Removing membrane locking in quadratic NURBS-based discretizations of linear plane Kirchhoff rods: CAS elements
Removing membrane locking in quadratic NURBS-based discretizations of linear plane Kirchhoff rods: CAS elements
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DOI：
10.1016/j.cma.2022.115354
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发表时间：
2022-09
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影响因子：
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通讯作者：
Hugo Casquero;Mahmoud Golestanian
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文献类型：
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作者：
Hugo Casquero;Mahmoud Golestanian
当应用于弯曲薄壁结构的原始公式时，基于NURBS的Galerkin方法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，使用线性拉格朗日多项式插值膜应变，同时保持膜应变的C0单元间连续性。据作者所知，CAS单元是第一种基于NURBS的单元类型，能够在宽长比范围内消除膜锁定，并具有以下独特特征：（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。算例表明，CAS单元的收敛性在1 0 4以内与细长比无关，而二次NURBS单元、局部B插值单元和局部ANS单元的收敛性在很大程度上依赖于细长比，且随着网格的细化，误差还会增大.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，而具有完全和缩减积分的二次NURBS单元、局部B NURBS单元和局部ANS单元则遭受应力合成中的大幅度寄生振荡。简而言之，CAS元素是一种精确，鲁棒，计算效率高的数值方案，以克服膜锁定在二次NURBS为基础的离散。
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 1 0 4 while the convergence of quadratic NURBS elements with full and reduced integration, local 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 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.