Efficient Inverse Design of 2D Elastic Metamaterial Systems using Invertible Neural Networks

Locally resonant elastic metamaterials (LREM) can be designed, by optimizing the geometry of the constituent self-repeating unit cells, to potentially damp out vibration in selected frequency ranges, thus yielding desired bandgaps. However, it remains challenging to quickly arrive at unit cell designs that satisfy any requested bandgap specifications within a given global frequency range. This paper develops a computationally efficient framework for (fast) inverse design of LREM, by integrating a new type of machine learning models called invertible neural networks or INN. An INN can be trained to predict the bandgap bounds as a function of the unit cell design, and interestingly at the same time it learns to predict the unit cell design given a bandgap, when executed in reverse. In our case the unit cells are represented in terms of the width's of the outer matrix and middle soft filler layer of the unit cell. Training data on the frequency response of the unit cell is provided by Bloch dispersion analyses. The trained INN is used to instantaneously retrieve feasible (or near feasible) inverse designs given a specified bandgap constraint, which is then used to initialize a forward constrained optimization (based on sequential quadratic programming) to find the bandgap satisfying unit cell with minimum mass. Case studies show favorable performance of this approach, in terms of the bandgap characteristics and minimized mass, when compared to the median scenario over ten randomly initialized optimizations for the same specified bandgaps. Further analysis using FEA verify the bandgap performance of a finite structure comprised of $8\times 8$ arrangement of the unit cells obtained with INN-accelerated inverse design.