The bending strip method for isogeometric analysis of Kirchhoff–Love shell structures comprised of multiple patches J. Kiendl a, ⁎, Y. Bazilevs b , M.-C. Hsu b , R. Wüchner a , K.-U. Bletzinger a a Lehrstuhl für Statik, Technische Universität München, Arcisstr. 21, München 80333, Germany b Department of Structural Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA abstract article info Article history: Received 24 December 2009 Received in revised form 15 March 2010 Accepted 17 March 2010 Available online 9 April 2010 Keywords: Isogeometric analysis Rotation-free shells Kirchhoff–Love theory Multiple patches Wind turbine rotor Shell–solid coupling In this paper we present an isogeometric formulation for rotation-free thin shell analysis of structures comprised of multiple patches. The structural patches are C 1 - or higher-order continuous in the interior, and are joined with C 0 -continuity. The Kirchhoff–Love shell theory that relies on higher-order continuity of the basis functions is employed in the patch interior as presented in Kiendl et al. [36]. For the treatment of patch boundaries, a method is developed in which strips of fictitious material with unidirectional bending stiffness and zero membrane stiffness are added at patch interfaces. The direction of bending stiffness is chosen to be transverse to the patch interface. This choice leads to an approximate satisfaction of the appropriate kinematic constraints at patch interfaces without introducing additional stiffness to the shell structure. The attractive features of the method include simplicity of implementation and direct applicability to complex, multi-patch shell structures. The good performance of the bending strip method is demonstrated on a set of benchmark examples. Application to a wind turbine rotor subjected to realistic wind loads is also shown. Extension of the bending strip approach to the coupling of solids and shells is proposed and demonstrated numerically. © 2010 Elsevier B.V. All rights reserved. 1. Introduction Isogeometric Analysis was first proposed in [33] as a technology that has the potential to bridge the gap between design and analysis. Non- Uniform Rational B-Splines (NURBS) were employed as the first basis function technology in isogeometric analysis and is currently the most developed one. NURBS-based isogeometric analysis was applied with great success to the study of solids, structures, fluids, fluid–structure interaction, turbulence, phase field modeling, and structural optimiza- tion [1,3,5–8,10,12–14,26,29,31,32,34,45,47]. Mathematical theory of NURBS-based isogeometric analysis was originally developed in [9]. Further refinements and insights into approximation properties of NURBS were studied in [30] using the concepts of Kolmogorov n-widths theory. Recent developments in isogeometric analysis include modeling and simulation of shell structures [17,18,36], coordinated synthesis of hierarchical engineering systems [44], isogeometric model quality assessment and improvement [24,38,46], applications to incompress- ible elasticity [2], and T-splines [4,28]. This article further develops the application of isogeometric analysis to shell structures that make use of the Kirchhoff–Love shell theory. The Kirchhoff–Love shell theory assumes that a cross- section normal to the middle surface of the shell remains normal to the middle surface during the deformation, which implies that transverse shear strains are negligible. This theory is appropriate for thin shells (20 ≤ R/t, where R is the shell radius of curvature and t is its thickness) [19]. Most shell structures of practical engineering interest satisfy this criterion. Thin shells have an optimal load-carrying behavior and therefore allow the construction of highly efficient light-weight structures [20,22]. In the governing mechanical varia- tional equations of the Kirchoff–Love theory, second order derivatives appear, and therefore C 1 -continuity of the approximation functions is required for the discrete formulation to be conforming. NURBS basis functions have the necessary smoothness at the patch level. NURBS are inherently higher order, which also alleviates locking associated with low order shell discretizations. The attractive feature of the Kirchhoff–Love theory is that the formulation is purely displacement based and no rotational degrees of freedom are necessary [23,37,39]. In the previous works on rotation-free isogeometric shell analysis [17,36], the authors showed that NURBS are able to attain very good accuracy and are efficient for shell structures. However, the develop- ments were confined to single-patch shell structures, or a very limited class of multi-patch shell structures with simple linear constraints between control points to maintain a conforming discretization. In this paper, we propose a formulation that is rotation-free and is capable of handling a significantly larger class of structures composed of an arbitrary number of patches and their relative orientations. The method consists of adding strips of material in places where the NURBS patches are joined with C 0 -continuity and has similarities with Computer Methods in Applied Mechanics and Engineering 199 (2010) 2403–2416 ⁎ Corresponding author. E-mail address: kiendl@bv.tum (J. Kiendl). 0045-7825/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2010.03.029 Contents lists available at ScienceDirect Computer Methods in Applied Mechanics and Engineering journal homepage: www.elsevier.com/locate/cma