Do locking-free finite element schemes lock for holey Reissner-Mindlin plates with mixed boundary conditions?

TL;DR

This paper analyzes finite element schemes for holey Reissner-Mindlin plates with mixed boundary conditions, establishing conditions for locking-free convergence and confirming many existing schemes satisfy these conditions.

Contribution

It introduces new conditions based on the de Rham complex for locking-free schemes in complex domains, extending prior results to holey plates and mixed boundary conditions.

Findings

Many existing schemes are locking-free under new conditions.

Conditions naturally extend to non-simply connected domains.

Optimal convergence rates are achieved for various boundary conditions.

Abstract

We revisit finite element discretizations of the Reissner-Mindlin plate in the case of non-simply connected (holey) domains with mixed boundary conditions. Guided by the de Rham complex, we develop conditions under which schemes deliver locking-free, optimal rates of convergence. We naturally recover the typical assumptions arising for clamped, simply supported plates. More importantly, we also see new conditions arise naturally from the presence of holes in the domain or in the case of mixed boundary conditions. We show that, fortunately, many of the existing popularly used schemes do, in fact, satisfy all of the conditions, and thus are locking-free.