# Global Mechanical Response Sensing of Corrugated Compensators Based on Digital Twins

^{1}

^{2}

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## Abstract

**:**

^{2}and standardized average leave-one-out cross-validation CV

_{avg}of the digital twin satisfy the recommended threshold, which indicates that the digital twin has excellent predictive performance. The single prediction time of the digital twin is 0.76% of the time spent on finite element analysis, and the prediction result has good consistency with the true response under dynamic input, indicating that the digital twin can achieve fast and accurate stress field prediction. The important state information hidden in the multi-source data obtained by limited sensors is effectively mined to achieve the real-time prediction of the stress field. This paper provides a new approach for intelligent sensing and feedback of corrugated compensators in the piping system.

## 1. Introduction

## 2. Method

#### 2.1. Design of Experiments (DoE)

#### 2.2. Gaussian Process Regression Model

**x**) and the covariance function k(

**x**,

**x**′) of a real process f(

**x**) are defined as

**x**) is often assumed to be 0. For the regression problem, consider the following model:

**x**is the input vector, f and y denote the function value and the observation with noise, respectively, and $\epsilon ~N\left(0,{\sigma}_{n}^{2}\right)$ is the error function that obeys the Gaussian distribution. The prior distribution of observation

**y**and the joint distribution of

**y**and predicted value f

_{*}can be obtained as

**x**

_{i}and

**x**

_{j}; and $K\left({\mathit{x}}_{\ast},X\right)=K{\left(X,{\mathit{x}}_{\ast}\right)}^{\mathrm{T}}$ is an n × 1 covariance matrix for test points

**x**

_{*}and inputs of training set X. $K\left({\mathit{x}}_{\ast},{\mathit{x}}_{\ast}\right)$ is the own covariance of the test points

**x**

_{*}. I

_{n}is the n-dimensional unit matrix. From this, the posterior distribution of the predicted value f

_{*}can be calculated as

_{*}of the test points

**x**

_{*}. GPR can choose different covariance functions to evaluate the similarity between samples, the commonly used squared exponential (SE) covariance function is chosen in this paper:

**l**is the length scale and ${\sigma}_{f}^{2}$ is the signal variance. The optimum solution of the hyperparameters $\theta =\left\{\mathit{l},{\sigma}_{f},{\sigma}_{n}\right\}$ is generally obtained by the maximum likelihood method. The negative log-likelihood function of the conditional probabilities of the training samples and its partial derivatives w.r.t the hyperparameters

**θ**can be expressed as

_{*}and the variances ${\widehat{\sigma}}^{2}{}_{{f}_{\ast}}$ of the test points

**x**

_{*}can be calculated with Equations (7) and (8). In this paper, a GPR model is built for each element in the finite element (FE) model, and all GPR models form the digital twin for global mechanical response sensing.

#### 2.3. Performance Criterion

_{avg}) [24] criterion described in Equation 12 is employed to evaluate the performance of digital twin:

_{avg}< 0.1 as the target for a useful surrogate model [24].

^{2}) shown in Equation 13 is chosen as the global performance metric in this paper:

^{2}is to 1, the higher accuracy of the DT model will be. The surrogate model is generally considered to have good predictive ability for a value of R

^{2}greater than 0.8 [25].

#### 2.4. Finite Element Model

_{x}. The value ranges of the three displacements are 0 ≤ v ≤ 10 mm, 0 ≤ w ≤ 10 mm, and 0 ≤ θ

_{x}≤ 5°. The FE mesh and boundary conditions are shown in Figure 2. The FE model contains a total of 19,800 elements.

#### 2.5. Construction Process of Digital Twin

## 3. Results and Discussion

^{2}and CV

_{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

_{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

_{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

_{avg}were all less than the recommended threshold of 0.1, which indicates that the digital twins had an excellent predictive capability.

^{2}value and the minimum CV

_{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 12th Gen Intel

^{®}Core

^{™}i7-12700KF CPU and 32GB 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.

_{x}in the FEA, and the sine signal of v, w, and θ

_{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 of the timing signal at observation points A, B, C, and D, respectively. It can be observed that with the dynamic change of the load, the values of the stresses output from the DT model and the trend with time showed a satisfactory consistency with the FEA calculation results, which further illustrates the reliability of the predicted performance of the DT model developed in this paper and the applicability of the construction method of the DT model for global mechanical response sensing.

## 4. Conclusions

^{2}and standardized average leave-one-out cross-validation CV

_{avg}of the DT satisfy the recommended threshold, which indicates that the DT has excellent predictive performance. The single prediction time of DT is 0.76% of the time spent on FEA, and the prediction result shows a satisfactory consistency with the true response under dynamic change of the load, indicating that DT can achieve fast and accurate stress field prediction. The important state information hidden in the multi-source data obtained by limited sensors is effectively mined to achieve the real-time prediction of the stress field. The study of global mechanical response sensing based on digital twins provides a new approach for intelligent sensing and feedback of corrugated compensators in the piping system.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Comparison of Mises stress values of DT and FEA outputs at observation points A, B, C and D.

**Table 1.**Geometric parameters and material properties of the corrugated bellow [27].

Outer Diameter (mm) | Inner Diameter (mm) | Number of Layers | |
---|---|---|---|

GeometricParameters | 363 | 295 | 1 |

Wall thickness (mm) | Wave height (mm) | Effective length (mm) | |

1.2 | 34 | 352.8 | |

Wave distance (mm) | Wave crest (valley) radius (mm) | Number of waves | |

44 | 10.4 | 8 | |

Materialproperties | Material Type | Elastic modulus (MPa) | Poisson’s ratio |

304 Stainless Steel | 1.95 × 10^{5} | 0.3 |

No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Average |
---|---|---|---|---|---|---|---|---|---|---|---|

R^{2} | 0.9060 | 0.8000 | 0.8807 | 0.8735 | 0.9077 | 0.8206 | 0.8317 | 0.8065 | 0.8850 | 0.8287 | 0.8540 ± 0.0409 |

CV_{avg} | 0.0399 | 0.0481 | 0.0392 | 0.0415 | 0.0315 | 0.0469 | 0.0488 | 0.0505 | 0.0365 | 0.0395 | 0.0422 ± 0.0061 |

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## Share and Cite

**MDPI and ACS Style**

Zhou, R.; Jiang, J.; Qin, J.; Du, N.; Shi, H.; Wang, Y.
Global Mechanical Response Sensing of Corrugated Compensators Based on Digital Twins. *Appl. Sci.* **2023**, *13*, 4048.
https://doi.org/10.3390/app13064048

**AMA Style**

Zhou R, Jiang J, Qin J, Du N, Shi H, Wang Y.
Global Mechanical Response Sensing of Corrugated Compensators Based on Digital Twins. *Applied Sciences*. 2023; 13(6):4048.
https://doi.org/10.3390/app13064048

**Chicago/Turabian Style**

Zhou, Run, Jingyan Jiang, Jianhua Qin, Ning Du, Haoran Shi, and Ying Wang.
2023. "Global Mechanical Response Sensing of Corrugated Compensators Based on Digital Twins" *Applied Sciences* 13, no. 6: 4048.
https://doi.org/10.3390/app13064048