Numerical approximations of thin structure deformations

TL;DR

This paper reviews models for thin structure deformations focusing on bending, using LDG finite elements and discrete Hessians, and demonstrates their effectiveness through numerical experiments.

Contribution

It introduces a novel approach combining LDG finite elements with a discrete Hessian for modeling large deformations of thin structures.

Findings

Effective approximation of large deformations demonstrated

Discrete gradient flows improve robustness in minimization

Variety of achievable shapes shown through experiments

Abstract

We review different (reduced) models for thin structures using bending as principal mechanism to undergo large deformations. Each model consists in the minimization of a fourth order energy, potentially subject to a nonconvex constraint. Equilibrium deformations are approximated using local discontinuous Galerkin (LDG) finite elements. The design of the discrete energies relies on a discrete Hessian operator defined on discontinuous functions with better approximation properties than the piecewise Hessian. Discrete gradient flows are put in place to drive the minimization process. They are chosen for their robustness and ability to preserve the nonconvex constraint. Several numerical experiments are presented to showcase the large variety of shapes that can be achieved with these models.