Topological Derivative-Based Optimization of Micro-Structures Considering Different Multi-Scale Models

A recently proposed algorithm for micro-structural optimization, based on the concept of topological derivative and a level-set domain representation, is applied to the synthesis of elastic and heat conducting bi-material micro-structures. The macroscopic properties are estimated by means of a family of multi-scale constitutive theories where the macroscopic strain and stress tensors (temperature gradient and heat flux vector in the heat conducting case) are defined as volume averages of their microscopic counterparts over a Representative Volume Element (RVE). Several finite element-based examples of micro-structural optimization are presented. Three multi-scale models, providing an upper and a lower bound for the macroscopic properties as well as the classical periodic medium solution, are considered in the optimization process. These models differ only in the kinematical constraints (thermal constraints in the heat conducting case) imposed on the RVE. The examples show that, in general, the obtained optimum micro-structure topology depends on the particular model adopted.