## 6 Numerical Illustrative Examples
### Numerical Model of a Thin Plate  
In order to verify the effectiveness and reliability of the proposed method in identifying the flutter derivatives of the bridge deck, the numerical model of a theoretical thin plate section was first employed. The following parameters were used for the numerical model: the width of the thin plate model \(B\) was 0.45 m, the mass per unit length \(m\) was 11.25 kg/m; the mass moment of inertia per unit length \(l\) was 0.2828 kg\(\cdot\)m\({}^{2}\)/m; the frequency of vertical bending mode was 1.9274 Hz; the frequency of torsional mode was 3.0239 Hz; and the damping ratios of vertical bending and torsional modes were both 0.5%. Smooth oncoming wind with 0 deg angle of attack was assumed. The air density \(\rho\) was set to 1.293 kg/m\({}^{3}\). In total, 20 sets of free-decay time histories were obtained for each wind speed, and a random noise was superimposed to the simulated response. Three cases of noise level were considered: 2%, 5%, and 10%. These 20 sets of data formed the sample population for bootstrap sampling. The proposed method was used to identify all eight flutter derivatives of the thin plate model. The control parameters in the MABC-Powell for numerical \begin{table}
\begin{tabular}{l l l l l l} \hline \hline \multicolumn{1}{c}{**Name**} & \multicolumn{1}{c}{**Function**} & \multicolumn{1}{c}{**Search Range**} & \multicolumn{1}{c}{**Accepted Accuracy**} \\ \hline Sphere & \(f_{1}(X)=\sum\limits_{i=1}^{n}x_{i}^{2}\) & \([-100,\ 100]^{n}\) & \(1\times 10^{-8}\) \\ SumSquare & \(f_{2}(X)=\sum\limits_{i=1}^{n}i\cdot x_{i}^{2}\) & \([-100,\ 100]^{n}\) & \(1\times 10^{-8}\) \\ Step & \(f_{3}(X)=\sum\limits_{i=1}^{n}(\lfloor x_{i}+0.5\rfloor)^{2}\) & \([-100,\ 100]^{n}\) & \(1\times 10^{-8}\) \\ Exponential & \(f_{4}(X)=\exp(0.5\sum\limits_{i=1}^{n}x_{i}^{2})-1\) & \([-1.28,\ 1.28]^{n}\) & \(1\times 10^{-8}\) \\ Rosebrock & \(f_{5}(X)=\sum\limits_{i=1}^{n-1}\left[100\big{(}x_{i+1}-x_{i}^{2}\big{)}^{2}+( x_{i}-1)^{2}\right]\) & \([-5,\ 10]^{n}\) & \(5\times 10^{0}\) \\ Alpine & \(f_{6}(X)=\sum\limits_{i=1}^{n}\left|x_{i}\cdot\sin x_{i}+0.1\cdot x_{i}\right|\) & \([-10,\ 10]^{n}\) & \(1\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test suite with six benchmark functions. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Fun**} & \multirow{2}{*}{**Precise**} & \multicolumn{3}{c}{**ABC**} & \multicolumn{3}{c}{**MABC-Powell**} \\ \cline{3-8}  & & **Mean** & **Maximum** & **Minimum** & **Mean** & **Maximum** & **Minimum** \\ \hline Sphere & 0 & \(5.99\times 10^{-12}\) & \(2.10\times 10^{-11}\) & \(3.66\times 10^{-13}\) & \(2.95\times 10^{-24}\) & \(6.92\times 10^{-23}\) & \(8.88\times 10^{-37}\) \\ SumSquare & 0 & \(8.67\times 10^{-13}\) & \(3.97\times 10^{-12}\) & \(7.09\times 10^{-14}\) & \(1.18\times 10^{-24}\) & \(4.76\times 10^{-23}\) & \(2.52\times 10^{-35}\) \\ Step & 0 & \(5.43\times 10^{-12}\) & \(1.52\times 10^{-11}\) & \(6.40\times 10^{-13}\) & \(1.25\times 10^{-23}\) & \(5.76\times 10^{-22}\) & 0 \\ Exponential & 0 & \(1.32\times 10^{-15}\) & \(3.77\times 10^{-15}\) & \(4.44\times 10^{-16}\) & 0 & 0 & 0 \\ Rosebrock & 0 & 0.312669 & 1.297957 & 0.029511 & \(3.43\times 10^{-5}\) & \(3.47\times 10^{-4}\) & \(4.95\times 10^{-7}\) \\ Alpine & 0 & 0.000269 & 0.001712 & \(3.90\times 10^{-5}\) & \(8.67\times 10^{-7}\) & \(3.57\times 10^{-6}\) & \(7.29\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Accuracy comparison of standard ABC and MABC-Powell on the six benchmark functions with dimension D = 30.  
optimization were those specified in the numerical example with benchmark functions. A bootstrap sample consisting of three sets of vertical bending and torsional time histories was randomly selected from the sample population of 20 data sets.