The computation of dispersion relations for three-dimensional elastic waveguides using the Scaled Boundary Finite Element Method

In this paper, a numerical approach for the computation of dispersion relations for three-dimensional waveguides with arbitrary cross-section is proposed. The formulation is based on the Scaled Boundary Finite Element Method (SBFEM). It is an extension of the approach previously derived for plate structures. It is shown that the wavenumbers of guided waves in a waveguide can be obtained as the eigenvalues of the ZZ matrix, which is well known in the SBFEM. The Hamiltonian properties of this matrix are utilized to derive an efficient way to compute the group velocities of propagating waves as eigenvalue derivatives. The cross-section of the waveguide is discretized using higher-order spectral elements. It is discussed in detail how symmetry axes can be utilized to reduce computational costs. In order to sort the solutions at different frequencies, a mode-tracking algorithm is proposed, based on the Padé expansion.

► We present a numerical approach for the computation of dispersion relations for three-dimensional waveguides. ► A Hamiltonian eigenvalue problem is derived to obtain wavenumbers and modeshapes. ► The group velocities are computed as eigenvalue derivatives. ► Higher-order spectral elements are utilized to increase computational efficiency. ► An efficient mode-tracking algorithm is presented.