PSI - Issue 52

Ilias N. Giannakeas et al. / Procedia Structural Integrity 52 (2024) 655–666 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 5 Two Gaussian Processes (GP) are used to approximate Eq. 5 with ( )~ ( ( ), ( , ′ )) and ( )~ (0, ( , ′)) where ( ) = is a constant and ( , ′ ) and ( , ′) are the covariance functions. A squared exponential function with automatic relevance determination is used here (Rasmussen 2003). The No-U Turn (NUTS) Hamiltonian Monte Carlo sampler (Hoffman and Gelman 2014; Salvatier, Wiecki, and Fonnesbeck (6) where, ̂( ∗ ) and ̂( ∗ ) can be computed using standard GP operations and and are the combined input and outcome matrices, respectively that contain the training data from the numerical and experimental campaigns. The interested reader is referred to (Giannakeas et al. 2023) and the original contributions that proposed this multi fidelity approach in (Higdon et al. 2008; 2004), (Goh et al. 2013) for a more detailed discussion. As illustrated in Fig. 1, during the operation of the SHM system, the detection module will be executed first. If damage is detected, then the localization module will be executed. Let ̃ be the HI and ( ̃ , ̃ ) the location estimate. Then the aim is to find a plausible ∗ that for the given location estimate ( ̃ , ̃ ) , explains ̃ . Using Eq. 6, the likelihood of observing ̃ given ∗ =( ∗ , ̃ , ̃ ) can be written as: ℒ( ̃ | ∗ , , ) = ( ̂ ∗ ( ∗ ), ̂ ∗ ( ∗ )) . (7) Using Eq. 7, Bayesian Inference is used to estimate ̃ as: ( ̃ | ̃ )∝ℒ( ̃ | ̃ ) ( ̃ ) (8) where ( ̃ | ̃ ) is the posterior, ( ̃ )~ (0,4000) is the prior and ℒ( ̃ | ̃ ) is the likelihood function from Eq. 7. It is noted that as a simplification, all terms but ̃ have been dropped. 2.3. Definition of the Experimental and Numerical Campaign Three flat composite stiffened panels were tested to extract the damage sensitive health indicators. Each panel is 1.6m long with 4 horizontal omega stringers and 3 vertical aluminium frames. All panels have been manufactured using standard thermoset unidirectional prepreg material M21/194/34%/T800S with stacking sequence [±45/0 2 / 90/0] . In each bay, 4 PZTs are surface mounted for the actuation and sensing of guided waves. Signal actuation and sensing is performed with a National Instrument waveform generator and an PXI 5105 Oscilloscope. First, all panels are tested under pristine conditions to establish the baseline response between each sensor pair. Environmental and Operational Conditions (EOC) can severely affect the robustness and reliability of the SHM system. To account for variations in the recorded signals due to temperature differences, pristine signals have been collected at various temperatures in the range 25℃ ≤ ≤ 40℃ , using a TVC J2235 environmental chamber. After the collection of the baselines, each panel is impacted, and signal acquisition is repeated. We limit the study to impacts performed at the stiffener foot that produced clear delamination between the skin and the stringer. The first two panels (denoted FP1 and FP2) are impacted at only one location while the third (panel FP3) is impacted at two locations. NDT using hand-held Dolphi Cam was performed after each impact to evaluate the damage area. The impacts are summarized in Table 1. Table 1: Impact Summary ID Energy [ ] [ ] [ ] [ ] FP1‑1 35 1638 −20.4 −90 0.743 FP2‑1 30 1421 −20.4 −90 0.609 FP3‑1 35 2146 −5.80 −163 0.812 FP3‑2 35 2697 −22.4 −163 0.989 The geometry of the flat panel and the locations of the impacts are illustrated in Fig. 3A. In typical GWSHM systems, window function are implemented to reduce boundary reflections while paths that cross multiple stringers are not considered due to excessive attenuation and mode changes that take place (Yue, Khodaei, and Aliabadi 2021). It is therefore easy to consider that the sensor network can be broken down to smaller repeating sub networks. This simplifies considerably the required analysis effort. A subnetwork consisting of 4 sensors is 659 2016) is used to explore the posterior distributions of the hyperparameters. Given a new input vector, ∗ , predictions can be made using: | ∗ , , ~ ( ̂ ∗ ( ∗ ), ̂ ∗ ( ∗ ))