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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This reorganization is made possible by the fact that the structural and material properties at the critical location, represented by x ∗ B , remain unchanged across the di ff erent load cases. As a consequence, n c points (from the perspective of damage d ) are selected by the U -learning function and must be evaluated to enrich the DOE used to calibrate the Kriging surrogate ˆ d . Applying the AK-SM imputation criterion individually to each of these n c selected points could lead to n c evaluations of the entire m B ◦ m A chain at each iteration, which is not desirable due to the high computational cost of the model m A . Hence, to maintain the computational e ffi ciency, at most one evaluation of the m B ◦ m A chain should be performed per iteration when the AK-SM imputation criterion does not apply. To achieve this, we propose identifying a critical load configuration , associated with a critical loading random vector X A critical , and evaluating m A only for realizations of X A critical during the enrichment process of AK-SM. In this study, the critical configuration is defined as the one associated with the loading random vector X A c of the highest magnitude. Configurations that occur most frequently over the system’s lifetime, or those identified through informed industrial feedback, are valid alternatives but are not discussed in this study. In the remainder of the paper, X A 1 denotes for the loading random vector with highest magnitude, followed by X A 2 with the second highest magnitude, and so on, up to X A nc which have the lowest magnitude. Therefore, the critical configuration is related to the random vector X A 1 and, at each enrichment iteration, the enrichment point ( x ∗ A 1 , x ∗ B ) is always selected among the n c candidates, see equation 16, and m A (¯ x A 1 ) is computed in place of the cost-intensive m A ( x ∗ A 1 ) when the imputation criterion applies; otherwise, only m A ( x ∗ A 1 ) is evaluated. One should note that systematically enriching with realizations of the random vector ( X A 1 , X B ) may however reduce the prediction accuracy of ˆ d for realizations drawn from the n c − 1 random vectors { ( X A c , X B ) } c ∈ 2 , n c associated to non-critical load configurations. To extend the representativeness of ˆ d to non-critical load, we propose to enrich the DOE with points corresponding to the n c − 1 non-critical configurations (i.e., points having x A -component of lower magnitude), without evaluating the expensive model m A . The sample points representing non-critical load configurations are obtained by projecting the points x , which are realizations of the critical random vector ( X A 1 , X B ), onto hyperplanes passing through the cut points { ¯ x c = (¯ x A c , ¯ x B ) } c ∈ 2 , n c . Figure 1 illustrates, these cut points, associated hyperplanes and the projection of two points onto hyperplanes. In this illustration, the x A -component of the critical cut point is chosen as the mean values of X A 1 , and the x A -component of non-critical cut points are computed via the centering operation ¯ x A c = ¯ x A 1 − α c σ A 1 . Finally, the x B -components are common to all cut points, and chosen as the mean value of X B . We now introduce the method Active Kriging for Sequential Model with Critical Configuration Approach (AK-SM-CCA). Figure 2 presents an overview of the proposed AK-SM-CCA approach in the form of a global flowchart. AK-SM-CCA follows the same general framework as AK-SM (see Appendix C) and shares the same sequence of steps. Below, we highlight the key di ff erences between AK-SM-CCA and the original AK-SM method at each stage of the procedure. In Step 2 , the design of experiments D LHS of size N D is generated using the random vector ( X A 1 , X B ), which is as sociated to the critical configuration. The datasets D A and D B are then obtained by projecting the D LHS onto the hyperplane passing through the critical cut point (¯ x A 1 , ¯ x B ), see figure 1. Additionally, n c − 1 datasets, denoted D c , must be computed by projecting D LHS onto the n c − 1 hyperplanes associated with the n c − 1 non-critical cut points { ¯ x A c , ¯ x B } c ∈ 2 , n c . Consequently, the initial design of experiments D , which gathers all initially computed points, be comes: D = D LHS ∪{ ¯ x c , d (¯ x c ) } c ∈ 1 , n c ∪ D A ∪ D B ∪ D 2 ∪ . . . ∪ D n c . The computational cost for generating D is now( N D + n c )( c A + c B ) + ( N D ( n c + 1)) c B , rather than ( N D + 1)( c A + c B ) + (2 N D ) c B in the original AK-SM. In Step3.a , ˆ d is calibrated from D , ˜ d A is calibrated from D A and ˜ d B is calibrated from D B . In Step3.b , ˆ d is used to compute ˆ g and σ ˆ g , see equation 15, and the U -learning function provide the enrichment point x ∗ , see equation 16. In Step3.d The imputation criterion described in section 3.1 is applied using ˆ d , ˜ d A and ˜ d B , and become :