A robust topological derivative-based multi-material optimization approach: Optimality condition and computational algorithm

Abstract

Topology optimization is a technique in the engineering area that allows obtaining optimal designs of devices, mechanisms, complex structures and even design the microstructure of an auxetic material. Classically, the focus was made on considering the problem of obtaining such designs comprised of only one material. More recently, some efforts have been made to incorporate more than one material to the optimization problem. The present work addresses the topology optimization problem considering multiple materials on a fixed project domain. For the geometric description of the material domains, the concept of multiple level-sets has been used, meanwhile, its nucleation and evolution are governed by functions constructed from the topological derivative of a functional to be minimized. The topological optimization algorithm developed to solve the multi-material topological optimization problem is described and its optimality condition is well established. Numerical examples applied to planar elasticity, bending plate, bi-dimensional steady-state heat transfer and optimal design of piezoelectric actuator problems are presented.