Isogeometric MITC shell

Abstract

This paper proposes an isogeometric formulation of the Reissner–Mindlin shell, using the Mixed Interpolation of Tensorial Components (MITC) technique to alleviate shear locking and membrane locking. Instead of over each Non-Uniform Rational B-Spline (NURBS) element, kinematics of the shell is directly formulated on the entire NURBS patch, with its displacements decoupled to the motions of control points and of director vectors tied to the control points. Since control points are in general not located on the surface, those control-point director vectors are interpolated from normal vectors at a set of pre-defined integration points on the patch. The assumed in-plane membrane strain field and the assumed transverse shear strain field are built as linear combinations of NURBS basis functions with degrees lower than those employed by the displacement interpolation, and tied to the original covariant strain fields at a set of well-selected tying points. Integration points for the definition of control-point director vectors and tying points of the assumed covariant strain fields are provided by the optimal and reduced macro-element quadrature rules, respectively. In this way, the classical MITC technique is fully incorporated into the isogeometric framework with arbitrary polynomial orders or patch configurations. Thanks to the high smoothness of NURBS basis functions, the geometry-preserving nature of isogeometric transformation, and the locking-resistive capability of MITC technique, the isogeometric MITC shell formulation shows superior convergence behavior, minimal geometrical error, and good robustness against patch distortion. In particular, its convergence property is insensitive to NURBS polynomial order or control point density. These advantages are demonstrated through a number of well-established benchmark problems, validating the proposed isogeometric MITC formulation being an accurate and efficient solution for linear analysis of shell structures.