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Selected Topics on Mixed/Enhanced Four-Node Shell Elements with Drilling Rotation

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Shell-like Structures

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 572))

Abstract

In these lecture notes, we describe basic features of the mixed/ enhanced four-node shell elements with six dofs/node based on the Hu-Washizu (HW) functional, developed for Green strain. The focus in on the following features:

  1. 1.

    Derivation of the so-called incomplete (partial) HW functionals for shells, with different treatment of the bending/twisting part and the transverse shear part of strain energy. This is an alternative to the derivation from the three-dimensional HW functional, and it allows to reduce the number of elemental parameters.

  2. 2.

    Selection of parameters of the assumed fields and selection of an enhancements for the HW shell elements, with the purpose to improve accuracy for distorted meshes. This includes also the use of the so-called skew coordinates, which are associated with the natural basis at the element center.

  3. 3.

    Numerical treatment of the drilling Rotation Constraint equation by the Perturbed Lagrange method. The faulty term resulting from the equal-order approximations of displacements and the drilling rotation is eliminated and one spurious mode is stabilized using the gamma method.

  4. 4.

    A simple additive/multiplicative scheme of treating finite rotations is described and tested on numerical examples. This simplified scheme is consistent with the typical update scheme used by FE codes.

The quality of the proposed formulation is demonstrated using the Hu-Washizu shell element with 29 parameters (HW29), which has a very good accuracy and is insensitive to the shape distortions for coarse meshes. Besides, it exhibits an excellent convergence and robustness in nonlinear examples.

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Notes

  1. 1.

    Note that the RC equation can also be used to constrain parameters of the 2nd order kinematics of shells, see Wisniewski and Turska (2000, 2001, 2002).

  2. 2.

    Note that the definition of Eq. (18) is not valid for the \( \mathbf {N}^* \) used here!

  3. 3.

    The tilde denotes the skew-symmetric tensor associated with the rotation vector, i.e. \( \tilde{\varvec{\theta }} \doteq \varvec{\theta }\times \mathbf {I}\).

  4. 4.

    The use of these programs is gratefully acknowledged.

  5. 5.

    S16 is the 16-node shell element based on bi-cubic Lagrangian shape functions.

  6. 6.

    2m is used in the circumferential direction.

References

  • Allman, D. J. (1984). A compatible triangular element including vertex rotations for plane elasticity analysis. Computers & Structures, 19(2), 1–8.

    Article  MATH  Google Scholar 

  • Altenbach, J., Altenbach, H., & Eremeyev, V. A. (2010). On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Archive of Applied Mechanics, 80, 73–92.

    Article  MATH  Google Scholar 

  • Altman, S. L. (1986). Rotations, quaternions and double groups. Oxford: Clarendon Press.

    Google Scholar 

  • Angeles, J. (1988). Rational kinematics (Vol. 34). Springer Tracts in Natural Philosophy New York: Springer.

    MATH  Google Scholar 

  • Atluri, S. N., & Cazzani, A. (1995). Rotations in computational solid mechanics. Archives of Computational Methods in Engineering, 2(1), 49–138.

    Article  MathSciNet  Google Scholar 

  • Badur, J., & Pietraszkiewicz, W. (1986). On geometrically non-linear theory of elastic shells derived from pseudo-Cosserat continuum with constrained micro-rotations. In W. Pietraszkiewicz (Ed.), Finite rotations in structural mechanics (pp. 19–32). Berlin: Springer.

    Chapter  Google Scholar 

  • Bathe, K.-J., & Dvorkin, E. N. (1985). A four-node plate bending element based on Mindlin-Reissner plate theory and mixed interpolation. International Journal for Numerical Methods in Engineering, 21, 367–383.

    Article  MATH  Google Scholar 

  • Bergan, P. G., & Felippa, C. A. (1985). A triangular membrane element with rotational degrees of freedom. Computer Methods in Applied Mechanics and Engineering, 50, 25–60.

    Article  MATH  Google Scholar 

  • Brendel, B., & Ramm, E. (1980). Linear and nonlinear stability analysis of cylindrical shell. Computers & Structures, 12, 549–558.

    Article  MATH  Google Scholar 

  • Buechter, N., & Ramm, E. (1992). Shell theory versus degeneration–A comparison in large rotation finite element analysis. International Journal for Numerical Methods in Engineering, 34, 39–59.

    Article  MATH  Google Scholar 

  • Cardona, A., & Geradin, M. (1988). A beam finite element nonlinear theory with finite rotations. International Journal for Numerical Methods in Engineering, 26, 2403–2438.

    Article  MathSciNet  MATH  Google Scholar 

  • Cartan, E. (1981). The theory of spinors. New York: Dover.

    MATH  Google Scholar 

  • Chróścielewski, J., & Witkowski, W. (2011). FEM analysis of Cosserat plates and shells based on some constitutive relations. Z. Angew. Math. Mech., 91, 400–412.

    Article  MathSciNet  MATH  Google Scholar 

  • Chróścielewski, J., Makowski, J., & Stumpf, H. (1992). Genuinely resultant shell finite elements accounting for geometric and material nonlinearity. International Journal for Numerical Methods in Engineering, 35, 63–94.

    Article  MathSciNet  MATH  Google Scholar 

  • Chróścielewski, J., Kreja, I., & Sabik, A. (2011). Modeling of composite shells in 6-parameter nonlinear theory with drilling degree of freedom. Mechanics of Advanced Materials and Structures, 18(6), 403–419.

    Article  Google Scholar 

  • Cook, R. D. (1986). On the Allman triangle and a related quadrilateral element. A plane hybrid element with rotational d.o.f. and adjustable stiffness. Computers & Structures, 22(6), 1065–1067.

    Article  Google Scholar 

  • deBoer, R. (1982). Vektor- und Tensorrechnung für Ingenieure. Berlin: Springer.

    MATH  Google Scholar 

  • Eremeyev, V. A., Lebedev, L. P., & Altenbach, H. (2012). Foundations of micropolar mechanics. Heidelberg: Springer.

    MATH  Google Scholar 

  • Flanagan, D. P., & Belytschko, T. (1981). A uniform strain hexahedron and quadrilateral with orthogonal hourglass control. International Journal for Numerical Methods in Engineering, 17, 679–706.

    Article  MATH  Google Scholar 

  • Geradin, M., & Cardona, A. (2001). Flexible multibody dynamics. New York: Wiley.

    MATH  Google Scholar 

  • Geradin, M., & Rixen, D. (1995). Parametrization of finite rotations in computational dynamics. A review. European Journal of Finite Elements, 4(Special Issue: Ibrahimbegovic A., Geradin M. (eds.)), 497–553.

    Google Scholar 

  • Goldstein, H. (1980). Classical mechanics (2nd ed.). Reading, MA: Addison Wesley.

    MATH  Google Scholar 

  • Gruttmann, F., & Wagner, W. (2006). Structural analysis of composite laminates using a mixed hybrid shell element. Computer Methods, 37, 479–497.

    MATH  Google Scholar 

  • Hughes, T. J. R., & Brezzi, F. (1989). On drilling degrees of freedom. Computer Methods in Applied Mechanics and Engineering, 72, 105–121.

    Article  MathSciNet  MATH  Google Scholar 

  • Korelc, J. (2002). Multi-language and multi-environment generation of nonlinear finite element codes. Engineering with Computers, 18, 312–327.

    Article  Google Scholar 

  • MacNeal, R. H., & Harder, R. L. (1988). A refined four-noded membrane element with rotational degrees of freedom. Computers & Structures, 28(1), 75–84.

    Article  MATH  Google Scholar 

  • Makowski, J., & Stumpf, H. (1995). On the “symmetry” of tangent operators in nonlinear mechanics. Z. Angew. Math. Mech., 75(3), 189–198.

    Article  MathSciNet  MATH  Google Scholar 

  • N.N. Abaqus Software, Ver. 6.13-3. Dassault Systemes Simulia, 2013.

    Google Scholar 

  • Panasz, P., & Wisniewski, K. (2008). Nine-node shell elements with 6 dofs/node based on two-level approximations. Finite Elements in Analysis and Design, 44, 784–796.

    Article  Google Scholar 

  • Pian, T. H. H., & Sumihara, K. (1984). Rational approach for assumed stress finite elements. International Journal for Numerical Methods in Engineering, 20, 1685–1695.

    Article  MATH  Google Scholar 

  • Rosenberg, R. M. (1977). Analytical dynamics of discrete systems. New York: Plenum Press.

    Book  MATH  Google Scholar 

  • Simo, J. C. (1992). The (symmetric) Hessian for geometrically nonlinear models in solid mechanics: Intrinsic definition and geometric interpretation. Computer Methods in Applied Mechanics and Engineering, 96, 189–200.

    Article  MathSciNet  MATH  Google Scholar 

  • Simo, J. C., & Armero, F. (1992). Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering, 33, 1413–1449.

    Article  MathSciNet  MATH  Google Scholar 

  • Simo, J. C., & Vu-Quoc, L. (1986). On the dynamics of 3-d finite strain rods. In Finite element methods for plate and shell structures (Vol. 2). Formulations and Algorithms (pp. 1–30). Swansea: Pineridge Press.

    Google Scholar 

  • Simo, J. C., & Wong, K. K. (1991). Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum. Computer Methods in Applied Mechanics and Engineering, 31, 19–52.

    MathSciNet  MATH  Google Scholar 

  • Simo, J. C., Fox, D. D., & Hughes, T. J. R. (1992). Formulations of finite elasticity with independent rotations. International Journal for Numerical Methods in Engineering, 95, 227–288.

    MathSciNet  MATH  Google Scholar 

  • Spring, K. W. (1986). Euler parameters and the use of quaternion algebra in the manipulation of finite rotations: a review. Mechanism and Machine Theory, 21(5), 365–373.

    Article  Google Scholar 

  • Stuelpnagel, J. (1964). On the parametrization of three-dimensional rotational group. SIAM Review, 6(4), 422–430.

    Article  MathSciNet  MATH  Google Scholar 

  • Taylor, R. L. (2014). Program FEAP, Ver.8.4. Berkeley: University of California.

    Google Scholar 

  • Taylor, R. L. (1988). Finite element analysis of linear shell problems. In J. R. Whiteman (Ed.), The mathematics of finite elements and applications VI (pp. 191–203)., MAFELAP 1987 London: Academic Press.

    Google Scholar 

  • Taylor, R. L., Beresford, P. J., & Wilson, E. L. (1976). A non-conforming element for stress analysis. International Journal for Numerical Methods in Engineering, 10, 1211–1220.

    Article  MATH  Google Scholar 

  • Wagner, W., & Gruttmann, F. (2005). A robust nonlinear mixed hybrid quadrilateral shell element. International Journal for Numerical Methods in Engineering, 64, 635–666.

    Article  MATH  Google Scholar 

  • Wilson, E. L., Taylor, R. L., Doherty, W. P., & Ghaboussi, J. (1973). Incompatible displacement models. In S. J. Fenves, N. Perrone, A. R. Robinson, & W. C. Schnobrich (Eds.), Numerical and computer methods in finite element analysis (pp. 43–57). New York: Academic Press.

    Google Scholar 

  • Wisniewski, K. (2010a). Recent improvements in formulation of mixed and mixed/enhanced shell elements. In W. Pietraszkiewicz & I. Kreja (Eds.), Shell structures. Theory and applications. Proceedings of 9th Conference SSTA’2009, Jurata, October 14–16, 2009, pp. 35–44. Milton Park, Abingdon, Oxford, Taylor & Francis.

    Google Scholar 

  • Wisniewski, K. (2010b). Finite rotation shells. Basic equations and finite elements for reissner kinematics. Heidelberg: Springer.

    MATH  Google Scholar 

  • Wisniewski, K., & Turska, E. (2000). Kinematics of finite rotation shells with in-plane twist parameter. Computer Methods in Applied Mechanics and Engineering, 190(8–10), 1117–1135.

    Article  MathSciNet  MATH  Google Scholar 

  • Wisniewski, K., & Turska, E. (2001). Warping and in-plane twist parameter in kinematics of finite rotation shells. Computer Methods in Applied Mechanics and Engineering, 190(43–44), 5739–5758.

    Article  MathSciNet  MATH  Google Scholar 

  • Wisniewski, K., & Turska, E. (2002). Second order shell kinematics implied by rotation constraint equation. Journal of Elasticity, 67, 229–246.

    Article  MathSciNet  MATH  Google Scholar 

  • Wisniewski, K., & Turska, E. (2006). Enhanced Allman quadrilateral for finite drilling rotations. Computer Methods in Applied Mechanics and Engineering, 195(44–47), 6086–6109.

    Article  MATH  Google Scholar 

  • Wisniewski, K., & Turska, E. (2008). Improved four-node Hellinger-Reissner elements based on skew coordinates. International Journal for Numerical Methods in Engineering, 76, 798–836.

    Article  MathSciNet  MATH  Google Scholar 

  • Wisniewski, K., & Turska, E. (2009). Improved four-node Hu-Washizu elements based on skew coordinates. Computers & Structures, 87, 407–424.

    Article  Google Scholar 

  • Wisniewski, K., & Turska, E. (2012). Four-node mixed Hu-Washizu shell element with drilling rotation. International Journal for Numerical Methods in Engineering, 90, 506–536.

    Article  MathSciNet  MATH  Google Scholar 

  • Wisniewski, K., Turska, E. (2013). On mixed/enhanced Hu-Washizu shell elements with drilling rotation. In W. Pietraszkiewicz & J. Górski, (Eds.), Shell structures. Theory and applications. Proceedings of 10th Conference SSTA’2013, Gdañsk, October 16–18, Vol. 3, pp. 469–472, Milton Park, Abingdon, Oxford, 2014. CRC Press/Balkema, Taylor & Francis Group.

    Google Scholar 

  • Wisniewski, K., Wagner, W., Turska, E., & Gruttmann, F. (2010). Four-node Hu-Washizu elements based on skew coordinates and contravariant assumed strain. Computers & Structures, 88, 1278–1284.

    Article  Google Scholar 

  • Wittenburg, J. (1977). Dynamics of systems of rigid bodies. Stuttgart: B.G. Teubner.

    Book  MATH  Google Scholar 

  • Zienkiewicz, O. C., & Taylor, R. L. (1989). The Finite element method (4th edn., Vol. 1). Basic Formulation and Linear Problems. New York: McGraw-Hill.

    Google Scholar 

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Authors and Affiliations

  1. IFTR, Polish Academy of Sciences (IPPT PAN), Warsaw, Poland

    K. Wisniewski

  2. Polish Japanese Institute of Information Technology (PJWSTK), Warsaw, Poland

    E. Turska

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  1. K. Wisniewski
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  2. E. Turska
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Correspondence to K. Wisniewski .

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Editors and Affiliations

  1. Faculty of Mechanical Engineering, Otto-von-Guericke-University Magdeburg, Magdeburg, Germany

    Holm Altenbach

  2. Technical Mechanics, Otto-von-Guericke-University Magdeburg, Magdeburg, Germany

    Victor Eremeyev

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Wisniewski, K., Turska, E. (2017). Selected Topics on Mixed/Enhanced Four-Node Shell Elements with Drilling Rotation. In: Altenbach, H., Eremeyev, V. (eds) Shell-like Structures. CISM International Centre for Mechanical Sciences, vol 572. Springer, Cham. https://doi.org/10.1007/978-3-319-42277-0_6

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  • DOI: https://doi.org/10.1007/978-3-319-42277-0_6

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