# Global Mechanical Response Sensing of Corrugated Compensators Based on Digital Twins
## 2 Method
### Construction Process of Digital Twin  
Figure 3 shows the construction process of a digital twin for global mechanical response sensing. The offline stage focuses on establishing a high-precision GPR model. [PERSON] et al. [24] suggest that the number of runs for an effective computer experiment should be about 10 times the input dimension \(d\). Thus, the number of training samples in this paper is set to 5\(d\), and the number of test samples is about one-quarter of the number of training samples. First, the input variables are generated by the LHS method, and the corresponding true response is obtained by simulation. Specifically, _llsdesign_, a built-in function of MATLAB, is used to generate the training samples, i.e., displacement load inputs for multiple sets of FE models. Then, the Input (Inp) file generated by MATLAB is submitted to ABAQUS Solver for solving. Extract the stress data, i.e., the corresponding true response values, from the output database (Odb) files by the Python secondary development scripts. So far, the input data and output data of the training set and test set can be obtained.  
[PERSON] and [PERSON] have developed the Gaussian Processes for Machine Learning (GPML) toolbox based on MATLAB [28], which is flexible, simple, and scalable. The GPR model is built with the GPML toolbox. The mean function of the GPR model is selected as the constant function, the Gauss likelihood function is chosen as the likelihood function, and the SE function with automatic relevance determination (ARD) distance measure is chosen as the kernel function. The results of [PERSON] et al. [29] reveal that once a kernel is chosen, different priors for the initial hyperparameters have no significant impact on the performance of GPR prediction. Therefore, the initial hyperparameters in Equations (8) and (9) are set as \(\Theta=\left\{I,\sigma_{f},\sigma_{n}\right\}=\left\{[0.5,0.5,0.5],1.0,0.1\right\}\) in this paper. The hyperparameters are trained by the conjugate gradient descent method. An independent GPR model is built and hyperparameter optimization is conducte  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Outer Diameter** & **Inner Diameter** & **Number of Layers** \\  & **(mm)** & **(mm)** & **Number of Layers** \\  & 363 & 295 & 1 \\ \cline{2-4}
**Geometric** & Wall thickness (mm) & Wave height (mm) & Effective length (mm) \\
**Parameters** & 1.2 & 34 & 352.8 \\ \cline{2-4}  & Wave distance (mm) & Wave crest (valley) & Number of waves \\  & 44 & 10.4 & 8 \\ \hline
**Material** & Material Type & Elastic modulus (MPa) & Poisson’s ratio \\
**properties** & 304 Stainless Steel & \(1.95\times 10^{5}\) & 0.3 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometric parameters and material properties of the corrugated bellow [27].  
Figure 2: FE model of the corrugated bellow.  
element. All GPR models form the digital twin that can calculate the structural stress field in real-time. In the online stage, the deformation information is collected by displacement sensors installed on the corrugated bellows and input into the digital twin. The digital twin feeds back the global mechanical response predictions through fast calculations.