  Quadratic NURBS-based discretizations of the Galerkin method 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 have been recently 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
generalize CAS elements to vanquish membrane locking in quadratic NURBS-based
discretizations of linear Kirchhoff-Love shells. CAS elements bilinearly
interpolate the membrane strains at the four corners of each element. Thus, the
assumed strains have C0 continuity across element boundaries. To the best of
the authors' knowledge, CAS elements are the first assumed-strain treatment to
effectively overcome membrane locking in quadratic NURBS-based discretizations
of Kirchhoff-Love shells while satisfying the following important
characteristics for computational efficiency: (1) No additional degrees of
freedom are added, (2) No additional systems of algebraic equations need to be
solved, (3) No matrix multiplications or matrix inversions are needed to obtain
the stiffness matrix, and (4) The nonzero pattern of the stiffness matrix is
preserved. The benchmark problems show that CAS elements, using either 2x2 or
3x3 Gauss-Legendre quadrature points per element, are an effective locking
treatment since this element type results in more accurate displacements for
coarse meshes and excises the spurious oscillations of the membrane forces. The
benchmark problems also show that CAS elements outperform state-of-the-art
element types based on Lagrange polynomials equipped with either assumed-strain
or reduced-integration locking treatments.
