## 3 Applications and Results
### Overtopping Reliability of a Breakwater
Even though errors reported in Figure 4(c) exhibit not as a smooth behavior as those observed in Figures 2 and 4 (obtained in an analogous way but for 1000 sets of \(n_{sim}\)), the qualitative trend is fairly similar, especially for mean values and not so small \(n_{sim}\). Indeed, power laws can be adequately fitted to \(\mu_{c}\) and \(\sigma_{c}\), as shown in Figure 4(d) by fitting the computed errors from \(1\times 10^{2}\) simulations and over; the mathematical functional form is analogous to that of Equations (23) and (24), except that the constant 1 in Equation (23) is omitted. The obtained fitted parameters are \(\delta=23.62\), \(\gamma=0.2342\), \(\kappa=47.94\) and \(\tau=0.4254\). As observed in Figure 4(d) the fitting is very adequate for \(\sigma_{c}\) and adequate for \(\mu_{c}\), albeit only 100 sets of \(n_{sim}\) were employed for the statistics. From Figure 4(c),d similar conclusions to those found before can be drawn, namely, that for decreasing \(n_{sim}\) the MCS tends to deviate more from the exact value than the normality polynomials (in terms of \(\varepsilon\)), that for decreasing number of simulations the error for the MCS can be unknown and that power laws are adequate to mathematically defined \(\mu_{c}\) and \(\sigma_{c}\) for the normality polynomial approach. Therefore, a designer could for instance use the normality polynomial method to compute the reliability index for a reduced number of simulations, whereas accepting an error in the estimation. However, such an error could be estimated if expressions of \(\mu_{\varepsilon}\) and \(\sigma_{\varepsilon}\) (like the power laws determined in this study) are known. As an example, in Figure 5a \(n_{sim}=7\times 10^{2}\) is used for the normality polynomial approach (black solid line) and, as it is shown, this leads to reasonably adequate results (using a fairly small number of simulations) when compared with the FORM and the MCS (also included in Figure 5a with a black dotted line). Moreover, the fitted equations shown in Figure 5d can be used to quantitatively compute the associated error and its standard deviation with respect to the exact reliability index, that is \(\mu_{\varepsilon}=5.09\%\) and \(\sigma_{\varepsilon}=2.95\%\). This is strictly applicable only to \(F_{c}=10\) m; however, comparable errors may be expected for a range of freeboard values by inspecting Figure 5a. Naturally, the contents of this paper could be extended to investigate how the error changes by varying one or more parameters of the LFSs. In such a case, one would expect that functional forms like those reported in this study can be used to assess \(\varepsilon\), but possibly with higher mean errors (and/or standard deviations) for higher reliability levels, because usually more simulations are required for lower probabilities of failure. This could be inferred from Figure 5a, where a last set of calculations is shown by increasing the breakwater slope to 1/2 (dashed and dotted grey lines for the normality polynomial and MCS techniques, respectively), where higher variations of the normality polynomial in relation to the FORM are observed; this higher reliability levels also have the effect of decreasing the ability of the MCS to capture the probability of failure, as also observed in Figure 5a for a wide range of \(F_{c}\). Additionally, although not shown in Figure 5a, it was observed that the minimum number of simulations required to adequately performed the multi-linear regression increases for higher reliability levels (e.g., larger breakwater slopes). Overall, results in this section indicate that the revisited simulation-based methods can be also effective for coastal engineering applications.