PSI - Issue 77

Douaa Benhaddouche et al. / Procedia Structural Integrity 77 (2026) 152–160 Douaa BENHADDOUCHE/ Structural Integrity Procedia 00 (2026) 000 – 000

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dynamic state at one location is not independent of others but is governed by shared mode shapes and wave propagation characteristics across the bridge component. In this paper, the dependency between the accelerometers is encoded by the adjacency matrix constructed using the DTW distance between them. The sensors that have DTW value under a fixed threshold value are considered similar and they shared comparable patterns. To capture these patterns and dependencies from DTW graph structure we employ an GCN layer. Let ∈ℝ × be the matrix of sensor features from a given time window, where M is the number of sensors and N the length of the time window, ̃ = + the adjacency matrix with self-loops, and = ∑ ̃ is the degree matrix. The single-layer GCN update is as following: = ( ̃ 1 2 × ̃ × ̃ 1 2 × × ) (2) where ∈ ℝ × is a learnable weight matrix and (. ) is the nonlinear activation ReLU . In this formulation, each sensor’s new representation is a normalized aggregate of its own features and those of its one -hop neighbors, transformed by W. The output H is then fed into a LSTM network to model temporal patterns. An LSTM is a recurrent neural network that uses a memory cell and three gates (forget, input, output) to decide what information to keep, update, or pass on, enabling it to learn long-term temporal dependencies in sequential data (Gers et al., 2000). The extracted spatial feature are fed sequentially into an LSTM unit. At each step, the LSTM unit computes the values at the forget gate f t , the input gate i t , a candidate cell state ̃ , and an output gate o t , that are defined by: f t = σ(W f H t + U f H t−1 + b f ) (3) Then, the cell state c t , and the hidden state are computed as following: c t = f t ⊙ c {t−1} + i t ⊙ ̃ h t = o t ⊙ (c t ) (5) Through this gating mechanism, the LSTM learns to retain relevant long-term vibration patterns while discarding noise or short-term fluctuations. A dense layer is added to compute the final prediction X̃ t+1 as following: X̃ t+1 = ℎ(h t ). (6) This combined GCN – LSTM model enables future sensor responses forecasting based on both spatial coupling and temporal evolution. 2.3 Damage identification and global structural state assessment Forecasting errors from undamaged and unknown states from different time period are used to identify damages and assess the global structural state. For damage detection a cumulative damage indicator (CDI) is calculated using forecasting errors of unknown states. The CDI accumulates the excess errors over time to give the trend of structure degradation, thus the appearance of damage. First, to identify this degradation, a threshold T is set as to the mean plus 1.5 standard deviations of the of undamaged prediction errors e . The CDI for undamaged and t unknown states is calculating as following: ( ) = ∑ ∑ max ( − ,0) = 1 =1 (7) where n is the length of measurement in the unknown state. The rate of CDI growth is proportional to both the frequency and the severity of threshold exceedances. A low value suggests that the system response remains within the expected undamaged domain, while a high value denotes abnormal behavior consistent with structural deterioration. To assess the difference between the prediction error of undamaged and damaged states, the Kolmogorov – Smirnov test is adopted as a global health index (GHI). This non parametric test measures the maximum vertical distance between the empirical cumulative distribution functions (CDFs) of two error distributions (Frank and MASSEY, (4) i t = σ(W i H t + U i H t−1 + b I ) ̃ = (W c H t + U c H t−1 + b c ) o t = σ(W o H t + U o H t−1 + b o )