Generalized Finite Element Method in Mixed Variational Formulation: A Study of Convergence and Solvability

  • W. Góis
  • S. P. B. Proença
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 5)


The Generalized Finite Element Method (GFEM) is first applied to hybrid-mixed stress formulations (HMSF). Generalized shape approximation functions are generated by means of polynomials of three independent approximation fields: stresses and displacements in the domain and displacements field on the static boundary. Firstly, the enrichment can independently be conducted over each of the three approximation fields. However, solvability and convergence problems are induced mainly due to spurious modes generated when enrichment is arbitrarily applied. With the aim of efficiently exploring enrichments in HMSF, an extension of the patch-test is proposed as a necessary condition to ensure enrichment, thus preserving convergence and solvability. In the present work, the inf-sup test based on Babuška-Brezzi condition was used to demonstrate the effectiveness of the Patch-Test. In particular, the inf-sup test was applied over selectively enriched quadrilateral bilinear and triangular finite element meshes. Numerical examples confirm the Patch-Test as a necessary but not sufficient condition for convergence and solvability.

Key words

Generalized finite element method hybrid-mixed stress formulation Babuška-Brezzi condition inf-sup test 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Babuška I., Error bounds for finite element methods. Numerische Mathematik 16:322–333, 1971.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Babuška I., On the inf-sup (Babuška-Brezzi) condition. The University of Texas at Austin. Technical Report #5, TICAM, 1996.Google Scholar
  3. 3.
    Brezzi F., On the existence, uniqueness and approximation of saddle point problems arising from Lagrange multipliers. RAIRD 8(r-2):127–151, 1974.MathSciNetGoogle Scholar
  4. 4.
    Chapelle D. and Bathe K.J., The inf-sup test. Computers & Structures, 47:537–545, 1993.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Duarte C.A., A review of some meshless methods to solve partial differential equations. The University of Texas at Austin. Technical Report, TICAM, 1995.Google Scholar
  6. 6.
    Freitas J.A.T., Almeida J.P.B.M. and Pereira E.M.B.R., Non-conventional formulations for the finite element method. Structural Engineering and Mechanics, 4:655–678, 1996.Google Scholar
  7. 7.
    Góis W., Generalized finite element method in hibryd mixed stress formulation, Master Dissertation. São Carlos School of Engineering. University of São Paulo, 2004 [in Portuguese]Google Scholar
  8. 8.
    Oden J.T., Duarte C.A. and Zienkiewicz O.C., A new cloud-based hp finite element method. Computer Methods in Applied Mechanics and Engineering, 153:117–126, 1998.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Pimenta P.M., Proença S.P.B. and Freitas J.A.T., Elementos finitos híbridos mistos com enriquecimento nodal. In: Proceedings of Métodos Numéricos em Ingeniería V, J.M. Gaicolea, C. Mota Soares, M. Pastor and G. Bugeda (Eds), SEMNI, 2002.Google Scholar
  10. 10.
    Schwab Ch., P-and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics, Oxford Science Publications, New York, 1998.MATHGoogle Scholar
  11. 11.
    Zienkiewicz O.C. et al., The patch test for mixed formulation. International Journal for Numerical Methods in Engineering, 23:1873–1882, 1986.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • W. Góis
    • 1
  • S. P. B. Proença
    • 1
  1. 1.Structural Engineering Department, São Carlos School of EngineeringUniversity of São PauloSão CarlosBrazil

Personalised recommendations