Topology optimization for microstructures of materials with macrostructure property constraint

To design periodic optimized microstructures of cellular steel or steel/aluminum composites, a topological optimization model was developed by independent continuous mapping method where minimized total mass of structure was taken as objective, and nodal displacement was referred to as constraint condition. It is assumed that macrostructure is made of cellular materials or composites whose effective properties are calculated through the homogenization theory. Topological variables in micromaterials were defined. Nodal displacement constraint was approximately formulated in terms of the first-order Taylor expansion. Various demands for designing materials were treated as constraint conditions in optimization model. The sensitivities expression of nodal displacement and total mass were derived. Filtering method by solving partial differential equation was adopted to eliminate numerical instabilities. A variety of optimal material microstructures which meet design requirements have been obtained in some 2D numerical examples. The results validate the feasibility and effectiveness of the proposed method for the topology optimization design for microstructures of materials.