Multiscale topology optimization for frequency domain response with bi-material interpolation schemes

In areas that require high performance components, such as the automotive, aeronautics and aerospace industries, optimization of the dynamic behavior of structures is sought through different approaches, such as the design of materials specific to the application, for instance through structural topology optimization. The bi-directional evolutionary structural optimization (BESO) method, in particular, has been used for the simultaneous design of hierarchical structures, which means that the structural domain consists not only of the macrostructure but also of the microstructural topology of the materials employed. The purpose of this work is to apply the BESO method to solve two-dimensional multiscale problems in order to minimize the response of structures subjected to forced vibrations in a given frequency range. The homogenization method is applied to integrate the different scales of the problem. In particular, the material interpolation model for two materials is used. The BESO method is applied to different cases of optimization, in macroscale, microscale, and multiscale structural domains. Numerical examples are presented to validate the optimization and demonstrate the potential of this approach. The numerical examples show that the multiscale bi-material topology optimization method implemented here is able to produce structures and microstructures for optimization of the frequency domain response, satisfying prescribed volume constraints.