A level set-based topology optimization method targeting metallic waveguide design problems

In this paper, we propose a level set-based topology optimization method targeting metallic waveguide design problems, where the skin effect must be taken into account since the metallic waveguides are generally used in the high-frequency range where this effect critically affects performance. One of the most reasonable approaches to represent the skin effect is to impose an electric field constraint condition on the surface of the metal. To implement this approach, we develop a boundary-tracking scheme for the arbitrary Lagrangian Eulerian (ALE) mesh pertaining to the zero iso-contour of the level set function that is given in an Eulerian mesh, and impose Dirichlet boundary conditions at the nodes on the zero iso-contour in the ALE mesh to compute the electric field. Since the ALE mesh accurately tracks the zero iso-contour at every optimization iteration, the electric field is always appropriately computed during optimization. For the sensitivity analysis, we compute the nodal coordinate sensitivities in the ALE mesh and smooth them by solving a Helmholtz-type partial differential equation. The obtained smoothed sensitivities are used to compute the normal velocity in the level set equation that is solved using the Eulerian mesh, and the level set function is updated based on the computed normal velocity. Finally, the utility of the proposed method is discussed through several numerical examples.