PSI - Issue 75

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

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Author name / Structural Integrity Procedia 00 (2025) 000–000

Appendix C. Active Kriging for Sequential Model (AK-SM) Constant et al. (2025)

The objective of AK-SM is to built a Kriging surrogate model ˆ g for a sequential performance function such as g ( . ) = d th − m B ◦ m A ( . ), while minimizing the number of calls to the most expensive model m A . The main steps of the approach are described below: 1. Generate a Monte Carlo population P of N points to be classified into the failure or safe domains according to the sign of µ ˆ g . 2. Select an initial DOE by using Latin Hypercube Sampling (LHS). The whole performance function g is then computed for these selected points: D LHS = { x ( i ) , g ( x ( i ) ) } i ∈ 1 , N D . Then, select the cut point ¯ x = (¯ x A , ¯ x B ) and com pute the B-projected DOE D A = { x ( i ) A , g ( x ( i ) A , ¯ x B ) } i ∈ 1 , N D and the A-projected DOE D B = { x ( i ) B , g (¯ x A , x ( i ) B ) } i ∈ 1 , N D . Finally, group all the points to form the intial DOE D = D LHS ∪{ ¯ x , g (¯ x ) }∪ D A ∪ D B 3. Iteratively enrich the DOE D : (a) Calibrate the Kriging surrogate model ˆ g from the DOE D , ˜ g A from D A and ˜ g B from D B . (b) Evaluate the U -learning function on the whole population P , x ∈ P : U ( x ) = | µ ˆ g ( x ) | σ ˆ g ( x ) . (c) Check the stopping criterion, therein the Error Based Stopping criterion (ESC), proposed by Wang and Shafieezadeh (2019). If this condition is satisfied, the surrogate model is considered accurate enough and the learning process is stopped (step 3 and 5). If not, the algorithm goes on step 3.d. (d) Check the imputation criterion as described in section 3.1 and decide whether to enrich with the full point x ∗ ∈ P , that minimizes the U -learning function and which requires the execution of the whole chain m B ◦ m A , or with the A-projected point (¯ x A , x ∗ B ), only requiring to run m B ( x ∗ B ). Then, the algorithm returns to step 3. 4. Evaluate the probability of failure p f with a crude Monte Carlo simulation using µ ˆ g instead of g . 5. Check the coe ffi cient of variation of p f . If it exceeds a predefined threshold, increase the size of P and go to step 3; otherwise end the process.