A newly-developed, dimensionally reduced, $hp$-type axisymmetric shell finite element model is extended to linear elastodynamic problems of thin shells of revolution. The $hp$-shell finite element relies on the hybridized version of a three-field dual-mixed variational formulation, the application of which dictates the obligate usage of the inverse three-dimensional constitutive relation for homogeneous and isotropic materials, thereby ensuring the volumetric locking-free characteristic of the shell model at theory level. The fundamental fields are the a priori non-symmetric stress tensor, the displacement vector, the infinitesimal rotation vector and the hybrid variable defined on the element interfaces. Since the dimensional reduction process guided by this hybrid-mixed formulation does not necessitate the use of any classical kinematic assumptions appearing in the scientific literature, the inverse 3D Hooke’s law does not have to be modified. The numerical performance of the shell finite element is analyzed comprehensively for natural frequency computations of clamped-free and simply supported, silicone, conoid, spherical and hyperboloid shells of revolution. From their relative error convergence behaviors it follows that the extended hybrid-mixed shell finite element is not sensitive to the decrease of the slenderness ratio, namely providing reliable, uniformly stable numerical results for both $h$- and $p$-approximation. From theoretical point of view, the beneficial properties of the hybrid-mixed $hp$ shell finite element model are as follows: (i) this is effectively applicable to modeling not only extremely thin but also moderately thick shell structures with transverse shear deformations, as well as (ii) both the through-the-thickness variation and the membrane stress normal to the shell mid-surface are retained as independent variables, making it much easier to upbuild shell model for contact problems. From numerical point of view, the nice feature of the hybrid-mixed $hp$ shell finite element is that the global flexibility matrix of the system can be inverted block-wise at element level at cheap computational cost during the assembling procedure because of the hybridization technique.

一种新开发的、尺寸减小的、 $H磷$型轴对称壳有限元模型扩展到旋转薄壳的线性弹性动力学问题。这$H磷$-shell 有限元依赖于三场双混合变分公式的混合版本，其应用决定了均匀和各向同性材料的逆三维本构关系的强制使用，从而确保了体积无锁定特性壳模型的理论水平。基本场是先验的非对称应力张量、位移矢量、无穷小旋转矢量和单元界面上定义的混合变量。由于由这种混合混合公式引导的降维过程不需要使用科学文献中出现的任何经典运动学假设，因此不必修改逆 3D 胡克定律。对壳有限元的数值性能进行了综合分析，用于无夹和简支、硅树脂、圆锥、球面和双曲面旋转壳的固有频率计算。从它们的相对误差收敛行为可以看出，扩展的混合混合壳有限元对长细比的减小不敏感，即为两者提供可靠、一致稳定的数值结果$H$- 和 $磷$-近似。从理论的角度来看，混合混合的有益特性$H磷$壳有限元模型如下：(i) 这不仅适用于模拟极薄而且具有横向剪切变形的中等厚壳结构，以及 (ii) 全厚度变化和膜应力壳中面的法线保留为自变量，从而更容易为接触问题建立壳模型。从数值上看，hybrid-mixed 的优点是$H磷$ 壳有限元是由于混合技术，系统的全局柔性矩阵可以在组装过程中以低廉的计算成本在单元级逐块反转。