Abstract
We present neat and efficient computer code implementation of two types of Discrete-Kirchhoff plate finite elements—the Discrete-Kirchhoff triangle and the Discrete-Kirchhoff quadrilateral—which can be used to model numerous thin plate problems in mechanics and biology. We also present an implicit a posteriori discretization error indicator computation, based on the superconvergent patch recovery technique, for the Discrete-Kirchhoff plate finite elements. This error indicator can drive an adaptive meshing algorithm providing the most suitable finite element mesh. For an illustration, some numerical results are given.
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References
Ainsworth, M., Oden, J.T.: A unified approach to a posteriori error estimation using element residual methods. Numer. Math. 65(1), 23–50 (1993)
Ainsworth, M, Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York, (2000)
Altenbach, H., Altenbach, J., Naumenko, K.: Ebene Flächentragwerke. Springer, Berlin (1998)
Altenbach, H., Eremeyev, V.: Direct approach based analysis of plates composed of functionally graded materials. Arch. Appl. Mech. 78(5), 775–794 (2008)
Argyris, J.H., Fried, I., Scharpf, D.W.: The TUBA family of plate elements for the matrix displacement method. Aeronaut. J. Royal Aeronaut. Soc. 72, 701–709 (1968)
Batoz, J.L., Bathe, K.J., Ho, L.W.: A study of three-node triangular plate bending elements. Int. J. Num. Meth. Eng. 15, 1771–1812 (1980)
Bohinc, U., Ibrahimbegovic, A., Brank, B.: Model adaptivity for finite element analysis of thin or thick plates based on equilibrated boundary stress resultants. Eng. Comput. 26(1/2), 69–99 (2009)
Bohinc, U., Brank, B., Ibrahimbegović, A.: Discretization error for the discrete Kirchhoff plate finite element approximation. Comput. Methods Appl. Mech. Eng. 269(1), 415–436 (2014)
Brank, B.: On boundary layer in the Mindlin plate model: Levy plates. Thin-walled Struct. 46(5), 451–465 (2008)
Clough, R.W, Tocher, J.L.: Finite element stiffness matrices for analysis of plate bending. Proceedings of Conference on Matrix Methods in Structural Mechanics, WPAFB, Ohio, pp. 66–80 (1965)
Felippa, C.A, Clough, R.: A refined quadrilateral element for analysis of plate bending. Proceedings of Conference on Matrix Methods in Structural Mechanics, WPAFB, Ohio, pp. 23–69 (1965)
Geuzaine, C., Remacle, J.F.: Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Num. Meth. Eng. 79(11), 1309–1331 (2009)
Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover publications, Minelo (2000)
Ibrahimbegovic, A.: Quadrilateral finite elements for analysis of thick and thin plates. Comput. Methods Appl. Mech. Eng. 110, 195–209 (1993)
Ibrahimbegovic, A.: Nonlinear Solid Mechanics: Theoretical Formulations and Finite Element Solution Methods. Springer, Berlin (2009)
Korelc, J.: AceGen and AceFEM user manuals (2006)
Ladevèze, P., Leguillon, D.: Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal. 20(3), 485–509 (1983)
Lee, C.K., Hobbs, R.E.: Automatic adaptive refinement for plate bending problems using reissner-mindlin plate bending elements. Int. J. Num. Meth. Eng. 41, 1–63 (1998)
Lee, K.H., Lim, G.T., Wang, C.M.: Thick Levy plates re-visited. Int. J. Solids Struct. 39, 127–144 (2002)
Morley, L.S.D.: Skew Plates and Structures. Pergamon Press, Oxford (1963)
Naumenko, K., Altenbach, J., Altenbach, H., Naumenko, V.: Closed and approximate analytical solutions for rectangular mindlin plates. Acta Mech. 147, 153–172 (2001)
Reddy, J., Wang, C.: An overview of the relationships between solutions of classical and shear deformation plate theories. Compos. Sci. Technol. 60, 2327–2335 (2000)
Taylor, R.: FEAP-A finite element analysis program, programmer manual. University of California, Berkeley, http://www.ce.berkeley.edu/rltedn (2013)
Taylor, R.L, Govindjee, S.: Solution of clamped rectangular plate problems. Report UCB/SEMM-2002/09 (2002)
Taylor, R.L.: FEAP-a finite element analysis program-Version 7.3. University of California, Berkeley (2000)
Timoshenko, S., Woinowsky-Krieger, S.: Theory of Plates and Shells. McGraw-Hill, New York (1959)
Yunus, S.M., Pawlak, T.P., Wheeler, M.J.: Application of the Zienkiewicz-Zhu error estimator for plate and shell analysis. Int. J. Num. Meth. Eng. 29(6), 1281–1298 (1990)
Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique. Int. J. Numer. Meth. Eng. 33, 1331–1364 (1992)
Zienkiewicz, O.C., Taylor, R.L.: Finite Element Method. Elsevier, London (2000)
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Brank, B., Ibrahimbegović, A., Bohinc, U. (2015). On Discrete-Kirchhoff Plate Finite Elements: Implementation and Discretization Error. In: Altenbach, H., Mikhasev, G. (eds) Shell and Membrane Theories in Mechanics and Biology. Advanced Structured Materials, vol 45. Springer, Cham. https://doi.org/10.1007/978-3-319-02535-3_6
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DOI: https://doi.org/10.1007/978-3-319-02535-3_6
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