Analytical sensitivities for statistically extrapolated extreme load constraints in structural optimization

For some applications in structural optimization, it is required to have constraints on the extreme loads that represent long term loading conditions. This usually involves a statistical extrapolation procedure that fits maxima from simulated load time series to short term extreme value distributions and then extrapolates to an n-year return value. Often such situations are highly simplified because of the apparent complexity involved in evaluating the sensitivity of such constraints. However, such simplification is not necessary. In this study, we present a method to evaluate the sensitivities of such extrapolated extreme load constraints in a semi-analytical way. The method uses the implicit function theorem to obtain local derivatives at the points defined by the solution of the maximum likelihood estimate that is used to calculate the parameters of the short term extreme value distributions. Comparing with high accuracy finite difference estimates, the method is shown to give reasonably accurate values. We also demonstrate how the method can be used to estimate the uncertainty of the estimated n-year return value caused by uncertainty in both the maximum likelihood estimate and inherent uncertainties in the data. The method then is applied to a simple optimization example and shown to perform very well compared with using finite difference estimates for the sensitivities. Finally, we note that the method is in principle fairly general and could be applied to similar problems that do not specifically involve statistical extrapolation.