# Structural Damage Detection through Dual-Channel Pseudo-Supervised Learning
## 3 Validation by a Numerical Model  
### IASC-ASCE SHM Benchmark Dataset  
The IASC-ASCE SHM research group established the IASC-ASCE SHM simulated benchmark structure [45] in 2003. This benchmark structure aimed to create a standardized platform for evaluating structural damage detection methods. It enables researchers to compare and analyze different structural damage detection methods. In this section, we used its numerical model for testing and evaluation. As shown in Figure 3, the IASC-ASCE SHM benchmark model is a four-story, 2 \(\times\) 2-span, steel-frame model with a floor height of 0.9 m and a plan size of 2.5 m \(\times\) 2.5 m and which is 1.25 m per span. The model has 9 steel columns on each floor and is set up with 8 diagonal supports. More information about this benchmark model can be found in reference [45]. Figure 2: Architecture of the proposed model. Gaussian white noise is a common method for simulating environmental excitation [46; 47]. In this paper, Gaussian white noise was used as the excitation signal to induce structural vibrations. This noise excitation was generated through a random process following a normal distribution. As a result, it has a flat power spectral density across the frequency domain and uniform energy distributions across all frequencies. Its power spectral density can be expressed as follows:  
\[S(f)=\frac{N_{0}}{2} \tag{20}\]  
where \(S(f)\) is the power spectral density at frequency \(f\), and \(N_{0}\) is the noise power spectral density. With an equal amount of energy in each frequency component, this noise signal effectively excites various vibrational modes within the structure, providing comprehensive excitation for accurately assessing the structure's dynamic response under various conditions. To evaluate the effects of noise on the proposed damage detection method, a series of Gaussian random pulses was introduced into the numerically calculated acceleration response data of the IASC-ASCE SHM benchmark structure [45]. \[\widetilde{x_{i}}=x_{i}+\left\{\frac{e}{100}\times RMS_{max}\times v_{i}\right\} \tag{21}\]  
where \(\widetilde{x_{i}}\) and \(x_{i}\) denote the acceleration responses of the \(i\)-th sensor with added noise and without noise, respectively. The variable \(e\) represents the noise percentage, for example, \(e\) = 20 means that 20% of the noise is considered. \(RMS_{max}\) is the maximum value of the root mean square (RMS) of the acceleration responses measured by all sensors on the structure. \(v_{i}\) denotes a set of Gaussian random variables generated using the MATLAB 2022b routine '_random_', with a mean value of zero and a standard deviation of one. In this study, three distinct levels of noise were taken into account, which were 0%, 20%, and 50%, respectively. As shown in Table 1, this study focuses on the four damage patterns of D.P.0, D.P.1, D.P.2, and D.P.4 defined in case 4 of the IASC-ASCE SHM benchmark. Figure 4 shows model diagrams of the three damage patterns of D.P.1, D.P.2, and D.P.4, and D.P.0 is shown in Figure 3.  
As shown in Figure 5, four acceleration sensors were deployed on each floor of the IASC-ASCE SHM simulated benchmark structure. Each sensor recorded 800 s of acceleration data at a sampling frequency of 250 Hz under the noise levels of 0%, 20%, and 50%, respectively. This resulted in each sensor collecting 200,000 data points for each damage pattern. The collected response data were preprocessed according to the methods described in Section 2.2. At every noise level, the subset with four damage patterns in any  
Figure 3: A model of the IASC-ASCE SHM benchmark structure. models. Among these, 1D CNN and LSTM are classical CNN and RNN architecture models, respectively, commonly used in structural damage detection research. The multi-head CNN is a variant of the CNN model. The details of the baseline models are described as follows:  
* One-dimensional CNN: This model can be used to process acceleration response data, where the convolution layer can capture the local features of the signal.