Numerical modeling and simulation of curved folding in thin elastic sheets

Abstract: This thesis investigates the numerical modeling and simulation of folding processes in thin elastic sheets. The modeling is based on a modified description for large plate bending isometries that can be derived via rigorous dimension reduction from 3D hyperelasticity and accounts for the presence of a crease curve dividing the plate into two segments.

A discontinuous Galerkin method is devised that allows for the natural description of foldable structures if contributions of gradients are neglected along a given crease in the corresponding penalty terms. The proposed method leads to a Γ-convergence result that establishes the validity of the discrete approach.

An isoparametric interior penalty discontinuous Galerkin method is rigorously derived for the corresponding linear model. It gives rise to the simulation of small foldable deflections that account for piecewise polynomial approximations of arbitrarily curved crease geometries. A priori and a posteriori error estimates are derived if the crease is accurately resolved by the isoparametric mesh. A bound on the geometric consistency error covers the case that the curved crease is approximated by polynomial mesh elements. The work includes various numerical experiments that validate the estimates and show that the proposed method avoids a Babuska-like paradox.

In the final part of this thesis, a Föppl-von Kármán model is modified to allow for the inclusion of piecewise linear crease lines. It is capable of describing the formation of folds and ridges in moderate deformations which is relevant in the analysis of material failure. The numerical approach consists of an energy-decreasing decoupled gradient flow which leads to a practical minimization of the corresponding energy functional. Numerical simulations predict the formation of singularities in a foldable cardboard box, whose distance to the boundary seem to match the length of cracks that appeared in a real cardboard after repeated actuation. This indicates a connection between the two phenomena. The model can further be adapted to describe deformations of foldable bilayer objects which lead to a realistic description of the actuating mechanism of Venus flytraps

Object Details

• Location: Deutsche Nationalbibliothek Frankfurt am Main
• Extent: Online-Ressource
• Language: Englisch
• Notes: Universität Freiburg, Dissertation, 2024

Classification and Topics

• Keyword: Numerisches Modell ; Numerisches Verfahren ; Mathematisches Modell ; Elastizität ; Finite-Elemente-Methode

Contributors, Places and Time

• Event: Veröffentlichung
• (where): Freiburg
• (who): Universität
• (when): 2024
• Creator: Tscherner, Philipp
• Contributor: Bartels, Sören ; Nochetto, Ricardo H. ; Albert-Ludwigs-Universität Freiburg. Abteilung für Angewandte Mathematik ; Albert-Ludwigs-Universität Freiburg. Fakultät für Mathematik und Physik

Further information

• DOI: 10.6094/UNIFR/248927
• URN: urn:nbn:de:bsz:25-freidok-2489272
• Rights: Open Access; Der Zugriff auf das Objekt ist unbeschränkt möglich.
• Last update: 14.08.2025, 10:50 AM CEST