## 1 Introduction  
For many years, composites have been used for a variety of applications, such as mechanical, aerospace, and civil engineering structures [1; 2; 3]. However, the dynamic behavior of composite structures has not been widely investigated: composite structures are slender and light compared with steel and concrete structures, and their dynamic behavior may be entirely different. Understanding this behavior is crucial to the effective application of composite materials in structural engineering [1; 4; 5; 6]. Thus, it is very important to study the dynamic behavior of composite structures in response to changes in material-specific factors (e.g., lamination fiber angles). The structural design process, demand, and the reliability of numerical analysis usually require the optimization of the shape, size, and/or placement of fibers within the material. However, this approach increases the difficulty and complexity of the design process [7] since the design of optimized composite laminate is combined with the search for the best fiber orientation in each layer of composite material. The optimization procedure of laminated composite structures is usually more complex than the one associated with isotropic material structures because of the large number of variables involved and the intrinsic anisotropic behavior of individual layers [8; 9]. In optimization problems associated with the design of a composite structure, two scenarios are possible [7; 10]: (i) a constant-stiffness design in which the lamination parameters are constant over the whole domain; (ii) a variable-stiffness design in which the lamination parameters may vary in a continuous manner over the domain. The constant-stiffness design scenario is simpler than the variable-stiffness design since there are generally fewer design variables involved [7]. In this study, a constant-stiffness design is applied for a one-segment shell structure, in which the same stacking layup is used over the whole structure, and a variable-stiffness design is applied for a three-segment shell, in which a different stacking layup is used in segments. In many engineering problems involving dynamic phenomena, the optimization of selected parameters is necessary. Such optimization is usually performed through the minimization or maximization of a carefully selected _objective function_ that describes the distance of particular structure parameters from their desired values (e.g., overall mass, strength, or fundamental natural frequency). The objective function is usually repeatedly solved during the optimization procedure. Moreover, the application of first-order optimization algorithms also involves derivatives of the objective function. In composite laminate design, the precise calculation or even approximation of objective function derivatives is often computationally costly or, in some cases, even impossible. One of the advantages of direct search algorithms (or zero-order optimization algorithms, e.g., Genetic Algorithms, GAs) is that they require only objective function values; no information about the gradient of the objective function is necessary. The single calculation of the objective function is neither difficult nor time-consuming. However, it is extremely computationally demanding when it has to be repeated thousands of times (e.g., when a zero-order search algorithm is applied). Artificial Neural Networks (ANNs) are employed to reduce this demand [11; 12]. The task of objective function calculation is herein improved and accelerated by using ANN as a tool to replace the Finite Element Method (FEM) for the prediction of dynamic parameters. The authors of [13] considered a reliability problem for laminated four-ply composite plates in free vibration. The design variables were the material properties, stacking sequence, and thickness of the plies, and the goal of the procedure was to obtain a fundamental natural frequency that was higher than the assumed excitation frequency.