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Some Applications of Discontinuous Galerkin Methods in Solid Mechanics

  • Adrian Lew
  • Alex Ten Eyck
  • Ramsharan Rangarajan
Part of the IUTAM BookSeries book series (IUTAMBOOK, volume 11)

Abstract

We provide a brief overview of our recent work on applications of discontinuous Galerkin methods in solid mechanics. The discussion is light in technical details, and rather emphasizes key ideas, advantages and disadvantages of the approach, illustrating these with several numerical examples.

Key words

discontinuous Galerkin nonlinear elasticity immersed boundary methods locking 

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References

  1. 1.
    D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39:1749–1779, 2002.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    D.N. Arnold, F. Brezzi, and L.D. Marini. A family of discontinuous Galerkin Finite Elements for the Reissner—Mindlin plate. Journal of Scientific Computing, 22(1):25–45, 2005.CrossRefMathSciNetGoogle Scholar
  3. 3.
    A. Buffa and C. Ortner. Variational convergence of IP-DGFEM. Technical Report 07/10, Oxford University Computing Laboratory, Numerical Analysis Group, Wolfson Building, Parks Road, Oxford, England OX1 3QD, April 2007.Google Scholar
  4. 4.
    F. Celiker, B. Cockburn, and H.K. Stolarski. Locking-free optimal discontinuous Galerkin methods for Timoshenko beams. SIAM Journal on Numerical Analysis, 44:2297, 2006.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    B. Cockburn and C.W. Shu. Runge-Kutta discontinuous Galerkin methods for convectiondominated problems. J. Sci. Comput, 16(3):173–261, 2001.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    J.K. Djoko, F. Ebobisse, A.T. McBride, and B.D. Reddy. A discontinuous Galerkin formulation for classical and gradient plasticity — Part 1: Formulation and analysis. Computer Methods in Applied Mechanics and Engineering, 196(37–40):3881–3897, 2007.CrossRefMathSciNetGoogle Scholar
  7. 7.
    J.K. Djoko, F. Ebobisse, A.T. McBride, and B.D. Reddy. A discontinuous Galerkin formulation for classical and gradient plasticity. Part 2: Algorithms and numerical analysis. Computer Methods in Applied Mechanics and Engineering, 197(1–4):1–21, 2007.CrossRefMathSciNetGoogle Scholar
  8. 8.
    G. Engel, K. Garikipati, T.J.R. Hughes, M.G. Larson, L. Mazzei, and R.L. Taylor. Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, strain gradient elasticity. Computer Methods in Applied Mechanics and Engineering, 191:3669–3750, 2002.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    A. Ten Eyck and A. Lew. Discontinuous Galerkin methods for nonlinear elasticity. International Journal for Numerical Methods in Engineering, 67:1204–1243, 2006.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    S. Guzey, B. Cockburn, and H.K. Stolarski. The embedded discontinuous Galerkin method: Application to linear shell problems. International Journal for Numerical Methods in Engineering, 70:757–790, 2007.CrossRefMathSciNetGoogle Scholar
  11. 11.
    P. Hansbo and M.G. Larson. Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Computer Methods in Applied Mechanics and Engineering, 191(17):1895–1908, 2002.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    A. Lew and G. Buscaglia. A discontinuous-Galerkin-based immersed boundary method. International Journal for Numerical Methods in Engineering, 2008 (in press).Google Scholar
  13. 13.
    A. Lew, P. Neff, D. Sulsky, and M. Ortiz. Optimal BV estimates for a discontinuous Galerkin method in linear elasticity. Applied Mathematics Research Express, 3:73–106, 2004.Google Scholar
  14. 14.
    L. Noels and R. Radovitzky. A new discontinuous Galerkin method for Kirchhoff-Love shells. http://asap.mit.edu/publications/journal/cmame-2007.pdf, 2007.
  15. 15.
    M. O’Connel and C. Taylor. Personal communication, 2007. Stanford University.Google Scholar
  16. 16.
    A. Ten Eyck, F. Celiker, and A. Lew. Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: Analytical estimates. Computer Methods in Applied Mechanics and Engineering, 197(33–40):2989–3000, 2008.CrossRefGoogle Scholar
  17. 17.
    A. Ten Eyck, F. Celiker, and A. Lew. Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: Motivation, formulation and numerical examples. Computer Methods in Applied Mechanics and Engineering, 2008 (in press).Google Scholar
  18. 18.
    G.N. Wells and N.T. Dung. A CO discontinuous Galerkin formulation for Kirchhoff plates. Computer Methods in Applied Mechanics and Engineering, 196(35–36):3370–3380, 2007.CrossRefMathSciNetGoogle Scholar
  19. 19.
    G.N. Wells, E. Kuhl, and K. Garikipati. A discontinuous Galerkin method for the Cahn—Hilliard equation. Journal of Computational Physics, 218(2):860–877, 2006.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    T.P. Wihler. Locking-free DGFEM for elasticity problems in polygons. IMA Journal of Numerical Analysis, 24:45–75, 2004.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V 2008

Authors and Affiliations

  • Adrian Lew
    • 1
  • Alex Ten Eyck
    • 1
  • Ramsharan Rangarajan
    • 1
  1. 1.Mechanical EngineeringStanford UniversityStanfordUSA

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