Application of topology optimization for the optimal design of periodic cellular materials employing the finite-volume theory

Abstract

Topology optimization is a technique employed in engineering to determine the optimal distribution of material within a defined design domain for various applications. This technique has been applied to the design of cellular materials to create an efficient microstructural arrangement that meets the necessary performance criteria and minimizes the weight of the material. This work presents an innovative approach that combines principles from finite-volume theory, homogenization methods, and density-based topology optimization to design highly efficient periodic cellular materials with optimized macroscopic elastic properties. The homogenization methods applied to periodic materials based on the unit cell concept determine the effective elastic properties of cellular material as an intermediate step in the topology optimization procedure. To find the optimal microstructural topology, specific linear combinations of components of the effective constitutive matrix are considered to obtain extreme elastic properties, such as maximum shear and bulk moduli or negative Poisson’s ratio, considering a prescribed volume constraint. The proposed approach uses a strategy based on potential functions of material stiffness to penalize intermediate material densities and apply the continuation method to minimize instability arising from the extreme local problem. Numerical examples involving materials with periodic cellular microstructures are analyzed, and the results demonstrate the excellent performance of the finite-volume theory for the optimal design of cellular materials.