# A Multi-Fidelity Successive Response Surface Method for Crashworthiness Optimization Problems
## 6 Results and Discussion
### Engineering Use Case: Crash Box Design
#### 6.3.2 Problem Formulation  
An overview of the structural system to be analyzed is given in Figure 12. Note that there are two V-notch crush initiators (sometimes called triggers) \(T_{1}\) and \(T_{2}\) at the top and bottom of the crash box. These are strategically placed engineering features introduced into the crash box components of the vehicle structure to control the deformation path during a collision.  
As design variables, we consider the thicknesses of the crash box (\(t_{U}\) and \(t_{S}\)), the flanges (\(t_{F}\)), and the bumper cross member (\(t_{B}\)). Additionally, we consider the distance from the flanges to the first trigger (\(d_{T_{1}}\)), the distance between the two triggers (\(d_{T_{2}}\)), and the angle (\(\alpha\)), which symmetrically adjusts the tilt of both the top and bottom faces of the crash box with respect to the horizontal plane. The last three variables imply a change in the geometry of the crash box, classifying the problem as a mixture of size and shape optimization. A detailed description of the design variables and their respective bounds is given in Table 1.  
The main objective is to maximize the SEA within certain passive safety constraints. First, we set a value for the maximum peak deformation force \(F_{max}\) and a maximum value for the average deformation force \(F_{m}\). This is to ensure that the crash box deforms before the longitudinal members, thus ensuring a step-wise increasing deformation curve [58]. With reference to Figure 13 we set a threshold of 61.75 kN for \(F_{max}\) (5% safety margin) and a threshold of 50 kN for \(F_{m}\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable Design** & **Label** & **Unit** & \
\begin{tabular}{c} **Lower** \\ **Bound** \\ \end{tabular} &
\begin{tabular}{c} **Upper** \\ **Bound** \\ \end{tabular} \\ \hline Upper crash box thickness & \(t_{U}\) & (mm) & 1.0 & 3.5 \\ Side crash box thickness & \(t_{S}\) & (mm) & 1.0 & 3.5 \\ Bumper cross member thickness & \(t_{B}\) & (mm) & 1.0 & 3.0 \\ Flange thickness & \(t_{F}\) & (mm) & 1.0 & 4.0 \\ Flange to \(T_{1}\) distance & \(d_{T_{1}}\) & (mm) & 20.0 & 70.0 \\ \(T_{1}\) to \(T_{2}\) distance & \(d_{T_{2}}\) & (mm) & 30.0 & 100.0 \\ Angle to horizontal plane & \(\alpha\) & \((deg)\) & 1.0 & 3.5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of design variables.  
Figure 12: Overview of the FEA model components: impactor, crash box, and bumper beam. The design variables of the optimization problem are highlighted in blue.  
Figure 13: Example of a step-wise increasing force curve with the smallest possible differences in the force levels of the individual components. Figure inspired by [58].  
To ensure effective energy absorption, we set a minimum threshold for the energy absorption of the bumper cross member after 35 ms, aligning with the expected timing of the first peak force due to crash box deformation. This constraint prevents excessively stiff bumper configurations that could cause premature deformation of the crash box. Additionally, we require that \(CFC\geq 0.5\) and \(ULC\leq 0.5\). Although these constraints are typically applied to crash boxes alone, in this context, they are employed to ensure an adequate trade-off between force fluctuations and the initial peak relative to the mean force value. The complete formulation of this optimization problem is provided in Appendix C.1.