Finite element methods for the stretching and bending of thin structures with folding

In Bonito (J. Comput. Phys. 448:110719, 2022), a local discontinous Galerkin method was proposed for approximating the large bending of prestrained plates, and in Bonito (IMA J. Numer. Anal. 43:627-662, 2023) the numerical properties of this method were explored. These works considered deformations driven predominantly by bending. Thus, a bending energy with a metric constraint was considered. We extend these results to the case of an energy with both a bending component and a nonconvex stretching component, and we also consider folding across a crease. The proposed discretization of this energy features a continuous finite element space, as well as the discrete Hessian used in Bonito (J. Comput. Phys. 448:110719, 2022; IMA J. Numer. Anal. 43:627-662, 2023). We establish the \(\Gamma \) $\mathrm{\Gamma }$ -convergence of the discrete to the continuous energy and also present an energy-decreasing gradient flow for finding critical points of the discrete energy. Finally, we provide numerical simulations illustrating the convergence of minimizers and the capabilities of the model.