Towards the Formulation of Effective General Shell Elements

  • Dominique Chapelle
  • Klaus-Jürgen Bathe
Part of the Computational Fluid and Solid Mechanics book series (COMPFLUID)


In Chapter 7 we discussed the difficulties encountered in the formulation of reliable and effective shell elements. These difficulties are summarized in the synopsis of Figure 8.1 in correspondence with the various types of shell asymptotic behaviors that can be encountered, as addressed in Chapter 5. The objective of the present chapter is to propose some strategies to evaluate shell finite element discretizations in the search for improved schemes. With general analytical proofs not available for the convergence behavior, the numerical assessment is a key ingredient in these strategies. As an example we present the formulation of the MITC shell elements and demonstrate how the numerical assessment of these elements can be performed.


Shell Element Convergence Curve Nite Element Membrane Norm Quadratic Rate 
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  1. Lee, P.S., & Bathe, K.J. (2004). Development of MITC isotropic triangular shell finite elements. Comput. & Structures, 82, 945-962.CrossRefGoogle Scholar
  2. Chapelle, D., & Bathe, K.J. (1993). The inf-sup test. Comput. & Structures, 47(4/5), 537-545.CrossRefzbMATHMathSciNetGoogle Scholar
  3. Kim, D.N., & Bathe, K.J. (2008). A 4-node 3D-shell element to model shell surface tractions and incompressible behavior. Comput. & Structures, 86(21-22), 2027-2041.CrossRefGoogle Scholar
  4. Malinen, M. (2001). On the classical shell model underlying bilinear degenerated shell finite elements. Internat. J. Numer. Methods Engrg., 52, 389-416.CrossRefzbMATHGoogle Scholar
  5. Chapelle, D., & Paris Suarez, I. (2008). Detailed reliability assessment of triangular MITC elements for thin shells. Comput. & Structures, 86, 2192-2202. Doi:10.1016/j.compstruc.2008.06.001.CrossRefGoogle Scholar
  6. Bathe, K.J., & Lee, P.S. (201x). Measuring the convergence behavior of shell analysis schemes. In preparation.Google Scholar
  7. Malinen, M., & Pitkäranta, J. (2000). A benchmark study of reduced-strain shell finite elements: quadratic schemes. Internat. J. Numer. Methods Engrg., 48, 1637-1671.CrossRefzbMATHMathSciNetGoogle Scholar
  8. Pitkäranta, J., Matache, A.M., & Schwab, C. (2001). Fourier mode analysis of layers in shallow shell deformations. Comput. Methods Appl. Mech. Engrg., 190, 2943-2975.CrossRefzbMATHGoogle Scholar
  9. Lee, P.S., & Bathe, K.J. (2010). The quadratic MITC plate and MITC shell elements in plate bending. Advances in Engineering Software, in Press.Google Scholar
  10. Bathe, K.J. (1996). Finite Element Procedures. Englewood Cliffs: Prentice Hall.Google Scholar
  11. Havu, V., & Pitkäranta, J. (2002). Analysis of a bilinear finite element for shallow shells. I: Approximation of inextensional deformations. Math. Comp., 71, 923-943.CrossRefzbMATHMathSciNetGoogle Scholar
  12. Bucalem, M.L., & Bathe, K.J. (1993). Higher-order MITC general shell elements. Internat. J. Numer. Methods Engrg., 36, 3729-3754.CrossRefzbMATHMathSciNetGoogle Scholar
  13. Lee, P.S., Noh, H.C., & Bathe, K.J. (2007). Insight into 3-node triangular shell finite elements: the effect of element isotropy and mesh patterns. Comput. & Structures, 85, 404-418.CrossRefGoogle Scholar
  14. Bathe, K.J., Chapelle, D., & Lee, P.S. (2003a). A shell problem `highly Sensitive’ to thickness changes. Internat. J. Numer. Methods Engrg., 57, 1039-1052.CrossRefzbMATHGoogle Scholar
  15. Pitkäranta, J., Leino, Y., Ovaskainen, O., & Piila, J. (1995). Shell deformation states and the finite element method: a benchmark study of cylindrical shells. Comput. Methods Appl. Mech. Engrg., 128, 81-121.CrossRefzbMATHMathSciNetGoogle Scholar
  16. Bathe, K.J. (2009). The finite element method. In B.Wah (Ed.) Encyclopedia of Computer Science and Engineering, (pp. 1253-1264). John Wiley & Sons.Google Scholar
  17. Karamian, P., Sanchez-Hubert, J., & Sanchez-Palencia, E. (2000). A model problem for boundary layers of thin elastic shells. M2AN Math. Model. Numer. Anal., 34(1), 1-30.CrossRefzbMATHMathSciNetGoogle Scholar
  18. Dvorkin, E.N., & Bathe, K.J. (1984). A continuum mechanics based fournode shell element for general non-linear analysis. Eng. Comput., 1, 77-88.CrossRefGoogle Scholar
  19. Ciarlet, P.G. (1978). The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland.zbMATHGoogle Scholar
  20. Chapelle, D., & Bathe, K.J. (1998). Fundamental considerations for the finite element analysis of shell structures. Comput. & Structures, 66, 19-36, 711-712.CrossRefzbMATHGoogle Scholar
  21. Betsch, P., & Stein, E. (1995). An assumed strain approach avoiding artificial thickness straining for a non-linear 4-node shell element. Commun. Numer. Meth. Engng., 11, 899-909.CrossRefzbMATHGoogle Scholar
  22. Chapelle, D. (2001). Some new results and current challenges in the finite element analysis of shells. In Acta Numerica, (pp. 215-250). Cambridge: Cambridge University Press.Google Scholar
  23. Bathe, K.J., Hiller, J.F., & Zhang, H. (2002). On the finite element analysis of shells and their full interaction with Navier-Stokes fluid flows. In B. Topping, & Z. Bittnar (Eds.) Computational Structures Technology. Edinburgh: Civil-Comp Press.Google Scholar
  24. Arnold, D.N., & Falk, R.S. (1996). Asymptotic analysis of the boundary layer for the Reissner-Mindlin plate model. SIAM J. Math. Anal., 27(2), 486-514.CrossRefzbMATHMathSciNetGoogle Scholar
  25. Beirão da Veiga, L., Chapelle, D., & Paris Suarez, I. (2007). Towards improving the MITC6 triangular shell element. Comput. & Structures, 85, 1589-1610.CrossRefGoogle Scholar
  26. Dauge, M., & Yosibash, Z. (2000). Boundary layer realization in thin elastic 3D domains and 2D hierarchic plate models. Internat. J. Solids Structures, 37, 2443-2471.CrossRefzbMATHGoogle Scholar
  27. Niemi, A.H. (2009). A bilinear shell element based on a refied shallow shell model. Internat. J. Numer. Methods Engrg., 81(4), 485-512.MathSciNetGoogle Scholar
  28. Bathe, K.J., & Dvorkin, E.N. (1986). A formulation of general shell elements|the use of mixed interpolation of tensorial components. Internat. J. Numer. Methods Engrg., 22, 697-722.CrossRefzbMATHGoogle Scholar
  29. Bathe, K.J., Bucalem, M.L., & Brezzi, F. (1990). Displacement and stress convergence of our MITC plate bending elements. Eng. Comput., 7, 291-302.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Dominique Chapelle
    • 1
  • Klaus-Jürgen Bathe
    • 2
  1. 1.INRIA Paris-RocquencourtLe ChesnayFrance
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

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