PSI - Issue 48

Behrooz Keshtegar et al. / Procedia Structural Integrity 48 (2023) 348–355 Keshtegar et al / Structural Integrity Procedia 00 (2019) 000 – 000

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1. Introduction The first order reliability method (FORM) is a probabilistic framework that allows to compute on approximation of the failure probability, widely used in structural reliability analysis (Taiwo, El Amine Ben Seghier, and Zayed 2023). FORM is popular due to its suitable balance between accuracy and efficiency of the reliability analysis. Unlike the Monte Carlo Simulation (MCS) approaches (Jafari-Asl, Ben Seghier, et al. 2021; Seghier et al. 2018; Seghier, Mustaffa, and Zayed 2022), the main effort in FORM is to search the most probable point (MPP U*) by the following optimization model (Guillal et al. 2020; Keshtegar et al. 2021; Nie and Ellingwood 2000): U * =arg min (|| U || | G( U ) =0) (1) where, U is a normal standard vector which is determined as { ( )} 1 X U X F   and G( U ) =0 is the performance function/limit state function in the normal standard space. Iterative analytical formulations can be applied to search MPP such as (HL-RF), chaos control (Yang 2010), directional stability transformation (DSTM) (Meng et al. 2017), relaxed approaches (Keshtegar and Meng 2017), finite step size (Yi and Zhu 2016) and conjugate approaches (El Amine Ben Seghier, Keshtegar, and Elahmoune 2018; Keshtegar 2016; Keshtegar and Chakraborty 2018; Ben Seghier, Keshtegar, and Mahmoud 2021) . The main effort is to enhance the robustness and efficiency, because the traditional FORM formula may produce unstable results as for chaotic and periodic solutions of highly nonlinear performance functions (Bagheri et al. 2020; Keshtegar et al. 2019). Therefore, the sensitivity vector of the iterative FORM formula may be inaccurately computed in the suitable direction to achieve the stabilization of the MPP. Recently, the no-gradient approach using cross entropy optimization has been used to achieve stable results for complex problems thanks to the free sensitivity vector (Ghohani Arab et al. 2019; Zhu et al. 2022). The applicability of the non-gradient approaches is superior to the iterative FORM, because direct optimization approach-based intelligent tools can be utilized for discrete and non-continues performance functions (Jafari-Asl et al. 2022; Jafari Asl, Ohadi, et al. 2021). The accuracy of the approximated failure probability is the major issue of the non-gradient approaches to FORM. The gradient-based FORM formulas have several drawbacks as inefficient algorithms-based computational burden for high-dimensional reliability problems, the instability iterative formulations for highly nonlinear problems, and inaccurate searching approaches for problems with several MPP (Keshtegar et al. 2019). On the other hand, the FORM-based gradient sensitivity analyses my loss the capability for discrete reliability problems. Therefore, non-gradient algorithms can be utilized to solve some drawbacks of FORM, whereas their accuracy and efficiency are depended on their random search process. In this paper, the convergence performances for robustness, efficiency and accuracy of the iterative FORM formulas using the traditional probabilistic model in Eq. (1) are compared with the non-gradient approaches. Two iterative formulations using HL-RF and DSTM methods are applied for FORM-based gradient approaches. The FORM optimization probabilistic model is refined to improve accuracy in the non-gradient approach. The harmony search (HS) and particle swarm optimization (PSO) intelligent framework-based non-gradient approaches are applied and compared in term robustness and accuracy to the refined-FORM with FORM-based HL-RF and DSTM. Three nonlinear/complex reliability problems are used to compare the convergence performances as accuracy, efficiency and robustness of the four reliability methods. The comparative results demonstrate that the FORM – based refined probabilistic model provides more accurate and robust results compared to FORM for highly nonlinear problems, and the robustness of FORM is improved by the non-gradient approach. The PSO a more accurate meta-heuristic approach than HS, and is more robust than DSTM and HL-RF for complex problems. 2. First Order Reliability Method The failure probability is approximated in the FORM by the reliability index  using the following relation(Keshtegar 2017):   ( ) ( ) ... ( ) 0 ( ) 0         X X X X X f d P g g f (2)