PSI - Issue 75

Thomas Constant et al. / Procedia Structural Integrity 75 (2025) 660–676 Author name / Structural Integrity Procedia 00 (2025) 000–000

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external loading configurations expected to represent realistic operational conditions over the system’s lifetime must be simulated to comprehensively evaluate the cumulative damage. However, the resulting cumulative damage is of course inherently subject to uncertainties, due to the natural variability in loading conditions, structural dimensions, and material properties to name just a few. To account for these uncertainties, a common engineering practice relies on the use of empirical safety factors to computed lifetimes. While this approach is expected to ensure structural integrity, it provides no explicit insight into the actual safety margin and often results in overly conservative designs. Therefore, it is valuable to evaluate the probability of failure p f . It reflects the risk associated with an insu ffi cient structural capacity in the presence of uncertainties. Nevertheless, depending on factors such as the nature of the FE analysis (linear or nonlinear), mesh resolution, and boundary conditions, a sole FE simulation can require several hours—or even days—to complete, making uncertainty propagation within the durability numerical chain computa tionally intractable without relevant strategies. Then, a critical research challenge here lies in addressing structural reliability problems involving a computationally expensive numerical chain. In such a context, crude Monte Carlo Simulation (MCS) (Gaspar et al. (2014))—as well as advanced sampling-based methods such as Subset Simulation (Au and Beck (2001)) or Importance Sampling (Melchers (1989))—become impractical, as they require thousands to hundreds of thousands of calls to the numerical model. Alternative strategies, like the First-Order Reliability Method (FORM) (Hasofer and Lind (1974) )and Second-Order Reliability Method (SORM) (Kiureghian et al. (1987)), have proven e ff ective for some problems of limited dimension but rely on strong assumptions regarding the linearity and smoothness of the limit-state function associated with the reliability problem. An alternative approach involves cali brating a surrogate model of the costly numerical chain and leveraging its negligible computational cost to estimate the probability of failure using sampling techniques. To minimize the number of calls to the numerical model required to calibrate the surrogate, active learning frameworks have been proposed in the last decades. Among all, the combina tion of Kriging surrogates with simulation-based techniques, such as AK-MCS (Echard et al. (2011)), AK-SS (Huang et al. (2016)) and AK-IS (Li et al. (2022)), has emerged as one of the most popular and widely quoted solution. How ever, as the computational cost of the finite element models within the durability numerical chain is extremely high, those conventional active Kriging methods may struggle to converge within the limited simulation budgets typically encountered in industrial contexts. To address this issue, the authors have developed a novel active Kriging-based learning method named Active Kriging for Sequential Model (AK-SM) (Constant et al. (2025)). In this work, an im putation criterion—based on a local functional decomposition and the Kriging prediction variance—was introduced to avoid unnecessary evaluations of the full sequence of finite element simulations and e ffi ciently earmarked the avail able simulation budget. The study has primarily focused on the methodology and has considered an unique load case as input of the numerical durability chain. However, in industrial applications, loading is typically represented by a set of realistic operational configurations. Therefore, the objective of the present paper is to extend the previously published approach to solve an industrial problem, where multiple load configurations are used to describe the sys tem’s operating conditions. The remainder of this paper is organized as follows. First, a mathematical formalization of the durability numerical chain is provided as well as its associated reliability problem. Next, the new approach for AK-SM to remain e ff ective is presented. Finally, a reliability analysis of a transient bracket model subjected to a realistic loading scenario—characterized by a set of di ff erent configurations—is discussed, followed by concluding remarks.

Nomenclature

X , x Random vector and its realization X A , X B Input random vector for m A and m B , respectively m A Finite element model m B Fatigue model d = m B ◦ m A Accumulated structural damage g Performance function

¯ x = { ¯ x A , ¯ x B } Cut point used to evaluate the local functional decomposition of g or d ˆ g , ˆ d Kriging surrogates of the performance function and the damage, respectively ˜ g A , ˜ d A Kriging surrogates of the X A -component of the functional decomposition of g and d , respectively