1 Introduction

As a key part of an aeroengine, turbine blisk is applied to convert and transfer the energy, which significantly affects the efficiency and safety of aeroengine. As up to 25% of aeroengine failures are related to turbine blisk. Accurate design and estimation on the dynamic responses of turbine blisk are more critical for the growing demand for high-efficiency and safe aeroengines.

The mechanical evaluation of aeroengine components have emerged. Meguid et al. [1] studied bird strike resistance of an aeroengine applying explicit numerical approach and 3-D finite-element modeling. Results indicated that the significant effect on the impact force belongs to the first contact area between the bird and target of the first phase of the impulse. Qi et al. presented a method to consider the boundary conditions and calculated the tip clearance of aeroengine considering the effects of rotation speed, pressure and temperature [2]. The main findings of this work include the main role of shaft speed on turbine disk displacement and temperature influence on turbine blade. Dong et al. investigated the effect of ice accretion and flowfield on the performance of an aeroengine structure. A comparison of the computational and experimental data showed acceptable agreement for strut temperature [3]. Szwaba et al. [4] examined the influence of aerodynamic characteristics and heat transfer of radial cooling passage walls of a turbine blade. Survey results indicated that transient method is easy to apply and its most important feature is that stable conditions in cooling walls are that stable conditions in wall temperature are not necessary to consider. Peschiulli et al. [5] proposed a methodology to investigate thermal characteristics and clearance control of complex mechanical system. Satish et al. [6] dynamically analyzed aeroengines blade-tip clearance (BTC) by an error identifying approaches to decrease system uncertainties.

For deterministic studies on the behavior and characteristics of aeroengine whole-body and components, various works on the probabilistic analysis of turbine blisk are conducted. Zhai et al. [7] developed a stochastic model updating (SMU) procedure using Monte Carlo (MC) method and an improved response surface method (RSM) for reliability analysis of aeroengines and complex structures, and showed that the SMU procedure could precisely simulate aeroengine system with desired efficiency. Lu et al. [8] developed Improved decomposed-coordinated Kriging modeling strategy for dynamic probabilistic analysis of multi-component structures with acceptable computational efficiency. Surrogate models are commonly used to optimization and RBDO of complex engineering problems [9, 10]. Machine-leaning basis nonlinear relations are successfully used to approximate the failure domains of probabilistic functions in structural reliability analyses. The call function as computational burden of the limit state function can be improved using hybrid reliability methods such as Kriging [11,12,13,14,15,16,17,18], response surface method [19], support vector machine [20,21,22,23,24], M5Tree [25, 26] and artificial neural network [27] in structural reliability design. Hu et al. [28] developed a low computational approach for the optimum design of turbine disk reliability using a single-loop-single-vector (SLSV) approach. Fei and Bai [29] proposed an efficient probabilistic numerical procedure based on support vector machine (SVM), and revealed the proposed approach with high efficiency and desirable accuracy for the optimization of gas turbines blade-tip radial running clearance. Wong et al. [30] applied Fourier series for the uncertain reliability analysis of turbine blade disks, and proved that its accuracy and efficiency are acceptable. Bai et al. [31] developed an efficient multi-stage multi-disciplinary model based on a dynamic substructure method for turbine blisk, and indicated that the computational efficiency and accuracy of the proposed approach are better than support vector regression. Lin et al. [32] proposed a model for reliability analysis of aeroengine blade considering fatigue failure mode, and demonstrated that the proposed model was reasonable and matched with real-system conditions. An et al. [33] used the stress–strength interference (SSI) model for the reliability analysis of aeroengine decelerator regarding random variables. Fei et al. [24] applied extremum response surface method (ERSM) -based support vector machine (SVM) for optimum design of blade-tip radial running clearance in respect of nonlinear properties of materials and thermal loads. Lu et al. [34] developed a multi-extremum response surface method (MERSM) for the reliability analysis of aeroengine turbine blisk considering multiple failure modes, and illustrated the good performance of MERSM in accuracy and efficiency. Zhang et al. [35, 36] applied improved RSM considering the effect of strain and stress, gas temperature and gas velocity, to study on low cycle fatigue and high-temperature creep of turbine blisk. Fei et al. [15] proposed a surrogate model method for the reliability analysis and robust design of turbine blisk. Wang et al. [37] investigated the parameters of BTC for different rotating speed and active clearance control (ACC) of aeroengines. Zuo et al. [38] studied the reliability of aeroengine oil system using evidential network model based fault tree analysis considering epistemic uncertainty. To reduce the computational burden of complex problems, the modeling method basis multi-extremum surface of failure regions is significantly reduced computational times to evaluate failure domains. The main change in modeling methods is to provide the accurate prediction of multi-extremum performance functions under complex uncertainties. Although the RSM is one of simple and efficient regression tools, the abilities of the RSM for accurate estimation of multi-extremum basis functions have some limitations due to the usage of second-order polynomial function. The flexibility of RSM with respect to the nonlinear relationship for nonlinear dynamic responses potentially enhance to reach the efficient and accurate multi-extremum nonlinear relationships.

The objective of this paper is to develop an accurate surrogate model to determine the nonlinear responses of multi-failure modes of complex structures. In respect of this purpose, two novel modified multi-extremum response surface models (MRSM) are developed for the dynamic response of aeroengine turbine blisk. Moreover, a sensitivity analysis using MRSM is adopted to investigate the variation of input variables on the nonlinear responses. The remaining of this paper is structured as follows. In Sect. 2, the fundamentals of the two MRSMs are discussed. A general overview of turbine blisk specifications and dynamic analysis are conducted in Sect. 3. Comparative modeling results are reported in Sect. 4. Section 5 summarizes some conclusions of this work.

2 Multi-extremum-modified response basis model

RSM is commonly applied to approximate to the failure modes of aeroengine structures under complex conditions in loads and manufacturing materials [15]. The ability of RSM to balance between accuracy and simplicity is one of the exciting characteristics in approximating complex engineering problems. The main effort in RSM is to provide a nonlinear relation as second-order basis polynomial functions between input variables (i.e., inlet temperature, material density, inlet velocity, and rotor speed) and outputs as multi-failure responses (i.e., blisk deformation, stress, and strain). The RSM cannot approximate highly nonlinear problems with high-correlation of input variables for complex problems due to approximating responses using polynomial function with quadratic form. Consequently, the accuracy of models, which is one of the major efforts in modeling process, is depended on the nonlinearity forms of approximated relation, input data points, training process, and parameters of models. In the current work, a multi-extremum-modified response model is proposed based on two calibrating processes. In the first modeling approach, each input variable is transferred based on a nonlinear mapping with power or exponential function, while the calibrating data points are provided by the first step, which is used for regressing the multi-extremum third-order polynomial functions. Figure 1 represents the modeling framework of two regressed RSM, denoted as MRSM. In this modeling approach, four layers of 1–4 are applied, including input layer (layer 1), normalized layer (layer 2), nonlinear mapping layer (layer 3) and calibrating layer (layer 4). The calibrating procedures using the nonlinear response of turbine blisk under fluid-thermal uncertainties are given in layer 3 and layer 4.

Fig. 1
figure 1

The framework of multi-extremum-modified response basis model

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As seen in Fig. 1, the results of MRSM are the approximated data (Y) of multi-failure modes of blisk deformation, stress, strain at layer 4 which are calibrated by input variables (X = (x1, x2, …, xn)), i.e., inlet temperature, inlet velocity, material density and rotor speed at layer 1. The details of layers 1–4 are expressed as follows:

Layer 1: Give input variable (x1, x2, …, xn) which are simulated using LHS with 150 sample in the train and 44 sample points in the test phases.

Layer 2: Normalize variables using the statistical properties of input variables [39], i.e.,

$$N\left( X \right) = \frac{X - \mu }{\sigma },$$
(1)

where \(N\left( X \right)\) presents the normal variable of X with mean of µ and standard diversion of σ, including temperature (T) (µT = 1164.94 K, σT = 26.01 K), material density (ρ) (µρ = 4450 kg/m3, σρ = 57.8 kg/m3), inlet velocity (v) (µv = 160 m/s, σv = 11.55 m/s) and rotor speed (ω) (µω = 1150.1 rad/s, σω = 86.6 rad/s), where µ and σ are computed based on simulated data points in training phase. In the normalized data, the standard deviation is a scale factor to control the range of the normalized input data. By normalizing input data, the free-dimensions of input data are used in the calibrating process. The large and small input bounds could be provided by the factor σ. The input data have a narrow bound when σ is minimal value, while a high-bound for input data is obtained as σ is large value.

The layers 1 and 2 are determined based on the input variables which are connected with the final layer. This connection is structured based on two significant calibrations, which are created in layers 3 and 4.

Layer 3: Mapping the normalized data based on the first regression procedure.

In the current work, two nonlinear mapping data points are presented to provide the nonlinear effects of input variables. In the first map, the linear function in the following equation is adopted to improve the RSM in MRSM-LN:

$$Y_{i}^{{}} = a_{0i}^{{}} + a_{1i}^{{}} N\left( {X_{i} } \right).$$
(2)

In the second map, the exponential function, namely MRSM-EXP, is used as follows:

$$Y_{i} = a_{0i} + a_{1i} \exp \left[ {N\left( {X_{i} } \right)} \right],$$
(3)

where \(a_{0i}\) and \(a_{1i}\) are the unknown coefficients.

The linear and exponential mapping data are used in the first calibrating process, which is provided by a connection between input data and output responses. Based on the calibrating, the input data are considered based on its influence that is determined by the coefficients \(a_{1i}\). The unknown coefficients are calculated using the dimensionless input variables, which are computed in layer 2. Therefore, a1 and a2 have the same dimensions for each input variable. Thus, the sensitivity of input variable is suggested by

$$\alpha_{i} = \frac{{a_{1i} }}{{\sqrt {\sum\nolimits_{i = 1}^{n} {\left( {a_{1i} } \right)^{2} } } }},$$
(4)

where ai (i = 1, 2, …, n) denotes the sensitivity degree for variable xi. In the proposed MRSM, the sensitivity analysis of input data is computed while the traditional RSM cannot handle input data as uncertainties. Moreover, the proposed model can be used as an acceptable guideline for robust design, controlling, and reliability analysis of complex structures. In this work, effective input variables are considered as the nonlinear response of dynamic turbine blisk failure modes.

The sensitivity degree of input variables is determined by the calibrating coefficients in the first stage. The flexibility of RSM is enhanced based on the transformation of input variables using linear and exponential maps.

Layer 4: Calibrate the nonlinear response of turbine blisk using the input mapping variables.

The approximated function of MRSMs using the linear and nonlinear maps are determined as [40]

$$f = a_{0} + \sum\limits_{i = 1}^{n} {a_{i} } Y_{i} + \sum\limits_{i = 1}^{n} {\sum\limits_{j = i}^{n} {a_{ij} } } Y_{i}^{{}} Y_{j}^{{}} ,$$
(5)

where Yi is the input data which are computed in the first calibrating process that is determined in layers 1–3. It is essential to compute the unknown coefficient \({\varvec{a}} \in \{ a_{0} ,a_{i} ,a_{ij} \,\,,\,\,ij = 1,2,...,n\}\) that is commonly determined by least squared methodology via minimizing the error between computed nonlinear response (O) and approximated function (f)\(e = \left[ {{\varvec{O}} - {\varvec{f}}} \right]^{T} \left[ {{\varvec{O}} - {\varvec{f}}} \right]\) [15, 24], in which \({\varvec{f}} = {\varvec{Pa}}\) and \({\varvec{P}}\) is the polynomial vector which is determined based on the mapping input data \(Y \in \left\{ {Y_{1} ,Y_{2} ,...,Y_{n} } \right\}\) as below:

$${\varvec{P}} = \left[ {1,\,\,Y_{1} ,\,Y_{2} ,...Y_{n} ,\,Y_{1}^{2} ,\,Y_{1}^{{}} Y_{2}^{{}} ,...,Y_{n - 1}^{{}} Y_{n}^{{}} ,Y_{n}^{2} } \right].$$
(6)

The computed responses are determined based on the nonlinear dynamic finite-element model in respect of train dataset comprising 150 samples, denoted by \({\varvec{O}} = \{ o_{1} ,o_{2} ,...o_{150} \}\). Thus, four input variables with 150 samples is used to train the models. The mapping input data points \(\{ Y_{1} ,Y_{2} ,...,Y_{n} \}\) are computed by

$${\varvec{Y}} = {\varvec{P}}_{xt}^{T} \left( {{\varvec{P}}_{x}^{T} {\varvec{P}}_{x} } \right)^{ - 1} \,\left( {{\varvec{P}}_{x}^{T} {\varvec{O}}} \right),$$
(7)

where \({\varvec{P}}_{x} = \left[ {1,N\left( X \right)} \right]\) for MRSM-LN and \({\varvec{P}}_{x} = \left[ {1,\exp \left( {N\left( X \right)} \right)} \right]\) for MRSM-EXP are train data, and \({\varvec{P}}_{xt}\) is for test data points. The factors \(a_{0} ,\,a_{1}\) are computed using 150 train samples as \(\left( {{\varvec{P}}_{x}^{T} {\varvec{P}}_{x} } \right)^{ - 1} \,\left( {{\varvec{P}}_{x}^{T} {\varvec{O}}} \right)\), by determining the first calibrating data in Eq. (7). The predicted value of a nonlinear response without the dynamic FE analysis is approximated by

$${\varvec{f}} = {\varvec{P}}_{t}^{T} \left( {{\varvec{P}}^{T} {\varvec{P}}} \right)^{ - 1} \left( {{\varvec{P}}^{T} {\varvec{O}}} \right),$$
(8)

where Pt indicates the polynomial basis function for test data. Unlike the RSM, the proposed approach is established by the two main regressions procedures, while the MRSM is simply provided by sensitivity factors for each input variable. The accurate prediction of the nonlinear response for complex dynamic problems with nonlinear analysis is the main effort in modeling process.

Figure 2 illustrates the framework of the current work using MRSM. The computational effort in the structure to build a nonlinear model is provided by the simple framework without complicated relations. The high-flexibility as the nonlinearity of the input variables is enhanced based on the first calibrating process in layer 3. The nonlinear mapping data using the exponential function can be improved by the nonlinearity of relations between input data and computed dynamic responses.

Fig. 2
figure 2

The framework of MRSM basis nonlinear input map

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3 Two-way fluid-thermal–structural analysis of turbine blisk

A high gas pressure turbine blisk of an aeroengine (in Fig. 3) under high temperature–speed is considered as the case study of this work. To completely simulate the performance of aeroengine, the flight profile and computing range are selected from start to cruise state [15, 41]. The Superalloy (Titanium alloy) is selected for the material of blisk. The analysis of blisk structure is performed by the fluid-thermal–structural coupling method considering nonlinear properties of materials and the dynamics of temperature and rotational speed. The time-dependent temperature load T and rotational speed ω are reported in Table 1.

Fig. 3
figure 3

Aeroengine turbine blisk

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Table 1 Temperature and speed value change with time
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3.1 Finite-element model

Turbine blisk is symmetrically typical. Thus, a single blisk could be considered as the object of simulation in Fig. 4. As shown in Figs. 4 and 5, the cooling hole of blade and convex platform of disk are selected to create a FE model of turbine blisk. The FE model of blisk consists of hexahedron and tetrahedron with 9 577 elements and 25 015 nodes, whiles the finite-volume (FV) model of flow field is built by hexahedron with 1 044 528 elements and 1 009 035 nodes.

Fig. 4
figure 4

FE model of the turbine blisk

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Fig. 5
figure 5

FE model of the flow field

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Statistical properties of input variables for train and test data points, including rotating speed (ω), gas temperature (T), material density (ρ), and inlet velocity (v), are reported in Table 2, where Xmin, Xmax, XMean, and STD are, respectively, minimum, maximum, average, and standard deviation of input variables. The data in the train and test phases are simulated by optimum Latin hypercube sampling with 150 points in train and 45 samples in test phases.

Table 2 Statistical characteristics of input variables for FE analysis
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3.2 Nonlinear dynamic analysis of blisk

The deterministic analysis of blisk within the time domain [0, 215s] is summarized as follows:

  1. (1)

    FV method is employed to investigate the fluid features of turbine blisk as the standard k-ε turbulence model.

  2. (2)

    The law of energy conservation is used to resolve the thermodynamic property of flow flied in fluid-thermal analysis.

  3. (3)

    FE method is employed to accomplish the structural analysis of turbine blisk and displacement function, geometric equation and constitutive equation are used to obtain analytical results based on shape equations of tetrahedron and hexahedron elements.

  4. (4)

    The system coupling is used to actualize the solution of turbine blisk with fluid-thermal–solid interaction by multiple iterations and coupling information updating.

From the two-way fluid-thermal–structural analysis of turbine blisk, the change curves of deformation, stress, and strain during the time domain are presented in Fig. 6.

Fig. 6
figure 6

Variation of deformation with time for turbine blisk

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As seen in Fig. 6, the maximum deformation, stress, and strain of turbine blisk are gained simultaneously at the time domain [160, 200 s]. The time point t = 200 s is considered as the critical point and computing point to estimate the safety of turbine blisk during the flight. Thus, the pressure distribution on the fluid-thermal–structural interface, the temperature distribution on turbine blisk, the nephograms of turbine blisk deformation, stress and strain, are shown in Figs. 7, 8, 9, respectively, in which p, Tb, u, σ, and ε indicate the pressure, temperature, deformation, stress, and strain values of turbine blisk, respectively.

Fig. 7
figure 7

Pressure distribution on the fluid-thermal–structure interface

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Fig. 8
figure 8

Temperature distribution on turbine blisk

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Fig. 9
figure 9

The nephograms of turbine blisk deformation

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As seen in Fig. 9, the maximum of blisk deformation emerges at the top of turbine blade, and the maximum values of blisk stress and strain are located at the bottom of the turbine blade.

To obtain modeling samples as train data and validated samples as test data generated by Latin hypercube sampling, three failure modes of turbine blisk, including maximum deformation, stress and strain are determined by nonlinear FE analysis regarding input variables. The data points of output responses of turbine blisk corresponding to the obtained input samples are acquired by two-way fluid-thermal–structural analyses. The statistical properties in training and test phases are listed in Table 3. The models are calibrated using training data, while the models performances are validated using the test dataset.

Table 3 Statistical characteristics of train and test data points for nonlinear responses of turbine blisk
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4 Comparative modeling results

The accuracy and agreement of the MRSM (i.e., MRSM-EXP and MRSM-LN) are compared with RSM, ANN models using four comparative statistics such as mean absolute error (MAE), root mean square error (RMSE), modified agreement index (d), and modified Nash and Sutcliffe efficiency (NSE) [42, 43], i.e.,

$${\text{RMSE}} = \frac{1}{N}\sqrt {\sum\limits_{i = 1}^{N} {[O_{i}^{{}} - f_{i}^{{}} ]^{2} } } ,$$
(9)
$${\text{MAE}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {|O_{i}^{{}} - f_{i}^{{}} |} ,$$
(10)
$$d = 1 - \frac{{\sum\nolimits_{i = 1}^{N} {|O_{i}^{{}} - f_{i}^{{}} |} }}{{\sum\nolimits_{i = 1}^{N} {|O_{i}^{{}} - \overline{O}| + |f_{i}^{{}} - \overline{O}|} }}\,\,,\,\,\,0 < d \le 1,$$
(11)
$${\text{NSE}} = 1 - \frac{{\sum\nolimits_{i = 1}^{N} {|O_{i}^{{}} - f_{i}^{{}} |} }}{{\sum\nolimits_{i = 1}^{N} {|O_{i}^{{}} - \overline{O}|} }}\,\,,\,\,\, - \infty < {\text{NSE}} \le 1,$$
(12)

where N denotes the number of data; Oi and Pi are, respectively, i-th data point for computed FE responses and predicted models; \(\overline{O}\) is the average of computed data using nonlinear dynamic FE. The RMSE and MAE values for each model are tended to zero. Thus, it can be induced that the predicted model relates to accurate predictions with minimum errors. d varies from 0 to 1 with no-correlation by the perfect fitness. The NSE presents the goodness-of-fitness of the model where NSE = 1 indicates perfect agreement predictions. These different statistics are determined for the train and test data points for blisk deformation, stress, and strain which are presented in Table 4. The ANN models is structured with 4 nodes in input layers, 4–10 nodes in the hidden layer which is obtained by trial and errors as 5, 7 and 8 nodes for blisk deformation, stress and strain, respectively. The bold numbers in the table are the best model. The results presented in Table 4 indicated that two modified models using linear and exponential maps are enhanced the performances of RSM for both accuracy (lowest RMSE, MAE) and agreement (highest d and NSE) for train and test phases. It can be conducted that ANN model is performed with worst abilities for all failure modes. Using MRSM-EXP, MAE (RMES) is enhanced about 82% (85%) and 77% (78%) for blisk deformation and about 24% (29%) and 10% (9%) for stress by comparing RSM and MRSM-LN models, respectively. The two regression processes in proposed MRSM models are improves the abilities of RSM for approximating the failure modes of this complex problem. It is suggested that the MRSM-LR and MRSM-EXP models are applied to simulate the performances of structure reliability design analyses in the future. Commonly, three failure modes of turbine blisk are predicted with the accurate results and acceptable agreement using MRSM-EXP compared to other models.

Table 4 Comparative statistics for different models of multi-nonlinear response in test and train datasets
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The scatterplot of test data between FE response and predicted models for blisk deformation are presented in Fig. 10. In this figure, R2 is the correlations between the computed and predicted data points for different models. The larger value for R2 denotes to higher tendency between predicted data and computed points. The scatterplot diagram for different models in the approximation of the test data for stress and strain are presented in Figs. 11 and 12, respectively. From the results of Fig. 10, the best and worst models are the MRSM-LN and ANN, respectively, against the other models. Moreover, the ability of MRSM-EXM is similar to the MRSM-LR and it is more robust and accurate than the RSM.

Fig. 10
figure 10

Scatterplot for blisk deformation in test data

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Fig. 11
figure 11

Scatterplot for stress in test data

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Fig. 12
figure 12

Scatterplot for strain in test data

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The scatterplot of test data for stress and strain in Figs. 11 and 12 demonstrated that the MRSM models with nonlinear exponential map enhances the abilities of MRSM to provide a nonlinear relation, compared to the RMS and ANN. The linear cross-correlations using second-order terms in RSM, i.e., YiYj can improve the accuracy of the models while it is not applied in ANN. It can be conducted that artificial intelligent models basis ANN relations are required to improve their structures in training phase such as nonlinear active functions or input data as cross-terms, in the future.

The Taylor diagram basis standard deviations and NSE is illustrated in Fig. 13 for the test database of three failure modes of (a) blisk deformation, (b) stress, and (c) strain. As seen in Fig. 13, the computed data point using nonlinear dynamic FE is shown with the gray point on the horizontal line. The Taylor diagram in Fig. 13 indicates that the ANN models generally provide less performances than RSM and MRSM. However, the MRSM showed close predictions for blisk deformation and stress with the highest agreement index. The MRSM-EXP provided most accurate and robust predictions for strain data compared to RSM and MRSM-LN. Form the results of the Taylor diagram, it can be conducted that the models from best to worst predictions could be listed by following models of MRSM-EXP, MRSM-LN, RSM and ANN.

Fig. 13
figure 13

Taylor diagram for test data (a) blisk deformation, (b) stress and (c) strain

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The effects of input variables can be evaluated based on the correlation coefficient with responses of failure modes in turbine blisk. Thus, it is defined as the sensitivity factor for each input data [44]:

$$\left\{ \begin{gathered} \alpha_{i} = \frac{{r_{i} }}{{\sum\nolimits_{i = 1}^{n} {\left| {r_{i} } \right|} }}\,\,\,\, \hfill \\ r = \frac{{N\sum\nolimits_{i = 1}^{N} {O_{i} x_{i} - \left[ {\sum\nolimits_{i = 1}^{N} {O_{i} } } \right]\left[ {\sum\nolimits_{i = 1}^{N} {x_{i} } } \right]} }}{{\sqrt {N\sum\nolimits_{i = 1}^{N} {O_{i}^{2} - \left[ {\sum\nolimits_{i = 1}^{N} {O_{i} } } \right]^{2} } } \sqrt {N\sum\nolimits_{i = 1}^{N} {x_{i}^{2} - \left[ {\sum\nolimits_{i = 1}^{N} {x_{i} } } \right]^{2} } } }} \hfill \\ \end{gathered} \right.\,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ,\,\,i = 1,2,...,n$$

where \(\alpha\) is sensitivity degree; N and n indicate the numbers of data points and input variables, respectively; O is the response of turbine blisk; x is the input variable. The sensitivity degree based on correlation and the proposed MRSM is compared with respect of the same input variables (ω, v, ρ, and T). The results for sensitivity degree of input variables are presented in Fig. 14 for relative failure modes. It can be seen that the strong agreements are provided by the exponential and linear relations with the correlation sensitivity degree. The sensitivity frameworks in the proposed modeling approach can accurately evaluate the importance of input variables. By comparing the sensitivity of blisk deformation, stress, and strain performance functions, the temperature provides highest effects on the nonlinear responses of the failure modes, while rotor speed as the load of the problem is positively affected in Layer 2 for deformation and strain failure modes. Inlet velocity adversely effects on the performances of studied failure modes, while other input variables of ω, ρ and T hold positive effects. The different sensitivity values of inlet velocity are computed for each model. It is obtained that inlet velocity significantly influences the performances of blisk deformation and stress, while it is insensitive on the strain and blisk deformation is more sensitive than v.

Fig. 14
figure 14

Sensitivity degree of input variables using correlation and MRSM. a Blisk deformation, b stress and c strain

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5 Conclusions

The main effort in this work was to develop a modified multi-extremum response (MRSM) to approximate the multi-failure nonlinear dynamic responses of aeroengine components under high-temperature and rotating speed loads, with regard to fluid-thermal–structure interaction. Two multi-extremum-modified response basis models are established based on two regressing processes. In the first regression phase, the exponential and linear functions are used to calibrate the input variable. In respect of the acquired calibrating results, it is a fast and straightforward sensitivity analysis during the modeling process. The influences of input variables on multi-failure modes are investigated using the proposed sensitivity factors. The abilities of the proposed MRSM models coupled with linear (MRSM-LN) and exponential (MRSM-EXP) mappings are compared with RSM and ANN. Compared results indicated that the proposed sensitivity vector for linear and exponential functions reveals the largest influence of gas temperature, following by angular speed and material density. At the same time, inlet velocity is an insensitive parameter for maximum strain. Two models of MRSM-LN and MRSM-EXP have superior performances compared to RSM and ANN. The proposed MRSM is a promising approach in approximating and simulating in optimization, reliability analysis and reliability-based design optimization of the complex structures under multi-failure response in future.