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Finite Element Methods for Linear Elasticity

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Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1939)

Keywords

  • Finite Element Method
  • Exact Sequence
  • Differential Form
  • Linear Elasticity
  • Element Space

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References

  1. S. Adams and B. Cockburn. A mixed finite element method for elasticity in three dimensions. J. Sci. Comput., 25:515–521, 2005.

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. M. Amara and J. M. Thomas. Equilibrium finite elements for the linear elastic problem. Numer. Math., 33:367–383, 1979.

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. D. N. Arnold, F. Brezzi, and J. Douglas, Jr. PEERS: a new mixed finite element for plane elasticity. Jpn. J. Appl. Math., 1:347–367, 1984.

    CrossRef  MATH  MathSciNet  Google Scholar 

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

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. D. N. Arnold and R. S. Falk. A new mixed formulation for elasticity. Numer. Math., 53:13–30, 1988.

    CrossRef  MATH  MathSciNet  Google Scholar 

  6. D. N. Arnold, R. S. Falk, and R. Winther. Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comput., 76(260):1699–1723, 2007.

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. D. N. Arnold, R. S. Falk, and R. Winther. Differential complexes and stability of finite element methods I: The de Rham complex. In Compatible Spatial Discretizations, The IMA Volumes in Mathematics and its Applications, vol. 142, pp. 23–46. Springer, Berlin Heidelberg New York, 2006.

    CrossRef  Google Scholar 

  8. D. N. Arnold, R. S. Falk, and R. Winther. Differential complexes and stability of finite element methods II: The elasticity complex. In Compatible Spatial Discretizations, The IMA Volumes in Mathematics and its Applications, vol. 142, pp. 47–68. Springer, Berlin Heidelberg New York, 2006.

    CrossRef  Google Scholar 

  9. D. N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus, homological techniques, and applications. Acta Numer., vol. 15, pp. 1–155, 2006.

    CrossRef  MATH  MathSciNet  Google Scholar 

  10. D. N. Arnold and R. Winther. Mixed finite elements for elasticity. Numer. Math., 92:401–419, 2002.

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. D. N. Arnold and R. Winther. Mixed finite elements for elasticity in the stress-displacement formulation. In Current trends in scientific computing (Xi’an, 2002), Contemp. Math., vol. 329, pp. 33–42. American Mathematical Society, Providence, RI, 2003.

    Google Scholar 

  12. D. N. Arnold and R. Winther. Nonconforming mixed elements for elasticity. Math. Models Methods Appl. Sci., 13(3):295–307, 2003. Dedicated to Jim Douglas, Jr. on the occasion of his 75th birthday.

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand. Differential operators on the base affine space and a study of 𝔤-modules. In Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) , pp. 21–64. Halsted, New York, 1975.

    Google Scholar 

  14. F. Brezzi. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 8:129–151, 1974.

    MathSciNet  Google Scholar 

  15. F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, vol. 15. Springer, Berlin Heidelberg New York, 1991.

    MATH  Google Scholar 

  16. A. Čap, J. Slovák, and V. Souček. Bernstein–Gelfand–Gelfand sequences. Ann. Math. (2), 154:97–113, 2001.

    MATH  Google Scholar 

  17. J. Douglas, Jr., T. Dupont, P. Percell, and L. R. Scott. A family of C 1 finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems. RAIRO Anal. Numér., 13(3):227–255, 1979.

    MATH  MathSciNet  Google Scholar 

  18. J. Douglas, Jr. and J. E. Roberts. Global estimates for mixed methods for second order elliptic equations. Math. Comp., 44:39–52, 1985.

    CrossRef  MATH  MathSciNet  Google Scholar 

  19. M. Eastwood. A complex from linear elasticity. Rend. Circ. Mat. Palermo (2) Suppl., (63):23–29, 2000.

    MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  21. M. Farhloul and M. Fortin. Dual hybrid methods for the elasticity and the Stokes problems: a unified approach. Numer. Math., 76:419–440, 1997.

    CrossRef  MATH  MathSciNet  Google Scholar 

  22. B. M. Fraeijs de Veubeke. Stress function approach. In Proc. of the World Congress on Finite Element Methods in Structural Mechanics, vol. 1, pp. J.1–J.51. Bournemouth, Dorset, England, 1975.

    Google Scholar 

  23. V. Girault and P.-A. Raviart. Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5. Springer, Berlin Heidelberg New York, 1986. Theory and Algorithms.

    MATH  Google Scholar 

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

    CrossRef  MATH  MathSciNet  Google Scholar 

  25. M. E. Morley. A family of mixed finite elements for linear elasticity. Numer. Math., 55:633–666, 1989.

    CrossRef  MATH  MathSciNet  Google Scholar 

  26. E. Stein and R. Rolfes. Mechanical conditions for stability and optimal convergence of mixed finite elements for linear plane elasticity. Comput. Methods Appl. Mech. Eng., 84:77–95, 1990.

    CrossRef  MATH  MathSciNet  Google Scholar 

  27. R. Stenberg. On the construction of optimal mixed finite element methods for the linear elasticity problem. Numer. Math., 48:447–462, 1986.

    CrossRef  MATH  MathSciNet  Google Scholar 

  28. R. Stenberg. A family of mixed finite elements for the elasticity problem. Numer. Math., 53:513–538, 1988.

    CrossRef  MATH  MathSciNet  Google Scholar 

  29. R. Stenberg. Two low-order mixed methods for the elasticity problem. In The Mathematics of Finite Elements and Applications, VI (Uxbridge, 1987) , pp. 271–280. Academic Press, London, 1988.

    Google Scholar 

  30. V. B. Watwood, Jr. and B. J. Hartz. An equilibrium stress field model for finite element solution of two-dimensional elastostatic problems. Int. J. Solids Struct., 4:857–873, 1968.

    CrossRef  MATH  Google Scholar 

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Falk, R.S. (2008). Finite Element Methods for Linear Elasticity. In: Boffi, D., Gastaldi, L. (eds) Mixed Finite Elements, Compatibility Conditions, and Applications. Lecture Notes in Mathematics, vol 1939. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78319-0_4

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