Numerische Mathematik

, Volume 53, Issue 5, pp 513–538 | Cite as

A family of mixed finite elements for the elasticity problem

  • Rolf Stenberg
Article

Summary

A new mixed finite element formulation for the equations of linear elasticity is considered. In the formulation the variables approximated are the displacement, the unsymmetric stress tensor and the rotation. The rotation act as a Lagrange multiplier introduced in order to enforce the symmetry of the stress tensor. Based on this formulation a new family of both two-and three-dimensional mixed methods is defined. Optimal error estimates, which are valid uniformly with respect to the Poisson ratio, are derived. Finally, a new postprocessing scheme for improving the displacement is introduced and analyzed.

Subject Classifications

AMS(MOS): 65 N 30 CR: G 1.8 

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References

  1. 1.
    Amara, M., Thomas, J.M.: Equilibrium finite elements for the linear elastic problem. Numer. Math.33, 367–383 (1979)Google Scholar
  2. 2.
    Arnold, D.N., Douglas, J., Gupta, C.P.: A family of higher order mixed finite element methods for plane elasticity. Numer. Math.45, 1–22 (1984)Google Scholar
  3. 3.
    Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Math. Mod. Anal. Numer.19, 7–32 (1985)Google Scholar
  4. 4.
    Arnold, D.N., Brezzi, F., Douglas, J.: PEERS: A new finite element for plane elasticity. Jap. J. Appl. Math.1, 347–367 (1984)Google Scholar
  5. 5.
    Arnold, D.N., Falk, R.S.: Well-posedness of the fundamental boundary value problems for constrained anisotropic elastic materials. Arch. Rat. Mech. Anal.98, 143–167 (1987)Google Scholar
  6. 6.
    Babuška, I., Osborn, J., Pitkäranta, J.: Analysis of mixed methods using mesh dependent norms. Math. Comput.35, 1039–1062 (1980)Google Scholar
  7. 7.
    Bercovier, M.: Perturbation of mixed variational problems: Application to mixed finite element methods. RAIRO Anal. Numer.12, 211–236 (1978)Google Scholar
  8. 8.
    Boland, J.M., Nicolaides, R. A.: Stability of finite elements under divergence constraints. SIAM J. Num. Anal.20, 722–731 (1983)Google Scholar
  9. 9.
    Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Ser. Rouge8, 129–151 (1974)Google Scholar
  10. 10.
    Brezzi, F., Douglas, J., Marini, L.D.: Two families of mixed finite element methods for second order elliptic problems. Numer. Math.47, 217–235 (1985)Google Scholar
  11. 11.
    Brezzi, F., Douglas, J., Marini, L.D.: Recent results on mixed finite element methods for second order elliptic problems. In: Vistas in Applied Mathematics. Numerical Analysis, Atmospheric Sciences and Immunology. pp. 25–43. Heidelberg Berlin New York: Springer 1986Google Scholar
  12. 12.
    Ciarlet, P.G.: The finite element method for elliptic problems.. Amsterdam: North-Holland 1978Google Scholar
  13. 13.
    Crouzeix, M., Raviart, P.A.: Conforming and nonconforming finite elements for solving the stationary Stokes equations. RAIRO Ser. Rouge7, 33–76 (1973)Google Scholar
  14. 14.
    Fraijs de Veubeke, B.X.: Displacement and equilibrium models in the finite element method. Stress Analysis. (O.C. Zienkiewicz, G. Holister, eds.), pp. 145–197. New York: Wiley 1965Google Scholar
  15. 15.
    Fraijs de Veubeke, B.X.: Stress function approach. Bournemouth, World Conference in Finite Elements 1975 pp. J.1–J.51Google Scholar
  16. 16.
    Johnson, C., Mercier, B.: Some equilibrium finite element methods for two-dimensional elasticity problems. Numer. Math.30, 103–116 (1978)Google Scholar
  17. 17.
    Karp, S.N., Karal, F.C.: The elastic field in the neighbouthood of a crack of arbitrary angle. CPAM15, 413–421 (1962)Google Scholar
  18. 18.
    Mansfield, L.: On mixed finite element methods for elliptic equations. Comput. Math. Appl.7, 59–66 (1981)Google Scholar
  19. 19.
    Mirza, F.A., Olson, M.D.: The mixed finite element method in plane elasticity. Int. J. Numer. Methods Eng.15, 273–290 (1980)Google Scholar
  20. 20.
    Nedelec, J.C.: Mixed finite elements inR 3. Numer. Math.35, 315–341 (1980)Google Scholar
  21. 21.
    Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. Proceedings of the Symposium on the mathematical aspects on the finite element method. Lecture Notes in Mathematics No.606, 292–315. Berlin Heidelberg New York: Springer 1977Google Scholar
  22. 22.
    Pitkäranta, J., Stenberg, R.: Analysis of some mixed finite element methods for plane elasticity equations. Math. Comput.41, 399–423 (1983)Google Scholar
  23. 23.
    Stenberg, R.: Analysis of mixed finite element methods for the Stokes problem: A unified approach. Math. Comput.42, 9–23 (1984)Google Scholar
  24. 24.
    Stenberg, R.: On the construction of optimal mixed finite element methods for the linear elasticity problem. Numer. Math.48, 447–462 (1986)Google Scholar
  25. 25.
    Stenberg, R.: On the postprocessing of mixed equilibrium finite element methods. In: Numerical techniques in continuum mechanics. Proceedings of the Second GAMM-Seminar, Kiel 1986. (W. Hackbusch, K. Witsch, ed.), pp. 102–109. Braunschweig: Vieweg 1987Google Scholar
  26. 26.
    Williams, M.L.: Stress singularities resulting from various boundary conditions in angular plates in extension. J. Appl. Mech.19, 526–528 (1952)Google Scholar
  27. 27.
    Zienkiewicz, O.C., Li, X.-K., Nakazawa S.: Iterative solution of mixed problems and the stress recovery procedures. Comm. Appl. Numer. Methods3, 3–9 (1985)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Rolf Stenberg
    • 1
  1. 1.Institute of MathematicsHelsinki University of TechnologyEspooFinland

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