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From the Hu-Washizu formulation to the average nodal strain formulation. (English) Zbl 1231.74424
Summary: We present a stabilized finite element method for the Hu-Washizu formulation of linear elasticity based on simplicial meshes leading to the stabilized nodal strain formulation or node-based uniform strain elements. We show that the finite element approximation converges uniformly to the exact solution for the nearly incompressible case.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74B05 Classical linear elasticity 74G65 Energy minimization in equilibrium problems in solid mechanics 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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##### References:
 [1] D.N. Arnold, F. Brezzi, Some new elements for the Reissner-Mindlin plate model, in: Boundary Value Problems for Partial Differential Equations and Applications, Masson, Paris, 1993, pp. 287-292. · Zbl 0817.73058 [2] Bochev, P.B.; Dohrmann, C.R., A computational study of stabilized, low order $$C^0$$ finite element approximations of Darcy equations, Comput. mech., 38, 323-333, (2006) · Zbl 1177.76191 [3] Boffi, D.; Lovadina, C., Analysis of new augmented Lagrangian formulations for mixed finite element schemes, Numer. math., 75, 405-419, (1997) · Zbl 0874.65085 [4] Bonet, J.; Burton, A.J., A simple average nodal pressure tetrahedral element for incompressible and nearly incompressible dynamic explicit applications, Commun. numer. methods engrg., 14, 437-449, (1998) · Zbl 0906.73060 [5] Bonet, J.; Marriott, H.; Hassan, O., Stability and comparison of different linear tetrahedral formulations for nearly incompressible explicit dynamic applications, Int. J. numer. methods engrg., 50, 119-133, (2001) · Zbl 1082.74547 [6] Braess, D., Finite elements. theory, fast solver, and applications in solid mechanics, (2001), Cambridge University Press [7] Braess, D., Finite elements. theory, fast solver, and applications in solid mechanics, (2007), Cambridge University Press [8] Braess, D.; Carstensen, C.; Reddy, B.D., Uniform convergence and a posteriori error estimators for the enhanced strain finite element method, Numer. math., 96, 461-479, (2004) · Zbl 1050.65097 [9] Brenner, S.C.; Sung, L., Linear finite element methods for planar linear elasticity, Math. comput., 59, 321-338, (1992) · Zbl 0766.73060 [10] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer-Verlag New York · Zbl 0788.73002 [11] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North Holland Amsterdam · Zbl 0445.73043 [12] de Souza Neto, E.A.; Andrade Pires, F.M.; Owen, D.R.J., F-bar-based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. part I: formulation and benchmarking, Int. J. numer. methods engrg., 62, 353-383, (2005) · Zbl 1179.74159 [13] Dohrmann, C.R.; Heinstein, M.W.; Jung, J.; Key, S.W.; Witkowski, W.R., Node-based uniform strain elements for three-node triangular and four-node tetrahedral meshes, Int. J. numer. methods engrg., 47, 1549-1568, (2000) · Zbl 0989.74067 [14] Ewing, R.E.; Lin, T.; Lin, Y., On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J. numer. anal., 39, 1865-1888, (2002) · Zbl 1036.65084 [15] Girault, V.; Raviart, P.-A., Finite element methods for navier – stokes equations, (1986), Springer-Verlag Berlin · Zbl 0413.65081 [16] Hu, H., On some variational principles in the theory of elasticity and the theory of plasticity, Sci. sinica, 4, 33-54, (1955) · Zbl 0066.17903 [17] Kozlov, V.A.; Maz’ya, V.G.; Rossmann, J., Spectral problems associated with corner singularities of solutions to elliptic equations, Mathematical surveys and monographs, vol. 85, (2001), American Mathematical Society Providence, RI · Zbl 0965.35003 [18] Lamichhane, B.P., Inf – sup stable finite element pairs based on dual meshes and bases for nearly incompressible elasticity, IMA journal of numerical analysis, 29, 404-420, (2009) · Zbl 1160.74046 [19] Lamichhane, B.P.; Reddy, B.D.; Wohlmuth, B.I., Convergence in the incompressible limit of finite element approximations based on the hu – washizu formulation, Numer. math., 104, 151-175, (2006) · Zbl 1175.74083 [20] Masud, A.; Hughes, T.J.R., A stabilized mixed finite element method for Darcy flow, Comput. methods appl. mech. engrg., 191, 4341-4370, (2002) · Zbl 1015.76047 [21] Andrade Pires, F.M.; de Souza Neto, E.A.; de la Cuesta Padilla, J.L., An assessment of the average nodal volume formulation for the analysis of nearly incompressible solids under finite strains, Commun. numer. methods engrg., 20, 569-583, (2004) · Zbl 1302.74173 [22] Puso, M.A.; Solberg, J., A stabilized nodally integrated tetrahedral, Int. J. numer. methods engrg., 67, 841-867, (2006) · Zbl 1113.74075 [23] Washizu, K., Variational methods in elasticity and plasticity, (1982), Pergamon Press · Zbl 0164.26001
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