Size-dependent two-scale topological design for maximizing structural fundamental eigenfrequency

A two-scale concurrent topology optimization method based on the couple stress theory is proposed for maximizing structural fundamental eigenfrequency. Because of the fact that the classical mechanics theory cannot reveal the size effect because of neglecting the influence of microstructure, the theory of couple stress including the microscopic properties of materials can be used to describe the size effect in deformations. On the foundation of the couple stress theory, the two-scale optimization model for finding optimal configurations of macrostructures and their periodic composite material microstructures is built. And the fundamental eigenfrequency of the macrostructure is maximized. The effective macroscopic couple stress constitutive constants of macrostructures are calculated by the representative volume element method. And a modified solid isotropic material with a penalization model is used to effectively avoid the localized mode. The optimization algorithm based on the bidirectional evolutionary structural optimization method is proposed. The optimal results of numerical examples show that the optimal topologies and natural frequencies obtained by the couple stress theory may differ significantly from those obtained by the typical Cauchy theory. It is obvious that couple stress theory can effectively describe the size effect in topology optimization.