Dispersion analysis of displacement-based and TDNNS mixed finite elements for thin-walled elastodynamics

We compare several lowest-order finite element approximations to the problem of elastodynamics of thin-walled structures by means of dispersion analysis, which relates the parameter frequency-times-thickness (f d) and the wave speed. We restrict to analytical theory of harmonic front-crested waves that freely propagate in an infinite plate. Our study is formulated as a quasi-periodic eigenvalue problem on a single tensor-product element, which is eventually layered in the thickness direction. In the first part of the paper it is observed that the displacement-based finite elements align with the theory provided there are sufficiently many layers. In the second part we present novel anisotropic hexahedral tangential-displacement and normal- normal-stress continuous (TDNNS) mixed finite elements for Hellinger-Reissner formulation of elastodynamics. It turns out that one layer of such elements is sufficient for f d up to 2000 [kHz mm]. Nevertheless, due to a large amount of TDNNS degrees of freedom the computational complexity is only comparable to the multi-layer displacement-based element. This is not the case at low frequencies, where TDNNS is by far more efficient since it allows for rough anisotropic discretizations, contrary to the displacement-based elements that suffer from the shear locking effect. (C) 2021 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

Automated summary

In this study, several lowest-order finite element approximations to the problem of elastodynamics of thin-walled structures were compared using dispersion analysis, showing alignment with theory with a sufficient number of layers. Additionally, novel anisotropic hexahedral tangential-displacement and normal-normal-stress continuous (TDNNS) mixed finite elements were presented for Hellinger-Reissner formulation of elastodynamics, demonstrating efficiency up to a certain frequency parameter despite a large amount of degrees of freedom.