PSI - Issue 54

Arvid Trapp et al. / Procedia Structural Integrity 54 (2024) 521–535

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Arvid Trapp / Structural Integrity Procedia 00 (2023) 000–000

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Fig. 9. The artificial neural network architecture, comprising N m = 4 hidden layers, each containing N = 8neurons

invariant to diagonal rescaling of gradients, making it well-suited for most problems. During the search for the opti mal model, regularization methods such as L2-regularization and dropout were tested. However, they did not enhance performance, as the training data set was su ffi ciently large and the number of learnable parameters was minimized.

4. Validation

This section includes an analysis of the prediction errors of the model, followed by two distinct demonstrations of the proposed procedure. In Section 4.1, we present the validation results, highlighting the e ff ectiveness of our approach. These are compared with established spectral damage estimators, herein including the Dirlik model. Then, in Section 4.2, we provide an example using recorded in-service loading data and a transfer function from a basic finite element structure. The aim is to show how our proposal can be used for a statistical fatigue assessment approximating real-world situations. Before, we briefly evaluate the model’s stability and accuracy by two metrics — the mean absolute error (MAE) and the mean squared error (MSE). To test whether the data and the model’s architecture are capable of learning the underlying relationships, the model was trained ten times, with the training, validation, and test data sets being reshu ffl ed each time. Table 1 displays mean and variance of these metrics. Their low variance and consistency suggest that the data is adequate and the model’s architecture is robust. train validation test mean of MAE 1.31e-2 1.27e-2 1.29e-2 mean of MSE 3.98e-4 3.91e-4 3.90e-4 variance of MAE 2.02e-7 2.44e-7 2.54e-7 variance of MSE 4.24e-10 5.17e-10 3.80e-10 Table 1. Comparison of the mean absolute error (MAE) and mean squared error (MSE) across data sets, calculated from ten trainings data set train validation test The test data validation is summarized by two synthetic data sets, which are shown in the Figures 10 and 11. Both show the predicted damage ratios from our model compared to the Dirlik method, which assumes stationary Gaussian stresses. We base these ratios on the reference s ( RFC ) eq and depict them for kurtosis β y . These plots show strong correlation between the kurtosis of response time series and fatigue damage. However, if kurtosis would be a su ffi cient descriptor we would encounter a line in this plot. As this is not the case, this motivates for the herein suggested use of closely related descriptors, such as the spectral moments (Eq. 5). Figure 10 presents data generated using the procedure of the training data, while Figure 11 uses data where we already begin with the load’s statistical characterization, which is then processed using linear systems theory, i.e. include its e ff ects in their results. As such, the training data shows the capability of ANN model, while the test data procedure represents the capability of the full 4.1. Test data