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.

我们通过色散分析将几种最低阶有限元逼近与薄壁结构的弹性动力学问题进行了比较，该分析将参数频率-时间-厚度（$Fd$）和波速。我们只限于在无限大的平板中自由传播的谐波前沿波的分析理论。我们的研究被表述为单个张量积元素上的准周期特征值问题，最终在厚度方向上分层。在本文的第一部分中，观察到基于位移的有限元与理论一致，前提是存在足够多的层。在第二部分中，我们提出了新颖的各向异性六面体切向位移和法向-法向应力连续（TDNNS）混合有限元，用于弹性动力学的Hellinger-Reissner公式。事实证明，一层这样的元素足以满足$Fd$高达2000  [kHz  mm]。然而，由于大量的TDNNS自由度，计算复杂度仅可与基于多层位移的元素相提并论。低频情况并非如此，因为TDNNS允许粗略的各向异性离散化，因此与受剪切锁定效应影响的基于位移的元素相反，低频下TDNNS的效率要高得多。