Topology optimization for functionally graded cellular composites with metamaterials by level sets

The application of auxetic composites in practice often relies on a compromise between properties as auxetics are mostly too porous (not dense enough or not stiff enough) to bear structural loads. Hence, the focus of this paper is topological design optimization of new functionally graded cellular composites with auxetics using a level set method. Firstly, a new hierarchical multi-scale formulation is developed to account for both the auxetic behavior of the microstructure and the stiffness of the macrostructure. The composite, comprising multiple layers of periodic microstructures, is tailored to have functionally graded properties for stiffness and auxetic behaviors, subject to volumetric gradient constraints. Secondly, the microstructures underpinning composite layers are topologically designed under the consideration of boundary and loading conditions of the macrostructure. Finally, a level set method is applied to evolve the shape and topology of the microstructure for each layer, with the numerical homogenization method to evaluate the effective properties of the microstructures. Several numerical examples are used to demonstrate the effectiveness of the proposed method. It can be seen that such composites systematically gear together the features of the functionally graded materials, cellular composites, and metamaterials towards a new kind of man-made composites.