Research Article  Open Access
ReliabilityBased Fatigue Life Prediction for Complex Structure with TimeVarying Surrogate Modeling
Abstract
To improve the computational efficiency and accuracy of reliabilitybased fatigue life prediction for complex structure, a timevarying particle swarm optimization (PSO) based general regression neural network (GRNN) surrogate model (called as TV/PSOGRNN) is developed. By integrating the proposed spacefilling Latin hypercube sampling technique and PSOGRNN regression function, the mathematical model of TV/PSOGRNN is studied. The reliabilitybased fatigue life prediction framework is illustrated in respect of the TV/PSOGRNN surrogate model. Moreover, the reliabilitybased fatigue life prediction of an aircraft turbine blisk under multiphysics interaction is performed to validate the TV/PSOGRNN model. We obtain the distributional characteristics, reliability degree, and sensitivity degree of fatigue failure cycle, which are useful for the turbine blisk design. By comparing the direct simulation (FE/FV model), RSM, GRNN, PSOGRNN, and TV/PSOGRNN, we observe that the TV/PSOGRNN surrogate model is promising to perform the reliabilitybased fatigue life prediction of the turbine blisk and enhance the computational efficiency while ensuring an acceptable computational accuracy. The efforts of this study offer a useful insight for the reliabilitybased design optimization of complex structure.
1. Introduction
Complex structure possesses complex geometric modelling and endures multiple loads during operation in many mechanical systems, such as aircraft engine and spacecraft [1–3]. Because of significant cyclic stresses induced by fluid loads, thermal loads, and centrifugal load, as one key failure mode, the fatigue failure seriously affects the security performance of the complex structure [4]. The everincreasing demands for highreliability performance and low maintenance cost drive the rising attention to the life prediction approaches. To date, many efforts have quantified the fatigue life with numerical and experimental investigations via deterministic analyses, which assures the prediction security for fatigue life by relatively conservative results [5–7]. However, these efforts possess great blindness in fatigue life prediction because the randomness of various impact factors is not considered. In fact, the fatigue life shows obvious stochastic behavior in nature by multiple uncertainties, such as material properties, load fluctuations, model variabilities, and other uncontrolled stochastic variations in engineering [8–10]. Therefore, the uncertainties should be addressed directly for fatigue life prediction. One viable alternative is reliabilitybased fatigue life prediction approach, which does consider the uncertainties through the probabilisticbased model to predict the probability distribution of fatigue life to overcome the shortages of deterministic analysis. In view of these virtues, reliabilitybased fatigue life prediction has been widely used to account for the uncertainties of materials and structures, including reliabilitybased crack growth assessment [11–13], probabilistic strainlife fatigue modelling [14–16], probabilistic analysis for creepfatigue behavior [17], physics of failurebased fatigue life prediction accounting for model uncertainty [18], low cycle fatigue life prediction under material variability [19, 20], and probabilistic life assessment using mature commercial software tools [21–24]. From the aforementioned studies, the uncertainty factors in fatigue life prediction have been adequately explored, and the feasibility and effectiveness of the reliabilitybased fatigue life prediction were validated as well. However, a few works consider multiple uncertainties in one unified reliabilitybased analysis regime, which neglects the combined effects among these uncertainties and there are therefore large calculating deviations on the results of fatigue life assessment [9, 25]. Therefore, it is increasingly desired to establish unified reliabilitybased prediction techniques to process the uncertain factors in one uniform fatigue life prediction framework.
Under such circumstances, some reliabilitybased prediction techniques have emerged for the multisource uncertainty issues [26–28]. As one valuable analytical approach, Monte Carlo (MC) simulation can have high computational accuracy in reliability evaluation fields by enough samples. Owing to the requirements of performing thousands of substantial iterative calculations in analysis process [29], however, the MC simulation might incur a high computing cost in solving the nonlinear state functions of complex structure with high nonlinearity, time variation, and strong coupling. Accordingly, the MC method is generally impractical to conduct a reliabilitybased fatigue life prediction for complex structure. To address this issue, surrogate model is developed to avoid tremendous calculation tasks with an acceptable computational efficiency [30, 31]. Recently, the surrogate models, such as response surface model (RSM) [32, 33], artificial neural network (ANN) [34, 35], support vector machine [36, 37], and Kriging model [38, 39], had been widely investigated in probabilistic analysis and optimization. Compared with other surrogate models, the ANN surrogate model can accomplish strong nonlinear regression in highdimension input variables analysis and optimization problems. In this case, we propose ANNbased surrogate model to complete the reliabilitybased fatigue life prediction of complex structure. As an important ANN surrogate model, general regression neural network (GRNN) holds strong nonlinear mapping ability and good approximation ability, and possesses great potential to improve the computational efficiency and accuracy of reliabilitybased prediction [40, 41]. For complex structure, the reliabilitybased fatigue life prediction considering multiuncertainty variables leads to high nonlinearity, large timevarying, and strong coupling in state functions. Therefore, the traditional GRNN model still hardly satisfies the requirements of the computational efficiency and the prediction of complex structure reliabilitybased fatigue life.
To enhance the computing efficiency and accuracy of the traditional GRNN surrogate model, we first develop the PSOGRNN surrogate model by integrating both the global searching ability of dynamic PSO algorithm and the local precise description ability of GRNN model. The dynamic PSO algorithm is a highspeed and highaccuracy optimization algorithm with great parallel computational ability [42, 43], which is conductive to optimize the undetermined smooth factor for the GRNN surrogate model. Therefore, the accurate smooth factor of the surrogate model can be found by adopting the PSO algorithm. However, it is difficult that the PSOGRNN is directly applied to the nonlinear reliabilitybased fatigue life prediction of complex structure with timevarying characteristics. To address this issue, the timevarying PSOGRNN (TV/PSOGRNN) surrogate model is further proposed. Although treating timevarying fatigue life response as output random variable and generating the highquality learning samples by the proposed spacefilled Latin hypercube sampling (SLHS) technique, the mathematical regression model of TV/PSOGRNN is established for the highfidelity reliabilitybased fatigue life prediction.
The objective of this study is to present an efficient timevarying surrogate model, called as TV/PSOGRNN, to improve the computational efficiency and accuracy of reliabilitybased fatigue life prediction for complex structure. As for this method, we adopt the dynamic PSO algorithm to search the smooth factor of GRNN, use the timevarying data set processing approach to treat timevarying fatigue life response as output random variable, and use the SLHS technique to generate highquality learning samples. The feasibility and effectiveness of the proposed TV/PSOGRNN surrogate model is verified by the reliabilitybased fatigue life prediction of turbine blisk in an aircraft engine with respect to multiphysics interaction.
The rest of this study is organized as follows. Section 2 discusses the basic timevarying surrogate modelling theory, including PSOGRNN and TV/PSOGRNN, for the reliabilitybased fatigue life prediction. In Section 3, the essential methodology of the reliabilitybased framework for fatigue life prediction with the proposed TV/PSOGRNN is investigated. In Section 4, the deterministic analyses of fatigue life prediction are discussed. The reliabilitybased fatigue life prediction of turbine blisk with respect to multiphysics interaction is performed to validate the proposed TV/PSOGRNN in Section 5. Some conclusions and outlooks on this study are summarized in Section 6.
2. TimeVarying Surrogate Modelling Theory
By both the global searching ability of dynamic PSO algorithm and the local precise description ability of GRNN model, the two surrogate models of both PSOGRNN and timevarying PSOGRNN (TV/PSOGRNN) are developed for reliabilitybased fatigue life prediction. The basic architecture, spacefilled sampling techniques, and mathematical model of PSOGRNN and TV/PSOGRNN surrogate model are discussed below.
2.1. PSOGRNN
As an important surrogate model, general regression neural network (GRNN) is developed based on the intelligent statistical learning theory and holds high computational efficiency, good regularization ability, and strong robustness ability [41]. The complex weights training process is avoided, and the approximation ability and nonlinear mapping ability of GRNN model are only determined by one smooth factor . To enhance the computational accuracy of GRNN model, the PSOGRNN surrogate model is proposed by combining the dynamic PSO algorithm of searching the optimal smooth factor and the GRNN model of constructing highfidelity surrogate model. The basic thought and mathematical model of the PSOGRNN are summarized as follows.
With a certain distribution type of joint probability density function and the corresponding learning sample set , to retrieve the precise output responses, the output space is mapped from the input vector space , and the nonlinear regression function is denoted bywhere is the data set of undetermined smooth factors .
In this study, the nonlinear regression function will be fitted by PSOGRNN. The topology structure of PSOGRNN surrogate model is shown in Figure 1. As shown in Figure 1, the PSOGRNN surrogate model refers to four different layers of input layer, pattern layer, summation layer, and output layer. Primarily, the input layer consists of the node sources of the PSOGRNN model, and the neuron number of input layer is equal to the dimension of input random variables, then the input variables are directly transferred to pattern layer.
In the pattern layer, the total neuron number in the pattern layer is decided by the learning sample number . In view of the rapid decay characteristics, the Gaussian basis function is selected as the transfer function of pattern neurons, which leads to different mapping processing units in PSOGRNN model. In this process, multiple computation tasks simultaneously are processed to improve the fitting efficiency effectively in nonlinear mapping problems of multiple objective design. The continuous transfer function of pattern layer is expressed aswhere denotes the exponential function operator and represents the square of Euclid distance between input vector and th learning sample point .
After pattern output calculation, the summation operator will be performed in the summation layer. The arithmetical summation and weighted summation are completed in neuron and neuron, respectively. The transfer functions of neuron and neuron are defined asin which indicates the connection weights of th pattern neuron and neuron.
Hence, the output response and the nonlinear regression function can be retrieved with and :
The feasibility and effectiveness of the aforementioned PSOGRNN mathematical model heavily depends on the smooth factor σ. Obviously, the problem of fitting highfidelity regression function is transformed into obtaining optimal solution of the following learning model:where indicates the th estimated output response value and the th real output response value.
To solve the optimization model and enhance the approximation accuracy of PSOGRNN, the dynamic PSO algorithm is proposed in this study. PSO algorithm is a notable searching algorithm based on the collaborative searching of particle swarm, which holds advantages in searching accuracy and searching efficiency [42]. To further improve the searching efficiency and accuracy of PSO algorithm, the dynamic PSO algorithm with dynamic inertia weight and dynamic learning factors are adopted. The objective of design is to complete the dynamic searching of particle swarm and weight the relationship between global searching ability and local searching ability to acquire the better optimal solution set.
The basic thought of dynamic PSO algorithm is that the particle position is composed of smooth factors, and the fitness value adopts the learning error of PSOGRNN model. Each particle is a potential solution for initial smooth factor of the PSOGRNN model. In searching process, all particles search for the optimal solution in the solution space by current optimal particles and updating particle individual positions, individual extreme values and population extremum values. The renewal formula of particle position and velocity in dynamic PSO algorithm are determined bywhere is the th particle; the current iteration number; the maximum iteration number; the current particle velocity; the current particle position; the current individual extremum; the current population extremum; , the random numbers during time domain [0,1]; the dynamic nonlinear inertia weight; the initial inertia weight; the inertia weight at the largest number of iteration; and the dynamic nonlinear individual learning factor and dynamic nonlinear population learning factor in the th iteration, respectively; and the initial individual learning factor and initial population learning factor, respectively; and and the individual learning factor and population learning factor in the th iteration, respectively.
After acquiring the optimal smooth factor in the learning process, the PSOGRNN model is built. On account of the arbitrary shape property of network structure and the selfadaptation characteristics of dynamic PSO algorithm, the PSOGRNN model can effectively reduce the approximation error and deal with the regular nonlinear optimal problem with a relatively high computing efficiency and accuracy.
2.2. TV/PSOGRNN
For timevarying (transient or dynamic) reliability analysis and probabilistic prediction problem, the output response of each calculation loop is a random process. It is difficult to conduct the reliability analysis problem of complex structure with PSOGRNN because of the random process of output response. Facing this situation, the conventional approaches are to establish a plenty of surrogate models in the time domain and then choose one seemingly realistic response at one time point as the computational point of reliability analysis. However, in all time loops of calculations for reliability analysis, the results calculated from one single selected computational point are not so feasible and reasonable, which leads to unacceptable computing efficiency and accuracy. To address this issue, based on the PSOGRNN model and SLHS technique, this study develops the timevarying PSOGRNN (TV/PSOGRNN) surrogate model to calculate an extreme response rather than a series of dynamic output responses with different input variables during the time domain [0, T]. This process is equivalent to transform the complicated timevarying output response process into a random output variable in each stochastic analysis. In this case, the selected random extreme response can guarantee the analytical accuracy. Obviously, the TV/PSOGRNN is a heuristic way to improve the computational efficiency and enhance computational accuracy for reliability analysis. The analytical thought of TV/PSOGRNN is shown in Figure 2.
In view of the basic thought of TV/PSOGRNN in Figure 2, the extremum output response of all dynamic response corresponding with the th input random vector is obtained through a number of stochastic analyses within the time domain [0, T]. The data set consisting of the maximum output responses is used to construct the TV/PSOGRNN nonlinear regression function , and the extremum response curve ) is expressed by
Clearly, the computational performance (efficiency and accuracy) of TV/PSOGRNN surrogate model heavily depends on the feasibility and validity of data set , which should be generated by highreliability sampling technique. On this condition, the SLHS technique is introduced to generate samples points without overlap. In line with this method, the feasible and valid data set could be obtained.
Firstly, to control the data sparsity and avoid the scaling issues, all input variables should be normalized into the unit cube space . The normalized input variables can be obtained byin which indicates the th normalized input variable; the minimum and maximum of all normalized input variable; and the minimum and maximum of all input variables.
Then, split the design space into equal sized hypercubes and placing points in it, ensuring that from each occupied hypercube we could exit the design space along any direction parallel with any of the axes without encountering any other occupied hypercubes. The sampling result of SLHS technique is illustrated for three dimensions in Figure 3.
(a)
(b)
(c)
(d)
In the light of the nonlinear regression function of PSOGRNN (Equation (4)) and the generated data set, the TV/PSOGRNN nonlinear regression function is constructed as
When the TV/PSOGRNN nonlinear regression function is applied to evaluate the timevarying reliability of complex structure replacing the FE model, this mathematical model is called as TV/PSOGRNN method, which is suitable to fulfill the reliabilitybased fatigue life prediction of complex structures.
3. ReliabilityBased Fatigue Life Prediction Framework
In view of the elasticplastic behavior in the fatigue life prediction of complex structure is inherently affected by the multiple stochastics of input parameters (material properties, load fluctuations, and model variabilities), deterministic analysis with the specified input variables is unsuitable to investigate the multiuncertainty features of fatigue life assessment. In this case, the reliabilitybased fatigue life prediction, that is, a probabilistic analysis approach, is emerged to address the multiuncertainty issues by considering the material properties, load fluctuations, and model variabilities as random variables. As for complex structure, the reliabilitybased fatigue life prediction is promising to quantify the structural reliability and determine the incidence of uncertainty parameters on the failure or reliability. In this section, the reliabilitybased fatigue life prediction framework with TV/PSOGRNN model is introduced.
Assume that the allowable output structure response is [], based on the TV/PSOGRNN model, the limit state function of structural fatigue life is calculated by
As shown in Equation (10), can ensure the safety of complex structure. With the random sampling on the established limit state function, the structural fatigue reliability degree is expressed aswhere is the indicator function of the whole reliability domain; expresses the number of sample points in the secure domain; and indicates the number of the total sample points.
Accordingly, the sensitivity analysis is performed to provide insights into the influence level of the variables mean values on the failure probability. For variables with higher sensitivity, changes in their values will lead to a greater change in the failure probability, and vice versa. In particular, we compute the expected failure probability and sensitivity degree by the variable distribution characteristics from failed samples. Thus, the failure probability and the sensitivity degree of variables mean values on the failure probability are obtained bywhere represents the failure probability; the sensitivity degree of the input variables effecting the output response; is the indicator function of the whole failure domain; indicates the number of sample points in the failure domain; and expresses the th input sample point. By comparing Equations (11) and (12), we can see that the failure probability complements the reliability degree , as their sum must be equal to one. This is due to the fact that a sample can only be deemed as a success or a failure, but not “none” or “both.”
To improve the computational efficiency and accuracy of the reliabilitybased fatigue life prediction of complex structure, the reliabilitybased fatigue life prediction framework is constructed based on the timevarying powerful mapping ability of TV/PSOGRNN surrogate model, which is illustrated in Figure 4.
4. Deterministic Analysis with Multiphysics Interaction
As a vital component of an aircraft engine, the highpressure turbine blisk shown in Figure 5 endures multidisciplinary loads (gas pressure loads, heat loads, and centrifugal force) from multiple physical fields (fluid field, thermal field, and structural field). The multidisciplinary loads are timevarying and strong coupling in operating process, so that it is easy to lead to tensile stresses and serious low cycle fatigue damage. The load spectrum of an aircraft engine is shown in Figure 6 [44]. Furthermore, the specific values of nonlinear material parameters, including elastic modulus , heat transfer coefficient , expansion coefficient , and Poisson’s ratio , are varying with temperature, and its detailed nonlinear variation characteristics are shown in Table 1.

To reasonably assess the reliability and performance of turbine components and whole aircraft engine, it is necessary to predict the fatigue life of turbine blisk. Therefore, the low cycle fatigue life prediction of an aeroengine turbine blisk is regarded as the objective of study. The FE model of the turbine blisk is drawn in Figure 7.
To simplify the simulation complexity and cut the computational task, we decompose the multidisciplinary coupling system into several simple singledisciplinary subsystems (fluid subsystem, thermal subsystem, and structure subsystem). Each subsystem is assumed to be independent mutually in operation. Besides, the loads and responses are transferred among subsystems by multiphysics interaction (MPI) surface. The MPI sketch of turbine blisk is drawn in Figure 8. In fluid subsystem, the related parameters are 168 m/s for inlet fluid velocity, 600,000 Pa for inlet pressure, 11756 W/m^{2}K for heat transfer coefficient in MPI surface, and 1 atm for outlet pressure. During the fluid dynamics analysis, the pressure distribution on MPI surface is obtained in Figure 9. After acquiring the initial temperature loads from fluid subsystem, we select the related thermal parameters of 21.2 W/(m°C) for thermal conductivity coefficient and 14.8 × 10^{−6}°C^{−1} for thermal expansion coefficient. According to the temperature consistency principle, the thermal analysis is completed to gain the body temperature distribution on the MPI surface in Figure 10. The fluid pressure distribution loads and temperature distribution loads are transmitted to the structure subsystem. And we select material density 8.24 g/cm^{3}, elastic modulus 160 GPa, and the flight profile parameters in Figure 4. Thereby, the deterministic analysis of turbine blisk is performed with MPI, in which the maximum stress distribution and strain range distribution are shown in Figure 11.
(a)
(b)
As illustrated in the analytical results, the maximum stress and strain range of turbine blisk reach at the peak values at the back of blade root. Considering the main stress cycle 00, the fatigue life can be obtained as 3228 flight cycles by the improved MassonCoffin model [29]:where represents the mean stress; the fatigue strength coefficient; the fatigue ductility coefficient; the fatigue strength exponent; the fatigue ductility exponent; and the failure cycle number.
5. Transient ReliabilityBased Fatigue Life Prediction
In this section, the transient reliabilitybased fatigue life prediction of turbine blisk is performed with the proposed TV/PSOGRNN model, by considering the nonlinearities of material properties and multiphysics loads in operation. All computations are performed on an Inter(R) Core(TM) Desktop Computer (3 GHz CPU and 16 GB RAM).
5.1. Random Variables Selection
Under aircraft engine operation, the material properties, multiphysical loads, and model uncertainty possess evidently inherent randomness and seriously influence the fatigue life of turbine blisk [45, 46]. Therefore, we regarded rotor speed , gas temperature , fluid velocity , elastic modulus , and thermal conductivity coefficient as random variables. The distribution characteristics of physical variables are listed in Table 2. The distribution characteristics of model uncertainty parameters such as the fatigue strength exponent , fatigue ductility exponent , fatigue strength coefficient , and fatigue ductility coefficient are also considered as random variables as listed in Table 3 [47]. Assuming that the fatigue ductility coefficient obeys lognormal distribution and other random variables obey normal distribution, and all of selected random variables are reciprocally independent, respectively.


5.2. TV/PSOGRNNM Surrogate Modelling
The uncertainty parameters in Tables 2 and 3 are regarded as the input variables, and the failure cycle number of turbine blisk is taken as the output response. In light of Latin hypercube sampling and FE simulations, 20 groups of training samples and 100 groups of test samples are extracted. The TV/PSOGRNN surrogate model is constructed by training the surrogate model. The test result of TV/PSOGRNN surrogate model is drawn in Figure 12. To quantify approximation error and evaluate the fitting performance, the mean relative error (MRE) metric and root mean squared error (RMSE) matrix in Equation (14) are adopted to reveal. The comparison results with response surface model (RSM) [33], GRNN, PSOGRNN, and TV/PSOGRNN surrogate model are shown in Table 4:where is the real output response; estimated output response; th test sample point; and number of test samples.

As shown in Figure 10 and Table 4, even if the fatigue life of turbine blisk possesses large dispersion, the TV/PSOGRNN surrogate model still fit each test points with almost zero approximation errors. Thus, the approximation performance of TV/PSOGRNN model is superior to RSM, GRNN, and PSOGRNN. Therefore, as surrogate model the TV/PSOGRNN model is suitable to fulfill the reliabilitybased fatigue life prediction.
5.3. ReliabilityBased Fatigue Life Prediction
In view of the distribution types listed in Tables 1 and 2 and MC simulation, 10,000 groups of input variables samples are obtained. The detailed probabilistic distribution characteristics are revealed in Figure 13. According to the extracted input variables and TV/PSOGRNN model, 10,000 simulations are executed to predict the failure cycle number of turbine blisk. The simulation history and probabilistic distribution of fatigue life are revealed in Figure 14, which indicates that the output response of turbine blisk obeys a lognormal distribution and the fatigue life of turbine blisk under reliability 99.87% is 3265 cycles.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(a)
(b)
Through the sensitivity analysis based on Equation (12), the sensitivities and effect probability of input variables on turbine blisk fatigue failure are revealed in Figure 15. The analysis results explain that rotor speed is the most important factors and plays a leading role for the fatigue life failure with the effect probability of 32%. Besides, gas temperature and fatigue strength coefficient are also main factors with the effect probabilities of 25% and 21%, respectively, while the influences of other factors are weak relatively. In the fatigue failure design of turbine blisk, the rotor speed , gas temperature , and fatigue strength coefficient should be controlled preferentially. The fluid velocity and thermal conductivity coefficient are negatively correlated with the fatigue failure probability. The other variables are positively correlated with the fatigue failure probability, which basically agree with engineering practice.
5.4. TV/PSOGRNN Method Validation
To support the proposed TV/PSOGRNN model, the reliabilitybased fatigue life prediction of the blisk is performed with MC method, RSM, GRNN, and PSOGRNN, respectively. The simulation consumptions with different methods are listed in Table 5. To validate the feasibility and effectiveness of the proposed TV/PSOGRNN model, the uniform computational efficiency (UCE) and uniform computational accuracy (UCA) are introduced in Equation (15). The comparison results of the four methods under different simulations are listed in Table 6:where and are fitting time and computing time for surrogate models, respectively; the time of MC simulations; and and the failure cycle number obtained by surrogate models and direct simulation (FE/FV), with MC method, respectively.


5.5. Discussion
As depicted in Figure 14, the fatigue life of turbine blisk obeys a lognormal distribution and the fatigue life of turbine blisk under the reliability 99.87% is 3265 cycles. As revealed in Figure 15, the rotor speed , gas temperature , and fatigue strength coefficient are the leading factors on the fatigue life of turbine blisk as the effect probabilities are 32%, 25%, and 21%, respectively. The influences from other factors are relatively weak. Therefore, and should be considered with the priority in the reliabilitybased fatigue life design of turbine blisk. The reduction of inlet gas velocity and fatigue ductility coefficient causes the increase of blisk failure probability, while the increase of other parameters leads to the increase of blisk failure probability.
As revealed in Table 4 and Table 5, the fitting time and fitting number of TV/PSOGRNN are less than RSM, GRNN, and PSOGRNN, and the simulation consumption and total computational efficiency of TV/PSOGRNN is superior than RSM, GRNN, PSOGRNN, and direct simulation model as well. Moreover, the computational efficiency benefits of the proposed TV/PSOGRNN are more obvious with increasing simulations. Hence, the proposed TV/PSOGRNN holds high computing efficiency due to low time consumption. The main reasons are (1) TV/PSOGRNN only focus on the extremum value of the response process for each calculation within a time domain in the reliabilitybased fatigue life prediction, rather than the whole dynamic response process; (2) the TV/PSOGRNN model can rapidly fit the nonlinear regression functions because the GRNN reduces the network complexity of surrogate model, and the dynamic PSO algorithm effectively avoid the blind searching and poor initial training values in learning process by searching optimal smooth factor with the proposed dynamic inertia weight and dynamic learning factors; (3) SLHS technique also brings more effective learning data sets for establishing satisfactory surrogate model to avoid numerous loop iterations in learning process. Therefore, based on the aforementioned strengths, the optimal smooth factor of TV/PSOGRNN are quickly obtained, which save time and improve computing efficiency in reliabilitybased fatigue life prediction. Therefore, the TV/PSOGRNN model possesses high computational efficiency in reliabilitybased fatigue life prediction of complex structure.
As unveiled in Figure 12 and Table 4, the established TV/PSOGRNN surrogate model can approximate every test sample so that the proposed TV/PSOGRNN possesses the lowest fitting error. As illustrated in Table 6, the TV/PSOGRNN is more precise than the RSM, GRNN, and PSOGRNN and is almost consistent with the direct simulation (FE/FV) method. The TV/PSOGRNN thus holds good generalization ability and highaccuracy computational ability. The reasons are listed as follows: (1) the TV/PSOGRNN regression function is fitted based on GRNN with strong nonlinear mapping ability, (2) the global optimal smooth factor is obtained rather than local optimal solution by the dynamic PSO algorithm balance ability of global searching and local searching to ensure the computing accuracy, and (3) SLHS technique generates highquality learning data set with less data noise, which also provides a good way to enhance the fitting ability of surrogate model. Therefore, the TV/PSOGRNN model holds high computational accuracy in reliabilitybased fatigue life prediction of complex structure.
In summary, the proposed TV/PSOGRNN greatly saves computing time and improves computational efficiency while keeping computational accuracy. Therefore, the TV/PSOGRNN is a feasible and effective way in the reliabilitybased fatigue life prediction of complex structures.
6. Conclusions and Outlooks
To improve the computational efficiency and accuracy of reliabilitybased fatigue life prediction for complex structure, an efficient timevarying surrogate model (termed “TV/PSOGRNN”) is developed. The reliabilitybased fatigue life prediction of turbine blisk was taken as a case to support the validation and feasibility of TV/PSOGRNN. Some conclusions are drawn as follows:(1)Through the reliabilitybased fatigue life prediction of turbine blisk with the TV/PSOGRNN, the simulation histories, and distribution features of reliabilitybased fatigue life are obtained. Moreover, the high sensitivity parameters (i.e., rotor speed, gas temperature, and fatigue strength coefficient) are also retrieved, which provides a valuable reference for the design and optimization of turbine blisk.(2)By comparing several methods (i.e., direct simulation, RSM, GRNN, and PSOGRNN), the developed TV/PSOGRNN possesses the highest computational efficiency and accuracy. We also demonstrated that the TV/PSOGRNN surrogate model is an effective approach for the reliabilitybased fatigue life prediction of complex structure.(3)The efforts of this study enrich mechanical reliability theory from a surrogate modelling perspective, and shed light on the further applications in the reliabilitybased design optimization as well.
Although this investigation provides a novel timevarying surrogate model to improve the modelling accuracy and simulation efficiency for the reliabilitybased fatigue life prediction of complex structure, there are still some limitations that need to be addressed in future. Most deviations from the expected solution are likely to be attributed to inaccurate information caused by the parameters of reliabilitybased fatigue life prediction framework. To further develop the fatigue life prediction of complex structure, more reasonable analysis and design techniques should be developed. Moreover, advanced timevarying regression functions based on different surrogate models should be established in future to accomplish the reliabilitybased design optimization for fatigue life prediction of complex structure.
Acronyms
ANN:  Artificial neural network 
GRNN:  General regression neural network 
MPI:  Multiphysics interaction 
MRE:  Mean relative error 
PSO:  Particle swarm optimization 
PSOGRNN:  PSObased GRNN 
RMSE:  Root mean squared error 
RSM:  Response surface model 
SLHS:  Spacefilled Latin hypercube sampling 
TV/PSOGRNN:  Timevarying PSOGRNN 
UCA:  Uniform computational accuracy 
UCE:  Uniform computational efficiency. 
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interests regarding the publication of this article.
Acknowledgments
This paper was cosupported by the National Natural Science Foundation of China (Grant nos. 51475026, 51335003, and 51605016). The authors would like to thank them.
References
 S. Beretta, S. Foletti, M. Madia, and E. Cavalleri, “Structural integrity assessment of turbine discs in presence of potential defects: probabilistic analysis and implementation,” Fatigue & Fracture of Engineering Materials & Structures, vol. 38, no. 9, pp. 1042–1055, 2015. View at: Publisher Site  Google Scholar
 L. K. Song, C. W. Fei, J. Wen, and G. C. Bai, “Multiobjective reliabilitybased design optimization approach of complex structure with multifailure modes,” Aerospace Science and Technology, vol. 64, pp. 52–62, 2017. View at: Publisher Site  Google Scholar
 C. Y. Zhang, C. Lu, C. W. Fei, L. J. Liu, Y. S. Choy, and X. G. Su, “Multiobject reliability analysis of turbine blisk with multidiscipline under multiphysical field interaction,” Advances in Materials Science and Engineering, vol. 2015, Article ID 649046, 10 pages, 2015. View at: Publisher Site  Google Scholar
 N. Kolarevic, M. Ognjanovic, and M. Milos, “Failures of multifunctional bulkhead caused by high gradient of temperature, pressure and speed of rotation,” Engineering Failure Analysis, vol. 89, pp. 100–117, 2018. View at: Publisher Site  Google Scholar
 R. P. Skelton, “The energy density exhaustion method for assessing the creepfatigue lives of specimens and components,” Materials at High Temperatures, vol. 30, no. 3, pp. 183–201, 2013. View at: Publisher Site  Google Scholar
 K. Prasad, R. Sarkar, P. Ghosal, V. Kumar, and M. Sundararaman, “High temperature low cycle fatigue deformation behaviour of forged IN 718 superalloy turbine disc,” Materials Science and Engineering AStructural Materials Properties Microstructure and Processing, vol. 568, pp. 239–245, 2013. View at: Publisher Site  Google Scholar
 J. Wertz, T. Letcher, M. M. H. Shen, O. ScottEmuakpor, T. George, and C. Cross, “An energybased axial isothermalmechanical fatigue lifing method,” Journal of Engineering for Gas Turbines and PowerTransactions of the ASME, vol. 134, no. 10, Article ID 102502, 2012. View at: Publisher Site  Google Scholar
 D. Y. Hu, J. X. Mao, and R. Q. Wang, “Optimization strategy for a shrouded turbine blade using variablecomplexity modeling methodology,” AIAA Journal, vol. 54, no. 9, pp. 2808–2818, 2016. View at: Publisher Site  Google Scholar
 S. P. Zhu, H. Z. Huang, V. Ontiveros, L. P. He, and M. Modarres, “Probabilistic low cycle fatigue life prediction using an energybased damage parameter and accounting for model uncertainty,” International Journal of Damage Mechanics, vol. 21, no. 8, pp. 1128–1153, 2012. View at: Publisher Site  Google Scholar
 J. Fan, H. M. Liao, H. Wang et al., “Local maximumentropy based surrogate model and its application to structural reliability analysis,” Structural and Multidisciplinary Optimization, vol. 57, no. 1, pp. 373–392, 2018. View at: Publisher Site  Google Scholar
 Z. G. Wei, F. L. Yang, B. Lin, L. M. Luo, D. Konson, and K. Nikbin, “Deterministic and probabilistic creepfatigueoxidation crack growth modeling,” Probabilistic Engineering Mechanics, vol. 33, pp. 126–134, 2013. View at: Publisher Site  Google Scholar
 X. Wang, M. Rabiei, J. Hurtado, M. Modarres, and P. Hoffman, “A probabilisticbased airframe integrity management model,” Reliability Engineering and System Safety, vol. 94, no. 5, pp. 932–941, 2009. View at: Publisher Site  Google Scholar
 S. Sankararaman, Y. Ling, and S. Mahadevan, “Uncertainty quantification and model validation of fatigue crack growth prediction,” Engineering Fracture Mechanics, vol. 78, no. 7, pp. 1487–1504, 2011. View at: Publisher Site  Google Scholar
 M. Guida and F. Penta, “A Bayesian analysis of fatigue data,” Structural Safety, vol. 32, no. 1, pp. 64–76, 2010. View at: Publisher Site  Google Scholar
 Y. Zhao, B. Yang, and Z. Zhai, “The framework for a strainbased fatigue reliability analysis,” International Journal of Fatigue, vol. 30, no. 3, pp. 493–501, 2008. View at: Publisher Site  Google Scholar
 Y. Zhao, “A fatigue reliability analysis method including super long life regime,” International Journal of Fatigue, vol. 35, no. 1, pp. 79–90, 2012. View at: Publisher Site  Google Scholar
 D. Y. Hu, Q. H. Ma, L. H. Shang, Y. Gao, and R. .Q. Wang, “Creepfatigue behavior of turbine disc of superalloy GH720Li at 650^{°}C and probabilistic creepfatigue modeling,” Materials Science and Engineering AStructural Materials Properties Microstructure and Processing, vol. 670, pp. 17–25, 2016. View at: Publisher Site  Google Scholar
 S. P. Zhu, H. Z. Huang, W. W. Peng, H. K. Wang, and S. Mahadevan, “Probabilistic physics of failurebased framework for fatigue life prediction of aircraft gas turbine discs under uncertainty,” Reliability Engineering and System Safety, vol. 146, pp. 1–12, 2016. View at: Publisher Site  Google Scholar
 S. P. Zhu, S. Foletti, and S. Berrtta, “Probabilistic framework for multiaxial LCF assessment under material variability,” International Journal of Fatigue, vol. 103, pp. 371–385, 2017. View at: Publisher Site  Google Scholar
 R. Q. Wang, X. Liu, D. Y. Hu, F. C. Meng, D. Li, and B. Li, “Zonebased reliability analysis on fatigue life of GH720Li turbine disk concerning uncertainty quantification,” Aerospace Science and Technology, vol. 70, pp. 300–309, 2017. View at: Publisher Site  Google Scholar
 G. R. Leverant, H. R. Millwater, R. C. McClung, and M. P. Enright, “A new tool for design and certification of aircraft turbine rotors,” Journal of Engineering for Gas Turbines and PowerTransactions of the ASME, vol. 126, no. 1, pp. 155–159, 2004. View at: Publisher Site  Google Scholar
 Y. T. Wu, M. P. Enright, and H. R. Millwater, “Probabilistic methods for design assessment of reliability with inspection,” AIAA Journal, vol. 40, no. 5, pp. 937–946, 2002. View at: Publisher Site  Google Scholar
 M. P. Enright, S. J. Hudak, R. C. McCung, and H. R. Millwater, “Application of probabilistic fracture mechanics to prognosis of aircraft engine components,” AIAA Journal, vol. 44, no. 2, pp. 311–316, 2006. View at: Publisher Site  Google Scholar
 W. A. Grell and P. J. Laz, “Probabilistic fatigue life prediction using AFGROW and accounting for material variability,” International Journal of Fatigue, vol. 32, no. 7, pp. 1042–1049, 2010. View at: Publisher Site  Google Scholar
 Z. C. Peng, H. Z. Huang, H. K. Wang, S. P. Zhu, and Z. Q. Lv, “A new approach to the investigation of load interaction effects and its application in residual fatigue life prediction,” International Journal of Damage Mechanics, vol. 25, no. 5, pp. 672–690, 2016. View at: Publisher Site  Google Scholar
 C. Y. Zhang, L. K. Song, C. W. Fei, C. Lu, and Y. M. Xie, “Advanced multiple response surface method of sensitivity analysis for turbine blisk reliability with multiphysics coupling,” Chinese Journal of Aeronautics, vol. 29, no. 4, pp. 962–971, 2016. View at: Publisher Site  Google Scholar
 S. Sankararaman, Y. Ling, C. Shantz, and S. Mahadevan, “Inference of equivalent initial flaw size under multiple sources of uncertainty,” International Journal of Fatigue, vol. 33, no. 2, pp. 75–89, 2011. View at: Publisher Site  Google Scholar
 C. W. Fei, G. C. Bai, W. Z. Tang, and Y. S. Choy, “Optimum control for nonlinear dynamic radial deformation of turbine casing with timevarying LSSVM,” Advances in Materials Science and Engineering, vol. 2015, Article ID 680406, 9 pages, 2015. View at: Publisher Site  Google Scholar
 J. X. Mao, D. Y. Hu, D. Li, R. Q. Wang, and J. Song, “Novel adaptive surrogate model based on LRPIM for probabilistic analysis of turbine disc,” Aerospace Science and Technology, vol. 70, pp. 76–87, 2017. View at: Publisher Site  Google Scholar
 J. Li and D. B. Xiu, “Evaluation of failure probability via surrogate models,” Journal of Computational Physics, vol. 299, no. 23, pp. 8966–8980, 2010. View at: Publisher Site  Google Scholar
 S. Adhikari, R. Chowdhury, and M. I. Friswell, “High dimensional model representation method for fuzzy structural dynamics,” Journal of Sound and Vibration, vol. 330, no. 7, pp. 1516–1529, 2011. View at: Publisher Site  Google Scholar
 F. Duprat and A. Sellier, “Probabilistic approach to corrosion risk due to carbonation via an adaptive response surface method,” Probabilistic Engineering Mechanics, vol. 21, no. 3, pp. 207–216, 2006. View at: Publisher Site  Google Scholar
 C. W. Fei, G. C. Bai, and W. Z. Tang, “Probabilistic design of HPT bladetip radial running clearance with distributed collaborative response surface method,” Journal of Aerospace Engineering, vol. 28, no. 2, Article ID 04014069, 2015. View at: Publisher Site  Google Scholar
 D. G. Giovanis, I. Papaioannou, and D. Straub, “Bayesian updating with subset simulation using artificial neural networks,” Computer Methods in Applied Mechanics and Engineering, vol. 319, pp. 124–145, 2017. View at: Publisher Site  Google Scholar
 L. K. Song, J. Wen, C. W. Fei, and G. C. Bai, “Distributed collaborative probabilistic design of multifailure structure with fluidstructure interaction using fuzzy neural network of regression,” Mechanical Systems and Signal Processing, vol. 104, pp. 72–86, 2018. View at: Publisher Site  Google Scholar
 C. W. Fei and G. C. Bai, “Distributed collaborative probabilistic design for turbine bladetip radial running clearance using support vector machine of regression,” Mechanical Systems and Signal Processing, vol. 49, no. 12, pp. 196–208, 2014. View at: Publisher Site  Google Scholar
 R. E. Stern, J. Song, and D. B. Work, “Accelerated Monte Carlo system reliability analysis through machine learningbased surrogate models of network connectivity,” Reliability Engineering and System Safety, vol. 164, pp. 1–9, 2017. View at: Publisher Site  Google Scholar
 I. Couckuyt, A. Forrester, D. Gorissen, F. D. Turck, and T. Dhaene, “Blind Kriging: implementation and performance analysis,” Advances in Engineering Software, vol. 49, pp. 1–13, 2012. View at: Publisher Site  Google Scholar
 T. W. Simpson and F. Mistree, “Kriging models for global approximation in simulationbased multidisciplinary design optimization,” AIAA Journal, vol. 39, no. 12, pp. 2233–2241, 2001. View at: Publisher Site  Google Scholar
 L. Y. Xiao, F. Qian, and W. Shao, “Multistep wind speed forecasting based on a hybrid forecasting architecture and an improved bat algorithm,” Energy Conversion and Management, vol. 143, pp. 410–430, 2017. View at: Publisher Site  Google Scholar
 M. Masikos, K. Demestichas, and E. Adamopoulou, “Mesoscopic forecasting of vehicular consumption using neural networks,” Soft Computing, vol. 19, no. 1, pp. 145–156, 2015. View at: Publisher Site  Google Scholar
 L. K. Song, C. W. Fei, G. C. Bai, and L. C. Yu, “Dynamic neural network methodbased improved PSO and BR algorithms for transient probabilistic analysis of flexible mechanism,” Advanced Engineering Informatics, vol. 33, pp. 144–153, 2017. View at: Publisher Site  Google Scholar
 A. Sedki and D. Ouazar, “Hybrid particle swarm optimization and differential evolution for optimal design of water distribution systems,” Advanced Engineering Informatics, vol. 26, no. 3, pp. 582–591, 2012. View at: Publisher Site  Google Scholar
 C. W. Fei, Y. S. Choy, D. Y. Hu, G. C. Bai, and W. Z. Tang, “Dynamic probabilistic design approach of highpressure turbine bladetip radial running clearance,” Nonlinear Dynamics, vol. 86, no. 1, pp. 205–223, 2016. View at: Publisher Site  Google Scholar
 S. P. Zhu, Q. Liu, W. W. Peng, and X. C. Zhang, “Computationalexperimental approaches for fatigue reliability assessment of turbine bladed disks,” International Journal of Mechanical Sciences, vol. 142143, pp. 502–517, 2018. View at: Publisher Site  Google Scholar
 S. P. Zhu, Q. Liu, J. Zhou, and Z. Y. Yu, “Fatigue reliability assessment of turbine discs under multisource uncertainties,” Fatigue and Fracture of Engineering Materials and Structures, vol. 41, no. 6, pp. 1291–1305, 2018. View at: Publisher Site  Google Scholar
 China Zhijian Publishing House and Standards Press of China, Academic Committee of the Superalloys, China Superalloys Handbook, China Zhijian Publishing House and Standards Press of China, Beijing, China, 2012, in Chinese.
Copyright
Copyright © 2018 LuKai Song et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.