# Global Mechanical Response Sensing of Corrugated Compensators Based on Digital Twins
## 3 Results and Discussion  
Digital twins need to be accurate and real-time, both of which were verified by numerical simulations below. Since the input dimension \(d=3\), the number of initial samples was \(10d=30\), and the number of test samples was set as 8. To avoid the chance of calculation results caused by the training samples, 10 sets of training samples and test samples were randomly selected by LHS. Figure 4 shows the sampling results of the first simulation, where the red dots are the training samples and the blue triangles are the test samples. The samples have good randomness and space-filling. For each of the ten sets of samples, the digital twin model was built by the training samples, and then the performance metrics \(R^{2}\) and \(CV_{\text{avg}}\) were calculated according to the method in Section 2.3 to test the accuracy of the digital twin. The calculated results of the performance metrics for the ten simulations and their average values are shown in Table 2. The average values of \(R^{2}\) and \(CV_{\text{avg}}\) were 0.8540 and 0.0422, respectively. The data of Table 2 were plotted in Figure 5, where the red dots were \(R^{2}\), the blue triangles were \(CV_{\text{avg}}\), and the dashed lines indicated the recommended thresholds for the performance metrics. From Figure 5, it can be observed that the values of \(R^{2}\) were no less than the recommended threshold of 0.8 and the values of \(CV_{\text{avg}}\) were all less than the recommended threshold of 0.1, which indicates that the digital twins had an excellent predictive capability. The optimized calculations of the fifth simulation with the maximum \(R^{2}\) value and the minimum \(CV_{\text{avg}}\) value were chosen as the final DT model, which included 19,800 sets of GPR hyperparameters corresponding to 19,800 elements. In this paper, we examine the real-time computing capability of the digital twin by comparing the computational delay time of the FEA and digital twin under the same working condition. When the program with a randomly selected working condition was run on a PC with 12 th Gen Intel(r) Core(tm) i7-12700 KF CPU and 32 GB RAM, the computation time of the FEA was about 85 s, while the prediction time of the GPR model was about 0.65 s, which is 0.76% of the time spent of FEA. It was shown that the GPR model can achieve rapid prediction of the bellows stress field in quasi-real time. Figure 3: The construction process of a digital twin for global mechanical response sensing. The main purpose of this study is to verify the reasonableness and applicability of the technical framework of the digital twin by comparing the predicted values with the theoretical values obtained through finite element analysis. Therefore, the applicability of the DT model to the dynamic monitoring data was verified by comparing the prediction results of the DT model with the FEM calculation results by using the time-course data as the input of the DT model. As shown in Figure 6, the sine signal was used as the loading curve of three displacements \(v\), \(w\), and \(\theta_{x}\) in the FEA, and the sine signal of \(v\), \(w\), and \(\theta_{x}\) with a sampling frequency of 1 Hz and containing noise was used as the input data of DT model, to simulate the real-time monitoring data acquired by the corresponding displacement sensors. Set up four observation points A, B, C, and D as shown in Figure 2. As shown in Figure 7, the red line represents the predicted response value of the digital twin, and the blue line represents the true response value calculated by FEA. The FEA-calculated stress values and the stress values predicted by the DT model were output at each sampling point