A discontinuous Galerkin method for the Naghdi shell model

TL;DR

This paper introduces a mixed discontinuous Galerkin finite element method for the Naghdi shell model, providing error analysis and guidelines for mesh refinement to ensure optimal accuracy regardless of shell thickness.

Contribution

It develops a novel mixed discontinuous Galerkin method for Naghdi shells with comprehensive error estimates and practical refinement strategies for improved accuracy.

Findings

Optimal order of accuracy when curvature and Christoffel components are piecewise linear

Error estimates depend on shell geometry and thickness

Mesh refinement is crucial in regions with significant geometric changes

Abstract

We propose a mixed discontinuous Galerkin method for the bending problem of Naghdi shell, and present an analysis for its accuracy. The error estimate shows that when components of the curvature tensor and Christoffel symbols are piecewise linear functions, 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. These are the results for a balanced method in which the primary displacement components and rotation components are approximated by discontinuous piecewise quadratic polynomials, while components of the scaled membrane stress tensor and shear stress vector are approximated by continuous piecewise linear functions. On elements that have edges on the free boundary of the shell, finite element space for displacement components needs to be enriched slightly, for stability purpose. Results on higher order finite elements are also included.