evidence-based multidisciplinary design optimization with

Research ArticleEvidence-Based Multidisciplinary Design Optimization with theActive Global Kriging Model

Fan Yang 123 Ming Liu 3 Lei Li 4 Hu Ren 5 and Jianbo Wu 5

1State Key Laboratory of Mechanics and Control of Mechanical Structures Nanjing University of Aeronautics and AstronauticsNanjing 210016 China2College of Aerospace Engineering Nanjing University of Aeronautics and Astronautics Nanjing 210016 China3State Key Laboratory for Strength and Vibration of Mechanical Structures Xirsquoan Jiaotong University Xirsquoan 710049 China4Department of Engineering Mechanics Northwestern Polytechnincal University Xirsquoan 710072 China5Wuxi Hengding Supercomputing Center Ltd Wuxi 214135 China

Correspondence should be addressed to Fan Yang fanyangnuaaeducn

Received 25 March 2019 Revised 11 September 2019 Accepted 26 September 2019 Published 15 November 2019

Academic Editor Eulalia Martınez

Copyright copy 2019 Fan Yang et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

is article presents an approach that combines the active global Kriging method and multidisciplinary strategy to investigate theproblem of evidence-based multidisciplinary design optimization e global Kriging model is constructed by introducing a so-called learning function and using actively selected samples in the entire optimization space With the Kriging model theplausibility Pl of failure is obtained with evidence theory e multidisciplinary feasible and collaborative optimization strategiesof multidisciplinary design optimization are combined with the evidence-based reliability analysis Numerical examples areprovided to illustrate the eciency and accuracy of the proposed method e numerical results show that the proposed algorithmis eective and valuable which is valuable in engineering application

1 Introduction

e performance of structures in engineering system is ofteninuenced by uncertainties e optimal solution obtainedby conventional deterministic design frequently falls on theboundary of the feasible region [1] e deterministic designmay fall into the failure region if the system is disturbed byuncertainties In the multidisciplinary coupling systems thepropagation of uncertainty factors between dierent subjectsmakes the search for an optimal design more complicated aswell as dicult than the search for an optimal individualdiscipline design due to the coupling eect [2 3] e in-uence of uncertainty factors must be considered to ensurethe reliability of the optimized design results and reliability-based multidisciplinary design optimization (RBMDO)needs to be performed [4 5]

Uncertainties can be categorized as aleatory and epi-stemic [6 7] Aleatory or objective uncertainties arise fromthe inherent randomness of a system Probability theory can

be adopted to handle random uncertainties Howeversucient information is required to construct the distri-bution function of uncertain variables Epistemic or sub-jective uncertainties stem from the lack of sucientinformation during modeling and optimization Severaltheories including probability box models fuzzy setsBayesian approaches and evidence theory have been de-veloped to deal with epistemic uncertainties [8ndash15] Evi-dence theory provides a general modeling of epistemicuncertainty and can be reduced to other theories When theinterval of evidence theory is inumlnite it is equivalent totraditional probability theory When no conict exists be-tween uncertain pieces of information the theory isequivalent to possibility theory When the subinterval of theevidence variable is unique the theory degenerates intoconvex model theory

Several methods have been proposed for reliability an-alyses based on evidence theory and these approaches in-clude Cartesian product method (CPM) [16] Monte Carlo

HindawiComplexityVolume 2019 Article ID 8390865 13 pageshttpsdoiorg10115520198390865

simulation (MCS) [17ndash20] polynomial chaos expansion [21]and surrogate-based models [22ndash24] Design optimizationbased on evidence theory has also been studied [25ndash28]Zhang et al [29] adopted two-stage framework to handle theevidence-based design optimization problem where the ev-idence variables were transformed into random variables andthe sequential optimization and reliability assessment wereused He and Qu [30] gave an overview of possibility andevidence theory-based design optimization However limitedresearch has been conducted on evidence-based multidisci-plinary design optimization (EBMDO) [31ndash33] ConventionalEBMDO has a three-level loop architecture +e outer loop ismultidisciplinary optimization the middle loop is reliabilityanalysis based on evidence theory and the inner loop isdiscipline analysis +e computational cost of EBMDO ishigh especially in the inner discipline analysis Practicalengineering problems require repeatedly calling the simula-tion model to evaluate limit state and objective functions andthis process is time consuming

To address this issue this study proposes a surrogatemodel-based method with an active global learning strategywhich constructs surrogate models in entire design spacewith intelligent sampling approach to reduce the amount ofcomputation [34] +e active learning Kriging [35] model isutilized as a surrogate model to evaluate the limit statefunction in evidence-based reliability analysis Since theoriginal active Kriging model is utilized for reliabilityanalysis a model in local region here the Kriging model isconstructed in the entire design space called active globalKriging model +en two multidisciplinary integrationframeworks namely Multidisciplinary Feasible (MDF) andCollaborative Optimization (CO) are adopted and com-bined with the Kriging model Using the global Krigingmodel the computational cost of EBMDO is reduced to theapproximate equivalent in deterministic multidisciplinarydesign optimization (MDO)

+e rest of the article is organized as follows MDF andCO strategies for EBMDO are briefly introduced in Section2 An evidence-based reliability analysis using the globalkriging model for MDO is presented in detail in Section 3Numerical examples are provided and discussed in Section4 and the conclusions are presented in Section 5

2 Methodology

21 EBMDO +e general EBMDO model is expressed as

find ds d μXs μX1113966 1113967

min f dsd μXs μX μK1113872 1113873

st Pl Gi ds diXsXiKbulli( 1113857lt 01113858 1113859lt 1 minus Φ(β) i 1 n

(1)

where d pertains to the deterministic variables ds pertains toshared design variables that are common among all disci-plines di pertains to the design variables of the ith disciplineX pertains to the uncertain variables μX denotes the mean ofthe X which is treated as the uncertain design variables Xspertains to uncertain shared evidence variables that are the

common input variables for all disciplines andXi pertains tothe local evidence variables of the ith discipline +e sharedvariable Xs and the input variable X are independent var-iables K encompasses coupling variables and f is the ob-jective function Pl[middot] is the upper bound of failureprobability which represents the worst-case and is calledplausibility Pl[Gi(middot)lt 0]lt 1 minus Φ(β) is the failure probabilityconstraint of the ith discipline and Gi(middot) is the limit state ofthe ith discipline +e failure model is defined by Gi(middot)lt 0 βis the reliability index and Φ(middot) denotes the cumulativedistribution function (CDF) of the standard normal dis-tribution +e coupling variables between design and un-certain variables are expressed as

Kbulli Kji ds diXsXi( 1113857 j 1 n jne i (2)

where the first subscript j and the second subscript i of Kji

denote the values from the jth discipline to the ith discipline+e meaning of Kji can be explained in detail that thecoupling variables are inner shared variables in the jthdiscipline and the ith discipline and Kji is the output valuesof the jth discipline and the input values for the ith disciplineFor example in fluid structure thermal coupling analysis thepressure values from fluid analysis should be transferred tothe structure discipline for finite element analysis +econventional structure of EBMDO contains a triple loop asshown in Figure 1 [36 37]

As shown in Figure 1 due to the nested architecturefailure plausibility must be calculated at every design pointduring optimization +erefore the multidisciplinaryanalysis is repeatedly called to evaluate the limit statefunction Furthermore according to evidence theory theoptimization problem should be implemented to determinethe upper bound of the limit state +is nested structuremakes EBMDO costly

22 Multidisciplinary Feasible Method +e framework ofMDO aims to manage coupled variables and subsystems andimprove calculation efficiency +e MDO framework has twomain forms namely single-level and multilevel structures

MDF is a traditional single-level optimization method Asystematic analysis is performed for each iteration duringoptimization and a multidisciplinary analysis is performedrepeatedly to obtain consistent solutions of the couplingvariables among disciplines +is optimization frameworkincludes a system of integrated and subsystem analysismodules Only one interface exists for the input and outputbetween the system module and subsystem modules asshown in Figure 2 +e system module is an optimizer tosearch for the optimal solution +e subsystem modules arefor multidisciplinary and reliability analyses In each opti-mization iteration the coupling variables should be consistentand the constraints should be also satisfied +erefore whenthe optimal solution is obtained the consistency constraintsof coupling variables and reliability constraints are satisfied

23 Collaborative Optimization Method CO is a bilevelframework of MDO +e framework of MDO is designed to

2 Complexity

be top level (system level) and low level (discipline level) byCO +e top level is a system optimizer responsible forassigning target values for system-level state variables todisciplines +e low level is a parallel distributed multiple-discipline subsystem In addition to satisfying the con-straints of the subsystems the objective function is the

smallest difference between the coupling state variables ofthe top-level and low-level systems After low-level systemoptimization the objective function is fed back to the top-level system which constitutes the top-level consistencyconstraint Optimization of the top-level system is per-formed to address the uncoordinated state variables amongthe subsystems +e framework for CO is defined asfollows

min f ds sl dsl μXs sl μXsl

μPsl μKsl

1113872 1113873

st Jlowasti 0 i 1 N

dLs le ds sl le d

Us

dL le dle dU

XLs le μXs sl

leXUs

XL le μX leXU

(3)

where f is the objective function of system-level problemsds_sl denotes the shared deterministic design variables ofsystem-level problems dsl denotes the deterministic designvariables of discipline-level problems Xs_sl denotes theshared uncertain design variables of system-level problemsXsl denotes the uncertain design variables of the discipline-

K1i Kin

Ki1 KniKn1

K1n

Optimal designInitial designOptimization

Design variables

Uncertain variables

Reliability constraints


Optimization loop

Evidence analysis

Reliability loop

Multidisciplinary analysis

Output parametersInput parameters

Discipline 1 Discipline i Discipline n

Figure 1 Conventional procedure of RBMDO

Optimization

Evidence analysis

Pl

f Pl

Discipline nDiscipline 1


Discipline iK1i Kin

Ki1 KniKn1K1n

ds dXs X

ds dμXs μX

Figure 2 MDF architecture of EBMDO

Complexity 3

level problems Psl denotes the global uncertain parametersand Jlowast is the constraint of the discipline-level problems

+e ith discipline subproblem is

min Ji ds sl minus dsi

2

+ dsl minus di

2

+ μXs slminus μXsi

2

+ μXslminus μXi

2

st Plij gij XsXP( 1113857lt 01113872 1113873lt 1 minus Φ(β) j 1 m

XL le μX leXU

(4)

where Ji is the objective of the discipline-level problems dsldenotes the shared design variables di denotes the designvariables of the ith discipline si denotes the shared uncertaindesign variables and Xi denotes the uncertain design var-iables of the ith discipline

+e diagram of CO for EBMDO is shown in Figure 3

3 Reliability Analysis

31 Reliability Analysis Based on Evidence lteory +e limitstate function is denoted asG(x) Failure domain F is definedas F G G(x)lt 0 where the evidence variables arex [x1 x2 xnX] +e evidence variable xj can be de-scribed by evidence space (XEjmEj) where XEj is the set ofall focal elements [21] and XEj A1

j A2j A

nj

j1113966 1113967 whereAi

j is the ith focal element In the evidence analysis the focal

element is usually an interval mEj is the basic probabilityassignment (BPA) of each focal element +e joint focalelement (JFE) is defined by CPM

Ak A1 A2 AnE| Ak x

nX

j1A(i)j A

(i)j isin Aj1113966 1113967 (5)

+e BPA associated with JFE is calculated by

mE Ak( 1113857 1113945

nX

j1mEj A

(i)j1113872 1113873 (6)

where nX is the number of JFEs In the evidence spacefailure probability Pf is an interval +e lower and upperbounds of the interval are called belief (Bel) and plausibility(Pl) respectively Bel means that the system is in the absolutefailure state and Pl means that the system is in the possiblefailure state +e belief and plausibility measures of failureprobability are calculated by

Bel(F) 1113944

nE

k1I Ak subeF( 1113857m Ak( 1113857 (7)

Pl(F) 1113944

nE

k1I Ak capFneempty( 1113857m Ak( 1113857 (8)

where I(middot) is the indicator function

I(middot) 1 true

0 false1113896 (9)

+e equivalent form of equations (7) and (8) are

Bel(F) 1113944

nE

k1I max

xisinAk

G(x)lt 01113888 1113889m Ak( 1113857

Pl(F) 1113944

nE

k1I min

xisinAk

G(x)lt 01113888 1113889m Ak( 1113857

(10)

+is method involves a combinatorial explosion prob-lem because it needs to explore the extrema of G(x) in nE

JFEs According to equations (7) and (8) if n1 variables andn2 focal elements exist for each variable then the totalnumber of JFEs will be n

n12 +e Monte Carlo method is

adopted to randomly sample the focal elements and over-come the obstacle of dimensions JFEs are formed by thesefocal elements An extremum analysis is performed based onJFEs and Bel and Pl can be estimated using statisticalmethods shown as

Bel(F)1N

1113944

N

k1I max

xisinAk

G(x)lt 01113888 1113889 (11)

Pl(F) 1N

1113944

N

k1I min

xisinAk

G(x)lt 01113888 1113889 (12)

where N is the number of samples for the Monte Carlosimulation (MCS)

32 Evidence Analysis with the Kriging Model Although thedimensionality challenge is addressed the computationalcost remains large for MCS To further improve the com-putational efficiency the Kriging model is proposed toevaluate the limit state function in a multidisciplinarycoupling system +e Kriging model is briefly introduced asfollows

Suppose that a set of N samples called design of ex-periments (DoE) has the form (x G) where x is an n-di-mensional vector [x(1) x(2) x(n)]T and G is thecorresponding output [G(x(1)) G(x(2)) G(x(n))]T +epredicted value G(x) and variance s2(x) are given as

G(x) β + r(x)TRminus 1

(G minus β)

s2(x) σ2 1 +

1TRminus 1r minus 1( 11138572

1TRminus 11minus rTRminus 1r⎡⎣ ⎤⎦

(13)

where 1 is an n-dimensional unit vector r is the vector of thecorrelation function of x and points in DoE R is the matrixof the correlation function of points in DoE and β σ2 are theparameters to be determined +e methods of the fittingparameters are adopted as modified DIRECTalgorithm [38]Further details about the Kriging model can be found in Ref[39]

For evidence-based design optimization the Krigingmodel is used to approximate the limit state function in theentire design space +e idea here is that if the sign given bythe Kriging model is the same to that of the true limit statefunction then the result of MCS combined with the Krigingmodel can be considered trustworthy +e following

4 Complexity

property can ensure that the sign of the extremum value bythe Kriging model is consistent with that of the true limitstate function

Property In the entire design space Ω if sign(1113954G(X))

sign(G(X)) then in the local design space Ωlocalsign (min(1113954G(X))) sign (min(G(X))) and sign (max(1113954G(X))) sign (max(G(X))) where (sign (middot)) is the signfunction

Proof Consider the case of the minimum Suppose Xlowastmin isthe minimum point of 1113954G(X) and Xlowastmin is the minimumpoint of G(X) namely 1113954G(Xlowastmin) min(1113954G(X)) andG(Xlowastmin) min((X)) We will discuss in two cases

Case 1 If 1113954G(Xlowastmin)lt 0 because sign (1113954G(X)) sign (G(X))then G(Xlowastmin)lt 0 because G(Xlowastmin) min(G(X)) soG(Xlowastmin)leG(Xlowastmin)lt 0

Case 2 If 1113954G(Xlowastmin)gt 0 because 1113954G(Xlowastmin) min(1113954G(X)) then0lt 1113954G(Xlowastmin)le 1113954G(Xlowastmin) because sign (1113954G(X)) sign (G(X))so G(Xlowastmin)gt 0

+erefore when the limit state function has a minimumvalue the sign by the Kriging model and the true limit statefunction are the same+e case of themaximum value can beproven in a similar way Suppose that the Kriging model hasalready been constructed to correctly estimate the sign of thelimit state function equations (11) and (12) can be written as

Bel(F) 1N

1113944

N

k1I max

xisinAk

G(x)lt 01113888 1113889

Pl(F) 1N

1113944

N

k1I min

xisinAk

G(x)lt 01113888 1113889

(14)

To reduce the sample size we use the expected riskfunction (ERF) [23 39] as learning function to adaptivelyselect the samples

E[R(X)] minus sign(G(X))G(X)Φ minus sign(G(X))G(X)

1113954s(X)1113888 1113889

+ 1113954s(X)ϕG(X)

1113954s(X)1113888 1113889

(15)

where ϕ(middot) andΦ(middot) are the probability density function andCDF of standard normal distribution respectively +econvergence criterion for the Kriging model is written as

E R Xlowast( )[ ]

(G(X) + ε)lt 10minus 4 (16)

where Xlowast is the value for maximum of ERF X is the mean ofthe X and ε is 10minus 6

33 Validation of Reliability Analysis with the EvidenceVariables +e following numerical example is adopted tovalidate the proposed reliability analysis method +e limitstate function is written as [13]

g(x) x21x2

αminus 1 (17)

where α is a deterministic design variable and x1 and x2 areevidence variables When glt 0 the system is in the failureregion +e BPA structures of x1 and x2 are listed in Table 1

+ere are 13 points selected to construct the Krigingmodel +en 105 Monte Carlo simulation is performed tocalculate the Pl of the failure probability +e results of thereliability analysis are shown in Table 2 As can be seen theresult is close to the result from literature [13] which meansthe proposed evidence analysis method has a high accuracy

34 Summary of EBMDO with Kriging-Based ReliabilityAnalysis First a large population S that contains nMC pointsis generated Points are randomly selected from populationS +e limit state function is evaluated based on these points

System optimization

stmin

Jilowastlt ε i = 1 2 n

f

Optimization 1 Optimization i Optimization n

Analysis 1 Analysis i Analysis n

Evidence analysis 1 Evidence analysis i Evidence analysis n

J1 Plj1

J1 Plj1

Plj1

Ji Plji

Ji Plji

Plji

Jn Pljn

Jn Pljn

Pljn

ds_sl dsl

microXs_sl microXslds_sl dsl

microXs_sl microXsl

dsi di

microXsi microXi

dsi diXsi Xi

dsn dnXsn Xn

ds1 d1Xs1 X1

dsn dn

microXsn microXn

ds1 d1

microXs1 microX1

ds_sl dsl

microXs_sl microXsl

Figure 3 CO architecture of EBMDO

Complexity 5

thus forming the initial DoE SDoE for the Kriging model+eKrigingmodel is then constructed and the values of ERFin population S are calculated +e point with the largestvalue of ERF is selected and added to SDoE +e Krigingmodel is updated by SDoE until convergence is achieved

Second a reliability constraint is applied during opti-mization by the Kriging model +e framework of MDF andCO is adopted to achieve an optimal design

+e procedure of EBMDO is summarized as follows

Step 1 Generate nMC points S according to the range ofthe variablesStep 2 Randomly select n0 points from S evaluate thelimit state function and form the initial SDoEStep 3 Construct the Kriging model based on SDoEStep 4 Calculate the ERF on each point in S and pickout point xlowast with the maximum value of ERFStep 5 Evaluate the limit state function on the point xlowastand add the sample in SDoEStep 6 If the Kriging model is converged go to nextstep otherwise go to Step (3)Step 7 Set the initial value of the optimizationStep 8 At each optimal point during optimizationevaluate the reliability of the system by the Kriging modelStep 9 If the reliability constraint is satisfied and theobjective function is converged stop otherwise go toStep (8)

During Kriging modeling MDA should be performed toevaluate the limit state function During optimization MDA

is not required for reliability analysis because the Krigingmodel is constructed to approximate the limit state functionin the entire design space +erefore computational effi-ciency is considerably improved

4 Application Examples

+e proposed methodology is validated with two examplesof EBMDO For the inequality constraints of failure prob-ability evidence theory is used to estimate the epistemicuncertainty in terms of failure plausibility

41 Case 1 +e first EMBDO problem is a simple mathe-matical example +e problem is defined as

find d d1 d21113858 1113859

min f d21 + d

22

st Pl gi(x)lt 01113858 1113859ltPf

g1(x) 4 minus βx1 minus x2

g2(x) x1 + βx2 minus 2

0ledi le 5

i 1 2 β 05

(18)

where d1 d2 are the design variables and x1 x2 are theevidence variables and xi d + Δxi +e system utilizes twodesign variables and outputs two states +ey are not sharedvariables thus the systems are uncoupled +e objectivefunction is nonlinear with two constraints Pf is the con-straint of failure probability +e BPA structure of theuncertain variables is shown in Table 3 and the corre-sponding bar chart of epistemic variables is illustrated inFigure 4

+e CO architecture of this problem can be written asone system optimization and two discipline optimizations+e system optimization is written as

find d d1sl d2sl1113858 1113859

min f d21sl + d

22sl

st J1 d1sl minus d11( 11138572

+ d2sl minus d21( 11138572 le 10minus 5

J2 d1sl minus d12( 11138572


(19)

where d1sl d2sl are the system-level design variables dij

denotes the ith variables in the jth discipline and J1 J2 aretwo auxiliary variables and the objective function of thediscipline optimization problem +e auxiliary variables aimto reduce the difference of the system and discipline vari-ables to 10minus 5 +e optimization problem of discipline 1 isdefined as

find d d11 d211113858 1113859

min J1 d1sl minus d11( 11138572

+ d2sl minus d21( 11138572

st Pl g1(x)lt 0( 1113857ltPf


(20)

+e optimization problem of discipline 2 is defined as

Table 2 Reliability analysis results for validation case

α +e proposed method Literature [13]Pl(G) Pl(G)

800 10000 10000900 10000 100001000 10000 100001100 09995 099951200 09975 098901300 09857 098601400 09821 098181500 09310 091021600 08863 083131700 08524 08047

Table 1 BPA structure of epistemic variables of validation case

x1 x2

Interval BPA () Interval BPA ()

[30 40] 22 [50 64] 22[40 45] 136 [64 72] 136[45 475] 150 [72 76] 150[475 50] 192 [76 80] 192[50 525] 192 [80 84] 192[525 55] 150 [84 88] 150[55 60] 136 [88 96] 136[60 70] 22 [96 106] 22

6 Complexity

find d d12 d221113858 1113859


+ d2sl minus d22( 11138572



(21)

+e initial value of the design variables are set to [1608] which is the result of deterministic optimization (DO)To verify the accuracy of the proposed approach MCScombined with theMDF architecture is used to directly solveEBMDO +e nonlinear sequential quadratic programming(NLPQL) is adopted as the optimization algorithm for bothsystem and discipline optimization

Firstly the 105 candidate points are generated by Latinhypercube sampling +en according to experience 12points are randomly selected and evaluated by executing thedisciplinary analysis +e Kriging model is constructed andupdated by the proposed active global learning methodSubsequently the MDF and CO framework are performedbased on the Kriging model to search the optimal solutions

+e results are summarized and compared with those ofthe deterministic design andMCS in Table 4+e second rowshows the deterministic results EBMDO is performed withthree failure probability constraints namely Pf

02 01 00013 +e results show that the objective functionof MDO is smaller than that of EBMDO +e constraints offailure probability are not satisfied For a small failureprobability constraint the design variables and the objectivefunction must be large to meet the requirement of failure

plausibility +e obtained results are close to those of MCSand only the second constraint is active +e outer opti-mization loop has 100 iterations and for each design point105 times ofMCS are performed in the evidence analysis Foreach evidence analysis the limit state function is called forabout 10 times to obtain the minimum value Hence thelimit state function is called for approximately 2 times 108 times+e number of limit state function calls by the proposedmethod is 26 times for this example which is only 13 times 10minus 7

of MCS

42 Case 2 +e second example contains two disciplineswith three design and two coupled variables +e constraintfunction is nonlinear +e EBMDO problem is defined by

find d d1 d2 d31113858 1113859

min f d22 + d3 + y1 + e

minus y2

st Pl gi(x)lt 0( 1113857ltPf i 1 2

g1 y1 minus 316

g2 24 minus y2

y1 x21 + x2 + x3 minus 02y2

y2 y1

radic+ x1 + x3

minus 8led1 le 8 2le d2 d3 le 8

(22)

where di(i 1 2 3) are design variables yi(i 1 2) arecoupled variables xi(i 1 2) are uncertainty variablesand Pf is the constraint of failure probability Pf 001 Inthis example the coupled variables make the problemhighly complex Probability theory is a special case ofevidence theory To compare the EBMDO design with theRBMDO design the BPA for each interval of the un-certain variables is considered to be similar to the areaunder the probability density function used in RBMDOas shown in Figure 5

In RBMDO xi pertains to normally distributed randomparameters with xi sim N(di 052) +e BPA structure of theuncertain variables for EBMDO is shown in Table 5

+e samples are generated in the entire design spacenamely [minus 8 8] times [2 8] times [2 8] Subsequently 12 points arerandomly selected to construct the initial DoE +en thelearning strategy mentioned in Section 3 is utilized to selectthe points for DoE In each selected point the multidisci-plinary analysis must be performed to obtain the response ofthe limit state function +e training history curve of theKriging model is shown in Figure 6

+e Kriging model converges in 9 and 13 iterations forthe first and second limit state function respectively Toverify the accuracy of the Kriging model the signs of 105samples that are randomly generated in the design space arecompared with that of the true limit state function +ehistogram of the sign is illustrated in Figure 7 +e predictedsigns have a good match with that of the true function +enumber of signs that are incorrectly predicted is only fourwhile the relative error is 4 times 10minus 5

+en the problem is solved using the CO strategy +eEBMDO problem can be decomposed into one system

Table 3 BPA structure of epistemic variables of Case 1

xi BPA ()

[di minus 01 di + 01] 100[di minus 01 di minus 005] 10[di minus 005 di] 40[di di + 005] 40[di + 005 di + 01] 10

ndash01 ndash005 0 005 010

005

01

015

02

025

03

035

04

045

05

BPA

Δxi

Figure 4 Illustration of BPA of epistemic variables

Complexity 7

problem and two corresponding discipline problems be-cause of the presence of two disciplines +e system opti-mization problem is defined as

find d d1sl d2sl d3sl y1sl y2sl1113858 1113859

min f d22sl + d3sl + y1sl + e

minus y2sl

st J1 d11 minus d1SL( 11138572

+ d21 minus d2SL( 11138572

+ d31 minus d3SL( 11138572

+ y11 minus y1SL( 11138572 le 10minus 5

J2 d12 minus d1SL( 11138572

+ d32 minus d3SL( 11138572

+ d22 minus d2SL( 11138572 le 10minus 5

(23)

where disl(i 1 2 3) contains the system design variablesand d1i d2i d3i denotes the design variables of the ith dis-cipline +e coupled variables y1 y2 are considered thedesign variables to decouple the disciplines

+e optimization problem of the first discipline is de-fined by

Table 4 Computational results of Case 1

Probability constraints Methods d1 d2 Pl1 Pl2 Objective Ncall of PF

mdash DO 1600 0800 0 06670 3200 mdash

Pf 02EBMDO-MDF 1712 0754 0 01299 34525 26EBMDO-CO 1703 0752 0 012876 34657 26

MCS 1700 0750 0 01230 34525 2 times 108


MCS 1700 0800 0 00502 3530 2 times 108


MCS 1725 0850 0 0 369813 2 times 108

0

01

02

03

04

05

06

07

08

f (x)

0

005

01

015

02

025

03

035

04

BPA

ndash1 0 1 2ndash2x

ndash1 0 1 2ndash2x

Figure 5 PDF and BPA for Case 2


xi BPA

[di minus 2 di + 2] 1[di minus 20 di minus 15] 00014[di minus 15 di minus 10] 00206[di minus 10 di minus 05] 0136[di minus 05 di] 0342[di di + 05] 0342[di + 05 di + 10] 0136[di + 10 di + 15] 00206[di + 15 di + 20] 00014

8 Complexity

find d d11 d21 d31 y11 y211113858 1113859

min J1 d11 minus d1sl( 11138572

+ d21 minus d2sl( 11138572

+ d31 minus d3sl )2

1113872 1113873

+ y11 minus y1sl( 11138572

st Pl g1 lt 0( 1113857ltPf

g1 y11 minus 316

y11 x211 + x21 + x31 minus 02y2sl

(24)+e optimization problem of the second discipline is

defined by

find d d12 d22 d32 y12 y221113858 1113859


+ d32 minus d3sl( 11138572

+ y22 minus y2sl( 11138572


g2 24 minus y22

y22 y1sl

radic+ x12 + x32

(25)+e data flowchart is presented in Figure 8 to show the

solving architecture of cooperative optimization +e blueand red lines indicate the input and output respectively +esystem variables are passed to the discipline analysis and theobjective functions of disciplines are returned to systemoptimization +e objective functions of disciplines arethe constraints of system optimization +e reliability

constraints are evaluated by evidence analysis with the globalKriging models

Table 6 compares the DO EBMDO byMDF EBMDO byCO RBMDO and MCS results +e conclusions are similarto those in the previous example +e evaluation number ofthe limit state function by the proposed approach is 46which is about 23 times 10minus 7 that by MCS +is result indicatesthat the proposed method can ensure accuracy while havinga great advantage over MCS

43 Case 3 +e third case is multidisciplinary design op-timization for cooling blade which considers the aero-dynamic heat transfer and strength analysis revised from[40] +e geometry model of the cooling turbine blade andthe design parameters are shown in Figure 9 wherexi(i 1 4)is the thickness of ribs and xi(i 5 10)

is the thickness of blade wall at different blade profile+e EBMDO problem is defined by

find x x1 middot middot middot x101113858 1113859

min f w1D(x) + w2Tave(x)

st Tmax le 1500K

σmax le 830MPa

P D minus 032ge 0 ge 999

xl le x le xu

(26)

Nor

mal

ized

lear

ning

func

tion

Nor

mal

ized

lear

ning

func

tion

Performance function 1


0

0005

001

0

1

2

3

2 4 6 8 10 12 140Iteration

85 7 932 41 6Iteration

Figure 6 History of the training Kriging model

Complexity 9

where x is the uncertainty design variables f is the objectivefunction and D is the damage of cooling turbine bladewhich is the function of the design variables In this case thedamage D is the whole damage which is constituted by twoparts namely the creep damage and the fatigue damage+ecreep damage is predicted by the LarsonndashMiller equation[41] and the fatigue damage is calculated by the nominalstress method [42] +e fluid-thermal-solid interactionanalysis is carried out to obtain the stress and the tem-perature inMDF framework Also in the CO framework thethree disciplines are decoupled and performed independentlyTave is the average temperature w1 3000 and w2 1 are theweight factor Tmax is the maximum temperature of coolingturbine blade σmax is the maximum stress P is the reliability

and x is the uncertainty design variables includingx1 x2 x10 +e design variables are considered as epi-stemic parameters +e lower and upper limit of variables areshown in Table 7 +e thickness difference Δx obeys three-parameter Weibull distribution with the location parameterthe shape parameter and the scale parameter as minus 0154022979 and 02923 respectively According to the Weibulldistribution andMonte Carlo sampling the BPA of the designvariables are shown in Table 8

+e EBMDO results compared with the RBMDO andMDO by [40] are listed in Table 9 +e multidisciplinaryfeasible approach is adopted in this case As can be seen theresult obtained by EBMDO is very close to RBMDO whichindicates the proposed method is effective

Evidence analysis 2Evidence analysis 1

Kriging model 1 Kriging model 2

Pl (g1 lt 0)

Pl (g1 lt 0) lt Pf

J1 Pl (g1 lt 0) J2 Pl (g2 lt 0)

Pl (g2 lt 0)d11 d21 d31 y11 y21

d1sl d2sl d3sl y1sl y2sld1sl d2sl d3sl y1sl y2sl

d12 d22 d32 y12 y22

System optimization

Discipline 1 Discipline 2

findminst

findminst Pl (g2 lt 0) lt Pfst

d = [d1sl d2sl d3sl y1sl y2sl]f = [d2

2sl + d3sl + y1sl + endashy2sl

J1 = (d11 ndash d1SL)2 + (d21 ndash d2SL)2 + (d31 ndash d3SL)2 + (y11 ndash y1SL)2 le 10ndash5

J2 = (d12 ndash d1SL)2 + (d31 ndash d3SL)2 + (d22 ndash d2SL)2 le 10ndash5

d = [d11 d21 d31 y11 y21] J1 = (d11 ndash d1sl)2 + (d21 ndash d2sl)2 + (d31 ndash d3sl)2 + (y11 ndash y1sl)2

findmin

d = [d12 d22 d32 y12 y22] J2 = (d12 ndash d1sl)2 + (d32 ndash d3sl)2 + (y22 ndash y2sl )2

g1 = y11 ndash 316 g2 = 24 ndash y22y11 = x2

11 + x21 + x31 ndash 02y2sl y22 = radicy1sl + x12 + x32

Figure 8 CO architecture of Case 2

Negative sign Positive sign0

1

2

3

4

5

6

7

True functionKriging predict

Num

ber

times104

Figure 7 Accuracy validation of the global Kriging model

10 Complexity

5 Conclusions

+is article presents a novel approach for EBMDO thatcombines the active global Kriging model MDF and CO By

Table 7 Lower and upper limit of variables

Design variables Original (mm) Boundary of variables+ickness of rib 1 x1 10 [10 250]+ickness of rib 2 x2 10 [10 250]+ickness of rib 3 x3 10 [10 250]+ickness of rib 4 x4 10 [10 250]

Blade wall thickness at rootof blade cross section

x5 170 [169 171]x6 169 [169 171]x7 169 [169 171]

Blade wall thickness at topof blade cross section

x8 12 [120 125]x9 12 [120 125]x10 10 [10 105]

Table 8 BPA structures of epistemic variables

Epistemic variables Interval BPA

+ickness difference Δx

[minus 015 0] 01544[0 015] 04430

[015 030] 03423[030 045] 00604


Methods d1 d2 d3 Pl1 Pl2 Objective Ncall of PF

DO 011167 20 20 06125 0 92507 EBMDO-MDF 187236 2486 2483 00093 0 157359 46EBMDO-CO 187562 2483 2482 00091 0 157275 46RBMDO 187167 20 224 000696 0 126556 46MCS 187167 2480 2480 00099 0 156927 2times108

x1

x2

x3

x4

x58

x69

x710

Rib4

Rib3

Rib2Rib1

Figure 9 Geometry model and the design parameters of the blade [40]

Table 9 Results of EBMDO and RBMDO [40]

Design variables MDO RBMDO EBMDO+ickness of rib 1 x1 250 24633 24814+ickness of rib 2 x2 20138 24316 24906+ickness of rib 3 x3 24275 22403 22127+ickness of rib 4 x4 24975 16709 16913

Blade wall thickness at root ofblade cross section

x5 170 17073 17032x6 1709 16939 16997x7 16917 17068 17085

Blade wall thickness at top ofblade cross section

x8 123 12276 12447x9 12 12023 12458x10 105 10464 10965

Complexity 11

introducing a learning function the global Kriging model isadaptively constructed to replace the limit state function+en the evidence analysis is performed by MCS and theKriging model +e computational cost of EBMDO is re-duced to approximately that of DO +ree examples areprovided to illustrate the proposed approach Comparedwith MCS the proposed method can obtain accurate resultsand hold a significant advantage in terms of computationalefficiency +erefore the proposed method is expected to beof great value in engineering applications

As part of further work some other architectures ofmultidisciplinary optimization can be taken into account inEBMDO such as concurrent subspace optimization (CSSO)and bi-level integrated system synthesis (BLISS)

Data Availability

All relevant data used to support the findings of this studyare included within the article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work has been supported by the Natural ScienceFoundation of Jiangsu Province (BK20190424) the NaturalScience Foundation of Shaanxi Province (2019JQ-4702019JQ-609) the National Key RampD Program of China(grant no 2017YFB0202702) and the Center for HighPerformance Computing and System Simulation of PilotNational Laboratory for Marine Science and Technology(Qingdao)

References

[1] B D Youn K K Choi R-J Yang and L Gu ldquoReliability-based design optimization for crashworthiness of vehicle sideimpactrdquo Structural and Multidisciplinary Optimizationvol 26 no 3-4 pp 272ndash283 2004

[2] T W Simpson T M Mauery J Korte and F MistreeldquoKriging models for global approximation in simulation-based multidisciplinary design optimizationrdquo AIAA Journalvol 39 no 12 pp 2233ndash2241 2001

[3] F Zhang S Tan L Zhang Y Wang and Y Gao ldquoFault treeinterval analysis of complex systems based on universal greyoperationrdquoComplexity vol 2019 Article ID 1046054 8 pages2019

[4] W Yao X ChenW Luo M van Tooren and J Guo ldquoReviewof uncertainty-based multidisciplinary design optimizationmethods for aerospace vehiclesrdquo Progress in Aerospace Sci-ences vol 47 no 6 pp 450ndash479 2011

[5] D Meng Y-F Li H-Z Huang Z Wang and Y Liu ldquoRe-liability-based multidisciplinary design optimization usingsubset simulation analysis and its application in the hydraulictransmission mechanism designrdquo Journal of MechanicalDesign vol 137 no 5 Article ID 051402 2015

[6] J C Helton ldquoUncertainty and sensitivity analysis in thepresence of stochastic and subjective uncertaintyrdquo Journal ofStatistical Computation and Simulation vol 57 no 1ndash4pp 3ndash76 1997

[7] H Agarwal J E Renaud E L Preston and D PadmanabhanldquoUncertainty quantification using evidence theory in multi-disciplinary design optimizationrdquo Reliability Engineering ampSystem Safety vol 85 no 1ndash3 pp 281ndash294 2004

[8] C Jiang Z Zhang X Han and J Liu ldquoA novel evidence-theory-based reliability analysis method for structures withepistemic uncertaintyrdquo Computers amp Structures vol 129pp 1ndash12 2013

[9] J Zhou Z P Mourelatos and C Ellis ldquoDesign under un-certainty using a combination of evidence theory and aBayesian approachrdquo SAE International Journal of Materialsand Manufacturing vol 1 no 1 pp 122ndash135 2008

[10] D A Alvarez ldquoOn the calculation of the bounds of probabilityof events using infinite random setsrdquo International Journal ofApproximate Reasoning vol 43 no 3 pp 241ndash267 2006

[11] Z Wang H-Z Huang Y Li Y Pang and N-C Xiao ldquoAnapproach to system reliability analysis with fuzzy randomvariablesrdquo Mechanism and Machine lteory vol 52 pp 35ndash46 2012

[12] S Salehghaffari M Rais-Rohani E B Marin andD J Bammann ldquoOptimization of structures under materialparameter uncertainty using evidence theoryrdquo EngineeringOptimization vol 45 no 9 pp 1027ndash1041 2013

[13] M Xiao L Gao H Xiong and Z Luo ldquoAn efficient methodfor reliability analysis under epistemic uncertainty based onevidence theory and support vector regressionrdquo Journal ofEngineering Design vol 26 no 10ndash12 pp 340ndash364 2015

[14] Z Zhang C Jiang G GWang and X Han ldquoFirst and secondorder approximate reliability analysis methods using evidencetheoryrdquo Reliability Engineering amp System Safety vol 137pp 40ndash49 2015

[15] R Kang Q Zhang Z Zeng E Zio and X Li ldquoMeasuringreliability under epistemic uncertainty review on non-probabilistic reliability metricsrdquo Chinese Journal of Aero-nautics vol 29 no 3 pp 571ndash579 2016

[16] J Helton J D Johnson W L Oberkampf and C B StorlieldquoA sampling-based computational strategy for the represen-tation of epistemic uncertainty in model predictions withevidence theoryrdquo Computer Methods in Applied Mechanicsand Engineering vol 196 no 37ndash40 pp 3980ndash3998 2007

[17] V Y Kreinovich A Bernat W Borrett Y Mariscal andE Villa Monte-Carlo Methods Make Dempster-Shafer For-malism Feasible NASA Johnson Space Center Houston TXUSA 1991

[18] P Limbourg and E De Rocquigny ldquoUncertainty analysisusing evidence theorymdashconfronting level-1 and level-2 ap-proaches with data availability and computational con-straintsrdquo Reliability Engineering amp System Safety vol 95no 5 pp 550ndash564 2010

[19] N Wilson ldquo+e combination of belief when and how fastrdquoInternational Journal of Approximate Reasoning vol 6 no 3pp 377ndash388 1992

[20] C Joslyn and V Kreinovich ldquoConvergence properties of aninterval probabilistic approach to system reliability estima-tionrdquo International Journal of General Systems vol 34 no 4pp 465ndash482 2005

[21] C Wang and H G Matthies ldquoEvidence theory-based re-liability optimization design using polynomial chaos expan-sionrdquo Computer Methods in Applied Mechanics andEngineering vol 341 pp 640ndash657 2018

[22] H-R Bae R V Grandhi and R A Canfield ldquoUncertaintyquantification of structural response using evidence theoryrdquoAIAA Journal vol 41 no 10 pp 2062ndash2068 2003

12 Complexity

[23] X Yang Y Liu and P Ma ldquoStructural reliability analysisunder evidence theory using the active learning Krigingmodelrdquo Engineering Optimization vol 49 no 11 pp 1922ndash1938 2017

[24] Y C Bai X Han C Jiang and J Liu ldquoComparative study ofmetamodeling techniques for reliability analysis using evi-dence theoryrdquo Advances in Engineering Software vol 53pp 61ndash71 2012

[25] Z P Mourelatos and J Zhou ldquoA design optimization methodusing evidence theoryrdquo Journal of Mechanical Design vol 128no 4 pp 901ndash908 2006

[26] Z L Huang C Jiang Z Zhang T Fang and X Han ldquoAdecoupling approach for evidence-theory-based reliabilitydesign optimizationrdquo Structural and Multidisciplinary Opti-mization vol 56 no 3 pp 647ndash661 2017

[27] M Xiao H Xiong L Gao and Q Yao ldquoAn efficient methodfor structural reliability analysis using evidence theoryrdquo inProceedings of the 2014 IEEE 17th International Conference onComputational Science and Engineering (CSE) December2014

[28] R K Srivastava and K Deb ldquoAn EA-based approach to designoptimization using evidence theoryrdquo in Proceedings of the13th Annual Conference on Genetic and EvolutionaryComputation July 2011

[29] J Zhang M Xiao L Gao H Qiu and Z Yang ldquoAn improvedtwo-stage framework of evidence-based design optimizationrdquoStructural and Multidisciplinary Optimization vol 58 no 4pp 1673ndash1693 2018

[30] L P He and F Z Qu ldquoPossibility and evidence theory-baseddesign optimization an overviewrdquo Kybernetes vol 37 no 910 pp 1322ndash1330 2008

[31] W Yao X Chen Q Ouyang and M van Tooren ldquoA re-liability-based multidisciplinary design optimization pro-cedure based on combined probability and evidence theoryrdquoStructural and Multidisciplinary Optimization vol 48 no 2pp 339ndash354 2013

[32] X Hu X Chen G T Parks andW Yao ldquoReview of improvedMonte Carlo methods in uncertainty-based design optimi-zation for aerospace vehiclesrdquo Progress in Aerospace Sciencesvol 86 pp 20ndash27 2016

[33] K Zaman and S Mahadevan ldquoReliability-based design op-timization of multidisciplinary system under aleatory andepistemic uncertaintyrdquo Structural and Multidisciplinary Op-timization vol 55 no 2 pp 681ndash699 2017

[34] C Lin F Gao and Y Bai ldquoAn intelligent sampling approachfor metamodel-based multi-objective optimization withguidance of the adaptive weighted-sum methodrdquo Structuraland Multidisciplinary Optimization vol 57 no 3 pp 1047ndash1060 2018

[35] B Echard N Gayton and M Lemaire ldquoAK-MCS an activelearning reliability method combining Kriging and MonteCarlo simulationrdquo Structural Safety vol 33 no 2 pp 145ndash154 2011

[36] X Du A Sudjianto and B Huang ldquoReliability-based designwith the mixture of random and interval variablesrdquo Journal ofMechanical Design vol 127 no 6 pp 1068ndash1076 2005

[37] F Yang Z Yue L Li and D Guan ldquoHybrid reliability-basedmultidisciplinary design optimization with random and in-terval variablesrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 232no 1 pp 52ndash64 2018

[38] A Tavassoli K H Hajikolaei S Sadeqi G G Wang andE Kjeang ldquoModification of DIRECT for high-dimensional

design problemsrdquo Engineering Optimization vol 46 no 6pp 810ndash823 2014

[39] X Yang Y Liu and Y Gao ldquoUnified reliability analysis byactive learning Kriging model combining with random-setbasedMonte Carlo simulation methodrdquo International Journalfor Numerical Methods in Engineering vol 108 no 11pp 1343ndash1361 2016

[40] L Li H Wan W Gao F Tong and H Li ldquoReliability basedmultidisciplinary design optimization of cooling turbine bladeconsidering uncertainty data statisticsrdquo Structural and Mul-tidisciplinary Optimization vol 59 no 2 pp 659ndash673 2019

[41] A K Gupta and M R Haider ldquoCreep life estimation of lowpressure reaction turbine bladerdquo International Journal ofTechnological Exploration and Learning (IJTEL) vol 3 no 2pp 402ndash404 2014

[42] H-Z Huang C-G Huang Z Peng Y-F Li and H YinldquoFatigue life prediction of fan blade using nominal stressmethod and cumulative fatigue damage theoryrdquo InternationalJournal of Turbo amp Jet-Engines 2017

Complexity 13

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Volume 2018

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Decision SciencesAdvances in


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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

simulation (MCS) [17ndash20] polynomial chaos expansion [21]and surrogate-based models [22ndash24] Design optimizationbased on evidence theory has also been studied [25ndash28]Zhang et al [29] adopted two-stage framework to handle theevidence-based design optimization problem where the ev-idence variables were transformed into random variables andthe sequential optimization and reliability assessment wereused He and Qu [30] gave an overview of possibility andevidence theory-based design optimization However limitedresearch has been conducted on evidence-based multidisci-plinary design optimization (EBMDO) [31ndash33] ConventionalEBMDO has a three-level loop architecture +e outer loop ismultidisciplinary optimization the middle loop is reliabilityanalysis based on evidence theory and the inner loop isdiscipline analysis +e computational cost of EBMDO ishigh especially in the inner discipline analysis Practicalengineering problems require repeatedly calling the simula-tion model to evaluate limit state and objective functions andthis process is time consuming

To address this issue this study proposes a surrogatemodel-based method with an active global learning strategywhich constructs surrogate models in entire design spacewith intelligent sampling approach to reduce the amount ofcomputation [34] +e active learning Kriging [35] model isutilized as a surrogate model to evaluate the limit statefunction in evidence-based reliability analysis Since theoriginal active Kriging model is utilized for reliabilityanalysis a model in local region here the Kriging model isconstructed in the entire design space called active globalKriging model +en two multidisciplinary integrationframeworks namely Multidisciplinary Feasible (MDF) andCollaborative Optimization (CO) are adopted and com-bined with the Kriging model Using the global Krigingmodel the computational cost of EBMDO is reduced to theapproximate equivalent in deterministic multidisciplinarydesign optimization (MDO)

+e rest of the article is organized as follows MDF andCO strategies for EBMDO are briefly introduced in Section2 An evidence-based reliability analysis using the globalkriging model for MDO is presented in detail in Section 3Numerical examples are provided and discussed in Section4 and the conclusions are presented in Section 5

2 Methodology

21 EBMDO +e general EBMDO model is expressed as

find ds d μXs μX1113966 1113967

min f dsd μXs μX μK1113872 1113873

st Pl Gi ds diXsXiKbulli( 1113857lt 01113858 1113859lt 1 minus Φ(β) i 1 n

(1)

where d pertains to the deterministic variables ds pertains toshared design variables that are common among all disci-plines di pertains to the design variables of the ith disciplineX pertains to the uncertain variables μX denotes the mean ofthe X which is treated as the uncertain design variables Xspertains to uncertain shared evidence variables that are the

common input variables for all disciplines andXi pertains tothe local evidence variables of the ith discipline +e sharedvariable Xs and the input variable X are independent var-iables K encompasses coupling variables and f is the ob-jective function Pl[middot] is the upper bound of failureprobability which represents the worst-case and is calledplausibility Pl[Gi(middot)lt 0]lt 1 minus Φ(β) is the failure probabilityconstraint of the ith discipline and Gi(middot) is the limit state ofthe ith discipline +e failure model is defined by Gi(middot)lt 0 βis the reliability index and Φ(middot) denotes the cumulativedistribution function (CDF) of the standard normal dis-tribution +e coupling variables between design and un-certain variables are expressed as

Kbulli Kji ds diXsXi( 1113857 j 1 n jne i (2)

where the first subscript j and the second subscript i of Kji

denote the values from the jth discipline to the ith discipline+e meaning of Kji can be explained in detail that thecoupling variables are inner shared variables in the jthdiscipline and the ith discipline and Kji is the output valuesof the jth discipline and the input values for the ith disciplineFor example in fluid structure thermal coupling analysis thepressure values from fluid analysis should be transferred tothe structure discipline for finite element analysis +econventional structure of EBMDO contains a triple loop asshown in Figure 1 [36 37]

As shown in Figure 1 due to the nested architecturefailure plausibility must be calculated at every design pointduring optimization +erefore the multidisciplinaryanalysis is repeatedly called to evaluate the limit statefunction Furthermore according to evidence theory theoptimization problem should be implemented to determinethe upper bound of the limit state +is nested structuremakes EBMDO costly

22 Multidisciplinary Feasible Method +e framework ofMDO aims to manage coupled variables and subsystems andimprove calculation efficiency +e MDO framework has twomain forms namely single-level and multilevel structures

MDF is a traditional single-level optimization method Asystematic analysis is performed for each iteration duringoptimization and a multidisciplinary analysis is performedrepeatedly to obtain consistent solutions of the couplingvariables among disciplines +is optimization frameworkincludes a system of integrated and subsystem analysismodules Only one interface exists for the input and outputbetween the system module and subsystem modules asshown in Figure 2 +e system module is an optimizer tosearch for the optimal solution +e subsystem modules arefor multidisciplinary and reliability analyses In each opti-mization iteration the coupling variables should be consistentand the constraints should be also satisfied +erefore whenthe optimal solution is obtained the consistency constraintsof coupling variables and reliability constraints are satisfied

23 Collaborative Optimization Method CO is a bilevelframework of MDO +e framework of MDO is designed to

2 Complexity




μPsl μKsl

1113872 1113873

st Jlowasti 0 i 1 N

dLs le ds sl le d

Us

dL le dle dU

XLs le μXs sl

leXUs

XL le μX leXU

(3)


K1i Kin

Ki1 KniKn1

K1n


Design variables

Uncertain variables



Optimization loop

Evidence analysis

Reliability loop





Optimization

Evidence analysis

Pl

f Pl



Discipline iK1i Kin

Ki1 KniKn1K1n

ds dXs X

ds dμXs μX


Complexity 3




2

+ dsl minus di

2


2

+ μXslminus μXi

2


XL le μX leXU

(4)





j A2j A

nj

j1113966 1113967 whereAi



Ak A1 A2 AnE| Ak x

nX

j1A(i)j A

(i)j isin Aj1113966 1113967 (5)


mE Ak( 1113857 1113945

nX

j1mEj A

(i)j1113872 1113873 (6)


Bel(F) 1113944

nE

k1I Ak subeF( 1113857m Ak( 1113857 (7)

Pl(F) 1113944

nE



I(middot) 1 true

0 false1113896 (9)


Bel(F) 1113944

nE

k1I max

xisinAk

G(x)lt 01113888 1113889m Ak( 1113857

Pl(F) 1113944

nE

k1I min

xisinAk

G(x)lt 01113888 1113889m Ak( 1113857

(10)





Bel(F)1N

1113944

N

k1I max

xisinAk

G(x)lt 01113888 1113889 (11)

Pl(F) 1N

1113944

N

k1I min

xisinAk

G(x)lt 01113888 1113889 (12)





(G minus β)

s2(x) σ2 1 +



(13)



4 Complexity








Bel(F) 1N

1113944

N

k1I max

xisinAk

G(x)lt 01113888 1113889

Pl(F) 1N

1113944

N

k1I min

xisinAk

G(x)lt 01113888 1113889

(14)



1113954s(X)1113888 1113889

+ 1113954s(X)ϕG(X)

1113954s(X)1113888 1113889

(15)


E R Xlowast( )[ ]

(G(X) + ε)lt 10minus 4 (16)



g(x) x21x2

αminus 1 (17)




System optimization

stmin


f




J1 Plj1

J1 Plj1

Plj1

Ji Plji

Ji Plji

Plji

Jn Pljn

Jn Pljn

Pljn

ds_sl dsl


microXs_sl microXsl

dsi di

microXsi microXi

dsi diXsi Xi

dsn dnXsn Xn

ds1 d1Xs1 X1

dsn dn

microXsn microXn

ds1 d1

microXs1 microX1

ds_sl dsl

microXs_sl microXsl


Complexity 5










find d d1 d21113858 1113859

min f d21 + d

22

st Pl gi(x)lt 01113858 1113859ltPf



0ledi le 5

i 1 2 β 05

(18)



find d d1sl d2sl1113858 1113859

min f d21sl + d

22sl



J2 d1sl minus d12( 11138572


(19)



find d d11 d211113858 1113859


+ d2sl minus d21( 11138572



(20)




800 10000 10000900 10000 100001000 10000 100001100 09995 099951200 09975 098901300 09857 098601400 09821 098181500 09310 091021600 08863 083131700 08524 08047


x1 x2


[30 40] 22 [50 64] 22[40 45] 136 [64 72] 136[45 475] 150 [72 76] 150[475 50] 192 [76 80] 192[50 525] 192 [80 84] 192[525 55] 150 [84 88] 150[55 60] 136 [88 96] 136[60 70] 22 [96 106] 22

6 Complexity

find d d12 d221113858 1113859


+ d2sl minus d22( 11138572



(21)






of MCS


find d d1 d2 d31113858 1113859

min f d22 + d3 + y1 + e

minus y2


g1 y1 minus 316

g2 24 minus y2

y1 x21 + x2 + x3 minus 02y2

y2 y1

radic+ x1 + x3


(22)







xi BPA ()


ndash01 ndash005 0 005 010

005

01

015

02

025

03

035

04

045

05

BPA

Δxi


Complexity 7




minus y2sl


+ d21 minus d2SL( 11138572

+ d31 minus d3SL( 11138572


J2 d12 minus d1SL( 11138572

+ d32 minus d3SL( 11138572


(23)





mdash DO 1600 0800 0 06670 3200 mdash


MCS 1700 0750 0 01230 34525 2 times 108


MCS 1700 0800 0 00502 3530 2 times 108


MCS 1725 0850 0 0 369813 2 times 108

0

01

02

03

04

05

06

07

08

f (x)

0

005

01

015

02

025

03

035

04

BPA

ndash1 0 1 2ndash2x

ndash1 0 1 2ndash2x



xi BPA


8 Complexity

find d d11 d21 d31 y11 y211113858 1113859


+ d21 minus d2sl( 11138572

+ d31 minus d3sl )2

1113872 1113873

+ y11 minus y1sl( 11138572


g1 y11 minus 316

y11 x211 + x21 + x31 minus 02y2sl


defined by

find d d12 d22 d32 y12 y221113858 1113859


+ d32 minus d3sl( 11138572

+ y22 minus y2sl( 11138572


g2 24 minus y22

y22 y1sl

radic+ x12 + x32









st Tmax le 1500K

σmax le 830MPa


xl le x le xu

(26)

Nor

mal

ized

lear

ning

func

tion

Nor

mal

ized

lear

ning

func

tion



0

0005

001

0

1

2

3

2 4 6 8 10 12 140Iteration



Complexity 9






Pl (g1 lt 0)

Pl (g1 lt 0) lt Pf

J1 Pl (g1 lt 0) J2 Pl (g2 lt 0)

Pl (g2 lt 0)d11 d21 d31 y11 y21


d12 d22 d32 y12 y22

System optimization


findminst







findmin






1

2

3

4

5

6

7


Num

ber

times104


10 Complexity

5 Conclusions





x5 170 [169 171]x6 169 [169 171]x7 169 [169 171]


x8 12 [120 125]x9 12 [120 125]x10 10 [10 105]




[minus 015 0] 01544[0 015] 04430

[015 030] 03423[030 045] 00604




x1

x2

x3

x4

x58

x69

x710

Rib4

Rib3

Rib2Rib1





x5 170 17073 17032x6 1709 16939 16997x7 16917 17068 17085


x8 123 12276 12447x9 12 12023 12458x10 105 10464 10965

Complexity 11



Data Availability




Acknowledgments


References























12 Complexity






















Complexity 13








Journal of












Journal of







Volume 2018






Volume 2018











μPsl μKsl

1113872 1113873

st Jlowasti 0 i 1 N

dLs le ds sl le d

Us

dL le dle dU

XLs le μXs sl

leXUs

XL le μX leXU

(3)


K1i Kin

Ki1 KniKn1

K1n


Design variables

Uncertain variables



Optimization loop

Evidence analysis

Reliability loop





Optimization

Evidence analysis

Pl

f Pl



Discipline iK1i Kin

Ki1 KniKn1K1n

ds dXs X

ds dμXs μX


Complexity 3




2

+ dsl minus di

2


2

+ μXslminus μXi

2


XL le μX leXU

(4)





j A2j A

nj

j1113966 1113967 whereAi



Ak A1 A2 AnE| Ak x

nX

j1A(i)j A

(i)j isin Aj1113966 1113967 (5)


mE Ak( 1113857 1113945

nX

j1mEj A

(i)j1113872 1113873 (6)


Bel(F) 1113944

nE

k1I Ak subeF( 1113857m Ak( 1113857 (7)

Pl(F) 1113944

nE



I(middot) 1 true

0 false1113896 (9)


Bel(F) 1113944

nE

k1I max

xisinAk

G(x)lt 01113888 1113889m Ak( 1113857

Pl(F) 1113944

nE

k1I min

xisinAk

G(x)lt 01113888 1113889m Ak( 1113857

(10)





Bel(F)1N

1113944

N

k1I max

xisinAk

G(x)lt 01113888 1113889 (11)

Pl(F) 1N

1113944

N

k1I min

xisinAk

G(x)lt 01113888 1113889 (12)





(G minus β)

s2(x) σ2 1 +



(13)



4 Complexity








Bel(F) 1N

1113944

N

k1I max

xisinAk

G(x)lt 01113888 1113889

Pl(F) 1N

1113944

N

k1I min

xisinAk

G(x)lt 01113888 1113889

(14)



1113954s(X)1113888 1113889

+ 1113954s(X)ϕG(X)

1113954s(X)1113888 1113889

(15)


E R Xlowast( )[ ]

(G(X) + ε)lt 10minus 4 (16)



g(x) x21x2

αminus 1 (17)




System optimization

stmin


f




J1 Plj1

J1 Plj1

Plj1

Ji Plji

Ji Plji

Plji

Jn Pljn

Jn Pljn

Pljn

ds_sl dsl


microXs_sl microXsl

dsi di

microXsi microXi

dsi diXsi Xi

dsn dnXsn Xn

ds1 d1Xs1 X1

dsn dn

microXsn microXn

ds1 d1

microXs1 microX1

ds_sl dsl

microXs_sl microXsl


Complexity 5










find d d1 d21113858 1113859

min f d21 + d

22

st Pl gi(x)lt 01113858 1113859ltPf



0ledi le 5

i 1 2 β 05

(18)



find d d1sl d2sl1113858 1113859

min f d21sl + d

22sl



J2 d1sl minus d12( 11138572


(19)



find d d11 d211113858 1113859


+ d2sl minus d21( 11138572



(20)




800 10000 10000900 10000 100001000 10000 100001100 09995 099951200 09975 098901300 09857 098601400 09821 098181500 09310 091021600 08863 083131700 08524 08047


x1 x2


[30 40] 22 [50 64] 22[40 45] 136 [64 72] 136[45 475] 150 [72 76] 150[475 50] 192 [76 80] 192[50 525] 192 [80 84] 192[525 55] 150 [84 88] 150[55 60] 136 [88 96] 136[60 70] 22 [96 106] 22

6 Complexity

find d d12 d221113858 1113859


+ d2sl minus d22( 11138572



(21)






of MCS


find d d1 d2 d31113858 1113859

min f d22 + d3 + y1 + e

minus y2


g1 y1 minus 316

g2 24 minus y2

y1 x21 + x2 + x3 minus 02y2

y2 y1

radic+ x1 + x3


(22)







xi BPA ()


ndash01 ndash005 0 005 010

005

01

015

02

025

03

035

04

045

05

BPA

Δxi


Complexity 7




minus y2sl


+ d21 minus d2SL( 11138572

+ d31 minus d3SL( 11138572


J2 d12 minus d1SL( 11138572

+ d32 minus d3SL( 11138572


(23)





mdash DO 1600 0800 0 06670 3200 mdash


MCS 1700 0750 0 01230 34525 2 times 108


MCS 1700 0800 0 00502 3530 2 times 108


MCS 1725 0850 0 0 369813 2 times 108

0

01

02

03

04

05

06

07

08

f (x)

0

005

01

015

02

025

03

035

04

BPA

ndash1 0 1 2ndash2x

ndash1 0 1 2ndash2x



xi BPA


8 Complexity

find d d11 d21 d31 y11 y211113858 1113859


+ d21 minus d2sl( 11138572

+ d31 minus d3sl )2

1113872 1113873

+ y11 minus y1sl( 11138572


g1 y11 minus 316

y11 x211 + x21 + x31 minus 02y2sl


defined by

find d d12 d22 d32 y12 y221113858 1113859


+ d32 minus d3sl( 11138572

+ y22 minus y2sl( 11138572


g2 24 minus y22

y22 y1sl

radic+ x12 + x32









st Tmax le 1500K

σmax le 830MPa


xl le x le xu

(26)

Nor

mal

ized

lear

ning

func

tion

Nor

mal

ized

lear

ning

func

tion



0

0005

001

0

1

2

3

2 4 6 8 10 12 140Iteration



Complexity 9






Pl (g1 lt 0)

Pl (g1 lt 0) lt Pf

J1 Pl (g1 lt 0) J2 Pl (g2 lt 0)

Pl (g2 lt 0)d11 d21 d31 y11 y21


d12 d22 d32 y12 y22

System optimization


findminst







findmin






1

2

3

4

5

6

7


Num

ber

times104


10 Complexity

5 Conclusions





x5 170 [169 171]x6 169 [169 171]x7 169 [169 171]


x8 12 [120 125]x9 12 [120 125]x10 10 [10 105]




[minus 015 0] 01544[0 015] 04430

[015 030] 03423[030 045] 00604




x1

x2

x3

x4

x58

x69

x710

Rib4

Rib3

Rib2Rib1





x5 170 17073 17032x6 1709 16939 16997x7 16917 17068 17085


x8 123 12276 12447x9 12 12023 12458x10 105 10464 10965

Complexity 11



Data Availability




Acknowledgments


References























12 Complexity






















Complexity 13








Journal of












Journal of







Volume 2018






Volume 2018











2

+ dsl minus di

2


2

+ μXslminus μXi

2


XL le μX leXU

(4)





j A2j A

nj

j1113966 1113967 whereAi



Ak A1 A2 AnE| Ak x

nX

j1A(i)j A

(i)j isin Aj1113966 1113967 (5)


mE Ak( 1113857 1113945

nX

j1mEj A

(i)j1113872 1113873 (6)


Bel(F) 1113944

nE

k1I Ak subeF( 1113857m Ak( 1113857 (7)

Pl(F) 1113944

nE



I(middot) 1 true

0 false1113896 (9)


Bel(F) 1113944

nE

k1I max

xisinAk

G(x)lt 01113888 1113889m Ak( 1113857

Pl(F) 1113944

nE

k1I min

xisinAk

G(x)lt 01113888 1113889m Ak( 1113857

(10)





Bel(F)1N

1113944

N

k1I max

xisinAk

G(x)lt 01113888 1113889 (11)

Pl(F) 1N

1113944

N

k1I min

xisinAk

G(x)lt 01113888 1113889 (12)





(G minus β)

s2(x) σ2 1 +



(13)



4 Complexity








Bel(F) 1N

1113944

N

k1I max

xisinAk

G(x)lt 01113888 1113889

Pl(F) 1N

1113944

N

k1I min

xisinAk

G(x)lt 01113888 1113889

(14)



1113954s(X)1113888 1113889

+ 1113954s(X)ϕG(X)

1113954s(X)1113888 1113889

(15)


E R Xlowast( )[ ]

(G(X) + ε)lt 10minus 4 (16)



g(x) x21x2

αminus 1 (17)




System optimization

stmin


f




J1 Plj1

J1 Plj1

Plj1

Ji Plji

Ji Plji

Plji

Jn Pljn

Jn Pljn

Pljn

ds_sl dsl


microXs_sl microXsl

dsi di

microXsi microXi

dsi diXsi Xi

dsn dnXsn Xn

ds1 d1Xs1 X1

dsn dn

microXsn microXn

ds1 d1

microXs1 microX1

ds_sl dsl

microXs_sl microXsl


Complexity 5










find d d1 d21113858 1113859

min f d21 + d

22

st Pl gi(x)lt 01113858 1113859ltPf



0ledi le 5

i 1 2 β 05

(18)



find d d1sl d2sl1113858 1113859

min f d21sl + d

22sl



J2 d1sl minus d12( 11138572


(19)



find d d11 d211113858 1113859


+ d2sl minus d21( 11138572



(20)




800 10000 10000900 10000 100001000 10000 100001100 09995 099951200 09975 098901300 09857 098601400 09821 098181500 09310 091021600 08863 083131700 08524 08047


x1 x2


[30 40] 22 [50 64] 22[40 45] 136 [64 72] 136[45 475] 150 [72 76] 150[475 50] 192 [76 80] 192[50 525] 192 [80 84] 192[525 55] 150 [84 88] 150[55 60] 136 [88 96] 136[60 70] 22 [96 106] 22

6 Complexity

find d d12 d221113858 1113859


+ d2sl minus d22( 11138572



(21)






of MCS


find d d1 d2 d31113858 1113859

min f d22 + d3 + y1 + e

minus y2


g1 y1 minus 316

g2 24 minus y2

y1 x21 + x2 + x3 minus 02y2

y2 y1

radic+ x1 + x3


(22)







xi BPA ()


ndash01 ndash005 0 005 010

005

01

015

02

025

03

035

04

045

05

BPA

Δxi


Complexity 7




minus y2sl


+ d21 minus d2SL( 11138572

+ d31 minus d3SL( 11138572


J2 d12 minus d1SL( 11138572

+ d32 minus d3SL( 11138572


(23)





mdash DO 1600 0800 0 06670 3200 mdash


MCS 1700 0750 0 01230 34525 2 times 108


MCS 1700 0800 0 00502 3530 2 times 108


MCS 1725 0850 0 0 369813 2 times 108

0

01

02

03

04

05

06

07

08

f (x)

0

005

01

015

02

025

03

035

04

BPA

ndash1 0 1 2ndash2x

ndash1 0 1 2ndash2x



xi BPA


8 Complexity

find d d11 d21 d31 y11 y211113858 1113859


+ d21 minus d2sl( 11138572

+ d31 minus d3sl )2

1113872 1113873

+ y11 minus y1sl( 11138572


g1 y11 minus 316

y11 x211 + x21 + x31 minus 02y2sl


defined by

find d d12 d22 d32 y12 y221113858 1113859


+ d32 minus d3sl( 11138572

+ y22 minus y2sl( 11138572


g2 24 minus y22

y22 y1sl

radic+ x12 + x32









st Tmax le 1500K

σmax le 830MPa


xl le x le xu

(26)

Nor

mal

ized

lear

ning

func

tion

Nor

mal

ized

lear

ning

func

tion



0

0005

001

0

1

2

3

2 4 6 8 10 12 140Iteration



Complexity 9






Pl (g1 lt 0)

Pl (g1 lt 0) lt Pf

J1 Pl (g1 lt 0) J2 Pl (g2 lt 0)

Pl (g2 lt 0)d11 d21 d31 y11 y21


d12 d22 d32 y12 y22

System optimization


findminst







findmin






1

2

3

4

5

6

7


Num

ber

times104


10 Complexity

5 Conclusions





x5 170 [169 171]x6 169 [169 171]x7 169 [169 171]


x8 12 [120 125]x9 12 [120 125]x10 10 [10 105]




[minus 015 0] 01544[0 015] 04430

[015 030] 03423[030 045] 00604




x1

x2

x3

x4

x58

x69

x710

Rib4

Rib3

Rib2Rib1





x5 170 17073 17032x6 1709 16939 16997x7 16917 17068 17085


x8 123 12276 12447x9 12 12023 12458x10 105 10464 10965

Complexity 11



Data Availability




Acknowledgments


References























12 Complexity






















Complexity 13








Journal of












Journal of







Volume 2018






Volume 2018















Bel(F) 1N

1113944

N

k1I max

xisinAk

G(x)lt 01113888 1113889

Pl(F) 1N

1113944

N

k1I min

xisinAk

G(x)lt 01113888 1113889

(14)



1113954s(X)1113888 1113889

+ 1113954s(X)ϕG(X)

1113954s(X)1113888 1113889

(15)


E R Xlowast( )[ ]

(G(X) + ε)lt 10minus 4 (16)



g(x) x21x2

αminus 1 (17)




System optimization

stmin


f




J1 Plj1

J1 Plj1

Plj1

Ji Plji

Ji Plji

Plji

Jn Pljn

Jn Pljn

Pljn

ds_sl dsl


microXs_sl microXsl

dsi di

microXsi microXi

dsi diXsi Xi

dsn dnXsn Xn

ds1 d1Xs1 X1

dsn dn

microXsn microXn

ds1 d1

microXs1 microX1

ds_sl dsl

microXs_sl microXsl


Complexity 5










find d d1 d21113858 1113859

min f d21 + d

22

st Pl gi(x)lt 01113858 1113859ltPf



0ledi le 5

i 1 2 β 05

(18)



find d d1sl d2sl1113858 1113859

min f d21sl + d

22sl



J2 d1sl minus d12( 11138572


(19)



find d d11 d211113858 1113859


+ d2sl minus d21( 11138572



(20)




800 10000 10000900 10000 100001000 10000 100001100 09995 099951200 09975 098901300 09857 098601400 09821 098181500 09310 091021600 08863 083131700 08524 08047


x1 x2


[30 40] 22 [50 64] 22[40 45] 136 [64 72] 136[45 475] 150 [72 76] 150[475 50] 192 [76 80] 192[50 525] 192 [80 84] 192[525 55] 150 [84 88] 150[55 60] 136 [88 96] 136[60 70] 22 [96 106] 22

6 Complexity

find d d12 d221113858 1113859


+ d2sl minus d22( 11138572



(21)






of MCS


find d d1 d2 d31113858 1113859

min f d22 + d3 + y1 + e

minus y2


g1 y1 minus 316

g2 24 minus y2

y1 x21 + x2 + x3 minus 02y2

y2 y1

radic+ x1 + x3


(22)







xi BPA ()


ndash01 ndash005 0 005 010

005

01

015

02

025

03

035

04

045

05

BPA

Δxi


Complexity 7




minus y2sl


+ d21 minus d2SL( 11138572

+ d31 minus d3SL( 11138572


J2 d12 minus d1SL( 11138572

+ d32 minus d3SL( 11138572


(23)





mdash DO 1600 0800 0 06670 3200 mdash


MCS 1700 0750 0 01230 34525 2 times 108


MCS 1700 0800 0 00502 3530 2 times 108


MCS 1725 0850 0 0 369813 2 times 108

0

01

02

03

04

05

06

07

08

f (x)

0

005

01

015

02

025

03

035

04

BPA

ndash1 0 1 2ndash2x

ndash1 0 1 2ndash2x



xi BPA


8 Complexity

find d d11 d21 d31 y11 y211113858 1113859


+ d21 minus d2sl( 11138572

+ d31 minus d3sl )2

1113872 1113873

+ y11 minus y1sl( 11138572


g1 y11 minus 316

y11 x211 + x21 + x31 minus 02y2sl


defined by

find d d12 d22 d32 y12 y221113858 1113859


+ d32 minus d3sl( 11138572

+ y22 minus y2sl( 11138572


g2 24 minus y22

y22 y1sl

radic+ x12 + x32









st Tmax le 1500K

σmax le 830MPa


xl le x le xu

(26)

Nor

mal

ized

lear

ning

func

tion

Nor

mal

ized

lear

ning

func

tion



0

0005

001

0

1

2

3

2 4 6 8 10 12 140Iteration



Complexity 9






Pl (g1 lt 0)

Pl (g1 lt 0) lt Pf

J1 Pl (g1 lt 0) J2 Pl (g2 lt 0)

Pl (g2 lt 0)d11 d21 d31 y11 y21


d12 d22 d32 y12 y22

System optimization


findminst







findmin






1

2

3

4

5

6

7


Num

ber

times104


10 Complexity

5 Conclusions





x5 170 [169 171]x6 169 [169 171]x7 169 [169 171]


x8 12 [120 125]x9 12 [120 125]x10 10 [10 105]




[minus 015 0] 01544[0 015] 04430

[015 030] 03423[030 045] 00604




x1

x2

x3

x4

x58

x69

x710

Rib4

Rib3

Rib2Rib1





x5 170 17073 17032x6 1709 16939 16997x7 16917 17068 17085


x8 123 12276 12447x9 12 12023 12458x10 105 10464 10965

Complexity 11



Data Availability




Acknowledgments


References























12 Complexity






















Complexity 13








Journal of












Journal of







Volume 2018






Volume 2018

















find d d1 d21113858 1113859

min f d21 + d

22

st Pl gi(x)lt 01113858 1113859ltPf



0ledi le 5

i 1 2 β 05

(18)



find d d1sl d2sl1113858 1113859

min f d21sl + d

22sl



J2 d1sl minus d12( 11138572


(19)



find d d11 d211113858 1113859


+ d2sl minus d21( 11138572



(20)




800 10000 10000900 10000 100001000 10000 100001100 09995 099951200 09975 098901300 09857 098601400 09821 098181500 09310 091021600 08863 083131700 08524 08047


x1 x2


[30 40] 22 [50 64] 22[40 45] 136 [64 72] 136[45 475] 150 [72 76] 150[475 50] 192 [76 80] 192[50 525] 192 [80 84] 192[525 55] 150 [84 88] 150[55 60] 136 [88 96] 136[60 70] 22 [96 106] 22

6 Complexity

find d d12 d221113858 1113859


+ d2sl minus d22( 11138572



(21)






of MCS


find d d1 d2 d31113858 1113859

min f d22 + d3 + y1 + e

minus y2


g1 y1 minus 316

g2 24 minus y2

y1 x21 + x2 + x3 minus 02y2

y2 y1

radic+ x1 + x3


(22)







xi BPA ()


ndash01 ndash005 0 005 010

005

01

015

02

025

03

035

04

045

05

BPA

Δxi


Complexity 7




minus y2sl


+ d21 minus d2SL( 11138572

+ d31 minus d3SL( 11138572


J2 d12 minus d1SL( 11138572

+ d32 minus d3SL( 11138572


(23)





mdash DO 1600 0800 0 06670 3200 mdash


MCS 1700 0750 0 01230 34525 2 times 108


MCS 1700 0800 0 00502 3530 2 times 108


MCS 1725 0850 0 0 369813 2 times 108

0

01

02

03

04

05

06

07

08

f (x)

0

005

01

015

02

025

03

035

04

BPA

ndash1 0 1 2ndash2x

ndash1 0 1 2ndash2x



xi BPA


8 Complexity

find d d11 d21 d31 y11 y211113858 1113859


+ d21 minus d2sl( 11138572

+ d31 minus d3sl )2

1113872 1113873

+ y11 minus y1sl( 11138572


g1 y11 minus 316

y11 x211 + x21 + x31 minus 02y2sl


defined by

find d d12 d22 d32 y12 y221113858 1113859


+ d32 minus d3sl( 11138572

+ y22 minus y2sl( 11138572


g2 24 minus y22

y22 y1sl

radic+ x12 + x32









st Tmax le 1500K

σmax le 830MPa


xl le x le xu

(26)

Nor

mal

ized

lear

ning

func

tion

Nor

mal

ized

lear

ning

func

tion



0

0005

001

0

1

2

3

2 4 6 8 10 12 140Iteration



Complexity 9






Pl (g1 lt 0)

Pl (g1 lt 0) lt Pf

J1 Pl (g1 lt 0) J2 Pl (g2 lt 0)

Pl (g2 lt 0)d11 d21 d31 y11 y21


d12 d22 d32 y12 y22

System optimization


findminst







findmin






1

2

3

4

5

6

7


Num

ber

times104


10 Complexity

5 Conclusions





x5 170 [169 171]x6 169 [169 171]x7 169 [169 171]


x8 12 [120 125]x9 12 [120 125]x10 10 [10 105]




[minus 015 0] 01544[0 015] 04430

[015 030] 03423[030 045] 00604




x1

x2

x3

x4

x58

x69

x710

Rib4

Rib3

Rib2Rib1





x5 170 17073 17032x6 1709 16939 16997x7 16917 17068 17085


x8 123 12276 12447x9 12 12023 12458x10 105 10464 10965

Complexity 11



Data Availability




Acknowledgments


References























12 Complexity






















Complexity 13








Journal of












Journal of







Volume 2018






Volume 2018








find d d12 d221113858 1113859


+ d2sl minus d22( 11138572



(21)






of MCS


find d d1 d2 d31113858 1113859

min f d22 + d3 + y1 + e

minus y2


g1 y1 minus 316

g2 24 minus y2

y1 x21 + x2 + x3 minus 02y2

y2 y1

radic+ x1 + x3


(22)







xi BPA ()


ndash01 ndash005 0 005 010

005

01

015

02

025

03

035

04

045

05

BPA

Δxi


Complexity 7




minus y2sl


+ d21 minus d2SL( 11138572

+ d31 minus d3SL( 11138572


J2 d12 minus d1SL( 11138572

+ d32 minus d3SL( 11138572


(23)





mdash DO 1600 0800 0 06670 3200 mdash


MCS 1700 0750 0 01230 34525 2 times 108


MCS 1700 0800 0 00502 3530 2 times 108


MCS 1725 0850 0 0 369813 2 times 108

0

01

02

03

04

05

06

07

08

f (x)

0

005

01

015

02

025

03

035

04

BPA

ndash1 0 1 2ndash2x

ndash1 0 1 2ndash2x



xi BPA


8 Complexity

find d d11 d21 d31 y11 y211113858 1113859


+ d21 minus d2sl( 11138572

+ d31 minus d3sl )2

1113872 1113873

+ y11 minus y1sl( 11138572


g1 y11 minus 316

y11 x211 + x21 + x31 minus 02y2sl


defined by

find d d12 d22 d32 y12 y221113858 1113859


+ d32 minus d3sl( 11138572

+ y22 minus y2sl( 11138572


g2 24 minus y22

y22 y1sl

radic+ x12 + x32









st Tmax le 1500K

σmax le 830MPa


xl le x le xu

(26)

Nor

mal

ized

lear

ning

func

tion

Nor

mal

ized

lear

ning

func

tion



0

0005

001

0

1

2

3

2 4 6 8 10 12 140Iteration



Complexity 9






Pl (g1 lt 0)

Pl (g1 lt 0) lt Pf

J1 Pl (g1 lt 0) J2 Pl (g2 lt 0)

Pl (g2 lt 0)d11 d21 d31 y11 y21


d12 d22 d32 y12 y22

System optimization


findminst







findmin






1

2

3

4

5

6

7


Num

ber

times104


10 Complexity

5 Conclusions





x5 170 [169 171]x6 169 [169 171]x7 169 [169 171]


x8 12 [120 125]x9 12 [120 125]x10 10 [10 105]




[minus 015 0] 01544[0 015] 04430

[015 030] 03423[030 045] 00604




x1

x2

x3

x4

x58

x69

x710

Rib4

Rib3

Rib2Rib1





x5 170 17073 17032x6 1709 16939 16997x7 16917 17068 17085


x8 123 12276 12447x9 12 12023 12458x10 105 10464 10965

Complexity 11



Data Availability




Acknowledgments


References























12 Complexity






















Complexity 13








Journal of












Journal of







Volume 2018






Volume 2018











minus y2sl


+ d21 minus d2SL( 11138572

+ d31 minus d3SL( 11138572


J2 d12 minus d1SL( 11138572

+ d32 minus d3SL( 11138572


(23)





mdash DO 1600 0800 0 06670 3200 mdash


MCS 1700 0750 0 01230 34525 2 times 108


MCS 1700 0800 0 00502 3530 2 times 108


MCS 1725 0850 0 0 369813 2 times 108

0

01

02

03

04

05

06

07

08

f (x)

0

005

01

015

02

025

03

035

04

BPA

ndash1 0 1 2ndash2x

ndash1 0 1 2ndash2x



xi BPA


8 Complexity

find d d11 d21 d31 y11 y211113858 1113859


+ d21 minus d2sl( 11138572

+ d31 minus d3sl )2

1113872 1113873

+ y11 minus y1sl( 11138572


g1 y11 minus 316

y11 x211 + x21 + x31 minus 02y2sl


defined by

find d d12 d22 d32 y12 y221113858 1113859


+ d32 minus d3sl( 11138572

+ y22 minus y2sl( 11138572


g2 24 minus y22

y22 y1sl

radic+ x12 + x32









st Tmax le 1500K

σmax le 830MPa


xl le x le xu

(26)

Nor

mal

ized

lear

ning

func

tion

Nor

mal

ized

lear

ning

func

tion



0

0005

001

0

1

2

3

2 4 6 8 10 12 140Iteration



Complexity 9






Pl (g1 lt 0)

Pl (g1 lt 0) lt Pf

J1 Pl (g1 lt 0) J2 Pl (g2 lt 0)

Pl (g2 lt 0)d11 d21 d31 y11 y21


d12 d22 d32 y12 y22

System optimization


findminst







findmin






1

2

3

4

5

6

7


Num

ber

times104


10 Complexity

5 Conclusions





x5 170 [169 171]x6 169 [169 171]x7 169 [169 171]


x8 12 [120 125]x9 12 [120 125]x10 10 [10 105]




[minus 015 0] 01544[0 015] 04430

[015 030] 03423[030 045] 00604




x1

x2

x3

x4

x58

x69

x710

Rib4

Rib3

Rib2Rib1





x5 170 17073 17032x6 1709 16939 16997x7 16917 17068 17085


x8 123 12276 12447x9 12 12023 12458x10 105 10464 10965

Complexity 11



Data Availability




Acknowledgments


References























12 Complexity






















Complexity 13








Journal of












Journal of







Volume 2018






Volume 2018








find d d11 d21 d31 y11 y211113858 1113859


+ d21 minus d2sl( 11138572

+ d31 minus d3sl )2

1113872 1113873

+ y11 minus y1sl( 11138572


g1 y11 minus 316

y11 x211 + x21 + x31 minus 02y2sl


defined by

find d d12 d22 d32 y12 y221113858 1113859


+ d32 minus d3sl( 11138572

+ y22 minus y2sl( 11138572


g2 24 minus y22

y22 y1sl

radic+ x12 + x32









st Tmax le 1500K

σmax le 830MPa


xl le x le xu

(26)

Nor

mal

ized

lear

ning

func

tion

Nor

mal

ized

lear

ning

func

tion



0

0005

001

0

1

2

3

2 4 6 8 10 12 140Iteration



Complexity 9






Pl (g1 lt 0)

Pl (g1 lt 0) lt Pf

J1 Pl (g1 lt 0) J2 Pl (g2 lt 0)

Pl (g2 lt 0)d11 d21 d31 y11 y21


d12 d22 d32 y12 y22

System optimization


findminst







findmin






1

2

3

4

5

6

7


Num

ber

times104


10 Complexity

5 Conclusions





x5 170 [169 171]x6 169 [169 171]x7 169 [169 171]


x8 12 [120 125]x9 12 [120 125]x10 10 [10 105]




[minus 015 0] 01544[0 015] 04430

[015 030] 03423[030 045] 00604




x1

x2

x3

x4

x58

x69

x710

Rib4

Rib3

Rib2Rib1





x5 170 17073 17032x6 1709 16939 16997x7 16917 17068 17085


x8 123 12276 12447x9 12 12023 12458x10 105 10464 10965

Complexity 11



Data Availability




Acknowledgments


References























12 Complexity






















Complexity 13








Journal of












Journal of







Volume 2018






Volume 2018













Pl (g1 lt 0)

Pl (g1 lt 0) lt Pf

J1 Pl (g1 lt 0) J2 Pl (g2 lt 0)

Pl (g2 lt 0)d11 d21 d31 y11 y21


d12 d22 d32 y12 y22

System optimization


findminst







findmin






1

2

3

4

5

6

7


Num

ber

times104


10 Complexity

5 Conclusions





x5 170 [169 171]x6 169 [169 171]x7 169 [169 171]


x8 12 [120 125]x9 12 [120 125]x10 10 [10 105]




[minus 015 0] 01544[0 015] 04430

[015 030] 03423[030 045] 00604




x1

x2

x3

x4

x58

x69

x710

Rib4

Rib3

Rib2Rib1





x5 170 17073 17032x6 1709 16939 16997x7 16917 17068 17085


x8 123 12276 12447x9 12 12023 12458x10 105 10464 10965

Complexity 11



Data Availability




Acknowledgments


References























12 Complexity






















Complexity 13








Journal of












Journal of







Volume 2018






Volume 2018








5 Conclusions





x5 170 [169 171]x6 169 [169 171]x7 169 [169 171]


x8 12 [120 125]x9 12 [120 125]x10 10 [10 105]




[minus 015 0] 01544[0 015] 04430

[015 030] 03423[030 045] 00604




x1

x2

x3

x4

x58

x69

x710

Rib4

Rib3

Rib2Rib1





x5 170 17073 17032x6 1709 16939 16997x7 16917 17068 17085


x8 123 12276 12447x9 12 12023 12458x10 105 10464 10965

Complexity 11



Data Availability




Acknowledgments


References























12 Complexity






















Complexity 13








Journal of












Journal of







Volume 2018






Volume 2018










Data Availability




Acknowledgments


References























12 Complexity






















Complexity 13








Journal of












Journal of







Volume 2018






Volume 2018





























Complexity 13








Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








evidence-based multidisciplinary design optimization with

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