An hp-adaptive analysis of some linear free vibration problems

This paper concerns an hp-adaptive finite element analysis of the free vibration problems of linear elasticity. We have implemented the 3D formulation of the Reissner-Mindlin shell theory, higher order hierarchical shell models and the 3D-elasticity to analyse complex (shell-solid) structures.

The idea is to use modified methods derived for elastostatics adaptive solutions i.e. Equilibrated Residual Method to estimate local error and Texas 3-step strategy (solutions for an initial mesh, h-refined mesh and p-enriched mesh) to reach the finite element space of desired properties.

Basically the procedure for error estimation is as follows. Firstly, we solve the eigenproblem for a given h and p. Then, for a given frequency: we calculate equilibrated interelement stresses and solve local (elemental) problems with these stresses as boundary conditions with h, p+1. With the latter we estimate local error in the energy norm. Having local errors one can perform finite element space update.

In this paper we show how the hp-adaptive technique for linear elastostatics can be used in case of free vibrations. Numerical examples conclude the paper.