## 6 Validation of CFD-Evaluated Force Coefficients and Flutter Derivatives
### Surrogate Model of the Flutter Derivatives
Figure 19: Surrogate model for the \(H_{2}^{*}\) flutter derivative as a function of the geometric design variables \(C/C_{\text{ref}}\). \(D/D_{\text{ref}}\) and \(G/D_{\text{ref}}\) and the reduced velocity \(U_{r}\). Figure 20: Surrogate model for the \(H_{3}^{*}\) flutter derivative as a function of the geometric design variables \(C/C_{\text{ref}}\). \(D/D_{\text{ref}}\) and \(G/D_{\text{ref}}\) and the reduced velocity \(U_{r}\). In [38], it is stated that coupled flutter stability might happen in slender deck cross-sections featuring flutter derivatives \(H_{1}^{*}<0\) and \(A_{2}^{*}<0\), which corresponds with the values for the flutter derivatives in Figure 14. In the next paragraphs, the impact of the changes in the deck geometry in the flutter performance will be explained, based on the role played by each flutter derivative in the stabilization or destabilization in coupled flutter, as proposed in [38]. Focusing on how the flutter derivatives related to the aeroelastic moment \(A_{i}^{*}\) (\(i=1,\,2,\,3\)) respond to changes in the deck geometry, the surrogate model for the \(A_{1}^{*}\) flutter derivative, in Figure 15, shows a trend for higher values of \(A_{1}^{*}\) as the \(G/D_{\text{ref}}\) ratio increases; furthermore, for a given gap-to-depth ratio and reduced velocity, the flutter derivative \(A_{1}^{*}\) increases with higher \(C/C_{\text{ref}}\) and \(D/D_{\text{ref}}\) ratios. According to [PERSON] and [PERSON] (2011) [8], low absolute values in \(A_{1}^{*}\) are linked with coupled flutter stabilization; thus, low values for the box width and depth (\(C/C_{\text{ref}}\) and \(D/D_{\text{ref}}\) ratios) should be beneficial for flutter performance. Larger slots are linked with higher values for \(A_{1}^{*}\), explaining the lower critical flutter speed obtained, for instance, in [PERSON] et al. (2015) [9] in the upper range of the studied gap distances in that reference. The \(A_{2}^{*}\) flutter derivative follows the oposite trend (see Figure 16), as it increases with shorter slots between boxes. Similarly, \(A_{2}^{*}\) increases as the \(D/D_{\text{ref}}\) ratio decreases and the \(C/C_{\text{ref}}\) ratio increases. Negative values for \(A_{2}^{*}\) show a stabilizing effect, thus larger gaps, and values for the box width and depth close to the ones adopted as reference for this study are related to a better flutter performance. The impact of the \(A_{2}^{*}\) flutter derivative in the coupled flutter stabilization has been further stressed in [8]. The \(A_{3}^{*}\) flutter derivative is relatively insensitive to changes in the twin-box deck geometry within the considered design domain (Figure 17), as can be also noticed in Figure 14. For medium and high reduced velocities, \(A_{3}^{*}\) tends to reach maximum values for box widths close to \(C_{\text{ref}}\) and depths slightly higher than \(D_{\text{ref}}\). For the aeroelastic forces related to the aeroelastic lift force, the surrogate model for the \(H_{1}^{*}\) flutter derivative, depicted in Figure 18, shows a strong sensitivity to the gap distance, with lower values of \(H_{1}^{*}\) as the gap-to-depth ratio increases, therefore improving the coupled flutter stability of the bridge. On the other hand, for medium and high reduced velocities, shorter box widths \(C/C_{\text{ref}}\) are associated with higher values of \(H_{1}^{*}\), with destabilizing effect on flutter performance. For this flutter derivative, the impact of changes in the box depth is minor. In Figure 19, the \(H_{2}^{*}\) flutter derivate shows a strong sensitivity to the gap to depth ratio \(G/D_{\text{ref}}\), as larger gaps produce higher values for the flutter derivatives. Additionally, for small reduced velocities, the maximum in \(H_{2}^{*}\) is located in the vicinity of the reference box design (\(C_{\text{ref}}\), \(D_{\text{ref}}\)).