Hierarchical design of structures and multiphase material cells

• A hierarchical model of structures and multiphase material cells is proposed.
• Aiming is to find the minimum system thermal compliance under double constraints.
• Multiphase materials have superiority in hierarchical optimization.
• The advantage of hierarchical porous structure is its multifunctional properties.
• The volume fraction constraint and mass constraint play different roles.

This paper presents a hierarchical topology optimization method to simultaneously achieve the optimum structures and multiphase material cells for minimum system thermal compliance. Macro design variables and micro phase design variables are introduced independently, and coupled through elemental phase relative density. Based on uniform interpolation scheme with multiple materials, the sensitivities of thermal compliance with respect to the design variables on the two scales are derived. Correspondingly, the hierarchical optimization model of structures and multiphase material cells is built under prescribed volume fraction and mass constraints. The proposed method and computational model are validated by several 2D numerical examples. The superiority of multiphase materials in hierarchical optimization is presented through the comparison of single phase materials. The optimized results of periodic structure, hierarchical structure and traditional continuous structure are compared and analyzed. At last, the effects of volume fraction and mass constraints are discussed.