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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Table 3. Median results of the transient modal bracket reliability analysis. ”Number of m A calls” refers to the number of calls to m A required to generate the initial DOE D + the number of executions of the model m A during the enrichment process. Same logic applies to ”Number of calls to m B ”. Method p f δ p f (%) Number of m A calls Number of m B calls N iter AK-MCS-ME 4 . 299 × 10 − 2 2 . 18 5 + 24 5 + 24 6 AK-MCS-CCA 4 . 305 × 10 − 2 2 . 18 9 + 6 34 + 20 6 AK-SM-CCA 4 . 305 × 10 − 2 2 . 18 9 + 1 34 + 24 7 All three methods yield similar failure probability estimates, with a coe ffi cient of variation of 2 . 18%. In terms of computational cost, AK-SM-CCA reduces the number of m A calls by 33% compared to AK-MCS-CCA and by 65% compared to AK-MCS-ME, o ff ering substantial savings. This e ffi ciency comes with two main trade-o ff s: On one side, AK-SM-CCA may requires more iterations to converge, as enrichment often uses { ¯ x A c 1 , x B } instead of the U -selected { x ∗ A c 1 , x ∗ B } owing to the imputation criterion. However, since this mainly adds m B calls, the overall cost remains low. On the other hand, fewer m A evaluations come at the cost of more m B evaluations. This is explained by the imputation strategy associated with the critical configuration approach. However, given that c A = 329 c B here, all the additional m B call costs only 11% of a sole m A call, making the trade-o ff highly favourable. Therefore, the enrichment guidance step (see figure 2 and section 3.1) implemented in the AK-SM-CCA method significantly impacts the total numerical cost of evaluating p f , especially in industrial settings where c A ≥ 1000 c B . Let N iter denote the number of iterations of an AK algorithm, and let us analyze the theoretical cost associated with each of the studied AK strategies, namely AK-MCS-ME, AK-MCS-CCA, and AK-SM-CCA. First, AK-MCS-ME does not require any cut-point selection or projection evaluation, but it necessitates n c evaluations of m B ◦ m A per iteration. The total computational cost is:

C AK-MCS-ME = N LHS ( c A + c B ) + n c N iter ( c A + c B ) ≈ N LHS c A + n c N iter c A ,

since c A ≫ c B . Then, AK-MCS-CCA requires computing cut-points and performing projections onto hyperplanes, but it involves only one evaluation of the full numerical chain per iteration. Its total computational cost is approximately:

C AK-MCS-CCA ≈ ( N LHS + n c ) c A + N iter c A ,

as c B can be neglected. Finally, resorting to the proposed AK-SM-CCA entails reusing a proportion p ∈ [0 , 1] of the m A evaluations from the single computation m A (¯ x A 1 ) already performed during the selection of the critical cut-point (see Figure 2). The total computational cost of p f is:

C AK-SM-CCA ≈ ( N LHS + n c ) c A + pN iter c A ,

Clearly, we have:

C AK-SM-CCA ≤ C AK-MCS-CCA ≪ C AK-MCS-ME .

When the sensitivity of the limit state to the x A component decreases, p → 0 and the computational gain is maximized. Conversely, p → 1 if g depends solely on x A . In this case, AK-SM-CCA degenerates into AK-MCS-CCA.