Finite Element Approximation of Large-Scale Isometric Deformations of Parametrized Surfaces

In this paper, the numerical approximation of isometric deformations of thin elastic shells is discussed. To this end, for a thin shell represented by a parametrized surface, it is shown how to transform the stored elastic energy for an isometric deformation such that the highest order term is quadr...

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Bibliographic Details
Published in:arXiv.org 2022-06
Main Authors: Rumpf, Martin, Simon, Stefan, Smoch, Christoph
Format: Article
Language:English
Subjects:
Approximation
Bending
Convergence
Elastic deformation
Elastic shells
Mathematical analysis
Parameterization
Thin walled shells
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Summary:In this paper, the numerical approximation of isometric deformations of thin elastic shells is discussed. To this end, for a thin shell represented by a parametrized surface, it is shown how to transform the stored elastic energy for an isometric deformation such that the highest order term is quadratic. For this reformulated model, existence of optimal isometric deformations is shown. A finite element approximation is obtained using the Discrete Kirchhoff Triangle (DKT) approach and the convergence of discrete minimizers to a continuous minimizer is demonstrated. In that respect, this paper generalizes the results by Bartels for the approximation of bending isometries of plates. A Newton scheme is derived to numerically simulate large bending isometries of shells. The proven convergence properties are experimentally verified and characteristics of isometric deformations are discussed.
ISSN:2331-8422