Dimension reduction and surrogate based topology optimization of periodic structures

The frequency bandgap of periodic structures has many potential applications. Topology optimization of the unit cell of periodic structures offers great potential to design periodic structures with desirable bandgap characteristics. However, topology optimization typically involves many design variables stemming from discretization of the unit cell. This creates computational challenges for both optimization (i.e., high-dimensional discrete design variables) and the calculation of frequency bandgaps for a given design (i.e., finer discretization requires higher computational effort). To address these challenges, this paper proposes an efficient dimension reduction and surrogate based approach for topology optimization of periodic structures. Using information from a set of reference topologies with more desirable bandgap characteristics, dimension reduction technique (i.e., logistic principal component analysis) is used to establish a low-dimensional representation of different topologies in latent design space. To reduce computational effort in calculation of bandgap, Kriging surrogate model is built with respect to the low-dimensional latent continuous design variables, and used within efficient global optimization to efficiently and adaptively identify the optimal topology. The effectiveness and great efficiency of the proposed approach are verified through an example on topology optimization of 2D periodic structures to maximize the in-plane frequency bandgaps.