On numerical issues for the wave/finite element method

Abstract

The waveguide finite element (WFE) method is a numerical method to investigate wave motion in a uniform waveguide. Numerical issues for the WFE method are specifically illustrated in this report. The method starts from finite element mass and stiffness matrices of only one element of the section of the waveguide. The matrices may be derived from commercial FE software such that existing element libraries can be used to model complex general structures. The transfer matrix, and hence the eigenvalue problem, is formed from the dynamic stiffness matrix in conjunction with a periodicity condition. The results of the eigenvalue problem represent the free wave characteristics in the waveguide. This report
concerns numerical errors occurring in the WFE results and proposing approaches to improve the errors.
In the WFE method, numerical errors arise because of (1) the FE discretisation error, (2) round-off errors due to the inertia term and (3) ill-conditioning. The FE discretisation error becomes large when element length becomes large enough compared to the wavelength. However, the round-off error due to the inertia term becomes large for small element lengths when the dynamic stiffness matrix is formed. This tendency is illustrated by numerical examples for one-dimensional structures.
Ill-conditioning occurs when the eigenvalue problem is formed and solved and the resulting errors can become large, especially for complex structures. Zhong’s method is used to improve the conditioning of the eigenvalue problem in this report. Errors in the eigenvalue problem are first mathematically discussed and Zhong’s method validated. In addition, singular value decomposition is proposed to reduce errors in numerically determining the
eigenvectors. For waveguides with a one-dimensional cross-section, the effect of the aspect ratio of the elements on the conditioning is also illustrated. For general structures, there is a crude trade-off between the conditioning, the FE discretisation error and the round-off error due to the inertia term. To alleviate the trade-off, the model with internal nodes is applied. At low frequencies, the approximate condensation formulation is derived and significant error
reduction in the force eigenvector components is observed.
Three approaches to numerically calculate the group velocity are compared and the finite difference and the power and energy relationship are shown to be efficient approaches for general structures.

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ORCID for B.R. Mace: orcid.org/0000-0003-3312-4918

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