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.

翻译：伽辽金法的二次NURBS离散在应用于Kirchhoff-Love壳公式时会出现膜锁闭现象。膜锁闭不仅导致位移小于预期值，还会引起膜力的大幅值伪振荡。连续假定应变（CAS）单元最近被引入以消除线性平面曲梁二次NURBS离散中的膜锁闭（Casquero等人，CMAME，2022）。本研究将CAS单元推广至线性Kirchhoff-Love壳的二次NURBS离散，以消除膜锁闭。CAS单元对每个单元的四角膜应变进行双线性插值，因此假定应变在单元边界上具有C0连续性。据作者所知，CAS单元是首个满足以下计算效率重要特征、有效克服Kirchhoff-Love壳二次NURBS离散中膜锁闭的假定应变处理方法：(1) 不增加额外自由度，(2) 无需求解额外代数方程组，(3) 无需矩阵乘法或求逆即可获得刚度矩阵，(4) 保留刚度矩阵的非零模式。基准算例表明，采用每单元2×2或3×3高斯-勒让德积分点的CAS单元能有效消除锁闭——该单元类型在粗网格下可得到更精确的位移，并消除膜力的伪振荡。基准算例还表明，CAS单元的性能优于采用假定应变或减缩积分锁闭处理的基于拉格朗日多项式的先进单元类型。