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Locking-Free Optimal Discontinuous Galerkin Methods for a Naghdi-Type Arch Model

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Abstract

In this paper, we introduce and analyze discontinuous Galerkin methods for a Naghdi type arch model. We prove that, when the numerical traces are properly chosen, the methods display optimal convergence uniformly with respect to the thickness of the arch. These methods are thus free from membrane and shear locking. We also prove that, when polynomials of degree k are used, all the numerical traces superconverge with a rate of order h 2k+1. Numerical experiments verifying the above-mentioned theoretical results are displayed.

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References

  1. Arnold, D.N.: Discretization by finite elements of a model parameter dependent problem. Numer. Math. 37, 405–421 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold, D.N., Brezzi, F., Falk, R., Marini, L.D.: Locking-free Reissner-Mindlin elements without reduced integration. Comput. Methods Appl. Mech. Eng. 196, 3660–3671 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Arnold, D.N., Falk, R.: A uniformly accurate finite element method for the Reissner-Mindlin plate. SIAM J. Numer. Anal. 26(6), 1276–1290 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arnold, D.N., Falk, R.S.: The boundary layer for the Reissner-Mindlin plate model. SIAM J. Math. Anal. 21(2), 281–312 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Arnold, D.N., Falk, R.S.: Analysis of a linear-linear finite element for the Reissner-Mindlin plate model Math. Models Methods Appl. Sci. 7(2), 217–238 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Arnold, D.N., Liu, X.: Interior estimates for a low order finite element method for the Reissner-Mindlin plate model. Adv. Comput. Math. 7(3), 337–360 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Babuška, I., Suri, M.: Locking effects in the finite element approximation of elasticity problems. Numer. Math. 62, 439–463 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Babuška, I., Suri, M.: On locking and robustness in the finite element method. SIAM J. Numer. Anal. 29, 1261–1293 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brezzi, F., Fortin, M.: Numerical approximation of Mindlin-Reissner plates. Math. Comp. 47(175)

  12. Castillo, P., Cockburn, B., Schötzau, D., Schwab, C.: Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems. Math. Comput. 71, 455–478 (2002)

    MATH  Google Scholar 

  13. Celiker, F.: Discontinuous Galerkin methods for structural mechanics. Ph.D. thesis, University of Minnesota, Minneapolis (2005)

  14. Celiker, F., Cockburn, B.: Element-by-element post-processing of discontinuous Galerkin methods for Timoshenko beams. J. Sci. Comput. 27(1–3), 177–187 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Celiker, F., Cockburn, B., Güzey, S., Kanapady, R., Soon, S.-C., Stolarski, H.K., Tamma, K.K.: Discontinuous Galerkin methods for Timoshenko beams. In: Numerical Mathematics and Advanced Applications, ENUMATH 2003, pp. 221–231. Springer, Berlin (2003)

    Google Scholar 

  16. Celiker, F., Cockburn, B., Shi, K.: A projection-based error analysis of HDG methods for Timoshenko beams. Math. Comp., to appear

  17. Celiker, F., Cockburn, B., Shi, K.: Hybridizable discontinuous Galerkin methods for Timoshenko beams. J. Sci. Comput. 44, 1–37 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Celiker, F., Cockburn, B., Stolarski, H.K.: Locking-free optimal discontinuous Galerkin methods for Timoshenko beams. SIAM J. Numer. Anal. 44(6), 2297–2325 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chinosi, C., Lovadina, C., Marini, L.D.: Nonconforming locking-free finite elements for Reissner-Mindlin plates. Comput. Methods Appl. Mech. Eng. 195, 3448–3460 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  21. Durán, R., Ghioldi, A., Wolanski, N.: A finite element method for the Mindlin-Reissner plate model. SIAM J. Numer. Anal. 28, 1004–1014 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Durán, R., Herhández, E., Hervella-Nieto, L., Liberman, E., Rodríguez, R.: Error estimates for low-order quadrilateral finite elements for plates. SIAM J. Numer. Anal. 41, 1751–1772 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Falk, R.S., Tu, T.: Locking-free finite elements for the Reissner-Mindlin plate. Math. Comput. 69(231), 911–928 (2000)

    MathSciNet  MATH  Google Scholar 

  24. Houston, P., Schwab, C., Süli, E.: Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39, 2133–2163 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hughes, T.J.R., Taylor, R.L., Kanoknukulchai, W.: A simple and efficient element for plate bending. Int. J. Numer. Methods Eng. 11, 1529–1543 (1977)

    Article  MATH  Google Scholar 

  26. Li, L.: Discretization of the Timoshenko beam problem by the p and the h–p versions of the finite element method. Numer. Math. 57, 413–420 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lovadina, C.: A low-order nonconforming finite element for Reissner-Mindlin plates. SIAM J. Numer. Anal. 42(6), 2688–2701 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Malkus, D.S., R Hughes, T.J.: Mixed finite element methods-reduced integration and selective integration techniques: a unification of concepts. Comput. Methods Appl. Mech. Eng. 15, 63–81 (1978)

    Article  MATH  Google Scholar 

  29. Pitkäranta, J., Suri, M.: Upper and lower bounds for plate-bending finite elements. Numer. Math. 84, 611–648 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. Schwab, C.: p- and hp-FEM. Theory and Application to Solid and Fluid Mechanics. Oxford University Press, New York (1998)

    Google Scholar 

  31. Suri, M., Babuška, I., Schwab, C.: Locking effects in the finite element approximation of plate models. Math. Comput. 210, 461–482 (1995)

    Google Scholar 

  32. Zhang, S.: An asymptotic analysis on the form of Naghdi type arch model. Math. Models Methods Appl. Sci. 18(3) (2008)

  33. Zhang, Z.: Arch beam models: finite element analysis and superconvergence. Numer. Math. 61, 117–143 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang, Z.: A note on the hybrid-mixed C 0 curved beam elements. Comput. Methods Appl. Mech. Eng. 95, 243–252 (1992)

    Article  Google Scholar 

  35. Zhang, Z.: Locking and robustness in the finite element method for circular arch problem. Numer. Math. 69, 509–522 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  37. Zhang, Z., Zhang, S.: Derivative superconvergence of rectangular finite elements for the Reissner-Mindlin plate. Comput. Methods Appl. Mech. Eng. 134, 1–16 (1996)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Fatih Celiker, Li Fan, Sheng Zhang & Zhimin Zhang

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  1. Fatih Celiker
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  2. Li Fan
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  3. Sheng Zhang
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  4. Zhimin Zhang
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Correspondence to Fatih Celiker.

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Celiker, F., Fan, L., Zhang, S. et al. Locking-Free Optimal Discontinuous Galerkin Methods for a Naghdi-Type Arch Model. J Sci Comput 52, 49–84 (2012). https://doi.org/10.1007/s10915-011-9532-0

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  • DOI: https://doi.org/10.1007/s10915-011-9532-0

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