# A Novel Hybrid Whale-Chimp Optimization Algorithm for Structural Damage Detection
## 4 Numerical Simulations
### Structural Damage Detection of Both Asymmetric Damage and Symmetric Damage  
To verify the universality of the proposed method in both asymmetric damage and symmetric damage, a simply supported beam is shown in Figure 8. Physical parameters of the beam are as follows: The beam length \(L=2.5\) m, the section width \(b=0.15\) m, the section height \(h=0.02\) m, the elastic modulus \(E=7\times 10^{10}\)Pa, and the material density \(\rho=2750\) kg/m\({}^{3}\). The simply supported beam is divided into 10 elements, each element has two nodes and four degrees of freedom. Assume that the axial deformation is ignored and only the vertical degrees of freedom of each element are extracted from the modal shape during damage detection. In this simulation example, the search dimension of the W-ChOA is set to 10 and other parameter settings are the same as in the previous simulation example. As shown in Table 5, the stiffness of the elements is reduced by 10% to simulate different damage cases of the simply supported beam. Random noise is also added to the measured modal parameters as shown in Equation (19). Figure 9 shows the single structural damage detection results of the simply supported beam by the ChOA and the W-ChOA under different noise levels. In this case, both the ChOA and the W-ChOA have perfect damage identification results without any misjudgment. Figure 8: Finite element model of the simply supported beam. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & **Type** & **Element** & **Degree of Damage** \\ \hline
1 & 3 & 10\% \\
2 & asymmetric & 3,7 & 10\%, 10\% \\
3 & 3,7,9 & 10\%, 10\% \\ \hline
4 & symmetric & 2,9 & 10\%, 10\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Different damage cases of the simply supported beam. Figure 10 shows the double damage detection results of the simply supported beam by the ChOA and the W-ChOA under different noise levels. As shown in Figure 10, if the damage percent is 10% and the noise level is 5% or 10%, the W-ChOA can accurately identify damage location and damage degree at element 3 and element 7 without any misjudgment. In contrast, the identification results of the ChOA have more than two misjudgments in this case. Figure 10: _Cont._  
Figure 9: Single damage detection results of the simply supported beam by two chimp-based optimization algorithms. (**a**) 5% noise level. (**b**) 10% noise level. Figure 11 shows the triple damage detection results of the simply supported beam by the ChOA and the W-ChOA under different noise levels. As shown in Figure 11a, if the damage percent is 10% and the noise level is 5%, the W-ChOA can accurately identify damage location and damage degree at element 3, element 7, and element 9 with only one misjudgment at element 4. In contrast, the identification results of the ChOA have four misjudgments in element 1, element 2, element 4, and element 5. As shown in Figure 11b, if the damage percentage is 10% and the noise level is 10%, the ChOA results have five misjudgments. The identification results of the W-ChOA are much better with only two misjudgments in element 1 and element 6. Figure 11: Triple damages detection results of the simply supported beam by two chimp-based optimization algorithms. (**a**) 5% noise level. (**b**) 10% noise level. Figure 10: Double damages detection results of the simply supported beam by two chimp-based optimization algorithms. (**a**) 5% noise level. (**b**) 10% noise level. Figures 9-11 show the damage detection results of the simply supported beam in asymmetric damage cases. To satisfy the integrity of numerical simulation, Figure 12 illustrates the damage detection results of the simply supported beam in the symmetric damage case. As shown in Figure 12, if the damage percent is 10% and the noise level is 5% or 10% in the symmetric damage case, the W-ChOA can accurately identify damage location and damage degree at element 2 and element 9 without any misjudgment. In contrast, the identification results of the ChOA have more than two misjudgments in this case.