### Natural frequency analysis of shells of revolution based on hybrid dual-mixed hp-finite element formulation

Balázs Tóth
2021 Applied Mathematical Modelling
Please cite this article as: Balázs Tóth, Natural frequency analysis of shells of revolution based on hybrid dual-mixed hp-finite element formulation, Applied Mathematical Modelling (2021), doi: https://doi. Highlights • A newly-developed hp-finite element is extended to linear elastodynamic problems of thin shells of revolution. • The hp-shell-finite element is based on the hybridization of a three-field dual-mixed variational formulation. • The theoretical model does not rely on the standard
more » ... ypotheses used in Naghdi-and Koiter-type shell models. • The unmodified, inverse three-dimensional constitutive relation is applied to homogeneous and isotropic materials. • The convergence behavior of the locking-free hp-shell element is tested comprehensively through natural frequency analyzes. Abstract 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 hybridmixed 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.