# Modal Tracking Based on Group Theory

## TL;DR

This paper introduces a group theory-based method for improved modal tracking in eigenvalue problems, effectively handling crossings and degeneracies, with applications demonstrated in characteristic mode analysis and computational acceleration.

## Contribution

It presents a novel modal tracking approach using point group theory, enhancing accuracy and efficiency in eigenvalue problems with symmetry considerations.

## Key findings

- Improved modal tracking accuracy in crossing scenarios.
- Effective handling of modal degeneracies.
- Accelerated computations using symmetry-adapted bases.

## 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.