Generalized Finite Element Method in Mixed Variational Formulation: A Study of Convergence and Solvability
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 wordsGeneralized finite element method hybrid-mixed stress formulation Babuška-Brezzi condition inf-sup test
Unable to display preview. Download preview PDF.
- 2.Babuška I., On the inf-sup (Babuška-Brezzi) condition. The University of Texas at Austin. Technical Report #5, TICAM, 1996.Google Scholar
- 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.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.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
- 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