Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Mixed finite element methods for linear elasticity with weakly imposed symmetry

Authors: Douglas N. Arnold, Richard S. Falk and Ragnar Winther
Journal: Math. Comp. 76 (2007), 1699-1723
MSC (2000): Primary 65N30; Secondary 74S05
Published electronically: May 9, 2007
MathSciNet review: 2336264
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger-Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.

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

  • 1. Scot Adams and Bernardo Cockburn, A mixed finite element method for elasticity in three dimensions, Journal of Scientific Computing 25 (2005), 515-521. MR 2221175 (2006m:65251)
  • 2. Mohamed Amara and Jean-Marie Thomas, Equilibrium finite elements for the linear elastic problem, Numer. Math. 33 (1979), no. 4, 367-383. MR 553347 (81b:65096)
  • 3. Douglas N. Arnold, Gerard Awanou, and Ragnar Winther, Finite elements for symmetric tensors in three dimensions, Submitted to Math. Comp.
  • 4. Douglas N. Arnold, Franco Brezzi, and Jim Douglas, Jr., PEERS: a new mixed finite element for plane elasticity, Japan J. Appl. Math. 1 (1984), no. 2, 347-367. MR 840802 (87h:65189)
  • 5. Douglas N. Arnold, Jim Douglas, Jr., and Chaitan P. Gupta, A family of higher order mixed finite element methods for plane elasticity, Numer. Math. 45 (1984), no. 1, 1-22. MR 761879 (86a:65112)
  • 6. Douglas N. Arnold and Richard S. Falk, A new mixed formulation for elasticity, Numer. Math. 53 (1988), no. 1-2, 13-30. MR 946367 (89f:73020)
  • 7. Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Differential complexes and stability of finite element methods. I: The de Rham complex, in Compatible Spatial Discretizations, D. Arnold, P. Bochev, R. Lehoucq, R. Nicolaides, and M. Shashkov, eds., IMA Volumes in Mathematics and its Applications 142, Springer-Verlag 2005, 23-46. MR 2249344
  • 8. Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Differential complexes and stability of finite element methods. II: The elasticity complex, in Compatible Spatial Discretizations, D. Arnold, P. Bochev, R. Lehoucq, R. Nicolaides, and M. Shashkov, eds., IMA Volumes in Mathematics and its Applications 142, Springer-Verlag 2005, 47-67. MR 2249345
  • 9. Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and application, Acta Numerica (2006), 1-155. MR 2269741
  • 10. Douglas N. Arnold and Ragnar Winther, Mixed finite elements for elasticity, Numer. Math. 92 (2002), no. 3, 401-419. MR 1930384 (2003i:65103)
  • 11. I.N. Bernstein, I.M. Gelfand, and S.I. Gelfand, Differential operators on the base affine space and a study of $ \mathfrak{g}$-modules, Lie groups and their representation, I.M. Gelfand, ed., (1975), 21-64. MR 0578996 (58:28285)
  • 12. Franco Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129-151. MR 0365287 (51:1540)
  • 13. Franco Brezzi and Michel Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
  • 14. Andreas Cap, Jan Slovák, and Vladimír Soucek, Bernstein-Gelfand-Gelfand sequences, Ann. Math. (2) 154 (2001), 97-113. MR 1847589 (2002h:58034)
  • 15. Richard M. Christensen, Theory of Viscoelasticity, Dover Publications, 1982.
  • 16. Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, 1978. MR 0520174 (58:25001)
  • 17. Jim Douglas, Jr. and Jean E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), no. 169, 39-52. MR 771029 (86b:65122)
  • 18. Michael Eastwood, A complex from linear elasticity, Rend. Circ. Mat. Palermo (2) Suppl. (2000), no. 63, 23-29. MR 1758075 (2001j:58033)
  • 19. Richard S. Falk, Finite element methods for linear elasticity, to appear in Mixed Finite Elements, Compatibility Conditions, and Applications, Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 26-July 1, 2006, Lecture Notes in Mathematics, Springer-Verlag.
  • 20. Richard S. Falk and John E. Osborn, Error estimates for mixed methods, R.A.I.R.O. Analyse numérique/Numerical Analysis, 14 (1980), no. 3, 249-277. MR 592753 (82j:65076)
  • 21. Baudoiun M. Fraejis de Veubeke, Stress function approach, Proc. of the World Congress on Finite Element Methods in Structural Mechanics, Vol. 1, Bournemouth, Dorset, England (Oct. 12-17, 1975), J.1-J.51.
  • 22. V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations. Theory and algorithms, Springer Series in Computational Mathematics, 5, Springer-Verlag, Berlin, 1986. MR 851383 (88b:65129)
  • 23. Claes Johnson and Bertrand Mercier, Some equilibrium finite element methods for two-dimensional elasticity problems, Numer. Math. 30 (1978), no. 1, 103-116. MR 0483904 (58:3856)
  • 24. Mary E. Morley, A family of mixed finite elements for linear elasticity, Numer. Math. 55 (1989), no. 6, 633-666. MR 1005064 (90f:73006)
  • 25. Jean-Claude Nédélec, Mixed finite elements in $ R^{3}$, Numer. Math. 35 (1980), no. 3, 315-341. MR 592160 (81k:65125)
  • 26. Jean-Claude Nédélec, A new family of mixed finite elements in $ R^3$, Numer. Math. 50 (1986), no. 1, 57-81. MR 864305 (88e:65145)
  • 27. Erwin Stein and Raimund Rolfes, Mechanical conditions for stability and optimal convergence of mixed finite elements for linear plane elasticity, Comput. Methods Appl. Mech. Engrg. 84 (1990), no. 1, 77-95. MR 1082821 (91i:73045)
  • 28. Rolf Stenberg, On the construction of optimal mixed finite element methods for the linear elasticity problem, Numer. Math. 48 (1986), no. 4, 447-462. MR 834332 (87i:73062)
  • 29. -, A family of mixed finite elements for the elasticity problem, Numer. Math. 53 (1988), no. 5, 513-538. MR 954768 (89h:65192)
  • 30. -, Two low-order mixed methods for the elasticity problem, The mathematics of finite elements and applications, VI (Uxbridge, 1987), Academic Press, London, 1988, pp. 271-280. MR 956898 (89j:73074)
  • 31. Vernon B. Watwood Jr. and B. J. Hartz, An equilibrium stress field model for finite element solution of two-dimensional elastostatic problems, Internat. Jour. Solids and Structures 4 (1968), 857-873.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 74S05

Retrieve articles in all journals with MSC (2000): 65N30, 74S05

Additional Information

Douglas N. Arnold
Affiliation: Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455

Richard S. Falk
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854-8019

Ragnar Winther
Affiliation: Centre of Mathematics for Applications and Department of Informatics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway

Keywords: Mixed method, finite element, elasticity
Received by editor(s): October 31, 2005
Received by editor(s) in revised form: September 11, 2006
Published electronically: May 9, 2007
Additional Notes: The work of the first author was supported in part by NSF grant DMS-0411388
The work of the second author was supported in part by NSF grant DMS03-08347
The work of the third author was supported by the Norwegian Research Council
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society