Data-driven metamaterial design with Laplace-Beltrami spectrum as “shape-DNA”

Abstract

The design of multiscale metamaterial systems often suffers from high computational cost and incompatible boundaries between unit cells. As a result, unit cells are either assumed to be repeated (periodic) everywhere or limited to a small number of shapes. To address these limitations, this work proposes a data-driven design framework consisting of a metamaterial genome with a reduced-order geometrical representation as well as methods for the efficient design and analysis of 2D aperiodic metamaterials with compatible boundaries. To collect a large amount of designs, a set of unit cells generated by topology optimization is taken as initial seeds for the genome, and then expanded iteratively through random shape perturbations to form a rich database that covers a wide range of properties. For a reduced-order representation, the Laplace-Beltrami (LB) spectrum is adopted to describe complex unit cell shapes using a low number of descriptors, therefore significantly reducing the design dimensionality. Moreover, the physical and geometrical information contained in the LB spectrum is revealed through both quantitative and theoretical analysis. This information as well as the lower dimensionality allows the genome to be effectively leveraged to build a neural network model of structure-property relations for the rapid design of new unit cells. Finally, the combination of the metamaterial genome with an efficient optimization method based on the Markov random field (MRF) model is proposed to ensure connected boundaries between unit cells in multiscale aperiodic microstructure designs.