Inverse design of nonlocal lattices with arbitrary multiband dispersion relations

We present a numerical algorithm to implement arbitrary multiband dispersion relations in one-dimensional periodic lattices with N[jls-end-space/]-degree-of-freedom unit cells using nonlocal interactions. First, we present a general model to calculate the dispersion relation for periodic lattices with N[jls-end-space/]-DOF unit cells with linear, mass–spring–damper-type interactions. We show that the dispersion relation is obtained from a quadratic eigenvalue problem, in which nonlocal interactions yield wavenumber-dependent mass, stiffness, and damping matrices via a Fourier series. In principle, because these properties can be specified at every wavenumber, nonlocal interactions can yield any set of dispersion curves. We then develop a generalized numerical framework that enables the inversion of dispersion relations for metamaterials with arbitrary band structures, including active and damped components, variable mass distributions, and inertial interactions. By formulating the inverse problem as an optimization task, we incorporate constraints ensuring physical realizability while employing L1 regularization to promote sparsity in interaction matrices. Our approach successfully reconstructs desired dispersion relations for various test cases, demonstrating the ability to design lattices with customized multiband wave characteristics. The results highlight the robustness and versatility of our methodology, providing a powerful toolkit for advanced wave manipulation and nonreciprocal wave control.