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

¹ Abteilung für angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 10, 79104 Freiburg im Breisgau, Germany
² Institut für Mathematik, Universität Augsburg, Universitätsstr. 14, 86159 Augsburg, Germany

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 24 February 2025
Accepted: 4 April 2026

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.