Analysis and finite element approximation of compressible and incompressible linear isotropic elasticity based upon a variational principle. (English) Zbl 0688.73044

Summary: We propose a variational principle for compressible and incompressible linear isotropic elasticity involving four dependent variables: the strain tensor, the augmented stress tensor, pressure and displacement. The variational formulation derived from the principle is analyzed employing a general result due to F. Brezzi [Revue Franç. Automat. Inform. Rech. Opérat. 8, R-2, 129-151 (1974; Zbl 0338.90047)]. Finite element methods using the Galerkin and SBB methods are shown to converge for a wide family of finite element interpolations.


74S05 Finite element methods applied to problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
65K10 Numerical optimization and variational techniques
49M15 Newton-type methods


Zbl 0338.90047
Full Text: DOI


[1] Herrmann, L.R., Elasticity equations for incompressible and nearly incompressible materials by a variational theorem, Aiaa j., 3, 1896-1900, (1965)
[2] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO anal. numér., R-2, 129-151, (1974) · Zbl 0338.90047
[3] Franca, L.P.; Loula, A.F.D., A new mixed finite element method for the Timoshenko beam problem, () · Zbl 0737.76044
[4] Franca, L.P.; Hughes, T.J.R., Two classes of mixed finite element methods, Comput. methods appl. mech., engrg., 69, 89-129, (1988) · Zbl 0651.65078
[5] Girault, V.; Raviart, P.A., Finite element methods for Navier-Stokes equations, () · Zbl 0396.65070
[6] Franca, L.P.; Hughes, T.J.R.; Loula, A.F.D.; Miranda, I., A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element method, Numer. math., 53, 123-141, (1988) · Zbl 0656.73036
[7] Fortin, M., Old and new finite element methods for incompressible flows, Internat. J. numer. methods fluids, 1, 347-364, (1981) · Zbl 0467.76030
[8] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (1987), Prentice-Hall Englewood Cliffs, NJ
[9] Oden, J.T., R.I.P. methods for Stokesian flows, () · Zbl 0447.76029
[10] 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
[11] Verfürth, R., Error estimates for a mixed finite element approximation of the Stokes equations, RAIRO anal. numer., 18, 2, 175-182, (1984) · Zbl 0557.76037
[12] Franca, L.P., New mixed finite element methods, () · Zbl 0651.65078
[13] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[14] Hughes, T.J.R.; Franca, L.P.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. circumventing the babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. methods appl. mech. engrg., 59, 85-99, (1986) · Zbl 0622.76077
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