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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Appendix B. Active Kriging Monte Carlo Simulation (AK-MCS) Echard et al. (2011)

AK-MCS is an active learning reliability method combining Kriging and Monte-Carlo. It proposes to replace the true performance function g by its mean Kriging counterpart µ ˆ g . The main steps of AK-MCS are as follows: 1. Generate a Monte Carlo population P of N points to be classified into the failure ( g ( x ) ≤ 0) or safe domains ( g ( x ) > 0) according to the sign of µ ˆ g . 2. Select an initial DOE by using several possible techniques, such as random sampling or Latin Hypercube Sampling (LHS). The true performance function g is then computed for these selected points and gathered in D = { x ( i ) , g ( x ( i ) ) } i ∈ 1 , N D . 3. Iteratively enrich the DOE: (a) Calibrate the Kriging surrogate model ˆ g from the DOE. (b) Evaluate the U -learning function on the whole population P , x ∈ P : U ( x ) = | µ ˆ g ( x ) | σ ˆ g ( x ) . U ( x ) > 2. If this condition is satisfied, the surrogate model is consid ered accurate enough from a reliability point of view and the learning process is stopped (step 3 and 5). If not, the algorithm goes on step 3.d. (d) Add the point x ∗ ∈ P to the DOE that minimizes the U -learning function in 3.b. It here requires the evalua tion of the expensive performance function g ( x ∗ ). Then, the algorithm return to step 3. 4. Evaluate the probability of failure p f with a crude Monte Carlo simulation using µ ˆ g insteadof 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. (c) Check the stopping criterion: min x ∈ P