A group theory based topology optimization scheme for the design of inhomogeneous waveguides with dihedral group symmetries

In this paper, we introduce a novel topology optimization scheme dedicated to designing waveguides with inhomogeneities holding rotation and reflection symmetries. To fully exploit the symmetry features of the guide, we build the scheme by first developing a new computational algorithm that combines the Finite Element Method (FEM) with the group representation theory (GRT), i.e., the GRT – FEM algorithm. This combination allows the decomposition of the eigenvalue problem in a classic FEM to orthogonal (decoupled) subproblems. The subproblems are of much smaller sizes and can be treated in parallel, which therefore leads to an improved solver efficiency by an order of magnitude. For a targeted mode profile, the involvement of the GRT does not only enable a pre-selection of the subproblems that are to be optimized, which further improves the computational speed, but also completely avoids potential mode degeneracies, which adversely affects the convergence of the optimization. To validate, the GRT – FEM algorithm is first tested against the results from a full FEM simulation for a guide with inhomogeneities with six-fold rotation and reflection symmetries. Then, as an illustration, the entire scheme is applied to achieve a desired eigenmode with a targeted propagation constant. The proposed scheme does not only provide an efficient design tool for exploiting interesting wave phenomena in waveguiding structures possessing rotation and reflection symmetries, but also lays down a theoretical foundation for systematically integrating symmetries with classic Computational Electromagnetics (CEM) algorithms.