First Order Reliability Method

In the present work, the probability of failure Pf of the wing structure is computed. It describes the probability that the structure does not to comply with the predefined requirements. Thus, the term failure has to be distinguished from other terms, like e. g. crash or disaster. Since the coupled fluid-structure analyses are very time con­suming, the first order reliability method (FORM) was implemented to calculate the stochastic characteristics of the wing [10]. FORM introduces the reliability index в to describe the reliability of the structure. The main input to the method is the limit state function G(X), where X is the vector of stochastic variables that influence the structure. By definition, the limit state function is positive, if the structure fulfils its requirements. Negative values are returned, if at least one requirement is violated.

In order to generate unique results for every problem, the vector of stochastic variables is transformed into a vector of standard normal random variables X. This leads to a limit state function G(X’) which is analyzed using the FORM routine. The FORM is a gradient based optimization procedure which calculates the minimum distance в between the limit state function defined by G(X’) = 0 and the origin of the standard normal variable space spanned by the normalized stochastic variables.

At the beginning of the FORM algorithm, a віпШаї has to be estimated. The bet­ter the estimation of this initial value factor the fewer iterations are needed in the algorithm to get the final в. With the вмш and the limit state function value, all parameters are defined to start the main iteration of the FORM algorithm consisting of three main steps: (cp. Haldar, Mahadevan [11])

• Transformation of stochastic variables into standard normal variable space. In order to get unique results, all non-standard normal variables have to be trans­formed. For normal variables, a general conversion can be applied, for other variables, the Method of Rackwitz and Fiessler [12] has to be used.

• Generation of derivatives of the limit state function with respect to the standard normal variables. The coupled fluid-structure model can not be solved algeb­raically. Thus, the derivatives have to be estimated by finite differences in the neighbourhood of the design point.

• Calculation of the direction, where the steepest trend in the limit state function occurs and estimation of a new design point and the corresponding в value

This iteration is repeated until the limit state function value is zero and the в value converges. The resulting в value is then transferred to the fitness value calculation routine of the optimization.