PSI - Issue 75

Thomas Constant et al. / Procedia Structural Integrity 75 (2025) 660–676 Author name / Structural Integrity Procedia 00 (2025) 000–000

665

6

where x A c governs the external excitation of the loading configuration c ∈ 1 , n c , the latter being supposed to occur with a fixed proportion p c suchas n c c = 1 p c = 1. If uncertainties about x A 1 , . . . , x A nc and x B come into play, the individual damages d c and total damage D in turn become random and the failure event appears if D exceeds a deterministic threshold d th . Then, the performance function of the durability of system subjected to multiple loading configurations reads:

n c c = 1

g ( X A 1 , · · · , X A nc , X B ) = d th −

p c m B ◦ m A ( X A c , X B )

(11)

In its general formulation, the probability of failure associated to g is given by:

p f = Prob( g ( X ) ≤ 0) =

f X ( x )d x

(12)

g ( x ) ≤ 0

with f X the joint probability distribution function (PDF) of the random vector X = ( X A 1 , · · · , X A nc , X B ). By convention, the system is assumed to be in a failure (resp. safe) state when g ( x ) < 0 (resp. g ( x ) > 0), and g ( x ) = 0 is the limit state. The remainder of the paper is dedicated to addressing this reliability problem.

3. Extension of the AK-SM reliability method to industrial fatigue problem with several loading configurations

3.1. Brief presentation of AK-SM

AK-SM (Constant et al. (2025)) has been initially intended to solve reliability problems in which the underlying physical modelling is represented by a numerical chain of models, with the early stages being significantly more computationally expensive than the latter ones. From this previous work, only a single loading configuration may be considered, and Equation 11 reduces to:

g ( X A , X B ) = d th − m B ◦ m A ( X A , X B )

(13)

The core concept of AK-SM is to view the intermediate output m A ( X A ) as a sometimes missing value, due to its pro hibitive computational cost c A . An imputation criterion has therefore been introduced in the framework of AK-MCS (see Apprendix B) to avoid unnecessary evaluations of m A during the enrichment process whenever possible. The im putation criterion is based on a local first-order truncated functional decomposition inspired from High Dimensional Model Representation (Rabitz and Alis (1999)) of the performance function, evaluated at a cut point ¯ x = (¯ x A , ¯ x B ) optimally chosen for reliability analysis. The proposed decomposition allows us to separate the e ff ect of variables x A and x B on g. It reads:

g ( x A , x B ) = g 0 + g A ( x A ) + g B ( x B ) with: g 0 = g (¯ x A , ¯ x B ) = m B ( m A (¯ x A ) , ¯ x B )

(14)

g A ( x A ) = g ( x A , ¯ x B ) − g 0 = m B ( m A ( x A ) , ¯ x B ) − g 0 g B ( x B ) = g (¯ x A , x B ) − g 0 = m B ( m A (¯ x A ) , x B ) − g 0