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

Abstract

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