Computationally-efficient locking-free isogeometric discretizations of geometrically nonlinear Kirchhoff–Love shells

Abstract

Discretizations based on the Bubnov-Galerkin method and the isoparametric concept suffer from membrane locking when applied to Kirchhoff–Love shell formulations. Membrane locking causes not only smaller displacements than expected, but also large-amplitude spurious oscillations of the membrane forces. Continuous-assumed-strain (CAS) elements were originally introduced to remove membrane locking in quadratic NURBS-based discretizations of linear plane curved Kirchhoff rods (Casquero et al., CMAME, 2022). In this work, we propose ${\text{CAS}}^{\text{s}}$ and ${\text{CAS}}^{\text{ns}}$ elements to overcome membrane locking in quadratic NURBS-based discretizations of geometrically nonlinear Kirchhoff–Love shells. ${\text{CAS}}^{\text{s}}$ and ${\text{CAS}}^{\text{ns}}$ elements are interpolation-based assumed-strain locking treatments. The assumed strains have $C^0$ continuity across element boundaries and different components of the membrane strains are interpolated at different interpolation points. ${\text{CAS}}^{\text{s}}$ elements use the assumed strains to obtain both the physical strains and the virtual strains, which results in a global tangent matrix which is a symmetric matrix. ${\text{CAS}}^{\text{ns}}$ elements use the assumed strains to obtain only the physical strains, which results in a global tangent matrix which is a non-symmetric matrix. To the best of the authors’ knowledge, ${\text{CAS}}^{\text{s}}$ and ${\text{CAS}}^{\text{ns}}$ elements are the first assumed-strain treatments to effectively overcome membrane locking in quadratic NURBS-based discretizations of geometrically nonlinear Kirchhoff–Love shells while satisfying the following important characteristics for computational efficiency: (1) No additional unknowns are added, (2) No additional systems of algebraic equations need to be solved, (3) The same elements are used to approximate the displacements and the assumed strains, (4) No additional matrix operations such as matrix inversions or matrix multiplications are needed to obtain the stiffness matrix, and (5) The nonzero pattern of the stiffness matrix is preserved. Analogously to the interpolation-based assumed-strain locking treatments for Lagrange polynomials that are widely used in commercial FEA software, the implementation of ${\text{CAS}}^{\text{s}}$ and ${\text{CAS}}^{\text{ns}}$ elements only requires to modify the subroutine that computes the element residual vector and the element tangent matrix. The benchmark problems show that ${\text{CAS}}^{\text{s}}$ and ${\text{CAS}}^{\text{ns}}$ elements, using either 2 × 2 or 3 × 3 Gauss–Legendre quadrature points per element, are effective locking treatments since these element types result in more accurate displacements for coarse meshes and excise the spurious oscillations of the membrane forces.