## 4 Application
### Method Comparison  
In this section, we compare both morphing approaches and their results for the optimization at 777 Hz. First, we perform a design optimization using the metamodels in Figure 10. As described earlier, we use small \(x_{1}\)-values, great \(x_{2}\)-values and great \(x_{3}\)-values. We use  
\[\mathbf{x}_{DoE}=\left(-10\,\mathrm{mm},10\,\mathrm{mm},10\,\mathrm{mm}\right)^{T} \tag{15}\]  
and transform it onto the design space surface using Equation (14) and obtain  
\[\mathbf{x}_{border}=\left(-5.77\,\mathrm{mm},5.77\,\mathrm{mm},5.77\,\mathrm{mm} \right)^{T} \tag{16}\]  
as the first optimized design. A black triangle represents the design in Figure 10. We create this design using both morphing approaches. The metamodels in Figure 10 hint a resulting amplitude of approximately \(5.15\,\mathrm{N}\,\mathrm{N}^{-1}\), when averaging the 4 metamodels at the chosen point. Thus the peak is reduced by \(23\,\mathrm{\char 37}\). The results for the manual and numerical optimization are listed in Table 1. Both designs resulting from numerical optimization reside on the design space border. We create all three designs in Table 1 using both morphing approaches and compare the amplitude results and the prediction qualities. The resulting FRFs are shown in Figure 11. The FRF of the original design is marked gray, and the simulation results using Direct and Box Morphing are marked black and blue respectively. The predicted value at 777 Hz is the amplitude value of the corresponding metamodel. In Figure 11a, the manually predicted amplitude value from Figure 10 is too high and both morphing approaches hint a stronger amplitude improvement than the two metamodels predict. In Figure 11b,c, the corresponding metamodel matches the simulation result. In each case, the Direct Morphing approach overestimates the amplitude. This is due to the morphing boundary condition for both methods. The Box Morphing approach enforces stronger boundary conditions like bushing axes or machined surfaces to follow the imposed displacements. In contrast, the Direct Morphing approach only enforces single nodes to fit the imposed displacement. Larger imposed areas could be implemented in the Direct Morphing approach, too. Since the method is designed to be fast and easy to use, we decided to impose the displacement at the kinematic hard points only. Figure 11 clearly shows that both morphing approaches detect the same influences, so both methods are suitable to lead design choices. Both the Direct Morphing and the Box Morphing approach proved suitable for the prediction of FRF characteristics. The two-dimensional metamodels provide an easy and fast to use tool to direct design changes using visual hints for improvement. A component designer is able to approximate design changes without numerical calculation or optimization, while still being able to perform an optimization when necessary. The workflow to achieve the predictions and the accuracy differs though. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & \(\mathbf{x_{1}}\) & \(\mathbf{x_{2}}\) & \(\mathbf{x_{3}}\) & **Predicted Amplitude** \\ \hline Manually & \(-5.77\,\mathrm{mm}\) & \(5.77\,\mathrm{mm}\) & \(5.77\,\mathrm{mm}\) & \(5.15\,\mathrm{N}\,\mathrm{N}^{-1}\) \\ Optimization Direct & \(-4.42\,\mathrm{mm}\) & \(3.36\,\mathrm{mm}\) & \(8.32\,\mathrm{mm}\) & \(4.04\,\mathrm{N}\,\mathrm{N}^{-1}\) \\ Optimization Box & \(-0.30\,\mathrm{mm}\) & \(4.83\,\mathrm{mm}\) & \(8.75\,\mathrm{mm}\) & \(3.31\,\mathrm{N}\,\mathrm{N}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Designs, optimized by different metamodels. Reusing the tools and workflows created for this paper, predictions for another component are possible. The Direct Morphing approach only needs to receive the FE model of the new component as matrix \(\underline{x}_{FE}\) and the new displacement matrices \(\underline{x}_{d}\) and \(\underline{v}_{d}\). The generation of these matrices takes approximately one working hour and the tool is ready to morph and create metamodels.