## 3 Results
### Implementation for Surrogate-Based Optimization Schemes  
The preceding Sections 3.1 and 3.2 demonstrate that the surrogate-based GOMORS framework mostly has a superior performance to the NSGA-II in the case of classical shape optimization. However, penalty factors may cause surrogate models to map the behavior of the actual physical model poorly. This behavior was already predicted in [20] and is now validated in Section 3.3 of this article. Hence, the following section tests a second approach to validate the feasibility of the penalization scheme for surrogate-based processes, which are represented in this case by GOMORS. As the unsteady behavior of the penalization scheme may cause an inferior quality of the surrogate models [20], the penalty is therefore set as the third objective function. The approach benefits from the lower amount of needed mesh evaluations, as it does not manipulate the infeasible designs. Still, the number of required calculations increased. The objective function value of the infeasibility factor \(f(x)_{feas}\) is zero if no infeasibility is detected, and \(f(x)_{feas}\) is larger than zero if a surface penatration is detected. As GOMORS performed well on the unconstrained and provided worse results in the constrained approach, the approach is only validated for GOMORS under consideration of the following three objective functions: mass \(f_{m}(x)\), stiffness \(f_{s}(x)\) and feasibility \(f_{feas}(x)\):  
\[\min_{x}\qquad f_{m}(x),f_{s}(x),f_{feas}(x) \tag{6}\]  
With respect to the preceding optimizations, the sampling size is set to 200, and the number of iterations is set to 100. Figure 9 shows the Pareto-frontier, which is built during the optimization. The normalization is relative concerning the initial design, and it considers the outer sheet metal, the inner sheet metal, and the stiffener. Hence, the percentage relations are the same as in Figure 9.  
With the presented optimization setup, only a feasible design outperforms the initial setup in terms of Pareto-optimality. Still, the improvement is minimal, which means no increase in terms of mass and \(\sim\)10% in displacement. Conversely, there are potential lighter  
Figure 9: The Pareto-frontier for the GOMORS optimization concerning three objective functions: mass, displacement, and feasibility. All objectives are relative to the initial design. candidates, but they exhibit a more significant static deformation. This lack of potential candidates may result from the larger solution space (10 DV) and the strong influence of the outer and inner shell movement on the weight and resulting displacement due to the feature importance. The sampling size is increased to 1000 samples to improve the surrogate model quality, while 200 iterations allow a more detailed optimization run. Furthermore, the number of design variables is increased to 24. The 16 DVs for the outer and inner shells are activated to ensure that the influence of the DVs is evenly weighted and that more design freedom leads to candidates of higher potential. Furthermore, the feasibility approach with penalty constraint and the feasibility approach with three objective functions are directly compared for the chosen sampling and the NSGA-II and the GOMORS iterations, as shown in Table 1.  
According to Table 1, the three scenarios are optimized and visualized in the following paragraphs. Again, the b-pillar is analyzed, but the number of DVs is increased to 24. The optimization problem can be formulated as:  
\[ \begin{split}&\min_{x}f_{m}(x),\ f_{s}(x)\\ & subject\ to\ f_{feas}(x)=0\end{split}  \tag{7}\]  
For a better understanding of the outer shell movement, four possible feasible candidates are shown in Figure 10. The stiffener can move in between the upper and lower shell, as presented in Figure 4.  
Figure 11 displays the overall performance of the GOMORS three objective implementation. It plots the third objective function for feasible results beside the performance of the