Multiple bi-directional FGMs topology optimization approach with a preconditioned conjugate gradient multigrid

This article aims to introduce a multigrid preconditioned conjugate gradient (MPCG)-oriented topology optimization (TO) methodology using multiple bi-directional functionally graded (BFG) models for the first time. For that purpose, the MPCG paradigm is integrated into the TO procedure to more effectively and quickly resolve linear algebraic systems arising from the differential equations' discretization. In addition, a refined BFG material interpolation in Solid Isotropic Material with Penalization (SIMP), which is based on an alternating active-phase algorithm, is produced. In which continuously altered macroscopic material properties are represented by an explicit exponential function. This paper describes the well-founded mathematical formulations for multi-material topology optimization of BFG structures in great detail. Finally, several numerical examples are tested to demonstrate the proposed approach's capability and efficiency.