Shear‐flexible subdivision shells

Abstract

SUMMARY We present a new shell model and an accompanying discretisation scheme that is suitable for thin and thick shells. The deformed configuration of the shell is parameterised using the mid‐surface position vector and an additional shear vector for describing the out‐of‐plane shear deformations. In the limit of vanishing thickness, the shear vector is identically zero and the Kirchhoff–Love model is recovered. Importantly, there are no compatibility constraints to be satisfied by the shape functions used for discretising the mid‐surface and the shear vector. The mid‐surface has to be interpolated with smooth C1‐continuous shape functions, whereas the shear vector can be interpolated with C0‐continuous shape functions. In the present paper, the mid‐surface as well as the shear vector are interpolated with smooth subdivision shape functions. The resulting finite elements are suitable for thin and thick shells and do not exhibit shear locking. The good performance of the proposed formulation is demonstrated with a number of linear and geometrically non‐linear plate and shell examples. Copyright © 2012 John Wiley & Sons, Ltd.

Isogeometric collocation methods for the Reissner–Mindlin plate problem

“…For the primal formulation, we employ the standard approach of considering displacement and rotations as primal variables. An alternative approach could be to consider the displacement and the shear strains as primal variables, as shown in the context of Galerkin formulations for shells in [24,25]. For the mixed formulation, instead, shear forces are considered as independent variables additionally to displacement and rotations.…”

Isogeometric shape optimisation of shell structures using multiresolution subdivision surfaces

“…Specifically, Loop subdivision surfaces were used for discretising the Kirchhoff-Love shells and representing their geometry. As an extension of this approach, the treatment of industrially prevalent non-manifold shell * Corresponding author geometries and the inclusion of out-of-plane shear deformations relevant for thicker shells were proposed in [4] and [5], respectively. More recently the isogeometric analysis of shells and beams using NURBS basis functions were introduced in [6,7].…”

Virtual elements for a shear-deflection formulation of Reissner–Mindlin plates

“…Recently, a new approach to solve the Reissner-Mindlin bending problem has been presented in [10] by Beirão da Veiga et al (see also [33,36]). In this case a variational formulation of the plate bending problem is written in terms of shear strain and deflection with the advantage that the "shear locking phenomenon" is avoided.…”

“…An equivalent variational formulation. The variational formulation that will be considered here, was introduced in the context of shells in [33,36] and has been studied in [10] for Reissner-Mindlin plates using Isogeometric Analysis. Now, we note that the equivalent formulation is derived by simply considering the following change of variables: (3) (w, θ) ←→ (w, γ) with θ = ∇w + γ.…”