Concurrent topology optimization of macrostructures and material microstructures for natural frequency

Based on the bi-direction evolutionary structural optimization (BESO) method, a concurrent two-scale topology optimization algorithm is proposed for maximizing natural frequency of structures. The macro-scale structure is assumed to be constructed with a composite material, whose microstructure is represented by periodic unit cells (PUC). This optimization scheme aims to obtain the optimal topologies of the structure at the macro-scale level and microstructure of its material at the micro scale simultaneously, so that the resulting structure with a given weight has maximum natural frequency. The effective properties of a composite material with representative PUC are homogenized and integrated into the frequency analysis of the macrostructure. To implement topology optimization at both scales, the design variables are assigned for both the macrostructure and microstructure of its material. The sensitivity analysis with regard to the variation of design variables is conducted for iteratively updating the topologies at both scales synchronously. Numerical 2D and 3D examples are presented to demonstrate the validity of the proposed concurrent optimization algorithm for frequency optimization problems.