Bi-orthogonality relations: general formulation, conversion to the FE format, and application for analysis of homogeneous and periodic waveguides

Bi-orthogonality relations are a powerful tool to solve complicated wave propagation problems in a surprisingly simple way. For any symmetric waveguide, these relations identify so-called Class A and Class B boundary conditions, for which eigenfrequencies and eigenmodes of a slice of waveguide provide dispersion diagrams of propagating waves and (for periodic wave-guides) location of pass- and stop-bands. It is a straightforward matter to convert these 'class-consistent' boundary conditions to the FE format and use standard FE modal analysis software to study properties of waveguides of arbitrary complexity. However, an application of this methodology, especially for periodic waveguides, is challenged by the necessity to track mode shapes and pair Class A and B eigenfrequencies into their modal families. A novel algorithm to resolve this issue is proposed and tested on post-processing results of modal analysis in the AN-SYS environment. Underlying theory of bi-orthogonality, analytical solutions of benchmark problems, and FE case-studies are presented and explained.