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Two lower order nonconforming quadrilateral elements for the Reissner-Mindlin plate

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Abstract

This paper generalizes two nonconforming rectangular elements of the Reissner-Mindlin plate to the quadrilateral mesh. The first quadrilateral element uses the usual conforming bilinear element to approximate both components of the rotation, and the modified nonconforming rotated Q 1 element enriched with the intersected term on each element to approximate the displacement, whereas the second one uses the enriched modified nonconforming rotated Q 1 element to approximate both the rotation and the displacement. Both elements employ a more complicated shear force space to overcome the shear force locking, which will be described in detail in the introduction. We prove that both methods converge at optimal rates uniformly in the plate thickness t and the mesh distortion parameter in both the H 1-and the L 2-norms, and consequently they are locking free.

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References

  1. Hu J, Shi Z C. Two lower order nonconforming rectangular elements for the Reissner-Mindlin Plate. Math Comp, 76: 1771–1786 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Han H D. Nonconforming elements in the mixed finite element method. J Comput Math, 2: 223–233 (1984)

    MATH  MathSciNet  Google Scholar 

  3. Rannacher R, Turek S. Simple nonconforming quadrilateral Stokes element. Numer Methods Partial Differential Equations, 8: 97–111 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  4. Lin Q, Tobiska L, Zhou A. On the superconvergence of nonconforming low order finite elements applied to the Poisson equation. IMA J Numer Anal, 25: 160–181 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Arnold D N, Brezzi F, Marini L D. A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate. J Sci Comput, 22/23: 25–45 (2005)

    Article  MathSciNet  Google Scholar 

  6. Brezzi F, Marini L D. A nonconforming element for the Reissner-Mindlin plate. Computers and Structures, 81: 515–522 (2003)

    Article  MathSciNet  Google Scholar 

  7. Lovadina C. A low order nonconforming finite element for Reissner-Mindlin plates. SIAM J Numer Anal, 42: 2688–2705 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ming P B, Shi Z C. Quadrilateral mesh. Chinese Ann Math, Ser B, 23: 235–252 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brezzi F, Fortin M. Numerical approximation of Reissner-Mindlin plates. Math Comp, 47: 151–158 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  10. Arnold D N, Falk R S. A uniformly accurate finite element method for the Reissner-Mindlin plate. SIAM J Numer Anal, 26: 1276–1290 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ciarlet P G. The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland, 1978

    Book  MATH  Google Scholar 

  12. Shi Z C. A convergence condition for quadrilateral Wilson element. Numer Math, 44: 349–361 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ming P B, Shi Z C. Two nonconforming quadrilateral elements for the Reissner-Mindlin plate. Math Model Methods Appl Sci, 15: 1503–1518 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Duan H Y, Liang G P. Nonconforming elements in least-squares mixed finite element methods. Math Comp, 73: 1–18 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Stenberg R. Analysis of mixed finite element methods for the stokes problems: A unified approach. Math Comp, 42: 9–23 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  16. Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Berlin: Springer-Verlag, 1991

    MATH  Google Scholar 

  17. Brenner S C. Korn’s inequalities for piecewise H 1 vector fields. Math Comp, 73: 1067–1087 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Brenner S C. Poincare-Friedrichs inequalities for piecewise H 1 functions. SIAM J Numer Anal, 41: 306–324 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Arnold D N, Boffi D, Falk R S. Approximation by quadrilateral finite elements. Math Comp, 71: 909–922 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Bathe K J, Dvorkin E. A four-node plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation. Internat J Numer Methods Engrg, 21: 367–383 (1985)

    Article  MATH  Google Scholar 

  21. Durán R, Hernández E, Hervella-Nieto L, et al. Error estimates for lower-order isoparametric quadrilateral finite elements for plates. SIAM J Numer Anal, 41: 1751–1772 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hu J, Shi Z C. Error analysis of quadrilateralWilson element for the Reissner-Mindlin plate. Comput Methods Appl Mech Engrg, 197: 464–475 (2008)

    Article  MathSciNet  Google Scholar 

  23. Zhang Z, Zhang S. Wilson element for the Reissner-Mindlin plate. Comput Methods Appl Mech Engrg, 113: 55–65 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hu J, Ming P B, Shi Z C. Nonconforming quadrilateral rotated Q 1 element for Reissner-Mindlin plate. J Comput Math, 21: 25–32 (2003)

    MATH  MathSciNet  Google Scholar 

  25. Ming P. B, Shi Z C. Nonconforming rotated Q 1 finite element for Reissner-Mindlin plate. Math Models Methods Appl Sci, 11: 1311–1342 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hu J. Quadrilateral locking free elements in elasticity. Doctorate Dissertation, Institute of Computaional Mathematics, CAS, Beijing, 2004

    Google Scholar 

  27. Ye X. A rectangular element for the Reissner-Mindlin plate. Numer Methods Partial Differential Equations, 16: 184–192 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Jun Hu

  2. Institute of Computational Mathematics, Chinese Academy of Sciences, Beijing, 100080, China

    ZhongCi Shi

Authors
  1. Jun Hu
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  2. ZhongCi Shi
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Correspondence to Jun Hu.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant No. 10601003) and National Excellent Doctoral Dissertation of China (Grant No. 200718)

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Hu, J., Shi, Z. Two lower order nonconforming quadrilateral elements for the Reissner-Mindlin plate. Sci. China Ser. A-Math. 51, 2097–2114 (2008). https://doi.org/10.1007/s11425-008-0087-y

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  • DOI: https://doi.org/10.1007/s11425-008-0087-y

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