Spectral element computation of high-frequency leaky modes in three-dimensional solid waveguides

Abstract

A numerical method is proposed to compute high-frequency low-leakage modes in structural waveguides surrounded by infinite solid media. In order to model arbitrary shape structures, a waveguide formulation is used, which consists of applying to the elastodynamic equilibrium equations a space Fourier transform along the waveguide axis and then a discretization method to the cross-section coordinates. However several numerical issues must be faced related to the unbounded nature of the cross-section, the number of degrees of freedom required to achieve an acceptable error in the high-frequency regime as well as the number of modes to compute. In this paper, these issues are circumvented by applying perfectly matched layers (PML) along the cross-section directions, a high-order spectral element method for the discretization of the cross-section, and an eigensolver shift suited for the computation of low-leakage modes. First, computations are performed for an embedded cylindrical bar, for which literature results are available. The proposed PML waveguide formulation yields good agreement with literature results, even in the case of weak impedance contrast. Its performance with high-order spectral elements is assessed in terms of convergence and accuracy and compared to traditional low-order finite elements. Then, computations are performed for an embedded square bar. Dispersion curves exhibit strong similarities with cylinders. These results show that the properties of low-leakage modes observed in cylindrical bars can also occur in other types of geometry.