Numerical Treatment of a Geometrically Nonlinear Planar Cosserat Shell Model

TL;DR

This paper introduces a novel discretization method for a geometrically nonlinear planar Cosserat shell model using geodesic finite elements, enabling large rotations and solving the resulting nonlinear problem with a Riemannian trust-region method.

Contribution

It develops a new discretization approach for nonlinear Cosserat shells with geodesic finite elements and a Riemannian trust-region solver, allowing for large rotations and arbitrary approximation orders.

Findings

Objective discrete model with large rotations

Global convergence of the Riemannian trust-region method

Numerical examples including elastic sheet wrinkles

Abstract

We present a new way to discretize a geometrically nonlinear elastic planar Cosserat shell. The kinematical model is similar to the general 6-parameter resultant shell model with drilling rotations. The discretization uses geodesic finite elements, which leads to an objective discrete model which naturally allows arbitrarily large rotations. Finite elements of any approximation order can be constructed. The resulting algebraic problem is a minimization problem posed on a nonlinear finite-dimensional Riemannian manifold. We solve this problem using a Riemannian trust-region method, which is a generalization of Newton's method that converges globally without intermediate loading steps. We present the continuous model and the discretization, discuss the properties of the discrete model, and show several numerical examples, including wrinkles of thin elastic sheets in shear.