Multi-material topology optimization design of microstructures with extreme mechanical properties

Porous materials have found extensive application in engineering structural design, which are often constituted by periodic microstructures. Multi-material topology optimization can achieve periodic microstructures with better comprehensive performance through the combination of materials with different properties. A novel multi-material topology optimization method is proposed for microstructure design containing three or more phases. Energy-based homogenization approach is utilized to evaluate the equivalent elastic properties of 2D/3D microstructures. The combination of the linear material model and the floating projection constraint results in reasonable 0/1 solutions and eliminates the numerical instability caused by excessive material penalties. The optimality criterion is employed to update multi-material variables. The three-field density representation technique is introduced to accelerate the formation of topological structure. With the objective of maximum bulk modulus or maximum shear modulus, the optimization design of multi-material microstructure is carried out under multiple volume constraints. Several 2D and 3D examples indicate that the proposed multi-material optimization method can realize a reasonable distribution of multiple materials and design microstructures with extreme mechanical properties. The final structure has distinct boundaries and material interfaces by smooth design, which can be directly used for manufacturing. It can provide guidance for the design of engineering structures comprising multi-material microstructures, and can also be potentially applied to the design of multi-functional metamaterials.