论文标题
卸下二次NURBS基于二次NURBS的膜锁定线性平面的离散化Kirchhoff杆：CAS元素
Removing membrane locking in quadratic NURBS-based discretizations of linear plane Kirchhoff rods: CAS elements
论文作者
论文摘要
将基于NURB的离散化应用于弯曲的薄壁结构的原始配方时，会遭受膜锁定。我们认为线性平面弯曲的基尔chhoff杆是一个模型问题，可以研究如何从基于NURBS的离散化中去除膜锁定。在这项工作中，我们提出了连续构造的元素（CAS）元素，这是一种假定的应变处理，可去除二次NURB的膜锁定，以延长细长比率。 CAS元素利用了二次NURB给出的位移载体的C1间元素连续性，使用线性Lagrange多项式插入膜应变，同时保留膜菌株的C0元素间连续性。 CAS元素是第一个基于NURBS的元素类型，可以消除膜锁定的膜锁定，以结合以下特征的各种细长比率：（1）不添加额外的自由度，（2）不需要求解代数方程式的其他额外的自由度系统，并且（3）（3）刚度矩阵的非零模式保留。由于所提出的元素类型所需的唯一其他计算是评估基础函数的衍生物和打结处的单位切线向量，因此提出的方案几乎没有增加计算成本，而对于基于锁定的NURBS基于锁定的NURBS的离散化。基准问题表明，CAS元素的收敛性与细长比率无关，而二次NURBS元素，局部BBAR元素和局部ANS元素的收敛性在很大程度上取决于细长的比率。数值示例还显示了CAS元素如何消除由膜锁定引起的压力导致的虚假振荡，而二次NURBS元素，局部BBAR元件和局部ANS元素在压力产生剂中遭受了大振幅振荡。
NURBS-based discretizations 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 C1 inter-element continuity of the displacement vector given by quadratic NURBS to interpolate the membrane strain using linear Lagrange polynomials while preserving the C0 inter-element continuity of the membrane strain. 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 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 while the convergence of quadratic NURBS elements, local Bbar elements, and local ANS elements depends heavily on the slenderness ratio. The numerical examples also show how CAS elements remove the spurious oscillations in stress resultants caused by membrane locking while quadratic NURBS elements, local Bbar elements, and local ANS elements suffer from large-amplitude spurious oscillations in stress resultants.
