Two-scale concurrent topology optimization with multiple micro materials based on principal stress orientation

This paper studies two-scale concurrent topology optimization with multiple micro heterogeneous materials subjected to volume constraints. In previous work on concurrent two-scale optimization, either only one material with optimal micro-structure is assumed or multiple micro materials are included but are distributed in prescribed geometrical domains. Here the selection of micro heterogeneous materials is based on the criterion for principal stress orientation in the macro structure. To meet this requirement, an additional constraint, called misplaced material volume constraint, is introduced to constrain the volume fraction of material that is misplaced in macro structure to be less than a small parameter epsilon. This constraint comprises several piecewise smooth penalty functions, each of which is a proper modification of Heaviside function. One advantage of the misplaced material volume constraint is that, without much modification to the original formulation, the optimized macro material is distributed in line with the use criterion and the material microstructures automatically converge to different optimized topologies. Three numerical examples are presented to show the effectiveness of the proposed method.