A DG method for a stress formulation of the elasticity eigenproblem with strongly imposed symmetry

TL;DR

This paper presents a novel discontinuous Galerkin method for the elasticity eigenproblem that enforces stress symmetry strongly, achieving spectral correctness and optimal convergence rates, validated through numerical examples.

Contribution

It introduces a pure-stress formulation with a strongly symmetric stress discretization using an H(div)-based DG method, ensuring spectral correctness and optimal convergence.

Findings

Spectral correctness of the proposed scheme

Optimal convergence rates for eigenvalues and eigenfunctions

Numerical validation in 2D and 3D examples

Abstract

We introduce a pure--stress formulation of the elasticity eigenvalue problem with mixed boundary conditions. We propose an H(div)-based discontinuous Galerkin method that imposes strongly the symmetry of the stress for the discretization of the eigenproblem. Under appropriate assumptions on the mesh and the degree of polynomial approximation, we demonstrate the spectral correctness of the discrete scheme and derive optimal rates of convergence for eigenvalues and eigenfunctions. Finally, we provide numerical examples in two and three dimensions.