A topology optimization method for design of negative permeability metamaterials

A methodology based on topology optimization for the design of metamaterials with negative permeability is presented. The formulation is based on the design of a thin layer of copper printed on a dielectric, rectangular plate of fixed dimensions. An effective media theory is used to estimate the effective permeability, obtained after solving Maxwell's equations on a representative cell of a periodic arrangement using a full 3D finite element model. The effective permeability depends on the layout of copper, and the subject of the topology optimization problem is to find layouts that result in negative (real) permeability at a prescribed frequency. A SIMP-like model is invoked to represent the conductivity of regions of intermediate density. A number of different filtering strategies are invoked to facilitate convergence to binary solutions. Examples of designs for S-band applications are presented for illustration. New metamaterial concepts are uncovered, beyond the classical split-ring inspired layouts.