Computers and Structures 89 (2011) 27–37

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Computers and Structures

journal homepage: www.elsevier .com/locate /compstruc

Optimal design of aeroengine turbine disc based on kriging surrogate models

Zhangjun Huang a,b,⇑, Chengen Wang b, Jian Chen a, Hong Tian a

a School of Energy and Thermal Power Engineering, Changsha University of Science and Technology, Changsha 410114, Chinab Key Laboratory of Process Industry Automation, Northeastern University, Shenyang 110004, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 3 August 2009Accepted 30 July 2010

Keywords:Turbine discOptimal designSurrogate modelKriging methodDifferential evolutionary algorithmDesign of experiments

0045-7949/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.compstruc.2010.07.010

⇑ Corresponding author at: School of Energy andChangsha University of Science and Technology, Chan731 82309461; fax: +86 731 82618242.

E-mail address: zhangjunhuang@gmail.com (Z. Hu

A design optimization method based on kriging surrogate models is proposed and applied to the shapeoptimization of an aeroengine turbine disc. The kriging surrogate model is built to provide rapid approx-imations of time-consuming computations. For improving the accuracy of surrogate models without sig-nificantly increasing computational cost, a rigorous sample selection is employed to reduce additionaldesign samples based on design of experiments over a sequential trust region. The minimum-mass shapedesign of turbine discs under thermal and mechanical loads has demonstrated the effectiveness and effi-ciency of the presented optimization approach.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In advanced propulsive systems, a turbine disc bears vastmechanical and thermal loads under its working conditions of hightemperature gradients and high rotational velocity, which may in-duce intensive stresses and dangerous damages. A significantobjective of the shape optimization of the turbine disc is to mini-mize its mass subject to constraints on the stresses and some otherpractical conditions. Clearly, an effective optimization method willbe valuable to enhance the quality of the turbine disc and hence toimprove the engine specific thrust, thrust-to-weight ratio, and sys-tem reliability.

With the growth in computing power of current computers,computationally expensive finite element (FE) method hasbecome a common and important technique in the product devel-opment process, and a large number of FE codes, including com-mercial packages and in-house codes developed have beenmainly used for function evaluations (evaluations of the objectiveand/or constraint functions), such as stress analysis, thermal anal-ysis, vibration analysis and fatigue life estimates in the design andoptimization of aeroengine discs [1–4]. However, the optimizationdesign of a complex system like a gas turbine often involvesexploring a broad design space. This requires analyzing large num-bers of design points. If all evaluations of these designs are per-formed using computationally expensive FE method, it will lead

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ang).

to an excessive computational cost and therefore an impracticalruntime of the optimization process.

One alternative is to construct a simple surrogate model toapproximate the response of the costly FE solvers. The surrogatemodel expresses the relationship between the objective or con-straint functions and the design variables with simple form equa-tions. The surrogate model can be used to cover regions in thedesign space for which a solution cannot practically be obtained.In addition, the use of surrogate model often requires only a smallnumber of expensive FE analyses and can reduce significantly thecomputing time in obtaining the optimal design. Therefore, theapproximation approach has been widely applied to engineeringoptimization problems so as to reduce the computational cost.

There are several different categories surrogate models (alsocalled meta-models or approximate models) for engineering de-sign problems [5,6], including the polynomial-based response sur-face model [7,8], the neural networks (NNs) based surrogate model[9,10] and the kriging model [11–13].

A polynomial-based response surface model is a most widelyused surrogate model due to its simplicity and effectiveness. The re-sponse surface method uses least-squares regression analysis to fitlow-order polynomials to a set of experimental data. Because thepolynomial-based response surface model normally requires theassumption of the order of the approximated base function, the de-signer must evaluate the schematic shape of the objective functionover an entire solution space. This will sometimes be difficult sinceit requires an understanding of the qualitative tendency of the en-tire design space. Besides, the model function is typically chosento be first- or second-order polynomials, because a higher-orderpolynomial not only tends to show severe oscillations but also

28 Z. Huang et al. / Computers and Structures 89 (2011) 27–37

requires too many support points [14]. This may result in limitedaccuracy of the response surface model when the response data tobe modeled have multiple local extrema. Therefore, the polyno-mial-based response surface model is likely awkward when it isused for representing multi-modalities and non-linearity com-monly appeared in complex engineering problem [15].

Neural networks are inherently massive parallel computationalsystems comprised of simple nonlinear processing elements withadjustable interconnections. The predictive ability of the networkis stored in inter-unit connection strengths called weights obtainedby a process of adaptation and learning from a set of design train-ing data. Training of a network requires repeated cycling throughthe data and continues until the error target is met or until themaximum number of neurons is reached. So, neural networks arequite powerful and flexible when it comes to handling complexinteracting functions, irregular, non-smooth, discontinuous or non-linear design spaces. Due to the feature of neural networks, theNNs-based surrogate model has good performance in predictionaccuracy for complex engineering problems. However, NNs-basedmodel presents some practical difficulties. For example, this moderequires a good many sample points and much computation timefor the training of NNs [16].

Recently, kriging models have drawn much attention and beenwidely used in a variety of applications such as structural optimi-zation [17,18], multidisciplinary design optimization [19,20] andaerospace engineering [11,21]. This type of models predicts the va-lue of the unknown point using stochastic processes. Sample pointsare interpolated with the Gaussian random function to estimatethe trend of the stochastic processes. The model has a sufficientflexibility to model response data with multiple local extremaand to represent the nonlinear and multimodal functions, thoughthis flexibility is obtained at an increase in computational expenseand a decrease in ease of use [17,22].

One important issues associated with adaptive optimizationstrategies based on the knowledge obtained from the surrogatemodel is the sequential updating of an approximate model withadditional data, which is also called sequential sampling. Variousmethods for sequential sampling in the design space have beenstudied in recent years. Space-filling designs and the expectedimprovement concept were introduced by Jones et al. [23] as anefficient sequential sampling strategy in the efficient global opti-mization (EGO) algorithm, where the expected improvement crite-rion is used for the balance between local and global search. For theinfill sample selection in the global optimization of stochasticblack-box systems, Huang et al. [24] adopted an augmented ex-pected improvement function with desirable properties for sto-chastic responses. Besides, Lee and Kang [18] used other twosequential sampling strategies simultaneously for improving theaccuracy of kriging model. One was to select a new sample pointby maximizing the mean square error of the initial kriging model,and the other was to select the new sample point as a stationarypoint. What’s more, Farhang-Mehr and Azarm [25] introduced asequential maximum entropy design approach so that the re-sponse function behavior could be automatically adapted byemphasizing the irregular regions of the design space and design-ing the next set of experiments accordingly. In addition, Xiong et al.[26] introduced a sequential sampling procedure that use designconfidence as a metric to assist designers in making decisionregarding when to terminate the sampling process so that the cur-rent optimal design can be accepted as the ‘‘true” optimal solutionwith desired confidence.

The other important issue associated is the framework for themanagement of surrogate models. Frameworks based on trust re-gions and gradient-based search procedures have attracted muchattention in the past few decades [27–29]. These rigorous frame-works guarantee convergence to a model local optimum and

work with nonlinear programming techniques or direct-searchmethods. Besides, several attempts have been made to tacklethe problem of using surrogate models with evolutionary searchmethods. Ratle [30] proposed a simple local convergence criterionto decide when the exact model should be resorted to in a proce-dure integrating a genetic algorithm with kriging models. How-ever, this does not prevent the search from converging to falseoptima. Jin et al. [31] proposed a framework for coupling evolu-tionary strategy and NNs-based surrogate models. Two types ofevolution control methods were presented to decide the fre-quency at which the exact model should be used. Song and Keane[32] coupled a real-coded genetic algorithm with a kriging surro-gate model in order to reduce computational cost without sacri-ficing the ability of the GA in finding the global optimum forcomplex landscapes, using a new approach based on the posteriorvariance estimate to suggest new sample points for re-evaluationusing exact models. New sample points obtained were insertedinto an ordered database storing all the exact solutions evaluatedso far, and the surrogate model was updated when these newpoints felled into the section of the dataset used in the construc-tion of the surrogate model.

In this work, we explore a method combining a robust archiveddifferential evolution (RADE) algorithm [33] with kriging surrogatemodels so as to reduce the total number of expensive functionevaluations and therefore the computation effort in the shape opti-mization of aeroengine turbine disc. The kriging model is con-structed on the base of data collected by evaluating the objectiveand constraint functions at a few initial points, and afterward up-dated gradually with additional sample points. At each iterationthe kriging model constructed is invoked repeatedly by the robustarchived differential evolution algorithm to estimate the locationof the optimum and suggest points where additional function eval-uations may help improve this estimate. A new additional samplepoint will be analyzed and inserted into the design dataset for theupdate of the kriging model, until the kriging model is sufficientlyaccurate and the optimization process is converged.

2. Problem formulation

2.1. Optimization model

The turbine disc can be simplified as an axi-symmetric rotatingdisc with a centric bore. The circular cylindrical coordinates (r, h, z)are adopted for the convenience of description and analysis in thispaper, where the symmetric axis z and the axial direction of theturbine disc are consistent with each other. The half-axial crosssection of the turbine disc is shown in Fig. 1, where the disc shapeis defined with several geometric parameters, including the deadrim radius (R1), bore radius (R2), web outer radius (R3), web innerradius (R4), dead rim width (W1), bore width (W2), web outer width(W3), web inner width (W4), dead rim height (H1) and bore height(H2).

Among these geometric parameters above, the dead rim radius(R1), bore radius (R2), dead rim width (W1) and dead rim height(H1) can commonly be predetermined according to requirementson other components such as the turbine blade and the gas flowpassage. The other six geometric parameters are identified as de-sign variables and their ranges are given in Table 1.

Constraints for the turbine disc optimization consist of size con-straints and performance constraints. The size constraints limit thevariable ranges. The performance constraints, such as life-span,stress and structural transformation, are always set according toa design rule. In this paper, we consider the radial stress and cir-cumferential stress in the disc as the performance constraints forminimizing the disc mass.

Fig. 1. The half-axial cross section of the turbine disc.

Table 1Ranges of design variables (mm).

Design variable Lower bound Upper bound

Bore width (W2) 80.0 100.0Bore height (H2) 15.0 40.0Web outer width (W3) 12.0 17.0Web outer radius (R3) 202.0 210.0Web inner width (W4) 15.0 19.0Web inner radius (R4) 150.0 180.0

Z. Huang et al. / Computers and Structures 89 (2011) 27–37 29

Considering permissible stresses and limited variable ranges,we can set up the design optimization problem as a standard opti-mization problem of determining shape variables that minimizedisk mass subject to constraints related to material strength andshape. The optimization problem can be formulated as

Minimize Mass ¼ f ðW2;H2;W3;R3;W4;R4Þ

Subject to :

rrmax 6 ½rr�;rhmax 6 ½rh�;80:0 6W2 6 100:0; 15:0 6 H2 6 40:0;12:0 6W3 6 17:0; 202:0 6 R3 6 210:0;15:0 6W4 6 19:0; 150:0 6 R4 6 180:0;

8>>>>>><>>>>>>:

ð1Þ

where rrmax is the maximum radial stress, rhmax is the maximumcircumferential stress, [rr] denotes the maximum permissible radialstress, [rh] denotes the maximum permissible circumferentialstress, f denotes the objective function for calculating the disc mass.

2.2. Basic equations for stress-calculation

The displacement, strain and stress in an axi-symmetric turbinedisc can be described in circular cylindrical coordinates asd ¼ ½u; t�T ; e ¼ ½er; eh; ez; crz�

T and r ¼ ½rr ;rh;rz; srz�T respectively,where u denotes radial displacement, t denotes circumferentialdisplacement, u denotes radial displacement, er denotes radialstrain, eh denotes circumferential strain, ez denotes axial strain,crz denotes shearing strain in r–z plane, rr denotes radial stress,rh denotes circumferential stress, rz denotes axial stress and srz de-notes shearing stress in r–z plane. And the basic relation between

strains and displacements in the axi-symmetric turbine disc canbe given by

½er; eh; ez; crz�T ¼ @u

@r;@t@z;ur;@t@rþ @u@z

� �T

: ð2Þ

On the assumption that the material of the turbine disc hasthermal and elastic isotropy, the relationship between the strainand stress in the axi-symmetric turbine disc is

rr

rh

rz

srz

26664

37775¼ Eð1� mÞð1þ mÞð1�2mÞ

1 m1�m

m1�m 0

m1�m 1 m

1�m 0m

1�mm

1�m 1 00 0 0 1�2m

2ð1�mÞ

266664

377775

er

eh

ez

crz

26664

37775�

a �DT

a �DT

a �DT

0

26664

37775

0BBB@

1CCCA;

ð3Þ

where E, m and a denote Young’s modulus, Poisson’s ratio and ther-mal expansion coefficient of materials respectively. T denotes thetemperature field in the turbine disc, which will be calculated usingthe software MSC.NASTRAN according to the following method andconditions.

2.3. Calculation of the temperature field

Thermal conduction and convection are considered as two ma-jor factors effecting on the thermal loads, and thus the partial dif-ferential equation of steady heat conduction in cylindricalcoordinate system for the turbine disc can be formulated as

1r@

@rrk@T@r

� �þ 1

r2

@

@hk@T@h

� �þ @

@zk@T@z

� �¼ 0; ð4Þ

where T denotes the temperature, r is the radius coordinate, h is thecircumferential coordinate, z stands for the axial coordinate, k is theheat conduction coefficient of the turbine disc.

The heat exchange processes through boundaries of the turbinedisc include the thermal conduction between the dead rim andblade root, the thermal convection on the surface of turbine discweb and bore. The thermal convection on the surface of bore andthe thermal conduction between the dead rim and blade root arereplaced with equivalent first class of boundary conditions in thisstudy. The thermal convection on the surface of turbine disc webcan be calculated according to Newton’s law of cooling, whichhas the expression:

�kg@T@n

� �w¼ hðTw � Tf Þ; ð5Þ

where n denotes the outward normal direction of the boundary w, hdenotes the heat convection coefficient, Tw is the temperature ofwall surface, Tf is the temperature of main flow (beyond the bound-ary layer), kg is the heat conduction coefficient of gas flow and esti-mated by

kk0� T

T0

� �nk

; ð6Þ

where T0 is a constant temperature with the value of 273.15, T is theabsolute temperature of gas flow, k0 is the heat conduction coeffi-cient of gas flow at T0, nk is the temperature exponent for estimatingthe heat conduction coefficient of gas flow.

The heat convection coefficient h between the turbine disc weband the gas flow is generally determined by the Nusselt numberNu, which is calculated by

h ¼ Nu � kg=r: ð7Þ

Nusselt number Nu depends on the rotational Reynolds numberRex at the radius position r and the state of the gas flow. In thiswork the Nusselt number Nu is calculated by

30 Z. Huang et al. / Computers and Structures 89 (2011) 27–37

Nu ¼c1 � Re0:5

x ; Rex < Re0;

c2 � Re0:8x ; Rex P Re0;

(ð8Þ

where c1 and c2 are constant coefficients, Re0 is the critical rota-tional Reynolds number, Rex is the rotational Reynolds at a radiusposition r and calculated by

Rex ¼qgxr2

l; ð9Þ

where qg is the mass density of gas flow, x is the angular velocity ofrotation, l is the dynamic viscosity coefficient of gas flow and isestimated by

ll0� T

T0

� �nl

; ð10Þ

where l0 is the dynamic viscosity coefficient of gas flow at T0, nl isthe temperature exponent for estimating the dynamic viscositycoefficient of gas flow.

The turbine disc and blades have a relatively great mass densityand a quick working rotation, which gives rise to the inertial forcewithin them according to Newton’s second law. The effect from theinertial force of blades is considered as a radial tensile stress on thetop surface of the dead rim and replaced with an equivalent uni-form pressure p in this study. Because of the axial symmetry ofgeometry and boundary conditions, the circumferential displace-ment is zero. In addition, the material of this disc is the GH4169supper alloy [34]. Some important parameter settings are listedin Table 2.

3. Optimal design approach

3.1. Optimal design process

In the optimal design of aeroengine turbine disc in this work,the FE method is used for calculating the disc mass, the maximumradial stress rrmax and the maximum circumferential stress rhmax.In order to approximately estimate a design point in the three re-sponse items, the kriging model for each response item should beconstructed in the design optimization process. The optimizationprocess is shown in Fig. 2, where tmax is a user-specified maximumiteration number, S denotes a dataset of samples constructed fromobserved design points with exact responses, and t is the currentiteration number.

This optimization process starts with triggering the definition ofan optimization problem in block 1 where the designer specifiesthe inputs and outputs. And then, initial design points are gener-ated in block 2 according to the definition of input variables andtheir ranges. Because the initial design points dictate the distribu-tion of the information in design space and influence greatly thepredictive capability of kriging surrogate models, they arecommonly generated by design of experiments (DOE), such as frac-tional factorial [35], orthogonal array (OA) [36], central composite

Table 2Some important parameter settings.

Parameter Value

Temperature at outer surface of dead rim t1 (�C) 600Temperature at surface of bore t2 (�C) 300Temperature of gas flow t0 (�C) 500Rotational speed n (r/min) 12,885Critical rotational Reynolds number Re0 2.5 � 105

k0 (W/(m K)) 0.024l0 (Pa s) 0.716 � 10�4

nk 0.81nl 0.666

design (CCD) [37], face centered cubic design (FCCD) [5] and Latinhypercube design [7,23].

Each design point is evaluated subsequently in block 3 using theexpensive FE analysis so as to obtain the exact responses. These de-signs and their corresponding responses compose the initial sam-pled design dataset S (in block 4) when their evaluations arefinished. Considering that the expensive FE analysis may has twoor more outputs interested, a corresponding kriging model is con-structed in block 5 for each response variable interested using thedesign dataset S. And then a global optimization algorithm is per-formed in block 6 over the constructed kriging surrogate model(s)so as to find an approximate optimum X*.

After the approximate optimum is obtained, we examine if itbelongs to the preceding design dataset S in block 7. If the approx-imate optimum X* is very close to any existing sample point in S, itis considered to belongs to S, and a DOE plan is performed over anadjusted trust region to produce some new design points in block8. These new design points are evaluated over the current krigingmodel in block 9 and the best design point among them is selectedto replace X* in block 10. This selection of the best new designpoints is carried out according to the following criterions for thecomparison between two design points based on the estimationover kriging models:

(1) If one design point is feasible and the other is not, the formeris better than the other.

(2) If the two design points are both infeasible, the one with asmaller average of squared value(s) violated constraint(s)is better than the other.

(3) If two design points are both feasible solution, the one with alarger value of expected improvement [24] is considered as abetter design point.

What follows is to execute the exact expensive code at X* inblock 11 and examine the accuracy of the current kriging modelin block 12. If the accuracy of the current kriging model is accept-able (block 13) and the optimization process is converged (block14), we output the optimal solution by far (block 15) and terminatethe optimization process. Otherwise, we insert X* and its corre-sponding response(s) as a new observed design into the dataset S(block 18) and return to block 5 to repeat the construction of kri-ging models and optimization process unless the current iterationnumber t reaches its maximum tmax (blocks 16 and 17).

It should be noted that the current iteration number t is equal tothe number of additional designs analyzed using the FE method.Hence, the number of initial design points plus the iteration num-ber t is the total number of FE analyses, which indicates mostly thetotal computational cost.

3.2. Construction of kriging model

In the optimal design approach proposed, the kriging model[11,21,22] is used for approximate evaluation of design points.

Parameter Value

Dead rim radius R1 (mm) 237.0Bore radius R2 (mm) 83.0Dead rim width W1 (mm) 40.0Dead rim height H1 (mm) 7.5c1 0.350c2 0.018p (MPa) 131.85Maximum permissible radial stress [rr] (MPa) 680Maximum permissible circumferential stress [rh] (MPa) 790

Fig. 2. Flowchart of the proposed optimization method based on kriging models.

Z. Huang et al. / Computers and Structures 89 (2011) 27–37 31

The method for constructing and assessing kriging surrogate modelin this work is described below.

Assumed that data has been collected at n points denoted byXðiÞ ¼ ðxðiÞ1 ; . . . ; xðiÞd Þ, ði ¼ 1; . . . ;nÞ, where d is the dimension of designvector, and the associated response is denoted by y(X(i)). The un-known function may then be expressed as

yðXÞ ¼ f ðXÞ þ ZðXÞ; ð11Þ

where f(X) represents a global model, and Z(X) is a model of aGaussian and stationary stochastic process with mean of zero andvariance of r2.

The global model f(X) is always considered as an unknown con-stant b, and then Eq. (11) becomes

yðXÞ ¼ bþ ZðXÞ: ð12Þ

The covariance matrix of Z(X) is given by

covðZðXðiÞÞ; ZðXðjÞÞÞ ¼ r2M½RðXðiÞ;XðjÞÞ�; ð13Þ

where i,j e {1, . . . , n}, M denotes the n � n correlation matrix whose(i, j) entry comes from the correlation function R(X(i), X(j)) betweenthe two points X(i) and X(j).

The Gaussian function is the most commonly used in engineer-ing design as it provides a smooth and infinitely differentiable

surface. So, the Gaussian function with only a single correlationparameter h is used as the correlation function in this work. In thiscase, the correlation between X(i) and X(j) is calculated by [15]

RðXðiÞ;XðjÞÞ ¼ exp �hXd

k¼1

jxðiÞk � xðjÞk j2

!: ð14Þ

The unknown correlation parameter h can be estimated by max-imizing the log-likelihood function given by [23]

Lnðb;r2; hÞ ¼ �12

n lnð2pÞ þ n ln r2�þ ln jMj þ 1

r2 ðY � IbÞT M�1ðY � IbÞ�; ð15Þ

where I denotes an d-dimensional unit vector, and Y = [y(X(1)), . . . ,y(X(n))].

By differentiating the log-likelihood function with respect to band r2, respectively, and letting them be equal to 0, the closed-form solution for the optimal values of b and r2 can be obtainedand formulated as:

b ¼ ðIT M�1IÞ�1ðIT M�1IÞ; ð16Þ

and

32 Z. Huang et al. / Computers and Structures 89 (2011) 27–37

r2 ¼ 1nðY � IbÞT M�1ðY � IbÞ: ð17Þ

This model leads to a best linear unbiased predictor and anassociated mean squared error. The predictor of y at an untried Xcan be predicted by [32]

yðXÞ ¼ bþ rT M�1ðY � IbÞ; ð18Þ

where r(X) is the n � 1 vector of correlations whose ith element isR(X, X(i)) for i = 1, . . . , n between Z at X and at each of the n sampledpoints.And the mean squared error of the predictor is

s2ðXÞ ¼ r2 1� rT M�1rþ ð1� IT M�1rÞ2

IT M�1I

" #: ð19Þ

Once the kriging model has been constructed, its quality can beassessed according to the accuracy of predicting the original modelat unobserved locations. There are several commonly used mea-surements for the accuracy of the kriging model, including the rootmean squared error (RSME), the squared multiple correlation coef-ficient (R2) and the adjusted squared multiple correlation coeffi-cient (Adjusted R2) [5,18,38] that are used in this paper.

3.3. Searching method

Since the robust archived differential evolution (RADE) algo-rithm [33] has been demonstrated to be robust, effective, efficientand suitable for different type global optimization problems inour previous work, we use it in this paper as the global searchingmethod over the constructed kriging surrogate model(s). TheRADE algorithm uses a constraint-handling technique based ona dynamic penalty function and a fitness function of individualsto deal with various constraints effectively. Besides, an archiveof solutions and an iterative control operator are used for avoid-ing unnecessary and worthless search and speeding up the totalconvergence. A flexibility processing operator and an efficiencyprocessing operator are used in the RADE algorithm for extendingthe applicability of this method and enhancing the local searchingefficiency and the final searching quality. The flowchart of theRADE algorithm used in the optimal design approach proposedis illustrated in Fig. 3.

It is not necessary for us to tune the parameters of the RADEalgorithm when we use it, so all the parameters of the RADE algo-rithm in the optimal design approach in this work take their de-fault values suggested in [33].

Fig. 3. Flowchart of the robust archive

3.4. Design points sampling method

The choice of a proper set of sampling points for design ofexperiments is an important issue for constructing the kriging sur-rogate model. In this paper, we use the orthogonal array [36] as theDOE plan to arrange the design points, because this method canefficiently sample the design space by a very small number of de-sign points.

The DOE plan is performed over a sequential trust region thatdefines dynamically the sampling space in the optimal design ap-proach. The trust region is adjusted according to the current opti-mal point X�t and the scaling coefficient nt that is defined as

nt ¼0:618; if dt 6 0:25;1; if 0:25 < dt 6 0:75;1:618; otherwise;

8><>: ð20Þ

where t is the iteration number, and dt is calculated by

dt ¼yðX�t�1Þ � yðX�t ÞyðX�t�1Þ � yðX�t Þ

; ð21Þ

where yðX�t Þ denotes the observed objective value at the tth optimalpoint, and yðX�t Þ denotes the approximate objective value at the tthcurrent optimal point.

According to the scaling coefficient nt above, we can calculatethe trust region at the tth iteration by

Ut ¼ X�t�1 þ nt � ðUt�1 � Lt�1Þ;Lt ¼ X�t�1 � nt � ðUt�1 � Lt�1Þ;

�ð22Þ

where Lt and Ut denote the lower and upper limits of the designvariables at the tth iteration respectively.

4. Experimental study

The experimental study for assessing our proposed method con-sists of two stages. During the first stage of the experimental study,three numerical test functions [39] are employed to assess theaccuracy and efficiency of our method. The second stage includesthe optimization of the turbine disc using our proposed methodand the verification of the final optimal design obtained. All com-putations are performed on an Intel Pentium 4 desktop computer(2.93 GHz CPU and 512 MB RAM).

4.1. Numerical experiments

The first test problem, denoted by P1, is a mystery function, de-fined as

d differential evolution algorithm.

Z. Huang et al. / Computers and Structures 89 (2011) 27–37 33

Minimize f ðxÞ ¼ 2þ 0:01ðx2 � x21Þ

2 þ ð1� x1Þ2 þ 2ð2� x2Þ2

þ 7 sinð0:5x1Þ sinð0:7x1x2Þ ð23Þ

with xi 2 ½0;5�; ði ¼ 1;2Þ. It has three local optimal. One of them isthe global solution with a value of �1.4565 at x = (2.5044,2.5778).The second test problem, denoted by P2, is the Branin testfunction, defined as

Minimize f ðxÞ ¼ x2 �5:14p2 x2

1 þ5p

x1 � 6� �2

þ 10ð1� 18pÞ cosðx1Þ þ 10 ð24Þ

with x1 2 ½�5;10� and x2 2 ½0;15�. It has three global minima atx = (�3.1416, 12.2750), x = (3.1416, 2.2750) and x = (9.4248,2.4750) with an identical function value of 0.3979.

The third test problem, denoted by P3, is the six hump camel-back function, defined as

Minimize f ðxÞ ¼ 4� 2:1x21 þ

13

x32

� �x2

1 þ x1x2

þ ð�4þ 4x22Þx2

2 ð25Þ

with xi 2 ½�10;10�; ði ¼ 1;2Þ. It has six local optimal, two of whichare global solutions at x = (�0.0898, 0.7127) and x = (0.0898,�0.7127) with a value of �1.0316.

The algorithm using our optimal design approach was carriedout 10 independent runs for each of the three test problems. Whenthe objective function is evaluated as a simulation process, a 5-stime delay function is called though the time delay is much lessthan the elapsed time for most engineering simulations. The com-parison between results from approximate optimization methodand direct optimization method for the three test problems is givenin Table 3, where the approximate optimization method is ouroptimal design approach based on kriging surrogate models, the

Table 3Comparison between results from approximate optimization method and direct optimizat

Problem Method Objective value Re

P1 Approximate �1.455820 0.Direct �1.456526 0.

P2 Approximate 0.398023 0.Direct 0.397887 0.

P3 Approximate �1.031138 0.Direct �1.031628 0.

Table 4Results of the initial DOE.

No. W3 (mm) R3 (mm) W4 (mm) R4 (mm) W2 (

1 12.0 202.0 15.0 150.0 80.2 14.5 206.0 17.0 165.0 90.3 17.0 210.0 19.0 180.0 100.4 12.0 202.0 17.0 165.0 100.5 14.5 206.0 19.0 180.0 80.6 17.0 210.0 15.0 150.0 90.7 12.0 206.0 15.0 180.0 90.8 14.5 210.0 17.0 150.0 100.9 17.0 202.0 19.0 165.0 80.10 12.0 210.0 19.0 165.0 90.11 14.5 202.0 15.0 180.0 100.12 17.0 206.0 17.0 150.0 80.13 12.0 206.0 19.0 150.0 100.14 14.5 210.0 15.0 165.0 80.15 17.0 202.0 17.0 180.0 90.16 12.0 210.0 17.0 180.0 80.17 14.5 202.0 19.0 150.0 90.18 17.0 206.0 15.0 165.0 100.

direct optimization method is to search optimal solutions usingRADE algorithm directly, the objective value is the average objec-tive value of 10 optimal solutions for each test problem, the rela-tive error is the error between the average objective value andthe theoretic optimal value, and the function evaluations (FES) isthe total times that the objective functions are evaluated as simu-lations during searching for optimal solutions.

It can be seen from Table 3 that the average objective values ob-tained using approximate optimization method and direct optimi-zation method are quite close to each other for all the three testproblems. The relative errors between objective values and theo-retic optimal values are all very small (less than 0.05%). This meansthe approximate optimization method proposed has a good accu-racy for these test problems. Furthermore, the FES and elapsedtime for the three test problems by approximate optimizationmethod based on kriging surrogate models are much less than thatby direct optimization method. This indicates quite low computa-tional cost and high efficiency of the approximate optimizationmethod.

4.2. Optimization of the turbine disc

In the optimal design of the aeroengine turbine disc in this re-search, the business software MSC.PATRAN and MSC.NASTRANare used for the FE analysis of a design point when it is needed.The squared multiple correlation coefficient (R2) and the adjustedsquared multiple correlation coefficient (Adjusted R2) are bothadopted to assess kriging models. The accuracy of kriging surrogatemodel is considered to be acceptable only if the R2 and adjusted R2

are both larger than 0.9.According to the design variables defined for the optimization

problem of the turbine disc, there are six factors in the design ofexperiments. Results of the initial DOE are shown in Table 4, where

ion method.

lative error Function evaluations Time elapsed (s)

047% 27 171.00002% 1560 7817.83

031% 27 181.95003% 1512 7583.48

046% 22 122.39003% 1556 7795.40

mm) H2 (mm) Mass (kg) rrmax (MPa) rhmax (MPa)

0 15.0 32.58 714.0 639.00 27.5 41.98 578.0 593.00 40.0 53.37 598.0 642.00 40.0 47.31 796.0 691.00 15.0 41.68 623.0 720.00 27.5 38.30 735.0 677.00 40.0 47.39 832.0 648.00 15.0 36.36 715.0 720.00 27.5 42.23 617.0 729.00 15.0 38.90 733.0 777.00 27.5 46.62 714.0 682.00 40.0 41.09 641.0 704.00 27.5 40.22 872.0 712.00 40.0 41.54 658.0 732.00 15.0 42.39 640.0 699.00 27.5 42.16 798.0 717.00 40.0 43.17 663.0 655.00 15.0 39.35 762.0 794.0

Fig. 5. Half-axial cross section of the turbine disc before and after optimization.

34 Z. Huang et al. / Computers and Structures 89 (2011) 27–37

the 2nd experiment design is the baseline design with each designvariable at its middle level.

After eight iterations, the R2 and adjusted R2 are 0.9269 and0.90004, respectively, and the final optimal design is:W3 = 15.8 mm, R3 = 202.8 mm, W4 = 15.0 mm, R4 = 150.0 mm,W2 = 80.0 mm, H2 = 15.0 mm, Mass = 34.13 kg, rrmax = 649.0 MPa,rhmax = 789.0 MPa. The history of RSME, R2, adjusted R2 and discMass is illustrated in Fig. 4.

The half-axial cross sections of the turbine disc before and afteroptimization are shown in Fig. 5, where the left one is the half-axialcross section of the baseline design and the right one is the half-ax-ial cross section of the final optimal design. It can be seen fromFig. 5 that the right figure (i.e., the final optimal design Fig. 5(b))is obviously thinner than the left one (i.e., the baseline designFig. 5(a)), which accords with the results: The disc mass of the finaloptimal design is less than that of the baseline design for about18.7%.

The final optimal design is simulated using MSC.PATRAN andMSC.NASTRAN so as to calculate accurately the temperature fieldand stress distribution in the turbine disc. The distributions of tem-perature, temperature grade and thermal stress within the optimalturbine disc obtained are shown in Fig. 6. The distributions of ra-dial stress, circumferential stress and Von Mises stress under thecoupling effect of thermal loads and mechanical loads are shownin Fig. 7.

The temperature field shown in Fig. 6(a) is quite uneven at thedisc rim and bore, and the relatively high temperature gradesshown in Fig. 6(b) lie in the corners of the disc rim and bore, whichis primarily induced by the difference in temperature and thesharp corners. From Fig. 6(c) it can be seen that thermal stressesat the disc bore are relatively high and the maximum thermalstress is about 464.0 MPa.

The radial stresses within the disc rim are relatively high asshown in Fig. 7(a), and the maximum radial stress is about649.0 MPa. The circumferential stresses and Von Mises stresseswithin the disc bore are relatively high as shown in Fig. 7(b) and(c), and the maximum circumferential stress and maximum VonMises stress are about 789.0 MPa and 681.0 MPa, respectively.The maximum radial and circumferential stresses both meet thecorresponding requirements, and the maximum circumferential

Fig. 4. The history of RSME, R2, ad

stress is even about 99.9% of the maximum permissible circumfer-ential stress, which implies a high material utilization rate.

In order to assess the accuracy of the approximation based onour kriging surrogate model, we define the relative error asA�B

B

�� ��� 100%, where A denotes the value from kriging approxima-tion, and B is the value from FE analysis using MSC.NASTRAN. Com-parison between results from kriging approximation and FEanalysis for the final optimal design is given in Table 5. It can beseen from Table 5 that the relative errors at the disc mass, the max-imum radial stress and the maximum circumferential stress are allsmall (not more than 1.2%), which indicates the kriging approxima-tion for the turbine disc design has a good accuracy.

The FE analyses and elapsed time for the turbine disc design areshown in Fig. 8, where the approximate optimization method isour optimal design approach based on kriging surrogate models,and the direct optimization method is to search optimal solutionsusing RADE algorithm directly. It should be noted that the FE anal-yses and elapsed time by the approximate optimization methodare only 26 and 1489 s, respectively, while the FE analyses by thedirect optimization method will be much more than 100 accordingto our numerical experiments. Therefore, it can be seen clearlyfrom Fig. 8 that the FE analyses and elapsed time by the approxi-mate optimization method are much less than that by direct

justed R2 and disc Mass (kg).

Fig. 6. Distributions of temperature (�C) (a), temperature grade (�C/mm) (b) andthermal stress (Pa) (c).

Fig. 7. Distributions of radial stress (a), circumferential stress (b) and Von Misesstress (c).

Z. Huang et al. / Computers and Structures 89 (2011) 27–37 35

optimization method even though FE analyses by direct optimiza-tion method is underestimated to be 100 for the moment.

4.3. Discussion

From the numerical experiments and the optimization of theturbine disc above, we can make a summary according to the

experimental results. On the one hand, the approximate optimiza-tion method based on kriging surrogate models has a good accu-racy for these test problems and for the turbine disc design. Onthe other hand, the number of function evaluations (FE analyses)and elapsed time by approximate optimization method based onkriging models are much less than that by direct optimization

Table 5Comparison between results from kriging approximation and FE analysis for the finaloptimal design.

Method Mass rrmax rhmax

Kriging approximation 33.73 kg 654 MPa 790 MPaFE analysis 34.13 kg 649 MPa 789 MPaRelative error 1.2% 0.8% 0.1%

Fig. 8. The FE analyses and elapsed time for the turbine disc design.

36 Z. Huang et al. / Computers and Structures 89 (2011) 27–37

method, which indicates quite lower computational cost and high-er efficiency of the approximate optimization method since thenumber of function evaluations or FE analyses has a positive corre-lation with computational cost no matter the number of designvariables is large or not.

5. Conclusions

In this study an efficient optimization method for seeking theminimum-mass design of turbine discs under thermal andmechanical loads was proposed by combining the kriging methodand the robust archived differential evolution (RADE) algorithm.The major features and advantages of this proposed method canbe summarized as follows: First, this optimization method uses aflexible framework for the management of surrogate models sothat one or more surrogate model(s) can be simultaneously con-tributed for efficient evaluations of time-consuming objectiveand constraint functions. Thereby the proposed method is suitablefor engineering optimization problems with single output or multi-outputs. Second, this proposed method requires only a small num-ber of expensive FE analyses in the design optimization processdue to the flexible framework and a rigorous sample selectionbased on design of experiments, which can significantly reducethe total computational cost and design time. Third, the turbinedisc according to the optimal design obtained by the proposedmethod has desirable mass and reasonable distributions of tem-perature and stresses, which illustrates the feasibility and validityof the proposed method for the shape optimization problem.

Despite current work demonstrates the potential of the pro-posed method for design optimization problems with computa-tionally expensive objective and constraint functions, morestudies should be performed in our future work to compare thispromising method with other techniques in terms such as effi-ciency and robustness through solving various engineering optimi-zation problems, especially some complex design optimizationproblems with large number of design variables.

Acknowledgments

Authors of the present paper gratefully acknowledge the Pre-Research Foundation of Military Equipment of China and theNational Natural Science Foundation of China (No. 50775015) forsupporting this research.

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