×

zbMATH — the first resource for mathematics

New rotation-free finite element shell triangle accurately using geometrical data. (English) Zbl 1227.74100
Summary: A new triangle shell element is presented. The advantages of this element are threefold: simplicity, generality and geometrical accuracy. The formulation is free from rotation degrees of freedom. The triangle here presented can be used regardless of the mesh topology, thus generality is conserved for any mesh-represented surface.From an original first order approach we evolve to a third order geometric description. The higher degree geometric description is based on the Bézier triangles concept, a very well known geometry in the domain of CAGD. Using this concept we show the path to reconstruct a general third order interpolating surface using only the three coordinates at each node.This work takes as starting point the nodal implementation of a basic triangle shell element [E. Oñate and F. Zárate, Int. J. Numer. Methods Eng. 47, No. 1–3, 557–603 (2000; Zbl 0968.74070)]. In order to use an exact formula for the curvature, the normal directions at each node and the way to characterize them are proposed. Then, the geometrical properties and the mechanical behavior of the surface created are introduced. Finally, different examples are presented to depict the versatility and accuracy of the element.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74K25 Shells
65D17 Computer-aided design (modeling of curves and surfaces)
Software:
MAT-fem
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] G. Farin, Curves and Surfaces for CAGD. A Practical Guide, fifth ed., Morgan Kaufmann Publishers, San Francisco, CA, 2002.
[2] Oñate, E.; Zárate, F., Rotation-free triangular plate and shell elements, Int. J. numer. methods engrg., 47, 557-603, (2000) · Zbl 0968.74070
[3] Oñate, E., Structural analysis with the finite element method, plates and shells, vol. 2, (2009), CIMNE-Springer
[4] Zienkiewicz, O.; Taylor, R., Finite element method, (2000), Butterworth-Heinemann Oxford, UK · Zbl 0991.74002
[5] Cirak, F.; Ortiz, M.; Schröder, P., Subdivision surfaces: a new paradigm for thin shell finite-element analysis, Int. J. numer. methods engrg., 47, 12, 2039-2072, (2000) · Zbl 0983.74063
[6] Hughes, T.; Cottrell, J.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. methods appl. mech. engrg., 194, 39-41, 4135-4195, (2005) · Zbl 1151.74419
[7] Flores, F.G.; Oñate, E., Improvements in the membrane behaviour of the three node rotation-free BST shell triangle using an assumed strain approach, Comput. methods appl. mech. engrg., 194, 6-8, 907-932, (2005) · Zbl 1112.74510
[8] Oñate, E.; Flores, F., Advances in the formulation of the rotation-free basic shell triangle, Comput. methods appl. mech. engrg., 194, 21-24, 2406-2443, (2005) · Zbl 1083.74048
[9] P.-A. Ubach, E. Oñate, New rotation-free composite shell triangle using accurate geometrical data, in: WCCM VII. 7th World Congress on Computational Mechanics, vol. CD-ROM, UCLA, IACM, Northwestern University, Los Angeles, California, USA, 2006.
[10] P.-A. Ubach, E. Oñate, Advances of the new rotation-free finite element shell triangle using accurate geometrical data, in: 9th US National Congress on Computational Mechanics, vol. CD-ROM, University of California at Berkeley, USACM, San Francisco, California, USA, 2007.
[11] J. Linhard, K.-U. Bletzinger, M. Firl, Upgrading membranes to shells – the CEG rotation free shell element and its applications, in: 9th US National Congress on Computational Mechanics, vol. CD-ROM, University of California at Berkeley, USACM, San Francisco, California, USA, 2007.
[12] Kreyszig, E., Differential geometry, (1991), Dover
[13] Struik, D., Lectures on classical differential geometry, (1988), Dover · Zbl 0697.53002
[14] R. Taylor, E. Oñate, P.-A. Ubach, Textile composites and inflatable structures, Computational Methods in Applied Sciences, vol. 3, Springer, The Netherlands, 2005, Chapter Finite Element Analysis of Membrane Structures, pp. 47-68.
[15] P. de Casteljau, Outillages méthodes calcul, Tech. rep., A. Citroën, Paris, 1959.
[16] P. de Casteljau, Courbes et surfaces á pôles, Tech. rep., A. Citrodn, Paris, 1963.
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.