Topology optimization for negative permeability metamaterials using level-set algorithm

This paper aims to develop a level-set-based topology optimization approach for the design of negative permeability electromagnetic metamaterials, where the topological configuration of the base cell is represented by the zero-level contour of a higher-dimensional level-set function. Such an implicit expression enables us to create a distinct interface between the free space and conducting phase (metal). By seeking for an optimality of a Lagrangian functional in terms of the objective function and the governing wave equation, we derived an adjoint system. The normal velocity (sensitivity) of the level-set model is determined by making the Eulerian derivative of the Lagrangian functional non-positive. Both the governing and adjoint systems are solved by a powerful finite-difference time-domain algorithm. The solution to the adjoint system is separated into two parts, namely the self-adjoint part, which is linearly proportional to the solution of the governing equation; and the non-self-adjoint part, which is obtained by swapping the locations of the incident wave and the receiving planes in the simulation model. From the demonstrative examples, we found that the well-known U-shaped metamaterials might not be the best in terms of the minimal value of the imaginary part of the effective permeability. Following the present topology optimization procedure, some novel structures with desired negative permeability at the specified frequency are obtained.