Numerical simulation of a fine-tunable Föppl-von Kármán model for foldable and bilayer plates

TL;DR

The paper develops a robust numerical framework for the fine-tunable bilayer Föppl--von Kármán model, enabling accurate identification of low-energy configurations across regimes controlled by the prestrain parameter $\theta$. By combining $P_1$ finite elements for in-plane displacements with discrete Kirchhoff elements for bending, and a decoupled gradient-flow scheme with adaptive time stepping, the approach yields energy-decreasing iterations and convergence to stationary states. The authors establish $\Gamma$-convergence of the discrete energy to the continuous energy, providing a rigorous link between the numerical method and the variational model. Numerical experiments demonstrate sphere-to-cylinder transitions, curvature inversion, foldable cardboard behavior, and bilayer folding with creases, highlighting the model’s capacity to capture complex, bistable actuation and crease-mediated responses in thin-film systems. Overall, the work offers a practical, rigorously justified tool for analyzing prestrained bilayer plates and foldable devices with applications to engineering and biomimetic actuation.

Abstract

A numerical scheme is proposed to identify low energy configurations of a Föppl-von Kármán model for bilayer plates. The dependency of the corresponding elastic energy on the in-plane displacement $u$ and the out-of-plane deflection $w$ leads to a practical minimization of the functional via a decoupled gradient flow. In particular, the energies of the resulting iterates are shown to be monotonically decreasing. The discretization of the model relies on $P1$ finite elements for the horizontal part $u$ and utilizes the discrete Kirchhoff triangle for the vertical component $w$. The model allows for analysing various different problem settings via numerical simulation: (i) stable low-energy configurations are detected dependent on a specified prestrain described by elastic material properties, (ii) curvature inversions of spherical and cylindrical configurations are investigated, (iii) elastic responses of foldable cardboards for different spontaneous curvatures and crease geometries are compared.

Figures (9)

• Figure 1.1: Bistable mechanism of a foldable cardboard. After repeated actuation, a crack can be observed (left picture) which emerges from the boundary and spreads along the given crease.
• Figure 5.1: Energy development of the sequence $(w_h^k,u_h^k)_{k\geq1}$ and step sizes $(\tau_k)_{k\geq1}$ of the iterative scheme for the first fifty iterations with $\theta=1000$ and $h=0.05$.
• Figure 5.2: Numerical solutions of \ref{['discr_decoupled_grad_flow']} for an initially flat disk with $\theta=1$ (top left), $\theta=300$ (top right), $\theta=350$ (bottom left) and $\theta=1000$ (bottom right). The colors represent the vertical magnitude from dark (lowest) to bright (highest). A transition from spherical to cylindrical configurations for increasing values of $\theta$ can be observed.
• Figure 5.3: Development of the directional mean curvatures (left) and the quotient $q_{\text{sym}}$, see \ref{['qsym']}, of the directional in-plane deflections (right) of the final configurations for $\theta=1,2,...,400$ and $\theta=400,450,...,600$. A critical regime at around $\theta=300$ appears from which on a stark break of symmetry occurs.
• Figure 5.4: Numerical solutions of \ref{['discr_decoupled_grad_flow']} for $\alpha\in\{1,0.7,0.3,-1\}$ (top to bottom) with $\theta=0$ (left) and $\theta=1000$ (right). The colors represent the strain density from dark (lowest) to bright (highest). The transition from $\alpha=1$ to $\alpha=-1$ induces a curvature inversion, where the value $\alpha=0$ leads to flat configurations for arbitrary $\theta$.

Theorems & Definitions (15)

• Theorem 2.1
• Corollary 2.1
• Definition 2.1: See nonlinear_bartels
• Remark 2.1
• Lemma 2.1: See nonlinear_bartels
• Lemma 3.1: Energy decay
• proof
• Proposition 3.1: Uniqueness of minimizers
• proof
• Proposition 3.2: Newton scheme