Inverse design of arbitrary non-reciprocal acoustic dispersion in non-local phononic crystals and metamaterials

Abstract

We present inverse methods to obtain arbitrary non-reciprocal dispersion relations in periodic, multi-degree-of-freedom lattices with non-local interactions. Just as phase-delayed interactions such as local resonance govern the frequency-dependence of a lattice’s dispersion relation, spatially non-local interactions govern the wavenumber dependence of the coefficients of the lattice’s characteristic equation. Thus, the use of fully general non-local interactions can, in principle, specify every branch of the dispersion diagram at every wavenumber. However, calculating a viable set of non-local interactions for a given set of desired dispersion curves is challenging, as there are multiple possible sets of coefficients that yield the same polynomial roots. To solve this problem, we first present a general method to calculate the dispersion relation of multi-degree-of-freedom non-local lattices, showing that non-local interactions create a Fourier series expansion that governs the wavenumber dependence of each matrix element. Next, we present numerical techniques to solve the inverse problem—i.e., given the desired dispersion relation, calculate the required non-local interactions—and discuss the associated computational challenges. Finally, we discuss practical methods to realize non-local interactions leveraging piezoelectric sensors and actuators, and we highlight how non-local interactions can be introduced in unit-cell based finite-element dispersion calculations.