## 3 Method
### Metamodeling  
Both morphing approaches are completely scripted so the morphing process is automated with an original FE model and text based displacement inputs for the kinematic hard points. This allows us to automatically create a large data base with displacements as input parameters and--in our application--characteristic values of the calculated FRFs as output parameters. This enables us to create an analytical approximation, called metamodel, of the coherence between displacements of hard points and characteristic FRF values.  
We create the data base using a space filling latin hypercube approach using _Optislang_ by _[PERSON]_[31]. Depending on the number of geometric input coordinates, we create \(N\) designs. The selected morphing approach automatically creates \(N\) morphed FE models that are used as input for FE simulation. The morphing error of the Direct and Box Morphing approach is eliminated by using the effective hard point displacements instead of the given ones as input data for the metamodel. From the FRF results we extract characteristic values; for example, the amplitudes \(A\) at specific frequencies for specific FRFs of the component:  
\[\underline{A}=\left[A_{1},A_{2},\ldots,A_{N}\right] \tag{10}\]  
These values and the input parameters are the basis for a multivariate polynomial regression that predicts the amplitude \(\hat{A}\). We use cubic polynomials with coupled terms of the form  
\[\hat{A}\left(p,q\right)=\alpha_{0}+\alpha_{1}p+\alpha_{2}p^{2}+ \alpha_{3}p^{3}+\alpha_{4}q+\alpha_{5}pq+\alpha_{6}p^{2}q+\alpha_{7}p^{3}q+ \alpha_{8}q^{2}\\ +\alpha_{9}pq^{2}+\alpha_{10}p^{2}q^{2}+\alpha_{11}p^{3}q^{2}+ \alpha_{12}q^{3}+\alpha_{13}pq^{3}+\alpha_{14}p^{2}q^{3}+\alpha_{15}p^{3}q^{3} \tag{11}\]  
with two variables \(p\) and \(q\) and the parameters \(\alpha_{0}\) to \(\alpha_{15}\). The two dimensional approach enables simple plots that can lead the design process. We use a least squares approach to calculate the parameters \(\alpha\) for all combinations of two input parameters and the simulated result value. For example,  
Figure 6: Showcase of morphing structure necessary for displacement and smoothing operations. (**a**) Control points for general displacement. (**b**) Control points for 1 st smoothing. (**c**) Control points for 2 nd smoothing.  
the polynomial \(\hat{A}\left(v_{c,3,1},v_{c,3,3}\right)\) describes the regression between the characteristic value \(A\) and the imposed displacements \(v_{c,3,1}\) at position 3 in direction 1 and \(v_{c,3,3}\) at position 3 in direction 3.  
The coefficient of determination  
\[R^{2}=1-\frac{\sum_{i=1}^{N}\left(A_{i}-\hat{A}_{i}\right)^{2}}{\sum_{i=1}^{N} \left(A_{i}-\bar{A}\right)^{2}}=1-\frac{\sum_{i=1}^{N}\left(A_{i}-\hat{A} \left(p_{i},q_{i}\right)\right)^{2}}{\sum_{i=1}^{N}\left(A_{i}-\bar{A}\right)^ {2}} \tag{12}\]  
with the mean value \(\bar{A}\) of the result values \(A_{i}\) for the design \(i\) of \(N\), is an indicator for the quality of the polynomial fit [32] (p. 398). An \(R^{2}\) near one hints a good fit.  
A combination of well fitted polynomials can then perform as an input for an optimization of \(A\). The found optimum is then validated by a separate geometry morph. This procedure fastens optimization as the optimization is performed on an analytical metamodel [20] (p. 231). Additionally, the characteristic parameter \(A\) can vary without re-simulating a single design, as \(A\) is extracted from the stored results. A changed optimization target and its resulting design can thus be investigated in a matter of minutes.  
Additionally to the two dimensional approach, we fit higher dimensional polynomials for numerical optimizations. Again, the optimization parameters can change after performing the FE simulation.