Inverse design of phononic topological pumping in continuous solids

Topological insulators have been widely studied for their unique properties, particularly their ability to propagate energy with minimal losses in a manner that is robust to structural defects. More recently, topological pumping, which provides a mechanism to transport energy from one location to another in a structure without the need for direct coupling between the locations, has emerged as a phenomena of interest. However, previous studies on topological pumping of phonons have been performed without developing an understanding of how the efficiency of the pumping, as well as control over the pumping pathway in continuous solids, can be systematically controlled. Therefore, in this work we introduce a novel framework for the inverse design of continuous structures that can exhibit topological pumping of phonons, that is based on two key steps: (I) shape design of unit cells that not only exhibit topologically non-trivial edge states, but whose edge states span a wide range of phase values and wavenumbers at the excitation frequency to achieve a robust pumping effect; (II) optimizing the functional form to enable nonlinear modulation of the phase, which enables control both over the pumping path, and also the efficiency of the energy transport along the desired pumping pathway. Using this approach, we are able to establish connections between the dynamical properties of the unit cell, and various properties that impact the pumping efficiency, including the bandgap width, wavevector range, unit cell truncation, and the path of the phase modulation. We further demonstrate the ability to perform pumping for both out-of-plane and in-plane elastic waves, as well as for quantum valley Hall-based topological insulators.