Determining stop band locations in periodic waveguides by conventional finite element modal analysis

The use of metamaterials for the mitigation of structural wave propagation shows relevance in a wide range of engineering solutions. At the present time, the design of such metamaterials is often based on iterative design and numerical optimization around the Wave Finite Element Method. This approach is computationally heavy. In this paper, it is demonstrated how post-processing of mode shape results from conventional finite element modal analysis can identify partial stop bands in periodic waveguides. This provides a foundation for iterative design and efficient numerical optimization of periodic waveguides. When the modal analysis is performed on a symmetric unit periodicity cell, using ‘class-consistent’ boundary conditions, the eigenfrequency-locations coincide with the beginning and end of partial stop bands. Identifying which eigenfrequencies pair up, to indicate a partial stop band, is a matter of determining which modeshapes belong to the same branch in the dispersion diagrams. A novel algorithm is developed and tested, able to pair-wise match modal solutions belonging to the same branch of the dispersion diagrams, only based on mode shape information, available from any commercial finite element code.