Abstract
This paper proposes a reliability-based design optimization (RBDO) approach that adopts the second-order reliability method (SORM) and complex-step (CS) derivative approximation. The failure probabilities are estimated using the SORM, with Breitung’s formula and the technique established by Hohenbichler and Rackwitz, and their sensitivities are analytically derived. The CS derivative approximation is used to perform the sensitivity analysis based on derivations. Given that an imaginary number is used as a step size to compute the first derivative in the CS derivative method, the calculation stability and accuracy are enhanced with elimination of the subtractive cancellation error, which is commonly encountered when using the traditional finite difference method. The proposed approach unifies the CS approximation and SORM to enhance the estimation of the probability and its sensitivity. The sensitivity analysis facilitates the use of gradient-based optimization algorithms in the RBDO framework. The proposed RBDO/CS–SORM method is tested on structural optimization problems with a range of statistical variations. The results demonstrate that the performance can be enhanced while satisfying precisely probabilistic constraints, thereby increasing the efficiency and efficacy of the optimal design identification. The numerical optimization results obtained using different optimization approaches are compared to validate this enhancement.
Highlights
IntroductionPublisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
The first-order reliability method (FORM) is an analytical approximation of the probability integral, obtained by linearizing a limit-state function transformed into the standard normal space at an optimal point
An efficient and accurate reliability-based design optimization approach was developed by integrating the second-order reliability method and CS derivative approximation
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Many algorithms and methods have been proposed to address these challenges and increase the efficacy of the reliability analysis, including simulation-based [1,2], surrogate model [3,4], matrix-based [5], first-order linearization [6,7], and quadratic approximation approaches [8,9,10] Among these techniques, the first-order reliability method (FORM) linearizes the limit-state equations to approximate the probability, whereas the second-order reliability method (SORM). To identify the optimal structure that satisfies a certain level of reliability, it is crucial to perform the sensitivity analysis of the failure probability estimated using the SORM, which is generally more accurate than that obtained using the FORM. The performance and applicability of the proposed method are demonstrated by numerical examples
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