## 2 Materials and Methods
### FEM Model Description and Optimization Framework  
An analysis based on a simplified two-dimensional model of a vane with casing is performed. The nozzle guide vane is modeled by means of plane stress with thickness elements. The casing is modeled by means of axisymmetric elements. Appropriate thicknesses are established based on the perimeters derived from the radius of the area's center of gravity (CoG), as reported in Figure 12. The aerodynamic profile was additionally scaled based on the empty to full volume ratio. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Memory cells & 3 \\ Clone number & 6 \\ Termination criteria—number of iterations & 10 \\ Mutation probability & 0.75 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Algorithm parameters. Figure 11: Bohachevsky’s function—comparison of standard artificial immune system (AIS) with costimulated artificial immune system (CAIS). The FEM model is subdivided into different areas to reflect the different behavior (axisymmetric and plane stress) and model thickness. These areas are merged by means of contact elements defined as presented in Figure 13.  
The defined contacts restrain relative movement and guarantee the absence of separation between the surfaces. The turbine casing is fixed in the radial and axial directions on the forward cut face and rear flange. The temperature of the casing and nozzle is set to ambient temperature. The parameters for the optimization process are defined based on the strain energy distribution for the first system mode in the baseline (reference) configuration. Figure 14 highlights the zones with energy accumulation due to the considered mode shape deformation. A change in the stiffness in the highlighted zone will impact the natural mode frequency and therefore qualitatively defines the area of interest for parameterization. A parametric model is defined by the casing shell thicknesses and the nozzle outer band hook's geometrical characteristics, as presented in Figure 15. The parameters related to the hook position and leaning also affect the casing geometry due to the need for casing rail adjustment (the rails follow the hook to ensure a contact interface). Tables 3 and 4 define the lower and upper bounds of the geometrical characteristics applied in the optimization. Figure 12: Approach for two-dimensional analysis. Figure 13: FEM model constraints and contact pair definition. The characteristic of the first system mode is presented in Figure 16. The nozzle is moving as a pendulum (an inner band traveling back and forth), being bonded to the turbine casing. The second and the following system modes of the structure are not within the scope of interest since they are outside the operating range and are not in interaction with any excitation source from shaft imbalance. The optimization process is performed by means of an artificial immune system with the framework presented in Figure 17 (green box item represents optimization algorithm routine, blue box items identify CAE software operations). The process starts with the definition of a parametric model considering the set of geometrical characteristics selected from the strain energy distribution. The model is verified to ensure the geometry's feasibility within the established design variable ranges (Tables 3 and 4). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Lower Bound, deg.** & **Upper Bound, deg.** \\ \hline Forward hook leaning & 80 & 100 \\ Rear hook leaning & 80 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Geometrical angular parameters. Figure 14: Strain energy distribution for pendulum mode shape. Figure 15: Geometrical parameters dedicated to optimization. The optimization loop uses a static metamodel [8] that is an approximation of the real phenomena concerning the model area and natural frequency.