PSI - Issue 75

Mohamed El Yazrhi et al. / Procedia Structural Integrity 75 (2025) 262–275 Mohamed El Yazrhi , Jean-Yves Disson / Structural Integrity Procedia (2025)

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This approach is extremely lightweight — just a few simple calculations — However, because it treats damage as always growing at the same rate, it can’t show how confident we are in its predictions or adjust if conditions change. That’s why we turn to the more advanced methods next. 4.2. Monte Carlo Simulation with GMM-Based Block Damage Modelling To incorporate variability in loading and usage patterns, this method uses a probabilistic way to estimate how long it will last (RUL). It’s based on Monte Carlo simulation, which means it runs many random tests to see how damage builds up over time. Each block of damage is treated like a random event, and by running many simulations, the method predicts how damage might grow in the future. The main idea is to use past data to create a model of how damage happens, then simulate many future scenarios to estimate when failure might occur. Block Damage Distribution Modeling During the observation period of duration , the system records for each block and each frequency j. We split the training record into M non-overlapping blocks of length ; with = [ ] The cumulated ( ) for the training duration for frequency j is ( ) = ∑ ( ) =1 Rather than assuming a single deterministic damage rate, we fit a statistical model to the distribution of ( ) across the observed blocks. In this method, we use a Gaussian Mixture Model (GMM) for this purpose. A GMM is a probabilistic model that represents an arbitrary distribution as a weighted sum of multiple Gaussian (normal) distributions: ( ) = ∑ ( | , 2 ), ∑ = 1, ≥0. = 1 = 1 Each component k has weight , mean , and variance . To select ∈{1,2} , we fit both models and choose the one with the lower Bayesian Information Criterion (BIC): = −2 (∁( )) + , Where ∁( ) is the maximized likelihood and p the total number of free parameters. This can be useful because the block damage data may be multi-modal or skewed – for example, some blocks may see high damage (e.g., during periods of high load) while others see low damage. The GMM can capture such mixed behaviour by effectively clustering the block damages into a few representative modes, each with its own mean and variance. The model is trained on the observed block damage values to find parameters for several Gaussian components. This fitting is done during an initialization phase, and the result is a statistical distribution that approximates how much damage a future block is likely to accumulate. The following figure illustrates the advantage of using a Gaussian Mixture Model (GMM) over a standard normal distribution. For certain oscillators and specific signal types (as in this case, road driving conditions), the distribution of the logarithm of the SDF becomes complex and non-Gaussian. The normal distribution fails to capture this complexity, whereas the GMM effectively models the multi-modal nature of the data.