Quadratic NURBS-based discretizations of the Galerkin method suffer from volumetric locking when applied to nearly-incompressible linear elasticity. Volumetric locking causes not only smaller displacements than expected, but also large-amplitude spurious oscillations of normal stresses. 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 propose two generalizations of CAS elements (named CAS1 and CAS2 elements) to overcome volumetric locking in quadratic NURBS-based discretizations of nearly-incompressible linear elasticity. CAS1 elements linearly interpolate the strains at the knots in each direction for the term in the variational form involving the first Lam\'e parameter while CAS2 elements linearly interpolate the dilatational strains at the knots in each direction. For both element types, a displacement vector with C1 continuity across element boundaries results in assumed strains with C0 continuity across element boundaries. In addition, the implementation of the two locking treatments proposed in this work does not require any additional global or element matrix operations such as matrix inversions or matrix multiplications. The locking treatments are applied at the element level and the nonzero pattern of the global stiffness matrix is preserved. The numerical examples solved in this work show that CAS1 and CAS2 elements, using either two or three Gauss-Legrendre quadrature points per direction, are effective locking treatments since they not only result in more accurate displacements for coarse meshes, but also remove the spurious oscillations of normal stresses.

翻译：伽辽金法的二次NURBS离散化在应用于近似不可压线弹性问题时会出现体积锁闭。体积锁闭不仅导致位移小于预期值，还会引起法向应力的大振幅伪振荡。连续假定应变（CAS）单元近期被引入以消除基于二次NURBS的线性平面曲率基尔霍夫杆离散化中的薄膜锁闭（Casquero等人，CMAME，2022）。本文提出两种CAS单元的推广形式（命名为CAS1和CAS2单元），以克服近似不可压线弹性二次NURBS离散化中的体积锁闭。CAS1单元在变分形式中涉及第一拉梅参数的项中，对各方向的节点应变进行线性插值；而CAS2单元则对各方向的节点体应变进行线性插值。对于两种单元类型，跨单元边界具有C1连续性的位移向量将生成跨单元边界具有C0连续性的假定应变。此外，本文提出的两种锁闭处理方法在实现过程中无需任何额外的全局或单元矩阵运算（如矩阵求逆或矩阵乘法）。锁闭处理在单元层面实施，且全局刚度矩阵的非零模式得以保持。本文求解的数值算例表明：无论每个方向采用两点还是三点高斯-勒让德求积，CAS1和CAS2单元均为有效的锁闭处理方法，其不仅能在粗网格下获得更精确的位移，还能消除法向应力的伪振荡。