A hierarchical model for concurrent material and topology optimisation of three-dimensional structures

This paper presents an extension of the hierarchical model for topology optimisation to three-dimensional structures. The problem addressed covers the simultaneous characterisation of the optimal topology of the structure and the optimal design of the cellular material used in its construction. In this study, hierarchical suggests that the optimisation model works at two interconnected levels, the global and local levels identified, respectively, with the structure and its material. The class of cellular materials, defining the material microstructure, is restricted to single scale cellular materials, with the cell geometry locally optimised for the given objective function and constraints. The model uses the asymptotic homogenisation model to obtain the equivalent material properties for the specific local microstructures designed using a SIMP based approach. The necessary optimality conditions for the hierarchical optimal design problem are discussed and approximated numerically by a proper finite element discretisation of the global and local analysis and design problems. Examples to explore and demonstrate the model developed are presented.