Shape optimization of periodic-microstructures for stiffness maximization of a macrostructure

In this study, we present a shape optimization approach for designing the shapes of periodic microstructures using the homogenization method and the H1 gradient method. The compliance of a macrostructure is minimized under the constraint conditions of the total area of the microstructures distributed in the macrostructure, the elastic equation of the macrostructure and the homogenization equation of the unit cells. The shape optimization problem is formulated as a distributed-parameter optimization problem, and the shape gradient function involving the state and adjoint variables for both the macro- and micro-structures is theoretically derived. Clear and smooth boundary shapes of the unit cells can be determined with the H1 gradient method. The proposed method is applied to multiscale structures, in which the numbers of domains with the microstructures are varied and the optimized shapes of the unit cells and the compliances obtained are compared. The numerical results confirm the effectiveness of the proposed method for creating the optimal shapes of microstructures distributed in macrostructures One of the solutions that sustainably balances environmental problems such as resource saving and CO2 emissions with the issues of improving the performance of industrial products is to reduce the weight of the product, and the demand for it becomes stricter with each development. Weight reduction is a vital issue for transportation equipment that requires fuel efficiency and kinetic performance, and designers are always on the lookout for solutions. One of the solutions that is attracting attention is a multi-scale structure in which microstructures made of various porous materials are distributed and arranged in appropriate places, and a design method thereof (see Fig. 1). The use of porous materials, which are often found in biological structures, has been around for a long time, but the recent innovation in 3D printing technology [1] for metal and resin materials has made this a reality. However, there is a limit to the design of microstructures through testing based on intuition and experience. As result, much attention has been paid to multiscale structure optimization, which concurrently