Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



A posteriori error analysis for conforming MITC elements for Reissner-Mindlin plates

Authors: C. Carstensen and Jun Hu
Journal: Math. Comp. 77 (2008), 611-632
MSC (2000): Primary 65N30, 65N15, 35J25
Published electronically: October 18, 2007
MathSciNet review: 2373172
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper establishes a unified a posteriori error estimator for a large class of conforming finite element methods for the Reissner-Mindlin plate problem. The analysis is based on some assumption (H) on the consistency of the reduction integration to avoid shear locking. The reliable and efficient a posteriori error estimator is robust in the sense that the reliability and efficiency constants are independent of the plate thickness $ t$. The presented analysis applies to all conforming MITC elements and all conforming finite element methods without reduced integration from the literature.

References [Enhancements On Off] (What's this?)

  • 1. M. Ainsworth and J.T. Oden, A posteriori error estimation in finite element analysis, Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308 (2003b:65001)
  • 2. D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, M$ ^2$AN, 19(1985), pp. 7-32. MR 813687 (87g:65126)
  • 3. D.N. Arnold and F. Brezzi, Some new elements for the Reissner-Mindlin plate model, J.L.Lions, C.Baiocchi(eds.), Boundary value problem for partial differential equations and applications, Masson, 1993, pp. 287-292. MR 1260452 (94k:73066)
  • 4. D.N. Arnold and R.S. Falk, A uniformly accurate finite element method for Reissner-Mindlin plates, SIAM. J.Numer. Anal., 26(1989), pp. 1276-1290. MR 1025088 (91c:65068)
  • 5. D.N. Arnold and R.S. Falk, The boundary layer for the Reissner-Mindlin plate model, SIAM J.Math.Anal. 21(1990), pp. 281-312. MR 1038893 (91c:73053)
  • 6. D.N. Arnold, D. Boffi and R.S. Falk, Quadrilateral H( $ \mathrm{div}$) finite elements, SIAM J. Numer. Anal. 42 (2005), pp. 2429-2451. MR 2139400 (2006d:65129)
  • 7. I. Babuska, A. Miller, A feedback finite element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator, Comp. Meth. Appl. Mech. Engrg., 61 (1987), pp. 1-40 MR 880421 (88d:73036)
  • 8. I. Babuška and W.C. Rheinboldt, A posteriori error analysis of finite element solutions for one-dimensional problems, SIAM J.Numer. Anal., 18 (1981), pp. 565-589. MR 615532 (82j:65082)
  • 9. I. Babuška and T. Strouboulis, The Finite Element Method and its Reliability, The Clarendon Press Oxford University Press, 2001. MR 1857191 (2002k:65001)
  • 10. K. J. Bathe, F. Brezzi and M. Fortin, A simplified analysis of two-plate elements: The MITC4 and MITC9 element, G.N. Pande and J. Middleton (eds), Numeta 87 Vol. 1, Numerical Techniques for Engineering Analysis and Design,Martinus Nijhoff, Amsterdam.
  • 11. K. J. Bathe, F. Brezzi and M. Fortin, Mixed-interpolated elements for Reissner-Mindlin plates, Int. J. Num. Meths. Engrg., 28(1989), pp. 1787-1801. MR 1008138 (90g:73090)
  • 12. K. J. Bathe and E. Dvorkin, A four-node plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation, Int. J. Num. Meths. Engrg., 21(1985), pp. 367-383.
  • 13. D. Braess, and P. Peisker, Uniform convergence of mixed interpolated elements for Reissner-Mindlin plates, M$ ^2$AN., 26(1992), pp. 557-574. MR 1177387 (93j:73070)
  • 14. D. Braess, Finite Elements, Cambridge University Press, 1997. MR 1463151 (98f:65002)
  • 15. S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer Verlag, Second Edition, 2002. MR 1894376 (2003a:65103)
  • 16. F. Brezzi and M. Fortin, Numerical approximation of Reissner-Mindlin plates, Math Comp., 47(1986), pp. 151-158. MR 842127 (87g:73057)
  • 17. F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, Berlin, 1991. MR 1115205 (92d:65187)
  • 18. F. Brezzi, M. Fortin and R. Stenberg, Error analysis of mixed-interpolated elements for Reissner-Mindlin plate, Math. Models. Meth. Appl. Sci., 1 (1991), pp. 125-151. MR 1115287 (92e:73030)
  • 19. C. Bernardi, V. Girault, A local regularisation operator for triangular and quadrilateral finite elements, SIAM J. Numer. Anal. 35 (1998), pp. 1893-1916. MR 1639966 (99g:65107)
  • 20. C. Carstensen, Quasi-interpolation and a posteriori error analysis in finite element methods, Math. Model. Numer. Anal., 33 (1999), 1187-1202. MR 1736895 (2001a:65135)
  • 21. Carsten Carstensen, Residual-Based a posteriori error estimate for a nonconforming Reissner-Mindlin plate finite element, SIAM J. Numer. Anal., 39(2002), pp. 2034-2044. MR 1897948 (2003e:65213)
  • 22. C. Carstensen and J. Schöberl, Residual-Based a posteriori error estimate for a mixed Reissner-Mindlin plate finite element, Numer.Math., 103 (2006), pp. 225-250. MR 2222809 (2007c:74055)
  • 23. D. Chapelle and R. Stenberg, An optimal low-order locking-free finite element method for Reissner-Mindlin plates, Math. Model Meth. Appl. Sci. 8 (1998), pp. 407-430. MR 1624871 (99d:73088)
  • 24. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, 1978; reprinted as SIAM Classics in Applied Mathematics, 2002. MR 0520174 (58:25001)
  • 25. P. Clément, Approximation by finite element functions using local regularization, RAIRO Anal. Numér., 9 (1975), pp. 77-84. MR 0400739 (53:4569)
  • 26. R. Durán and E. Liberman, On mixed finite element methods for Reissner-Mindlin plate model, Math. Comp., 58 (1992), pp. 561-573. MR 1106965 (92f:65135)
  • 27. R. Durán, E. Hernández, L. Hervella-Nieto, E. Liberman, and R. Rodríguez, Error estimates for lower-order isoparametric quadrilateral finite elements for plates, SIAM. J. Numer. Anal., 41 (2003), pp. 1751-1772. MR 2035005 (2004m:65192)
  • 28. I. Fried and S.K. Yang, Triangular, nine-degrees-of-freedoms, plate bending element of quadratic accuracy, Quart. Appl. Math., 31 (1973), pp. 303-312.
  • 29. J. Hu, Quadrilateral locking free elements in elasticity, Doctorate Dissertation (in Chinese), Institute of Computational Mathematics, Chinese Academy of Science (2004).
  • 30. J. Hu and Z.C. Shi, Analysis for quadrilateral MITC elements for Reissner-Mindlin plate, Preprint 2003-12, Institute of Computational Mathematics, Chinese Academy of Sciences,, Submitted to Math. Comp., 2005.
  • 31. T.J.R. Hughes, R.L. Taylor and W. Kanoknukuchai, A simple and efficient element for plate bending, Int. J. Numer. Meth. Engrg., 11 (1977), pp. 1529-1543.
  • 32. T.J.R. Hughes, M. Cohen and M. Haroun, Reduced and selective integration techniques in the finite element analysis of plates, Nuclear Engineering and Design, 46 (1978), pp. 203-222.
  • 33. T.J.R. Hughes, The finite element method: Linear static and dynamic finite element analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1987. MR 1008473 (90i:65001)
  • 34. T.J.R. Hughes and R.L. Taylor, The linear Triangular plate bending element, In J.R. Whiteman, editor, The Mathematics of Finite Elements and Applications IV, MAFELAP, 1981, pp. 127-142, Academic Press, 1982.
  • 35. F. Kikuchi and K. Ishii, An improved 4-node quadrilateral plate bending element of the Reissner-Mindlin type, Comput. Mech., 23 (1999), pp. 240-249.
  • 36. E. Liberman, A posteriori error estimator for a mixed finite element method for Reissner-Mindlin plate, Math. Comp., 70 (2001), pp. 1383-1396. MR 1836909 (2003c:74094)
  • 37. C. Lovadina and R. Stenberg, A posteriori error analysis of the linked interpolation technique for plate bending problems, SIAM J. Numer. Anal., 43 (2005), 2227-2249. MR 2192338 (2006i:65198)
  • 38. M. Lyly, On the connection between some linear triangular Reissner-Mindlin plate bending elements, Numer. Math., 85 (2000), pp. 77-107. MR 1751364 (2001b:65127)
  • 39. P.B. Ming and Z.C. Shi, Quadrilateral mesh, Chinese Annals of Mathematics, 23B (2002), pp. 1-18.
  • 40. J. Pitkåranta and M. Süri, Design principles and error analysis for reduced-shear plate bending finite elements, Numer. Math., 75 (1996), pp. 223-266. MR 1421988 (98c:73078)
  • 41. L. R. Scott, S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), pp. 483-493. MR 1011446 (90j:65021)
  • 42. Z.C. Shi, A convergence condition for quadrilateral Wilson element, Numer. Math. 44 (1984), pp. 349-361. MR 757491 (86d:65151)
  • 43. R. Stenberg and M. Süri, An hp error analysis of MITC plate elements, SIAM. J. Numer. Anal., 34 (1997), pp. 544-568. MR 1442928 (98g:65112)
  • 44. M. Süri and I. Babuška, C. Schwab, Locking effects in the finite element approximation of plate models, Math. Comp., 64 (1995), pp. 461-482. MR 1277772 (95f:65207)
  • 45. A. Tessler and T. J. R. Hughes, An improved treatment of transverse shear in the Mindlin type four-node quadrilateral element, Comp. Meths. Appl. Mech. Engrg., 39 (1983), pp. 311-335.
  • 46. P.A.Raviart and J.M.Thomas, A mixed finite element method for second order elliptic problems, Proc. Sympos. Mathematical Aspects of the Finite Element Method (Rome, 1975), Lecture Notes in Math., 606 (1977), pp. 292-315, Springer-Verlag. MR 0483555 (58:3547)
  • 47. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, 1996.
  • 48. O.C. Zienkiewicz, R.L. Taylor and J.M. Too, Reduced integration Technique in general analysis of plates and shells, Int. J. Numer. Meth. Engrg., 3 (1971), pp. 275-290.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 65N15, 35J25

Retrieve articles in all journals with MSC (2000): 65N30, 65N15, 35J25

Additional Information

C. Carstensen
Affiliation: Institut für Mathematik, Humboldt Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany

Jun Hu
Affiliation: LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China

Keywords: A posteriori, error analysis, Reissner-Mindlin Plate, MITC element
Received by editor(s): March 3, 2006
Received by editor(s) in revised form: November 11, 2006
Published electronically: October 18, 2007
Additional Notes: The first author was supported by DFG Research Center MATHEON “Mathematics for key technologies” in Berlin
The second author was partially supported by Natural Science Foundation of China under Grant 10601003 A Foundation for the Author of Excellent Doctoral Dissertation of PR China 200718
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society