zbMATH — the first resource for mathematics

On a stress resultant geometrically exact shell model. II: The linear theory; computational aspects. (English) Zbl 0724.73138
Summary: Computational aspects of a linear stress resultant (classical) shell theory obtained by systematic linearization of the geometrically exact nonlinear theory, considered in Part I of this work [the first two authors, Comput. Methods Appl. Mech. Eng. 72, No.3, 267-304 (1989; Zbl 0692.73062)], are examined in detail. In particular, finite element interpolations for the reference director field and the linearized rotation field are constructed such that the underlying geometric structure of the continuum theory is preserved exactly by the discrete approximation. A discrete canonical, singularity-free mapping between the five and the six degree of freedom formulation is constructed by exploiting the geometric connection between the orthogonal group (SO(3)) and the unit sphere \((S^ 2).\)
The proposed numerical treatment of the membrane and bending fields, based on a mixed Hellinger-Reissner formulation, provides excellent results for the 4-node bilinear isoparametric element. As an example, convergent results are obtained for rather coarse meshes in fairly demanding, singularity-dominated problems such as the classical rhombic plate test. The proposed theory and finite element implementation are evaluated through an extensive set of benchmark problems. The results obtained with the present approach exactly match previous solutions obtained with state-of-the-art implementations based on the so-called degenerated solid approach.

74K15 Membranes
74S05 Finite element methods applied to problems in solid mechanics
74P99 Optimization problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
Full Text: DOI
[1] Arnold, D.N.; Falk, R.S., A uniformly accurate finite element method for the Mindlin-Reissner plate, SIAM J. numer. anal., (1988), (to appear)
[2] Bathe, K.J., Finite element procedures in engineering analysis, (1982), Prentice-Hall Englewood Cliff, NJ · Zbl 0528.65053
[3] Bathe, K.J.; Brezzi, F.; Bathe, K.J.; Brezzi, F., On the convergence of a four-node plate bending element based on mindlin—reissner plate theory and a mixed interpolation, (), 491-503 · Zbl 0589.73068
[4] Bathe, K.J.; Brezzi, F., A simplified analysis of two plate bending elements—the MITC4 and MITC9 elements, () · Zbl 0589.73068
[5] Bathe, K.J.; Dvorkin, E.N., A continuum mechanics based four-node shell element for general non-linear analysis, Internat. J. comput. aided engrg. software, 1, (1984)
[6] Belytschko, T.; Stolarski, H.; Liu, W.K.; Carpenter, N.; Ong, J.S.-J., Stress projection for membrane and shear locking in shell finite elements, Comput. meths. appl. mech. engrg., 51, 221-258, (1985) · Zbl 0581.73091
[7] Belytschko, T.; Tsay, C.S., A stabilization procedure for the quadrilateral plate element with one-point quadrature, Internat. J. numer. meths. engrg., 19, 405-420, (1983) · Zbl 0502.73058
[8] Belytschko, T.; Ong, J.S.-J.; Liu, W.K.; Kennedy, J.M., Hourglass control in linear and nonlinear problems, Comput. meths. appl. mech. engrg., 43, 251-276, (1984) · Zbl 0522.73063
[9] Brezzi, F.; Fortin, M., Numerical approximation of Mindlin-Reissner plates, Math. comput., 42, 175, 151-158, (1985) · Zbl 0596.73058
[10] Budiansky, B.; Sanders, J.L., On the “best” first-order linear shell theory, Prog. appl. mech., 20, 129-140, (1963)
[11] Huang, H.C.; Hinton, E., A nine-node Lagrangian plate element with enhanced shear interpolation, Engrg. comput., 1, 369-379, (1984)
[12] Hughes, T.J.R., Generalization of selective integgration procedures to anisotropic and nonlinear media, Internat. J. numer. meths. engrg., 15, 9, 1413-1418, (1980) · Zbl 0437.73053
[13] Hughes, T.J.R.; Cohen, M.; Haroun, M., Reduced and selective integration techniques in the finite element analysis of plates, Nucl. engrg. design, 46, 203-222, (1978)
[14] Hughes, T.J.R.; Liu, W.K., Nonlinear finite element analysis of shells: part I. three-dimensional shells, Comput. meths. appl. mech. engrg., 26, 331-362, (1981) · Zbl 0461.73061
[15] Hughes, T.J.R.; Liu, W.K., Nonlinear finite element analysis of shells: part II. two-dimensional shells, Comput. meths. appl. mech. engrg., 27, 167-182, (1981)
[16] Hughes, T.J.R.; Tezduyar, T.E., Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element, J. appl. mech., 587-596, (1981) · Zbl 0459.73069
[17] Hughes, T.J.R., The finite element method, (1987), Prentice-Hall Englewood-Cliffs, NJ
[18] J.H. Jang and P. Pinsky, A nine-node assumed covariant strain shell element, Internat. J. Numer. Meths. Engrg. · Zbl 0623.73090
[19] Liu, K.K.; Law, E.S.; Lam, D.; Belytschko, T., Resultant-stress degenerated-shell element, Comput. meths. appl. mech. engrg., 55, 259-300, (1986) · Zbl 0587.73113
[20] MacNeal, R.H., A simple quadrilateral shell element, Comput. & structures, 8, 175-183, (1978) · Zbl 0369.73085
[21] MacNeal, R.H., Derivation of element stiffness matrices by assumed strain distribution, Nucl. engrg. design, 70, 3-12, (1982)
[22] MacNeal, R.H.; Harder, R.L., A proposed standard set of problems to test finite element accuracy, ()
[23] Morely, L.S.D., Skew plates and structures, international series of monographs in aeronautics and astronautics, (1963), MacMillan New York
[24] Morely, L.S.D.; Morris, A.J., Conflict between finite elements and shell theory, (1978), Royal Aircraft Establishment London, Rept.
[25] Naghdi, P.M., The theory of shells, () · Zbl 0154.22602
[26] Niordson, F.I., Shell theory, north-holland series in applied mathematics and mechanics, (1985), North-Holland Amsterdam
[27] Parks, K.C.; Stanley, G.M., A curved \(C\^{}\{0\}\) shell element based on assumed natural-coordinate strains, J. appl. mech., 53, 2, 278-290, (1986) · Zbl 0588.73137
[28] Parisch, H., A critical survey of the 9-node degenerated shell element with special emphasis on thin shell application and reduced integration, Comput. meths. appl. mech. engrg., 20, 323-350, (1979) · Zbl 0419.73076
[29] Pian, T.H.H.; Sumihara, K., Rational approach for assumed stress finite elements, Internat. J. numer. meths. engrg., 20, 1685-1695, (1984) · Zbl 0544.73095
[30] Sanders, J.L., An improved first-approximation theory for thin shells, ()
[31] Scordellis, A.C.; Lo, K.S., Computer analysis of cylindrical shells, J. amer. concr. inst., 61, 539-561, (1969)
[32] Simo, J.C.; Fox, D.D., On a stress resultant geometrically exact shell model. part I: formulation and optimal parametrization, Comput. meths. appl. mech. engrg., (1988) · Zbl 0692.73062
[33] Simo, J.C.; Marsden, J.E.; Krishnaprassad, P.S., The Hamiltonian structure of elasticity. the convective representation of solids, rods and plates, Arch. rat. mech. anal., (1987), (to appear)
[34] Simo, J.C.; Quoc, L.V., A three-dimensional finite strain rod model. part II: geometric and computational aspects, Comput. meths. appl. mech. engrg., 58, 79-116, (1986) · Zbl 0608.73070
[35] Simo, J.C.; Vu-Quoc, L., A beam model including shear and torsional warping distortions based on an exact geometric description of nonlinear deformations, Internat. J. solids and structures, (1987), (to appear)
[36] Simo, J.C.; Vu-Quoc, L., On the dynamics in space of rods undergoing large motions—a geometrically exact approach, Comput. meths. appl. mech. engrg., (1987), (to appear) · Zbl 0618.73100
[37] Simo, J.C.; Hughes, T.J.R., On the variational foundations of assumed strain methods, J. appl. mech., 53, 1, 51-54, (1986) · Zbl 0592.73019
[38] Stanley, G., Continuum-based shell elements, ()
[39] C.R. Steele, Private Communication, 1987.
[40] Stolarski, H.; Belytschko, T., Membrane locking and reduced integration for curved elements, J. appl. mech., 49, 172-176, (1982) · Zbl 0482.73060
[41] Stolarski, H.; Belytschko, T., Shear and membrane locking in curved \(C\^{}\{0\}\) elements, Comput. meths. appl. mech. engrg., 41, 279-296, (1983) · Zbl 0509.73072
[42] Taylor, R.L., Finite element analysis of linear shell problems, () · Zbl 0682.73048
[43] Taylor, R.L.; Simo, J.C.; Zienkiewicz, O.C.; Chan, A.C., The patch test: A condition for assessing finite element convergence, Internat. J. numer. meths. engrg., 22, 1, 39-62, (1986) · Zbl 0593.73072
[44] Vu-Quoc, L.; Mora, J.A., A class of simple and efficient degenerated shell elements, () · Zbl 0687.73071
[45] Zienkiewicz, O.C., The finite element method, (1978), McGraw-Hill New York · Zbl 0391.76004
[46] Zienkiewicz, O.C.; Taylor, R.L.; Too, J.M., Reduced integration technique in general analysis of plates and shells, Internat. J. numer. meths. engrg., 3, 275-290, (1971) · Zbl 0253.73048
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.