Discretizations based on the Bubnov-Galerkin method and the isoparametric concept 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 were originally 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 propose and elements to overcome membrane locking in quadratic NURBS-based discretizations of geometrically nonlinear Kirchhoff–Love shells. and elements are interpolation-based assumed-strain locking treatments. The assumed strains have continuity across element boundaries and different components of the membrane strains are interpolated at different interpolation points. elements use the assumed strains to obtain both the physical strains and the virtual strains, which results in a global tangent matrix which is a symmetric matrix. elements use the assumed strains to obtain only the physical strains, which results in a global tangent matrix which is a non-symmetric matrix. To the best of the authors’ knowledge, and elements are the first assumed-strain treatments to effectively overcome membrane locking in quadratic NURBS-based discretizations of geometrically nonlinear Kirchhoff–Love shells while satisfying the following important characteristics for computational efficiency: (1) No additional unknowns are added, (2) No additional systems of algebraic equations need to be solved, (3) The same elements are used to approximate the displacements and the assumed strains, (4) No additional matrix operations such as matrix inversions or matrix multiplications are needed to obtain the stiffness matrix, and (5) The nonzero pattern of the stiffness matrix is preserved. Analogously to the interpolation-based assumed-strain locking treatments for Lagrange polynomials that are widely used in commercial FEA software, the implementation of and elements only requires to modify the subroutine that computes the element residual vector and the element tangent matrix. The benchmark problems show that and elements, using either 2 × 2 or 3 × 3 Gauss–Legendre quadrature points per element, are effective locking treatments since these element types result in more accurate displacements for coarse meshes and excise the spurious oscillations of the membrane forces.

中文翻译：

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几何非线性基尔霍夫-洛夫壳的计算高效无锁等几何离散化


当应用于基尔霍夫-洛夫壳公式时，基于 Bubnov-Galerkin 方法和等参概念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。连续假设应变 (CAS) 单元最初是为了消除基于二次 NURBS 的线性平面弯曲基尔霍夫杆离散化中的膜锁定而引入的（Casquero 等人，CMAME，2022）。在这项工作中，我们提出了克服几何非线性基尔霍夫-洛夫壳的二次 NURBS 离散化中膜锁定的元素。和元素是基于插值的假定应变锁定处理。假设的应变在单元边界上具有连续性，并且膜应变的不同分量在不同的插值点处插值。单元使用假设应变来获得物理应变和虚拟应变，从而产生一个对称矩阵的全局正切矩阵。元素使用假设应变仅获得物理应变，这会产生一个非对称矩阵的全局正切矩阵。 据作者所知， 和 单元是第一个有效克服基于二次 NURBS 的几何非线性 Kirchhoff-Love 壳离散化中膜锁定的假设应变处理，同时满足以下重要的计算效率特征： (1) 否添加额外的未知数，(2) 无需求解额外的代数方程组，(3) 使用相同的元素来近似位移和假设应变，(4) 无需额外的矩阵运算，例如矩阵求逆或矩阵乘法需要获得刚度矩阵，并且 (5) 保留刚度矩阵的非零模式。与商业有限元分析软件中广泛使用的拉格朗日多项式基于插值的假设应变锁定处理类似， 和 elements 的实现只需要修改计算单元残差向量和单元正切矩阵的子程序。基准问题表明，每个单元使用 2 × 2 或 3 × 3 高斯-勒让德正交点的单元是有效的锁定处理，因为这些单元类型会导致粗网格的更准确的位移，并消除膜力的寄生振荡。