Analysis of a discontinuous Galerkin method for Koiter shell

TL;DR

This paper analyzes a mixed finite element method for Koiter shell bending problems, providing error estimates that depend on shell geometry and thickness, and highlighting the importance of mesh refinement for accuracy.

Contribution

It offers the first detailed error analysis for a balanced mixed finite element method applied to Koiter shells, including conditions for optimal convergence.

Findings

Error estimates are uniform with respect to shell thickness.

Optimal accuracy depends on shell geometry and mesh refinement.

The method achieves optimal order of accuracy under certain geometric conditions.

Abstract

We present an analysis for a mixed finite element method for the bending problem of Koiter shell. We derive an error estimate showing that when the geometrical coefficients of the shell mid-surface satisfy certain conditions the finite element method has the optimal order of accuracy, which is uniform with respect to the shell thickness. Generally, the error estimate shows how the accuracy is affected by the shell geometry and thickness. It suggests that to achieve optimal rate of convergence, the triangulation should be properly refined in regions where the shell geometry changes dramatically. The analysis is carried out for a balanced method in which the normal component of displacement is approximated by discontinuous piecewise cubic polynomials, while the tangential components are approximated by discontinuous piecewise quadratic polynomials, with some enrichment on elements that have edges on the free boundary. Components of the membrane stress are approximated by continuous piecewise linear functions.