In this paper, we propose a geometrically nonlinear spectral shell element based on Reissner--Mindlin kinematics using a rotation-based formulation with additive update of the discrete nodal rotation vector. The formulation is provided in matrix notation in detail. The use of a director vector, as opposed to multi-parameter shell models, significantly reduces the computational cost by minimizing the number of degrees of freedom. Additionally, we highlight the advantages of the spectral element method (SEM) in combination with Gauss-Lobatto-Legendre quadrature regarding the computational costs to generate the element stiffness matrix. To assess the performance of the new formulation for large deformation analysis, we compare it to three other numerical methods. One of these methods is a non-isoparametric SEM shell using the geometry definition of isogeometric analysis (IGA), while the other two are IGA shell formulations which differ in the rotation interpolation. All formulations base on Rodrigues' rotation tensor. Through the solution of various challenging numerical examples, it is demonstrated that although IGA benefits from an exact geometric representation, its influence on solution accuracy is less significant than that of shape function characteristics and rotational formulations. Furthermore, we show that the proposed SEM shell, despite its simpler rotational formulation, can produce results comparable to the most accurate and complex version of IGA. Finally, we discuss the optimal SEM strategy, emphasizing the effectiveness of employing coarser meshes with higher-order elements.

翻译：本文提出了一种基于Reissner-Mindlin运动学的几何非线性谱壳单元，采用基于旋转的表述方式，并通过对离散节点旋转向量进行加法更新。该公式以矩阵符号详细给出。与多参数壳模型相比，采用方向向量的方法通过最小化自由度数量显著降低了计算成本。此外，我们强调了谱元法（SEM）结合Gauss-Lobatto-Legendre积分在生成单元刚度矩阵计算成本方面的优势。为评估新公式在大变形分析中的性能，我们将其与其他三种数值方法进行了比较。其中一种方法是采用等几何分析（IGA）几何定义的非等参谱元壳，另外两种是IGA壳公式，它们在旋转插值方式上有所不同。所有公式均基于Rodrigues旋转张量。通过求解多个具有挑战性的数值算例，结果表明尽管IGA受益于精确的几何表示，但其对求解精度的影响小于形函数特性和旋转公式的影响。此外，我们证明所提出的谱元壳尽管采用更简单的旋转公式，仍能产生与最精确且复杂的IGA版本相当的结果。最后，我们讨论了最优谱元策略，强调了采用较粗网格配合高阶单元的有效性。