We investigate discretizations of a geometrically nonlinear elastic Cosserat shell with nonplanar reference configuration originally introduced by B\^irsan, Ghiba, Martin, and Neff in 2019. The shell model includes curvature terms up to order 5 in the shell thickness, which are crucial to reliably simulate high-curvature deformations such as near-folds or creases. The original model is generalized to shells that are not homeomorphic to a subset of $\mathbb{R}^2$. For this, we replace the originally planar parameter domain by an abstract two-dimensional manifold, and verify that the hyperelastic shell energy and three-dimensional reconstruction are invariant under changes of the local coordinate systems. This general approach allows to determine the elastic response for even non-orientable surfaces like the M\"obius strip and the Klein bottle. We discretize the model with a geometric finite element method and, using that geometric finite elements are $H^1$-conforming, prove that the discrete shell model has a solution. Numerical tests then show the general performance and versatility of the model and discretization method.

翻译：我们研究了一类具有非平面参考构型的几何非线性弹性Cosserat壳的离散化方法，该模型最初由Bîrsan、Ghiba、Martin和Neff于2019年提出。该壳模型包含厚度方向高达五阶的曲率项，这些项对于可靠模拟高曲率变形（如近折叠或折痕）至关重要。原模型被推广至非与ℝ²子集同胚的壳结构。为此，我们将原本的平面参数域替换为抽象二维流形，并验证了超弹性壳能量及三维重构在局部坐标系变换下的不变性。这种通用方法甚至能确定不可定向曲面（如莫比乌斯带和克莱因瓶）的弹性响应。我们采用几何有限元法对模型进行离散化，并利用几何有限元具有H¹相容性这一特性，证明了离散壳模型解的存在性。数值实验进一步展示了该模型及离散化方法的普适性能与通用性。