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Hybrid-mixed shell finite elements and implicit dynamic schemes for shell post-buckling. (English) Zbl 07270538
Altenbach, Holm (ed.) et al., Recent developments in the theory of shells. Dedicated to Wojciech Pietraszkiewicz on the occasion of his 80th birthday. Cham: Springer (ISBN 978-3-030-17746-1/hbk; 978-3-030-17747-8/ebook). Advanced Structured Materials 110, 383-412 (2019).
Summary: Two topics are addressed: (i) hybrid-mixed formulations for geometrically exact shell models, and (ii) post-buckling analysis of shells by implicit dynamics schemes. As for the hybrid-mixed elements, seven formulations are compared. The one with the assumed-natural-strain interpolation of membrane strains shows very little sensitivity to mesh distortion for curved shells. Another one, which is based on the Hu-Washizu three-field functional, allows for very large solution increments. Hence, a new element is proposed that combines positive features of both mentioned formulations. As for the post-buckling analysis of shells, we use implicit dynamics. In particular, five time-stepping schemes are tested for shell stability problems that include mode jumping. These are trapezoidal rule, schemes with numerical dissipation in the high-frequency range, and energy-momentum conserving method. Numerical examples show that the dissipative schemes are suitable for simulation of complex phenomena that appear in shell stability.
For the entire collection see [Zbl 1429.74005].
74S05 Finite element methods applied to problems in solid mechanics
74K25 Shells
74H60 Dynamical bifurcation of solutions to dynamical problems in solid mechanics
74H55 Stability of dynamical problems in solid mechanics
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[1] Andelfinger, U., Ramm, E.: EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements. Int. J. Numer. Meth. Eng. 36, 1311-1337 (1993) · Zbl 0772.73071
[2] Armero, F., Romero, I.: On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part I: low-order methods for two model problems and nonlinear elastodynamics. Comput. Methods Appl. Mech. Eng. 190, 2603-2649 (2001) · Zbl 1008.74035
[3] Betsch, P., Stein, E.: An assumed strain approach avoiding artificial thickness straining for a nonlinear 4-node shell element. Commun. Numer. Methods Eng. 11, 899-909 (1995) · Zbl 0833.73051
[4] Betsch, P., Gruttmann, F., Stein, E.: A 4-node finite shell element for the implementation of general hyperelastic 3D-elsticity at finite strains. Comput. Methods Appl. Mech. Eng. 130(1-2), 57-79 (1996) · Zbl 0861.73068
[5] Brank, B., Ibrahimbegovic, A.: On the relation between different parametrizations of finite rotations for shells. Eng. Comput. 18, 950-973 (2001) · Zbl 1017.74069
[6] Brank, B.: Assessment of 4-node EAS-ANS shell elements for large deformation analysis. Comput. Mech. 42, 39-51 (2008) · Zbl 1161.74049
[7] Brank, B., Carrera, E.: Multilayered shell finite element with interlaminar continuous shear stresses: a refinement of the Reissner-Mindlin formulation. Int. J. Numer. Meth. Eng. 48(6), 843-874 (2000) · Zbl 0991.74066
[8] Brank, B., Korelc, J., Ibrahimbegovic, A.: Dynamics and time-stepping schemes for elastic shells undergoing finite rotations. Comput. Struct. 81, 1193-1210 (2003)
[9] Brank, B.: Nonlinear shell models with seven kinematic parameters. Comput. Methods Appl. Mech. Eng. 194, 2336-2362 (2005) · Zbl 1082.74050
[10] Brank, B., Briseghella, L., Tonello, N., Damjanić, F.B.: On non-linear dynamics of shells: Implementation of energy-momentum conserving algorithm for a finite rotation shell model. Int. J. Numer. Meth. Eng. 42, 409-442 (1998) · Zbl 0926.74071
[11] Brank, B., Mamouri, S., Ibrahimbegović, A.: Constrained finite rotations in dynamics of shells and Newmark implicit time-stepping schemes. Eng. Comput. 22(5/6), 505-535 (2005) · Zbl 1186.74111
[12] Brank, B., Perić, D., Damjanić, F.B.: On large deformations of thin elasto-plastic shells: Implementation of a finite rotation model for quadrilateral shell element. Int. J. Numer. Meth. Eng. 40, 689-726 (1997) · Zbl 0892.73055
[13] Choi, C.K., Paik, J.G.: An effective four node degenerated shell element for geometrically nonlinear analysis. Thin-Walled Struct. 24(3), 261-283 (1996)
[14] Chung, J., Hulbert, G.M.: A time integration algorithm for structural dynamics with improved numerical dissipation—the generalized-alpha method. J. Appl. Mech.-Trans. ASME 60, 371-375 (1993) · Zbl 0775.73337
[15] Crisfield, M.A., Peng, X.: Instabilities induced by coarse meshes for a nonlinear shell problem. Eng. Comput. 13(6), 110-114 (1996) · Zbl 0983.74575
[16] Crisfield, M.A.: Non-linear Finite Element Analysis of Solids and Structures, Vol. 2, Advanced topics. Wiley (1996)
[17] Dvorkin, E.N., Bathe, K.J.: A continuum mechanics based four-node shell element for general nonlinear analysis. Eng. Comput. 1, 77-88 (1984)
[18] Goto, Y., Watanabe, Y., Kasugai, T., Obata, M.: Elastic buckling phenomenon applicable to deployable rings. Int. J. Solids Struct. 29(7), 893-909 (1992)
[19] Gruttmann, F., Wagner, W.: A linear quadrilateral shell element with fast stiffness computation. Comp. Methods Appl. Mech. Eng. 194, 4279-4300 (2005) · Zbl 1151.74418
[20] Gruttmann, F., Wagner, W.: Structural analysis of composite laminates using a mixed hybrid shell element. Comput. Mech. 37, 479-497 (2006) · Zbl 1158.74358
[21] Ibrahimbegović, A., Brank, B., Courtois, P.: Stress resultant geometrically exact form of classical shell model and vector‐like parameterization of constrained finite rotations. Int. J. Numer. Methods Eng. 52(11), 1235-1252 (2001) · Zbl 1112.74420
[22] Klinkel, S., Gruttmann, F., Wagner, W.: A mixed shell dormulation accounting for thickness strains and finite strain 3d material models. Int. J. Numer. Meth. Eng. 74, 945-970 (2008) · Zbl 1158.74491
[23] Ko, Y., Lee, P.S., Bathe, K.J.: A new MITC4+ shell element. Comput. Struct. 182, 404-418 (2017)
[24] Ko, Y., Lee, P.S., Bathe, K.J.: The MITC4+ shell element in geometric nonlinear analysis. Comput. Struct. 185, 1-14 (2017)
[25] Kobayashi, T., Mihara, Y., Fujii, F.: Path-tracing analysis for post-buckling process of elastic cylindrical shells under axial compression. Thin-walled Struct. 61, 180-187 (2012)
[26] Korelc, J., Wriggers, P.: Automation of Finite Element Methods. Springer International Publishing (2016) · Zbl 1367.74001
[27] Kuhl, D., Ramm, E.: Generalized energy-momentum method for non-linear adaptive shell dynamics. Comput. Methods Appl. Mech. Eng. 178, 343-366 (1999) · Zbl 0968.74030
[28] Kuhl, D., Crisfield, M.A.: Energy-conserving and decaying algorithms in non-linear structural dynamics. Int. J. Numer. Meth. Eng. 45, 569-599 (1999) · Zbl 0946.74078
[29] Kulikov, G.M., Plotnikova, S.V.: A family of ANS four-node exact geometry shell elements in general convected curvilinear coordinates. Int. J. Numer. Meth. Eng. 83(10), 1376-1406 (2010) · Zbl 1202.74174
[30] Lavrenčič, M., Brank, B.: Simulation of shell buckling by implicit dynamics and numerically dissipative schemes. Thin-walled Struct. 132, 682-699 (2018)
[31] Lavrenčič, M.: Complete animations of buckling processes available from: http://fgg-web.fgg.uni-lj.si/ /mlavrenc/ (2018)
[32] Newmark, N.M.: Method of computation for structural dynamics. Press. Vessel. Piping Des. Anal. 2, 1235-1264 (1972)
[33] Oesterle, B., Sachse, R., Ramm, E., Bischoff, M.: Hierarchic isogeometric large rotation shell elements including linearized transverse shear parametrization. Comput. Methods Appl. Mech. Eng. 321, 383-405 (2017) · Zbl 1439.74457
[34] Pian, T.H.H., Sumihara, K.: Rational approach for assumed stress finite elements. Int. J. Numer. Meth. Eng. 20(9), 1685-1695 (1984) · Zbl 0544.73095
[35] Pietraszkiewicz, W.: Lagrangian description and incremental formulation in the non-linear theory of thin shells. Int. J. Non-Linear Mech. 19(2), 115-140 (1984) · Zbl 0537.73050
[36] Pietraszkiewicz, W., Eremeyev, V.A.: On vectorially parameterized natural strain measures of the nonlinear Cosserat continuum. Int. J. Solids Struct. 46, 2477-2480 (2009) · Zbl 1217.74012
[37] Schieck, B., Pietraszkiewicz, W., Stumpf, H.: Theory and numerical analysis of shells undergoing large elastic strains. Int. J. Solids Struct. 29(6), 689-709 (1992) · Zbl 0760.73044
[38] Simo, J.C., Fox D.D.: On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parametrization. Comput. Methods Appl. Mech. Eng. 72, 267-304 (1989) · Zbl 0692.73062
[39] Simo, J.C., Fox, D.D., Rifai, M.S.: On a stress resultant geometrically exact shell model. Part III: computational aspects of the nonlinear theory. Comput. Methods Appl. Mech. Eng. 79, 21-70 (1990) · Zbl 0746.73015
[40] Simo, J.C., Rifai, M.S.: A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Meth. Eng. 29, 1595-1638 (1990) · Zbl 0724.73222
[41] 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
[42] Simo, J.C., Rifai, M.S., Fox, D.D.: On a stress resultant geometrically exact shell model. Part IV: variable thickness shells with through-the-thickness stretching. Comput. Methods Appl. Mech. Eng. 81, 91-126 (1990) · Zbl 0746.73016
[43] Simo, J.C., Tarnow, N.: A new energy and momentum conserving algorithm for the nonlinear dynamics of shells. Int. J. Numer. Meth. Eng. 37, 2527-2549 (1994) · Zbl 0808.73072
[44] Stanić, A., Brank, B., Korelc, J.: On path-following methods for structural failure problems. Comput. Mech. 58, 281-306 (2016) · Zbl 1398.74407
[45] Sze, K.Y., Liu, X.H., Lo, S.H.: Popular benchmark problems for geometric nonlinear analysis of shells. Finite Elem. Anal. Des. 40, 1551-1569 (2004)
[46] Wagner, W., Gruttmann, F.: A robust non-linear mixed hybrid quadrilateral shell element. Int. J. Numer. Meth. Eng. 64, 635-666 (2005) · Zbl 1122.74526
[47] Wiśniewski, K.: Finite Rotation Shells, Basic Equations and Finite Elements for Reissner Kinematics. Springer, Netherlands (2010) · Zbl 1201.74004
[48] Wisniewski, K., Turska, E.: Improved 4-node Hu-Washizu elements based on skew coordinates. Comput. Struct. 87, 407-424 (2009)
[49] Xu, F., Potier-Ferry, M.: On axisymmetric/diamond-like mode transitions in axially compressed core-shell cylinders. J. Mech. Phys. Solids 94, 68-87 (2016)
[50] Yamaki, N.: Elastic Stability of Circular Cylindrical Shells. North-Holland, Netherlands (1984) · Zbl 0544.73062
[51] Zhao, Y. · Zbl 1349.74255
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