A novel self-adaptive step size first-order method for structural reliability analysis based on modified Sigmoid function and Armijo rule

Abstract

Among the first-order reliability methods (FORM), the Hasofer-Lind and Rackwitz-Fiessler method (HL-RF), and the chaos control method (CC) rely on a fixed step size and cannot generate an adaptive one, making it difficult to achieve a satisfactory trade-off between efficiency and robustness when dealing with highly nonlinear problems. To address these drawbacks, this paper proposes two innovative first-order reliability methods: the Sigmoid-based self-adaptive step size adjustment method (SSA) and the hybrid Sigmoid-based self-adaptive step size adjustment method (HSSA). The iterative rotation angle is first determined for both proposed methods. In the SSA method, a modified Sigmoid function is employed to enable nonlinear adaptive adjustments of the step size based on changes in the iterative turning angles, allowing for rapid convergence. Subsequently, the HSSA method incorporates the Armijo rule to further explore a more effective solution. Both proposed methods demonstrate strong computational merits, favorable performance, and user-friendly procedures, providing self-adaptive step sizes suitable for engineering problems, thus offering a broad range of applications. The paper introduces eight examples to showcase the remarkable performance of the two proposed methods. The results indicate that both methods exhibit significantly superior efficiency and robustness compared to other comparative analytical FORM methods when addressing highly nonlinear engineering challenges. Finally, a discussion is presented.