## 2 Materials and Methods
### Metamodeling with Genetic Aggregation  
A metamodel is the approximation of the real phenomena by mathematical model to support computationally expensive simulations. In the presented study it provides dependances between geometrical features, the model area and the natural frequency of the system.  
\[f_{1}=y(x)\cong\hat{y}(x) \tag{19}\]  
where:  
\(y(x)\)--real response for given \(x\),  
\(\hat{y}(x)\)--approximated response for given \(x\),  
The mathematical model can be estimated using three methods. The first one is the analytical method for fundamental physical phenomena, where laws and formulas are well described in the literature (i.e., bending stress, natural frequency of harmonic oscillator etc.). The second method is a passive experiment where the metamodel is based on observed process results. It is applied in cases where the analyzed process cannot be interrupted or influenced. The metamodel is obtained by correlation analysis and statistical tests. The third method is an active experiment planning so called design of experiment (DOE) where output parameters are evaluated for defined cases. This method can be stationary (all cases defined at the beginning of the experiment) or non-stationary (the first set defined, and the next set of input parameters driven by results from the previous set).  
In the article, the stationary design of the experiment method and the applied metamodel with genetic aggregation response surface approach is used. The model can be written as an ensemble [15; 16; 17] using a weighted average of different metamodels:  
\[\hat{y}_{ens}(x)=\sum_{i=1}^{N_{m}}w_{i}\hat{y}_{i}(x) \tag{20}\]  
where:  
\(\hat{y}_{ens}\)--prediction of the ensemble,  
\(\hat{y}_{i}\)--prediction of the \(i\)th response surface,  
\(N_{m}\)--number of metamodel used, \(Nm\geq 1\),  
\(w\)--weight factor of the \(i\)th response surface,  
The weight factors need to satisfy the following equation  
\[\sum_{i=1}^{N_{m}}w_{i}=1\text{ {and} }w_{i}\geq 1,\text{ }1\leq i\text{ }\leq N_{m} \tag{21}\]  
To find the best weight factors, algorithm is minimizing a root mean square error (RMSE) and predicted sum of squares (PRESS) as reported by Equations (22) and (23) respectively.  
\[RMSE(\hat{y}_{ens})=\sqrt{\frac{1}{N}\sum_{j=1}^{N}\bigl{(}y\bigl{(}x_{j} \bigr{)}-\hat{y}_{ens}\bigl{(}x_{j}\bigr{)}\bigr{)}^{2}} \tag{22}\]  
\[PRESS_{RMSE}(\hat{y}_{ens})=\sqrt{\frac{1}{N}\sum_{j=1}^{N}\bigl{(}y\bigl{(}x_ {j}\bigr{)}-\hat{y}_{ens,-j}\bigl{(}x_{j}\bigr{)}\bigr{)}^{2}} \tag{23}\]  
where:  
\(\hat{y}_{ens,-j}(x)=\sum_{i=1}^{N_{m}}w_{i}\hat{y}_{i,-j}(x)\)--prediction of the metamodels ensemble build without \(j\)th design point,  
\(x_{j}\)--design point \(j\)th,\(y(x_{j})\)--output parameter value at \(x_{j}\) design point,  
\(\hat{y}_{i,-j}(x)\)--prediction of the metamodel \(i\) build without \(j\)th design point,  
N--number of design points.  
The implementation example of the mentioned aggregation \(\hat{y}_{ens}(x)\) for 1D function \(y(x)\) using 5 metamodels; \(\hat{y}_{A}\)(linear regression), \(\hat{y}_{B}\), \(\hat{y}_{C}\), \(\hat{y}_{D}\), \(\hat{y}_{E}\) (polynomial with orders from 2 to 5) is reported in Figure 9. Metamodels were fitted using a set of six design points.  
\[y(x)=\frac{1}{e^{x}}\sin x-0.1x+1 \tag{24}\]  
The aggregated metamodel reveals a better fit than each metamodel separately, as reported in Table 1:  
Noticed lower coefficient of determination for higher order polynomial function (metamodel E) that is even worse than linear regression model.