Topological design of microstructures using periodic material-field series-expansion and gradient-free optimization algorithm

Light-weight cellular materials with periodic repetitive microstructures are widely used in various fields due to their superior mechanical/multi-physical performances. As the microstructural design problem is known to have multiple local minima, most gradient-based topology optimization methods significantly depend on the initial guess of the microstructural geometry, thus requiring the designer's experiences. This paper presents an effective gradient-free framework for periodic microstructure design, which exhibits powerful global searching capabilities and requires no sensitivity information. The proposed framework combines the material-field series-expansion (MFSE) topology representation of periodic microstructures and the sequential Kriging-based optimization algorithm. The MFSE method decouples the topological representation from the finite element discretization, and describes a relatively complex microstructural topology with high-quality boundary description using a greatly reduced number of design variables. Based on the Kriging surrogate model, a solution scheme is suggested to solve material microstructural topology optimization successively in a sequence of sub-optimization problems with self-adaptive design spaces. With the present gradient-free optimization method, high-performance cellular materials that approach the H-S upper bound with porosities from 0.2 to 0.6, or that achieve negative Poisson's ratios of -0.94 in the principal directions for materials with square symmetry, are obtained without prior knowledge of the optimum microstructural topology.