Overcoming membrane locking in quadratic NURBS-based discretizations of linear Kirchhoff–Love shells: CAS elements
Overcoming membrane locking in quadratic NURBS-based discretizations of linear Kirchhoff–Love shells: CAS elements
复制标题
克服基于二次 NURBS 的线性 Kirchhoff Love 壳离散化中的膜锁定：CAS 元素
DOI：
10.1016/j.cma.2023.116523
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发表时间：
2023
影响因子：
7.2
通讯作者：
Mathews, Kyle Dakota
中科院分区：
文献类型：
--
作者：
Casquero, Hugo;Mathews, Kyle Dakota
当应用于基尔霍夫-洛夫壳公式时，伽辽金方法的基于二次 NURBS 的离散化会受到膜锁定的影响。膜锁定不仅导致比预期更小的位移，而且还导致膜力的大幅寄生振荡。最近引入了连续假设应变 (CAS) 单元，用于消除线性平面弯曲基尔霍夫杆的基于二次 NURBS 的离散化中的膜锁定（Casquero 等人，CMAME，2022）。在这项工作中，我们推广了 CAS 元素，以克服线性 Kirchhoff-Love 壳的二次 NURBS 离散化中的膜锁定。 CAS 单元在每个单元的四个角处对膜应变进行双线性插值。因此，假设的应变在单元边界上具有 C 0 连续性。据作者所知，CAS 单元是第一个有效克服基尔霍夫-洛夫壳二次 NURBS 离散化中膜锁定的假设应变处理，同时满足计算效率的以下重要特征：(1) 不添加额外的自由度，(2) 不需要求解额外的代数方程组，(3) 不需要矩阵乘法或矩阵求逆来获得刚度矩阵，(4) 刚度矩阵的非零模式刚度矩阵被保留。基准问题表明，每个单元使用 2×2 或 3×3 高斯-勒让德求积点的 CAS 单元是一种有效的锁定处理，因为这种单元类型可以使粗网格产生更准确的位移，并消除膜力的寄生振荡。基准问题还表明，CAS 单元的性能优于基于配备假定应变或缩减积分锁定处理的拉格朗日多项式的最先进单元类型。
Quadratic NURBS-based discretizations of the Galerkin method suffer from membrane locking when applied to Kirchhoff–Love shell formulations. Membrane locking causes not only smaller displacements than expected, but also large-amplitude spurious oscillations of the membrane forces. Continuous-assumed-strain (CAS) elements have been recently introduced to remove membrane locking in quadratic NURBS-based discretizations of linear plane curved Kirchhoff rods (Casquero et al., CMAME, 2022). In this work, we generalize CAS elements to vanquish membrane locking in quadratic NURBS-based discretizations of linear Kirchhoff–Love shells. CAS elements bilinearly interpolate the membrane strains at the four corners of each element. Thus, the assumed strains have C 0 continuity across element boundaries. To the best of the authors’ knowledge, CAS elements are the first assumed-strain treatment to effectively overcome membrane locking in quadratic NURBS-based discretizations of Kirchhoff–Love shells while satisfying the following important characteristics for computational efficiency:(1) No additional degrees of freedom are added,(2) No additional systems of algebraic equations need to be solved,(3) No matrix multiplications or matrix inversions are needed to obtain the stiffness matrix, and (4) The nonzero pattern of the stiffness matrix is preserved. The benchmark problems show that CAS elements, using either 2× 2 or 3× 3 Gauss–Legendre quadrature points per element, are an effective locking treatment since this element type results in more accurate displacements for coarse meshes and excises the spurious oscillations of the membrane forces. The benchmark problems also show that CAS elements outperform state-of-the-art element types based on Lagrange polynomials equipped with either assumed-strain or reduced-integration locking treatments.