Global Mechanical Response Sensing of Corrugated Compensators Based on Digital Twins

Abstract

The corrugated compensators are important components in the piping system, absorbing mechanical deformation flexibly. To reduce the risk of the piping system with corrugated compensators and improve the safety and stability of industrial equipment, condition monitoring and fault diagnosis of bellows is necessary. However, the stress monitoring method of corrugated compensators with limited localized sensors lack real-time and full-domain sensing. Therefore, this paper proposes a digital twin construction method for global mechanical response sensing of corrugated compensators, combining Gaussian process regression in machine learning and finite element analysis. The sensing data of three types of displacements are used as the associated information of a finite element model with 19,800 elements and its digital twin. The results show that the values of performance metrics correlation of determination R² 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

Corrugated compensators, also named expansion joints, are widely used on thermal pipelines and pressure vessels, and other equipment in metallurgical, chemical, and electric power industries for displacement compensation, vibration, and noise reduction [1]. Corrugated compensators consist of one or more flexible bellows elements with end fittings such as flanges to allow connection to the adjacent piping or equipment. The flexibility of the bellows is achieved by bending the convolution sidewalls and flexing within their crest and root radii. Figure 1 illustrates four deformation modes of the bellows, including a reduction of the bellow’s length due to piping expansion (axial compression), an increase of the bellows length due to pipe contraction (axial extension), bending about the longitudinal center line of the expansion joint (angular rotation), and transverse motion which is perpendicular to the plane of the pipe with the corrugated compensators ends remaining parallel (lateral offset). In general, the bellows are affected by the combination of internal pressure, temperature change, corrosive environment, and displacement load, and compensate for pipeline deformation under alternating load [2]. Fatigue crack caused by the alternating load is corrugated compensators’ most significant failure mode [3].

For the industrial equipment system, the local accidental failure of mechatronic production equipment can cause the whole production system to be difficult to control, and can also easily cause a series of chain reactions, endangering the reliability of industrial equipment operation [4]. Condition monitoring and fault diagnosis of corrugated compensators are required to reduce the production risk of modern piping systems with corrugated compensators and improve the safety and stability of industrial equipment operation. The global service state information of corrugated compensators such as deformation, pressure, and temperature are easy to obtain based on the corresponding sensors [5]. However, for the acquisition of the stress field, we can only obtain local stress data through finite strain gauges. Pagar et al. [6,7] investigated the effect of angular rotational misalignment in pipe structure on the deflection-based convolution stresses by mounting the strain gauges at the prescribed locations of U-shape bellows specimens. Failures of the corrugated compensators are very unpredictable during the functioning [8], hence in-depth stress monitoring and analysis of the corrugated portion are required. Therefore, to achieve the effect of advanced prediction and timely maintenance, it is of great significance to study the effective strategy of mechanical response sensing of the corrugated compensators and to accurately obtain complete information representing the time domain and space domain of the equipment by limited sensing devices, to ensure the safe and stable operation of the equipment [9].

With the current rapid development and improvement of sensing technology, numerical simulation methods, and Artificial Intelligence (AI) algorithms, it is possible to apply the actual monitoring data in the digital model to make full-domain and real-time performance predictions of the equipment according to different load conditions. In this background, digital twin (DT) is gradually applied to the structural analysis of complex mechanical equipment to realize the interaction between the real world and the virtual world [9]. As an important way to realize the real-time connection and dynamic interaction between the physical entity and the virtual twin, the digital twin can map the working state and mechanical properties of the physical entity to the virtual space [10]. In the last decade, NASA and the U.S. Air Force have continued the exploration of grounded applications of the digital twin in the aerospace field [11]. Tuegel et al. [12], based on the concept of the digital twin, envisaged the construction of a high-fidelity DT model of the vehicle to improve the life prediction and management capabilities of the vehicle and discussed the difficulties and potentially feasible technical solutions. Zakrajsek et al. [13] established and applied a digital twin for aircraft tires to improve the prediction accuracy of tire landing wear, and constructed a nonlinear landing wear response model based on high-fidelity test data.

In terms of digital twin applications that integrate real-time numerical computation and performance prediction, Haag et al. [14] demonstrated how the digital twin can be applied to a real cantilever beam model, which was based on a simulation model and sensor data to calculate the displacement field of the cantilever beam in real-time. Although the structural form of this case is simple, it demonstrates the application effect of structural digital twin vividly. Moi et al. [15] proposed a digital twin solution for condition monitoring of a small-scale knuckle boom crane and developed an inverse method based on strain gauges to calculate force vector weights, which can calculate the real-time stress, strain, and load at specific locations of the crane, thus enabling real-time condition monitoring of the crane. Lai et al. [16,17] developed a digital twin technology framework for shape-performance integration with multi-model fusion and introduced the DT model construction method for structural analysis of mechanical equipment using a boom crane and a 2D truss as examples. The prediction of mechanical response fields of cranes and trusses is achieved by using multi-source dynamic sensing data as model inputs and by real-time computation of analytical, numerical, and AI models. Song et al. [18,19] constructed a digital twin of the human lumbar spine based on this framework and realized real-time monitoring of the mechanical properties of the human lumbar spine by combining data-filling technology and inverse kinematics technology. With multi-source sensing data as model input, the prediction of mechanical response can be achieved by the real-time calculation of the analytic model, numerical model, and AI model. However, the digital twin model for equipment of piping systems such as corrugated compensators, which are under multiple loads, is rarely involved.

It is important to study the effective strategy of global mechanical response sensing of corrugated compensators and to accurately obtain complete state information of the equipment with limited sensing devices to ensure the safe and stable operation of the equipment. Given this, this paper proposes a construction method of the DT for global mechanical response sensing of corrugated compensators, which combines Gaussian process regression (GPR) in machine learning and finite element analysis (FEA). The training samples and test samples are sampled by the Latin hypercube sampling (LHS) method, the GPR is performed by MATLAB, and the FEA is carried out by the commercial software ABAQUS (2020) and Python secondary development scripts. The sensing data of three types of displacements are used as the associated information of a finite element model with 19,800 elements and its digital twin. 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.

2. Method

2.1. Design of Experiments (DoE)

For the sampling of training and test sets without prior knowledge, it is important to ensure uniform distribution in the design space. In this paper, Latin hypercube sampling (LHS), a DoE method with good space-filling and widely used in modern design, is employed. Assume that there are n samples in a d-dimensional space and the d-dimensional space is divided into n cells with equal probability. A random value is taken for each cell so that each dimension has n values. Then, the n variable values in d-dimensions are randomly combined into an n × d matrix, i.e., n samples. LHS ensures that when all samples are arbitrarily projected into a dimension, there is only one sample in each cell, thus avoiding the aggregation of samples in a small area. Specifically, lhsdesign, a built-in function of MATLAB, is used to generate training samples and test samples.

2.2. Gaussian Process Regression Model

The irregular shape of corrugated compensators can lead to problems such as high nonlinearity and large data size when calculating the mechanical properties, which leads to high calculation costs. The key to the fast prediction of the structural stress field is to build its surrogate model. In this paper, Gaussian process regression (GPR) is chosen to build the surrogate model of the corrugated compensator. As a machine learning regression method developed in recent years, GPR is well adapted to handle complex problems with high dimensionality, small samples, and nonlinearities [20]. GPR has the advantages of easy implementation, adaptive acquisition of hyperparameters, probabilistic significance of the output, etc. Görtler et al. [21] demonstrated how the Gaussian process (GP) works in solving regression problems with a few vivid visual examples. The following is the inference process of GPR in function space [22].

A GP is a collection of random variables, any finite number of which have a joint Gaussian distribution. A GP is completely specified by its mean function and covariance function. The mean function m(x) and the covariance function k(x,x′) of a real process f(x) are defined as

$$\begin{array}{c}\hspace{1em}m(\mathit{x})=E[f(\mathit{x})]\hfill \\ k\left(\mathit{x},\mathit{x}\prime \right)=E\left[(f(\mathit{x})-m(\mathit{x}))\left(f\left(\mathit{x}\prime -m\left(\mathit{x}\prime \right)\right)\right)\right]\end{array}$$

(1)

and the GP can be written as

$$f(\mathit{x})\sim \mathrm{GP}\left(m(\mathit{x}),k\left(\mathit{x},\mathit{x}\prime \right)\right)$$

(2)

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

$$y=f(\mathit{x})+\epsilon$$

(3)

where 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

$$\mathit{y}~N\left(0,K\left(X,X\right)+{\sigma }_{n }^2{I}_n\right)$$

(4)

$$\left[\begin{array}{c}\mathit{y}\\ f_{\ast }\end{array}\right]~N\left(0,\left[\begin{array}{cc}K\left(X,X\right)+{\sigma }_{n }^2{I}_n& K\left(X,{\mathit{x}}_{\ast }\right)\\ K\left({\mathit{x}}_{\ast },X\right)& K\left({\mathit{x}}_{\ast },{\mathit{x}}_{\ast }\right)\end{array}\right]\right)$$

(5)

where $K\left(X,X\right)=K_n=\left(k_{ij}\right)$ is an n-dimensional symmetric and positive covariance matrix, whose matrix elements $k_{ij}=k\left({\mathit{x}}_i,{\mathit{x}}_j\right)$ are used to measure the correlation between 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

$$f_{\ast }|X,\mathit{y},{\mathit{x}}_{\ast }~N\left({\overline{f}}_{\ast },\mathrm{cov}\left({\overline{f}}_{\ast }\right)\right)$$

(6)

where

$${\overline{f}}_{\ast }=K\left({\mathit{x}}_{\ast },X\right){\left[K\left(X,X\right)+{\sigma }_{n }^2{I}_n\right]}^{-1}\mathit{y}$$

(7)

$$\mathrm{cov}\left({\overline{f}}_{\ast }\right)=K\left({\mathit{x}}_{\ast },{\mathit{x}}_{\ast }\right)-K\left({\mathit{x}}_{\ast },X\right){\left[K\left(X,X\right)+{\sigma }_{n }^2{I}_n\right]}^{-1}K\left(X,{\mathit{x}}_{\ast }\right)$$

(8)

So ${\widehat{\mu }}_{\ast }={\overline{f}}_{\ast },{\widehat{\sigma }}^2{}_{f_{\ast }}=\mathrm{cov}\left({\overline{f}}_{\ast }\right)$ are the mean and variance of the corresponding predicted value f_* 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:

$$k_{\mathrm{SE}}\left({\mathit{x}}_p,{\mathit{x}}_q\right)={\sigma }_f^2\exp \left(-\frac{{‖{\mathit{x}}_p-{\mathit{x}}_q‖}^2}{2{\mathit{l}}^2}\right)$$

(9)

where 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

$$L\left(\theta \right)=\frac{1}{2}{\mathit{y}}^{\mathrm{T}}{C}^{-1}\mathit{y}+\frac{1}{2}\log \left|C\right|+\frac{n}{2}\log 2\pi$$

(10)

$$\frac{\partial L\left(\theta \right)}{\partial {\theta }_i}=\frac{1}{2}\mathrm{tr}\left(\left(\mathit{\alpha }{\mathit{\alpha }}^{\mathrm{T}}-C^{-1}\right)\frac{\partial C}{\partial {\theta }_i}\right)$$

(11)

where $C=K_n+{\sigma }_{n }^2{I}_n$ and $\mathit{\alpha }=C^{-1}\mathit{y}$. The optimal hyperparameters can be obtained by minimizing Equation (10). In addition, the predicted value f_* 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

Once the digital twin model has been built, the performance criterion is to evaluate its similarity to the physical entity. Only digital twins with qualified accuracy can be applied to real engineering problems as a valid alternative to simulation calculations. As the test set of leave-one-out cross-validation (LOO CV) is sampled from the training set and does not need additional expensive simulations, LOO CV is widely used to validate the performance of the surrogate model [23]. For the training set with n samples, LOO CV uses the model built by n − 1 samples to predict the response of the remaining sample and repeats the above calculation process until the responses of n samples are calculated. In this paper, the standardized average LOO CV (CV_avg) [24] criterion described in Equation 12 is employed to evaluate the performance of digital twin:

$$C{V}_{\mathrm{avg} }=\frac{\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{\left({\widehat{y}}_{-i}^{(i)}-y^{(i)}\right)}^2}}}{ \mathrm{range} \mathrm{of} y^{(1)},\dots ,y^{(n)}}$$

(12)

where ${\widehat{y}}_{-i}^{(i)}$ and $y^{(i)}$ are the predicted response value and true response value of the i-th omitted sample, respectively. The tolerable level of inaccuracy is application-specific but typically takes CV_avg < 0.1 as the target for a useful surrogate model [24].

For further quantitative evaluation of DT performance, the correlation of determination (R²) shown in Equation 13 is chosen as the global performance metric in this paper:

$$R^2=1-\frac{{\displaystyle {\sum }_{i=1}^{n_t}{\left(y^{\left(i\right)}-{\widehat{y}}^{\left(i\right)}\right)}^2}}{{\displaystyle {\sum }_{i=1}^{n_t}{\left(y^{\left(i\right)}-\overline{y}\right)}^2}}$$

(13)

where $n_t$ is the number of test samples, $y^{\left(i\right)}$ is the true response at the i-th test sample, ${\widehat{y}}^{\left(i\right)}$ is the corresponding predicted value, and $\overline{y}$ is the mean of the response values. The closer R² 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 R² greater than 0.8 [25].

2.4. Finite Element Model

A commonly used single-layer unreinforced corrugated bellow with U-shape corrugations is the study object, whose geometric parameters and material properties are listed in

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

	Outer Diameter (mm)	Inner Diameter (mm)	Number of Layers
Geometric Parameters	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
Material properties	Material Type	Elastic modulus (MPa)	Poisson’s ratio
	304 Stainless Steel	1.95 × 105	0.3

. The digital twin is constructed with the high-fidelity FE model as the basis. The FEA is calculated in the commercial software ABAQUS. For the thin shell structure of the corrugated bellow, the S4 element type in ABAQUS [26] is used to establish the three-dimensional FE model, and the geometric nonlinearity is considered. One end of the corrugated bellow is fixed and displacement loads are applied to the other end. The movement capabilities shown in Figure 1 are achieved by three types of displacements demonstrated in Figure 2: v, w, and θ_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

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. Loeppky 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 5d, 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, lhsdesign, 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.

Carl Edward Rasmussen and Hannes Nickisch 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 Chen 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 $\theta =\left\{\mathit{l},{\sigma }_f,{\sigma }_n\right\}=\left\{\left[0.5,0.5,0.5\right],1.0,0.1\right\}$ in this paper. The hyperparameters are trained by the conjugate gradient descent method. An independent GPR model is built and hyperparameter optimization is conducted separately for each 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.

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 R² 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. Calculated results of the performance criterion for the simulations.

No.	1	2	3	4	5	6	7	8	9	10	Average
R2	0.9060	0.8000	0.8807	0.8735	0.9077	0.8206	0.8317	0.8065	0.8850	0.8287	0.8540 ± 0.0409
CVavg	0.0399	0.0481	0.0392	0.0415	0.0315	0.0469	0.0488	0.0505	0.0365	0.0395	0.0422 ± 0.0061

Note: The values after “±” represent the standard deviations.

. The average values of R² 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 R², 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 R² 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.

The optimized calculations of the fifth simulation with the maximum R² 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.

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 θ_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

The corrugated compensators are important components in the piping system, absorbing mechanical deformation flexibly. It is important to study the effective strategy of global mechanical response sensing of corrugated compensators and to accurately obtain complete state information of the equipment with limited sensing devices to reduce the risk of the piping system with corrugated compensators and improve the safety and stability of industrial equipment. However, the stress monitoring method of corrugated compensators with limited localized sensors lack real-time and full-domain sensing. Given this, this paper proposes a construction method of the DT for global mechanical response sensing of corrugated compensators, which combines Gaussian process regression (GPR) in machine learning and finite element analysis (FEA). The training samples and test samples are sampled by the Latin hypercube sampling (LHS) method, the GPR is performed by MATLAB, and the FEA is carried out by the commercial software ABAQUS and Python secondary development scripts. The sensing data of three types of displacements are used as the associated information of a finite element model with 19,800 elements and its digital twin.

The results show that the values of performance metrics correlation of determination R² 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.