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Complements on Plate Problems

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Mixed Finite Element Methods and Applications

Part of the book series: Springer Series in Computational Mathematics ((SSCM,volume 44))

Abstract

In this chapter, we shall present a few among many applications of mixed methods to plate problems. In the first section, we shall describe a mixed method for the linear thin plates theory and in the second, a dual hybrid method. In the last section, we shall report some recent results on the discretisation of the Mindlin-Reissner formulation for moderately thick plates.

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References

  1. D. N. Arnold, F. Brezzi, and L. D. Marini. Locking-free Reissner-Mindlin elements without reduced integration. Comput. Methods Appl. Mech. Engrg, 196:3660–3671, 2007.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. D.N. Arnold and F. Brezzi. Mixed and non-conforming finite element methods: implementation, post-processing and error estimates. Math. Modelling Numer. Anal., 19:7–35, 1985.

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  5. D.N. Arnold, J. Douglas, and C.P. Gupta. A family of higher order mixed finite element methods for plane elasticity. Numer. Math., 45:1–22, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  6. D.N. Arnold and J.S. Falk. A uniformly accurate finite element method for the Mindlin-Reissner plate. SIAM. J. Num. Anal., 26:1276–1290, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  7. D.N. Arnold and J.S. Falk. The boundary layer for the Reissner-Mindlin plate model. SIAM J. Math. Anal., 21:281–312, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  8. D.N. Arnold and A.L. Madureira. Asymptotic estimates of hierarchical modeling. Math. Models Methods Appl. Sci, 13:1325–1350, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  9. I. Babuška, J.E. Osborn, and J. Pitkaranta. Analysis of mixed methods using mesh-dependent norms. Math. Comp., 35:1039–1079, 1980.

    MathSciNet  MATH  Google Scholar 

  10. K.J. Bathe and F. Brezzi. The convergence of a four-node plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation. In J.R. Whiteman, editor, Proceedings of the Conference on Mathematics of Finite Elements and applications V , pages 491–503. Academic Press, New-York, 1985.

    Chapter  Google Scholar 

  11. K.J. Bathe and E. Dvorkin. A continuum mechanics based four-node shell element for general non-linear analysis. J. Eng. Comp., 1:77–78, 1984.

    Article  Google Scholar 

  12. J.H. Bramble and R.S. Falk. Two mixed finite element methods for the simply supported plate problem. R.A.I.R.O. Anal. Numer., 17:337–384, 1983.

    Google Scholar 

  13. F. Brezzi. Sur la méthode des éléments finis hybrides pour le problème biharmonique. Numer. Math., 24:103–131, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  14. F. Brezzi. Hybrid approximations of nonlinear plate-bending problems. In Hybrid and mixed finite element methods (Atlanta, Ga., 1981), pages 267–280. Wiley, Chichester, 1983.

    Google Scholar 

  15. F. Brezzi, K.J. Bathe, and M. Fortin. Mixed-interpolated elements for Reissner-Mindlin plates. Int. J. Num. Meth. Eng., 28:1787–1801, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Springer-Verlag, New York, 1991.

    Book  MATH  Google Scholar 

  17. F. Brezzi, M. Fortin, and R. Stenberg. Error analysis of mixed-interpolated elements for Reissner-Mindlin plates. Math. Models Methods Appl. Sci., 1(2):125–151, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  18. F. Brezzi and L.D. Marini. On the numerical solution of plate bending problems by hybrid methods. R.A.I.R.O. Anal. Numer., pages 5–50, 1975.

    Google Scholar 

  19. F. Brezzi and P.A. Raviart. Mixed finite element methods for 4th order elliptic equations. In J. Miller, editor, Topics in Numerical Analysis III. Academic Press, New-York, 1978.

    Google Scholar 

  20. P.G. Ciarlet and R. Glowinski. Dual iterative techniques for solving a finite element approximation of the biharmonic equation. Comp. Meth. Appl. Mech. Eng., 5:227–295, 1975.

    Article  MathSciNet  Google Scholar 

  21. P.G. Ciarlet and P.A. Raviart. A mixed finite element method for the biharmonic equation. In C. de Boor, editor, Mathematical Aspects of Finite Element in Partial Differential Equations. Academic Press, New York, 1974.

    Google Scholar 

  22. L. Comodi. The Hellan-Hermann-Johnson method: error estimates for the Lagrange multipliers and post processing. Math. Comp., 52:17–30, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Destuynder. Une théorie asymptotique des plaques minces en élasticité linéaire. Masson, Paris, 1986.

    MATH  Google Scholar 

  24. R. Duran and E. Liberman. On mixed finite element methods for the Reissner-Mindlin plate model. Math. Comp, 58(198):561–573, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. G. Duran and E. Liberman. On the convergence of a triangular mixed finite element method for Reissner-Mindlin plates. Math. Models Methods Appl. Sci, 6:339–352, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  26. R.S. Falk. Approximation of the biharmonic equation by a mixed finite element method. SIAM J. Numer. Anal., 15, 556–567 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  27. R.S. Falk. Finite elements for the Reissner-Mindlin plate. In D. Boffi and L. Gastaldi, editors, Mixed Finite Elements,Compatibility Conditions and Applications. Springer-Verlag, Berlin, 2008.

    Google Scholar 

  28. R.S. Falk and J.E. Osborn. Error estimates for mixed methods. RAIRO Anal. Numér., 14(3):249–277, 1980.

    MathSciNet  MATH  Google Scholar 

  29. G.J. Fix, M.D. Gunzburger, and R.A. Nicolaides. Theory and applications of mixed finite element methods. In C.V. Coffman and G.J. Fix, editors, Constructive Approaches to Mathematical Models, pages 375–393. Academic Press, New York, 1979.

    Google Scholar 

  30. V. Girault and P.A. Raviart. Finite Element Approximation of Navier-Stokes Equations, volume 749 of Lectures Notes in Math. Springer-Verlag, Berlin, 1979.

    Google Scholar 

  31. R. Glowinski. Approximations externes par éléments finis de Lagrange d’ordre un et deux, du problème de Dirichlet pour l’opérateur biharmonique. Méthodes itératives de résolution des problèmes approchés. In J. Miller, editor, Topics in Numerical Analysis, pages 123–171, New York, 1973. Academic Press.

    Google Scholar 

  32. R. Glowinski. Approximations externes par éléments finis d’ordre 1 et 2 du problème de Dirichlet pour l’opérateur biharmonique,methode itérative de résolution des problèmes approchés. In J. Miller, editor, Topics in Numerical Analysis. Academic Press, New York, 1973.

    Google Scholar 

  33. R. Glowinski. Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, Berlin, 1984.

    Book  MATH  Google Scholar 

  34. R. Glowinski and O. Pironneau. Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Review, 17:167–212, 1979.

    Article  MathSciNet  Google Scholar 

  35. P. Grisvard. Elliptic Problems in Non-Smooth Domains. Pitman, London, 1985.

    Google Scholar 

  36. P. Grisvard. Singularities in Boundary Value Problems. Masson, Paris, 1992.

    MATH  Google Scholar 

  37. T.J.R. Hughes and T.E. Tezduyar. Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element. Jour. of App. Mech., 48:587–596, 1981.

    Article  MATH  Google Scholar 

  38. C. Johnson. On the convergence of a mixed finite element method for plate bending problems. Numer. Math., 21:43–62, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  39. C. Johnson and B. Mercier. Some equilibrium finite element methods for two-dimensional elasticity problems. Numer. Math., 30:103–116, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  40. J.L. Lions and E. Magenes. Problèmes aux limites non-homogènes et applications. Dunod, Paris, 1968.

    MATH  Google Scholar 

  41. L.D. Marini. Implementation of hybrid finite element methods and associated numerical problems, part 1. Technical Report 36, IAN-CNR, Pavia, 1976.

    Google Scholar 

  42. L.D. Marini. Implementation of hybrid finite element methods and associated numerical problems, part 2. Technical Report 182, IAN-CNR, Pavia, 1978.

    Google Scholar 

  43. B. Mercier. Numerical solution of the biharmonic problem by mixed finite elements of class C 0. Boll. U.M.I., 10:133–149, 1974.

    Google Scholar 

  44. T. Miyoshi. A finite element method for the solution of fourth order partial differential equations. Kumamoto J. Sci. Math., 9:87–116, 1973.

    MathSciNet  MATH  Google Scholar 

  45. T.H.H. Pian and P. Tong. Basis of finite element methods for solid continua. Int. J. Num. Meth. Eng., 1:3–28, 1969.

    Article  MATH  Google Scholar 

  46. J. Pitkäranta. Analysis of some low-order finite element schemes for Mindlin-Reissner and Kirchhoff plates. Numer. Math., 53:237–254, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  47. R. Scholz. Approximation von Sattelpunkten mit Finiten Elementen. Bonner Mathematischen Schriften, 89:54–66, 1976.

    Google Scholar 

  48. R. Scholz. A mixed method for fourth order problems using linear finite elements. R.A.I.R.O. Anal. Numer., 12:85–90, 1978.

    Google Scholar 

  49. R. Scholz. A remark on the rate of convergence for a mixed finite element method for second order problems. Numer. Funct. Anal. Optim., 4:269–277, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  50. R. Temam. Navier-Stokes Equations. North-Holland, Amsterdam, 1977.

    MATH  Google Scholar 

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Boffi, D., Brezzi, F., Fortin, M. (2013). Complements on Plate Problems. In: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36519-5_10

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