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.

Keywords

• multiscale optimization
• conccurent optimization
• homogenization method
• shape optimization
• h1 gradient method
• periodic microstructure