Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Babuška I., Error bounds for finite element methods. Numerische Mathematik 16:322–333, 1971.
Babuška I., On the inf-sup (Babuška-Brezzi) condition. The University of Texas at Austin. Technical Report #5, TICAM, 1996.
Brezzi F., On the existence, uniqueness and approximation of saddle point problems arising from Lagrange multipliers. RAIRD 8(r-2):127–151, 1974.
Chapelle D. and Bathe K.J., The inf-sup test. Computers & Structures, 47:537–545, 1993.
Duarte C.A., A review of some meshless methods to solve partial differential equations. The University of Texas at Austin. Technical Report, TICAM, 1995.
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.
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]
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.
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.
Schwab Ch., P-and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics, Oxford Science Publications, New York, 1998.
Zienkiewicz O.C. et al., The patch test for mixed formulation. International Journal for Numerical Methods in Engineering, 23:1873–1882, 1986.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer
About this paper
Cite this paper
Góis, W., Proença, S.P.B. (2007). Generalized Finite Element Method in Mixed Variational Formulation: A Study of Convergence and Solvability. In: Leitão, V.M.A., Alves, C.J.S., Armando Duarte, C. (eds) Advances in Meshfree Techniques. Computational Methods in Applied Sciences, vol 5. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-6095-3_3
Download citation
DOI: https://doi.org/10.1007/978-1-4020-6095-3_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-6094-6
Online ISBN: 978-1-4020-6095-3
eBook Packages: EngineeringEngineering (R0)