Recent advancements in additive manufacturing have facilitated the widespread use of functionally graded materials (FGMs), creating new opportunities for optimizing structural performance. When the goal is to minimize both stresses and structural weight, the problem often becomes an inverse optimization task. While there are analytical or semi-analytical solutions available for one-dimensional (1D) cases with geometric, material, and loading symmetries, extending these solutions to two-dimensional (2D) problems is still quite challenging. This study introduces a finite element model with element-wise variable material properties, aimed at optimizing material distributions. The main contribution of this work is the implementation of a numerical algorithm within ANSYS APDL that optimizes functionally graded material distributions to achieve uniform stress states. This implementation reduces simplifying assumptions and accommodates complex loading and boundary conditions. The proposed algorithm has demonstrated effectiveness for both 1D and 2D problems involving isotropic and spherically isotropic linear elastic materials. The paper offers theoretical insights into the inverse problem by identifying the conditions under which explicit analytical solutions for Young's modulus distribution can be derived. Case studies further illustrate that the use of FGMs under specific thermo-mechanical loadings can lead to significant weight reductions, with mass savings of up to a factor of five. • Spherically isotropic FGMs under stationary thermo-mechanical loads are tailored • 1D and 2D inverse problems for pressurized spheres with thermal gradients are solved • An axisymmetric FEM has been developed to address these inverse problems • An efficient numerical algorithm is proposed for solving by using FEA • Compared to homogeneous materials FGMs help reduce maximum stress and structural mass
Sorohan et al. (2026) studied this question.