In this study, we propose a novel multiscale optimization method for determining the optimal dimensions of microscopic structures composed of beams and shells. These microstructures are embedded within a macroscopic structure and are connected to it through the NIAH (Novel Numerical Implementation of Asymptotic Homogenization) method, which enables efficient coupling between different scales. The objective of the optimization is to maximize the overall stiffness of the macroscopic structure while satisfying a global volume constraint that includes the volume of the microstructures. The optimization problem is formulated as a size optimization problem based on the variational method, allowing for a mathematically rigorous treatment of distributed design variables. A sensitivity function is analytically derived using the method of Lagrange multipliers and the adjoint method, providing the gradient information required for efficient optimization. The scalar-type H1 gradient method is then employed to update the design variables, specifically the diameters of the beam elements and the thicknesses of the shell elements, in a way that guarantees smoothness and convergence. Through numerical examples, we demonstrate the effectiveness and applicability of the proposed method in optimizing the size distribution of complex microstructures. The results indicate that the method can handle intricate geometries and provide improved structural performance through optimized microstructural design.
SHINOJIMA et al. (Wed,) studied this question.