×

zbMATH — the first resource for mathematics

A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element formulation. (English) Zbl 0656.73036
This paper presents a mixed Petrov-Galerkin finite element method for the Hellinger-Reissner principle. The usual isoparametric elements can be used and the equal-order interpolation is shown to be stable within the present method. Numerical examples are given.
Reviewer: F.Chatelin

MSC:
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65K10 Numerical optimization and variational techniques
PDF BibTeX Cite
Full Text: DOI EuDML
References:
[1] Arnold, D.N., Brezzi, F., Fortin, M.: A Stable Finite Element for the Stokes Equations. CALCOLOXXI 337-344 (1984) · Zbl 0593.76039
[2] Arnold, D.N., Falk, R.S.: A New Mixed Formulation for Elasticity. Preprint 1986 · Zbl 0621.73102
[3] Babu?ka, I.: Error Bounds for Finite Element Method. Numer. Math.16, 322-333 (1971) · Zbl 0214.42001
[4] Bercovier, M.: Perturbation of Mixed Variational Problems. Application to Mixed Finite Element Methods. Rev. Française d’Automatique Informatique et Recherche Opérationnelle, Ser. Rouge Anal. Numér.12, 211-236 (1978)
[5] Bercovier, M., Pironneau, O.: Error Estimates for Finite Element Method Solution of the Stokes Problem in the Primitive Variables. Numer. Math.33, 211-224 (1979) · Zbl 0423.65058
[6] Brezzi, F.: On the Existence, Uniqueness and Approximation of Saddle-point Problems Arising From Lagrange Multipliers. Rev. Française d’Automatique Informatique et Recherche Opérationnelle, Ser. Rouge Anal. Numér.8, 129-151 (1974)
[7] Crouzeix, M., Raviart, P.A.: Conforming and Non-conforming Finite Element Methods for Solving the Stationary Stokes Equations. Rev. Française d’Automatique Informatique et Recherche Opérationnelle, Ser. Rouge Anal. Numér.7, 33-76 (1973
[8] Engelman, M.S., Sani, R.L., Gresho, P.M., Bercovier, M.: Consistent vs. Reduced Integration Penalty Methods for Incompressible Media using several Old and New Elements. Int. J. Numer. Methods Fluids2, 25-42 (1982) · Zbl 0483.76013
[9] Fortin, A., Fortin, M.: Experiments with Several Elements for Viscous Incompressible Flows. EPM/RT-85-9. Ecole Polytechnique de Montréal, Montreal, Canada 1985 · Zbl 0573.76032
[10] Fortin, M.: An Analysis of the Convergence of Mixed Finite Element Methods. Rev. Française d’Automatique Informatique et Recherche Opérationnelle, Ser. Rouge Anal. Numér.11, 341-354 (1977)
[11] Fortin, M.: Old and New Finite Element Methods for Incompressible Flows. Int. J. Numer. Methods Fluids1, 347-364 (1981) · Zbl 0467.76030
[12] Franca, L.P.: New Mixed Finite Element Methods. Ph.D. Thesis, Division of Applied Mechanics, Stanford University, Stanford, California 1987
[13] Hellinger, E.: Der Allgemeine Ansatz der Mechanik der Kontinua. Encyclopädie Math. Wiss.4, 602-694 (1914)
[14] Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Englewood Cliffs, NJ: Prentice-Hall 1987 · Zbl 0634.73056
[15] Hughes, T.J.R., Franca, L.P., Balestra, M.: Circumventing the Babu?ka-Brezzi Condition: A Stable Petrov-Galerkin Formulation of the Stokes Problem Accomodating Equal-Order Interpolations. Comput. Methods Appl. Mech. Eng.59, 85-99 (1986) · Zbl 0622.76077
[16] Hughes, T.J.R., Liu, W.K., Brooks, A.: Review of Finite Element Analysis of Incompressible Viscous Flows by the Penalty Function Formulation. J. Comput. Phys.30, 1-60 (1979) · Zbl 0412.76023
[17] Loula, A.F.D., Hughes, T.J.R., Franca, L.P., Miranda, I.: Mixed Petrov-Galerkin Methods for the Timoshenko Beam Problem. Comput. Meth. Appl. Mech. Eng.63, 133-154 (1987) · Zbl 0607.73076
[18] Loula, A.F.D., Miranda, I., Hughes, T.J.R., Franca, L.P.: A Successful Mixed Formulation for Axisymmetric Shell Analysis Employing Discontinuous Stress Fields of the Same Order as the Displacement Field. Fourth Brazilian Symposium on Piping and Pressure Vessels, Salvador, Brazil, October 1986
[19] Oden, J.T., Jacquotte, O.P.: Stability of Some Mixed Finite Element Methods for Stokesian Flows. Comput. Methods Appl. Mech. Eng.43, 231-247 (1984) · Zbl 0598.76033
[20] Oden, J.T., Kikuchi, N., Song, Y.J.: Penalty Finite Element Methods for the Analysis of Stokesian Flows. Comput. Methods Appl. Mech. Eng.31, 297-329 (1982) · Zbl 0487.76040
[21] Reissner, E.: On a Variational Theorem in Elasticity. J. Math. Phys.29, 90-95 (1950) · Zbl 0039.40502
[22] Sani, R., Gresho, P.M., Lee, R.L., Griffiths, D.F.: The Cause and Cure (?) of the Spurious Pressures Generated by Certain FEM Solutions of the Incompressible Navier-Stokes Equations, Part 1. Int. J. Numer. Methods Fluids1, 17-43 (1981) · Zbl 0461.76021
[23] Sani, R., Gresho, P.M., Lee, R.L., Griffiths, D.F., Engelman, M.: The Cause and Cure (?) of the Spurious Pressures Generated by Certain FEM Solutions of the Incompressible Navier-Stokes Equations, Part 2. Int. J. Numer. Methods Fluids1, 171-204 (1981) · Zbl 0461.76022
[24] Taylor, C., Hood, P.: Numerical Solution of the Navier-Stokes Equations Using the Finite Element Technique. Comput. Fluids1, 1-28 (1973) · Zbl 0328.76020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.