Finite Element Methods for the Stretching and Bending of Thin Structures with Folding

TL;DR

The proposed discretization of this energy features a continuous finite element space, as well as a discrete Hessian operator, and establishes the $\Gamma$-convergence of the discrete to the continuous energy and also presents an energy-decreasing gradient flow for finding critical points ofThe discrete energy.

Abstract

In [Bonito et al., J. Comput. Phys. (2022)], a local discontinuous Galerkin method was proposed for approximating the large bending of prestrained plates, and in [Bonito et al., IMA J. Numer. Anal. (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 a discrete Hessian operator. We establish the $Γ$-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.

Figures (6)

• Figure 1: Evolution of the flapping device as the ends are squeezed. The top row shows the (physically correct) evolution using the preasymptotic model. The bottom row is the evolution using the algorithm proposed in Prestrain_BGNY_2022 for the minimization of bending energies with isometry constraint.
• Figure 2: Element-wise bending energy of the final deformation using the second fundamental form (left) and the Hessian (right). The same color map is used for both plot and the ranges of values are [2.53e-8,6.82e-8] and [3.69e-8,6.84e-8], respectively.
• Figure 3: Final configurations for the oscillating boundary experiment. Left to right and top to bottom: $\theta^2 = 10^{-1}$, $10^{-2}$, $10^{-3}$, and $0$.
• Figure 4: Final configuration for the oscillating boundary experiment with $\theta = 10^{-3}$, where we use the linearization $a_h^S({\bf y}_h^n; \cdot, \cdot)$ instead of $a_h^S({\bf w}_h^n; \cdot, \cdot)$.
• Figure 5: Computational domain (left) and mesh (right) for the flapping device experiment.

Theorems & Definitions (26)

• Lemma 3.1: Discrete Poincaré--Friedrichs inequality
• Lemma 3.2: Compactness
• proof
• Lemma 3.3: Discrete Sobolev inequality
• proof
• Lemma 3.4: Weak convergence of $H_h$
• proof
• Lemma 3.5: Strong convergence of $H_h$
• proof
• Theorem 3.6: Partial coercivity of $E_h$