Concurrent multi-scale shape optimization for micro and macro shape design of structures

Abstract

In this study, we propose a novel shape optimization method for designing micro- and macro-structures concurrently. We assume the macro-structure consists of several arbitrary domains, which have different periodic micro-structures. The macro-structure and the micro-structures are connected by the homogenized elastic tensors, which are calculated by applying the homogenization method to the unit cells of the micro-structures. Defining the boundary shapes of the macro-, the micro-structures and the interface shapes between the domains as design variable, the compliance of the macrostructure is minimized. The volume of the macro-structure considering the whole holes in the micro-structures is used as the constraint. The homogenization equations for the micro-structures and the equilibrium equation for the macro-structure are also used as the constraint. This design problem is formulated as a distributed-parameter optimization problem, and the shape sensitivity functions are theoretically derived. The optimum boundary and the interface shapes of the macro- and the micro-structures are determined by applying the shape sensitivity functions to the H¹ gradient method. The proposed concurrent shape optimization method is applied to several numerical examples to confirm the effectiveness of the proposed method for designing the shapes of multi-scale structures. Also, the compliance and the shapes optimized are, compared and discussed for the different domains.