A geometrically nonlinear Cosserat shell model for orientable and non-orientable surfaces: Discretization with geometric finite elements

TL;DR

This paper develops a geometrically nonlinear Cosserat shell model applicable to both orientable and non-orientable surfaces, discretized with geometric finite elements, enabling simulation of complex high-curvature deformations.

Contribution

It generalizes the Cosserat shell model to non-orientable surfaces and proves the existence of solutions for its discretization using geometric finite elements.

Findings

Model successfully simulates high-curvature deformations.

Discretization with geometric finite elements is mathematically validated.

Numerical tests demonstrate the model's versatility.

Abstract

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.