Study on Mode Repulsion of Ultrasonic-Guided Waves in Pipe Bends

The wave motion of guided waves in pipe bends is still veiled in some mystery, which hinders the application of guided-wave techniques in the inspection of pipelines with bends. Mode repulsion, which exists in the wavenumber versus frequency dispersion curves of guided waves in pipe bends, is an intriguing phenomenon deserving in depth study. The governing equation of wave motion in pipe bends, deduced by the semi-analytical finite element (SAFE) method, can be regarded as an eigenvalue problem. The eigenvalue derivatives, with respect to the wavenumber, are investigated to determine whether mode repulsion will occur or not. A term in the second derivative of the eigenvalue is identified to determine the mode repulsions. With respect to symmetry, it is found that mode repulsion only occurs between modes of one and the same type, such as symmetric or antisymmetric modes, and does not occur between modes of different type, like between symmetric and antisymmetric modes. A specific case of mode repulsion in a small-bore thin-walled pipe in the low-frequency range, where relatively fewer modes exist, is further studied, and the interactions between these modes are clarified. The evolutions of mode shapes before and after mode repulsion are further illustrated.

Higher-order sensitivity analysis of periodic 3-D eigenvalue problems for electromagnetic field calculations

An algorithm to perform a higher-order sensitivity analysis for electromagnetic eigenvalue problems is presented. By computing the eigenvalue and eigenvector derivatives, the Brillouin Diagram for periodic structures can be calculated. The discrete model is described using the Finite Integration Technique (FIT) with periodic boundaries, and the sensitivity analysis is performed with respect to the phase shift φ between the periodic boundaries. For validation, a reference solution is calculated by solving multiple eigenvalue problems (EVP). Furthermore, the eigenvalue derivatives are compared to reference values using finite difference (FD) formulas.

Eigenvector Derivatives of Generalized Nondefective Eigenproblems With Repeated Eigenvalues

A new method is presented for computation of eigenvalue and eigenvector derivatives associated with repeated eigenvalues of the generalized nondefective eigenproblem. This approach is an extension of recent work by Daily and by Juang et al. and is applicable to symmetric or nonsymmetric systems. The extended phases read as follows. The differentiable eigenvectors and their derivatives associated with repeated eigenvalues are determined for a generalized eigenproblem, requiring the knowledge of only those eigenvectors to be differentiated. Moreover, formulations for computing eigenvector derivatives have been presented covering the case where multigroups of repeated first eigenvalue derivatives occur. Numerical examples are given to demonstrate the effectiveness of the proposed method.