Multi-scale topology optimization of structures with multi-material microstructures using stiffness and mass design criteria

Nowadays, there is a great interest on the part of the automotive and aerospace industry to design environmentally-friendly structures. To that purpose, stiffness-oriented designs are proposed here by extending previous work on multi-scale topology optimization to the multi-material setting reformulating the problem to include appropriately mass constraints and discussing different design domain parametrizations and algorithmic strategies. On the macroscale, the problem of minimizing the compliance subject to a global mass constraint is addressed. On the microstructure scale, the multi-material design is carried out by solving the problem of minimizing the local complementary strain energy density with mass density constraint. As a result, very efficient structures composed of spatially varying porous and multi-material microstructures are obtained. The optimal design of the multi-material microstructure can be done either in a pointwise manner or in larger subdomains, to promote design uniformity. These parametrizations are here compared and discussed. Moreover, two different algorithmic strategies to solve the multi-scale problem are proposed, and their pros and cons discussed. They differ in the way the macro and micro design variables are related and updated. The macro design variables consider the mass density distribution along the structure, while the micro variables define the microstructure's topology using a multi-material SIMP interpolation scheme. The results show very efficient structures with locally optimized multi-material microstructures, which can outperform their single-material counterparts with regards to stiffness while maintaining the same mass. Additionally, the maximum stress verified on multi-material structures tends to be lower than the one obtained in the single-material counterparts.