Topology optimization of anisotropic metamaterials tracing the target EFC and field polarization

Equi-frequency contours (EFCs) can be used to predict certain intrinsic wave behaviors of a metamaterial, and are therefore frequently used for analysis and design. However, information about field polarization (FP) dictating whether particle motion along an EFC is longitudinal, shear, or longitudinal-shear coupled should also be considered to accurately predict the actual wave phenomena of a metamaterial. For instance, anomalous polarization can be used to create mode converters characterized only when both EFCs and FPs are simultaneously considered. In this work, we aim to develop a topology optimization method to design anisotropic metamaterials that trace target EFCs and FPs. Earlier studies on the topology optimization method have considered EFCs or dispersion curves; however, there is no topology optimization that considers the EFC and FP simultaneously. As a means to trace the FP, we introduce a scalar measure equivalent to the modal assurance criterion and derive its sensitivity with respect to density design variables. We present various case studies to design metamaterial unit cells to trace target EFCs and FPs corresponding to anomalous polarization. Furthermore, mode converters made of the metamaterials designed by the developed topology optimization method are created and their performance in converting a longitudinal wave mode to a transverse wave mode is evaluated.