## 3 Kriging-Based Adaptive Surrogate Modeling for Limit Capacity of the Tower
### Example Study
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material**[32] & **Mean (\(\mu\))** & **C.O.V (\(\delta\))** & **Distribution** \\ \hline Yield strength (\(f_{y,Q345}\)) & 387 MPa & 0.07 & Lognormal \\ Yield strength (\(f_{y,Q235}\)) & 264 MPa & 0.07 & Lognormal \\ Elastic modulus (\(E_{s}\)) & 206,000 MPa & 0.03 & Lognormal \\ Poisson ratio (\(\
u\)) & 0.3 & 0.03 & Lognormal \\ \hline Geometry [17] & Mean */standard deviation (\(\mu\)/\(\sigma\)) & C.O.V (\(\delta\)) & Distribution \\ \hline Thickness of angle steel & & & \\ members (\(t\)) & 0.985 & 0.032 & Normal \\ Length of angle steel & & & \\ members (\(l\)) & 1.001 & 0.008 & Normal \\ \hline \hline \end{tabular}  
* The mean value of geometrical parameters is determined by the design size.  
\end{table} 
Table 2: Probabilistic features of structural parameters. As can be seen in Figure 6a, the coefficient of determination (RD) grew quickly to a value of 0.98 and then kept stable. The mean error percent (ME) continuously decreased with the scale of DoE increasing. When the value of ME reached 2%, we terminated the adaptive modeling. Then, the modeling process ended when the number of points used to construct the kriging model was close to 3500. Concerning the local performance of the kriging surrogate, Figure 6c,d show the trends of the local error measures RMSE and MAE. In this process, RMSE fluctuated around the value of 3.5 and the fluctuation weakened with the scale of DoE increasing. MAE shows a decreasing trend with the final value approximating to 2, and the fluctuation of MAE became less obvious in the latter process. Figure 7 further gives the comparison result between the kriging prediction and the observation result in the last loop of the adaptive modeling. Most of the predicted values were located in the range of \(\pm\)5% error, and the prediction yielded satisfactory results when the limit wind speed was less than 100 m/s. The kriging model is expected to predict an assembly of capacity surfaces for transmission towers that subjected to winds, and these capacity surfaces are characterized by probabilities. The prediction on limit wind speed aims at a probability distribution rather than a single value. Hence, in the region of interest, the kriging prediction is the prediction of an assembly of probability distributions. At the issued-based validation step, we first generated 10,000 samples randomly, depending on probabilistic features of structural parameters (shown in Table 2). The 10,000 samples were taken as the set of prediction points, which was characterized by randomness. Then, we took 42 points of the capacity surface uniformly based on the ranges of the wind angle of attack and horizontal span, and the 42 surface points were taken as the set of validation points. In each loop, the constructed kriging model was used to predict the limit wind speed at each validation  
Figure 6: Accuracy/error measures of the kriging model. (**a**) the change of RD; (**b**) the change of ME; (**c**) the change of RMSE; (**d**) the change of MAE. Figure 7: The comparison between the predicted result and the observation result. point of the capacity surface and 10,000 samples of the tower were involved in each validation point. Then, the mean value and standard deviation of the limit wind speed at each validation point were calculated, which further yielded the change ratio of the mean value and standard deviation (Equations (22) and (23)) with the loop number increasing:  
\[R_{\mu}^{i,j}=\frac{\left|\mu_{i}^{j}-\mu_{i-1}^{j}\right|}{\mu_{i-1}^{j}} \tag{22}\]  
\[R_{\sigma}^{i,j}=\frac{\left|\sigma_{i}^{j}-\sigma_{i-1}^{j}\right|}{\sigma_{i -1}^{j}} \tag{23}\]  
where \(\mu_{i}^{j}\) and \(\sigma_{i}^{j}\) are the mean and standard deviation of the limit wind speed of the \(j\)-th validation point in the \(i\)-th loop.