Locking-free HDG methods for Reissner-Mindlin plates equations on polygonal meshes

TL;DR

This paper introduces a locking-free HDG method for Reissner-Mindlin plates that achieves optimal convergence on polygonal meshes, independent of plate thickness, and includes analysis and numerical validation.

Contribution

A novel HDG scheme based on Helmholtz Decomposition for Reissner-Mindlin plates, providing uniform convergence and effective preconditioning on general meshes.

Findings

Achieves optimal $(k+1)$th order convergence in L2 norm.

Method is locking-free and uniform with respect to plate thickness.

Numerical experiments confirm theoretical results.

Abstract

We present and analyze a new hybridizable discontinuous Galerkin method (HDG) for the Reissner-Mindlin plate bending system. Our method is based on the formulation utilizing Helmholtz Decomposition. Then the system is decomposed into three problems: two trivial Poisson problems and a perturbed saddle-point problem. We apply HDG scheme for these three problems fully. This scheme yields the optimal convergence rate ($(k+1)$th order in the $\mathrm{L}^2$ norm) which is uniform with respect to plate thickness (locking-free) on general meshes. We further analyze the matrix properties and precondition the new finite element system. Numerical experiments are presented to confirm our theoretical analysis.