We make use of linear plane curved Timoshenko rods as a model problem to study how to overcome shear and membrane locking in NURBS-based discretizations. In this work, we propose lumped-assumed-strain (LAS) elements, a projection-based assumed-strain treatment that removes shear and membrane locking for a very broad range of slenderness ratios. For a NURBS patch with basis functions of degree and continuity across element boundaries, the space of the assumed strains is defined on a B-spline patch with the same elements, but using basis functions of degree and continuity across element boundaries. The assumed strains are obtained by performing a projection of the compatible strains at the patch level in which the consistent mass matrix is substituted with the lumped mass matrix. Our numerical investigations suggest that LAS elements need either the same or slightly finer mesh resolutions than elements in order for the relative errors in norm of the unknowns and the stress resultants to be all below 1%. However, LAS elements are significantly more computationally efficient than elements for a given mesh. The use of -continuous quadratic LAS elements with 2 Gauss–Legendre quadrature points per element exhibit superconvergence of the stress resultants for most slenderness ratios, making it a particularly cost-effective choice of obtaining accurate results. When compared with linear Lagrange elements with 2 Gauss–Legendre quadrature points per element, -continuous quadratic LAS elements with 2 Gauss–Legendre quadrature points per element require far fewer elements in order to obtain relative errors in norm of the unknowns and the stress resultants smaller than 1%.

中文翻译：

---

线性平面 Timoshenko 杆的无锁定等几何离散化：LAS 单元

我们利用线性平面弯曲 Timoshenko 杆作为模型问题来研究如何克服基于 NURBS 的离散化中的剪切和膜锁定。在这项工作中，我们提出了集总假设应变（LAS）单元，这是一种基于投影的假设应变处理，可以消除非常广泛的长细比范围内的剪切和膜锁定。对于具有跨单元边界的次数和连续性的基函数的 NURBS 面片，假设应变的空间在具有相同单元的 B 样条面片上定义，但使用跨单元边界的次数和连续性的基函数。假设的应变是通过在斑块水平上执行相容应变的投影来获得的，其中一致的质量矩阵被集中质量矩阵取代。我们的数值研究表明，LAS 单元需要与单元相同或稍细的网格分辨率，以便使未知数范数和应力结果的相对误差均低于 1%。然而，LAS 元素的计算效率明显高于给定网格的元素。使用每个单元有 2 个高斯-勒让德求积点的连续二次 LAS 单元，对于大多数长细比而言，应力合成结果具有超收敛性，这使其成为获得准确结果的特别经济有效的选择。与每个单元具有 2 个高斯 - 勒让德求积点的线性拉格朗日单元相比，每个单元具有 2 个高斯 - 勒让德求积点的连续二次 LAS 单元需要少得多的单元才能获得未知数范数的相对误差，并且应力结果更小超过1%。