# CFD-Based Metamodeling of the Propagation Distribution of Styrene Spilled from a Ship
## 3 Results and Discussion
### Hidden-Point Tests for the Evaluation  
Mostly, comparison with experimental data has been widely adopted for model evaluation in many relevant numerical studies. Unfortunately, there is no available data for direct comparison of styrene under the specified conditions. Instead, we used another method, the so-called hidden point (or layer) method, which has been mainly introduced in machine learning and neural networks. More clearly, the hidden point (or condition) is not included in 80 different CFD cases that are used in making the current metamodel. Hidden points are new scenarios. Based on them, the CFD simulation was first conducted and then the metamodel ran separately, and finally, the two results were compared. We also \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(V_{s}\) (m/s) & \(V_{d}/V_{s}\) & \(D_{s}\) (m) & \(L_{c}\) & \(R^{2}\) & RMSE \\ \hline
0.1 & 0.5 & 30 & Bottom & 0.9773 & 2.17 \\
0.2 & 0.5 & 30 & Bottom & 0.9335 & 6.43 \\
0.5 & 0.5 & 30 & Bottom & 0.9575 & 6.91 \\
1.0 & 0.5 & 30 & Bottom & 0.9494 & 12.83 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The summary of fit with using \(R^{2}\) and RMSE. Figure 6: Interface of the spilled styrene predicted using the metamodel (black dots) and computational fluid dynamics (CFD) simulation (blue dots) in the case of \(t\) = 1000 s, \(v_{d}/v_{s}\) = 0.5, \(d_{s}\) = 30 m, and \(L_{c}\) = bottom crack. The criterion for the interface was defined as a mass fraction corresponding to the exposure limit (100 ppm). selected three additional parameters, the first of which was the boundary arrival time \(t_{\text{b}}\), defined as a time at which the styrene interface reached the outmost boundary face of the computational domain. The two other parameters were the \(x\) and \(y\) coordinates of the styrene interface at \(t=1000\) s after the spill started. Figure 7 shows the contours calculated by the metamodel at the given \(D_{s}\) of 30 m in the case of a bottom crack. The surface represents the response of the metamodel for the inputs, the black dot represents the CFD simulation result from 80 cases taken in the present study, and the red dot indicates the metamodel prediction based on the hidden point. The comparison result is summarized in Table 2. The CFD simulations were performed for eight additional cases, which were regarded as the hidden points. Those eight cases are listed in Table 2 were not included in the 80 cases used for making the metamodel. For measuring the relative error between the CFD result and the metamodel prediction, the quantitative error of boundary arrival time could be estimated using:  
\[\delta_{1}=\frac{V^{\prime}-V}{V}\times 100\,(\%) \tag{14}\]  
where \(V^{\prime}\) denotes the metamodel prediction at the hidden point and \(V\) represents the CFD result at the same point. For the \(x\)- and \(y\)-coordinates of the styrene interface, the relative error was obtained using:  
\[\delta_{2}=\frac{\sum\limits_{i=1}^{3}\sum\limits_{j=1}^{36}\sum\limits_{k=1} ^{42}\left(\frac{V^{\prime}-V}{V}\times 100\right)}{i\cdot j\cdot k}(\%) \tag{15}\]  
where \(i\) denotes the coordinate, and \(j\) and \(k\) represent the angle and the time, respectively. The estimated error of the styrene interface was less than 9.2%, which shows a relatively good performance. As shown in Table 2, the created metamodel showed relatively good for qualitatively predicting how the spatial distribution changed with time, though it provided a big difference and showed a low accuracy.