An approach for the concurrent homogenization-based microstructure type and topology optimization problem

This work proposes a methodology for the concurrent homogenization-based optimization of the type and the configuration of the microstructure of the structural domain, based on a list of pre-defined composite microstructures. The candidate microstructures are represented on this list by their homogenized mechanical properties, as predicted by means of the 3D homogenization theory. As such, each candidate property is tied to a specific microstructure -the one it has been derived from- and is conditioned on its topology, i.e. the geometric configuration of its unit cell. Similar to the standard discrete multi-material optimization problem (DMOP), in order to identify per element/patch in the structural domain the optimal microstructure from that list, weights are assigned to the candidate homogenized properties, with respect to which the final discrete microstructure type optimization problem (DMTOP) is posed. The topology of the optimal microstructure is determined by the volume constraint(s) imposed on one or more of its material components in the homogenization-based topology optimization problem (HTOP). The aim of this work is to combine the DMTOP with the HTOP in a unique mathematical framework in order to determine a unique microstructure type per element/patch in the structural domain, concurrently optimized in its topology. The proposed methodology is built step-by-step through the introduction of four microstructure types of two distinct constituent materials. Upon these auxiliary microstructures, the generalized concurrent homogenization-based topology and discrete microstructure type optimization problem (HTDMOP) is formulated for compliance minimization of the structure. Further, it is illustrated how the DMOP can be perceived as a DMTOP, and hence be combined in a similar fashion with the HTOP for the concurrent homogenization-based topology and material optimization (HTMO) of the structural domain. The paper concludes with demonstrating the developed methodology on the benchmark academic case study of the 3D Messerchmitt-Bölkow-Blohm (MBB) beam for the case where the auxiliary microstructures are considered as candidates for the domain.