A linear finite element procedure for the Naghdi shell model

TL;DR

This paper introduces a mixed finite element method for Naghdi shell models that adaptively balances accuracy and computational efficiency, effectively handling various shell geometries and thicknesses.

Contribution

It develops a unified finite element approach with a parameter to switch between membrane/shear locking reduction and a consistent discontinuous Galerkin method, ensuring optimal accuracy.

Findings

Achieves optimal order of accuracy for general shells

Robust accuracy with respect to shell thickness

Uniform accuracy for shells with piecewise constant coefficients

Abstract

We prove the accuracy of a mixed finite element method for bending dominated shells in which a major part of the membrane/shear strain is reduced, to free up membrane/shear locking. When no part of the membrane/shear strain is reduced, the method becomes a consistent discontinuous Galerkin method that is proven accurate for membrane/shear dominated shells and intermediate shells. The two methods can be coded in a single program by using a parameter. We propose a procedure of numerically detecting the asymptotic behavior of a shell, choosing the parameter value in the method, and producing accurate approximation for a given shell problem. The method uses piecewise linear functions to approximate all the variables. The analysis is carried out for shells whose middle surfaces have the most general geometries, which shows that the method has the optimal order of accuracy for general shells and the accuracy is robust with respect to the shell thickness. In the particular case that the geometrical coefficients of the shell middle surface are piecewise constants the accuracy is uniform with respect to the shell thickness.