Topology Optimization of Periodic Cellular Materials through the Progressive Directional Selection Method and Finite-Volume Theory

Resumo

This investigation presents the Progressive Directional Selection (PDS) method to optimize the topology of periodic cellular materials, achieving high performance and minimal weight. The homogenized elastic properties of cellular materials are determined using the homogenization method applied to periodic materials based on the unit cell concept as an intermediate step of the topology optimization procedure. The literature often constructs the design domain to conduct a finite element analysis. However, some problems are related to numerical issues, such as the checkerboard pattern and mesh dependence. The checkerboard effect is related to the assumptions of the finite element method, as the satisfaction of equilibrium and continuity conditions at the element nodes, particularly for linear triangular and quadrilateral discretizations in the absence of regularization schemes. This problem can be overcome by the Finite-Volume Theory (FVT), which satisfies the equilibrium equations at the subvolume level, and the compatibility conditions are established through the adjacent subvolume interfaces. The PDS method is inspired by natural selection processes found in biology and employs a strategy to meet the objective function of a discretized analyzed domain subject to a volume constraint. Population selection is based on performance criteria specific to the problem through an iterative process that concludes when the optimized topology ceases to evolve. Numerical examples of topology optimization for materials with periodic cellular microstructures are analyzed using PDS and FVT.