## 3 Applications and Results
### Overtopping Reliability of a Breakwater
Therefore, \(\beta=2.2\) is adopted as the exact value of the reliability index for the breakwater under overtopping. This value is to be used to assess the error by estimating the reliability index with the normality polynomial approach, and to compare versus the results obtained with the multi-linear regression approach. In fact, the results from these two approaches are also shown in Figure 4(b) (black solid line for the normality polynomial; grey dashed line for the multi-linear regression approach), where it is observed that the normality polynomial approach converges to a stable value (approximately \(\beta=2.12\)) from about \(2\times 10^{3}\) simulations on, leading in average to a slightly smaller reliability index (i.e., in the conservative side) but closer to the exact value than by using the FORM. The multi-linear regression approach (like the MCS and unlike the normality polynomial) requires a minimum number of simulations to be carried out, being this number \(1\times 10^{3}\) for Figure 4(b), but sometimes more simulations are required; nevertheless, when a sufficient large number of simulations is performed (e.g., about \(3\times 10^{4}\) or more in Figure 4(b)), the results of the multi-linear regression leads to practically the exact \(\beta\), and the design point and sensitivity factors can also be determined. For brevity, the coefficients of the polynomials and multi-linear regression, design points and sensitivity factors are not extensively listed in this section, but as an example values are given for a single case of \(2\times 10^{4}\) simulations, which led to coefficients for the normality polynomial of \(a_{0}=-2.164\), \(a_{1}=0.3175\), \(a_{2}=-0.0110\) and \(a_{3}\)= 0.0028, and for the multi-linear regression of \(c=1.4760\), \(b_{1}=-0.3382\), \(b_{2}=0.1235\), \(b_{3}=-0.5550\) and \(b_{4}=-0.0633\), as well as sensitivity factors equal to \(\alpha_{Au}=0.5089\), \(\alpha_{Bu}=-0.1859\), \(\alpha_{H}=0.8351\), and \(\alpha_{T}=0.0953\) and design points equal to \(x_{Au}=1.2874\), \(x_{Bu}=-0.7263\), \(x_{H}=9.3045\) m, and \(x_{T}=10.2051\) s, which compares very well with the corresponding sensitivity factors computed with the FORM, that are equal to 0.4959, \(-0.1712\), 0.8466, and 0.0900, respectively, and also very well to the design points from FORM equal to 1.2689, \(-0.7182\), 9.0796 m, and 10.1933 s, respectively. These values of sensitivity factors and design points are also very similar with those reported in [10]. It is noted that Figure 4(b) corresponds to only one set of simulations for every \(n_{sim}\), which may vary for different sets of generated random numbers (as shown in Figures 1 and 3), implying an uncertainty in the deviation from the exact value for different number of simulations. This uncertainty is assessed as for the classical and structural LFSs in previous sections, i.e., by computing the errors in the reliability index as per Equation (18) and fitting them to power laws with the mathematical functional form represented in Equations (19) and (20) (or Equations (23) and (24)), but with different values of the parameters. To do so, and unlike the case of the classical LSF and the reinforced concrete beam under flexure moment, not 1000 but only 100 sets of simulations are computed for each \(n_{sim}\), due to the more extensive required time and computational resources referred to earlier (A comparison in terms of computing time (CPU time), a description and a discussion are given in Appendix B and Figure A2 of the appendix). This leads to the mean errors shown in Figure 4(c) with a black solid line for the normality polynomial case (including mean values \(\pm\) one standard deviation indicated in black dashed lines), and with a grey dashed line for the MCS case (including mean values \(\pm\) one standard deviation indicated in grey dotted lines).