Phononic bandgap metamaterials, which consist of periodic cellular structures, are capable of absorbing energy within a certain frequency range. Designing metamaterials that trap waves across a wide wave frequency range is still a challenging task. In this paper, we present a deep feature learning-based design framework for both unsupervised generative design and supervised learning-based exploitative optimization. The Gaussian mixture beta variational autoencoder (GM-βVAE) is employed to extract latent features as design variables. Gaussian Process (GP) regression models are trained to predict the relationship between latent features and the properties for the property-driven optimization. The optimal structural designs are reconstructed by mapping the optimized latent feature values to the original image space. Compared with the regular variational autoencoder (VAE), we demonstrate that GM-βVAE has better learning capability and is able to generate a more diversified design set in unsupervised generative design. Furthermore, we propose an iterative GM-βVAE model updating-based design framework. In each iteration, the optimal designs found property-driven optimization are used to update the training dataset. The GM-βVAE model is re-trained with the updated dataset for the optimization search in the next iteration. A comparative study between the traditional single-loop and the iterative approaches is presented to demonstrate the effectiveness of the iterative framework. The caveats to designing phonic bandgap metamaterials are summarized.

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Phononic bandgap metamaterials, which consist of periodic cellular structures, are capable of absorbing energy within a certain frequency range. Designing metamaterials that trap waves across a wide wave frequency range is still a challenging task. In this study, we proposed a deep feature learning-based framework to design cellular metamaterial structures considering two design objectives: bandgap width and stiffness. A Gaussian mixture variational autoencoder (GM-VAE) is employed to extract structural features and a Gaussian Process (GP) model is employed to enable property-driven structure optimization. By comparing the GM-VAE and a regular variational autoencoder (VAE), we demonstrate that (i) GM-VAE has the advantage of learning capability, and (ii) GM-VAE discovers a more diversified design set (in terms of the distribution in the performance space) in the unsupervised learning-based generative design. Two supervised learning strategies, building independent single-response GP models for each output and building an all-in-one multi-response GP model for all outputs, are employed and compared to establish the relationship between the latent features and the properties of interest. Multi-objective design optimization is conducted to obtain the Pareto frontier with respect to bandgap width and stiffness. The effectiveness of the proposed design framework is validated by comparing the performances of newly discovered designs with existing designs. The caveats to designing phonic bandgap metamaterials are summarized.