Maio 2014 vol. 1 num. 1 - 10th World Congress on Computational Mechanics

Full Article - Open Access.

Idioma principal

ON THE CHOICE OF THE DIRECTOR FOR ISOGEOMETRIC REISSNER-MINDLIN SHELL ANALYSIS

Dornisch, W. ; Klinkel, S. ;

Full Article:

The contribution deals with the treatment of the director vector and the rotational degrees of freedom in isogeometric Reissner-Mindlin shell analysis. In NURBS-based formulations the director vectors and the nodal coordinate systems are defined at the control points. The interpolation with the basis functions maps them to the shell reference surface. For curved geometries the interpolated director at an arbitrary point is in general neither perpendicular with respect to the shell surface nor has the correct length. Thus, basic kinematic assumptions are violated. It is shown that this is one reason for the observed degradation of numerical results for shells when the polynomial order gets higher than five. In the present work a method to overcome these problems is provided. A geometrically exact Reissner-Mindlin shell formulation with five degrees of freedom, three displacements and two rotations, is presented. The two rotations are defined with respect to the orthogonal tangent vectors at the shell reference surface. In standard formulations the nodal coordinate system which also defines the director is constructed from the derivatives in the closest point projection of the control point. It is observed that error norms of angle and length of the interpolated director increase with the order of the basis functions. To improve the accuracy the nodal coordinate systems and directors are chosen in a way, that the interpolated basis systems in all integration points are orthonormal and in correct orientation. The existence of an unique solution is shown. This leads to nodal coordinate systems which are no longer orthonormal. An interpolation, which ensures orthonormality at the integration points, is introduced. The finite element implementation is tested with standard benchmark problems. In contrast to computations with nodal directors being the normal to the shell surface, deformations converge monotonically. In all computed examples the accuracy of the solution for a given mesh increases with the order of the basis functions. Simplifications in the underlying shell theory inhibit exponential convergence rates for curved domains. However, the proposed measures significantly improve the accuracy of computations with high order basis functions.

Full Article:

Download (PDF)

Palavras-chave: Isogeometric Analysis, Reissner-Mindlin Shells, Approximation of the Director, NURBS.,

Palavras-chave:

DOI: 10.5151/meceng-wccm2012-18100

Referências bibliográficas
  • [1] Bazilevs Y., Beirão da Veiga L., Cottrell J. A., Hughes T. J. R., Sangalli G., “Isogeometric analysis: Approximation, stability and error estimates for h-refined meshes”. Math. Mod. Methods Appl. Sci. 16, 1-60, 2006.
  • [2] Belytschko T., Stolarski H., Liu W. K., Carpenter N., Ong J. S.-J., “Stress projection for membrane and shear locking in shell finite elements”. Comput. Meth. Appl. Mech. Engrg. 51, 221-258, 1985.
  • [3] Benson D. J., Bazilevs Y., Hsu M.-C., Hughes T. J. R., “A large deformation, rotationfree, isogeometric shell”. Comput. Meth. Appl. Mech. Engrg. 200, 1367-1378, 2011.
  • [4] Benson D. J., Bazilevs Y., Hsu M.-C., Hughes T. J. R., “Isogeometric shell analysis: The Reissner-Mindlin shell”. Comput. Meth. Appl. Mech. Engrg. 199, 276-289, 2010.
  • [5] Bischoff M., Bletzinger K.-U., Wall W. A., Ramm E., “Models and finite elements for thin-walled structure”. Stein E.; Borst R. de; Hughes T. J. R. (Ed.): Encyclopedia of computational mechanics, Vol. 2. Wiley, Chichester, 2004.
  • [6] Cottrell J. A., Hughes T. J. R., Bazilevs Y., “Isogeometric analysis: Toward integration of CAD and FEA”, Wiley, Chichester, 2009.
  • [7] Gruttmann F., Wagner W., “Structural analysis of composite laminates using a mixed hybrid shell element”. Comput. Mech. 37, 479-497, 2006.
  • [8] Hughes T. J. R., Cottrell J. A., Bazilevs Y., “Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement”. Comput. Meth. Appl. Mech. Engrg. 194, 4135-4195, 2005.
  • [9] Kiendl J., Bletzinger K.-U., Linhard J., Wüchner R., “Isogeometric shell analysis with Kirchhoff-Love elements”. Comput. Meth. Appl. Mech. Engrg. 198, 3902-3914, 200
  • [10] Lawson C. L., Hanson R. J., “Solving least squares problems (Volume 15 of Classics in applied mathematics)”, SIAM, Philadelphia, 1995.
  • [11] Simo J. C., Fox D. D., “On a stress resultant geometrically exact shell model. part I: Formulation and optimal parametrization”. Comput. Meth. Appl. Mech. Engrg. 72, 267-304, 1989.
  • [12] 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. Meth. Appl. Mech. Engrg. 79, 21-70, 1990.
  • [13] Piegl, L., Tiller W., “The NURBS book (Monographs in visual communications)”, Springer, Berlin, second edition, 1997.
Como citar:

Dornisch, W.; Klinkel, S.; "ON THE CHOICE OF THE DIRECTOR FOR ISOGEOMETRIC REISSNER-MINDLIN SHELL ANALYSIS", p. 727-739 . In: In Proceedings of the 10th World Congress on Computational Mechanics [= Blucher Mechanical Engineering Proceedings, v. 1, n. 1]. São Paulo: Blucher, 2014.
ISSN 2358-0828, DOI 10.5151/meceng-wccm2012-18100

últimos 30 dias | último ano | desde a publicação


downloads


visualizações


indexações