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

Topological optimization has established as an efficient tool for the design of structures with high complexity and rational use of material, especially in problems involving multiple constraints and conflicting objectives. This work presents a new multi-material topology optimization approach based on the ESO smoothing method (SESO), formulated as a multi-objective optimization problem in a MATLAB environment. The multi-objective formulation simultaneously considers the minimization of the maximum von Mises equivalent stress and the maximum displacement, fundamental criteria for structural engineering design. The proposed methodology also incorporates a reliability analysis using the First-Order Reliability Method (FORM). The uncertainties associated with the applied force, volume fraction, and elastic modulus are modelled using normal and lognormal probability distributions, with a target reliability index of β_target=3.0. The consistency of the reliability analysis was evaluated through Monte Carlo simulations, used to validate the reliability indices obtained by the FORM method. The approach was applied to two classic three-dimensional numerical examples, a bottom-loaded cantilever beam and center-loaded cantilever beam, considering two widely used commercial materials, steel and concrete. The results indicated better multi-material distribution in the design domain and increased structural robustness against unfavorable loading planes, elastic modulus, and volume constraints imposed by the FORM formulation. Furthermore, the minimum yield stress is calibrated to incorporate the uncertainties inherent in the design process, establishing the minimum value required to achieve the target reliability index β. Thus, this method highlights the integration of the SESO method with multi-material, multi-objective, and reliability-based optimization as a consistent and robust strategy with potential for future applications in structural engineering design.