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Stability, convergence, and accuracy of a new finite element method for the circular arch problem. (English) Zbl 0607.73077
The arch problem with shear deformation based upon the Hellinger-Reissner variational formulation is studied in a parameter-dependent form. A mixed Petrov-Galerkin method is used to construct a discrete approximation. Finite elements with equal-order discontinuous stress and continuous displacement interpolations, unstable in the Galerkin method, are proved to be stable in the new formulation. Error estimates indicate optimal rates of convergence for displacements and suboptimal rates, with gap one, for stresses. Numerical experiments confirm these estimates. The good accuracy of the mixed Petrov-Galerkin method is illustrated in some deep and shallow thin arch examples. No shear or membrane locking is present using full integration schemes.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
65K10 Numerical optimization and variational techniques
74K15 Membranes
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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[1] Ashwell, D.G.; Gallagher, R.H., Finite element for thin shell and curved members, (1976), Wiley New York · Zbl 0397.73062
[2] Ashwell, D.G.; Sabir, A.B., Limitations of certain curved finite elements when applied to arches, Internat. J. mech. sci., 13, 133-139, (1971)
[3] Ashwell, D.G.; Sabir, A.B.; Roberts, T.M., Further studies in the application of curved finite elements to arches, Internat. J. mech. sci., 13, 507-517, (1971)
[4] Arnold, D.N., Discretization by finite elements of a model parameter dependent problem, Numer. math., 37, 405-421, (1981) · Zbl 0446.73066
[5] Babuška, I., Error bounds for finite element method, Numer. math., 16, 322-333, (1971) · Zbl 0214.42001
[6] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO ser. rouge anal. numér., 8, R-2, 129-151, (1974) · Zbl 0338.90047
[7] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[8] Bathe, K.J.; Almeida, C.A., A simple and effective pipe elbow element—interaction effects, J. appl. mech., 49, 165-171, (1982) · Zbl 0482.73061
[9] Cantin, G.; Clough, R.W.; curved, A, Cylindrical shell, finite element, Aiaa j., 6, 1057-1062, (1968) · Zbl 0159.27002
[10] Dawe, D.J., Numerical studies using circular arch finite element, Comput. & structures, 4, 729-740, (1974)
[11] L.P. Franca, T.J.R. Hughes, A.F.D. Loula and I. Miranda, A new family of stable elements for nearly incompressible elasticity based on a Mixed Petrov-Galerkin finite element method, Numer. Math. (to appear). · Zbl 0656.73036
[12] Girault, V.; Raviart, P.A., Finite element approximation of the Navier-Stokes equations, () · Zbl 0396.65070
[13] Hughes, T.J.R.; Franca, L.P.; Balestra, M., Circumventing the babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accomodating equal-order interpolations, Comput. meths. appl. mech. engrg., 59, 86-99, (1986) · Zbl 0622.76077
[14] Kikuchi, F., Accuracy of some finite element models for arch problems, Comput. meths. appl. mech. engrg., 35, 315-345, (1982) · Zbl 0499.73068
[15] Loula, A.F.D.; Hughes, T.J.R.; Franca, L.P.; Miranda, I., Mixed Petrov-Galerkin method for the Timoshenko beam, Comput. meths. appl. mech. engrg., (1987), (to appear).
[16] Loula, A.F.D.; Guerreiro, J.N., Influence of internal pressure in bending of pipes, (), (in Portuguese).
[17] 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, ()
[18] Meck, H.R., An accurate polynomial displacement function for finite ring elements, Comput. & structures, 11, 265-269, (1980) · Zbl 0433.73060
[19] Noor, A.K.; Peters, J.M., Mixed models and reduced/selective integration displacement models for non-linear analysis of curved beams, Internat. J. numer. meths. engrg., 17, 615-631, (1981) · Zbl 0459.73067
[20] Prathap, G., The curved beam/deep arch/finite ring element revisited, Internat. J. numer. meths. engrg., 21, 389-407, (1985) · Zbl 0559.73078
[21] Stolarski, H.; Belytschko, T., Membrane locking and reduced integration of curved elements, J. appl. mech., 49, 172-176, (1982) · Zbl 0482.73060
[22] Stolarski, H.; Belytschko, T., Shear and membrane locking in curved C0 elements, Comput. meths. appl. mech. engrg., 41, 279-296, (1983) · Zbl 0509.73072
[23] Yamada, Y.; Ezawa, Y., On curved finite elements for the analysis of circular arches, Internat. J. numer. meths. engrg., 11, 1635-1661, (1977) · Zbl 0367.73077
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