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Goal-Oriented a Posteriori Error Estimates in Nearly Incompressible Linear Elasticity

  • Dustin Kumor
  • Andreas RademacherEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

In this article, we consider linear elastic problems, where Poisson’s ratio is close to 0.5 leading to nearly incompressible material behavior. The use of standard linear or d-linear finite elements involves locking phenomena in the considered problem type. One way to overcome this difficulties is given by selective reduced integration. However, the discrete problem differs from the continuous one using this approach. This fact has especially to be taken into account, when deriving a posteriori error estimates. Here, we present goal-oriented estimates based on the dual weighted residual method using only the primal residual due to the linear problem considered. The major challenge is given by the construction of an appropriate numerical approximation of the error identity. Numerical results substantiate the accuracy of the presented estimator and the efficiency of the adaptive method based on it.

Notes

Acknowledgements

The authors gratefully acknowledge the financial support by the German Research Foundation (DFG) within the subproject A5 of the transregional collaborative research centre (Transregio) 73 “Sheet-Bulk-Metal-Forming”.

References

  1. 1.
    W. Bangerth, R. Rannacher, Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics, ETH Zürich (Birkhäuser, Basel, 2003)Google Scholar
  2. 2.
    R. Becker, R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10, 1–102 (2001)MathSciNetCrossRefGoogle Scholar
  3. 3.
    R. Becker, E. Estecahandy, D. Trujillo, Weighted marking for goal-oriented adaptive finite element methods. SIAM J. Numer. Anal. 49, 2451–2469 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. Braack, A. Ern, A posteriori control of modeling errors and discretization errors. Multiscale Model. Simul. 1, 221–238 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Feischl, D. Praetorius, K.G. Van der Zee, An abstract analysis of optimal goal-oriented adaptivity. SIAM J. Numer. Anal. 54, 1423–1448 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    M.B. Giles, E. Süli, Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numer. 11, 145–236 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    T.J.R. Hughes, The Finite Element Method. Linear Static and Dynamic Finite Element Analysis (Dover Publications, Mineola, 2000)Google Scholar
  8. 8.
    H. Melzer, R. Rannacher, Spannungskonzentration in Eckpunkten der Kirchhoffschen Platte. Bauingenieur 55, 181–184 (1980)Google Scholar
  9. 9.
    M.S. Mommer, R. Stevenson, A goal-oriented adaptive finite element method with convergence rates. SIAM J. Numer. Anal. 47, 861–886 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    R.H. Nochetto, A. Veeser, M. Verani, A safeguarded dual weighted residual method. IMA J. Numer. Anal. 29, 126–140 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    M. Paraschivoiu, J. Peraire, A.T. Patera, A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations. Comput. Methods Appl. Mech. Eng. 150, 289–312 (1997)MathSciNetCrossRefGoogle Scholar
  12. 12.
    S. Prudhomme, J.T. Oden, On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Comput. Methods Appl. Mech. Eng. 176, 313–331 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    R. Rannacher, J. Vihharev, Adaptive finite element analysis of nonlinear problems: balancing of discretization and iteration errors. J. Numer. Math. 21, 23–61 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    T. Richter, T. Wick, Variational localizations of the dual-weighted residual estimator. J. Comput. Appl. Math. 279, 192–208 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Technische Universität DortmundInstitute of Applied MathematicsDortmundGermany

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