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

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 s.t. 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.

A Homogenized Bending Theory for Prestrained Plates

“…Nonconforming discretizations based on the MINI-and Crouzeix-Raviart elements have been proposed in Bartels (2013b). Alternative discretizations using discrete Kirchhoff triangles or Discontinuous Galerkin (DG) finite elements appear in Bartels (2013a); Rumpf et al (2021) and Bonito et al (2021b), respectively. An approach using spline functions (which are in H 2 (S; R 3 ), but are not pointwise isometries) appears in Mohan et al (2022).…”

“…This leads to a (controllable) algebraic violation of the constraints beyond the one introduced by the discretization. Rumpf et al (2021) use a Lagrange multiplier formulation and a Newton method instead, and preserve the exact isometry constraints at the grid vertices. Note that we do not numerically minimize (2) in the present manuscript.…”