Optimization of one-dimensional Willis materials with hidden lumped elements

Willis coupling is a recently recognized material property that couples the pressure-strain and momentum-velocity equations and can lead to novel applications of metamaterials. We present an approach for optimizing the asymmetry factor (a non-dimensional measure of Willis coupling's effect on the specific acoustic impedance) in a metamaterial with lumped-element hidden degrees of freedom. The representative material element in the system we examine is a length of tube, which could be considered as a unit cell in a periodic system. Lumped-element features including side-branch resonators and membranes may be attached to the tube at various locations. We aim to determine optimal locations for each type of lumped-element feature, as well as optimal parameters in the design of the features themselves (e.g., for a resonator: volume, neck length, and neck cross-sectional surface area). An ideal design for a metamaterial exhibiting significant Willis coupling would result in a broadband non-negligible asymmetry factor.