Reliability-Based Fatigue Life Prediction for Complex Structure with Time-Varying Surrogate Modeling

To improve the computational efficiency and accuracy of reliability-based fatigue life prediction for complex structure, a timevarying particle swarm optimization(PSO-) based general regression neural network (GRNN) surrogate model (called as TV/PSO-GRNN) is developed. By integrating the proposed space-filling Latin hypercube sampling technique and PSO-GRNN regression function, the mathematical model of TV/PSO-GRNN is studied.1e reliability-based fatigue life prediction framework is illustrated in respect of the TV/PSO-GRNN surrogate model. Moreover, the reliability-based fatigue life prediction of an aircraft turbine blisk under multiphysics interaction is performed to validate the TV/PSO-GRNN 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, PSO-GRNN, and TV/PSO-GRNN, we observe that the TV/PSOGRNN surrogate model is promising to perform the reliability-based fatigue life prediction of the turbine blisk and enhance the computational efficiency while ensuring an acceptable computational accuracy. 1e efforts of this study offer a useful insight for the reliability-based design optimization of complex structure.


Introduction
Complex structure possesses complex geometric modelling and endures multiple loads during operation in many mechanical systems, such as aircraft engine and spacecraft [1][2][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].e ever-increasing demands for high-reliability 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][6][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][9][10].
erefore, the uncertainties should be addressed directly for fatigue life prediction.One viable alternative is reliability-based fatigue life prediction approach, which does consider the uncertainties through the probabilistic-based model to predict the probability distribution of fatigue life to overcome the shortages of deterministic analysis.In view of these virtues, reliability-based fatigue life prediction has been widely used to account for the uncertainties of materials and structures, including reliability-based crack growth assessment [11][12][13], probabilistic strain-life fatigue modelling [14][15][16], probabilistic analysis for creep-fatigue behavior [17], physics of failure-based 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][22][23][24].From the aforementioned studies, the uncertainty factors in fatigue life prediction have been adequately explored, and the feasibility and effectiveness of the reliability-based fatigue life prediction were validated as well.However, a few works consider multiple uncertainties in one unified reliability-based 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].erefore, it is increasingly desired to establish unified reliability-based prediction techniques to process the uncertain factors in one uniform fatigue life prediction framework.
Under such circumstances, some reliability-based prediction techniques have emerged for the multisource uncertainty issues [26][27][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 reliability-based 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 high-dimension input variables analysis and optimization problems.In this case, we propose ANN-based surrogate model to complete the reliability-based 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 reliability-based prediction [40,41].For complex structure, the reliability-based fatigue life prediction considering multiuncertainty variables leads to high nonlinearity, large time-varying, and strong coupling in state functions.erefore, 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 PSO-GRNN surrogate model by integrating both the global searching ability of dynamic PSO algorithm and the local precise description ability of GRNN model.e dynamic PSO algorithm is a high-speed and high-accuracy optimization algorithm with great parallel computational ability [42,43], which is conductive to optimize the undetermined smooth factor for the GRNN surrogate model.erefore, the accurate smooth factor of the surrogate model can be found by adopting the PSO algorithm.However, it is difficult that the PSO-GRNN is directly applied to the nonlinear reliability-based fatigue life prediction of complex structure with time-varying characteristics.To address this issue, the time-varying PSO-GRNN (TV/PSO-GRNN) surrogate model is further proposed.Although treating time-varying fatigue life response as output random variable and generating the high-quality learning samples by the proposed space-filled Latin hypercube sampling (SLHS) technique, the mathematical regression model of TV/PSO-GRNN is established for the high-fidelity reliability-based fatigue life prediction.
e objective of this study is to present an efficient time-varying surrogate model, called as TV/PSO-GRNN, to improve the computational efficiency and accuracy of reliability-based 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 time-varying data set processing approach to treat timevarying fatigue life response as output random variable, and use the SLHS technique to generate high-quality learning samples.
e feasibility and effectiveness of the proposed TV/PSO-GRNN surrogate model is verified by the reliability-based fatigue life prediction of turbine blisk in an aircraft engine with respect to multiphysics interaction.
e rest of this study is organized as follows.Section 2 discusses the basic time-varying surrogate modelling theory, including PSO-GRNN and TV/PSO-GRNN, for the reliability-based fatigue life prediction.In Section 3, the essential methodology of the reliability-based framework for fatigue life prediction with the proposed TV/PSO-GRNN is investigated.In Section 4, the deterministic analyses of fatigue life prediction are discussed.e reliability-based fatigue life prediction of turbine blisk with respect to multiphysics interaction is performed to validate the proposed TV/PSO-GRNN in Section 5. Some conclusions and outlooks on this study are summarized in Section 6.

Time-Varying 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 PSO-GRNN and time-varying PSO-GRNN (TV/PSO-GRNN) are developed for reliability-based fatigue life prediction.e basic architecture, space-filled sampling techniques, and mathematical model of PSO-GRNN and TV/PSO-GRNN surrogate model are discussed below.

PSO-GRNN.
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].e complex weights training process is avoided, and the approximation ability and nonlinear mapping ability of GRNN model are only determined by one smooth factor σ. To enhance the computational accuracy of GRNN model, the PSO-GRNN surrogate model is proposed by combining the dynamic PSO 2 Advances in Materials Science and Engineering algorithm of searching the optimal smooth factor and the GRNN model of constructing high-fidelity surrogate model.e basic thought and mathematical model of the PSO-GRNN are summarized as follows.
With a certain distribution type of joint probability density function F(x, y) (x ∈ R l , y(x) ∈ R) and the corresponding learning sample set ( x j ,  y)  j � 1, 2, . . ., m, to retrieve the precise output responses, the output space R is mapped from the input vector space R l , and the nonlinear regression function f σ (x) is denoted by where Λ is the data set of undetermined smooth factors σ.
In this study, the nonlinear regression function f σ (x) will be fitted by PSO-GRNN.e topology structure of PSO-GRNN surrogate model is shown in Figure 1.As shown in Figure 1, the PSO-GRNN 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 PSO-GRNN 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 m.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 PSO-GRNN model.In this process, multiple computation tasks simultaneously are processed to improve the fitting efficiency effectively in nonlinear mapping problems of multiple objective design.e continuous transfer function of pattern layer is expressed as where exp(•) denotes the exponential function operator and (x −  x j ) T (x −  x j ) represents the square of Euclid distance between input vector x and j-th learning sample point  x j .After pattern output calculation, the summation operator will be performed in the summation layer.e arithmetical summation and weighted summation are completed in S D neuron and S N neuron, respectively.e transfer functions of S D neuron and S N neuron are defined as in which w j indicates the connection weights of j-th pattern neuron and S N neuron.Hence, the output response y(x) and the nonlinear regression function f σ (x) can be retrieved with S N and S D : e feasibility and effectiveness of the aforementioned PSO-GRNN 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  y( x j ) indicates the j-th estimated output response value and  y j the j-th real output response value.To solve the optimization model and enhance the approximation accuracy of PSO-GRNN, 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.e 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.
e 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 PSO-GRNN model.Each particle is a potential solution for initial smooth factor of the PSO-GRNN 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 e renewal formula of particle position and velocity in dynamic PSO algorithm are determined by where s is the s-th particle; k the current iteration number; K the maximum iteration number; V s the current particle velocity; X s the current particle position; P s the current individual extremum; P g the current population extremum; r 1 , r 2 the random numbers during time domain [0,1]; w p (k) the dynamic nonlinear inertia weight; w p0 the initial inertia weight; w pK the inertia weight at the largest number of iteration; c 1 (k) and c 2 (k) the dynamic nonlinear individual learning factor and dynamic nonlinear population learning factor in the k-th iteration, respectively; c 1o and c 2o the initial individual learning factor and initial population learning factor, respectively; and c 1K and c 2K the individual learning factor and population learning factor in the Kth iteration, respectively.
After acquiring the optimal smooth factor σ in the learning process, the PSO-GRNN model is built.On account of the arbitrary shape property of network structure and the self-adaptation characteristics of dynamic PSO algorithm, the PSO-GRNN model can effectively reduce the approximation error and deal with the regular nonlinear optimal problem with a relatively high computing efficiency and accuracy.

TV/PSO-GRNN.
For time-varying (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 PSO-GRNN 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 PSO-GRNN model and SLHS technique, this study develops the time-varying PSO-GRNN (TV/PSO-GRNN) 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]. is process is equivalent to transform the complicated time-varying 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/PSO-GRNN is a heuristic way to improve the computational efficiency and enhance computational accuracy for reliability analysis.
e analytical thought of TV/PSO-GRNN is shown in Figure 2.
In view of the basic thought of TV/PSO-GRNN in Figure 2, the extremum output response Y i, max (x i ) of all dynamic response Y i (t, x i ) corresponding with the i-th input random vector x i is obtained through a number of stochastic analyses within the time domain consisting of the maximum output responses is used to construct the TV/PSO-GRNN nonlinear regression function f σ (x), and the extremum response curve y(x)) is expressed by Clearly, the computational performance (efficiency and accuracy) of TV/PSO-GRNN surrogate model heavily depends on the feasibility and validity of data set Y i, max (x i ), which should be generated by high-reliability 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 x should be normalized into the unit cube space [x N min , x N max ]. e normalized input variables x N can be obtained by in which x N i indicates the i-th normalized input variable; x N min , x N max the minimum and maximum of all normalized input variable; and x min , x max the minimum and maximum of all input variables.en, 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.
e sampling result of SLHS technique is illustrated for three dimensions in Figure 3.
In the light of the nonlinear regression function of PSO-GRNN (Equation ( 4)) and the generated data set, the TV/PSO-GRNN nonlinear regression function is constructed as   Advances in Materials Science and Engineering When the TV/PSO-GRNN nonlinear regression function is applied to evaluate the time-varying reliability of complex structure replacing the FE model, this mathematical model is called as TV/PSO-GRNN method, which is suitable to fulfill the reliability-based fatigue life prediction of complex structures.

Reliability-Based Fatigue Life Prediction Framework
In view of the elastic-plastic 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 reliability-based 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 reliability-based fatigue life prediction framework with TV/PSO-GRNN model is introduced.Assume that the allowable output structure response is [Y], based on the TV/PSO-GRNN model, the limit state function g(x) of structural fatigue life is calculated by As shown in Equation (10), g(x) ≥ 0 can ensure the safety of complex structure.With the random sampling on the established limit state function, the structural fatigue reliability degree is expressed as where I r [g(x i )] is the indicator function of the whole reliability domain; N r expresses the number of sample points in the secure domain; and N 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.us, the failure probability P and the sensitivity degree S of variables mean values on the failure probability are obtained by where P represents the failure probability; S the sensitivity degree of the input variables effecting the output response; is the indicator function of the whole failure domain; N f indicates the number of sample points in the failure domain; and x j expresses the j-th input sample point.By comparing Equations ( 11) and ( 12), we can see that the failure probability P complements the reliability degree R, as their sum must be equal to one. is 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 reliability-based fatigue life prediction of complex structure, the reliability-based fatigue life prediction framework is constructed based on the time-varying powerful mapping ability of TV/PSO-GRNN surrogate model, which is illustrated in Figure 4.

Deterministic Analysis with Multiphysics Interaction
As a vital component of an aircraft engine, the high-pressure 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).e 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.e load spectrum of an aircraft engine is shown in Figure 6

Advances in Materials Science and Engineering
Poisson's ratio μ, 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.erefore, the low cycle fatigue life prediction of an aeroengine turbine blisk is regarded as the objective of study.e 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 single-disciplinary subsystems ( uid 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.e MPI sketch of turbine blisk is drawn in Figure 8.In uid subsystem, the related parameters are 168 m/s for inlet uid velocity, 600,000 Pa for inlet pressure, 11756 W/m 2 K for heat transfer coe cient in MPI surface, and 1 atm for outlet pressure.During the uid dynamics analysis, the pressure distribution on MPI surface is obtained in Figure 9.After acquiring the initial temperature loads from uid subsystem, we select the related thermal parameters of 21.2 W/(m °C) for thermal conductivity coe cient and 14.8 × 10 −6 °C−1 for thermal expansion coe cient.According to the temperature consistency principle, the thermal analysis is completed to gain the body temperature distribution on the MPI surface in Figure 10.
e uid 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 ight pro le parameters in Figure 4.
ereby, 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.
As illustrated in the analytical results, the maximum stress σ max and strain range Δε t of turbine blisk reach at the peak values at the back of blade root.Considering the main stress cycle 0-σ max -0, the fatigue life can be obtained as 3228 ight cycles by the improved Masson-Co n model [29]: where σ m represents the mean stress; σ f ′ the fatigue strength coe cient; ε f ′ the fatigue ductility coe cient; b the fatigue strength exponent; c the fatigue ductility exponent; and N f the failure cycle number.

Random Variables Selection.
Under aircraft engine operation, the material properties, multiphysical loads, and model uncertainty possess evidently inherent randomness and seriously in uence the fatigue life of turbine blisk [45,46].erefore, we regarded rotor speed ω, gas temperature T, uid velocity ρ, elastic modulus E, and thermal conductivity coe cient λ as random variables.e distribution characteristics of physical variables are listed in Table 2.
e distribution characteristics of model uncertainty parameters such as the fatigue strength exponent b, fatigue ductility exponent c, fatigue strength coe cient σ ′ f , and fatigue ductility coe cient f are also considered as random variables as listed in Table 3 [47].Assuming that the fatigue ductility coe cient ε f ′ obeys lognormal distribution and other random variables obey normal distribution, and all of selected random variables are reciprocally independent, respectively.

TV/PSO-GRNNM Surrogate Modelling.
e uncertainty parameters in Tables 2 and 3 12.To quantify approximation error and evaluate the tting performance, the mean relative error (MRE) metric and root mean squared error (RMSE) matrix in Equation ( 14) are adopted to reveal.e comparison results with response surface model (RSM) [33], GRNN, PSO-GRNN, and TV/PSO-GRNN surrogate model are shown in Table 4: where y is the real output response; y estimated output response; i i-th test sample point; and n t 100 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/PSO-GRNN surrogate model still t each test points with almost zero approximation errors.us, the approximation performance of TV/PSO-GRNN model is superior to RSM, GRNN, and PSO-GRNN.erefore, as surrogate model the TV/PSO-GRNN model is suitable to ful ll the reliabilitybased fatigue life prediction.

Reliability-Based 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.e detailed probabilistic distribution characteristics are revealed in Figure 13.According to the extracted input variables and TV/PSO-GRNN model, 10,000 simulations are executed to predict the failure cycle number N f of turbine blisk.e simulation history and probabilistic distribution of fatigue life are revealed in Figure 14, which indicates that the output response N f of turbine blisk obeys a lognormal distribution and the fatigue life of turbine blisk under reliability 99.87% is 3265 cycles.
rough the sensitivity analysis based on Equation ( 12), the sensitivities and e ect probability of input variables on turbine blisk fatigue failure are revealed in Figure 15.e analysis results explain that rotor speed w is the most Advances in Materials Science and Engineering important factors and plays a leading role for the fatigue life failure with the e ect probability of 32%.Besides, gas temperature T and fatigue strength coe cient σ ′ f are also main factors with the e ect probabilities of 25% and 21%, respectively, while the in uences of other factors are weak relatively.In the fatigue failure design of turbine blisk, the rotor speed w, gas temperature T, and fatigue strength coe cient σ f ′ should be controlled preferentially.e uid velocity v and thermal conductivity coe cient λ are negatively correlated with the fatigue failure probability.e other variables are positively correlated with the fatigue failure probability, which basically agree with engineering practice.
TV/PSO-GRNN Method Validation.To support the proposed TV/PSO-GRNN model, the reliability-based fatigue life prediction of the blisk is performed with MC method, RSM, GRNN, and PSO-GRNN, respectively.e simulation consumptions with di erent methods are listed in Table 5.To validate the feasibility and e ectiveness of the proposed TV/PSO-GRNN model, the uniform computational e ciency (UCE) and uniform computational accuracy (UCA) are introduced in Equation ( 15). e comparison results of the four methods under di erent simulations are listed in Table 6:      where FT and CT are tting time and computing time for surrogate models, respectively; CT the time of MC simulations; and N f and N f 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 N f of turbine blisk obeys a log-normal distribution and the fatigue life of turbine blisk under the reliability 99.87% is 3265 cycles.As revealed in Figure 15, the rotor speed w, gas temperature T, and fatigue strength coe cient σ ′ f are the leading factors on the fatigue life of turbine blisk as the e ect probabilities are 32%, 25%, and 21%, respectively.e inuences from other factors are relatively weak.erefore, ω and T should be considered with the priority in the reliability-based fatigue life design of turbine blisk.e reduction of inlet gas velocity v and fatigue ductility coe cient ε ′ f 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 tting time and tting number of TV/PSO-GRNN are less than RSM, GRNN, and PSO-GRNN, and the simulation consumption and total computational e ciency of TV/PSO-GRNN is superior than RSM, GRNN, PSO-GRNN, and direct simulation model as well.Moreover, the computational eciency bene ts of the proposed TV/PSO-GRNN are more obvious with increasing simulations.Hence, the proposed TV/PSO-GRNN holds high computing e ciency due to low time consumption.
e main reasons are (1) TV/PSO-GRNN only focus on the extremum value of the response  As unveiled in Figure 12 and Table 4, the established TV/PSO-GRNN surrogate model can approximate every test sample so that the proposed TV/PSO-GRNN possesses the lowest fitting error.As illustrated in Table 6, the TV/PSO-GRNN is more precise than the RSM, GRNN, and PSO-GRNN and is almost consistent with the direct simulation (FE/FV) method.
e TV/PSO-GRNN thus holds good generalization ability and high-accuracy computational ability.e reasons are listed as follows: (1) the TV/PSO-GRNN 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 high-quality learning data set with less data noise, which also provides a good way to enhance the fitting ability of surrogate model.erefore, the TV/PSO-GRNN model holds high computational accuracy in reliability-based fatigue life prediction of complex structure.
In summary, the proposed TV/PSO-GRNN greatly saves computing time and improves computational efficiency while keeping computational accuracy.erefore, the TV/PSO-GRNN is a feasible and effective way in the reliabilitybased fatigue life prediction of complex structures.

Conclusions and Outlooks
To improve the computational efficiency and accuracy of reliability-based fatigue life prediction for complex structure, an efficient time-varying surrogate model (termed "TV/PSO-GRNN") is developed.
e reliability-based fatigue life prediction of turbine blisk was taken as a case to support the validation and feasibility of TV/PSO-GRNN.Some conclusions are drawn as follows: (1) rough the reliability-based fatigue life prediction of turbine blisk with the TV/PSO-GRNN, the simulation histories, and distribution features of reliability-based 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 PSO-GRNN), the developed TV/PSO-GRNN possesses the highest computational efficiency and accuracy.We also demonstrated that the TV/PSO-GRNN surrogate model is an effective approach for the reliability-based fatigue life prediction of complex structure.(3) e efforts of this study enrich mechanical reliability theory from a surrogate modelling perspective, and shed light on the further applications in the reliability-based design optimization as well.
Although this investigation provides a novel time-varying surrogate model to improve the modelling accuracy and simulation efficiency for the reliability-based 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 time-varying regression functions based on different surrogate models should be established in future to accomplish the reliability-based design optimization for fatigue life prediction of complex structure.

Figure 6 :
Figure 6: e loads spectrum of an aircraft engine.

Figure 8 :
Figure 8: e MPI sketch of turbine blisk are regarded as the input variables, and the failure cycle number N f 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.e TV/PSO-GRNN surrogate model is constructed by training the model.e test result of TV/PSO-GRNN surrogate model is drawn in Figure

Figure 11 :
Figure 11: e maximum stress (a) and strain range (b) distributions on turbine blisk.

Figure 15 :
Figure 15: e sensitivities and e ect probabilities of fatigue life.
[44].Furthermore, the specific values of nonlinear material parameters, including elastic modulus E, heat transfer coefficient λ, expansion coefficient α, and 6Advances in Materials Science and Engineering If MSE ≤ ε ?If k = K ?Rectify V s (k), X s (k), and P s (k), P g (k) k = k + 1

Table 2 :
e distribution characteristics of physical variables.

Table 3 :
e distribution characteristics of model variables.

Table 4 :
e approximation performance with di erent surrogate models.

Table 5 :
e simulation time of the reliability-based fatigue life prediction of turbine blisk.

Table 6 :
Uniform computational e ciency and accuracy of four surrogate models.Advances in Materials Science and Engineering process for each calculation within a time domain in the reliability-based fatigue life prediction, rather than the whole dynamic response process; (2) the TV/PSO-GRNN 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.erefore, based on the aforementioned strengths, the optimal smooth factor of TV/PSO-GRNN are quickly obtained, which save time and improve computing efficiency in reliability-based fatigue life prediction.erefore, the TV/PSO-GRNN model possesses high computational efficiency in reliability-based fatigue life prediction of complex structure.