Keywords: isogeometric analysis, shells, NURBS, benchmark study.

This paper constitutes the first steps in our work concerning isogeometric shell analysis. An isogeometric shell model of the Reissner-Mindlin type is introduced and a study of its accuracy in the classical pinched cylinder benchmark problem presented. In contrast to earlier works [1,2,3,4], the formulation is based on a shell model where the displacement, strain and stress fields are defined in terms of a curvilinear coordinate system arising from the NURBS description of the shell middle surface.

The isogeometric shell formulation is implemented using the PetIGA and igakit software packages developed by the authors. The igakit package is a Python package used to generate NURBS representations of geometries that can be utilised by the PetIGA finite element framework. The latter utilises data structures and routines of the portable, extensible toolkit for scientific computation (PETSc), [5,6]. The current shell implementation is valid for static, linear problems only, but the software package is well suited for future extensions to geometrically and materially nonlinear regime as well as to dynamic problems.

The accuracy of the approach in the pinched cylinder benchmark problem and present comparisons against the h-version of the finite element method with bilinear elements. Quadratic, cubic and quartic NURBS discretizations are compared against the isoparametric bilinear discretization introduced in [7]. The results show that the quadratic and cubic NURBS approximations exhibit notably slower convergence under uniform mesh refinement as the thickness decreases but the quartic approximation converges relatively quickly within the standard variational framework.

The authors future work is concerned with building an isogeometric finite element method for modelling nonlinear structural response of thin-walled shells undergoing large rigid-body motions. The aim is to use the model in a aeroelastic framework for the simulation of flapping wings.

1

T. Hughes, J. Cottrell, Y. Bazilevs, "Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement", Computer Methods in Applied Mechanics and Engineering, 194(39-41), 4135-4195, 2005. doi:10.1016/j.cma.2004.10.008

2

Y. Bazilevs, V.M. Calo, J.A. Cotrell, J.A. Evans, S. Lipton, T.J.R. Hughes, M.A. Scott, T.W. Sederberg, "Isogeometric analysis using T-splines", Computer Methods in Applied Mechanics and Engineering, 199, 229-263, 2010. doi:10.1016/j.cma.2009.02.036

3

D. Benson, Y. Bazilevs, M. Hsu, T. Hughes, "Isogeometric shell analysis: The Reissner-Mindlin shell", Computer Methods in Applied Mechanics and Engineering, 199(5-8), 276-289, 2010. doi:10.1016/j.cma.2009.05.011

4

D. Benson, Y. Bazilevs, M.C. Hsu, T. Hughes, "A large deformation, rotation-free, isogeometric shell", Computer Methods in Applied Mechanics and Engineering, 200(13-16), 1367-1378, 2011. doi:10.1016/j.cma.2010.12.003

5

S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, "PETSc Web page", 2010. http://www.mcs.anl.gov/petsc

6

S. Balay, K. Buschelman, V. Eijkhout, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, "PETSc Users Manual", Technical Report ANL-95/11 - Revision 3.0.0, Argonne National Laboratory, 2008.

7

A.H. Niemi, "A bilinear shell element based on a refined shallow shell model", International Journal for Numerical Methods in Engineering, 81, 485-512, 2010.

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