线性平面Kirchhoff杆离散化的二次NURBS膜锁定消除：CAS单元
Removing membrane locking in quadratic NURBS-based discretizations of linear plane Kirchhoff rods: CAS elements
摘要：
当应用于弯曲薄壁结构的原始公式时，基于NURBS的离散化遭受膜锁定。我们考虑线性平面弯曲基尔霍夫杆作为一个模型问题，研究如何消除膜锁定从基于NURBS的离散。在这项工作中，我们提出了连续假设应变（CAS）元素，一个假设的应变处理，消除膜锁定从二次NURBS的一个足够的范围内的细长比。CAS单元利用二次NURBS给出的位移向量的C1单元间连续性，使用线性Lagrange多项式插值膜应变，同时保持膜应变的C0单元间连续性。CAS单元是第一种基于ANSYS的单元类型，它能够在大范围的细长比下消除膜锁定，并具有以下特点：（1）不增加额外的自由度，（2）不需要求解额外的代数方程组，（3）保持刚度矩阵的非零模式。由于所提出的元素类型所需的唯一额外计算是评估基函数的导数和节点处的单位切向量，因此所提出的方案几乎没有增加计算成本，相对于锁定倾向的基于NURBS的原始公式的离散化。基准问题表明，CAS元的收敛性与细长比无关，而二次NURBS元、局部Bbar元和局部ANS元的收敛性则严重依赖于细长比。数值算例还显示了CAS单元如何消除由膜锁定引起的应力合成中的寄生振荡，而二次NURBS单元、局部Bbar单元和局部ANS单元则遭受应力合成中的大幅度寄生振荡。
英文摘要：
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.
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