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

摘要

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 (fd) 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 fd 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.