  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.
