The most commonly used objective function in structural optimization is weight minimization. Nodal displacements, compliance, the first natural frequency of vibration, the critical load factor concerning global stability, and others can also be considered additional objective functions. This paper aims to propose seven innovative many-objective structural optimization problems (MOSOPs) applied to 25-, 56-, 72-, 120-, and 582-bar trusses, not yet presented in the literature, in which the main objectives, in addition to the structure’s weight, refer to the structures’ vibrational and stability aspects. These characteristics are essential in designing structural models, such as the natural frequencies of vibration and load factors concerning global stability. Such new MOSOPs have more than three objective functions and are called many-objective structural optimization problems. The chosen objective functions refer to the structure’s weight, the natural frequencies of vibration, the difference between some of the natural frequencies of vibration, the critical load factor concerning the structure’s global stability, and the difference between some of its load factors. The sizing design variables are the cross-sectional areas of the bars (continuous or discrete). The methodology involves the finite element method (FEM) to obtain the objective functions and constraints and multi-objective evolutionary algorithms (MOEAs) based on differential evolution to solve the MOSOPs analyzed in this study. In addition, multi-criteria decision-making (MCDM) is adopted to extract the solutions from the Pareto fronts according to the artificial decision-maker’s (DM) preference scenarios, and the complete data for each chosen solution are provided. For the MOSOP with seven objective functions, it is possible to observe variations in the final weights of the optimum designs, considering the hypothetic scenarios, of 21.09% (25-bar truss), 289.73% (56-bar truss), 70.46% (72-bar truss), 45.35% (120-bar truss), and 74.92% (582-bar truss).

Computation-intensive structural design optimization problems are common in engineering. The computation burden is often caused by expensive simulation models and becomes a problem in practice. To address such a challenge, metamodeling techniques are often chosen to improve the optimization efficiency. Metamodels, or surrogate models, are very popular methods and are considered a valuable tool to support a wide scope of activities in modern engineering design. The paper discusses strategies to obtain suitable metamodels to assess their quality concerning prediction, considering machine learning techniques used in the literature. A multi-objective structural optimization problem is used to illustrate the applicability of the approach, considering as objective functions the structure’s weight, and critical load factor concerning the structure’s global stability. The numerical results demonstrate the efficiency and computational advantages of the proposed methodology.

The literature has broadly discussed developing and solving multi-objective structural optimization problems (MOSOPs) with two objectives. The conflicting objective functions commonly addressed are minimizing the structure's weight and maximum displacements. In this paper, the two objective functions considered are minimizing the weight and maximizing the first critical load factor related to the structure's global stability. The constraints are related to the maximum stresses in the bars, the maximum allowed nodal displacements, and the minimum value determined for the first natural frequency of vibration. The analyzed structure is a 25-bar truss. When defining the displacements and deformed configurations of the structure, a geometrically nonlinear analysis is applied using the arc-length method. This analysis allows the decision-maker to obtain more realistic and accurate values regarding the objective functions and constraints. The evolutionary algorithm used is the multi-objective meta-heuristic with iterative parameter distribution estimation (MM-IPDE). The Pareto fronts obtained in the proposed problems are presented, as it is possible to observe, for example, how the growth of the truss' weight causes increases in the first critical load factor. Finally, optimized solutions are extracted from the Pareto fronts.

Most structural optimization problems involving shallow domes with geometrically nonlinear behavior are formulated as single-objective, and the structure’s weight is the objective function to be minimized. However, other objective functions, such as the structure’s weight, can be incorporated into these formulations, introducing more realistic aspects into the desired optimal designs and significantly assisting the decision-maker in making their preferred choices. This paper extends the usual single-objective formulations by proposing new multi-objective formulations for structural optimization problems. Therefore, new structural optimization problems are proposed and solved, with objective functions including the structure’s weight, the first natural frequency of vibration, the first critical load factor for global stability, and compliance. Three multi-objective evolutionary algorithms are adopted to solve the proposed optimization problems and analyze several benchmark problems. Pareto fronts present the set of non-dominated solutions for the multi-objective formulations, and solutions are extracted from these sets according to multi-criteria decision-making.