A non-homogeneous multi-material topology optimization approach for functionally graded structures with cracks

This article presents an effective non-homogeneous multi-material topology optimization paradigm for functionally graded (FG) structures, considering both cracked and non-cracked cases for the first time. For that aim, an enrichment finite element concept known as the extended finite element method (X-FEM) is employed to analyze strong discontinuity states' critical mechanical behavior. The preconditioned block-conjugate gradient (pb-CG) is considered to deal with the X-FEM's natural block system form for saving computational efforts. In addition, a block Gauss-Seidel-based alternating active-phase algorithm is utilized to convert a multiphase topology optimization problem subjected to multiple constraints to many binary phase topology optimization sub-problems with only one constraint. Consequently, the current topology optimization methodology can dramatically reduce the number of design variables regardless of the number of material phases. Optimality Criteria (OC) optimizer is then utilized to update optimized design variables for such sub-problems. The study formulates in great detail mathematical expressions for topology optimization of cracked structures with multiple FG materials. Several numerical examples are tested to verify the efficiency and reliability of the current paradigm.