Modal Tracking Based on Group Theory

Abstract

Issues in modal tracking in the presence of crossings and crossing avoidances between eigenvalue traces are solved via the theory of point groups. The von Neumann-Wigner theorem is used as a key factor in predictively determining mode behavior over arbitrary frequency ranges. The implementation and capabilities of the proposed procedure are demonstrated using characteristic mode decomposition as a motivating example. The procedure is, nevertheless, general and can be applied to an arbitrarily parametrized eigenvalue problems. A treatment of modal degeneracies is included and several examples are presented to illustrate modal tracking improvements and the immediate consequences of improper modal tracking. An approach leveraging a symmetry-adapted basis to accelerate computation is also discussed. A relationship between geometrical and physical symmetries is demonstrated on a practical example.

Characteristic Modes: Progress, overview, and emerging topics

“…The same applies for excitation of multiband resonances [37]. Another benefit of tracking is based on time-domain characteristics inferred from modal quantities [38], where well-tracked traces embody causal response.…”

“…However, the tracking procedure is opened to interpretations induced by von Neumann-Wigner theorem [38], [39] and that the tracking becomes irrelevant when all the modal currents are superposed after excitation is added. Moreover, one should not deny that correct modal tracking can be challenging for certain structures where modal properties may change significantly over a small frequency interval even for the same mode.…”

Unified Theory of Characteristic Modes: Part II -- Tracking, Losses, and FEM Evaluation

“…Characteristic mode decomposition is solved independently for each frequency, i.e., the characteristic numbers and characteristic modes are initially uncorrelated at discrete frequency points, though it is often desirable to apply modal tracking to order and associate modal quantities such that they may be interpreted as continuous functions of frequency. Many methods have been proposed for numerical [4]- [7], [14], [15] or analytical symmetry-based [8], [16] tracking of characteristic mode quantities, however the small size of the transition matrix T [1], [17] and the high accuracy of its corresponding characteristic mode decomposition with eigenvectors related to the far field, see Part I [1], make it exceptionally well suited for this task.…”

“…with R 0 and X 0 being the real and imaginary parts of impedance matrix [20], in the 14 sample points using eigs() function in MATLAB. Reducing the discretization to 3120 basis functions reduces the aforementioned evaluation time to 6 s using matrix T and to 8 s using the decomposition (8).…”

“…Modal tracking [4]- [9] is a challenging post-processing step required both for visual inspection of the eigentraces [9] (characteristic numbers parameterized by frequency) and interpretation of physical properties of characteristic modes [8], [10]. In this work, decomposition via ordinary eigenvalue problems involving matrices with favorable algebraic properties is used to develop a tracking algorithm which operates natively over far fields and utilizes approximations of poles Manuscript received October 6, 2021; revised October 6, 2021.…”