Design of lattice structures with direct multiscale topology optimization

The structure of latticed/cellular materials is often designed with the lack of information about macro-material. Material information of each macro-element is realized by reducing the scale, homogenizing the microstructure, and calculating the properties of an equivalent material for the macro-element. The lattice structure is simultaneously optimized at both the macro- and microstructural levels with additional connectivity constraints, while finite element analysis (FEA) and design variable updates are required twice (at the macro- and micro-levels) for each optimization loop. This approach requires significant storage and has a substantial computational cost. In addition, when the size of the unit cell is quite large compared to the macrostructure, the homogenization method could fail to provide sufficient accuracy. To deal with these issues, in this work, we propose a new multiscale topology optimization approach for the direct and simultaneous design of lattice materials, without material homogenization at the microscale, using adaptive geometric components. The adaptive geometric components are projected onto macro- and micro-element density fields to calculate the effective densities of grid elements. Macro-and microstructures are simultaneously optimized, considering the load and boundary conditions of the overall structure without any additional constraints. FEA and design variable updates are required only once for each optimization loop. Furthermore, the minimum length scales of the macrostructure and the length scales of microstructures can be simultaneously controlled explicitly by simply adjusting the bounds of the size parameters. Some benchmark structures are topologically optimized with different types of lattice materials (such as square, diamond, and triangle) to verify the effectiveness of the proposed method.