## 1 Introduction  
The optimization process is crucial for successful design and providing balance between technical and program requirements like durability, weight, performance, maintainability, or component cost. In last year's optimization become extensively used due to development of computation power and need to provide competitive products. Modal analysis [1, 2] is one of the key elements in the design process of the nozzle guide vane in a low pressure turbine. The component aspect ratio is defined as nozzle height to axial width is much higher than high pressure turbine nozzle therefore is more prone for vibration. The aim of the analysis is to verify natural frequency and modal shapes of the system against the operating range and excitation (forcing function). Such verification is fundamental to prevent a high cycle fatigue failure mode or excessive wear at contact interfaces. Finite element method analyses [3] are typically used in the new product introduction phase before product validation through certification. In this article an approach of modal analysis for a two-dimensional finite element model is presented. The optimization process uses a meta-model (mathematical transfer function) determined on response surface (design of experiment) method to pass from geometrical parameters to natural frequency value and cross section area. Geometrical parametric optimization on the airfoils component is widely used to tune airfoil performance [4] as well as the lift coefficient [5]. The impact of the structural parameters on blade natural frequency reported in the article [6] mainly considers mainly the variation of various thicknesses. Presented study reveals the effect of thicknesses of the nozzle and casing as well as the hook position and leaning. Aviation turbofan engines uses a gas turbine aimed to extract energy from hot flow-path gas released from the combustion system and that propel the fan and compressor. In a typical two spool architecture presented in Figure 1, the high pressure (HP) turbine drives the compressor while the low pressure (LP) turbine drives the fan. In order to achieve the desired torque, several turbine stages are considered. One stage means the row of vanes and the row of moving blades attached to the disk and transmitting the torque to the propeller shaft. The number of stages depends on the relationship between the requested power from gas flow, the rotational speed at which power is produced, and the permitted diameter of the turbine. The Brayton cycle describes basic engine theory and the purpose of the turbine (Figure 2) [7]. Charts report two diagrams of the cycle considering pressure (P) vs. volume (V) and temperature (T) vs. entropy (S) relationships. The cycle is going through the air intake at atmospheric pressure (point 1), and then compression process increases the air temperature and pressure, reaching point 2. The combustion process further increases temperature up to point 3, where power extraction by the turbine module starts. The first stage vanes are converting high entropy to high velocity gas toward moving blades. Engines operating at higher turbine inlet temperatures (T3) are thermally more efficient and have an improved power to weight ratio; however, these operate in a more demanding  
Figure 1: Diagram of two spool turbofan engine [7]. Figure 2: P-v and T-s diagrams of an ideal Brayton cycle. environment. The operating temperature is limited by a component's material capability and the requirements for what concern durability, maintainability, and cost. The design of the nozzle guide vane airfoils and turbine blades are mainly driven by aerodynamic requirements in order to achieve optimum efficiency and to be compatible with the remaining engine modules, compressor and combustor. The nozzle guide vane is a static component and does not endure any centrifugal loads. The main failure modes are related to low and high cycle fatigue (LCF, HCF), creep and environmental attack (oxidation, corrosion).