# An Efficient and Fast Hybrid GWO-JAYA Algorithm for Design Optimization
## 3 Test Problems and Optimization Results
### Optimal Crashworthiness Design of a Vehicle Subject to Side Impact  
The test case (vi) solved in this study regarded the optimal crashworthiness design of a vehicle. The goal was to minimize the weight of the vehicle subject to side impact. Crashworthiness optimization must ensure that the vehicle effectively absorbs and dissipates impact energy, minimizing forces transferred to occupants. The vehicle structure must be optimized to provide maximum protection without significantly increasing weight or compromising other performance aspects [107; 108; 109]. Figure 13a shows the typical finite element schematization of a vehicle subject to side impact by the barrier [108], while Figure 13b illustrates the vehicle frame parts to be optimized. Since in side impact tests the B-pillar zone often is most affected, design variables are correlated with this zone. The optimal crashworthiness design problem solved in this study included 11 optimization variables of which 7 related with the vehicle's structural geometry: B-pillar inner thickness (\(x_{1}\)), B-pillar reinforcement thickness (\(x_{2}\)), floor side inner thickness (\(x_{3}\)), cross members thickness (\(x_{4}\)), door beam thickness (\(x_{5}\)), door beltline reinforcement (\(x_{6}\)), and roof rail thickness (\(x_{7}\)). Two discrete variables refer to material selection and quantify the efficiency of the structure in absorbing energy during impact: shape factor for the B-pillar inner (\(x_{8}\)) and shape factor for the floor side inner (\(x_{9}\)). The last two variables are related with the presence of the obstacle: the barrier height (\(x_{10}\)), and the hitting position of the barrier with respect to the center of mass of the vehicle (\(x_{11}\)). In summary, the optimization problem includes nine continuous and two discrete variables. Here, the optimal crashworthiness design problem was stated as follows:  
\[\textit{Minimize }\textit{W}(\overset{\rightarrow}{X})=1.98+4.90x_{1}+6.67x_{ 2}+6.98x_{3}+4.01x_{4}+1.78x_{5}+2.73x_{7} \tag{26}\]  
Figure 12: Comparison of convergence curves of FHGWJA and its competitors for the 25-variable hydrologic model calibration problem. \[\left\{ \begin{array}{ll}G_{1}(\overset{\rightarrow}{X})=&1.16-0.3717\chi_{2}x_{4 }-0.00931\chi_{2}x_{10}-0.484x_{3}x_{9}+0.01343x_{6}x_{10}\leq 1\\ G_{2}(\overset{\rightarrow}{X})=&0.261-0.0159x_{1}x_{2}-0.188x_{1}x_{8}-0.019x_{2 }x_{7}+0.0144x_{3}x_{5}+0.0008757x_{5}x_{10}+\\ &0.08045x_{6}x_{9}+0.00139x_{8}x_{11}+0.00001575x_{10}x_{11}\leq 0.32\\ G_{3}(\overset{\rightarrow}{X})=&0.214+0.00817x_{5}-0.131x_{1}x_{8}-0.0704x_{1 }x_{9}+0.0309x_{2}x_{6}-0.018x_{2}x_{7}+\\ &0.0208x_{3}x_{8}+0.121x_{3}x_{9}-0.00364x_{5}x_{6}+0.0007715x_{5}x_{10}-0.0005 354x_{6}x_{10}+\\ &0.00121x_{8}x_{11}+0.00184x_{9}x_{10}-0.02x_{2}{}^{2}\leq 0.32\\ G_{4}(\overset{\rightarrow}{X})=&0.074-0.61x_{2}-0.163x_{3}x_{8}+0.001232x_{3} x_{10}-0.166x_{7}x_{9}+0.227x_{2}{}^{2}\leq 0.32\\ G_{5}(\overset{\rightarrow}{X})=&28.98+3.818x_{3}-4.2x_{1}x_{2}+0.0207x_{5}x_{ 10}+6.63x_{6}x_{9}-7.7x_{7}x_{8}+0.32x_{9}x_{10}\leq 32\\ G_{6}(\overset{\rightarrow}{X})=&33.86+2.95x_{3}+0.1792x_{10}-5.057x_{1}x_{2 }-11.0x_{2}x_{8}-0.0215x_{5}x_{10}-9.98x_{7}x_{8}+\\ &22.0x_{8}x_{9}\leq 32\\ G_{7}(\overset{\rightarrow}{X})=&46.36-9.9x_{2}-12.9x_{1}x_{8}+0.1107x_{3}x_{ 10}\leq 32\\ G_{8}(\overset{\rightarrow}{X})=&4.72-0.5x_{4}-0.19x_{2}x_{3}-0.0122x_{4}x_{ 10}+0.009325x_{6}x_{10}+0.000191x_{11}{}^{2}\leq 4\\ G_{9}(\overset{\rightarrow}{X})=&10.58-0.674x_{1}x_{2}-1.95x_{2}x_{8}+0.02054x _{3}x_{10}-0.0198x_{4}x_{10}+0.028x_{6}x_{10}\leq 9.9\\ G_{10}(\overset{\rightarrow}{X})=&16.45-0.489x_{3}x_{7}-0.843x_{5}x_{6}+0.0432x _{9}x_{10}-0.0556x_{9}x_{11}-0.000786x_{11}{}^{2}\leq 15.7\end{array} \right. \tag{27}\]  
The expressions written above for the objective function \(W(\overset{\rightarrow}{X})\) (i.e., the structural weight of vehicle parts that must be optimized) and the constraints \(G(\overset{\rightarrow}{X})\) were fitted in [108] using response surface models.