NURBS-based discretizations of the Galerkin method suffer from membrane locking when applied to primal formulations of curved thin-walled structures. We consider linear plane curved Kirchhoff rods as a model problem to study how to remove membrane locking from NURBS-based discretizations. In this work, we propose continuous-assumed-strain (CAS) elements, an assumed strain treatment that removes membrane locking from quadratic NURBS for an ample range of slenderness ratios. CAS elements take advantage of the $C^1$ inter-element continuity of the displacement vector given by quadratic NURBS to interpolate the membrane strain using linear Lagrange polynomials while preserving the $C^0$ inter-element continuity of the membrane strain. To the authors’ knowledge, CAS elements are the first NURBS-based element type able to remove membrane locking for a broad range of slenderness ratios that combines the following distinctive characteristics: (1) No additional degrees of freedom are added, (2) No additional systems of algebraic equations need to be solved, and (3) The nonzero pattern of the stiffness matrix is preserved. Since the only additional computations required by the proposed element type are to evaluate the derivatives of the basis functions and the unit tangent vector at the knots, the proposed scheme barely increases the computational cost with respect to the locking-prone NURBS-based discretization of the primal formulation. The benchmark problems show that the convergence of CAS elements is independent of the slenderness ratio up to $10^4$ while the convergence of quadratic NURBS elements with full and reduced integration, local $\bar{B}$ elements, and local ANS elements depends heavily on the slenderness ratio and the error can even increase as the mesh is refined. The numerical examples also show how CAS elements remove the spurious oscillations in stress resultants caused by membrane locking while quadratic NURBS elements with full and reduced integration, local $\bar{B}$ elements, and local ANS elements suffer from large-amplitude spurious oscillations in stress resultants. In short, CAS elements are an accurate, robust, and computationally efficient numerical scheme to overcome membrane locking in quadratic NURBS-based discretizations.

Galerkin 方法的基于 NURBS 的离散化在应用于弯曲薄壁结构的原始公式时会受到膜锁定的影响。我们将线性平面弯曲基尔霍夫杆作为模型问题来研究如何从基于 NURBS 的离散化中去除膜锁定。在这项工作中，我们提出了连续假设应变 (CAS) 元素，这是一种假设的应变处理，可以从二次 NURBS 中消除膜锁定，以实现宽泛的细长比范围。CAS 元素利用$C^1$由二次 NURBS 给出的位移向量的元素间连续性，以使用线性拉格朗日多项式对膜应变进行插值，同时保留$C^0$膜应变的元素间连续性。据作者所知，CAS 单元是第一个基于 NURBS 的单元类型，能够针对广泛的细长比去除膜锁定，并结合了以下独特特征：(1) 没有添加额外的自由度，(2) 没有需要求解其他代数方程组，并且 (3)保留刚度矩阵的非零模式。由于所提议的元素类型所需的唯一额外计算是评估基函数和单位的导数在节点处的切线向量，所提出的方案几乎不会增加关于原始公式的基于锁定倾向的 NURBS 离散化的计算成本。基准问题表明，CAS 元素的收敛与长细比无关，直到$10^4$而二次 NURBS 元素的收敛与完全和减少的积分，局部$\overline{乙}$单元和局部 ANS 单元在很大程度上取决于细长比，随着网格的细化，误差甚至会增加。数值示例还显示了 CAS 单元如何消除由膜锁定引起的应力合成中的虚假振荡，而具有完全和减少积分的二次 NURBS 单元，局部$\overline{乙}$单元和局部 ANS 单元在应力合成中遭受大振幅寄生振荡。简而言之，CAS 元素是一种准确、稳健且计算效率高的数值方案，用于克服基于二次 NURBS 的离散化中的膜锁定。