Discontinuous Galerkin and the Crouzeix–Raviart element: Application to elasticity

Peter Hansbo; Mats G. Larson

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 37, Issue: 1, page 63-72
  • ISSN: 0764-583X

Abstract

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We propose a discontinuous Galerkin method for linear elasticity, based on discontinuous piecewise linear approximation of the displacements. We show optimal order a priori error estimates, uniform in the incompressible limit, and thus locking is avoided. The discontinuous Galerkin method is closely related to the non-conforming Crouzeix–Raviart (CR) element, which in fact is obtained when one of the stabilizing parameters tends to infinity. In the case of the elasticity operator, for which the CR element is not stable in that it does not fulfill a discrete Korn's inequality, the discontinuous framework naturally suggests the appearance of (weakly consistent) stabilization terms. Thus, a stabilized version of the CR element, which does not lock, can be used for both compressible and (nearly) incompressible elasticity. Numerical results supporting these assertions are included. The analysis directly extends to higher order elements and three spatial dimensions.

How to cite

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Hansbo, Peter, and Larson, Mats G.. "Discontinuous Galerkin and the Crouzeix–Raviart element: Application to elasticity." ESAIM: Mathematical Modelling and Numerical Analysis 37.1 (2010): 63-72. <http://eudml.org/doc/194156>.

@article{Hansbo2010,
abstract = { We propose a discontinuous Galerkin method for linear elasticity, based on discontinuous piecewise linear approximation of the displacements. We show optimal order a priori error estimates, uniform in the incompressible limit, and thus locking is avoided. The discontinuous Galerkin method is closely related to the non-conforming Crouzeix–Raviart (CR) element, which in fact is obtained when one of the stabilizing parameters tends to infinity. In the case of the elasticity operator, for which the CR element is not stable in that it does not fulfill a discrete Korn's inequality, the discontinuous framework naturally suggests the appearance of (weakly consistent) stabilization terms. Thus, a stabilized version of the CR element, which does not lock, can be used for both compressible and (nearly) incompressible elasticity. Numerical results supporting these assertions are included. The analysis directly extends to higher order elements and three spatial dimensions. },
author = {Hansbo, Peter, Larson, Mats G.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Crouzeix–Raviart element; Nitsche's method; discontinuous Galerkin; incompressible elasticity.; Crouzeix-Raviart element; incompressible elasticity},
language = {eng},
month = {3},
number = {1},
pages = {63-72},
publisher = {EDP Sciences},
title = {Discontinuous Galerkin and the Crouzeix–Raviart element: Application to elasticity},
url = {http://eudml.org/doc/194156},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Hansbo, Peter
AU - Larson, Mats G.
TI - Discontinuous Galerkin and the Crouzeix–Raviart element: Application to elasticity
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 1
SP - 63
EP - 72
AB - We propose a discontinuous Galerkin method for linear elasticity, based on discontinuous piecewise linear approximation of the displacements. We show optimal order a priori error estimates, uniform in the incompressible limit, and thus locking is avoided. The discontinuous Galerkin method is closely related to the non-conforming Crouzeix–Raviart (CR) element, which in fact is obtained when one of the stabilizing parameters tends to infinity. In the case of the elasticity operator, for which the CR element is not stable in that it does not fulfill a discrete Korn's inequality, the discontinuous framework naturally suggests the appearance of (weakly consistent) stabilization terms. Thus, a stabilized version of the CR element, which does not lock, can be used for both compressible and (nearly) incompressible elasticity. Numerical results supporting these assertions are included. The analysis directly extends to higher order elements and three spatial dimensions.
LA - eng
KW - Crouzeix–Raviart element; Nitsche's method; discontinuous Galerkin; incompressible elasticity.; Crouzeix-Raviart element; incompressible elasticity
UR - http://eudml.org/doc/194156
ER -

References

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  9. P. Hansbo and M.G. Larson, A simple nonconforming bilinear element for the elasticity problem. Trends in Computational Structural Mechanics, W.A. Wall et al. Eds., CIMNE (2001) 317-327.  
  10. T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, New Jersey (1987).  
  11. J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg36 (1971) 9-15.  
  12. R. Rannacher and S. Turek, A simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differential Equations8 (1992) 97-111.  
  13. F. Thomasset, Implementation of Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, New York (1981).  
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  15. B. Cockburn, K.E. Karniadakis and C.-W. Shu Eds., Discontinuous Galerkin Methods: Theory, Computation, and Applications. Lecture Notes Comput. Sci. Eng., Springer Verlag (1999).  

Citations in EuDML Documents

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  1. Blanca Ayuso de Dios, Ivan Georgiev, Johannes Kraus, Ludmil Zikatanov, A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations
  2. Yongxing Shen, Adrian J. Lew, A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity
  3. Yongxing Shen, Adrian J. Lew, A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity
  4. Yongxing Shen, Adrian J. Lew, A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity

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