## Convergence in the incompressible limit of finite element approximations based on the Hu-Washizu formulation.(English)Zbl 1175.74083

Summary: The classical Hu-Washizu mixed formulation for plane problems in elasticity is examined afresh, with the emphasis on behavior in the incompressible limit. The classical continuous problem is embedded in a family of Hu-Washizu problems parametrized by a scalar $$\alpha$$ for which $$\alpha = \lambda/\mu$$ corresponds to the classical formulation, with $$\lambda$$ and $$\mu$$ being the Lamé parameters. Uniform well-posedness in the incompressible limit of the continuous problem is established for $$\alpha \not= -1$$. Finite element approximations are based on the choice of piecewise bilinear approximations for the displacements on quadrilateral meshes. Conditions for uniform convergence are made explicit. These conditions are shown to be met by particular choices of bases for stresses and strains, and include bases that are well known, as well as newly constructed bases. Though a discrete version of the spherical part of the stress exhibits checkerboard modes, it is shown that a $$\lambda$$-independent a priori error estimate for the displacement can be established. Furthermore, a $$\lambda$$-independent estimate is established for the post-processed stress. The theoretical results are explored further through selected numerical examples.

### MSC:

 74S05 Finite element methods applied to problems in solid mechanics 74B05 Classical linear elasticity 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

UG
Full Text:

### References:

 [1] Andelfinger U., Ramm E.(1993): EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements. Int. J. Numer. Methods. Eng. 36: 1311–1337 · Zbl 0772.73071 [2] Arnold D.N., Scott L.R., Vogelius M.(1988): Regular inversion of the divergence operator with dirichlet boundary conditions on a polygon. Annali Scuola Norm. Sup. Pisa, Serie 4(15): 169–192 · Zbl 0702.35208 [3] Bastian P., Birken K., Johannsen K., Lang S., Neuß N., Rentz–Reichert H., Wieners C. (1997): UG – a flexible software toolbox for solving partial differential equations. Comput. Visual. Sci. 1: 27–40 · Zbl 0970.65129 [4] Braess D.(1996): Stability of saddle point problems with penalty. M 2 AN. 30: 731–742 · Zbl 0860.65054 [5] Braess D.(1998): Enhanced assumed strain elements and locking in membrane problems. Comp. Methods. Appl. Mech. Eng. 165: 155–174 · Zbl 0949.74065 [6] Braess D., Carstensen C., Reddy B.D.(2004): Uniform convergence and a posteriori error estimators for the enhanced strain finite element method. Numer. Math. 96: 461–479 · Zbl 1050.65097 [7] Brenner S.C., Sung L.(1992): Linear finite element methods for planar linear elasticity. Math. Comp. 59: 321–338 · Zbl 0766.73060 [8] Brezzi F., Fortin M.(1991): Mixed and Hybrid Finite Element Methods. Springer, Berlin Heidelberg New York · Zbl 0788.73002 [9] Brezzi F., Fortin M.(2001): A minimal stabilization procedure for mixed finite element methods. Numer. Math. 89: 457–491 · Zbl 1009.65067 [10] Djoko J.K., Lamichhane B.P., Reddy B.D., Wohlmuth B.I.(2006): Conditions for equivalence between the Hu-Washizu and related formulations, and computational behavior in the incompressible limit. Comp. Methods. Appl. Mech. Eng. 195: 4161–4178 · Zbl 1123.74020 [11] Fortin M.(1977): An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numer. 11: 341–354 · Zbl 0373.65055 [12] Fraeijs de Veubeke B.M.(2001): Displacement and equilibrium models. Int. J. Numer. Methods. Eng. 52: 287–342 · Zbl 1065.74625 [13] Girault V., Raviart P.-A.(1986): Finite Element Methods for Navier–Stokes Equations. Springer, Berlin Heidelberg New York · Zbl 0585.65077 [14] Hu H.(1955): On some variational principles in the theory of elasticity and the theory of plasticity. Sci. Sin, 4: 33–54 · Zbl 0066.17903 [15] Kasper E.P., Taylor R.L.(2000): A mixed-enhanced strain method Part I: geometrically linear problems. Comp. Struct. 75: 237–250 [16] Kasper E.P., Taylor R.L.(2000): A mixed-enhanced strain method Part II: geometrically nonlinear problems. Comp. Struct. 75: 251–260 [17] Küssner M., Reddy B.D.(2001): The equivalent parallelogram and parallelepiped, and their application to stabilized finite elements in two and three dimensions. Comp. Methods. Appl. Mech. Eng. 190: 1967–1983 · Zbl 1049.74047 [18] Pian T.H.H., Sumihara K.(1984): Rational approach for assumed stress finite elements. Int. J. Numer. Methods. Eng. 20: 1685–1695 · Zbl 0544.73095 [19] Reddy B.D., Simo J.C.(1995): Stability and convergence of a class of enhanced strain methods. SIAM J. Numer. Anal. 32: 1705–1728 · Zbl 0855.73073 [20] Simo J.C., Rifai M.S.(1990): A class of assumed strain methods and the method of incompatible modes. Int. J. Numer. Methods. Eng. 29: 1595–1638 · Zbl 0724.73222 [21] Vogelius M.(1983): An analysis of the p-version of the finite element method for nearly incompressible materials Uniformly valid, optimal error estimates. Numer. Math. 41, 39–53 · Zbl 0504.65061 [22] Washizu K.(1982): Variational Methods in Elasticity and Plasticity, 3rd edn. Pergamon Press, Oxford · Zbl 0498.73014 [23] Zhou T.-X., Nie Y.-F.(2001): Combined hybrid approach to finite element schemes of high performance. Int. J. Numer. Methods. Eng. 51: 181–202 · Zbl 0983.74078
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.