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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Then, three Kriging surrogate models are calibrated:

• ˆ g , the Kriging surrogate model of the true performance function, g = m B ◦ m A which is used to estimate p f . • ˜ g A and ˜ g B , which are auxiliary Kriging surrogates of g A and g B respectively. These two surrogates are only used to decide whether the full numerical chain m B ◦ m A must be evaluated in the enrichment process, or if m A ( x A ) can be replaced by an already computed quantity m A (¯ x A ), therefore requiring the sole evaluation of m B at a cost c B << c A . ˆ g , e.g. see Marrel and Iooss (2024) for an outstanding review of Kriging surrogate modelling, the imputation criterion reads, for the selected point x ∗ by the U -learning function, see appendix B: By leveraging the variance of the Kriging prediction σ 2

1. If σ 2 ˜ g A 2. If σ 2 ˜ g A

( x ∗ A ) ≥ σ 2 ˜ g B ( x ∗ A ) <σ 2 ˜ g B

( x ∗ B ) then a full numerical evaluation of m B ◦ m A has to be done at a cost c A + c B . ( x ∗ B ), the imputation applies and m A ( x A ) is replaced by m A (¯ x A ). This reduces the cost c A + c B to

c B .

Appendix C describes the main steps of the AK-SM method, including the generation of the Design of Experiments (DOE). AK-SM has proven e ff ective in optimally metamodeling sequential performance functions such as Equa tion 13. However, if distinct loading scenarios are considered, both the dimensionality and the complexity of the performance function g increases, making its direct application challenging. This calls for an updated version of the AK-SM method, able to e ffi ciently handling such multi-configuration loading. The next subsection introduces the proposed extension developed to address this challenge.

3.2. The proposed extension of AK-SM to multiple load configurations: AK-SM-CCA

Let us remember that the objective of the paper is to address the fatigue reliability of systems subjected to several load configurations. The dimension of such a problem, see equation 11, is directly a ff ected by the number n c of configurations. If we assume that the random vectors X A 1 , . . . , X A nc that described fatigue loading are of the same nature, the total dimension of the problem now reaches n c n A + n B instead of n A + n B . Then, a direct application of AK-SM becomes challenging whilst increasing n c without any appropriate strategy. In some industrial contexts, dozens load configurations may arise, say n c > 10. Calibrating a single Kriging metamodel ˆ g of g would thus be intractable due to the potentially high dimensionality of the problem. Moreover, such an approach would lack flexibility, as any change of the usage proportions p c during the system lifetime, would require recalibrating the metamodel, thereby leading to a suboptimal use of the available simulations. As a solution, the following approach suggests calibrating a Kriging surrogate ˆ d for the individual damage d = m B ◦ m A instead of g , thus preserving the original dimension of the problem. The prediction and the variance of the performance function, of dimension n c n A + n B , now becomes: ˆ g ( x A 1 , . . . , x A nc , x B ) = d th − n c c = 1 p c ˆ d ( x A c , x B ) σ 2 ˆ g ( x A 1 , . . . , x A nc , x B ) = n c i = 1 n c j = 1 p i p j k ˆ d (( x A i , x B ) , ( x A j , x B )) (15) with k ˆ d the covariance function associated with the ˆ d Kriging surrogate model of d . During the active Kriging enrichment step of AK-SM (see Appendix C, step 3), the U -learning function will select the most valuable point x ∗ A 1 , . . . , x ∗ A nc , x ∗ B , which can be reorganized as a set of n c vectors, each referring to one of the configuration loads x ∗ A c applied to a same given system of properties x ∗ B : x ∗ = x ∗ A 1 , x ∗ B , . . . , x ∗ A nc , x ∗ B (16)