reliability based multidisciplinary design and optimization for twin web disk using adaptive kriging surrogate model

Special Issue Article Reliability-based multidisciplinary design and optimization for twin-web disk using adaptive Kriging surrogate model Advances in Mechanical Engineering 2016, Vol 8(9) 1–12 Ó The Author(s) 2016 DOI: 10.1177/1687814016671448 aime.sagepub.com Mengchuang Zhang, Wenxuan Gou and Qin Yao Abstract Compared with the conventional single web disk, the twin-web disk has been designed as the future trend of the highpressure turbine disk by the US Integrated High Performance Turbine Engine Technology program due to its breakthrough in weight loss, strength, and heat transfer efficiency However, as a crucial component, the high-pressure turbine disk of aerocraft needs a high reliability and a steady quality at the same time The traditional deterministic multidisciplinary design of optimization method sometimes could not be able to satisfy both the two requirements and depends heavily on the selection strategy of safety factor In this article, reliability-based multidisciplinary design optimization has been performed to find a proper shape of twin-web disk with the minimum weight The structural strength reliability analysis is performed using Monte Carlo simulation and set as the constraints in order to ensure the stability and safety Kriging approximation is performed to reduce the computational cost Then, the optimal points obtained by reliability-based multidisciplinary design optimization and common multidisciplinary design optimization are compared The results show that the reliability-based multidisciplinary design optimization can obtain a better performance and less weight, which could be a reference in designing the twin-web disk for industry Keywords Twin-web disk, reliability-based multidisciplinary design and optimization, Monte Carlo simulation, adaptive Kriging surrogate model Date received: 15 June 2016; accepted: September 2016 Academic Editor: Yongming Liu Introduction The twin-web high-pressure turbine disk (TWD) has advantages in internal cooling and weight loss.1 As the future substitute of the conventional single web turbine disk (SWD), the TWD still lacks scientific design technique Previous investigation of TWD design focused on shape optimization.2–4 But in these studies, the thermal load was ignored or just obtained by empirical formula, which is unpractical when suffers an extreme high turbine inlet temperature (TIT) And the scatter in dimensions, material properties, and loading can also degrade the stability and safety of the TWD Multidisciplinary design optimization (MDO) is a suitable technique solving the problems especially in the case of multi-physics working conditions and intense coupling of multiple disciplines.5 For decades, MDO has obtained success in many aero industrial products.6–8 As a deterministic optimization, MDO is driven to the limit of the deterministic constraints However, designs without consideration of the model and Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an, P.R China Corresponding author: Mengchuang Zhang, Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710129, P.R China Email: zmc.olisadebe@163.com Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage) 2 Advances in Mechanical Engineering physical uncertainty are unreliable and could lead to systematic failure At this point, the reliability-based multidisciplinary design optimization (RBMDO) has been performed for the evaluations of performance probabilities and the formulations of the probabilistic constraints RBMDO has been widely studied in recent years.9–14 However, traditional RBMDO method is inefficient which often requires a huge computational resource The design of experiment (DOE) procedure is a way to develop the scientific strategy in design variables selection and reduce the design space.15 Then, surrogate model, which is used to approximate the unknown implicit function or high-fidelity finite element analysis (FEM) process, is also known as a way to reduce the computation cost Kriging approximation is widely applied as an efficient and accuracy method and makes it possible for reliability analysis.9,16–20 In the optimization process, parameterization provides a rapid and automated manipulation of the analysis model A high-quality parameterization has two conflicting objects: (1) ensure a bigger design space and (2) avoid any failure in establishing model A bigger design space could lead to a higher possibility of modeling failure Through a further study of the geometry characteristic of the TWD, a new parameterization approach is proposed in this article and could reduce the error rate of modeling to 0% and amplify the design space by at least 50% compared to our previous work.21 Therefore, we obtained an even better optimal result In this article, a developed parameterization with a series of methods used in RBMDO, including DOE analysis, Kriging approximation, and MCS for reliability analysis, are developed to search the optimal shape of TWD with objective of minimum weight under the probabilistic constraints The thermal data are transferred to the structural analysis by the inverse distance weighted (IDW) interpolation method Then, the deterministic MDO is also conducted as a comparison The RBMDO procedure proposed in this article can be an inspiration and reference for researchers and designers in designing of the TWD disk Proposed methodology Figure The deterministic MDO framework subject to : giR ðd, p, yðd, pÞÞ ! 0, i = 1, , Nhard gjD ðd, p, yðd, pÞÞ ! 0, j = 1, , Nsoft dl d du ð1Þ where d are the design variables and p are the constant parameters giR is the ith hard constraint that models the ith critical failure mechanism of the system (e.g stress, deflection, and loads) giD is the jth soft constraint that models the jth deterministic constraint (e.g cost and marketing) The design space is limited by dl and du For the example of the three-discipline system, the framework is shown in Figure This method is also known as multidisciplinary feasible (MDF) method A conservative safety margin of the deterministic designs is required to ensure design safety However, these measures may be not sufficient to provide information on design reliability Based on this MDO method, the hard constraints are replaced with reliability constraints in RBMDO minimize : f ðd, p, yðd, pÞÞ, d = ðd1 , d2 , , dn Þ À Á Pf , i , i = 1, , Nhard subject to : PriR giR ðdà , pÃ Þ ! gjD ðd, p, yðd, pÞÞ ! 0, j = 1, , Nsoft dl d du ð2Þ Ã where d andpà are the random vectors for design variables and system parameters, respectively; PriR is ith probabilistic constraint; and Pf , i is probability of allowable failure of ith constraints, as shown in Figure Review of the RBMDO Multidisciplinary systems are characterized by two or more disciplinary analyses The solution of these coupled disciplinary analyses is referred to as a system analysis A typical deterministic MDO problem can be formulated as follows22 minimize : f ðd, p, yðd, pÞÞ, d = ðd1 , d2 , , dn Þ Monte Carlo simulation In reliability analysis, the state limit function g is defined as g ðX1 , X2 , , Xn Þ = ð3Þ where g is the performance function and X is the vector of random variable Failure event is therefore defined Zhang and Gou Figure The framework of the random RBMDO as g \ Monte Carlo simulation (MCS) is a powerful and simple tool for evaluating the reliability of complicated engineering problems, especially when the limit state function is implicit With MCS, performance function is executed in a considerable number N, then the probability of failure is expressed as Pf = Nf =N ð4Þ where Nf is the number of failure events The accuracy of MCS largely depends on the number of simulation cycles Its acceptance as a way to compute the failure probability depends mainly on its efficiency and accuracy According to Lian and Kim,23 there is a 95% probability that the probability of failure estimated with the MCS will fall into the range 1024 1025 with million simulations Kriging model Kriging surrogate model is widely used in approximating finite element model (FEM) It can be written as a combination of a regression model and a random process y ð x Þ = P ð x Þ + zð x Þ ð5Þ where y(x) is the unknown polynomial function of x, P(x) is a known polynomial function of the n-dimensional variable x, and Z(x) is the realization of a normally distributed stochastic process P(x) approximates the global design space, while Z(x) relates to the localized deviations However, the initial Kriging model cannot be used directly for its unacceptable error Therefore, additional study points located in the region of interest are selected for learning and then rebuild the surrogate model in each iteration This method increases the predictive accuracy of the surrogate model in the points of interest while sacrificing the accuracy in other region.24,25 In this article, the interest region is the region of lower weight of TWD The additional points for rebuilding the Kriging model are selected by the possibility of existing in this region According to the previous works,17,20 in order to set the convergence tolerance e without considering the magnitudes of responses, the convergence criterion is chosen as   wðxi Þ À w ^ ðxi Þ  MAEr = max \e i w ðx i Þ ð6Þ MAEr is the relative maximum average error For clarification, the overall procedure of constructing the Kriging surrogate model is organized as the following steps: Step Generate the initial sample points by Latin Hypercube technique Step Calculate the response at all the initial points using high-fidelity solver, such as FEM Step Construct the initial Kriging surrogate model based on all the sample points and its corresponding responses 4 Advances in Mechanical Engineering Table Deterministic design variables Parameters Symbols Lower bounds (%) Upper bounds (%) Height of the rim (removed) Height of the hub H_RIM H_HUBOUT H_HUB W_RIM W_HUBOUT W_GAP R_NECK AL_WEB AL_THWEB 210 210 210 210 210 210 210 210 210 10 10 10 10 10 10 10 10 10 Width of the rim Width of the hub gap Radius of the neck Angle of the webs After the correctness analysis of the parameterized model, the new parameterization approach could reduce the error rate of modeling to 0% and amplify the design space by at least 50% compared to our previous work In this article, model is established with the three kinds of parameters: (1) the design variables (shown in Table 1), (2) the random parameters (shown in Table 2), and (3) the constant parameters The model of the TWD for fluid, thermal, and structural analysis and all the design conditions are based on previous work.21 Random variables The schematic drawing of the TWD, including the solid and fluid region for fluid, thermal, and structural analysis, is shown in Figure and all the design conditions are based on previous work.21 A stochastic coefficient number is introduced by a linear formula ki = aki ði = 1, 2, , nÞ Figure The design variables of the TWD in developed parameterization Step Searching the optimal point using the constructed Kriging model A number of optimal points set can be then used as the additional learning points Step Calculate the actual responses of the additional points and check the convergence If satisfy, stop; otherwise, add these points into the sample points set and go to step ð7Þ where ki is the material value for the ith temperature point and n is the number of temperature point in material test aE , ap, aTC , aTE are the elastic modulus, Poisson ratio, thermal conductivity, and expansion, respectively Besides, the design variables are also the normal distribution, where the mean value is the value in each optimization iteration The material used in this model is GH4169 Table shows the random variables in material and operation conditions The coefficient of variation of all the random design variables is set as 5% Constraints and objective RBMDO for TWD Developed parameterization method of the TWD According to the concept of Brujic et al.,26 the developed parameterization method is shown in Figure Generally, the safety requirements of the aero turbine disk are set as follows:3 Maximum hoop stress at disk hub scmax ; Maximum radial stress of web srmax ; Zhang and Gou Table Random variables in material and operation conditions Probability variables Symbols m CV s Density, kg/m3 Blade tension, MPa Rotational speed, r/min Outlet temperature, °C Coolant temperature, °C Coolant mass flow rate, kg/h Elastic coefficient Poisson coefficient Thermal conductivity coefficient Thermal expansion coefficient r s n Tout Tair _ m kE kP kTC kTE 8240 170 9726 650 350 300 1 1 5% 5% 5% 5% 5% 5% – – – – 412 8.5 486.3 32.5 17.5 15 0.01 0.01 0.01 0.01 CV: coefficient of variation find : x = xðx1 , x2 , , xn Þ > > > > > > : W ð xÞ > > > > > s:t: : scmax \½s0:2cmax Š > > > < s \ ½s rmax 0:2rmax Š P: >s  bc Š  c \½s > > > > > s  r \ ½s  br Š > > > > > > smax \½smax Š > > : xlb x xub Figure Schematic drawing of the TWD disk optimization models White region: solid domain; gray region: fluid domain Average hoop stress on meridian plane s  c (to ensure the disk working under the burst speed in meridian plane); Average radial stress of web s  r (to ensure the disk working under the burst speed in cylindrical plane); Maximum von Mises stress smax In the deterministic optimization, the safety factors of the strength limits or the yield limits of the material are often considered as the deterministic constraints Therefore, the key factor that influences the optimal results is how to select the safety factors The bracket ‘‘[ ]’’ indicates the value with the consideration of safety factor The deterministic constraints of the MDO can be then set as follows ð8Þ where x stands for the design variables and W(x) is the weight of TWD Based on standard, the reliability of stress is required to be greater than 0.999 Therefore, the probabilistic optimization problem is find : x = xðx1 , x2 , , xn Þ > > > > > > : W ð xÞ > > > > > s:t: : P(scmax \s0:2 ) ! 0:999 > > > < P(s \s ) ! 0:999 rmax 0:2 P: ð9Þ > P( sc \sb ) ! 0:999 > > > > > sr \sb ) ! 0:999 > P( > > > > P(smax \s0:2 ) ! 0:999 > > > : xlb x xub Block process building DOE analysis for initialization of the start point can accelerate optimization convergence by decreasing the number of variables The commerce software CATIA is used for geometry parameterization The design variables, mainly related to the shape of the TWD, are changed by ISIGHT optimization software and FORTRAN The mesh is developed in ICEM for thermo-fluid analysis and in PATRAN for structural analysis ANSYS CFX and MSC Nastran are used for thermo-fluid and structural analysis IDW method is used for data transfer in coupling Advances in Mechanical Engineering Figure System optimization framework disciplines.27 Due to the small amounts of design variables, the MDF method is adopted as the MDO system In the reliability loop, the random variables and reliability constraints are obtained by MCS After the optimization, we calculate the probable optimal points using FEM Figure shows the overall optimization framework Results DOE analysis In this part, the sensitivities of both design variables and random variables are analyzed by DOE The design variables The design variables, which manipulate the shape of the TWD, are changed during the optimization A total of 25 sample points are selected by the Latin Hypercube method The Pareto effects of design variables on (a) max von Mises stress and (b) weight are shown in Figure The changes in the angle of web mostly influence the stress, and the width of disk rim has great effects on disk weight They instruct us to amplify the design space of these variables in optimization process The random variables The random variables, which mostly manipulate the working condition and the material properties of the TWD, are changed during the MCS A total of 25 sample points are selected by the Latin Hypercube method The Pareto effects of random variables on (a) max von Mises stress and (b) weight are shown in Figure The rotational speed of the disk and the density of the material have the positive effects on disk stress Because a higher speed means a larger centrifugal stress F = mv2 r ð10Þ And with the same volume V, higher density r means a higher mass m m = rV ð11Þ Zhang and Gou Figure Pareto effects of design variables on (a) max von Mises stress and (b) weight Figure Pareto effects of random variables on (a) max von Mises stress and (b) weight According to equation (4), the higher density also means a larger centrifugal stress Only the density value of the material influences the disk weight Based on equation (5), the density and mass are linear correlation Based on the results, the Poisson coefficient kp is removed from the random variables because it has almost no effect on neither the stress nor the weight of the TWD We keep it constant as its mean value during the whole MCS Kriging surrogate model error analysis Kriging surrogate model is established by the MATLAB toolbox DACE.28 After the DOE analysis, eight design variables and nine random variables are used as the inputs to establish the Kriging surrogate model The high-fidelity process includes mesh generation, thermal fluid analysis, temperature interpolation and structural analysis With 32 core central processing unit (CPU), each calculation lasts for about 11.5 The mean iteration number of the computation for thermal fluid using ANSYS CFX is about 400 A total number of calling FEM is 410 for constructing Kriging surrogate The initial sample points set are also generated by Latin hypercube method Then, the genetic algorithm (GA) is selected as the optimization technique After 11 optimized iterations, the final Kriging model is constructed It spends about 72 h on constructing the Kriging model Then, we select the most two effective variables on Mises stress and weight for error analysis, shown in Figures and The response surface is built by Kriging model A total of 40 actual points are selected in Latin Hypercube method shown as the dots in the following figures Density is the only one random variable that influences the disk weight, so Figure 9(b) shows them in a linesymbol two-dimensional (2D) figure All the figures show that the accuracy of Kriging model can be acceptable 8 Figure Kriging model error analysis for deterministic design variables: (a) effects on von Mises stress and (b) effects on weight Optimization results After 150 optimized iterations, the optimal results are obtained The optimal searching history by RBMDO is shown in Figure 10 Where the red dot indicates that this design point is infeasible and the green dot means the optimal point We can observe that RBMDO obtains the minimum weight of the disk It demonstrates that the common deterministic optimization is a conservative method Then, the high-fidelity FEM is used to calculate the optimal point and several feasible points around the optimal one obtained by Kriging surrogate model Among these probable design points obtained by surrogate model, the Max von Mises Stress of RBMDO is beyond the deterministic upper limit, which is infeasible in deterministic optimization But in fact, the stress is not beyond the material upper strength limits The probability-based optimization does not depend on a deterministic safety factor and shows its advantages in optimal point searching In this study, the stricter safety factor is used, so this RBMDO’s optimal point is not feasible in MDO One optimal point for each method Advances in Mechanical Engineering Figure Kriging model error analysis for random variables: (a) effects on von Mises stress and (b) effects on weight (RBMDO or MDO) is then obtained, respectively Then, three design points, including the start point, the MDO’s and the RBMDO’s optimal point, are studied in the following part Figures 11–13 show the stress distribution of the three design points The MDO and RBMDO both can decrease the maximum stress The figures of the MDO’s optimal point show that the web is the most probable failure region, especially in the region of junction between the rim and the web (shown in Figure 11(a) and (c)) Therefore, the uniform stress distribution and enhancement should be considered And the figures of the RBMDO’s optimal point show that the stress distribution is ameliorated in RBMDO’s optimal point RBMDO obtained a smaller maximum von Mises stress in the region of disk hub and a more uniform stress distribution in web This development is brought by the decrease in hub weight and the increase in web thickness Table shows the details of the optimal design results of the three points All the parameters are normalized in the further study The AL_THWEB and the H_HUB are the most different between the two optimal points With the thicker web and the higher hub, Zhang and Gou MDO and RBMDO method can be decreased by 36.06% and 44.57%, respectively Besides, the von Mises stress of the MDO and RBMDO can be decreased by 13.79% and 15.67% The reliability analysis results in start and the optimal points are obtained by MCS and Kriging surrogate model The maximum iterative number for Monte Carlo is 105 Table shows the mean value, standard deviation, and the reliability of the three points The reliability of maximum hoop stress and radial stress in the start point and the reliability of maximum radial stress in MDO’s optimal point are beyond the constraints while the reliability of all responses of the RBMDO’s optimal point satisfies the probability constraints It demonstrates that the deterministic optimal point would be infeasible for reliability requirement when the lower safety factor is selected Besides, the standard deviation of RBMDO’s optimal point is the lowest, which shows a steady product quality Conclusion Figure 10 Optimal searching history of (a) MDO and (b) RBMDO the RBMDO’s optimal point obtains the lighter weight, compared to the MDO method Table shows the rate of change in all the responses and objectives Based on the start point, the weight by The use of the novel TWD can decrease the weight up to a maximum of 44.47% based on this study, which is significant for the turbine performance However, as a crucial component, the high-pressure turbine disk of aerocraft needs a high reliability and a steady quality at the same time The traditional deterministic MDO method sometimes could not be capable to satisfy both the requirements and depends heavily on the selection strategy of safety factor In this article, the RBMDO method and common MDO method are both adopted to search the minimum TWD’s weight The probable and determinate constraints are integrated in the optimization process Using the Kriging model, we get several probable optimal points Then, after high-fidelity FEM calculation, the final optimal points are obtained Some important conclusions are listed as follows: (1) after DOE Figure 11 Stress distribution of the start point: (a) radial stress, (b) hoop stress, and (c) von Mises stress 10 Advances in Mechanical Engineering Figure 12 Stress distribution of the MDO’s optimal point: (a) radial stress, (b) hoop stress, and (c) von Mises stress Figure 13 Stress distribution of the RBMDO’s optimal point: (a) radial stress, (b) hoop stress, and (c) von Mises stress Table Optimum design results of RBMDO and MDO Design variables Constraints and response Object Parameters symbols Start point Optimal point (MDO) Optimal point (RBMDO) AL_WEB AL_THWEB H_HUBOUT H_HUB R_NECK W_RIM W_HUBOUT W_GAP s h s r shmax srmax smax Mass 0.5000 0.5000 0.5000 0.8333 0.5000 0.6667 0.7333 0.4444 0.2376 0.1617 0.4576 0.6343 0.4353 0.9393 0.6400 0.4800 0.5600 0.2733 0.7000 0.0867 0.6333 0.4444 0.1740 0.0642 0.0070 0.7239 0.0587 0.3000 0.5800 0.1800 0.3800 0.0933 0.5000 0.1467 0.4733 0.6244 0.0098 0.1795 0.2966 0.0125 0.0073 0.1492 RBMDO: reliability-based multidisciplinary design optimization; MDO: multidisciplinary design optimization Zhang and Gou 11 Table Optimum design results of RBMDO and MDO (baseline: start point) Optimal point (MDO) Optimal point (RBMDO) s h s r shmax srmax smax m # 8.16% # 29.21% # 14.11% " 2.58% # 25.70% # 10.74% " 13.59% # 10.08% # 13.79% # 15.67% # 36.06% # 44.57% RBMDO: reliability-based multidisciplinary design optimization; MDO: multidisciplinary design optimization Table Reliability analysis results Start point s h s r shmax srmax smax m Optimal point (MDO) Optimal point (RBMDO) Mean SD Pr Mean SD Pr Mean SD Pr 0.2147 0.0066 0.7136 0.8871 0.4997 0.9721 25.895 16.898 74.57 59.811 66.588 0.1756 1 0.9988 0.9946 – 0.1935 0.0292 0.2462 0.9707 0.3153 0.2500 19.695 12.464 85.029 66.354 53.946 0.24578 1 0.9747 – 0.2090 0.0275 0.3515 0.5810 0.2850 0.0417 15.978 9.6129 68.465 43.604 48.287 0.2921 1 1.0000 – RBMDO: reliability-based multidisciplinary design optimization; MDO: multidisciplinary design optimization Bold significance that, the reliability values is smaller than the safety value (0.999), which indicates that the design point locates in the failure domain analysis, some crucial design variables (the width of rim and the angle of webs) and random variables (the density and rotational speed) are obtained, which is of significant effects on alleviating the optimization cost; (2) the accuracy of the Kriging model proposed in this article is dependable according to the error analysis; (3) with the thicker web and the higher hub, the lighter weight is obtained by the RBMDO method, compared to the MDO method; (4) based on the start point, the weight by MDO and RBMDO method can be decreased by 36.06% and 44.57%, respectively Besides, the von Mises stress of the MDO and RBMDO can be decreased by 13.79% and 15.67%; and (5) the standard deviation of RBMDO’s optimal point is the lowest, which shows a steady product quality Because of the universality of MCS and the efficient surrogate model, the RBMDO method can be used in wide range without any particular restriction Above all, the RBMDO provides a better way to meet the requirement of optimal minimum weight and steady quality for the TWD design The probability design for TWD could be a reference for further industrial product design Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article Funding The author(s) received no financial support for the research, authorship, and/or publication of this article References Cairo RR and Sargent KA Twin web disk: a step beyond convention J Eng Gas Turb Power 2002; 124: 298–302 Shen XL and Dong SJ Structure optimization and welding residual stress analysis of twin-web turbine disc Adv Mat Res 2013; 622–623: 309–314 Lu S and Zhao L Structural optimization design method of twin-web turbine disk with tenon J Aero Power 2014; 29: 875–880 Zhang CQ, Huang WZ, Liu XW, et al Design and optimization of low inertia turbine rotor structure Gas Turbine Exp Res 2013; 26: 33–36 Sobieszczanski-Sobieski J and Haftka RT Multidisciplinary aerospace design optimization: survey of recent developments Struct Multidiscip O 1997; 14: 1–23 Arsenyev I, Duddeck F and Fischersworring-Bunk A Adaptive surrogate-based multi-disciplinary optimization for vane clusters In: ASME turbo expo, Montreal, QC, Canada, 15–19 June 2015 Ashakiran GB Multi-disciplinary design optimization of a axial turbine stage of aircraft auxiliary power unit In: AIAA/ISSMO multidisciplinary analysis and optimization conference, Victoria, BC, Canada, 10–12 September 2008 Barsi D, Perrone A, Ratto L, et al Radial inflow turbine design through multi-disciplinary optimisation technique In: ASME turbo expo 2015: turbine technical conference and exposition, Montreal, QC, Canada, 15–19 June 2015 New York: ASME Dubourg V, Sudret B and Bourinet JM Reliability-based design optimization using Kriging surrogates and subset simulation Struct Multidiscip O 2011; 44: 673–690 10 Meng D, Huang HZ, Wang Z, et al Reliability-based multidisciplinary design optimization using subset simulation analysis and its application in the hydraulic 12 11 12 13 14 15 16 17 18 Advances in Mechanical Engineering transmission mechanism design J Mech Design 2015; 137: V004T06A016 (10 pp.) Meng D, Zhang X, Yang Y, et al Interaction balance optimization in multidisciplinary design optimization problems Concur Eng 2016; 24: 48–57 Zhu SP, Huang HZ, Peng W, et al Probabilistic physics of failure-based framework for fatigue life prediction of aircraft gas turbine discs under uncertainty Reliab Eng Syst Safe 2016; 146: 1–12 Yao W, Chen X, Ouyang Q, et al A reliability-based multidisciplinary design optimization procedure based on combined probability and evidence theory Struct Multidiscip O 2013; 48: 339–354 Zhu SP, Huang H, Smith R, et al Bayesian framework for probabilistic low cycle fatigue life prediction and uncertainty modeling of aircraft turbine disk alloys Probabilist Eng Mech 2013; 34: 114–122 Madsen JI, Wei S and Haftka RT Response surface techniques for diffuser shape optimization AIAA J 2012; 38: 1512–1518 Elsayed K and Lacor C Robust parameter design optimization using Kriging, RBF and RBFNN with gradientbased and evolutionary optimization techniques Appl Math Comput 2014; 236: 325–344 Gao Y, Wang X, Wang Y, et al A sequential optimization method based on Kriging surrogate model In: Proceedings of the 2011 fourth international joint conference on computational sciences and optimization, New York, 15–19 April 2011 New York: IEEE Jeong S, Murayama M and Yamamoto K Efficient optimization design method using Kriging model J Aircraft 2005; 42: 413–420 19 Lam XB, Kim YS, Hoang AD, et al Coupled aerostructural design optimization using the Kriging model and integrated multiobjective optimization algorithm J Optimiz Theory App 2009; 142: 533–556 20 Wang H, Zhu X and Du Z Aerodynamic optimization for low pressure turbine exhaust hood using Kriging surrogate model Int Commun Heat Mass 2010; 37: 998–1003 21 Zhang M, Gou W, Li L, et al Multidisciplinary design and optimization of the twin-web turbine disk Struct Multidiscip O 2016; 53: 1129–1141 22 Agarwal H Decomposition approach for reliabilitybased multidisciplinary design optimization In: AIAA/ ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, Norfolk, VA, 7–10 April 2003 Reston, VA: AIAA 23 Lian Y and Kim N-H Reliability-based design optimization of a transonic compressor AIAA J 2006; 44: 368–375 24 Huang D, Allen TT, Notz WI, et al Sequential Kriging optimization using multiple-fidelity evaluations Struct Multidiscip O 2006; 32: 369–382 25 Huang D, Allen TT, Notz WI, et al Global optimization of stochastic black-box systems via sequential Kriging meta-models J Global Optim 2006; 34: 441–466 26 Brujic D, Ristic M, Mattone M, et al CAD based shape optimization for gas turbine component design Struct Multidiscip O 2010; 41: 647–659 27 Franke R Scattered data interpolation: tests of some methods Math Comput 1982; 38: 181–200 28 Lophaven SN, Nielsen HB and Søndergaard J DACE A Matlab Kriging toolbox, version 2.0, 2002, http:// www2.imm.dtu.dk/projects/dace/dace.pdf