Topological design of microstructured materials using energy-based homogenization theory and proportional topology optimization with the level-set method

In this study, we propose a novel approach to designing microstructures of porous material with extreme mechanical properties to maximize the bulk or shear modulus. The approach integrates energy-based homogenization theory and proportional topology optimization with the level-set method (HPTO-LSM). A level-set function (LSF) evolution strategy based on the nodal element mutual energies (EME) proportion is proposed and drives the LSF's evolution. An EME proportion filtering technique is proposed and reduces the redundant branch structure. And a linear scaling strategy is applied to the HPTO-LSM to enhance the topological change ability. The effectiveness of the HPTO-LSM for designing microstructures of cellular materials to maximize the bulk or shear modulus is demonstrated using several 2D and 3D cases. The effects (changes in objective function values or topologies) of the EME proportion filtering scheme and scaling strategy on the HPTO-LSM are discussed. In addition, we compared the HPTO-LSM method with the solid isotropic material with penalization (SIMP) and bidirectional evolutionary structural optimization (BESO) methods through numerical examples. As shown by the results, effective objective function values and 2D and 3D microstructural topologies with clear interfaces and smooth boundaries can be acquired using the new approach. Applying the EME proportion filtering scheme and scaling strategy in the HPTO-LSM not only enhanced the topological variation of microstructures but also facilitated the acquisition of microstructures conducive to engineering applications. Compared to the SIMP and BESO methods, the HPTO-LSM method has significant advantages in obtaining better objective function values and better microstructural topologies.