Keywords: waveguides, mid frequency, spectral element, finite element, higher-order modes, structural dynamics, propagation modes.

There is a demand for reliable numerical simulations of wave propagation through long structures of arbitrary cross-section in the mid frequency range. This is so in part to take advantage of the periodic characteristic of these structures, but also motivated by the advances in the oil industry that require increasingly longer risers. As analytical solutions do not exist for most of these structures and finite element simulation requires a prohibitive computational cost, hybrid methods have been proposed since 1970s by Dong and Nelson [1], and later by Thompson [2] and Gavric [3].

This paper uses a straightforward approach based on a hybrid finite-spectral element method, also called the wave spectral finite element method (WSFEM). This method was based on the periodic structure theory developed by Mead [4], and later employed by Thompson [2] in order to obtain the dispersion relations for a rail track by modelling a thin slice of the structure using conventional finite elements. It consists in extracting wavenumbers and wave modes from a thin slice of the substructure modelled using conventional finite elements, and, subsequently, writing kinematic variables and internal forces in the spectral form using, respectively, the wave displacement and the wave force components. Then, a mathematical manipulation enables the building of the numerical spectral matrix for an element of arbitrary length. Finally, the forced response of the structure can be easily obtained using a similar finite element analysis procedure to evaluate harmonic responses.

In this paper, a thin plate with bending capability was the application studied under different boundary conditions: simply supported at all edges and free of constraints. The type of problem has allowed the validation of the proposed approach when an extensive number of modes are included in the formulation. Thus, confirming the potential of such an approach in predicting higher-order behaviour. It was also possible to verify that regardless of the frequency range considered or the modal density, the WSFEM seems to be a reliable method for studying wave propagation problems. When the computational cost is considered, the WSFEM is shown to be advantageous when compared with the conventional finite element method because the mesh refinement is only required in the modelling of the structure slice. When the numerical dynamic stiffness matrix for the whole structure is recovered, mesh refinement is no longer necessary as the displacement and force fields are written in the frequency domain.

1

S.B. Dong, R.B. Nelson, "On Natural Vibrations and Waves in Laminated Orthotropic Plates", Journal of Applied Mechanics, 39(3), 739-745, 1972. doi:10.1115/1.3422782

2

D. Thompson, "Wheel-rail Noise Generation, Part III: Rail Vibration", Journal of Sound and Vibration, 161, 421-446, 1993. doi:10.1006/jsvi.1993.1084

3

L. Gavric, "Finite Element Computation of Dispersion Properties of Thin-Walled Waveguides", Journal of Sound and Vibration, 173, 113-124, 1994. doi:10.1006/jsvi.1994.1221

4

D.J. Mead, "A General Theory of Harmonic Wave Propagation in Linear Periodic Systems with Multiple Coupling", Journal of Sound and Vibration, 27(2), 235-260, 1973. doi:10.1016/0022-460X(73)90064-3

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