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

TL;DR

This paper develops a finite element method for approximating large isometric deformations of parametrized thin elastic shells, proving convergence and demonstrating numerical simulations of bending isometries.

Contribution

It introduces a reformulated elastic energy model for isometric deformations and proves existence and convergence of finite element approximations for shells.

Findings

Finite element approximation converges to continuous minimizers.

Numerical simulations successfully reproduce large bending isometries.

The approach generalizes previous results for plates to shells.

Abstract

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.