin method, and show that, when using piecewise-linear approximations and a moderate number of degrees of freedom, the current method turns out to be more efficient for the computation of the gradient.

Original languageEnglish (US)
Pages (from-to)303-329
Number of pages27
JournalComputer Methods in Applied Mechanics and Engineering
Volume283
DOIs
StatePublished - Jan 1 2015

Bibliographical note

Funding Information:
This work was supported by the Franklin P. Johnson Jr. Stanford Graduate Fellowship to Hardik Kabaria. Adrian J. Lew is grateful for the support for this work from the following sources: Department of the Army Research Grant , grant number: W911NF-07- 2-0027 ; NSF Career Award , grant number: CMMI-0747089 ; and NSF , grant number CMMI-1301396 . Bernardo Cockburn would like to acknowledge the support of the NSF through the DMS-1115331 grant.

Publisher Copyright:
© 2014 Elsevier B.V.

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

Keywords

  • Cavitation
  • Hybridized discontinuous Galerkin
  • Large shear deformations
  • Non-convergence
  • Non-linear elasticity
  • Stabilization

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