Multi-Objective Topological Optimization of 3D Multi-Material Structures Using the SESO Method with FORM

Topological optimization has established itself as an efficient tool for the design of highly complex structures and the rational use of materials, especially in problems involving multiple constraints and conflicting objectives. This work presents a new multi-material topological optimization approach based on the ESO smoothing method (SESO), formulated as a multi-objective optimization problem in a MATLAB R2021a environment. The multi-objective formulation simultaneously considers the minimization of the maximum von Mises equivalent stress (or minimum principal stress) and the maximum displacement, which are fundamental criteria for structural engineering design. The proposed methodology also incorporates a reliability analysis using the First-Order Reliability Method (FORM), modeling uncertainties associated with the applied force, volume fraction, and modulus of elasticity through normal and lognormal probability distributions, with a target reliability index of βtarget=3.0. The consistency of the reliability analysis was evaluated using Monte Carlo simulations, validating the reliability indices obtained via FORM. The approach was applied to two classical three-dimensional numerical examples: a cantilever beam under base and center loads and an MBB beam, considering two widely used engineering materials, steel and concrete. The results indicate improved multi-material distribution in the design domain and greater structural robustness against unfavorable loading planes, variations in the modulus of elasticity, and volume constraints imposed by FORM. Furthermore, the minimum yield stress of steel (σymin) and the compressive strength of concrete (fckmin) were calibrated, representing the minimum material strengths required to resist the maximum von Mises stress in steel and the minimum principal stress (σ3) in concrete, ensuring the target reliability index is achieved. This method, thus, highlights the integration of SESO with multi-material, multi-objective, and reliability-based optimization as a consistent, robust, and practically relevant strategy with potential for future applications in structural engineering projects.