Topology Optimization of Arc Unit Cells by Extending Energy-Based Homogenization and Parametric Level Set Methods

This paper extends the energy-based homogenization method (EBHM) to evaluate the effective elastic properties of two-dimensional arc unit cells, where the periodic boundary conditions are constructed based on the polar strain–displacement relationships, and the displacement field is solved by the polar finite element method. Combining the extended EBHM with the parametric level set method (PLSM) and the augmented Lagrangian multiplier method (ALM), it is able to design the topological configurations of arc unit cells for seeking superior effective elastic properties. In addition, the level set function is first filtered by compactly supported radial basis functions (CSRBFs) in order to eliminate small holes in the configuration. Numerical examples are performed to demonstrate the advantages of the proposed method. The periodic boundary condition performs well when the circumferential angle of the arc unit cell is in the range of [0.0002, 0.2]. Results indicate that the effective elastic properties of optimized arc unit cells override those of geometrically transformed optimal square unit cells. This superiority mainly benefits from the fact that unisymmetrical configurations are allowed in arc unit cells but not in square unit cells. Results also imply that the capability of nucleation of new holes is retained even though small holes are eliminated by filtering. It contributes to the trade-off between performance and manufacturability.