## 1 Introduction  
A balance between program requirements like weight or durability and component costs is crucial to create competitive products. The design process of nozzle vanes (Figure 1) includes a modal analysis [1, 2].  
The results of the modal analysis are the modal forms (eigenvectors) and associated natural frequencies (eigenvalues) of the system. This analysis is crucial to understand the dynamic behavior and avoid failure modes related to high cycle phenomena. The development phase extensively uses finite element method (FEM) analyses [3]. The reported approach is based on a metamodel established via 2D modal analysis. A surrogate model describes the relationship between the geometrical characteristics and the natural frequency and model area. Typically, an optimization process on airfoils is used to tune the performance [4], lift coefficient [5] and turbine blade aerodynamics [6]. Blade natural frequency optimization [7] accounts for the thickness as a geometrical characteristic. The study presented in [8] revealed the additional impacts of the thicknesses, hook positions and leaning on the casing nozzle assembly.  
In the optimization process, an artificial immune system and a novel version of the algorithm based on costimulation were used. The use of artificial intelligence methods in the optimization of engineering problems is becoming more and more common. The combination of artificial intelligence methods with algorithms in the numerical analysis of engineering problems, such as the finite element method, makes it possible to obtain effective tools to support the work of an engineer. Examples of the use of artificial intelligence in various technical problems are considered in [9; 10; 11].  
The nozzle geometry can be subdivided into several design zones, as presented in Figure 2. The primary criterion for zone A (aerodynamic profile) is to meet the requirements regarding the turbine performance and flow function through airfoil design. A secondary aspect in this location is related to the dynamic characteristics of the airfoil. Location B is the connection between the airfoil and the outer and inner bands. This zone is mainly optimized for the level of stress in order to maximize the components' durability. An example of such a study is presented in [12]. Zone D establishes proper sealing between the rotating components (bladed disks and interstage seals) and prevents hot gas from bypassing the main flow path (reducing the performance). Zone C is designed to ensure correct mounting with the casing rails; it is determined by the durability requirements and is the subject of the optimization.  
Typical failure modes for nozzle guide vanes are low cycle fatigue (LCF), high cycle fatigue (HCF), creep and environmental attack (oxidation and corrosion). The main contributors to the low and high cycle fatigue failure modes are thermal and pressure  
Figure 1: Nozzle guide vane of low-pressure turbine in aircraft engine.  
Figure 2: Nozzle guide vane—design zones; A—aerodynamic profile, B—hub and tip fillets, C—outer band, D—inner band.  
loads. A thermal load creates thermal stress due to the gradients between the aerodynamic profile and the inner and outer bands. A pressure load pushes the nozzle backward, which concentrates the stresses at the airfoil tip and outer band hooks (the simple supported beam concept). The HCF failure mode relates to the vibration of the structure due to excitation of a natural frequency within the operating range (high dynamic response; see Figure 3).