The Reliability Analysis of a Turbine Rotor Structure Based on the Kriging Surrogate Model

Abstract

The turbine rotor is a core component in many energy conversion systems, where it is subjected to loads such as aerodynamic and centrifugal forces that make it highly susceptible to damage. Consequently, the reliability of the turbine rotor ranks among the key aspects of concern. This paper proposes an efficient approach based on the kriging model to conduct the reliability analysis of a turbine rotor. First, a parametric model of the turbine rotor was established. This parametric model was subsequently applied in a multifactor fluid–structure interaction model used to analyze the working performance of the turbine rotor. Finally, a kriging surrogate model was built and applied using these data in combination with various reliability analysis methods to analyze the structural reliability and reliability sensitivities of the turbine rotor. Furthermore, the reliability sensitivity results indicated that the outlet pressure had the greatest impact on rotor reliability. Thus, the proposed method was shown to have practical application value in the reliability analysis of the rotor structure.

1. Introduction

A typical natural gas liquefaction device uses a throttling valve to reduce the high pressure of the handled liquids. However, this process is irreversible and extremely violent, wasting the energy in the high-pressure liquid medium and potentially vaporizing the liquid, which can cause cavitation damage to the equipment. Alternatively, a hydraulic turbine can be applied as an energy recovery device to effectively convert the pressure energy in the liquid into mechanical energy. This converted energy can be used to drive other mechanical equipment, such as pumps, significantly improving energy efficiency while reducing energy consumption.

The hydraulic turbine is an excellent high-pressure residual energy recovery device, and its rotor is the core component for energy conversion. However, the turbine rotor is subjected to various loads during operation, such as fluid and centrifugal forces, that promote damage, violent vibrations, and even resonance, all of which can lead to failures, economic losses, and casualties. Therefore, reliability studies must be conducted on hydraulic turbines, and many researchers have proposed methods for improving the efficiency and reliability of hydraulic turbines accordingly.

Research on the reliability [1] of hydraulic turbines has addressed multiple processes, including rotor structure design, internal flow field analysis, energy loss optimization, and reliability-based design optimization. In terms of rotor structure design, Senchun et al. [2] optimized the design of turbine rotors for centrifugal pumps using genetic algorithms and neural networks to improve energy conversion efficiency. Other researchers [3,4] have reported that rotors with forward-curved blades facilitate higher flow rates and head and shaft powers at the maximum efficiency point than rotors with backward-curved blades. Jiang [5] further improved the energy recovery efficiency and operational stability of hydraulic turbines by optimizing the design method of the rotor splitter blade.

In terms of energy loss, Senchun et al. [6] discovered that the inlet region is the most significant area for energy loss and proposed corresponding optimization measures, including the addition of guide vanes to improve internal flow, reduce the number and size of vortices, and increase turbine efficiency.

In terms of reliability-focused design optimization, Yu et al. [7] analyzed centrifugal rotors using coupled information transfer technology to obtain an optimal design through a reliability-based multidisciplinary design optimization method, improving computational efficiency and shortening the design cycle. Qu [8] demonstrated a method for discretizing the random process of rotor vibration; this method converts time-varying system reliability problems into traditional constant system reliability problems, improving computational efficiency with significant implications in terms of enhancing rotor lifecycle safety. Lei et al. [9] introduced a lifecycle reliability-based multidisciplinary design optimization method for centrifugal rotors, then applied it to optimize a rotor design using surrogate models considering various uncertainty factors. Huang et al. [10] investigated the fatigue reliability of aero engine turbine disks using an improved adaptive kriging–Monte Carlo simulation method to calculate the fuzzy failure probability and sensitivity; the resulting failure probability was greater than the conventionally determined failure probability. Zhang et al. [11] proposed a shared training sample point strategy and support vector decomposition machine learning function using a kriging model to improve the optimization of the fatigue reliability of turbine disks. Zheng et al. [12] conducted a reliability assessment of centrifugal compressor rotors using coupled fluid–structure numerical simulations to predict the lifespan of centrifugal compressor rotors under the different operating conditions present at pipeline compressor stations, providing a basis for maintenance management. Wang et al. [13] used Weibull distribution fitting and Monte Carlo sampling to obtain the lifecycle distributions and reliabilities of turbine rotors, verifying the accuracy of the Monte Carlo method and reporting that the rotor failure threshold decreased faster with increasing load amplitude. Dileep et al. [14] used different models and algorithms to estimate centrifugal rotor fatigue damage and lifespan. Furthermore, Tang et al. [15] proposed an aerodynamic robustness optimization method for centrifugal compressor rotors with splitter blades that was based on uncertainty quantification. Wei and Ruofu [16] proposed an optimization strategy employing three-dimensional inverse design, computational fluid dynamics, response surface methodology, and multi-objective genetic algorithms that used the blade load distribution to parameterize the blade geometry; this approach reduced the number of design parameters as well as the optimization time while effectively improving the efficiency and operational stability of the designed pump or turbine.

Approaches for conducting reliability analyses can be classified into the approximate analysis, numerical simulation, and surrogate model methods. Approximate analysis methods such as the first-order second-moment method and the advanced first-order second-moment method (AFOSM) [17] are conceptually simple; however, the key issue is solving the design point [18], and their computational efficiency is relatively low. Numerical simulation methods include Monte Carlo simulation (MCS) [19,20], importance sampling (IS) [21], subset simulation (SS) [22], and Line sampling (LS) [23]. Although MCS has been widely applied, it is computationally intensive; improved methods, including IS, have increased computational efficiency for small failure probability problems. Finally, surrogate model methods, such as response surface methodology [24], neural networks [25], kriging [26], and support vector machines [27], approximate implicit limit-state functions using explicit functions and integrate intelligent sampling strategies to improve computational efficiency and accuracy. As a result, they are widely used in engineering design, optimization, and reliability analyses.

This study addresses the challenges associated with conducting reliability analyses of rotor structures by constructing a multi-factor fluid–structure interaction model. Due to the enormous computational cost involved in the finite element analysis of this complex rotor structure, a kriging surrogate model is adopted to construct the relationship between the output and the basic input of this structure. The reliability analysis is then carried out using the constructed kriging surrogate model and various reliability analysis methods. In particular, the kriging surrogate model could significantly reduce the computational cost in the structural reliability and reliability sensitivity of the rotor by fitting the limit state function. The result shows that the adopted kriging model provides significant improvements in the efficiency and accuracy of rotor reliability assessment.

2. Hydraulic Turbine Rotor Structure

A turbine rotor typically comprises components such as a hub, blades, and flow channels [28]. The geometric and material designs of these components are quite complex and must satisfy the requirements for efficient energy transfer and long-term operational reliability.

The parametric modeling of a hydraulic turbine rotor structure considers parameters such as the flow rate, head power, and rotational speed to determine the meridional flow channel of the rotor [29]. The initially defined meridional flow channel provides a basis for the adjustment of blade structure parameters and installation angles to derive a parametric blade structure model. The turbine rotor structure is shown in Figure 1.

Various turbine rotor parameters, including the size, boundary conditions, material properties, and operating conditions, affect the aerodynamic forces and structural reliability of the rotor and must be considered during the reliability analysis of its structure accordingly [30]. Notably, a coupled relationship exists between the aerodynamic forces and rotor structure. For example, the rotor speed, inlet and outlet pressures, and size parameters affect the pressure distribution on the rotor surface via the airflow field formed around the rotor and by its deformation under the action of aerodynamic and centrifugal forces. This mutual interaction makes rotor structure analysis a multidisciplinary coupling problem which actually involves the iterative solution approach [31]. However, the process of finding the iterative solution is quite demanding, especially for reliability analysis of practical structure. Thus this study conducted the analysis of the decoupled fluid–structure relationship; that is, the fluid and solid problems are solved independently. This approach is simpler but neglects the mutual influence between the two domains.

The process for analyzing the rotor without considering the coupling of the fluid–structure interaction is simple. First, the pressure distribution on the blade surface was solved within the fluid domain using computational fluid dynamics. Next, the corresponding pressure loads were applied to the blade structure model (see the finite element model for turbine blades in Figure 2) to determine the resulting stresses and strains of the blade structure. Finally, the von Mises stress results of the rotor blade structure calculated by using ANSYS ANSYS2024 R2 are shown in Figure 3.

The boundary conditions for the rotor analysis were established as follows: a fixed support constraint was applied at the hub base to simulate the connection with the shaft, while a rotational velocity of 3000 rpm was imposed on the entire structural domain, accounting for the Earth’s gravitational field (9.81 m/s²). Additionally, pressure loads obtained from a CFD analysis of the fluid domain were mapped onto the blade surfaces to capture fluid–structure interaction effects.

This combined boundary condition setup enabled the structural analysis to comprehensively account for centrifugal forces, gravitational loading, and fluid pressure effects on the impeller blades, thereby determining the actual stress and strain states of the blade structure. The current analysis focused on structural responses under mechanical loads (centrifugal forces, fluid pressures, etc.), with thermal stresses excluded due to their negligible impact under the investigated operating conditions. Figure 4 illustrates the physical setup, 3D model, and boundary conditions, while Figure 5 presents the resulting stress distribution.

3. A Method for Reliability Analysis Using a Kriging Surrogate Model

Coupled fluid–structure interaction modeling for the reliability analysis of complex turbine rotor geometry typically incurs high computational costs and can be quite time-consuming. Therefore, the efficiency of the reliability analysis was improved in this study using the kriging surrogate model, a statistical learning method that can predict the response of unknown points based on existing sample points while estimating prediction uncertainty. The kriging model can be expressed as

$$y\left(x\right)=f\left(x\right){\beta }^{{\rm T}}+z\left(x\right)$$

(1)

where $x$ is the input parameter vector; $y(x)$ is the fitted response of the maximum equivalent blade stress and maximum equivalent blade elastic strain input parameters; ${\beta }^{\mathrm{T}}$ is the transposed coefficient vector for the regression basis functions, which can be represented as $\beta =[{\beta }_1,{\beta }_2,\dots ,{\beta }_p]$ (in which $p$ is the number of basis functions); $f(x)$ is the basis function vector of the regression function, which can be represented as $f(x)=[f_1(x),f_2(x),\dots ,f_p(x)]$; and $z(x)$ is a random term generated from a normal distribution with the following properties:

$$\begin{array}{c}E(z(x))=0\\ Var(z(x))={\sigma }^2\\ Cov(z(x^{(i)})z(x^{(j)}))={\sigma }^{2}R(\theta ,x^{(i)},x^{(j)})\end{array}$$

(2)

where ${\sigma }^2$ is the process variance of the sample, $\theta$ is the vector of the correlation parameters in the correlation function, and $R(\theta ,x^{(i)},x^{(j)})$ is the spatial correlation function between points $x^{(i)}$ and $x^{(j)}$.

Assuming that the target function is $g(x)$, where $x=[x_1,x_2,\dots ,x_n]$ (in which n is the number of variables), and all variables follow independent normal distributions, a reasonable experimental design can be applied to obtain the initial training sample set. The variable set of samples was defined as $S={\{s^{(1)},s^{(2)},\dots ,s^{(m)}\}}^{{\rm T}}$, where m is the number of training sample points. By substituting the design variables into the target function, the corresponding response values were obtained as $G={\{g^{(1)},g^{(2)},\dots ,g^{(m)}\}}^{{\rm T}}$, thereby forming an initial training sample set containing m samples.

The predicted mean value (i.e., the predicted value) of the response provided by the kriging model is given by

$$\begin{array}{c}{\mu }_{\widehat{g}(x)}=\widehat{g}(\mathit{x})=r{(\mathit{x})}^T{R}^{-1}G-({[{\mathit{F}}^T{R}^{-1}r(\mathit{x})]}^T-f(\mathit{x})){\mathit{\beta }}^{\ast T}\\ =f(\mathit{x}){\mathit{\beta }}^{\ast T}+r{(\mathit{x})}^T{R}^{-1}(G-\mathit{F}{\mathit{\beta }}^{\ast T})\\ =f(\mathit{x}){\mathit{\beta }}^{\ast T}+r{(\mathit{x})}^T{\mathit{r}}^{\ast }\\ r(\mathit{x})={[R(\mathit{\theta },{\mathit{s}}^{(1)},\mathit{x}),R(\mathit{\theta },{\mathit{s}}^{(2)},\mathit{x}),\dots ,R(\mathit{\theta },{\mathit{s}}^{(m)},\mathit{x})]}^{\mathrm{T}}\\ \mathit{F}={[f({\mathit{x}}^{(1)}),f({\mathit{x}}^{(2)}),\dots ,f({\mathit{x}}^{(m)})]}^{\mathrm{T}}\end{array}$$

(3)

where ${\beta }^{\ast }$ is the coefficient of the regression function, which can be obtained using the least squares method, i.e., ${\beta }^{*}=(F^{{\rm T}}{R}^{-1}F{)}^{-1}{F}^{{\rm T}}{R}^{-1}G$, and satisfies $R{r}^{\ast }=G-F{\beta }^{\ast }$. Therefore, $r^{\ast }$ can be determined once the training set is determined.

The prediction variance provided by the kriging model is given by

$${\sigma }_{\widehat{g}(x)}^2={\sigma }^2(1+a^{⊺}(\mathit{x}){({\mathit{F}}^{⊺}{R}^{-1}\mathit{F})}^{-1}a(\mathit{x})-r^{⊺}(\mathit{x})R^{-1}r(\mathit{x}))$$

(4)

where ${\sigma }^2={(G-F{\beta }^{*})}^{\mathrm{T}}{R}^{-1}(G-F{\beta }^{*})/m$ is the matrix of basis functions and $a(x)=F^{{\rm T}}{R}^{-1}r(x)-f(x)$ is the correlation matrix.

The general principle underlying the application of the kriging model for reliability analysis is presented in Figure 6.

4. Reliability Analysis of a Turbine Rotor Structure Using the Kriging Model

4.1. Determination of Parameters for Rotor Structural Reliability Analysis

Seven parameters are considered as variables in the reliability analysis of the hydraulic turbine rotor in this study: rotational speed (RS) (r/min), inlet temperature (IT) (°C), outlet pressure (OP) (MPa), inlet pressure (IP) (MPa), elastic modulus (EM) (GPa), blade length (BL) (mm), and blade width (BW) (mm). All of these variables followed normal distributions with the characteristics listed in

Table 1. The distribution characteristics of the basic rotor structure variables.

Variable	Mean Value	Standard Deviation	Distribution
RS (rev/min)	3000	200	Normal
IT (°C)	−160.985	10.7317	Normal
OP (MPa)	0.18	0.01333	Normal
IP (MPa)	4.52	0.3	Normal
EM (GPa)	200	13.3333	Normal
BL (mm)	112	3.6667	Normal
BW (mm)	30	1	Normal

.

This study selects blade length and width as the key structural parameters, which can effectively characterize the primary structural characteristics of the blade. Although other parameters such as hub-to-tip ratio, solidity, and load distribution significantly influence blade performance, the current parameterization strategy aims to balance model accuracy with computational efficiency.

This study employs the 7075 aluminum alloy as the rotor blade material, whose high yield strength (approximately 503 MPa) ensures that the operational stresses remain consistently within the elastic deformation range. Based on the linear elastic assumption, the elastic modulus is established as the key design parameter, while the influences of yield strength and Poisson’s ratio can be neglected. Although composite and anisotropic materials require consideration of more complex constitutive relationships, the homogeneous metallic material system investigated in this study can be accurately characterized using a linear elastic constitutive model.

4.2. Establishment of Kriging Model for Rotor Structural Reliability

Considering the numerical characteristics of the input variables, 20 sets of experimental data were obtained using Latin hypercube sampling [32], as listed in

Table 2. Experimental data obtained by Latin hypercube sampling.

Data Set	RS (rev/min)	OT (°C)	OP (MPa)	IT (°C)	IP (MPa)	EM (GPa)	BL (mm)	BW (mm)
DP 1	3249	153.87	0.146	159.87	5.245	184	103.4	32.1
DP 2	2646	176.42	0.172	182.42	3.916	214	119.1	30.4
DP 3	3527	161.94	0.213	167.94	4.009	193	101.7	27.5
DP 4	2575	130.62	0.164	136.62	5.064	218	120.6	27.1
DP 5	2792	124.42	0.203	130.42	4.09	190	121.3	31.3
DP 6	2950	136.88	0.182	142.88	3.745	201	111.7	29.2
DP 7	2912	159.75	0.193	165.75	4.615	196	105.3	29.8
DP 8	3155	143.21	0.198	149.21	5.055	174	106.4	29.4
DP 9	3030	183.24	0.143	189.24	4.226	164	114.1	27.7
DP 10	2872	146.56	0.154	152.56	4.881	161	118.1	30
DP 11	2632	141.18	0.186	147.18	4.811	230	103.1	32.9
DP 12	3188	150.09	0.168	156.09	4.387	176	115.9	28.8
DP 13	2724	173.43	0.216	179.43	4.549	221	122	32.7
DP 14	3104	167.66	0.189	173.66	5.382	208	109.4	31
DP 15	3428	185.63	0.177	191.63	3.676	225	113	30.8
DP 16	2517	128.92	0.157	134.92	3.871	234	110.3	28
DP 17	3573	155.69	0.148	161.69	4.306	170	106.6	28.3
DP 18	2417	170.67	0.212	176.67	4.787	180	117.1	32.2
DP 19	3340	132.71	0.167	138.71	5.197	240	115.3	28.9
DP 20	3396	179.02	0.208	185.02	4.49	207	108.5	31.6

. The corresponding blade stress, deformation, and other simulation results for each set of experimental data were determined using the fluid–structure interaction simulation with the results listed in

Table 3. Fluid–structure interaction analysis results for the Latin hypercube sampling data.

Data Set	Maximum Equivalent Stress (MPa)	Maximum Total Deformation (mm)	Maximum Equivalent Elastic Strain (mm mm−1)
DP 1	125.3218	0.018499	0.000683
DP 2	113.2349	0.014279	0.000531
DP 3	99.68007	0.014068	0.000518
DP 4	138.9216	0.017197	0.000639
DP 5	113.1518	0.016074	0.000597
DP 6	104.7671	0.014053	0.000523
DP 7	120.5042	0.01662	0.000617
DP 8	123.2774	0.019219	0.000711
DP 9	112.2253	0.018512	0.000687
DP 10	127.8148	0.021477	0.000796
DP 11	131.7743	0.015471	0.000575
DP 12	111.2072	0.017123	0.000634
DP 13	123.1395	0.015038	0.000559
DP 14	130.3776	0.016997	0.000629
DP 15	96.62382	0.011668	0.000431
DP 16	115.7093	0.013341	0.000496
DP 17	104.6039	0.016792	0.000617
DP 18	136.1921	0.020404	0.000759
DP 19	122.1141	0.013835	0.00051
DP 20	108.6447	0.01427	0.000527

.

Finally, the experimental data shown in Table 2 and Table 3 were applied to fit the relationship between the maximum equivalent blade stress (MPa) and the input parameter $x$ using the kriging model. The limit state function established by the kriging surrogate model is given by

$$\widehat{g}(x)=5800-\hat{y}(x)$$

(5)

where $\hat{y}$ denotes the maximum blade stress from the surrogate model and the stress threshold is 5800 MPa.

4.3. Reliability and Sensitivity Analyses of a Turbine Rotor

An advanced first-order second-moment method [17] was used to approximate the failure probability of the original nonlinear functional function by linearly expanding the nonlinear functional function and then using the failure probability of the linear functional function to approximate the failure probability of the original nonlinear functional function.

Monte Carlo simulation [19,20] was used for structural reliability analysis by means of stochastic simulations or statistical tests. Compared with Monte Carlo simulation, importance sampling [21], an improved Monte Carlo method, has no special requirements for the form and number of function functions, the dimensions of the variables and their distribution forms, and improves the sampling efficiency by shifting the sampling center to the design point, so that more samples fall into the failure domain, but the determination of the important sampling density function relies on the selection of the design point.

Subset simulation [22] expresses the small failure probability as the product of a series of larger conditional failure probabilities by introducing reasonable intermediate failure events, and the larger conditional failure probabilities can be efficiently estimated by using the conditional sample points of Markov chain simulation, which greatly improves the computational efficiency of the reliability analysis.

Line sampling [23] is a variation reduction technique commonly used in structural reliability analysis for the calculation of failure probability in high-dimensional complex systems. It effectively estimates the system failure probability by transforming a multidimensional problem into a one-dimensional one.

Five reliability analysis methods above were used to obtain the failure probability of the rotor structure: AFOSM, MCS, IS, subset simulation (SS), and line sampling (LS). The results are compared in

Table 4. Failure probability results obtained using different reliability analysis methods combined with the kriging model.

Method	Failure Probability	Number of Samples
AFOSM	2.2807 × 10−5	——
IS	3.5000 × 10−5	104
SS	3.3420 × 10−5	5 × 103
LS	1.6430 × 10−5	103
MCS	3.3500 × 10−5	107

.

The MCS method is quite accurate, exhibits strong universality, and its computational accuracy increases with the number of simulations; however, it can be time-consuming. Therefore, the failure probability obtained in this study using the MCS method with 10⁷ samples (3.35 × 10⁻⁵) was considered the precise result. As shown in Table 4, the failure probabilities calculated using the AFOSM, IS, SS, and LS methods were all close to the precise value, with differences below one order of magnitude. These results indicate that the kriging surrogate model provides a suitable fit to the data and confirms that the combination of the kriging surrogate model with any of these reliability analysis methods provides accurate results that can be applied to engineering analysis.

Next, the degree of influence of each parameter on blade reliability was clarified by analyzing the sensitivity of the blade reliability to the mean and standard deviation of each variable. These sensitivity analyses were conducted using the kriging model with the AFOSM and MCS methods (considering the latter to provide the precise solution), with the results for mean and standard deviation compared in

Table 5. Sensitivity of reliability to means of variables (Unit: 10⁻⁶).

Method	RS	IT	OP	IP	EM	BL	BW
AFOSM	−0.1352	−4.173	343.5	46.28	4.204	−0.1345	59.34
MCS	0.1673	−7.429	700.4	31.07	5.989	−0.6738	49.85

and

Table 6. Sensitivity of reliability to standard deviations of variables (Unit:10⁻⁶).

Method	RS	IT	OP	IP	EM	BL	BW
AFOSM	0.1519	7.771	65.38	26.72	9.799	0.002757	146.4
MCS	0.2357	16.09	−516.1	31.41	13.03	−1.567	72.07

, respectively.

Table 5 shows that both methods determined that the reliability exhibited negative sensitivities to the mean inlet temperature and blade length, indicating that an increase in either of these values decreases reliability. In contrast, the reliability sensitivities to the mean outlet pressure, inlet pressure, elastic modulus, and blade width were positive, indicating that an increase in any of these values increases reliability. Furthermore, Table 6 shows that the reliability sensitivities to the standard deviations of the rotational speed, inlet temperature, inlet pressure, elastic modulus, and blade width were all positive, indicating that an increase in any of these standard deviations increases reliability. Finally, the reliability sensitivities to the mean and standard deviation of the outlet pressure exhibited the largest absolute values, indicating that this parameter has the greatest impact on rotor reliability.

Notably, the reliability sensitivities for the mean rotational speed and standard deviations of the outlet pressure and blade length exhibited different signs according to which method was used to obtain them. This discrepancy may have been a result of the insufficient computational accuracy of the AFOSM when dealing with complex limit state equations.

5. Conclusions

This study addressed the issue of hydraulic turbine rotor failure owing to loading during operation by conducting reliability and sensitivity analyses of a turbine rotor based on the kriging surrogate model. The specific achievements and conclusions of this study are as follows.

(1) A multifactor fluid–structure interaction model was constructed to determine the maximum rotor stress and deformation corresponding to changes in seven input rotor parameters—rotor speed, elastic modulus, length, width, inlet and outlet pressures, and temperature. A kriging surrogate model was established to significantly reduce the computational workload required to conduct sufficient numerical solutions of rotor reliability using these parameters.

(2) The reliability of the turbine rotor was analyzed using the constructed kriging surrogate model together with the AFOSM, MCS, IS, SS, or LS reliability analysis methods. The failure probability results obtained by different methods are consistent, which are all close to 3.3500 × 10⁻⁵ computed by MCS.

(3) Considering the reliability analysis results, the sensitivities of the reliability to the variable means and standard deviations were determined using the AFOSM and MCS methods. The results indicated that the outlet pressure had the greatest impact on rotor reliability.

It is also worthy of note that the construction of the surrogate model can be further optimized by incorporating active learning techniques. Specifically, leveraging a learning function for model training represents a promising avenue for future research.