Research ArticleAn Enhanced Analytical Target Cascading and Kriging ModelCombined Approach for Multidisciplinary Design Optimization

Ping Jiang Jianzhuang Wang Qi Zhou and Xiaolin Zhang

The State Key Laboratory of Digital Manufacturing Equipment and Technology School of Mechanical Science and EngineeringHuazhong University of Science amp Technology Wuhan 430074 China

Correspondence should be addressed to Jianzhuang Wang wjzmailhusteducn

Received 20 November 2014 Revised 16 February 2015 Accepted 16 February 2015

Academic Editor Ricardo Aguilar-Lopez

Copyright copy 2015 Ping Jiang et al This 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

Multidisciplinary design optimization (MDO) has been applied widely in the design of complex engineering systems To easeMDOproblems analytical target cascading (ATC) organizes MDO process into multilevels according to the components of engineeringsystems which provides a promising way to deal with MDO problems ATC adopts a coordination strategy to coordinate thecouplings between two adjacent levels in the design optimization process however existing coordination strategies in ATC facethe obstacles of complicated coordination process and heavy computation cost In order to conquer this problem a quadraticexterior penalty function (QEPF) based ATC (QEPF-ATC) approach is proposed where QEPF is adopted as the coordinationstrategy Moreover approximate models are adopted widely to replace the expensive simulationmodels inMDO a QEPF-ATC andKriging model combined approach is further proposed to deal with MDO problems owing to the comprehensive performancehigh approximation accuracy and robustness of Kriging model Finally the geometric programming and reducer design cases aregiven to validate the applicability and efficiency of the proposed approach

1 Introduction

Multidisciplinary design optimization (MDO) is a method-ology to design complex engineering systems by exploitingthe interactions between disciplines and has been successfullyapplied in the design of complex engineering systems suchas aircraft automobiles and mechanical equipment [1 2]MDO method also known as MDO strategy is a hot topicin the research area of MDO currently [3] Depending onwhether the process of design optimization is hierarchicallydesigned the MDO method is divided into two categoriessingle-level MDO method and multilevels MDO methodCompared with single-level MDO method the multilevelsMDOmethod can better match the organization structure ofcomplex engineering systems and is beneficial in implement-ing concurrent design anddistributed computationThere arefour main multilevels methods collaborative optimization(CO) [4 5] concurrent subspace optimization (CSSO) [67] bilevel integrated system synthesis (BLISS) [8 9] andanalytical target cascading (ATC) [10 11] CO CSSO andBLISS are all two-level MDO methods Compared with

CO CSSO and BLISS methods ATC can organize MDOprocess into different levels according to the components ofengineering systems and the convergence of ATC methodhas been proofed strictly [10] so ATC provides a promisingway to deal with MDO problems

Currently ATC needs to adopt a coordination strategy tocoordinate the couplings between two adjacent levels in thedesign optimization process and an appropriate coordina-tion strategy can improve the computation efficiency of ATCTill now some coordination strategies have been adoptedin ATC such as augmented Lagrangian function (ALF)[12] diagonal quadratic approximation (DQA) [13] andLagrangian dual function (LDF) [14] However these coor-dination strategies based ATC approaches face the obstaclesof complicated coordination process and heavy computationcost in dealing withMDO problems In order to conquer thisproblem a quadratic exterior penalty function (QEPF) basedATC (QEPF-ATC) approach is proposed in this paper whereQEPF is adopted as the coordination strategy Comparedwithexiting ATC method the QEPF-ATC approach can simplifythe coordination process and reduce the computation cost

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 685958 11 pageshttpdxdoiorg1011552015685958

2 Mathematical Problems in Engineering

Component

System

Subsystem

Cascade down

Rebalance up

Figure 1 Schematic of the ATC process

In the design optimization of complex engineering sys-tems generally a large number of simulation models withhigh accuracy (such as the structural finite element analysismodels aerodynamic analysis models and computationalfluid dynamicmodels) are involvedThese simulationmodelsincrease the computation expense greatlyTherefore approx-imate models are adopted widely to replace the expensivesimulation models in MDO [15ndash18] In this paper QEPF-ATC and Kriging model [15 18] combined approach isfurther proposed to deal with MDO problems owing to thecomprehensive performance high approximation accuracyand robustness of Kriging model Finally the geometricprogramming and reducer design cases are given to vali-date usability and effectiveness of the proposed QEPF-ATCapproach and the QEPF-ATC and Kriging model combinedapproach respectively

The rest of this paper is organized as follows In Section 2ATC is introduced and the QEPF-ATC approach is pro-posed QEPF-ATC and Kriging model combined approachis implemented in Section 3 In Section 4 the geometricprogramming and reducer design cases are given to validatethe usability and effectiveness of the proposed approachConclusion and future work are discussed in Section 5

2 The Proposed QEPF-ATC Approach

21 ATC The design optimization problem of complexengineering systems is decomposed into a set of subproblemsin ATC approach where the desired design specificationsor targets are established at the top level and cascadeddown to each lower level subproblem The target transfor-mation is a decomposition process from up to down in alevel-by-level way Moreover subproblems need to get theresponses to match the cascaded targets If targets cascadedto subsystems are not achievable or compatible the feedbackfrom subsystems is required resulting in an iterative targetcascading process Hence targets and responses are updatedand coordinated iteratively to achieve global consistency (seeFigure 1) [10] Coordination strategy is needed to implementthe communication of target and response values betweenparents and children problems

ATC is a model-based optimization method There existtwo kinds of models optimal design models 119875 and analysismodels 119877 The former is just so-called optimization designproblem that includes optimization object design variables

Optimal design model

Analysis model

Optimal design model k1 Optimal design model k2

Level i

Level i minus 1

Level i + 1

Rijxij

Ri(i+1)j Ri(i+1)j

yi(i+1)j yi(i+1)j yi+1(i+1)k2

Ri+1(i+1)k2

yi+1(i+1)k1

Ri+1(i+1)k1Ri+1(i+1)k1

Ri+1(i+1)k2

Riij yiij Riminus1ij yiminus1ip

Figure 2 Information flow between children subproblem andparent subproblem

and constraints The latter is physical model or mathematicalmodel that is the foundation of the optimization designFigure 2 shows the data flow between subproblems (parentand children) [19]

In Figure 2 there exist consistency constraints for eachsubproblem In order tomake each subproblem independentconsistency constraint is introduced as a kind of equalityconstraints For the subproblem at a given level consistencyconstraints are as follows

1198881015840

1= (119905119894119895minus 119903119894119895) = 0 119888

1015840

1= (119877119894minus1

119894119895minus 119877119894

119894119895) = 0

1198881015840

1= (119910119894minus1

119894119901minus 119910119894

119894119895) = 0

1198881015840

2= (119905(119894+1)119896

minus 119903(119894+1)119896

) = 0

1198881015840

2= (119877119894

(119894+1)119895minus 119877119894+1

(119894+1)119896) = 0 119888

1015840

2= (119910119894

(119894+1)119895minus 119910119894+1

(119894+1)119896) = 0

(1)

The general mathematical notation for subproblems119875119894119895in

the hierarchy is then

119875119894119895 min 119891

119894119895

wrt 119909119894119895 119910119894

119894119895 119877119894

(119894+1)119895 119910119894

(119894+1)119895

where 119877119894119895= 119903119894119895(119877119894

(119894+1)119895 119909119894119895 119910119894

119894119895)

st 1198881= (119877119894minus1

119894119895minus 119877119894

119894119895) = 0 119888

2= (119910119894minus1

119894119901minus 119910119894

119894119895) = 0

1198883= (119877119894

(119894+1)119895minus 119877119894+1

(119894+1)119896) = 0

1198884= (119910119894

(119894+1)119895minus 119910119894+1

(119894+1)119896) = 0

119892119894119895(119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895) le 0

ℎ119894119895(119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895) = 0

(2)

where 119875119894119895is the problem formulation of element 119895 at level 119894

119901 is used to designate the parent of element 119895 119896 designateschildren of element 119895 119909

119894119895is local variables for subproblem

119875119894119895 119903119894119895is the function that calculates responses for 119875

119894119895 119877119894119894119895is

Mathematical Problems in Engineering 3

response of 119875119894119895 119877119894minus1119894119895

is the target for 119875119894119895cascaded from the

(119894 minus 1) level parent element 119877119894+1(119894+1)119896

is response of the (119894 + 1)level children element 119877119894

(119894+1)119895= (119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

) is the

target for the (119894 + 1) level children element 119910119894119894119895is response of

linking variable of 119875119894119895 119910119894minus1119894119901

is the target of linking variable for119875119894119895from the (119894 minus 1) level parent element 119910119894+1

(119894+1)119896is response of

linking variable of the (119894 + 1) level children element 119910119894(119894+1)119895

is the target of linking variable for the (119894 + 1) level childrenelement from119875

119894119895119891119894119895is the local objective for119875

119894119895119892119894119895is function

of inequality constraints for 119875119894119895in negative null form ℎ

119894119895is

function of equality constraints for 119875119894119895in null form

The coordination strategy in ATC deals with the defi-nition of deviation between targets and responses (namelyconsistency constraint) and updating weight coefficientstherein By setting appropriate norm and weight coefficientsto coordinate each subproblem the optimal design point canbe obtained The coordination strategy is the key research inATC and an appropriate coordination strategy can improvethe computation efficiency of ATC In recent years somecoordination strategies in ATC have been studied

211 The Coordination Strategy Based on ALF Accordingto the ALF the deviation between targets and responses ispresented by augmented Lagrangian function and weightcoefficients are presented by the augmented Lagrange mul-tiplier The coordination way based on ALF is shown as

min 119891119894119895+ 120582119877

119894119895(119877119894

119894119895minus 119877119894minus1

119894119895) +10038171003817100381710038171003817119908119877

119894119895∘ (119877119894

119894119895minus 119877119894minus1

119894119895)10038171003817100381710038171003817

2

+ 120582119910

119894119895(119910119894minus1

119894119901minus 119910119894

119894119895) +10038171003817100381710038171003817119908119910

119894119895∘ (119910119894minus1

119894119901minus 119910119894

119894119895)10038171003817100381710038171003817

2

+ 120582119877

(119894+1)119895(119877119894

(119894+1)119895minus 119877119894+1

(119894+1)119896)

+10038171003817100381710038171003817119908119877

(119894+1)119895∘ (119877119894

(119894+1)119895minus 119877119894+1

(119894+1)119896)10038171003817100381710038171003817

2

+ 120582119910

(119894+1)119895(119910119894

(119894+1)119895minus 119910119894+1

(119894+1)119896)

+100381710038171003817100381710038171003817119908119910

(119894+1)119895(119910119894

(119894+1)119895minus 119910119894+1

(119894+1)119896)100381710038171003817100381710038171003817

2

st 119892119894119895(119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895) le 0

ℎ119894119895(119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895) = 0

(3)

Weight coefficients 120582119877119894119895 120582119910119894119895119908119877119894119895 and119908119910

119894119895in (3) are updated

in accordance with the following equation

120582119877(119896+1)

119894119895= 120582119877(119896)

119894119895+ 2119908119877(119896)

119894119895(119877119894(119896)

119894119895minus 119877119894minus1(119896)

119894119895)

120582119910(119896+1)

119894119895= 120582119910(119896)

119894119895+ 2119908119910(119896)

119894119895(119910119894(119896)

119894119895minus 119910119894minus1(119896)

119894119895)

119908119877(119896+1)

119894119895= 120573119908119877(119896)

119894119895

119908119910(119896+1)

119894119895= 120573119908119910(119896)

119894119895

(4)

where the superscript 119896 indicates iterations and 120573 is steplength Usually 120573 ge 1 but 2 lt 120573 lt 3 can accelerateconvergence

212 The Coordination Strategy Based on DQA DQA isrealized by approximating the deviation between targets andresponses on the basis of ALF The specific approximation is10038171003817100381710038171003817119877119894

119894119895minus 119877119894minus1

119894119895

10038171003817100381710038171003817

2

=100381710038171003817100381710038171003817(119877119894

119894119895)2

minus 2119877119894

119894119895119877119894minus1

119894119895+ (119877119894minus1

119894119895)2100381710038171003817100381710038171003817

where

119877119894

119894119895119877119894minus1

119894119895cong 119877119894(119896minus1)

119894119895119877119894minus1(119896minus1)

119894119895+ 119877119894(119896minus1)

119894119895(119877119894

119894119895minus 119877119894minus1(119896minus1)

119894119895)

+ 119877119894minus1(119896minus1)

119894119895(119877119894

119894119895minus 119877119894(119896minus1)

119894119895) so

10038171003817100381710038171003817119877119894

119894119895minus 119877119894minus1

119894119895

10038171003817100381710038171003817

2

=10038171003817100381710038171003817119877119894minus1(119896minus1)

119894119895minus 119877119894

119894119895

10038171003817100381710038171003817

2

+10038171003817100381710038171003817119877119894

119894119895minus 119877119894minus1(119896minus1)

119894119895

10038171003817100381710038171003817

2

+ 119877119894(119896minus1)

119894119895119877119894minus1(119896minus1)

119894119895(constant value)

(5)

Based on DQA the cross term among the deviation isapproximated that is to say an order Taylor type expansion isconducted in the last design point Among them the weightcoefficient is updated according to (5) too

213 The Coordination Strategy Based on LDF According tothe LDF the deviation is dealt with by Lagrangian dualityfunction and weight coefficients are presented by the dualLagrange multiplier The coordination of LDF is shown as

min 119891119894119895+ 120582119877

119894119895(119877119894

119894119895minus 119877119894minus1

119894119895) +10038171003817100381710038171003817120583119877

119894119895∘ (119877119894

119894119895minus 119877119894minus1

119894119895)10038171003817100381710038171003817

2

+ 120582119910

119894119895(119910119894minus1

119894119901minus 119910119894

119894119895) +10038171003817100381710038171003817120583119910

119894119895∘ (119910119894minus1

119894119901minus 119910119894

119894119895)10038171003817100381710038171003817

2

st 119892119894119895(119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895) le 0

ℎ119894119895(119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895) = 0

(6)

Weight coefficients 120582119877119894119895 120582119910119894119895 120583119877119894119895 and 120583119910

119894119895in (6) are updated

in accordance with the following equation

120582119877(119896+1)

119894119895= 120582119877(119896)

119894119895+ 120572119877(119896)

119894119895(119877119894(119896)

119894119895minus 119877119894minus1(119896)

119894119895)

120582119910(119896+1)

119894119895= 120582119910(119896)

119894119895+ 120572119910(119896)

119894119895(119910119894(119896)

119894119895minus 119910119894minus1(119896)

119894119895)

120583119877(119896+1)

119894119895= radic

10038161003816100381610038161003816120582119877(119896)

119894119895

10038161003816100381610038161003816 120583

119910(119896+1)

119894119895= radic

100381610038161003816100381610038161003816120582119910(119896)

119894119895

100381610038161003816100381610038161003816

120572119877(119896+1)

119894119895=1 + 119898

119896 + 119898sdot

1

10038171003817100381710038171003817119877119894(119896)

119894119895minus 119877119894minus1(119896)

119894119895

10038171003817100381710038171003817

120572119910(119896+1)

119894119895=1 + 119898

119896 + 119898sdot

1

10038171003817100381710038171003817119910119894(119896)

119894119895minus 119910119894minus1(119896)

119894119895

10038171003817100381710038171003817

(7)

4 Mathematical Problems in Engineering

where the superscript 119896 indicates iterations and119898 is a positiveintegers

The difference of these coordination strategies is theway of dealing with consistency constraints and weightcoefficients There exist multiweight coefficients in the abovethree coordination strategies to increase computational costFor this reason the coordination strategy based on quadraticexterior penalty function among which only one weight coef-ficient exists is put forward which simplifies the expression ofoptimization problem greatly thus improving the efficiencyto make ATC have wider application The coordinationstrategy based on the quadratic exterior penalty function willbe described briefly

22 The QEPF-ATC Approach A general equality-con-strained optimization problem can be defined as follows

min 119891 (x)

st 119888119894(x) = 0 119894 = 1 2 119898

xlb le x le xup

(8)

where x is the vector of the design variables with xlb andxup as its lower and upper bounds respectively The function119891(x) represents the objective function 119888

119894(x) is the equality

constraintsAccording to QEPF consistency constraints are added

into the objective function by introducing a penalty factorand the penalty factor is used as the weighting coefficientto realize the coordination The quadratic exterior penaltyfunction is defined as

119865 (119909 120588) = 119891 (x) +120588

2

119898

sum

119894=1

119888119894(x)2 = 119891 (x) +

120588

2119888 (x)119879 119888 (x) (9)

where the weight coefficient (penalty factor) is updated inaccordance with (10) The value of 120582 influences the con-vergence efficiency and computational efficiency so that itis necessary to set up appropriate 120582 to obtain much bettersolution efficiency Consider

120588119896+1= 120582120588119896 120582 = 14sim10 (10)

The mathematical formulation of QEPF-ATC is

119875119894119895 min 119891

119894119895+120588

2(10038171003817100381710038171003817(119877119894minus1

119894119895minus 119877119894

119894119895)10038171003817100381710038171003817

2

+10038171003817100381710038171003817(119910119894minus1

119894119901minus 119910119894

119894119895)10038171003817100381710038171003817

2

+10038171003817100381710038171003817(119877119894

(119894+1)119895minus 119877119894+1

(119894+1)119896)10038171003817100381710038171003817

2

+10038171003817100381710038171003817(119910119894

(119894+1)119895minus 119910119894+1

(119894+1)119896)10038171003817100381710038171003817

2

)

st 119892119894119895le 0 ℎ

119894119895= 0

119877119894119895= 119903119894119895(119909119894119895)

119909119894119895= (119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895)

(11)

Generally the much bigger the penalty factor is the muchcloser the optimization results are to the optimal solution

Optimizing system subproblem

Yes

Converged

Solving subproblems

Decomposing the problem

Initializing

No Updating weight

Optimized results

Figure 3 The solving process of QEPF-ATC approach

of the original problem But in the practical applicationthe value of the penalty factor must be moderate whenconsidering the computational cost due to the fact that ifthe penalty factor is too small the optimization results mayget bigger error from the original problem to bring lowcomputational efficiency Otherwise too large penalty factorwould increase the computational expense

Figure 3 shows the solving process of the QEPF-ATCapproach and the main steps are as follows

(1) decompose the original problem and confirm designvariables and constraints of system and subsystems

(2) initialize design variables (including local designvariables and linking variables) and 120588(0) and transferthe corresponding initial value to the subsystem as thetarget

(3) solve each subsystem and pass back current results tosystem level

(4) optimize the system problem on the basis of the thirdstep and transfer new target to each subsystem

(5) obtain the optimization result if the convergence con-dition is met otherwise update the weight coefficient120588 according to (10) and go back to the third step untilthe convergence condition is met

3 The Proposed QEPF-ATC and KrigingModel Combined Approach

31 Kriging Model Kriging models [15 16] have their originsin mining and geostatistical applications involving spatially

Mathematical Problems in Engineering 5

Decomposing the problem

Subsystem analyzing

Subsystem optimizing

System optimizing

Converged

No

Yes

Yes

No

Creating the Kriging model

Sampling by LHS

Optimized results

Validating approximation model

Figure 4The solving process of the QEPF-ATC and Kriging modelcombined approach

and temporally correlated data Kriging models combine aglobal model with localized departures

119910 (119909) = 119891 (119909) + 119911 (119909) (12)

where 119910(119909) is the unknown function of interest 119891(119909) is theknown approximation (usually polynomial) function and119911(119909) is the realization of a stochastic process with mean zerovariance1205902 and nonzero covarianceThe119891(119909) term is similarto a polynomial response surface providing a ldquoglobalrdquo modelof the design space The covariance of 119911(119909) is given by

cov [119911 (119909119894) 119911 (119909

119895)] = 120590

2119877 (119909119894 119909119895) (13)

where119877 is the correlation119877(119909119894 119909119895) is the correlation function

between any two of the 119899 sampled data points 119909119894and 119909

119895 and

it can be calculated by

119877 (119909119894 119909119895) = exp[minus

119899

sum

119896=1

120579119896(119909119896

119894minus 119909119896

119895)2

] (14)

Generally 119891(119909) is treated as a constant 120573 then predictedestimates

_119910 (119909) of the response 119910(119909) at untried values of 119909

are given by_119910 (119909) =

_120573 +119903119879

(119909) 119877minus1(119884 minus 119865

_120573) (15)

where 119910 is the column vector of length 119899 that contains thesample values of the response and 119891 is a column vector oflength 119899 that is filled with ones when 119891(119909) is taken as aconstant In (10) 119903119879(119909) is the correlation vector of length 119899between an untried 119909 and the sampled data points

Consider

119903119879

(119909) = [119877 (119909 1199091) 119877 (119909 119909

2) 119877 (119909 119909

3) 119877 (119909 119909

119899)]119879

(16)

In (10)_120573 is estimated using (12)

_120573 = (119865

119879119877minus1119865)minus1

119865119879119877minus1119884 (17)

In 119877 120579119896= maxminus[119899 ln(1205902) + ln |119877|]2

1205902=

(119884 minus 119865

_120573)

119879

119877minus1(119884 minus 119865

_120573)

119899

(18)

32 The QEPF-ATC and Kriging Model Combined ApproachIn QEPF-ATC the deviation between the target and responseis processed in the form of sum of squares as shown in (11)The sum of squares function is usually not smooth and non-linear And discrepancy of variables in the neighborhood ofthe current solution will lead to fluctuation of the deviationwhich will increase the number of iterations and so forthKriging model is appropriate for high nonlinear problemwith better comprehensive performance high approximationaccuracy and high robustness [20ndash22] Moreover fewersample points also can get higher precision approximationmodel So Kriging model is adopted in this paper

The solving process of the QEPF-ATC and Kriging modelcombined approach is shown in Figure 4 and the main stepsare as follows

(1) decompose the original problem and confirm designvariables and constraints of system and subsystems

(2) select a set of sample points using the Latin hypercubesampling (LHS) method

(3) analyse each subsystem to obtain responses at allsampling points of the subsystems

(4) build Kriging model based on the sample points andthe corresponding responses of each subsystem

(5) evaluate the Kriging model by error analysis if theprecision conditions are not met go back to step (2)to build a new Kriging model by adding some newsample points

(6) initialize variables and transfer the corresponding ini-tial value to the subsystem as the target and optimizeeach subproblem based on the created Krigingmodel

6 Mathematical Problems in Engineering

g1 g2

x1 x2 x3 x4 x5 x6 x7 x11 ge 0

xU3

xU11B

xL3

xL11B

g3 g4

x3 x8 x9 x10 x11 ge 0

x11

xU6

xU11C

xL6

xL11C

g5 g6

x6 x11 x12 x13 x14 ge 0

st

st st

R1 = x1 = r1(x3 x4 x5) = (x23 + xminus24 + x25)12

R2 = x2 = r2(x5 x6 x7) = (x25 + x26 + x27)12

R3 = x3 = r3(x8 x9 x10 x11)= (x28 + xminus29 + xminus210 + x

211)

12

PC minx6x11 x12 x13 x14

f22 =120590

2x6 minus x

U6 )) 2

+ (x11 minus xU11C)2]]

PA minx1x2x3x4x5x6x7x11

f11 = x21 + x22 +

120590

2x3 minus x

L3)

2+ (x6 minus xL6)

2+ (x11 minus xL11B)

2+ (x11 minus xL11C)

2])]

R4 = x6 = r4(x11 x12 x13 x14)= (x211 + x212 + x213 + x214)

12

xPB min

x3x8x9x10f21 =

120590

2x3 minus x

U3 )

2+ (x11 minus xU11B)

2])]11

Figure 5 The solving framework of QEPF-ATC for geometric programming

(7) pass back current results to system level and imple-ment the system level optimization

(8) obtain results if the convergence condition is metotherwise go to step (6) until the convergence con-dition is met

4 Case Studies

41 The Geometric Programming The geometric program-ming [23] that has only the global optimal solution isemployed to verify the usability of QEPF-ATC Its mathemat-ical expression is shown as

min 119891 = 1199092

1+ 1199092

2

st 1198921119909minus2

3+ 1199092

4

1199092

5

le 1 11989221199092

5+ 119909minus2

6

1199092

7

le 1

11989231199092

8+ 1199092

9

1199092

11

le 1

1198924119909minus2

8+ 1199092

10

1199092

11

le 1 11989251199092

11+ 119909minus2

12

1199092

13

le 1

11989261199092

11+ 1199092

12

1199092

14

le 1

ℎ1= 1199092

1minus 1199092

3minus 119909minus2

4minus 1199092

5= 0

ℎ2= 1199092

2minus 1199092

5minus 1199092

6minus 1199092

7= 0

ℎ3= 1199092

3minus 1199092

8minus 119909minus2

9minus 119909minus2

10minus 1199092

11= 0

ℎ4= 1199092

6minus 1199092

11minus 1199092

12minus 1199092

13minus 1199092

14= 0

(19)

Table 1 The optimal results

QEPF-ATC ALF DQA LDF AAO1199091

29049 285 286 287 2841199092

30551 304 310 305 3091199093

24930 248 249 243 2361199094

08805 086 082 079 0761199095

09676 094 095 089 0871199096

27103 272 275 278 2811199097

10357 102 095 103 0941199098

09830 098 098 101 0971199099

07572 078 079 082 08711990910

07104 076 079 081 08011990911

12408 127 126 125 13011990912

08409 086 085 083 08411990913

17088 173 174 171 17611990914

14875 152 149 150 155119891 177721 1736 1778 1753 1759Runtime (s) 289 355 336 368

Figure 5 shows the solution framework of the QEPF-ATCapproach for the problem where 119909

11is the linking variable

(coupling variable) coordinated by the system level 11990911988011119861

119909119880

11119862 1199091198803 and 119909119880

6are the targets of each subproblem passed

from the system level 11990911987111119861

and 11990911987111119862

are the responses of thelinking variable coming from the subproblems 119861 and 119862 119909119871

3

and 1199091198716are the responses of 119861 and 119862

Solve the geometric programming problem according toQEPF-ATC Initialize the design point 1199090 = (1 1 1 1 1 1 11 1 1 1 1 1 1)

119879 and take 120582 = 10 Here the acceptableinconsistency tolerance is 001 for every response variable andlinking variable Table 1 shows the optimal results based on

Mathematical Problems in Engineering 7

Table 2 The optimal results based on 120582

120582 (1199091 1199092 1199093 1199094 1199095 1199096 1199097 11990911) (119909

3 1199098 1199099 11990910 11990911) (119909

6 11990911 11990912 11990913 11990914) 119891 Iterations

14 (27556 30804 22975 0813109171 27718 09840 12678) (23012 09688 08427 07340 12682) (27638 12732 07447 18411 14655) 175992 78

4 (29716 29567 25227 0759808570 26708 09352 12343) (25230 09911 07320 07072 12321) (26791 12298 08409 16923 14686) 175728 27

7 (29662 29757 25012 0720408231 27169 09014 12386) (25011 09863 07492 07115 12386) (27168 12385 08408 17122 14915) 176531 17

10 (29049 30551 24930 0880509676 27103 10357 12408) (24930 09830 07572 07104 12408) (27102 12409 08409 17088 14875) 177721 5

Table 3 Design variables for speed reducer

Variables Range Units1199091 Gear face width 26 le 119909

1le 36 cm

1199092 Teeth module 07 le 119909

2le 08 cm

1199093

Number of teeth of pinion(integer variable) 17 le 119909

3le 28

1199094

Distance between bearings1 73 le 119909

4le 83 cm

1199095

Distance between bearings2 73 le 119909

5le 83 cm

1199096 Diameter of shaft 1 29 le 119909

6le 39 cm

1199097 Diameter of shaft 1 50 le 119909

7le 55 cm

1199091 1199092 and 119909

3are the design variables of the gear subproblem 119909

4and 1199095are

the design variables of the shaft subproblem 1199096and 1199097are linking variables

(coupling variables)

QEPF-ATC three coordination strategies (ALF DQA andLDF) and AAO (All-At-Once)

According to Table 1 the results of the QEPF-ATCapproach are very close to the optimal value of the originalproblem and the runtime is less than the other threemethods120582 affects the convergence efficiency and computational

efficiency of the QEPF-ATC approach In order to obtainmore suitable 120582 to improve the efficiency of the QEPF-ATC approach different values of 120582 are selected to solve theproblem Table 2 shows the results when different values of 120582are selected

In Table 2 the QEPF-ATC approach implement fastconvergence when 120582 is set to different values and the optimalresults are close to the results of the initial problems Withthe increase of 120582 the number of iterations decreases andthe solution error gets bigger (excessive weight) Generally14 le 120582 le 10 is strictly necessary but typically 4 lt

120582 lt 7 is recommended to speed up convergence and reducecomputational cost and get better optimal results

42 The Speed Reducer Design The speed reducer designcase represents the design of a simple gearbox and is amultidisciplinary problem comprising the coupling betweengear design and shaft design This has been used as a testingproblem for MDO method in the literature from NASALangley research center [24] The multidisciplinary systemsand notations of the reducer are shown in Figure 6 and

x1

x2

x3

x4x5

x6

x7

Figure 6 The multidisciplinary systems and notations of reducer

Speed reducer

x6 x7 Shaftx4 x5

Gearx1 x2 x3

Figure 7 The decomposition of speed reducer problem

the modelling is shown in Figure 7 The design objectiveis to minimize the speed reducer weight while satisfying anumber of constraints posed by gear and shaft parts Thereare seven variables in the speed reducer design optimizationTable 3 lists a brief description of the variables and gives theirreference values

According to the decomposition method of ATC wherea system is partitioned by object the speed reducer problemis decomposed into gear subproblem and shaft subproblem(shown in Figure 7) The design optimization problem isdefined as follows

min 119891 = 1198911(119909) + 119891

2(119909)

1198911(119909) = 07854119909

11199092

2

sdot (333331199092

3+ 149334119909

3minus 430934)

minus 150791199091(1199092

6+ 1199092

7)

8 Mathematical Problems in Engineering

The optimizer of system level

The optimizer of subproblem 1 The optimizer of subproblem 2

DOE-LHS DOE-LHS

Subproblem 1

analysis

Kriging model 1

Subproblem 2

analysis

Kriging model 2

g3 g4 g5 g6

xU8 xU7B x

U6B xL8 x

L7B x

L6B xU9 x

U6C x

U7C xL9 x

L6C x

L7C

X1 X1

X1

X1 G1

G1X2 X2

X2

X2 G2

G2

st

st st

R1 = x10 = r1(x3 x4 x5) = x8 + x9

X0 = x6 x7

G1 = g1 g2 g9 g10 g11R2 = x8 = r2(X1) = f1(x)

X1 = x1 x2 x3 x6 x7

G2 = g7 g8R3 = x9 = r3(X2) = f2(x)

X2 = x4 x5 x6 x7

PA min f11 = x10 + x8 minus xL8)

2+ (x9 minus xL9)

2+ (x6 minus xL6B)

2+ (x6 minus xL6C)

2+ (x7 minus xL7B)

2+ (x7 minus xL7C)

2]120590

2)]

PB min f21 = x8 minus xU8 )

2+ (x6 minus xU6B)

2+ (x7 minus xU7B)

2]120590

2)] PC min f22 = x9 minus x

U9 )

2+ (x6 minus xU6C)

2+ (x7 minus xU7C)

2]120590

2)]

Figure 8 The solving framework of the QEPF-ATC and Kriging model combined approach for the gear reducer

1198912(119909) = 7477 (119909

3

6+ 1199093

7) + 07854 (119909

41199092

6+ 11990951199092

7)

st 1198921=

27

(11990911199092

21199093)minus 1 le 0 119892

2=

3975

(11990911199092

21199092

3)minus 1 le 0

1198923=1931199093

4

(119909211990931199094

6)minus 1 le 0 119892

4=1931199093

5

(119909211990931199094

7)minus 1 le 0

1198925= 10119909

minus3

6radic(745119909

minus1

2119909minus1

31199094) + 169 times 107

minus 1100 le 0

1198926= 10119909

minus3

7radic(745119909

minus1

2119909minus1

31199095) + 1575 times 108

minus 850 le 0

1198927=(151199096+ 19)

1199094

minus 1 le 0

1198928=(151199097+ 19)

1199095

minus 1 le 0

1198929= 11990921199093minus 40 le 0 119892

10= 5 minus

1199091

1199092

le 0

11989211=1199091

1199092

minus 12 le 0

(20)

where 1198911is the gear weight and 119891

2is the shaft weight And 119892

1

and 1198922 are respectively gear bending stress and contact stress

constraints 1198923and 119892

4are respectively torsional distortion

constraints of shaft 1 and shaft 2 1198925and 119892

6are stress

constraints of the shaft 1 and shaft 2 1198927and 119892

8are experience

constraints of shaft 1 and shaft 2 and 1198929 11989210 and 119892

11are gear

geometry constraints1199091 1199092 and 119909

3are the design variables of the gear

subproblem 1199094and 119909

5are the design variables of the

shaft subproblem 1199096and 119909

7are linking variables (coupling

variables) According to the QEPF-ATC and Kriging modelcombined approach three variables 119909

8 1199099 and 119909

10are

introduced and meet the following formula

ℎ0= 1199098+ 1199099minus 11990910= 0

ℎ1= 1199098minus 1198911(119909) = 0

ℎ2= 1199099minus 1198912(119909) = 0

(21)

where ℎ0represents the analysis model of system level and

ℎ1and ℎ

2represent the analysis models of subproblem 1 and

subproblem 2 respectivelyThe solving framework of the QEPF-ATC and Kriging

model combined approach for gear reducer is shown inFigure 8

Creating the Kriging models of the shaft subsystem isintroduced in detail Firstly select 45 sample points byLHS method to create the Kriging models to conduct the

Mathematical Problems in Engineering 9

Table 4 The optimal results of the QEPF-ATC and Kriging model combined approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299453432 26 138 (3504 0698 17 7300 7845 3410 5283)2 299453217 25 124 (3501 0700 17 7310 7763 3353 5286)3 299521246 18 113 (3498 0699 17 7306 7755 3312 5279)4 299564235 10 36 (3499 0702 17 7297 7718 3350 5282)5 299433263 24 121 (3502 0700 17 7298 7720 3347 5287)

Table 5 The optimal results of the QEPF-ATC approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299463432 42 272 (3500 0699 17 7300 7825 3392 5282)2 299551327 38 215 (3500 0703 17 7305 7746 3353 5287)3 299612356 30 198 (3497 0696 17 7300 7713 3326 5275)4 299575363 17 97 (3499 0700 17 7299 7715 3349 5282)5 299521554 52 306 (3500 0700 17 7300 7719 3350 5286)

subsystem analysis and the created models are shown inFigure 9

Then select 20 sample points by random samplingmethod to evaluate the createdKrigingmodels Error analysisfor Kriging models of the shaft subsystem is shown inFigure 10 and it can be seen that the predicted valuesobtained by Kriging models are basically equal to the actualvalues

There are four credibility evaluation standards for approx-imate models that is the average the maximum the rootmean square and the 119877-squared For the average the max-imum and the root mean square the much smaller thevalues are the much better the credibility of approximatemodels is The default threshold of the average and theroot mean square is 02 and the maximum is 03 Forthe 119877-squared the much bigger the value is the muchbetter the credibility of approximate models is and thedefault threshold is 09 Figure 11 describes the values of thefour credibility evaluation standards for the created Krigingmodels of the shaft subsystem and it can be seen that the fourcredibility evaluation standards are all met So the createdKrigingmodels are reliable enough and can be used to replacethe original models with high accuracy in the next solvingprocess

Five initial points are selected and the sequentialquadratic programming (SQP) method is adopted to solvethe gear reducer problem and the results provided by theQEPF-ATC and Kriging model combined approach andthe QEPF-ATC approach are shown in Tables 4 and 5respectively It can be seen that results of the QEPF-ATCand Kriging model combined approach and the QEPF-ATCapproach are very close and the best result provided bythe QEPF-ATC and Kriging model combined approach iscloser to the optimal value of the gear reducer problemcompared to that provided by the QEPF-ATC approachIn terms of the iterations and runtime the QEPF-ATCand Kriging model combined approach saves computingtime by more than one half compared with QEPF-ATCapproach

5 Conclusion

Because ATC approach can flexibly organize the design opti-mization process according to the components of engineeringsystems it has been widely applied in MDOThe QEPF-ATCapproach is proposed to reduce the complicated coordinationprocess and heavy computation cost faced by exiting ATCapproaches where quadratic exterior penalty function isadopted to be the replaced coordination strategy Furtherin order to deal with the design optimization of complexengineering systems the QEPF-ATC and Kriging modelcombined approach is implemented where Kriging modelsare used to replace simulation models with high accuracyin the solving process of MDO problems The QEPF-ATCand Kriging model combined approach can avoid the impactof numerical noise brought by using simulation modelsduring the iterative solving process and reduce computationcost A geometric programming case is given to validatethe usability and effectiveness of the QEPF-ATC approachThe QEPF-ATC and Kriging model combined approach hasbeen successfully applied to solve the speed reducer designproblem

It should be pointed out that usingmetamodels within theQEPF-ATC approach is not without its own set of potentialdrawbacks which are well known and beyond the scope ofthis work Prior to employing the metamodel assisted QEPF-ATC approach the user should become familiar with thestrengths and weaknesses associated with surrogate approxi-mation Variable-fidelity modeling provides a promising wayto flexibly use approximate models in dealing with MDOproblems The combination of the QEPF-ATC approach andvariable-fidelity modeling will be studied to design complexengineering systems in the future

Abbreviations and Acronyms

ALF Augmented Lagrangian functionAM Approximate modelsATC Analytical target cascadingBLISS Bilevel integrated system synthesis

10 Mathematical Problems in Engineering

002minus002minus006minus01minus014minus018minus022minus026minus03

002minus002minus006minus01minus014minus018minus022minus026minus03

8

8 8

8

75

75 75

75

7 7

77

g7 g7

02

01501005

0minus005minus01minus015minus02

02

015010050minus005minus01minus015minus02

8

8 8

8

75

75 75

75

7 7

77

g8g8

x 4

x 4

x5

x5

x 4

x 4

x5

x5

Figure 9 Kriging models of the shaft subsystem

minus006

minus016

minus026minus026 minus016 minus006

Actu

al

Actu

al

Predicted Predicted

005

minus005

minus015minus015 minus005 005

g8g7

Figure 10 Error analysis for Kriging models of the shaft subsystem

Figure 11 The error data of Kriging models of subsystem 2

Mathematical Problems in Engineering 11

CO Collaborative optimizationCSSO Concurrent subspace optimizationDQA Diagonal quadratic approximationLDF Lagrangian dual functionLHS Latin hypercube samplingMDO Multidisciplinary design

optimizationQEPF Quadratic exterior penalty functionQEPF-ATC Quadratic exterior penalty function

based analytical target cascadingSQP Sequential quadratic programming

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by theNational Natural Sci-ence Foundation of China (NSFC) under Grant no 51121002the National Basic Research Program (973 Program) ofChina under Grant no 2014CB046703 and the FundamentalResearch Funds for the Central Universities HUST Grantno 2014TS040 The authors also would like to thank theanonymous referees for their valuable comments

References

[1] J Sobieszczanski-Sobieski ldquoA linear decomposition method forlarge optimization problemsrdquo NASA Technical Memorandum83248 1982

[2] TW Simpson and J R R AMartins ldquoMultidisciplinary designoptimization for complex engineered systems report from anational science foundation workshoprdquo Journal of MechanicalDesign vol 133 no 10 Article ID 101002 2011

[3] M Balesdent N Berend P Depince and A Chriette ldquoA surveyof multidisciplinary design optimization methods in launchvehicle designrdquo Structural and Multidisciplinary Optimizationvol 45 no 5 pp 619ndash642 2012

[4] I Sobieski and I Kroo ldquoAircraft design using collaborativeoptimizationrdquo AIAA Paper 715 1996

[5] H-Z Huang H Yu X Zhang S Zeng and ZWang ldquoCollabo-rative optimization with inverse reliability for multidisciplinarysystems uncertainty analysisrdquoEngineeringOptimization vol 42no 8 pp 763ndash773 2010

[6] R S Sellar S M Batill and J E Renaud ldquoResponse surfacebased concurrent subspace optimization for multidisciplinarysystem designrdquo AIAA Paper 714 AIAA 1996

[7] L S Li J H Liu and S H Liu ldquoAn efficient strategy formultidisciplinary reliability design and optimization based onCSSO and PMA in SORA frameworkrdquo Structural and Multidis-ciplinary Optimization vol 49 no 2 pp 239ndash252 2014

[8] J Sobieszczanski-Sobieski ldquoBi-level integrated system synthesis(BLISS)rdquo AIAA Paper 4916 AIAA 1998

[9] J Sobieszczanski-Sobieski T D Altus M Phillips and RSandusky ldquoBi-level integrated system synthesis (BLISS) forconcurrent and distributed processingrdquo AIAA Paper 54092002

[10] N Michelena H Park and P Y Papalambros ldquoConvergenceproperties of analytical target cascadingrdquo AIAA Journal vol 41no 5 pp 897ndash905 2003

[11] M Gardenghi M M Wiecek and W Wang ldquoBiobjectiveoptimization for analytical target cascading optimality vsachievabilityrdquo Structural and Multidisciplinary Optimizationvol 47 no 1 pp 111ndash133 2013

[12] S Tosserams L F Etman P Y Papalambros and J E RoodaldquoAn augmented Lagrangian relaxation for analytical target cas-cading using the alternating direction method of multipliersrdquoStructural and Multidisciplinary Optimization vol 31 no 3 pp176ndash189 2006

[13] Y Li Z Lu and J J Michalek ldquoDiagonal quadratic approxima-tion for parallelization of analytical target cascadingrdquo Journalof Mechanical Design Transactions of the ASME vol 130 no 5Article ID 051402 2008

[14] H M Kim W Chen and M M Wiecek ldquoLagrangian coor-dination for enhancing the convergence of analytical targetcascadingrdquo AIAA Journal vol 44 no 10 pp 2197ndash2207 2006

[15] D W Kim and J Lee ldquoAn improvement of Kriging basedsequential approximate optimization method via extended useof design of experimentsrdquo Engineering Optimization vol 42 no12 pp 1133ndash1149 2010

[16] P B Backlund D W Shahan and C C Seepersad ldquoA com-parative study of the scalability of alternative metamodellingtechniquesrdquo Engineering Optimization vol 44 no 7 pp 767ndash786 2012

[17] J Zhang S Chowdhury and A Messac ldquoAn adaptive hybridsurrogate modelrdquo Structural and Multidisciplinary Optimiza-tion vol 46 no 2 pp 223ndash238 2012

[18] V Dubourg B Sudret and J-M Bourinet ldquoReliability-baseddesign optimization using kriging surrogates and subset simu-lationrdquo Structural and Multidisciplinary Optimization vol 44no 5 pp 673ndash690 2011

[19] H M Kim N F Michelena P Y Papalambros and T JiangldquoTarget cascading in optimal system designrdquo Transactions of theASMEmdashJournal of Mechanical Design vol 125 no 3 pp 474ndash480 2003

[20] J P Costa L Pronzato and E Thierry ldquoA comparison betweenKriging and radial basis function networks for nonlinearpredictionrdquo in Proceedings of the IEEE-EURASIP Workshop onNonlinear Signal and Image Processing (NSIP rsquo99) pp 726ndash7301999

[21] V Cherkassky D Gehring and F Mulier ldquoComparison ofadaptive methods for function estimation from samplesrdquo IEEETransactions on Neural Networks vol 7 no 4 pp 969ndash9841996

[22] M S Kim H Cho S G Lee J Choi and D Bae ldquoDFSS androbust optimization tool for multibody system with randomvariablesrdquo Journal of System Design and Dynamics vol 1 no 3pp 583ndash592 2007

[23] H M Kim Target cascading in optimal system design [PhDdissertation] University of Michigan 2001

[24] S L Padula N Alexandrov and L L Green ldquoMDO testsuite at NASA Langley Research Centerrdquo in Proceedings of the6th Symposium onMultidisciplinary Analysis and OptimizationAIAA Paper 96-4028 pp 410ndash420 Bellevue Wash USASeptember 1996

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Component

System

Subsystem

Cascade down

Rebalance up

Figure 1 Schematic of the ATC process

In the design optimization of complex engineering sys-tems generally a large number of simulation models withhigh accuracy (such as the structural finite element analysismodels aerodynamic analysis models and computationalfluid dynamicmodels) are involvedThese simulationmodelsincrease the computation expense greatlyTherefore approx-imate models are adopted widely to replace the expensivesimulation models in MDO [15ndash18] In this paper QEPF-ATC and Kriging model [15 18] combined approach isfurther proposed to deal with MDO problems owing to thecomprehensive performance high approximation accuracyand robustness of Kriging model Finally the geometricprogramming and reducer design cases are given to vali-date usability and effectiveness of the proposed QEPF-ATCapproach and the QEPF-ATC and Kriging model combinedapproach respectively

The rest of this paper is organized as follows In Section 2ATC is introduced and the QEPF-ATC approach is pro-posed QEPF-ATC and Kriging model combined approachis implemented in Section 3 In Section 4 the geometricprogramming and reducer design cases are given to validatethe usability and effectiveness of the proposed approachConclusion and future work are discussed in Section 5

2 The Proposed QEPF-ATC Approach

21 ATC The design optimization problem of complexengineering systems is decomposed into a set of subproblemsin ATC approach where the desired design specificationsor targets are established at the top level and cascadeddown to each lower level subproblem The target transfor-mation is a decomposition process from up to down in alevel-by-level way Moreover subproblems need to get theresponses to match the cascaded targets If targets cascadedto subsystems are not achievable or compatible the feedbackfrom subsystems is required resulting in an iterative targetcascading process Hence targets and responses are updatedand coordinated iteratively to achieve global consistency (seeFigure 1) [10] Coordination strategy is needed to implementthe communication of target and response values betweenparents and children problems

ATC is a model-based optimization method There existtwo kinds of models optimal design models 119875 and analysismodels 119877 The former is just so-called optimization designproblem that includes optimization object design variables

Optimal design model

Analysis model

Optimal design model k1 Optimal design model k2

Level i

Level i minus 1

Level i + 1

Rijxij

Ri(i+1)j Ri(i+1)j

yi(i+1)j yi(i+1)j yi+1(i+1)k2

Ri+1(i+1)k2

yi+1(i+1)k1

Ri+1(i+1)k1Ri+1(i+1)k1

Ri+1(i+1)k2

Riij yiij Riminus1ij yiminus1ip

Figure 2 Information flow between children subproblem andparent subproblem

and constraints The latter is physical model or mathematicalmodel that is the foundation of the optimization designFigure 2 shows the data flow between subproblems (parentand children) [19]

In Figure 2 there exist consistency constraints for eachsubproblem In order tomake each subproblem independentconsistency constraint is introduced as a kind of equalityconstraints For the subproblem at a given level consistencyconstraints are as follows

1198881015840

1= (119905119894119895minus 119903119894119895) = 0 119888

1015840

1= (119877119894minus1

119894119895minus 119877119894

119894119895) = 0

1198881015840

1= (119910119894minus1

119894119901minus 119910119894

119894119895) = 0

1198881015840

2= (119905(119894+1)119896

minus 119903(119894+1)119896

) = 0

1198881015840

2= (119877119894

(119894+1)119895minus 119877119894+1

(119894+1)119896) = 0 119888

1015840

2= (119910119894

(119894+1)119895minus 119910119894+1

(119894+1)119896) = 0

(1)

The general mathematical notation for subproblems119875119894119895in

the hierarchy is then

119875119894119895 min 119891

119894119895

wrt 119909119894119895 119910119894

119894119895 119877119894

(119894+1)119895 119910119894

(119894+1)119895

where 119877119894119895= 119903119894119895(119877119894

(119894+1)119895 119909119894119895 119910119894

119894119895)

st 1198881= (119877119894minus1

119894119895minus 119877119894

119894119895) = 0 119888

2= (119910119894minus1

119894119901minus 119910119894

119894119895) = 0

1198883= (119877119894

(119894+1)119895minus 119877119894+1

(119894+1)119896) = 0

1198884= (119910119894

(119894+1)119895minus 119910119894+1

(119894+1)119896) = 0

119892119894119895(119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895) le 0

ℎ119894119895(119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895) = 0

(2)

where 119875119894119895is the problem formulation of element 119895 at level 119894

119901 is used to designate the parent of element 119895 119896 designateschildren of element 119895 119909

119894119895is local variables for subproblem

119875119894119895 119903119894119895is the function that calculates responses for 119875

119894119895 119877119894119894119895is

Mathematical Problems in Engineering 3

response of 119875119894119895 119877119894minus1119894119895

is the target for 119875119894119895cascaded from the

(119894 minus 1) level parent element 119877119894+1(119894+1)119896

is response of the (119894 + 1)level children element 119877119894

(119894+1)119895= (119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

) is the

target for the (119894 + 1) level children element 119910119894119894119895is response of

linking variable of 119875119894119895 119910119894minus1119894119901

is the target of linking variable for119875119894119895from the (119894 minus 1) level parent element 119910119894+1

(119894+1)119896is response of

linking variable of the (119894 + 1) level children element 119910119894(119894+1)119895

is the target of linking variable for the (119894 + 1) level childrenelement from119875

119894119895119891119894119895is the local objective for119875

119894119895119892119894119895is function

of inequality constraints for 119875119894119895in negative null form ℎ

119894119895is

function of equality constraints for 119875119894119895in null form

The coordination strategy in ATC deals with the defi-nition of deviation between targets and responses (namelyconsistency constraint) and updating weight coefficientstherein By setting appropriate norm and weight coefficientsto coordinate each subproblem the optimal design point canbe obtained The coordination strategy is the key research inATC and an appropriate coordination strategy can improvethe computation efficiency of ATC In recent years somecoordination strategies in ATC have been studied

211 The Coordination Strategy Based on ALF Accordingto the ALF the deviation between targets and responses ispresented by augmented Lagrangian function and weightcoefficients are presented by the augmented Lagrange mul-tiplier The coordination way based on ALF is shown as

min 119891119894119895+ 120582119877

119894119895(119877119894

119894119895minus 119877119894minus1

119894119895) +10038171003817100381710038171003817119908119877

119894119895∘ (119877119894

119894119895minus 119877119894minus1

119894119895)10038171003817100381710038171003817

2

+ 120582119910

119894119895(119910119894minus1

119894119901minus 119910119894

119894119895) +10038171003817100381710038171003817119908119910

119894119895∘ (119910119894minus1

119894119901minus 119910119894

119894119895)10038171003817100381710038171003817

2

+ 120582119877

(119894+1)119895(119877119894

(119894+1)119895minus 119877119894+1

(119894+1)119896)

+10038171003817100381710038171003817119908119877

(119894+1)119895∘ (119877119894

(119894+1)119895minus 119877119894+1

(119894+1)119896)10038171003817100381710038171003817

2

+ 120582119910

(119894+1)119895(119910119894

(119894+1)119895minus 119910119894+1

(119894+1)119896)

+100381710038171003817100381710038171003817119908119910

(119894+1)119895(119910119894

(119894+1)119895minus 119910119894+1

(119894+1)119896)100381710038171003817100381710038171003817

2

st 119892119894119895(119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895) le 0

ℎ119894119895(119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895) = 0

(3)

Weight coefficients 120582119877119894119895 120582119910119894119895119908119877119894119895 and119908119910

119894119895in (3) are updated

in accordance with the following equation

120582119877(119896+1)

119894119895= 120582119877(119896)

119894119895+ 2119908119877(119896)

119894119895(119877119894(119896)

119894119895minus 119877119894minus1(119896)

119894119895)

120582119910(119896+1)

119894119895= 120582119910(119896)

119894119895+ 2119908119910(119896)

119894119895(119910119894(119896)

119894119895minus 119910119894minus1(119896)

119894119895)

119908119877(119896+1)

119894119895= 120573119908119877(119896)

119894119895

119908119910(119896+1)

119894119895= 120573119908119910(119896)

119894119895

(4)

where the superscript 119896 indicates iterations and 120573 is steplength Usually 120573 ge 1 but 2 lt 120573 lt 3 can accelerateconvergence

212 The Coordination Strategy Based on DQA DQA isrealized by approximating the deviation between targets andresponses on the basis of ALF The specific approximation is10038171003817100381710038171003817119877119894

119894119895minus 119877119894minus1

119894119895

10038171003817100381710038171003817

2

=100381710038171003817100381710038171003817(119877119894

119894119895)2

minus 2119877119894

119894119895119877119894minus1

119894119895+ (119877119894minus1

119894119895)2100381710038171003817100381710038171003817

where

119877119894

119894119895119877119894minus1

119894119895cong 119877119894(119896minus1)

119894119895119877119894minus1(119896minus1)

119894119895+ 119877119894(119896minus1)

119894119895(119877119894

119894119895minus 119877119894minus1(119896minus1)

119894119895)

+ 119877119894minus1(119896minus1)

119894119895(119877119894

119894119895minus 119877119894(119896minus1)

119894119895) so

10038171003817100381710038171003817119877119894

119894119895minus 119877119894minus1

119894119895

10038171003817100381710038171003817

2

=10038171003817100381710038171003817119877119894minus1(119896minus1)

119894119895minus 119877119894

119894119895

10038171003817100381710038171003817

2

+10038171003817100381710038171003817119877119894

119894119895minus 119877119894minus1(119896minus1)

119894119895

10038171003817100381710038171003817

2

+ 119877119894(119896minus1)

119894119895119877119894minus1(119896minus1)

119894119895(constant value)

(5)

Based on DQA the cross term among the deviation isapproximated that is to say an order Taylor type expansion isconducted in the last design point Among them the weightcoefficient is updated according to (5) too

213 The Coordination Strategy Based on LDF According tothe LDF the deviation is dealt with by Lagrangian dualityfunction and weight coefficients are presented by the dualLagrange multiplier The coordination of LDF is shown as

min 119891119894119895+ 120582119877

119894119895(119877119894

119894119895minus 119877119894minus1

119894119895) +10038171003817100381710038171003817120583119877

119894119895∘ (119877119894

119894119895minus 119877119894minus1

119894119895)10038171003817100381710038171003817

2

+ 120582119910

119894119895(119910119894minus1

119894119901minus 119910119894

119894119895) +10038171003817100381710038171003817120583119910

119894119895∘ (119910119894minus1

119894119901minus 119910119894

119894119895)10038171003817100381710038171003817

2

st 119892119894119895(119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895) le 0

ℎ119894119895(119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895) = 0

(6)

Weight coefficients 120582119877119894119895 120582119910119894119895 120583119877119894119895 and 120583119910

119894119895in (6) are updated

in accordance with the following equation

120582119877(119896+1)

119894119895= 120582119877(119896)

119894119895+ 120572119877(119896)

119894119895(119877119894(119896)

119894119895minus 119877119894minus1(119896)

119894119895)

120582119910(119896+1)

119894119895= 120582119910(119896)

119894119895+ 120572119910(119896)

119894119895(119910119894(119896)

119894119895minus 119910119894minus1(119896)

119894119895)

120583119877(119896+1)

119894119895= radic

10038161003816100381610038161003816120582119877(119896)

119894119895

10038161003816100381610038161003816 120583

119910(119896+1)

119894119895= radic

100381610038161003816100381610038161003816120582119910(119896)

119894119895

100381610038161003816100381610038161003816

120572119877(119896+1)

119894119895=1 + 119898

119896 + 119898sdot

1

10038171003817100381710038171003817119877119894(119896)

119894119895minus 119877119894minus1(119896)

119894119895

10038171003817100381710038171003817

120572119910(119896+1)

119894119895=1 + 119898

119896 + 119898sdot

1

10038171003817100381710038171003817119910119894(119896)

119894119895minus 119910119894minus1(119896)

119894119895

10038171003817100381710038171003817

(7)

4 Mathematical Problems in Engineering

where the superscript 119896 indicates iterations and119898 is a positiveintegers

The difference of these coordination strategies is theway of dealing with consistency constraints and weightcoefficients There exist multiweight coefficients in the abovethree coordination strategies to increase computational costFor this reason the coordination strategy based on quadraticexterior penalty function among which only one weight coef-ficient exists is put forward which simplifies the expression ofoptimization problem greatly thus improving the efficiencyto make ATC have wider application The coordinationstrategy based on the quadratic exterior penalty function willbe described briefly

22 The QEPF-ATC Approach A general equality-con-strained optimization problem can be defined as follows

min 119891 (x)

st 119888119894(x) = 0 119894 = 1 2 119898

xlb le x le xup

(8)

where x is the vector of the design variables with xlb andxup as its lower and upper bounds respectively The function119891(x) represents the objective function 119888

119894(x) is the equality

constraintsAccording to QEPF consistency constraints are added

into the objective function by introducing a penalty factorand the penalty factor is used as the weighting coefficientto realize the coordination The quadratic exterior penaltyfunction is defined as

119865 (119909 120588) = 119891 (x) +120588

2

119898

sum

119894=1

119888119894(x)2 = 119891 (x) +

120588

2119888 (x)119879 119888 (x) (9)

where the weight coefficient (penalty factor) is updated inaccordance with (10) The value of 120582 influences the con-vergence efficiency and computational efficiency so that itis necessary to set up appropriate 120582 to obtain much bettersolution efficiency Consider

120588119896+1= 120582120588119896 120582 = 14sim10 (10)

The mathematical formulation of QEPF-ATC is

119875119894119895 min 119891

119894119895+120588

2(10038171003817100381710038171003817(119877119894minus1

119894119895minus 119877119894

119894119895)10038171003817100381710038171003817

2

+10038171003817100381710038171003817(119910119894minus1

119894119901minus 119910119894

119894119895)10038171003817100381710038171003817

2

+10038171003817100381710038171003817(119877119894

(119894+1)119895minus 119877119894+1

(119894+1)119896)10038171003817100381710038171003817

2

+10038171003817100381710038171003817(119910119894

(119894+1)119895minus 119910119894+1

(119894+1)119896)10038171003817100381710038171003817

2

)

st 119892119894119895le 0 ℎ

119894119895= 0

119877119894119895= 119903119894119895(119909119894119895)

119909119894119895= (119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895)

(11)

Generally the much bigger the penalty factor is the muchcloser the optimization results are to the optimal solution

Optimizing system subproblem

Yes

Converged

Solving subproblems

Decomposing the problem

Initializing

No Updating weight

Optimized results

Figure 3 The solving process of QEPF-ATC approach

of the original problem But in the practical applicationthe value of the penalty factor must be moderate whenconsidering the computational cost due to the fact that ifthe penalty factor is too small the optimization results mayget bigger error from the original problem to bring lowcomputational efficiency Otherwise too large penalty factorwould increase the computational expense

Figure 3 shows the solving process of the QEPF-ATCapproach and the main steps are as follows

(1) decompose the original problem and confirm designvariables and constraints of system and subsystems

(2) initialize design variables (including local designvariables and linking variables) and 120588(0) and transferthe corresponding initial value to the subsystem as thetarget

(3) solve each subsystem and pass back current results tosystem level

(4) optimize the system problem on the basis of the thirdstep and transfer new target to each subsystem

(5) obtain the optimization result if the convergence con-dition is met otherwise update the weight coefficient120588 according to (10) and go back to the third step untilthe convergence condition is met

3 The Proposed QEPF-ATC and KrigingModel Combined Approach

31 Kriging Model Kriging models [15 16] have their originsin mining and geostatistical applications involving spatially

Mathematical Problems in Engineering 5

Decomposing the problem

Subsystem analyzing

Subsystem optimizing

System optimizing

Converged

No

Yes

Yes

No

Creating the Kriging model

Sampling by LHS

Optimized results

Validating approximation model

Figure 4The solving process of the QEPF-ATC and Kriging modelcombined approach

and temporally correlated data Kriging models combine aglobal model with localized departures

119910 (119909) = 119891 (119909) + 119911 (119909) (12)

where 119910(119909) is the unknown function of interest 119891(119909) is theknown approximation (usually polynomial) function and119911(119909) is the realization of a stochastic process with mean zerovariance1205902 and nonzero covarianceThe119891(119909) term is similarto a polynomial response surface providing a ldquoglobalrdquo modelof the design space The covariance of 119911(119909) is given by

cov [119911 (119909119894) 119911 (119909

119895)] = 120590

2119877 (119909119894 119909119895) (13)

where119877 is the correlation119877(119909119894 119909119895) is the correlation function

between any two of the 119899 sampled data points 119909119894and 119909

119895 and

it can be calculated by

119877 (119909119894 119909119895) = exp[minus

119899

sum

119896=1

120579119896(119909119896

119894minus 119909119896

119895)2

] (14)

Generally 119891(119909) is treated as a constant 120573 then predictedestimates

_119910 (119909) of the response 119910(119909) at untried values of 119909

are given by_119910 (119909) =

_120573 +119903119879

(119909) 119877minus1(119884 minus 119865

_120573) (15)

where 119910 is the column vector of length 119899 that contains thesample values of the response and 119891 is a column vector oflength 119899 that is filled with ones when 119891(119909) is taken as aconstant In (10) 119903119879(119909) is the correlation vector of length 119899between an untried 119909 and the sampled data points

Consider

119903119879

(119909) = [119877 (119909 1199091) 119877 (119909 119909

2) 119877 (119909 119909

3) 119877 (119909 119909

119899)]119879

(16)

In (10)_120573 is estimated using (12)

_120573 = (119865

119879119877minus1119865)minus1

119865119879119877minus1119884 (17)

In 119877 120579119896= maxminus[119899 ln(1205902) + ln |119877|]2

1205902=

(119884 minus 119865

_120573)

119879

119877minus1(119884 minus 119865

_120573)

119899

(18)

32 The QEPF-ATC and Kriging Model Combined ApproachIn QEPF-ATC the deviation between the target and responseis processed in the form of sum of squares as shown in (11)The sum of squares function is usually not smooth and non-linear And discrepancy of variables in the neighborhood ofthe current solution will lead to fluctuation of the deviationwhich will increase the number of iterations and so forthKriging model is appropriate for high nonlinear problemwith better comprehensive performance high approximationaccuracy and high robustness [20ndash22] Moreover fewersample points also can get higher precision approximationmodel So Kriging model is adopted in this paper

The solving process of the QEPF-ATC and Kriging modelcombined approach is shown in Figure 4 and the main stepsare as follows

(1) decompose the original problem and confirm designvariables and constraints of system and subsystems

(2) select a set of sample points using the Latin hypercubesampling (LHS) method

(3) analyse each subsystem to obtain responses at allsampling points of the subsystems

(4) build Kriging model based on the sample points andthe corresponding responses of each subsystem

(5) evaluate the Kriging model by error analysis if theprecision conditions are not met go back to step (2)to build a new Kriging model by adding some newsample points

(6) initialize variables and transfer the corresponding ini-tial value to the subsystem as the target and optimizeeach subproblem based on the created Krigingmodel

6 Mathematical Problems in Engineering

g1 g2

x1 x2 x3 x4 x5 x6 x7 x11 ge 0

xU3

xU11B

xL3

xL11B

g3 g4

x3 x8 x9 x10 x11 ge 0

x11

xU6

xU11C

xL6

xL11C

g5 g6

x6 x11 x12 x13 x14 ge 0

st

st st

R1 = x1 = r1(x3 x4 x5) = (x23 + xminus24 + x25)12

R2 = x2 = r2(x5 x6 x7) = (x25 + x26 + x27)12

R3 = x3 = r3(x8 x9 x10 x11)= (x28 + xminus29 + xminus210 + x

211)

12

PC minx6x11 x12 x13 x14

f22 =120590

2x6 minus x

U6 )) 2

+ (x11 minus xU11C)2]]

PA minx1x2x3x4x5x6x7x11

f11 = x21 + x22 +

120590

2x3 minus x

L3)

2+ (x6 minus xL6)

2+ (x11 minus xL11B)

2+ (x11 minus xL11C)

2])]

R4 = x6 = r4(x11 x12 x13 x14)= (x211 + x212 + x213 + x214)

12

xPB min

x3x8x9x10f21 =

120590

2x3 minus x

U3 )

2+ (x11 minus xU11B)

2])]11

Figure 5 The solving framework of QEPF-ATC for geometric programming

(7) pass back current results to system level and imple-ment the system level optimization

(8) obtain results if the convergence condition is metotherwise go to step (6) until the convergence con-dition is met

4 Case Studies

41 The Geometric Programming The geometric program-ming [23] that has only the global optimal solution isemployed to verify the usability of QEPF-ATC Its mathemat-ical expression is shown as

min 119891 = 1199092

1+ 1199092

2

st 1198921119909minus2

3+ 1199092

4

1199092

5

le 1 11989221199092

5+ 119909minus2

6

1199092

7

le 1

11989231199092

8+ 1199092

9

1199092

11

le 1

1198924119909minus2

8+ 1199092

10

1199092

11

le 1 11989251199092

11+ 119909minus2

12

1199092

13

le 1

11989261199092

11+ 1199092

12

1199092

14

le 1

ℎ1= 1199092

1minus 1199092

3minus 119909minus2

4minus 1199092

5= 0

ℎ2= 1199092

2minus 1199092

5minus 1199092

6minus 1199092

7= 0

ℎ3= 1199092

3minus 1199092

8minus 119909minus2

9minus 119909minus2

10minus 1199092

11= 0

ℎ4= 1199092

6minus 1199092

11minus 1199092

12minus 1199092

13minus 1199092

14= 0

(19)

Table 1 The optimal results

QEPF-ATC ALF DQA LDF AAO1199091

29049 285 286 287 2841199092

30551 304 310 305 3091199093

24930 248 249 243 2361199094

08805 086 082 079 0761199095

09676 094 095 089 0871199096

27103 272 275 278 2811199097

10357 102 095 103 0941199098

09830 098 098 101 0971199099

07572 078 079 082 08711990910

07104 076 079 081 08011990911

12408 127 126 125 13011990912

08409 086 085 083 08411990913

17088 173 174 171 17611990914

14875 152 149 150 155119891 177721 1736 1778 1753 1759Runtime (s) 289 355 336 368

Figure 5 shows the solution framework of the QEPF-ATCapproach for the problem where 119909

11is the linking variable

(coupling variable) coordinated by the system level 11990911988011119861

119909119880

11119862 1199091198803 and 119909119880

6are the targets of each subproblem passed

from the system level 11990911987111119861

and 11990911987111119862

are the responses of thelinking variable coming from the subproblems 119861 and 119862 119909119871

3

and 1199091198716are the responses of 119861 and 119862

Solve the geometric programming problem according toQEPF-ATC Initialize the design point 1199090 = (1 1 1 1 1 1 11 1 1 1 1 1 1)

119879 and take 120582 = 10 Here the acceptableinconsistency tolerance is 001 for every response variable andlinking variable Table 1 shows the optimal results based on

Mathematical Problems in Engineering 7

Table 2 The optimal results based on 120582

120582 (1199091 1199092 1199093 1199094 1199095 1199096 1199097 11990911) (119909

3 1199098 1199099 11990910 11990911) (119909

6 11990911 11990912 11990913 11990914) 119891 Iterations

14 (27556 30804 22975 0813109171 27718 09840 12678) (23012 09688 08427 07340 12682) (27638 12732 07447 18411 14655) 175992 78

4 (29716 29567 25227 0759808570 26708 09352 12343) (25230 09911 07320 07072 12321) (26791 12298 08409 16923 14686) 175728 27

7 (29662 29757 25012 0720408231 27169 09014 12386) (25011 09863 07492 07115 12386) (27168 12385 08408 17122 14915) 176531 17

10 (29049 30551 24930 0880509676 27103 10357 12408) (24930 09830 07572 07104 12408) (27102 12409 08409 17088 14875) 177721 5

Table 3 Design variables for speed reducer

Variables Range Units1199091 Gear face width 26 le 119909

1le 36 cm

1199092 Teeth module 07 le 119909

2le 08 cm

1199093

Number of teeth of pinion(integer variable) 17 le 119909

3le 28

1199094

Distance between bearings1 73 le 119909

4le 83 cm

1199095

Distance between bearings2 73 le 119909

5le 83 cm

1199096 Diameter of shaft 1 29 le 119909

6le 39 cm

1199097 Diameter of shaft 1 50 le 119909

7le 55 cm

1199091 1199092 and 119909

3are the design variables of the gear subproblem 119909

4and 1199095are

the design variables of the shaft subproblem 1199096and 1199097are linking variables

(coupling variables)

QEPF-ATC three coordination strategies (ALF DQA andLDF) and AAO (All-At-Once)

According to Table 1 the results of the QEPF-ATCapproach are very close to the optimal value of the originalproblem and the runtime is less than the other threemethods120582 affects the convergence efficiency and computational

efficiency of the QEPF-ATC approach In order to obtainmore suitable 120582 to improve the efficiency of the QEPF-ATC approach different values of 120582 are selected to solve theproblem Table 2 shows the results when different values of 120582are selected

In Table 2 the QEPF-ATC approach implement fastconvergence when 120582 is set to different values and the optimalresults are close to the results of the initial problems Withthe increase of 120582 the number of iterations decreases andthe solution error gets bigger (excessive weight) Generally14 le 120582 le 10 is strictly necessary but typically 4 lt

120582 lt 7 is recommended to speed up convergence and reducecomputational cost and get better optimal results

42 The Speed Reducer Design The speed reducer designcase represents the design of a simple gearbox and is amultidisciplinary problem comprising the coupling betweengear design and shaft design This has been used as a testingproblem for MDO method in the literature from NASALangley research center [24] The multidisciplinary systemsand notations of the reducer are shown in Figure 6 and

x1

x2

x3

x4x5

x6

x7

Figure 6 The multidisciplinary systems and notations of reducer

Speed reducer

x6 x7 Shaftx4 x5

Gearx1 x2 x3

Figure 7 The decomposition of speed reducer problem

the modelling is shown in Figure 7 The design objectiveis to minimize the speed reducer weight while satisfying anumber of constraints posed by gear and shaft parts Thereare seven variables in the speed reducer design optimizationTable 3 lists a brief description of the variables and gives theirreference values

According to the decomposition method of ATC wherea system is partitioned by object the speed reducer problemis decomposed into gear subproblem and shaft subproblem(shown in Figure 7) The design optimization problem isdefined as follows

min 119891 = 1198911(119909) + 119891

2(119909)

1198911(119909) = 07854119909

11199092

2

sdot (333331199092

3+ 149334119909

3minus 430934)

minus 150791199091(1199092

6+ 1199092

7)

8 Mathematical Problems in Engineering

The optimizer of system level

The optimizer of subproblem 1 The optimizer of subproblem 2

DOE-LHS DOE-LHS

Subproblem 1

analysis

Kriging model 1

Subproblem 2

analysis

Kriging model 2

g3 g4 g5 g6

xU8 xU7B x

U6B xL8 x

L7B x

L6B xU9 x

U6C x

U7C xL9 x

L6C x

L7C

X1 X1

X1

X1 G1

G1X2 X2

X2

X2 G2

G2

st

st st

R1 = x10 = r1(x3 x4 x5) = x8 + x9

X0 = x6 x7

G1 = g1 g2 g9 g10 g11R2 = x8 = r2(X1) = f1(x)

X1 = x1 x2 x3 x6 x7

G2 = g7 g8R3 = x9 = r3(X2) = f2(x)

X2 = x4 x5 x6 x7

PA min f11 = x10 + x8 minus xL8)

2+ (x9 minus xL9)

2+ (x6 minus xL6B)

2+ (x6 minus xL6C)

2+ (x7 minus xL7B)

2+ (x7 minus xL7C)

2]120590

2)]

PB min f21 = x8 minus xU8 )

2+ (x6 minus xU6B)

2+ (x7 minus xU7B)

2]120590

2)] PC min f22 = x9 minus x

U9 )

2+ (x6 minus xU6C)

2+ (x7 minus xU7C)

2]120590

2)]

Figure 8 The solving framework of the QEPF-ATC and Kriging model combined approach for the gear reducer

1198912(119909) = 7477 (119909

3

6+ 1199093

7) + 07854 (119909

41199092

6+ 11990951199092

7)

st 1198921=

27

(11990911199092

21199093)minus 1 le 0 119892

2=

3975

(11990911199092

21199092

3)minus 1 le 0

1198923=1931199093

4

(119909211990931199094

6)minus 1 le 0 119892

4=1931199093

5

(119909211990931199094

7)minus 1 le 0

1198925= 10119909

minus3

6radic(745119909

minus1

2119909minus1

31199094) + 169 times 107

minus 1100 le 0

1198926= 10119909

minus3

7radic(745119909

minus1

2119909minus1

31199095) + 1575 times 108

minus 850 le 0

1198927=(151199096+ 19)

1199094

minus 1 le 0

1198928=(151199097+ 19)

1199095

minus 1 le 0

1198929= 11990921199093minus 40 le 0 119892

10= 5 minus

1199091

1199092

le 0

11989211=1199091

1199092

minus 12 le 0

(20)

where 1198911is the gear weight and 119891

2is the shaft weight And 119892

1

and 1198922 are respectively gear bending stress and contact stress

constraints 1198923and 119892

4are respectively torsional distortion

constraints of shaft 1 and shaft 2 1198925and 119892

6are stress

constraints of the shaft 1 and shaft 2 1198927and 119892

8are experience

constraints of shaft 1 and shaft 2 and 1198929 11989210 and 119892

11are gear

geometry constraints1199091 1199092 and 119909

3are the design variables of the gear

subproblem 1199094and 119909

5are the design variables of the

shaft subproblem 1199096and 119909

7are linking variables (coupling

variables) According to the QEPF-ATC and Kriging modelcombined approach three variables 119909

8 1199099 and 119909

10are

introduced and meet the following formula

ℎ0= 1199098+ 1199099minus 11990910= 0

ℎ1= 1199098minus 1198911(119909) = 0

ℎ2= 1199099minus 1198912(119909) = 0

(21)

where ℎ0represents the analysis model of system level and

ℎ1and ℎ

2represent the analysis models of subproblem 1 and

subproblem 2 respectivelyThe solving framework of the QEPF-ATC and Kriging

model combined approach for gear reducer is shown inFigure 8

Creating the Kriging models of the shaft subsystem isintroduced in detail Firstly select 45 sample points byLHS method to create the Kriging models to conduct the

Mathematical Problems in Engineering 9

Table 4 The optimal results of the QEPF-ATC and Kriging model combined approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299453432 26 138 (3504 0698 17 7300 7845 3410 5283)2 299453217 25 124 (3501 0700 17 7310 7763 3353 5286)3 299521246 18 113 (3498 0699 17 7306 7755 3312 5279)4 299564235 10 36 (3499 0702 17 7297 7718 3350 5282)5 299433263 24 121 (3502 0700 17 7298 7720 3347 5287)

Table 5 The optimal results of the QEPF-ATC approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299463432 42 272 (3500 0699 17 7300 7825 3392 5282)2 299551327 38 215 (3500 0703 17 7305 7746 3353 5287)3 299612356 30 198 (3497 0696 17 7300 7713 3326 5275)4 299575363 17 97 (3499 0700 17 7299 7715 3349 5282)5 299521554 52 306 (3500 0700 17 7300 7719 3350 5286)

subsystem analysis and the created models are shown inFigure 9

Then select 20 sample points by random samplingmethod to evaluate the createdKrigingmodels Error analysisfor Kriging models of the shaft subsystem is shown inFigure 10 and it can be seen that the predicted valuesobtained by Kriging models are basically equal to the actualvalues

There are four credibility evaluation standards for approx-imate models that is the average the maximum the rootmean square and the 119877-squared For the average the max-imum and the root mean square the much smaller thevalues are the much better the credibility of approximatemodels is The default threshold of the average and theroot mean square is 02 and the maximum is 03 Forthe 119877-squared the much bigger the value is the muchbetter the credibility of approximate models is and thedefault threshold is 09 Figure 11 describes the values of thefour credibility evaluation standards for the created Krigingmodels of the shaft subsystem and it can be seen that the fourcredibility evaluation standards are all met So the createdKrigingmodels are reliable enough and can be used to replacethe original models with high accuracy in the next solvingprocess

Five initial points are selected and the sequentialquadratic programming (SQP) method is adopted to solvethe gear reducer problem and the results provided by theQEPF-ATC and Kriging model combined approach andthe QEPF-ATC approach are shown in Tables 4 and 5respectively It can be seen that results of the QEPF-ATCand Kriging model combined approach and the QEPF-ATCapproach are very close and the best result provided bythe QEPF-ATC and Kriging model combined approach iscloser to the optimal value of the gear reducer problemcompared to that provided by the QEPF-ATC approachIn terms of the iterations and runtime the QEPF-ATCand Kriging model combined approach saves computingtime by more than one half compared with QEPF-ATCapproach

5 Conclusion

Because ATC approach can flexibly organize the design opti-mization process according to the components of engineeringsystems it has been widely applied in MDOThe QEPF-ATCapproach is proposed to reduce the complicated coordinationprocess and heavy computation cost faced by exiting ATCapproaches where quadratic exterior penalty function isadopted to be the replaced coordination strategy Furtherin order to deal with the design optimization of complexengineering systems the QEPF-ATC and Kriging modelcombined approach is implemented where Kriging modelsare used to replace simulation models with high accuracyin the solving process of MDO problems The QEPF-ATCand Kriging model combined approach can avoid the impactof numerical noise brought by using simulation modelsduring the iterative solving process and reduce computationcost A geometric programming case is given to validatethe usability and effectiveness of the QEPF-ATC approachThe QEPF-ATC and Kriging model combined approach hasbeen successfully applied to solve the speed reducer designproblem

It should be pointed out that usingmetamodels within theQEPF-ATC approach is not without its own set of potentialdrawbacks which are well known and beyond the scope ofthis work Prior to employing the metamodel assisted QEPF-ATC approach the user should become familiar with thestrengths and weaknesses associated with surrogate approxi-mation Variable-fidelity modeling provides a promising wayto flexibly use approximate models in dealing with MDOproblems The combination of the QEPF-ATC approach andvariable-fidelity modeling will be studied to design complexengineering systems in the future

Abbreviations and Acronyms

ALF Augmented Lagrangian functionAM Approximate modelsATC Analytical target cascadingBLISS Bilevel integrated system synthesis

10 Mathematical Problems in Engineering

002minus002minus006minus01minus014minus018minus022minus026minus03

002minus002minus006minus01minus014minus018minus022minus026minus03

8

8 8

8

75

75 75

75

7 7

77

g7 g7

02

01501005

0minus005minus01minus015minus02

02

015010050minus005minus01minus015minus02

8

8 8

8

75

75 75

75

7 7

77

g8g8

x 4

x 4

x5

x5

x 4

x 4

x5

x5

Figure 9 Kriging models of the shaft subsystem

minus006

minus016

minus026minus026 minus016 minus006

Actu

al

Actu

al

Predicted Predicted

005

minus005

minus015minus015 minus005 005

g8g7

Figure 10 Error analysis for Kriging models of the shaft subsystem

Figure 11 The error data of Kriging models of subsystem 2

Mathematical Problems in Engineering 11

CO Collaborative optimizationCSSO Concurrent subspace optimizationDQA Diagonal quadratic approximationLDF Lagrangian dual functionLHS Latin hypercube samplingMDO Multidisciplinary design

optimizationQEPF Quadratic exterior penalty functionQEPF-ATC Quadratic exterior penalty function

based analytical target cascadingSQP Sequential quadratic programming

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by theNational Natural Sci-ence Foundation of China (NSFC) under Grant no 51121002the National Basic Research Program (973 Program) ofChina under Grant no 2014CB046703 and the FundamentalResearch Funds for the Central Universities HUST Grantno 2014TS040 The authors also would like to thank theanonymous referees for their valuable comments

References

[1] J Sobieszczanski-Sobieski ldquoA linear decomposition method forlarge optimization problemsrdquo NASA Technical Memorandum83248 1982

[2] TW Simpson and J R R AMartins ldquoMultidisciplinary designoptimization for complex engineered systems report from anational science foundation workshoprdquo Journal of MechanicalDesign vol 133 no 10 Article ID 101002 2011

[3] M Balesdent N Berend P Depince and A Chriette ldquoA surveyof multidisciplinary design optimization methods in launchvehicle designrdquo Structural and Multidisciplinary Optimizationvol 45 no 5 pp 619ndash642 2012

[4] I Sobieski and I Kroo ldquoAircraft design using collaborativeoptimizationrdquo AIAA Paper 715 1996

[5] H-Z Huang H Yu X Zhang S Zeng and ZWang ldquoCollabo-rative optimization with inverse reliability for multidisciplinarysystems uncertainty analysisrdquoEngineeringOptimization vol 42no 8 pp 763ndash773 2010

[6] R S Sellar S M Batill and J E Renaud ldquoResponse surfacebased concurrent subspace optimization for multidisciplinarysystem designrdquo AIAA Paper 714 AIAA 1996

[7] L S Li J H Liu and S H Liu ldquoAn efficient strategy formultidisciplinary reliability design and optimization based onCSSO and PMA in SORA frameworkrdquo Structural and Multidis-ciplinary Optimization vol 49 no 2 pp 239ndash252 2014

[8] J Sobieszczanski-Sobieski ldquoBi-level integrated system synthesis(BLISS)rdquo AIAA Paper 4916 AIAA 1998

[9] J Sobieszczanski-Sobieski T D Altus M Phillips and RSandusky ldquoBi-level integrated system synthesis (BLISS) forconcurrent and distributed processingrdquo AIAA Paper 54092002

[10] N Michelena H Park and P Y Papalambros ldquoConvergenceproperties of analytical target cascadingrdquo AIAA Journal vol 41no 5 pp 897ndash905 2003

[11] M Gardenghi M M Wiecek and W Wang ldquoBiobjectiveoptimization for analytical target cascading optimality vsachievabilityrdquo Structural and Multidisciplinary Optimizationvol 47 no 1 pp 111ndash133 2013

[12] S Tosserams L F Etman P Y Papalambros and J E RoodaldquoAn augmented Lagrangian relaxation for analytical target cas-cading using the alternating direction method of multipliersrdquoStructural and Multidisciplinary Optimization vol 31 no 3 pp176ndash189 2006

[13] Y Li Z Lu and J J Michalek ldquoDiagonal quadratic approxima-tion for parallelization of analytical target cascadingrdquo Journalof Mechanical Design Transactions of the ASME vol 130 no 5Article ID 051402 2008

[14] H M Kim W Chen and M M Wiecek ldquoLagrangian coor-dination for enhancing the convergence of analytical targetcascadingrdquo AIAA Journal vol 44 no 10 pp 2197ndash2207 2006

[15] D W Kim and J Lee ldquoAn improvement of Kriging basedsequential approximate optimization method via extended useof design of experimentsrdquo Engineering Optimization vol 42 no12 pp 1133ndash1149 2010

[16] P B Backlund D W Shahan and C C Seepersad ldquoA com-parative study of the scalability of alternative metamodellingtechniquesrdquo Engineering Optimization vol 44 no 7 pp 767ndash786 2012

[17] J Zhang S Chowdhury and A Messac ldquoAn adaptive hybridsurrogate modelrdquo Structural and Multidisciplinary Optimiza-tion vol 46 no 2 pp 223ndash238 2012

[18] V Dubourg B Sudret and J-M Bourinet ldquoReliability-baseddesign optimization using kriging surrogates and subset simu-lationrdquo Structural and Multidisciplinary Optimization vol 44no 5 pp 673ndash690 2011

[19] H M Kim N F Michelena P Y Papalambros and T JiangldquoTarget cascading in optimal system designrdquo Transactions of theASMEmdashJournal of Mechanical Design vol 125 no 3 pp 474ndash480 2003

[20] J P Costa L Pronzato and E Thierry ldquoA comparison betweenKriging and radial basis function networks for nonlinearpredictionrdquo in Proceedings of the IEEE-EURASIP Workshop onNonlinear Signal and Image Processing (NSIP rsquo99) pp 726ndash7301999

[21] V Cherkassky D Gehring and F Mulier ldquoComparison ofadaptive methods for function estimation from samplesrdquo IEEETransactions on Neural Networks vol 7 no 4 pp 969ndash9841996

[22] M S Kim H Cho S G Lee J Choi and D Bae ldquoDFSS androbust optimization tool for multibody system with randomvariablesrdquo Journal of System Design and Dynamics vol 1 no 3pp 583ndash592 2007

[23] H M Kim Target cascading in optimal system design [PhDdissertation] University of Michigan 2001

[24] S L Padula N Alexandrov and L L Green ldquoMDO testsuite at NASA Langley Research Centerrdquo in Proceedings of the6th Symposium onMultidisciplinary Analysis and OptimizationAIAA Paper 96-4028 pp 410ndash420 Bellevue Wash USASeptember 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

response of 119875119894119895 119877119894minus1119894119895

is the target for 119875119894119895cascaded from the

(119894 minus 1) level parent element 119877119894+1(119894+1)119896

is response of the (119894 + 1)level children element 119877119894

(119894+1)119895= (119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

) is the

target for the (119894 + 1) level children element 119910119894119894119895is response of

linking variable of 119875119894119895 119910119894minus1119894119901

is the target of linking variable for119875119894119895from the (119894 minus 1) level parent element 119910119894+1

(119894+1)119896is response of

linking variable of the (119894 + 1) level children element 119910119894(119894+1)119895

is the target of linking variable for the (119894 + 1) level childrenelement from119875

119894119895119891119894119895is the local objective for119875

119894119895119892119894119895is function

of inequality constraints for 119875119894119895in negative null form ℎ

119894119895is

function of equality constraints for 119875119894119895in null form

The coordination strategy in ATC deals with the defi-nition of deviation between targets and responses (namelyconsistency constraint) and updating weight coefficientstherein By setting appropriate norm and weight coefficientsto coordinate each subproblem the optimal design point canbe obtained The coordination strategy is the key research inATC and an appropriate coordination strategy can improvethe computation efficiency of ATC In recent years somecoordination strategies in ATC have been studied

211 The Coordination Strategy Based on ALF Accordingto the ALF the deviation between targets and responses ispresented by augmented Lagrangian function and weightcoefficients are presented by the augmented Lagrange mul-tiplier The coordination way based on ALF is shown as

min 119891119894119895+ 120582119877

119894119895(119877119894

119894119895minus 119877119894minus1

119894119895) +10038171003817100381710038171003817119908119877

119894119895∘ (119877119894

119894119895minus 119877119894minus1

119894119895)10038171003817100381710038171003817

2

+ 120582119910

119894119895(119910119894minus1

119894119901minus 119910119894

119894119895) +10038171003817100381710038171003817119908119910

119894119895∘ (119910119894minus1

119894119901minus 119910119894

119894119895)10038171003817100381710038171003817

2

+ 120582119877

(119894+1)119895(119877119894

(119894+1)119895minus 119877119894+1

(119894+1)119896)

+10038171003817100381710038171003817119908119877

(119894+1)119895∘ (119877119894

(119894+1)119895minus 119877119894+1

(119894+1)119896)10038171003817100381710038171003817

2

+ 120582119910

(119894+1)119895(119910119894

(119894+1)119895minus 119910119894+1

(119894+1)119896)

+100381710038171003817100381710038171003817119908119910

(119894+1)119895(119910119894

(119894+1)119895minus 119910119894+1

(119894+1)119896)100381710038171003817100381710038171003817

2

st 119892119894119895(119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895) le 0

ℎ119894119895(119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895) = 0

(3)

Weight coefficients 120582119877119894119895 120582119910119894119895119908119877119894119895 and119908119910

119894119895in (3) are updated

in accordance with the following equation

120582119877(119896+1)

119894119895= 120582119877(119896)

119894119895+ 2119908119877(119896)

119894119895(119877119894(119896)

119894119895minus 119877119894minus1(119896)

119894119895)

120582119910(119896+1)

119894119895= 120582119910(119896)

119894119895+ 2119908119910(119896)

119894119895(119910119894(119896)

119894119895minus 119910119894minus1(119896)

119894119895)

119908119877(119896+1)

119894119895= 120573119908119877(119896)

119894119895

119908119910(119896+1)

119894119895= 120573119908119910(119896)

119894119895

(4)

where the superscript 119896 indicates iterations and 120573 is steplength Usually 120573 ge 1 but 2 lt 120573 lt 3 can accelerateconvergence

212 The Coordination Strategy Based on DQA DQA isrealized by approximating the deviation between targets andresponses on the basis of ALF The specific approximation is10038171003817100381710038171003817119877119894

119894119895minus 119877119894minus1

119894119895

10038171003817100381710038171003817

2

=100381710038171003817100381710038171003817(119877119894

119894119895)2

minus 2119877119894

119894119895119877119894minus1

119894119895+ (119877119894minus1

119894119895)2100381710038171003817100381710038171003817

where

119877119894

119894119895119877119894minus1

119894119895cong 119877119894(119896minus1)

119894119895119877119894minus1(119896minus1)

119894119895+ 119877119894(119896minus1)

119894119895(119877119894

119894119895minus 119877119894minus1(119896minus1)

119894119895)

+ 119877119894minus1(119896minus1)

119894119895(119877119894

119894119895minus 119877119894(119896minus1)

119894119895) so

10038171003817100381710038171003817119877119894

119894119895minus 119877119894minus1

119894119895

10038171003817100381710038171003817

2

=10038171003817100381710038171003817119877119894minus1(119896minus1)

119894119895minus 119877119894

119894119895

10038171003817100381710038171003817

2

+10038171003817100381710038171003817119877119894

119894119895minus 119877119894minus1(119896minus1)

119894119895

10038171003817100381710038171003817

2

+ 119877119894(119896minus1)

119894119895119877119894minus1(119896minus1)

119894119895(constant value)

(5)

Based on DQA the cross term among the deviation isapproximated that is to say an order Taylor type expansion isconducted in the last design point Among them the weightcoefficient is updated according to (5) too

213 The Coordination Strategy Based on LDF According tothe LDF the deviation is dealt with by Lagrangian dualityfunction and weight coefficients are presented by the dualLagrange multiplier The coordination of LDF is shown as

min 119891119894119895+ 120582119877

119894119895(119877119894

119894119895minus 119877119894minus1

119894119895) +10038171003817100381710038171003817120583119877

119894119895∘ (119877119894

119894119895minus 119877119894minus1

119894119895)10038171003817100381710038171003817

2

+ 120582119910

119894119895(119910119894minus1

119894119901minus 119910119894

119894119895) +10038171003817100381710038171003817120583119910

119894119895∘ (119910119894minus1

119894119901minus 119910119894

119894119895)10038171003817100381710038171003817

2

st 119892119894119895(119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895) le 0

ℎ119894119895(119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895) = 0

(6)

Weight coefficients 120582119877119894119895 120582119910119894119895 120583119877119894119895 and 120583119910

119894119895in (6) are updated

in accordance with the following equation

120582119877(119896+1)

119894119895= 120582119877(119896)

119894119895+ 120572119877(119896)

119894119895(119877119894(119896)

119894119895minus 119877119894minus1(119896)

119894119895)

120582119910(119896+1)

119894119895= 120582119910(119896)

119894119895+ 120572119910(119896)

119894119895(119910119894(119896)

119894119895minus 119910119894minus1(119896)

119894119895)

120583119877(119896+1)

119894119895= radic

10038161003816100381610038161003816120582119877(119896)

119894119895

10038161003816100381610038161003816 120583

119910(119896+1)

119894119895= radic

100381610038161003816100381610038161003816120582119910(119896)

119894119895

100381610038161003816100381610038161003816

120572119877(119896+1)

119894119895=1 + 119898

119896 + 119898sdot

1

10038171003817100381710038171003817119877119894(119896)

119894119895minus 119877119894minus1(119896)

119894119895

10038171003817100381710038171003817

120572119910(119896+1)

119894119895=1 + 119898

119896 + 119898sdot

1

10038171003817100381710038171003817119910119894(119896)

119894119895minus 119910119894minus1(119896)

119894119895

10038171003817100381710038171003817

(7)

4 Mathematical Problems in Engineering

where the superscript 119896 indicates iterations and119898 is a positiveintegers

The difference of these coordination strategies is theway of dealing with consistency constraints and weightcoefficients There exist multiweight coefficients in the abovethree coordination strategies to increase computational costFor this reason the coordination strategy based on quadraticexterior penalty function among which only one weight coef-ficient exists is put forward which simplifies the expression ofoptimization problem greatly thus improving the efficiencyto make ATC have wider application The coordinationstrategy based on the quadratic exterior penalty function willbe described briefly

22 The QEPF-ATC Approach A general equality-con-strained optimization problem can be defined as follows

min 119891 (x)

st 119888119894(x) = 0 119894 = 1 2 119898

xlb le x le xup

(8)

where x is the vector of the design variables with xlb andxup as its lower and upper bounds respectively The function119891(x) represents the objective function 119888

119894(x) is the equality

constraintsAccording to QEPF consistency constraints are added

into the objective function by introducing a penalty factorand the penalty factor is used as the weighting coefficientto realize the coordination The quadratic exterior penaltyfunction is defined as

119865 (119909 120588) = 119891 (x) +120588

2

119898

sum

119894=1

119888119894(x)2 = 119891 (x) +

120588

2119888 (x)119879 119888 (x) (9)

where the weight coefficient (penalty factor) is updated inaccordance with (10) The value of 120582 influences the con-vergence efficiency and computational efficiency so that itis necessary to set up appropriate 120582 to obtain much bettersolution efficiency Consider

120588119896+1= 120582120588119896 120582 = 14sim10 (10)

The mathematical formulation of QEPF-ATC is

119875119894119895 min 119891

119894119895+120588

2(10038171003817100381710038171003817(119877119894minus1

119894119895minus 119877119894

119894119895)10038171003817100381710038171003817

2

+10038171003817100381710038171003817(119910119894minus1

119894119901minus 119910119894

119894119895)10038171003817100381710038171003817

2

+10038171003817100381710038171003817(119877119894

(119894+1)119895minus 119877119894+1

(119894+1)119896)10038171003817100381710038171003817

2

+10038171003817100381710038171003817(119910119894

(119894+1)119895minus 119910119894+1

(119894+1)119896)10038171003817100381710038171003817

2

)

st 119892119894119895le 0 ℎ

119894119895= 0

119877119894119895= 119903119894119895(119909119894119895)

119909119894119895= (119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895)

(11)

Generally the much bigger the penalty factor is the muchcloser the optimization results are to the optimal solution

Optimizing system subproblem

Yes

Converged

Solving subproblems

Decomposing the problem

Initializing

No Updating weight

Optimized results

Figure 3 The solving process of QEPF-ATC approach

of the original problem But in the practical applicationthe value of the penalty factor must be moderate whenconsidering the computational cost due to the fact that ifthe penalty factor is too small the optimization results mayget bigger error from the original problem to bring lowcomputational efficiency Otherwise too large penalty factorwould increase the computational expense

Figure 3 shows the solving process of the QEPF-ATCapproach and the main steps are as follows

(1) decompose the original problem and confirm designvariables and constraints of system and subsystems

(2) initialize design variables (including local designvariables and linking variables) and 120588(0) and transferthe corresponding initial value to the subsystem as thetarget

(3) solve each subsystem and pass back current results tosystem level

(4) optimize the system problem on the basis of the thirdstep and transfer new target to each subsystem

(5) obtain the optimization result if the convergence con-dition is met otherwise update the weight coefficient120588 according to (10) and go back to the third step untilthe convergence condition is met

3 The Proposed QEPF-ATC and KrigingModel Combined Approach

31 Kriging Model Kriging models [15 16] have their originsin mining and geostatistical applications involving spatially

Mathematical Problems in Engineering 5

Decomposing the problem

Subsystem analyzing

Subsystem optimizing

System optimizing

Converged

No

Yes

Yes

No

Creating the Kriging model

Sampling by LHS

Optimized results

Validating approximation model

Figure 4The solving process of the QEPF-ATC and Kriging modelcombined approach

and temporally correlated data Kriging models combine aglobal model with localized departures

119910 (119909) = 119891 (119909) + 119911 (119909) (12)

where 119910(119909) is the unknown function of interest 119891(119909) is theknown approximation (usually polynomial) function and119911(119909) is the realization of a stochastic process with mean zerovariance1205902 and nonzero covarianceThe119891(119909) term is similarto a polynomial response surface providing a ldquoglobalrdquo modelof the design space The covariance of 119911(119909) is given by

cov [119911 (119909119894) 119911 (119909

119895)] = 120590

2119877 (119909119894 119909119895) (13)

where119877 is the correlation119877(119909119894 119909119895) is the correlation function

between any two of the 119899 sampled data points 119909119894and 119909

119895 and

it can be calculated by

119877 (119909119894 119909119895) = exp[minus

119899

sum

119896=1

120579119896(119909119896

119894minus 119909119896

119895)2

] (14)

Generally 119891(119909) is treated as a constant 120573 then predictedestimates

_119910 (119909) of the response 119910(119909) at untried values of 119909

are given by_119910 (119909) =

_120573 +119903119879

(119909) 119877minus1(119884 minus 119865

_120573) (15)

where 119910 is the column vector of length 119899 that contains thesample values of the response and 119891 is a column vector oflength 119899 that is filled with ones when 119891(119909) is taken as aconstant In (10) 119903119879(119909) is the correlation vector of length 119899between an untried 119909 and the sampled data points

Consider

119903119879

(119909) = [119877 (119909 1199091) 119877 (119909 119909

2) 119877 (119909 119909

3) 119877 (119909 119909

119899)]119879

(16)

In (10)_120573 is estimated using (12)

_120573 = (119865

119879119877minus1119865)minus1

119865119879119877minus1119884 (17)

In 119877 120579119896= maxminus[119899 ln(1205902) + ln |119877|]2

1205902=

(119884 minus 119865

_120573)

119879

119877minus1(119884 minus 119865

_120573)

119899

(18)

32 The QEPF-ATC and Kriging Model Combined ApproachIn QEPF-ATC the deviation between the target and responseis processed in the form of sum of squares as shown in (11)The sum of squares function is usually not smooth and non-linear And discrepancy of variables in the neighborhood ofthe current solution will lead to fluctuation of the deviationwhich will increase the number of iterations and so forthKriging model is appropriate for high nonlinear problemwith better comprehensive performance high approximationaccuracy and high robustness [20ndash22] Moreover fewersample points also can get higher precision approximationmodel So Kriging model is adopted in this paper

The solving process of the QEPF-ATC and Kriging modelcombined approach is shown in Figure 4 and the main stepsare as follows

(1) decompose the original problem and confirm designvariables and constraints of system and subsystems

(2) select a set of sample points using the Latin hypercubesampling (LHS) method

(3) analyse each subsystem to obtain responses at allsampling points of the subsystems

(4) build Kriging model based on the sample points andthe corresponding responses of each subsystem

(5) evaluate the Kriging model by error analysis if theprecision conditions are not met go back to step (2)to build a new Kriging model by adding some newsample points

(6) initialize variables and transfer the corresponding ini-tial value to the subsystem as the target and optimizeeach subproblem based on the created Krigingmodel

6 Mathematical Problems in Engineering

g1 g2

x1 x2 x3 x4 x5 x6 x7 x11 ge 0

xU3

xU11B

xL3

xL11B

g3 g4

x3 x8 x9 x10 x11 ge 0

x11

xU6

xU11C

xL6

xL11C

g5 g6

x6 x11 x12 x13 x14 ge 0

st

st st

R1 = x1 = r1(x3 x4 x5) = (x23 + xminus24 + x25)12

R2 = x2 = r2(x5 x6 x7) = (x25 + x26 + x27)12

R3 = x3 = r3(x8 x9 x10 x11)= (x28 + xminus29 + xminus210 + x

211)

12

PC minx6x11 x12 x13 x14

f22 =120590

2x6 minus x

U6 )) 2

+ (x11 minus xU11C)2]]

PA minx1x2x3x4x5x6x7x11

f11 = x21 + x22 +

120590

2x3 minus x

L3)

2+ (x6 minus xL6)

2+ (x11 minus xL11B)

2+ (x11 minus xL11C)

2])]

R4 = x6 = r4(x11 x12 x13 x14)= (x211 + x212 + x213 + x214)

12

xPB min

x3x8x9x10f21 =

120590

2x3 minus x

U3 )

2+ (x11 minus xU11B)

2])]11

Figure 5 The solving framework of QEPF-ATC for geometric programming

(7) pass back current results to system level and imple-ment the system level optimization

(8) obtain results if the convergence condition is metotherwise go to step (6) until the convergence con-dition is met

4 Case Studies

41 The Geometric Programming The geometric program-ming [23] that has only the global optimal solution isemployed to verify the usability of QEPF-ATC Its mathemat-ical expression is shown as

min 119891 = 1199092

1+ 1199092

2

st 1198921119909minus2

3+ 1199092

4

1199092

5

le 1 11989221199092

5+ 119909minus2

6

1199092

7

le 1

11989231199092

8+ 1199092

9

1199092

11

le 1

1198924119909minus2

8+ 1199092

10

1199092

11

le 1 11989251199092

11+ 119909minus2

12

1199092

13

le 1

11989261199092

11+ 1199092

12

1199092

14

le 1

ℎ1= 1199092

1minus 1199092

3minus 119909minus2

4minus 1199092

5= 0

ℎ2= 1199092

2minus 1199092

5minus 1199092

6minus 1199092

7= 0

ℎ3= 1199092

3minus 1199092

8minus 119909minus2

9minus 119909minus2

10minus 1199092

11= 0

ℎ4= 1199092

6minus 1199092

11minus 1199092

12minus 1199092

13minus 1199092

14= 0

(19)

Table 1 The optimal results

QEPF-ATC ALF DQA LDF AAO1199091

29049 285 286 287 2841199092

30551 304 310 305 3091199093

24930 248 249 243 2361199094

08805 086 082 079 0761199095

09676 094 095 089 0871199096

27103 272 275 278 2811199097

10357 102 095 103 0941199098

09830 098 098 101 0971199099

07572 078 079 082 08711990910

07104 076 079 081 08011990911

12408 127 126 125 13011990912

08409 086 085 083 08411990913

17088 173 174 171 17611990914

14875 152 149 150 155119891 177721 1736 1778 1753 1759Runtime (s) 289 355 336 368

Figure 5 shows the solution framework of the QEPF-ATCapproach for the problem where 119909

11is the linking variable

(coupling variable) coordinated by the system level 11990911988011119861

119909119880

11119862 1199091198803 and 119909119880

6are the targets of each subproblem passed

from the system level 11990911987111119861

and 11990911987111119862

are the responses of thelinking variable coming from the subproblems 119861 and 119862 119909119871

3

and 1199091198716are the responses of 119861 and 119862

Solve the geometric programming problem according toQEPF-ATC Initialize the design point 1199090 = (1 1 1 1 1 1 11 1 1 1 1 1 1)

119879 and take 120582 = 10 Here the acceptableinconsistency tolerance is 001 for every response variable andlinking variable Table 1 shows the optimal results based on

Mathematical Problems in Engineering 7

Table 2 The optimal results based on 120582

120582 (1199091 1199092 1199093 1199094 1199095 1199096 1199097 11990911) (119909

3 1199098 1199099 11990910 11990911) (119909

6 11990911 11990912 11990913 11990914) 119891 Iterations

14 (27556 30804 22975 0813109171 27718 09840 12678) (23012 09688 08427 07340 12682) (27638 12732 07447 18411 14655) 175992 78

4 (29716 29567 25227 0759808570 26708 09352 12343) (25230 09911 07320 07072 12321) (26791 12298 08409 16923 14686) 175728 27

7 (29662 29757 25012 0720408231 27169 09014 12386) (25011 09863 07492 07115 12386) (27168 12385 08408 17122 14915) 176531 17

10 (29049 30551 24930 0880509676 27103 10357 12408) (24930 09830 07572 07104 12408) (27102 12409 08409 17088 14875) 177721 5

Table 3 Design variables for speed reducer

Variables Range Units1199091 Gear face width 26 le 119909

1le 36 cm

1199092 Teeth module 07 le 119909

2le 08 cm

1199093

Number of teeth of pinion(integer variable) 17 le 119909

3le 28

1199094

Distance between bearings1 73 le 119909

4le 83 cm

1199095

Distance between bearings2 73 le 119909

5le 83 cm

1199096 Diameter of shaft 1 29 le 119909

6le 39 cm

1199097 Diameter of shaft 1 50 le 119909

7le 55 cm

1199091 1199092 and 119909

3are the design variables of the gear subproblem 119909

4and 1199095are

the design variables of the shaft subproblem 1199096and 1199097are linking variables

(coupling variables)

QEPF-ATC three coordination strategies (ALF DQA andLDF) and AAO (All-At-Once)

According to Table 1 the results of the QEPF-ATCapproach are very close to the optimal value of the originalproblem and the runtime is less than the other threemethods120582 affects the convergence efficiency and computational

efficiency of the QEPF-ATC approach In order to obtainmore suitable 120582 to improve the efficiency of the QEPF-ATC approach different values of 120582 are selected to solve theproblem Table 2 shows the results when different values of 120582are selected

In Table 2 the QEPF-ATC approach implement fastconvergence when 120582 is set to different values and the optimalresults are close to the results of the initial problems Withthe increase of 120582 the number of iterations decreases andthe solution error gets bigger (excessive weight) Generally14 le 120582 le 10 is strictly necessary but typically 4 lt

120582 lt 7 is recommended to speed up convergence and reducecomputational cost and get better optimal results

42 The Speed Reducer Design The speed reducer designcase represents the design of a simple gearbox and is amultidisciplinary problem comprising the coupling betweengear design and shaft design This has been used as a testingproblem for MDO method in the literature from NASALangley research center [24] The multidisciplinary systemsand notations of the reducer are shown in Figure 6 and

x1

x2

x3

x4x5

x6

x7

Figure 6 The multidisciplinary systems and notations of reducer

Speed reducer

x6 x7 Shaftx4 x5

Gearx1 x2 x3

Figure 7 The decomposition of speed reducer problem

the modelling is shown in Figure 7 The design objectiveis to minimize the speed reducer weight while satisfying anumber of constraints posed by gear and shaft parts Thereare seven variables in the speed reducer design optimizationTable 3 lists a brief description of the variables and gives theirreference values

According to the decomposition method of ATC wherea system is partitioned by object the speed reducer problemis decomposed into gear subproblem and shaft subproblem(shown in Figure 7) The design optimization problem isdefined as follows

min 119891 = 1198911(119909) + 119891

2(119909)

1198911(119909) = 07854119909

11199092

2

sdot (333331199092

3+ 149334119909

3minus 430934)

minus 150791199091(1199092

6+ 1199092

7)

8 Mathematical Problems in Engineering

The optimizer of system level

The optimizer of subproblem 1 The optimizer of subproblem 2

DOE-LHS DOE-LHS

Subproblem 1

analysis

Kriging model 1

Subproblem 2

analysis

Kriging model 2

g3 g4 g5 g6

xU8 xU7B x

U6B xL8 x

L7B x

L6B xU9 x

U6C x

U7C xL9 x

L6C x

L7C

X1 X1

X1

X1 G1

G1X2 X2

X2

X2 G2

G2

st

st st

R1 = x10 = r1(x3 x4 x5) = x8 + x9

X0 = x6 x7

G1 = g1 g2 g9 g10 g11R2 = x8 = r2(X1) = f1(x)

X1 = x1 x2 x3 x6 x7

G2 = g7 g8R3 = x9 = r3(X2) = f2(x)

X2 = x4 x5 x6 x7

PA min f11 = x10 + x8 minus xL8)

2+ (x9 minus xL9)

2+ (x6 minus xL6B)

2+ (x6 minus xL6C)

2+ (x7 minus xL7B)

2+ (x7 minus xL7C)

2]120590

2)]

PB min f21 = x8 minus xU8 )

2+ (x6 minus xU6B)

2+ (x7 minus xU7B)

2]120590

2)] PC min f22 = x9 minus x

U9 )

2+ (x6 minus xU6C)

2+ (x7 minus xU7C)

2]120590

2)]

Figure 8 The solving framework of the QEPF-ATC and Kriging model combined approach for the gear reducer

1198912(119909) = 7477 (119909

3

6+ 1199093

7) + 07854 (119909

41199092

6+ 11990951199092

7)

st 1198921=

27

(11990911199092

21199093)minus 1 le 0 119892

2=

3975

(11990911199092

21199092

3)minus 1 le 0

1198923=1931199093

4

(119909211990931199094

6)minus 1 le 0 119892

4=1931199093

5

(119909211990931199094

7)minus 1 le 0

1198925= 10119909

minus3

6radic(745119909

minus1

2119909minus1

31199094) + 169 times 107

minus 1100 le 0

1198926= 10119909

minus3

7radic(745119909

minus1

2119909minus1

31199095) + 1575 times 108

minus 850 le 0

1198927=(151199096+ 19)

1199094

minus 1 le 0

1198928=(151199097+ 19)

1199095

minus 1 le 0

1198929= 11990921199093minus 40 le 0 119892

10= 5 minus

1199091

1199092

le 0

11989211=1199091

1199092

minus 12 le 0

(20)

where 1198911is the gear weight and 119891

2is the shaft weight And 119892

1

and 1198922 are respectively gear bending stress and contact stress

constraints 1198923and 119892

4are respectively torsional distortion

constraints of shaft 1 and shaft 2 1198925and 119892

6are stress

constraints of the shaft 1 and shaft 2 1198927and 119892

8are experience

constraints of shaft 1 and shaft 2 and 1198929 11989210 and 119892

11are gear

geometry constraints1199091 1199092 and 119909

3are the design variables of the gear

subproblem 1199094and 119909

5are the design variables of the

shaft subproblem 1199096and 119909

7are linking variables (coupling

variables) According to the QEPF-ATC and Kriging modelcombined approach three variables 119909

8 1199099 and 119909

10are

introduced and meet the following formula

ℎ0= 1199098+ 1199099minus 11990910= 0

ℎ1= 1199098minus 1198911(119909) = 0

ℎ2= 1199099minus 1198912(119909) = 0

(21)

where ℎ0represents the analysis model of system level and

ℎ1and ℎ

2represent the analysis models of subproblem 1 and

subproblem 2 respectivelyThe solving framework of the QEPF-ATC and Kriging

model combined approach for gear reducer is shown inFigure 8

Creating the Kriging models of the shaft subsystem isintroduced in detail Firstly select 45 sample points byLHS method to create the Kriging models to conduct the

Mathematical Problems in Engineering 9

Table 4 The optimal results of the QEPF-ATC and Kriging model combined approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299453432 26 138 (3504 0698 17 7300 7845 3410 5283)2 299453217 25 124 (3501 0700 17 7310 7763 3353 5286)3 299521246 18 113 (3498 0699 17 7306 7755 3312 5279)4 299564235 10 36 (3499 0702 17 7297 7718 3350 5282)5 299433263 24 121 (3502 0700 17 7298 7720 3347 5287)

Table 5 The optimal results of the QEPF-ATC approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299463432 42 272 (3500 0699 17 7300 7825 3392 5282)2 299551327 38 215 (3500 0703 17 7305 7746 3353 5287)3 299612356 30 198 (3497 0696 17 7300 7713 3326 5275)4 299575363 17 97 (3499 0700 17 7299 7715 3349 5282)5 299521554 52 306 (3500 0700 17 7300 7719 3350 5286)

subsystem analysis and the created models are shown inFigure 9

Then select 20 sample points by random samplingmethod to evaluate the createdKrigingmodels Error analysisfor Kriging models of the shaft subsystem is shown inFigure 10 and it can be seen that the predicted valuesobtained by Kriging models are basically equal to the actualvalues

There are four credibility evaluation standards for approx-imate models that is the average the maximum the rootmean square and the 119877-squared For the average the max-imum and the root mean square the much smaller thevalues are the much better the credibility of approximatemodels is The default threshold of the average and theroot mean square is 02 and the maximum is 03 Forthe 119877-squared the much bigger the value is the muchbetter the credibility of approximate models is and thedefault threshold is 09 Figure 11 describes the values of thefour credibility evaluation standards for the created Krigingmodels of the shaft subsystem and it can be seen that the fourcredibility evaluation standards are all met So the createdKrigingmodels are reliable enough and can be used to replacethe original models with high accuracy in the next solvingprocess

Five initial points are selected and the sequentialquadratic programming (SQP) method is adopted to solvethe gear reducer problem and the results provided by theQEPF-ATC and Kriging model combined approach andthe QEPF-ATC approach are shown in Tables 4 and 5respectively It can be seen that results of the QEPF-ATCand Kriging model combined approach and the QEPF-ATCapproach are very close and the best result provided bythe QEPF-ATC and Kriging model combined approach iscloser to the optimal value of the gear reducer problemcompared to that provided by the QEPF-ATC approachIn terms of the iterations and runtime the QEPF-ATCand Kriging model combined approach saves computingtime by more than one half compared with QEPF-ATCapproach

5 Conclusion

Because ATC approach can flexibly organize the design opti-mization process according to the components of engineeringsystems it has been widely applied in MDOThe QEPF-ATCapproach is proposed to reduce the complicated coordinationprocess and heavy computation cost faced by exiting ATCapproaches where quadratic exterior penalty function isadopted to be the replaced coordination strategy Furtherin order to deal with the design optimization of complexengineering systems the QEPF-ATC and Kriging modelcombined approach is implemented where Kriging modelsare used to replace simulation models with high accuracyin the solving process of MDO problems The QEPF-ATCand Kriging model combined approach can avoid the impactof numerical noise brought by using simulation modelsduring the iterative solving process and reduce computationcost A geometric programming case is given to validatethe usability and effectiveness of the QEPF-ATC approachThe QEPF-ATC and Kriging model combined approach hasbeen successfully applied to solve the speed reducer designproblem

It should be pointed out that usingmetamodels within theQEPF-ATC approach is not without its own set of potentialdrawbacks which are well known and beyond the scope ofthis work Prior to employing the metamodel assisted QEPF-ATC approach the user should become familiar with thestrengths and weaknesses associated with surrogate approxi-mation Variable-fidelity modeling provides a promising wayto flexibly use approximate models in dealing with MDOproblems The combination of the QEPF-ATC approach andvariable-fidelity modeling will be studied to design complexengineering systems in the future

Abbreviations and Acronyms

ALF Augmented Lagrangian functionAM Approximate modelsATC Analytical target cascadingBLISS Bilevel integrated system synthesis

10 Mathematical Problems in Engineering

002minus002minus006minus01minus014minus018minus022minus026minus03

002minus002minus006minus01minus014minus018minus022minus026minus03

8

8 8

8

75

75 75

75

7 7

77

g7 g7

02

01501005

0minus005minus01minus015minus02

02

015010050minus005minus01minus015minus02

8

8 8

8

75

75 75

75

7 7

77

g8g8

x 4

x 4

x5

x5

x 4

x 4

x5

x5

Figure 9 Kriging models of the shaft subsystem

minus006

minus016

minus026minus026 minus016 minus006

Actu

al

Actu

al

Predicted Predicted

005

minus005

minus015minus015 minus005 005

g8g7

Figure 10 Error analysis for Kriging models of the shaft subsystem

Figure 11 The error data of Kriging models of subsystem 2

Mathematical Problems in Engineering 11

CO Collaborative optimizationCSSO Concurrent subspace optimizationDQA Diagonal quadratic approximationLDF Lagrangian dual functionLHS Latin hypercube samplingMDO Multidisciplinary design

optimizationQEPF Quadratic exterior penalty functionQEPF-ATC Quadratic exterior penalty function

based analytical target cascadingSQP Sequential quadratic programming

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by theNational Natural Sci-ence Foundation of China (NSFC) under Grant no 51121002the National Basic Research Program (973 Program) ofChina under Grant no 2014CB046703 and the FundamentalResearch Funds for the Central Universities HUST Grantno 2014TS040 The authors also would like to thank theanonymous referees for their valuable comments

References

[1] J Sobieszczanski-Sobieski ldquoA linear decomposition method forlarge optimization problemsrdquo NASA Technical Memorandum83248 1982

[2] TW Simpson and J R R AMartins ldquoMultidisciplinary designoptimization for complex engineered systems report from anational science foundation workshoprdquo Journal of MechanicalDesign vol 133 no 10 Article ID 101002 2011

[3] M Balesdent N Berend P Depince and A Chriette ldquoA surveyof multidisciplinary design optimization methods in launchvehicle designrdquo Structural and Multidisciplinary Optimizationvol 45 no 5 pp 619ndash642 2012

[4] I Sobieski and I Kroo ldquoAircraft design using collaborativeoptimizationrdquo AIAA Paper 715 1996

[5] H-Z Huang H Yu X Zhang S Zeng and ZWang ldquoCollabo-rative optimization with inverse reliability for multidisciplinarysystems uncertainty analysisrdquoEngineeringOptimization vol 42no 8 pp 763ndash773 2010

[6] R S Sellar S M Batill and J E Renaud ldquoResponse surfacebased concurrent subspace optimization for multidisciplinarysystem designrdquo AIAA Paper 714 AIAA 1996

[7] L S Li J H Liu and S H Liu ldquoAn efficient strategy formultidisciplinary reliability design and optimization based onCSSO and PMA in SORA frameworkrdquo Structural and Multidis-ciplinary Optimization vol 49 no 2 pp 239ndash252 2014

[8] J Sobieszczanski-Sobieski ldquoBi-level integrated system synthesis(BLISS)rdquo AIAA Paper 4916 AIAA 1998

[9] J Sobieszczanski-Sobieski T D Altus M Phillips and RSandusky ldquoBi-level integrated system synthesis (BLISS) forconcurrent and distributed processingrdquo AIAA Paper 54092002

[10] N Michelena H Park and P Y Papalambros ldquoConvergenceproperties of analytical target cascadingrdquo AIAA Journal vol 41no 5 pp 897ndash905 2003

[11] M Gardenghi M M Wiecek and W Wang ldquoBiobjectiveoptimization for analytical target cascading optimality vsachievabilityrdquo Structural and Multidisciplinary Optimizationvol 47 no 1 pp 111ndash133 2013

[12] S Tosserams L F Etman P Y Papalambros and J E RoodaldquoAn augmented Lagrangian relaxation for analytical target cas-cading using the alternating direction method of multipliersrdquoStructural and Multidisciplinary Optimization vol 31 no 3 pp176ndash189 2006

[13] Y Li Z Lu and J J Michalek ldquoDiagonal quadratic approxima-tion for parallelization of analytical target cascadingrdquo Journalof Mechanical Design Transactions of the ASME vol 130 no 5Article ID 051402 2008

[14] H M Kim W Chen and M M Wiecek ldquoLagrangian coor-dination for enhancing the convergence of analytical targetcascadingrdquo AIAA Journal vol 44 no 10 pp 2197ndash2207 2006

[15] D W Kim and J Lee ldquoAn improvement of Kriging basedsequential approximate optimization method via extended useof design of experimentsrdquo Engineering Optimization vol 42 no12 pp 1133ndash1149 2010

[16] P B Backlund D W Shahan and C C Seepersad ldquoA com-parative study of the scalability of alternative metamodellingtechniquesrdquo Engineering Optimization vol 44 no 7 pp 767ndash786 2012

[17] J Zhang S Chowdhury and A Messac ldquoAn adaptive hybridsurrogate modelrdquo Structural and Multidisciplinary Optimiza-tion vol 46 no 2 pp 223ndash238 2012

[18] V Dubourg B Sudret and J-M Bourinet ldquoReliability-baseddesign optimization using kriging surrogates and subset simu-lationrdquo Structural and Multidisciplinary Optimization vol 44no 5 pp 673ndash690 2011

[19] H M Kim N F Michelena P Y Papalambros and T JiangldquoTarget cascading in optimal system designrdquo Transactions of theASMEmdashJournal of Mechanical Design vol 125 no 3 pp 474ndash480 2003

[20] J P Costa L Pronzato and E Thierry ldquoA comparison betweenKriging and radial basis function networks for nonlinearpredictionrdquo in Proceedings of the IEEE-EURASIP Workshop onNonlinear Signal and Image Processing (NSIP rsquo99) pp 726ndash7301999

[21] V Cherkassky D Gehring and F Mulier ldquoComparison ofadaptive methods for function estimation from samplesrdquo IEEETransactions on Neural Networks vol 7 no 4 pp 969ndash9841996

[22] M S Kim H Cho S G Lee J Choi and D Bae ldquoDFSS androbust optimization tool for multibody system with randomvariablesrdquo Journal of System Design and Dynamics vol 1 no 3pp 583ndash592 2007

[23] H M Kim Target cascading in optimal system design [PhDdissertation] University of Michigan 2001

[24] S L Padula N Alexandrov and L L Green ldquoMDO testsuite at NASA Langley Research Centerrdquo in Proceedings of the6th Symposium onMultidisciplinary Analysis and OptimizationAIAA Paper 96-4028 pp 410ndash420 Bellevue Wash USASeptember 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

where the superscript 119896 indicates iterations and119898 is a positiveintegers

The difference of these coordination strategies is theway of dealing with consistency constraints and weightcoefficients There exist multiweight coefficients in the abovethree coordination strategies to increase computational costFor this reason the coordination strategy based on quadraticexterior penalty function among which only one weight coef-ficient exists is put forward which simplifies the expression ofoptimization problem greatly thus improving the efficiencyto make ATC have wider application The coordinationstrategy based on the quadratic exterior penalty function willbe described briefly

22 The QEPF-ATC Approach A general equality-con-strained optimization problem can be defined as follows

min 119891 (x)

st 119888119894(x) = 0 119894 = 1 2 119898

xlb le x le xup

(8)

where x is the vector of the design variables with xlb andxup as its lower and upper bounds respectively The function119891(x) represents the objective function 119888

119894(x) is the equality

constraintsAccording to QEPF consistency constraints are added

into the objective function by introducing a penalty factorand the penalty factor is used as the weighting coefficientto realize the coordination The quadratic exterior penaltyfunction is defined as

119865 (119909 120588) = 119891 (x) +120588

2

119898

sum

119894=1

119888119894(x)2 = 119891 (x) +

120588

2119888 (x)119879 119888 (x) (9)

where the weight coefficient (penalty factor) is updated inaccordance with (10) The value of 120582 influences the con-vergence efficiency and computational efficiency so that itis necessary to set up appropriate 120582 to obtain much bettersolution efficiency Consider

120588119896+1= 120582120588119896 120582 = 14sim10 (10)

The mathematical formulation of QEPF-ATC is

119875119894119895 min 119891

119894119895+120588

2(10038171003817100381710038171003817(119877119894minus1

119894119895minus 119877119894

119894119895)10038171003817100381710038171003817

2

+10038171003817100381710038171003817(119910119894minus1

119894119901minus 119910119894

119894119895)10038171003817100381710038171003817

2

+10038171003817100381710038171003817(119877119894

(119894+1)119895minus 119877119894+1

(119894+1)119896)10038171003817100381710038171003817

2

+10038171003817100381710038171003817(119910119894

(119894+1)119895minus 119910119894+1

(119894+1)119896)10038171003817100381710038171003817

2

)

st 119892119894119895le 0 ℎ

119894119895= 0

119877119894119895= 119903119894119895(119909119894119895)

119909119894119895= (119877119894

(119894+1)1198961

119877119894

(119894+1)119896119888119894119895

119909119894119895 119910119894

119894119895)

(11)

Generally the much bigger the penalty factor is the muchcloser the optimization results are to the optimal solution

Optimizing system subproblem

Yes

Converged

Solving subproblems

Decomposing the problem

Initializing

No Updating weight

Optimized results

Figure 3 The solving process of QEPF-ATC approach

of the original problem But in the practical applicationthe value of the penalty factor must be moderate whenconsidering the computational cost due to the fact that ifthe penalty factor is too small the optimization results mayget bigger error from the original problem to bring lowcomputational efficiency Otherwise too large penalty factorwould increase the computational expense

Figure 3 shows the solving process of the QEPF-ATCapproach and the main steps are as follows

(1) decompose the original problem and confirm designvariables and constraints of system and subsystems

(2) initialize design variables (including local designvariables and linking variables) and 120588(0) and transferthe corresponding initial value to the subsystem as thetarget

(3) solve each subsystem and pass back current results tosystem level

(4) optimize the system problem on the basis of the thirdstep and transfer new target to each subsystem

(5) obtain the optimization result if the convergence con-dition is met otherwise update the weight coefficient120588 according to (10) and go back to the third step untilthe convergence condition is met

3 The Proposed QEPF-ATC and KrigingModel Combined Approach

31 Kriging Model Kriging models [15 16] have their originsin mining and geostatistical applications involving spatially

Mathematical Problems in Engineering 5

Decomposing the problem

Subsystem analyzing

Subsystem optimizing

System optimizing

Converged

No

Yes

Yes

No

Creating the Kriging model

Sampling by LHS

Optimized results

Validating approximation model

Figure 4The solving process of the QEPF-ATC and Kriging modelcombined approach

and temporally correlated data Kriging models combine aglobal model with localized departures

119910 (119909) = 119891 (119909) + 119911 (119909) (12)

where 119910(119909) is the unknown function of interest 119891(119909) is theknown approximation (usually polynomial) function and119911(119909) is the realization of a stochastic process with mean zerovariance1205902 and nonzero covarianceThe119891(119909) term is similarto a polynomial response surface providing a ldquoglobalrdquo modelof the design space The covariance of 119911(119909) is given by

cov [119911 (119909119894) 119911 (119909

119895)] = 120590

2119877 (119909119894 119909119895) (13)

where119877 is the correlation119877(119909119894 119909119895) is the correlation function

between any two of the 119899 sampled data points 119909119894and 119909

119895 and

it can be calculated by

119877 (119909119894 119909119895) = exp[minus

119899

sum

119896=1

120579119896(119909119896

119894minus 119909119896

119895)2

] (14)

Generally 119891(119909) is treated as a constant 120573 then predictedestimates

_119910 (119909) of the response 119910(119909) at untried values of 119909

are given by_119910 (119909) =

_120573 +119903119879

(119909) 119877minus1(119884 minus 119865

_120573) (15)

where 119910 is the column vector of length 119899 that contains thesample values of the response and 119891 is a column vector oflength 119899 that is filled with ones when 119891(119909) is taken as aconstant In (10) 119903119879(119909) is the correlation vector of length 119899between an untried 119909 and the sampled data points

Consider

119903119879

(119909) = [119877 (119909 1199091) 119877 (119909 119909

2) 119877 (119909 119909

3) 119877 (119909 119909

119899)]119879

(16)

In (10)_120573 is estimated using (12)

_120573 = (119865

119879119877minus1119865)minus1

119865119879119877minus1119884 (17)

In 119877 120579119896= maxminus[119899 ln(1205902) + ln |119877|]2

1205902=

(119884 minus 119865

_120573)

119879

119877minus1(119884 minus 119865

_120573)

119899

(18)

32 The QEPF-ATC and Kriging Model Combined ApproachIn QEPF-ATC the deviation between the target and responseis processed in the form of sum of squares as shown in (11)The sum of squares function is usually not smooth and non-linear And discrepancy of variables in the neighborhood ofthe current solution will lead to fluctuation of the deviationwhich will increase the number of iterations and so forthKriging model is appropriate for high nonlinear problemwith better comprehensive performance high approximationaccuracy and high robustness [20ndash22] Moreover fewersample points also can get higher precision approximationmodel So Kriging model is adopted in this paper

The solving process of the QEPF-ATC and Kriging modelcombined approach is shown in Figure 4 and the main stepsare as follows

(1) decompose the original problem and confirm designvariables and constraints of system and subsystems

(2) select a set of sample points using the Latin hypercubesampling (LHS) method

(3) analyse each subsystem to obtain responses at allsampling points of the subsystems

(4) build Kriging model based on the sample points andthe corresponding responses of each subsystem

(5) evaluate the Kriging model by error analysis if theprecision conditions are not met go back to step (2)to build a new Kriging model by adding some newsample points

(6) initialize variables and transfer the corresponding ini-tial value to the subsystem as the target and optimizeeach subproblem based on the created Krigingmodel

6 Mathematical Problems in Engineering

g1 g2

x1 x2 x3 x4 x5 x6 x7 x11 ge 0

xU3

xU11B

xL3

xL11B

g3 g4

x3 x8 x9 x10 x11 ge 0

x11

xU6

xU11C

xL6

xL11C

g5 g6

x6 x11 x12 x13 x14 ge 0

st

st st

R1 = x1 = r1(x3 x4 x5) = (x23 + xminus24 + x25)12

R2 = x2 = r2(x5 x6 x7) = (x25 + x26 + x27)12

R3 = x3 = r3(x8 x9 x10 x11)= (x28 + xminus29 + xminus210 + x

211)

12

PC minx6x11 x12 x13 x14

f22 =120590

2x6 minus x

U6 )) 2

+ (x11 minus xU11C)2]]

PA minx1x2x3x4x5x6x7x11

f11 = x21 + x22 +

120590

2x3 minus x

L3)

2+ (x6 minus xL6)

2+ (x11 minus xL11B)

2+ (x11 minus xL11C)

2])]

R4 = x6 = r4(x11 x12 x13 x14)= (x211 + x212 + x213 + x214)

12

xPB min

x3x8x9x10f21 =

120590

2x3 minus x

U3 )

2+ (x11 minus xU11B)

2])]11

Figure 5 The solving framework of QEPF-ATC for geometric programming

(7) pass back current results to system level and imple-ment the system level optimization

(8) obtain results if the convergence condition is metotherwise go to step (6) until the convergence con-dition is met

4 Case Studies

41 The Geometric Programming The geometric program-ming [23] that has only the global optimal solution isemployed to verify the usability of QEPF-ATC Its mathemat-ical expression is shown as

min 119891 = 1199092

1+ 1199092

2

st 1198921119909minus2

3+ 1199092

4

1199092

5

le 1 11989221199092

5+ 119909minus2

6

1199092

7

le 1

11989231199092

8+ 1199092

9

1199092

11

le 1

1198924119909minus2

8+ 1199092

10

1199092

11

le 1 11989251199092

11+ 119909minus2

12

1199092

13

le 1

11989261199092

11+ 1199092

12

1199092

14

le 1

ℎ1= 1199092

1minus 1199092

3minus 119909minus2

4minus 1199092

5= 0

ℎ2= 1199092

2minus 1199092

5minus 1199092

6minus 1199092

7= 0

ℎ3= 1199092

3minus 1199092

8minus 119909minus2

9minus 119909minus2

10minus 1199092

11= 0

ℎ4= 1199092

6minus 1199092

11minus 1199092

12minus 1199092

13minus 1199092

14= 0

(19)

Table 1 The optimal results

QEPF-ATC ALF DQA LDF AAO1199091

29049 285 286 287 2841199092

30551 304 310 305 3091199093

24930 248 249 243 2361199094

08805 086 082 079 0761199095

09676 094 095 089 0871199096

27103 272 275 278 2811199097

10357 102 095 103 0941199098

09830 098 098 101 0971199099

07572 078 079 082 08711990910

07104 076 079 081 08011990911

12408 127 126 125 13011990912

08409 086 085 083 08411990913

17088 173 174 171 17611990914

14875 152 149 150 155119891 177721 1736 1778 1753 1759Runtime (s) 289 355 336 368

Figure 5 shows the solution framework of the QEPF-ATCapproach for the problem where 119909

11is the linking variable

(coupling variable) coordinated by the system level 11990911988011119861

119909119880

11119862 1199091198803 and 119909119880

6are the targets of each subproblem passed

from the system level 11990911987111119861

and 11990911987111119862

are the responses of thelinking variable coming from the subproblems 119861 and 119862 119909119871

3

and 1199091198716are the responses of 119861 and 119862

Solve the geometric programming problem according toQEPF-ATC Initialize the design point 1199090 = (1 1 1 1 1 1 11 1 1 1 1 1 1)

119879 and take 120582 = 10 Here the acceptableinconsistency tolerance is 001 for every response variable andlinking variable Table 1 shows the optimal results based on

Mathematical Problems in Engineering 7

Table 2 The optimal results based on 120582

120582 (1199091 1199092 1199093 1199094 1199095 1199096 1199097 11990911) (119909

3 1199098 1199099 11990910 11990911) (119909

6 11990911 11990912 11990913 11990914) 119891 Iterations

14 (27556 30804 22975 0813109171 27718 09840 12678) (23012 09688 08427 07340 12682) (27638 12732 07447 18411 14655) 175992 78

4 (29716 29567 25227 0759808570 26708 09352 12343) (25230 09911 07320 07072 12321) (26791 12298 08409 16923 14686) 175728 27

7 (29662 29757 25012 0720408231 27169 09014 12386) (25011 09863 07492 07115 12386) (27168 12385 08408 17122 14915) 176531 17

10 (29049 30551 24930 0880509676 27103 10357 12408) (24930 09830 07572 07104 12408) (27102 12409 08409 17088 14875) 177721 5

Table 3 Design variables for speed reducer

Variables Range Units1199091 Gear face width 26 le 119909

1le 36 cm

1199092 Teeth module 07 le 119909

2le 08 cm

1199093

Number of teeth of pinion(integer variable) 17 le 119909

3le 28

1199094

Distance between bearings1 73 le 119909

4le 83 cm

1199095

Distance between bearings2 73 le 119909

5le 83 cm

1199096 Diameter of shaft 1 29 le 119909

6le 39 cm

1199097 Diameter of shaft 1 50 le 119909

7le 55 cm

1199091 1199092 and 119909

3are the design variables of the gear subproblem 119909

4and 1199095are

the design variables of the shaft subproblem 1199096and 1199097are linking variables

(coupling variables)

QEPF-ATC three coordination strategies (ALF DQA andLDF) and AAO (All-At-Once)

According to Table 1 the results of the QEPF-ATCapproach are very close to the optimal value of the originalproblem and the runtime is less than the other threemethods120582 affects the convergence efficiency and computational

efficiency of the QEPF-ATC approach In order to obtainmore suitable 120582 to improve the efficiency of the QEPF-ATC approach different values of 120582 are selected to solve theproblem Table 2 shows the results when different values of 120582are selected

In Table 2 the QEPF-ATC approach implement fastconvergence when 120582 is set to different values and the optimalresults are close to the results of the initial problems Withthe increase of 120582 the number of iterations decreases andthe solution error gets bigger (excessive weight) Generally14 le 120582 le 10 is strictly necessary but typically 4 lt

120582 lt 7 is recommended to speed up convergence and reducecomputational cost and get better optimal results

42 The Speed Reducer Design The speed reducer designcase represents the design of a simple gearbox and is amultidisciplinary problem comprising the coupling betweengear design and shaft design This has been used as a testingproblem for MDO method in the literature from NASALangley research center [24] The multidisciplinary systemsand notations of the reducer are shown in Figure 6 and

x1

x2

x3

x4x5

x6

x7

Figure 6 The multidisciplinary systems and notations of reducer

Speed reducer

x6 x7 Shaftx4 x5

Gearx1 x2 x3

Figure 7 The decomposition of speed reducer problem

the modelling is shown in Figure 7 The design objectiveis to minimize the speed reducer weight while satisfying anumber of constraints posed by gear and shaft parts Thereare seven variables in the speed reducer design optimizationTable 3 lists a brief description of the variables and gives theirreference values

According to the decomposition method of ATC wherea system is partitioned by object the speed reducer problemis decomposed into gear subproblem and shaft subproblem(shown in Figure 7) The design optimization problem isdefined as follows

min 119891 = 1198911(119909) + 119891

2(119909)

1198911(119909) = 07854119909

11199092

2

sdot (333331199092

3+ 149334119909

3minus 430934)

minus 150791199091(1199092

6+ 1199092

7)

8 Mathematical Problems in Engineering

The optimizer of system level

The optimizer of subproblem 1 The optimizer of subproblem 2

DOE-LHS DOE-LHS

Subproblem 1

analysis

Kriging model 1

Subproblem 2

analysis

Kriging model 2

g3 g4 g5 g6

xU8 xU7B x

U6B xL8 x

L7B x

L6B xU9 x

U6C x

U7C xL9 x

L6C x

L7C

X1 X1

X1

X1 G1

G1X2 X2

X2

X2 G2

G2

st

st st

R1 = x10 = r1(x3 x4 x5) = x8 + x9

X0 = x6 x7

G1 = g1 g2 g9 g10 g11R2 = x8 = r2(X1) = f1(x)

X1 = x1 x2 x3 x6 x7

G2 = g7 g8R3 = x9 = r3(X2) = f2(x)

X2 = x4 x5 x6 x7

PA min f11 = x10 + x8 minus xL8)

2+ (x9 minus xL9)

2+ (x6 minus xL6B)

2+ (x6 minus xL6C)

2+ (x7 minus xL7B)

2+ (x7 minus xL7C)

2]120590

2)]

PB min f21 = x8 minus xU8 )

2+ (x6 minus xU6B)

2+ (x7 minus xU7B)

2]120590

2)] PC min f22 = x9 minus x

U9 )

2+ (x6 minus xU6C)

2+ (x7 minus xU7C)

2]120590

2)]

Figure 8 The solving framework of the QEPF-ATC and Kriging model combined approach for the gear reducer

1198912(119909) = 7477 (119909

3

6+ 1199093

7) + 07854 (119909

41199092

6+ 11990951199092

7)

st 1198921=

27

(11990911199092

21199093)minus 1 le 0 119892

2=

3975

(11990911199092

21199092

3)minus 1 le 0

1198923=1931199093

4

(119909211990931199094

6)minus 1 le 0 119892

4=1931199093

5

(119909211990931199094

7)minus 1 le 0

1198925= 10119909

minus3

6radic(745119909

minus1

2119909minus1

31199094) + 169 times 107

minus 1100 le 0

1198926= 10119909

minus3

7radic(745119909

minus1

2119909minus1

31199095) + 1575 times 108

minus 850 le 0

1198927=(151199096+ 19)

1199094

minus 1 le 0

1198928=(151199097+ 19)

1199095

minus 1 le 0

1198929= 11990921199093minus 40 le 0 119892

10= 5 minus

1199091

1199092

le 0

11989211=1199091

1199092

minus 12 le 0

(20)

where 1198911is the gear weight and 119891

2is the shaft weight And 119892

1

and 1198922 are respectively gear bending stress and contact stress

constraints 1198923and 119892

4are respectively torsional distortion

constraints of shaft 1 and shaft 2 1198925and 119892

6are stress

constraints of the shaft 1 and shaft 2 1198927and 119892

8are experience

constraints of shaft 1 and shaft 2 and 1198929 11989210 and 119892

11are gear

geometry constraints1199091 1199092 and 119909

3are the design variables of the gear

subproblem 1199094and 119909

5are the design variables of the

shaft subproblem 1199096and 119909

7are linking variables (coupling

variables) According to the QEPF-ATC and Kriging modelcombined approach three variables 119909

8 1199099 and 119909

10are

introduced and meet the following formula

ℎ0= 1199098+ 1199099minus 11990910= 0

ℎ1= 1199098minus 1198911(119909) = 0

ℎ2= 1199099minus 1198912(119909) = 0

(21)

where ℎ0represents the analysis model of system level and

ℎ1and ℎ

2represent the analysis models of subproblem 1 and

subproblem 2 respectivelyThe solving framework of the QEPF-ATC and Kriging

model combined approach for gear reducer is shown inFigure 8

Creating the Kriging models of the shaft subsystem isintroduced in detail Firstly select 45 sample points byLHS method to create the Kriging models to conduct the

Mathematical Problems in Engineering 9

Table 4 The optimal results of the QEPF-ATC and Kriging model combined approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299453432 26 138 (3504 0698 17 7300 7845 3410 5283)2 299453217 25 124 (3501 0700 17 7310 7763 3353 5286)3 299521246 18 113 (3498 0699 17 7306 7755 3312 5279)4 299564235 10 36 (3499 0702 17 7297 7718 3350 5282)5 299433263 24 121 (3502 0700 17 7298 7720 3347 5287)

Table 5 The optimal results of the QEPF-ATC approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299463432 42 272 (3500 0699 17 7300 7825 3392 5282)2 299551327 38 215 (3500 0703 17 7305 7746 3353 5287)3 299612356 30 198 (3497 0696 17 7300 7713 3326 5275)4 299575363 17 97 (3499 0700 17 7299 7715 3349 5282)5 299521554 52 306 (3500 0700 17 7300 7719 3350 5286)

subsystem analysis and the created models are shown inFigure 9

Then select 20 sample points by random samplingmethod to evaluate the createdKrigingmodels Error analysisfor Kriging models of the shaft subsystem is shown inFigure 10 and it can be seen that the predicted valuesobtained by Kriging models are basically equal to the actualvalues

There are four credibility evaluation standards for approx-imate models that is the average the maximum the rootmean square and the 119877-squared For the average the max-imum and the root mean square the much smaller thevalues are the much better the credibility of approximatemodels is The default threshold of the average and theroot mean square is 02 and the maximum is 03 Forthe 119877-squared the much bigger the value is the muchbetter the credibility of approximate models is and thedefault threshold is 09 Figure 11 describes the values of thefour credibility evaluation standards for the created Krigingmodels of the shaft subsystem and it can be seen that the fourcredibility evaluation standards are all met So the createdKrigingmodels are reliable enough and can be used to replacethe original models with high accuracy in the next solvingprocess

Five initial points are selected and the sequentialquadratic programming (SQP) method is adopted to solvethe gear reducer problem and the results provided by theQEPF-ATC and Kriging model combined approach andthe QEPF-ATC approach are shown in Tables 4 and 5respectively It can be seen that results of the QEPF-ATCand Kriging model combined approach and the QEPF-ATCapproach are very close and the best result provided bythe QEPF-ATC and Kriging model combined approach iscloser to the optimal value of the gear reducer problemcompared to that provided by the QEPF-ATC approachIn terms of the iterations and runtime the QEPF-ATCand Kriging model combined approach saves computingtime by more than one half compared with QEPF-ATCapproach

5 Conclusion

Because ATC approach can flexibly organize the design opti-mization process according to the components of engineeringsystems it has been widely applied in MDOThe QEPF-ATCapproach is proposed to reduce the complicated coordinationprocess and heavy computation cost faced by exiting ATCapproaches where quadratic exterior penalty function isadopted to be the replaced coordination strategy Furtherin order to deal with the design optimization of complexengineering systems the QEPF-ATC and Kriging modelcombined approach is implemented where Kriging modelsare used to replace simulation models with high accuracyin the solving process of MDO problems The QEPF-ATCand Kriging model combined approach can avoid the impactof numerical noise brought by using simulation modelsduring the iterative solving process and reduce computationcost A geometric programming case is given to validatethe usability and effectiveness of the QEPF-ATC approachThe QEPF-ATC and Kriging model combined approach hasbeen successfully applied to solve the speed reducer designproblem

It should be pointed out that usingmetamodels within theQEPF-ATC approach is not without its own set of potentialdrawbacks which are well known and beyond the scope ofthis work Prior to employing the metamodel assisted QEPF-ATC approach the user should become familiar with thestrengths and weaknesses associated with surrogate approxi-mation Variable-fidelity modeling provides a promising wayto flexibly use approximate models in dealing with MDOproblems The combination of the QEPF-ATC approach andvariable-fidelity modeling will be studied to design complexengineering systems in the future

Abbreviations and Acronyms

ALF Augmented Lagrangian functionAM Approximate modelsATC Analytical target cascadingBLISS Bilevel integrated system synthesis

10 Mathematical Problems in Engineering

002minus002minus006minus01minus014minus018minus022minus026minus03

002minus002minus006minus01minus014minus018minus022minus026minus03

8

8 8

8

75

75 75

75

7 7

77

g7 g7

02

01501005

0minus005minus01minus015minus02

02

015010050minus005minus01minus015minus02

8

8 8

8

75

75 75

75

7 7

77

g8g8

x 4

x 4

x5

x5

x 4

x 4

x5

x5

Figure 9 Kriging models of the shaft subsystem

minus006

minus016

minus026minus026 minus016 minus006

Actu

al

Actu

al

Predicted Predicted

005

minus005

minus015minus015 minus005 005

g8g7

Figure 10 Error analysis for Kriging models of the shaft subsystem

Figure 11 The error data of Kriging models of subsystem 2

Mathematical Problems in Engineering 11

CO Collaborative optimizationCSSO Concurrent subspace optimizationDQA Diagonal quadratic approximationLDF Lagrangian dual functionLHS Latin hypercube samplingMDO Multidisciplinary design

optimizationQEPF Quadratic exterior penalty functionQEPF-ATC Quadratic exterior penalty function

based analytical target cascadingSQP Sequential quadratic programming

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by theNational Natural Sci-ence Foundation of China (NSFC) under Grant no 51121002the National Basic Research Program (973 Program) ofChina under Grant no 2014CB046703 and the FundamentalResearch Funds for the Central Universities HUST Grantno 2014TS040 The authors also would like to thank theanonymous referees for their valuable comments

References

[1] J Sobieszczanski-Sobieski ldquoA linear decomposition method forlarge optimization problemsrdquo NASA Technical Memorandum83248 1982

[2] TW Simpson and J R R AMartins ldquoMultidisciplinary designoptimization for complex engineered systems report from anational science foundation workshoprdquo Journal of MechanicalDesign vol 133 no 10 Article ID 101002 2011

[3] M Balesdent N Berend P Depince and A Chriette ldquoA surveyof multidisciplinary design optimization methods in launchvehicle designrdquo Structural and Multidisciplinary Optimizationvol 45 no 5 pp 619ndash642 2012

[4] I Sobieski and I Kroo ldquoAircraft design using collaborativeoptimizationrdquo AIAA Paper 715 1996

[5] H-Z Huang H Yu X Zhang S Zeng and ZWang ldquoCollabo-rative optimization with inverse reliability for multidisciplinarysystems uncertainty analysisrdquoEngineeringOptimization vol 42no 8 pp 763ndash773 2010

[6] R S Sellar S M Batill and J E Renaud ldquoResponse surfacebased concurrent subspace optimization for multidisciplinarysystem designrdquo AIAA Paper 714 AIAA 1996

[7] L S Li J H Liu and S H Liu ldquoAn efficient strategy formultidisciplinary reliability design and optimization based onCSSO and PMA in SORA frameworkrdquo Structural and Multidis-ciplinary Optimization vol 49 no 2 pp 239ndash252 2014

[8] J Sobieszczanski-Sobieski ldquoBi-level integrated system synthesis(BLISS)rdquo AIAA Paper 4916 AIAA 1998

[9] J Sobieszczanski-Sobieski T D Altus M Phillips and RSandusky ldquoBi-level integrated system synthesis (BLISS) forconcurrent and distributed processingrdquo AIAA Paper 54092002

[10] N Michelena H Park and P Y Papalambros ldquoConvergenceproperties of analytical target cascadingrdquo AIAA Journal vol 41no 5 pp 897ndash905 2003

[11] M Gardenghi M M Wiecek and W Wang ldquoBiobjectiveoptimization for analytical target cascading optimality vsachievabilityrdquo Structural and Multidisciplinary Optimizationvol 47 no 1 pp 111ndash133 2013

[12] S Tosserams L F Etman P Y Papalambros and J E RoodaldquoAn augmented Lagrangian relaxation for analytical target cas-cading using the alternating direction method of multipliersrdquoStructural and Multidisciplinary Optimization vol 31 no 3 pp176ndash189 2006

[13] Y Li Z Lu and J J Michalek ldquoDiagonal quadratic approxima-tion for parallelization of analytical target cascadingrdquo Journalof Mechanical Design Transactions of the ASME vol 130 no 5Article ID 051402 2008

[14] H M Kim W Chen and M M Wiecek ldquoLagrangian coor-dination for enhancing the convergence of analytical targetcascadingrdquo AIAA Journal vol 44 no 10 pp 2197ndash2207 2006

[15] D W Kim and J Lee ldquoAn improvement of Kriging basedsequential approximate optimization method via extended useof design of experimentsrdquo Engineering Optimization vol 42 no12 pp 1133ndash1149 2010

[16] P B Backlund D W Shahan and C C Seepersad ldquoA com-parative study of the scalability of alternative metamodellingtechniquesrdquo Engineering Optimization vol 44 no 7 pp 767ndash786 2012

[17] J Zhang S Chowdhury and A Messac ldquoAn adaptive hybridsurrogate modelrdquo Structural and Multidisciplinary Optimiza-tion vol 46 no 2 pp 223ndash238 2012

[18] V Dubourg B Sudret and J-M Bourinet ldquoReliability-baseddesign optimization using kriging surrogates and subset simu-lationrdquo Structural and Multidisciplinary Optimization vol 44no 5 pp 673ndash690 2011

[19] H M Kim N F Michelena P Y Papalambros and T JiangldquoTarget cascading in optimal system designrdquo Transactions of theASMEmdashJournal of Mechanical Design vol 125 no 3 pp 474ndash480 2003

[20] J P Costa L Pronzato and E Thierry ldquoA comparison betweenKriging and radial basis function networks for nonlinearpredictionrdquo in Proceedings of the IEEE-EURASIP Workshop onNonlinear Signal and Image Processing (NSIP rsquo99) pp 726ndash7301999

[21] V Cherkassky D Gehring and F Mulier ldquoComparison ofadaptive methods for function estimation from samplesrdquo IEEETransactions on Neural Networks vol 7 no 4 pp 969ndash9841996

[22] M S Kim H Cho S G Lee J Choi and D Bae ldquoDFSS androbust optimization tool for multibody system with randomvariablesrdquo Journal of System Design and Dynamics vol 1 no 3pp 583ndash592 2007

[23] H M Kim Target cascading in optimal system design [PhDdissertation] University of Michigan 2001

[24] S L Padula N Alexandrov and L L Green ldquoMDO testsuite at NASA Langley Research Centerrdquo in Proceedings of the6th Symposium onMultidisciplinary Analysis and OptimizationAIAA Paper 96-4028 pp 410ndash420 Bellevue Wash USASeptember 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Decomposing the problem

Subsystem analyzing

Subsystem optimizing

System optimizing

Converged

No

Yes

Yes

No

Creating the Kriging model

Sampling by LHS

Optimized results

Validating approximation model

Figure 4The solving process of the QEPF-ATC and Kriging modelcombined approach

and temporally correlated data Kriging models combine aglobal model with localized departures

119910 (119909) = 119891 (119909) + 119911 (119909) (12)

where 119910(119909) is the unknown function of interest 119891(119909) is theknown approximation (usually polynomial) function and119911(119909) is the realization of a stochastic process with mean zerovariance1205902 and nonzero covarianceThe119891(119909) term is similarto a polynomial response surface providing a ldquoglobalrdquo modelof the design space The covariance of 119911(119909) is given by

cov [119911 (119909119894) 119911 (119909

119895)] = 120590

2119877 (119909119894 119909119895) (13)

where119877 is the correlation119877(119909119894 119909119895) is the correlation function

between any two of the 119899 sampled data points 119909119894and 119909

119895 and

it can be calculated by

119877 (119909119894 119909119895) = exp[minus

119899

sum

119896=1

120579119896(119909119896

119894minus 119909119896

119895)2

] (14)

Generally 119891(119909) is treated as a constant 120573 then predictedestimates

_119910 (119909) of the response 119910(119909) at untried values of 119909

are given by_119910 (119909) =

_120573 +119903119879

(119909) 119877minus1(119884 minus 119865

_120573) (15)

where 119910 is the column vector of length 119899 that contains thesample values of the response and 119891 is a column vector oflength 119899 that is filled with ones when 119891(119909) is taken as aconstant In (10) 119903119879(119909) is the correlation vector of length 119899between an untried 119909 and the sampled data points

Consider

119903119879

(119909) = [119877 (119909 1199091) 119877 (119909 119909

2) 119877 (119909 119909

3) 119877 (119909 119909

119899)]119879

(16)

In (10)_120573 is estimated using (12)

_120573 = (119865

119879119877minus1119865)minus1

119865119879119877minus1119884 (17)

In 119877 120579119896= maxminus[119899 ln(1205902) + ln |119877|]2

1205902=

(119884 minus 119865

_120573)

119879

119877minus1(119884 minus 119865

_120573)

119899

(18)

32 The QEPF-ATC and Kriging Model Combined ApproachIn QEPF-ATC the deviation between the target and responseis processed in the form of sum of squares as shown in (11)The sum of squares function is usually not smooth and non-linear And discrepancy of variables in the neighborhood ofthe current solution will lead to fluctuation of the deviationwhich will increase the number of iterations and so forthKriging model is appropriate for high nonlinear problemwith better comprehensive performance high approximationaccuracy and high robustness [20ndash22] Moreover fewersample points also can get higher precision approximationmodel So Kriging model is adopted in this paper

The solving process of the QEPF-ATC and Kriging modelcombined approach is shown in Figure 4 and the main stepsare as follows

(1) decompose the original problem and confirm designvariables and constraints of system and subsystems

(2) select a set of sample points using the Latin hypercubesampling (LHS) method

(3) analyse each subsystem to obtain responses at allsampling points of the subsystems

(4) build Kriging model based on the sample points andthe corresponding responses of each subsystem

(5) evaluate the Kriging model by error analysis if theprecision conditions are not met go back to step (2)to build a new Kriging model by adding some newsample points

(6) initialize variables and transfer the corresponding ini-tial value to the subsystem as the target and optimizeeach subproblem based on the created Krigingmodel

6 Mathematical Problems in Engineering

g1 g2

x1 x2 x3 x4 x5 x6 x7 x11 ge 0

xU3

xU11B

xL3

xL11B

g3 g4

x3 x8 x9 x10 x11 ge 0

x11

xU6

xU11C

xL6

xL11C

g5 g6

x6 x11 x12 x13 x14 ge 0

st

st st

R1 = x1 = r1(x3 x4 x5) = (x23 + xminus24 + x25)12

R2 = x2 = r2(x5 x6 x7) = (x25 + x26 + x27)12

R3 = x3 = r3(x8 x9 x10 x11)= (x28 + xminus29 + xminus210 + x

211)

12

PC minx6x11 x12 x13 x14

f22 =120590

2x6 minus x

U6 )) 2

+ (x11 minus xU11C)2]]

PA minx1x2x3x4x5x6x7x11

f11 = x21 + x22 +

120590

2x3 minus x

L3)

2+ (x6 minus xL6)

2+ (x11 minus xL11B)

2+ (x11 minus xL11C)

2])]

R4 = x6 = r4(x11 x12 x13 x14)= (x211 + x212 + x213 + x214)

12

xPB min

x3x8x9x10f21 =

120590

2x3 minus x

U3 )

2+ (x11 minus xU11B)

2])]11

Figure 5 The solving framework of QEPF-ATC for geometric programming

(7) pass back current results to system level and imple-ment the system level optimization

(8) obtain results if the convergence condition is metotherwise go to step (6) until the convergence con-dition is met

4 Case Studies

41 The Geometric Programming The geometric program-ming [23] that has only the global optimal solution isemployed to verify the usability of QEPF-ATC Its mathemat-ical expression is shown as

min 119891 = 1199092

1+ 1199092

2

st 1198921119909minus2

3+ 1199092

4

1199092

5

le 1 11989221199092

5+ 119909minus2

6

1199092

7

le 1

11989231199092

8+ 1199092

9

1199092

11

le 1

1198924119909minus2

8+ 1199092

10

1199092

11

le 1 11989251199092

11+ 119909minus2

12

1199092

13

le 1

11989261199092

11+ 1199092

12

1199092

14

le 1

ℎ1= 1199092

1minus 1199092

3minus 119909minus2

4minus 1199092

5= 0

ℎ2= 1199092

2minus 1199092

5minus 1199092

6minus 1199092

7= 0

ℎ3= 1199092

3minus 1199092

8minus 119909minus2

9minus 119909minus2

10minus 1199092

11= 0

ℎ4= 1199092

6minus 1199092

11minus 1199092

12minus 1199092

13minus 1199092

14= 0

(19)

Table 1 The optimal results

QEPF-ATC ALF DQA LDF AAO1199091

29049 285 286 287 2841199092

30551 304 310 305 3091199093

24930 248 249 243 2361199094

08805 086 082 079 0761199095

09676 094 095 089 0871199096

27103 272 275 278 2811199097

10357 102 095 103 0941199098

09830 098 098 101 0971199099

07572 078 079 082 08711990910

07104 076 079 081 08011990911

12408 127 126 125 13011990912

08409 086 085 083 08411990913

17088 173 174 171 17611990914

14875 152 149 150 155119891 177721 1736 1778 1753 1759Runtime (s) 289 355 336 368

Figure 5 shows the solution framework of the QEPF-ATCapproach for the problem where 119909

11is the linking variable

(coupling variable) coordinated by the system level 11990911988011119861

119909119880

11119862 1199091198803 and 119909119880

6are the targets of each subproblem passed

from the system level 11990911987111119861

and 11990911987111119862

are the responses of thelinking variable coming from the subproblems 119861 and 119862 119909119871

3

and 1199091198716are the responses of 119861 and 119862

Solve the geometric programming problem according toQEPF-ATC Initialize the design point 1199090 = (1 1 1 1 1 1 11 1 1 1 1 1 1)

119879 and take 120582 = 10 Here the acceptableinconsistency tolerance is 001 for every response variable andlinking variable Table 1 shows the optimal results based on

Mathematical Problems in Engineering 7

Table 2 The optimal results based on 120582

120582 (1199091 1199092 1199093 1199094 1199095 1199096 1199097 11990911) (119909

3 1199098 1199099 11990910 11990911) (119909

6 11990911 11990912 11990913 11990914) 119891 Iterations

14 (27556 30804 22975 0813109171 27718 09840 12678) (23012 09688 08427 07340 12682) (27638 12732 07447 18411 14655) 175992 78

4 (29716 29567 25227 0759808570 26708 09352 12343) (25230 09911 07320 07072 12321) (26791 12298 08409 16923 14686) 175728 27

7 (29662 29757 25012 0720408231 27169 09014 12386) (25011 09863 07492 07115 12386) (27168 12385 08408 17122 14915) 176531 17

10 (29049 30551 24930 0880509676 27103 10357 12408) (24930 09830 07572 07104 12408) (27102 12409 08409 17088 14875) 177721 5

Table 3 Design variables for speed reducer

Variables Range Units1199091 Gear face width 26 le 119909

1le 36 cm

1199092 Teeth module 07 le 119909

2le 08 cm

1199093

Number of teeth of pinion(integer variable) 17 le 119909

3le 28

1199094

Distance between bearings1 73 le 119909

4le 83 cm

1199095

Distance between bearings2 73 le 119909

5le 83 cm

1199096 Diameter of shaft 1 29 le 119909

6le 39 cm

1199097 Diameter of shaft 1 50 le 119909

7le 55 cm

1199091 1199092 and 119909

3are the design variables of the gear subproblem 119909

4and 1199095are

the design variables of the shaft subproblem 1199096and 1199097are linking variables

(coupling variables)

QEPF-ATC three coordination strategies (ALF DQA andLDF) and AAO (All-At-Once)

According to Table 1 the results of the QEPF-ATCapproach are very close to the optimal value of the originalproblem and the runtime is less than the other threemethods120582 affects the convergence efficiency and computational

efficiency of the QEPF-ATC approach In order to obtainmore suitable 120582 to improve the efficiency of the QEPF-ATC approach different values of 120582 are selected to solve theproblem Table 2 shows the results when different values of 120582are selected

In Table 2 the QEPF-ATC approach implement fastconvergence when 120582 is set to different values and the optimalresults are close to the results of the initial problems Withthe increase of 120582 the number of iterations decreases andthe solution error gets bigger (excessive weight) Generally14 le 120582 le 10 is strictly necessary but typically 4 lt

120582 lt 7 is recommended to speed up convergence and reducecomputational cost and get better optimal results

42 The Speed Reducer Design The speed reducer designcase represents the design of a simple gearbox and is amultidisciplinary problem comprising the coupling betweengear design and shaft design This has been used as a testingproblem for MDO method in the literature from NASALangley research center [24] The multidisciplinary systemsand notations of the reducer are shown in Figure 6 and

x1

x2

x3

x4x5

x6

x7

Figure 6 The multidisciplinary systems and notations of reducer

Speed reducer

x6 x7 Shaftx4 x5

Gearx1 x2 x3

Figure 7 The decomposition of speed reducer problem

the modelling is shown in Figure 7 The design objectiveis to minimize the speed reducer weight while satisfying anumber of constraints posed by gear and shaft parts Thereare seven variables in the speed reducer design optimizationTable 3 lists a brief description of the variables and gives theirreference values

According to the decomposition method of ATC wherea system is partitioned by object the speed reducer problemis decomposed into gear subproblem and shaft subproblem(shown in Figure 7) The design optimization problem isdefined as follows

min 119891 = 1198911(119909) + 119891

2(119909)

1198911(119909) = 07854119909

11199092

2

sdot (333331199092

3+ 149334119909

3minus 430934)

minus 150791199091(1199092

6+ 1199092

7)

8 Mathematical Problems in Engineering

The optimizer of system level

The optimizer of subproblem 1 The optimizer of subproblem 2

DOE-LHS DOE-LHS

Subproblem 1

analysis

Kriging model 1

Subproblem 2

analysis

Kriging model 2

g3 g4 g5 g6

xU8 xU7B x

U6B xL8 x

L7B x

L6B xU9 x

U6C x

U7C xL9 x

L6C x

L7C

X1 X1

X1

X1 G1

G1X2 X2

X2

X2 G2

G2

st

st st

R1 = x10 = r1(x3 x4 x5) = x8 + x9

X0 = x6 x7

G1 = g1 g2 g9 g10 g11R2 = x8 = r2(X1) = f1(x)

X1 = x1 x2 x3 x6 x7

G2 = g7 g8R3 = x9 = r3(X2) = f2(x)

X2 = x4 x5 x6 x7

PA min f11 = x10 + x8 minus xL8)

2+ (x9 minus xL9)

2+ (x6 minus xL6B)

2+ (x6 minus xL6C)

2+ (x7 minus xL7B)

2+ (x7 minus xL7C)

2]120590

2)]

PB min f21 = x8 minus xU8 )

2+ (x6 minus xU6B)

2+ (x7 minus xU7B)

2]120590

2)] PC min f22 = x9 minus x

U9 )

2+ (x6 minus xU6C)

2+ (x7 minus xU7C)

2]120590

2)]

Figure 8 The solving framework of the QEPF-ATC and Kriging model combined approach for the gear reducer

1198912(119909) = 7477 (119909

3

6+ 1199093

7) + 07854 (119909

41199092

6+ 11990951199092

7)

st 1198921=

27

(11990911199092

21199093)minus 1 le 0 119892

2=

3975

(11990911199092

21199092

3)minus 1 le 0

1198923=1931199093

4

(119909211990931199094

6)minus 1 le 0 119892

4=1931199093

5

(119909211990931199094

7)minus 1 le 0

1198925= 10119909

minus3

6radic(745119909

minus1

2119909minus1

31199094) + 169 times 107

minus 1100 le 0

1198926= 10119909

minus3

7radic(745119909

minus1

2119909minus1

31199095) + 1575 times 108

minus 850 le 0

1198927=(151199096+ 19)

1199094

minus 1 le 0

1198928=(151199097+ 19)

1199095

minus 1 le 0

1198929= 11990921199093minus 40 le 0 119892

10= 5 minus

1199091

1199092

le 0

11989211=1199091

1199092

minus 12 le 0

(20)

where 1198911is the gear weight and 119891

2is the shaft weight And 119892

1

and 1198922 are respectively gear bending stress and contact stress

constraints 1198923and 119892

4are respectively torsional distortion

constraints of shaft 1 and shaft 2 1198925and 119892

6are stress

constraints of the shaft 1 and shaft 2 1198927and 119892

8are experience

constraints of shaft 1 and shaft 2 and 1198929 11989210 and 119892

11are gear

geometry constraints1199091 1199092 and 119909

3are the design variables of the gear

subproblem 1199094and 119909

5are the design variables of the

shaft subproblem 1199096and 119909

7are linking variables (coupling

variables) According to the QEPF-ATC and Kriging modelcombined approach three variables 119909

8 1199099 and 119909

10are

introduced and meet the following formula

ℎ0= 1199098+ 1199099minus 11990910= 0

ℎ1= 1199098minus 1198911(119909) = 0

ℎ2= 1199099minus 1198912(119909) = 0

(21)

where ℎ0represents the analysis model of system level and

ℎ1and ℎ

2represent the analysis models of subproblem 1 and

subproblem 2 respectivelyThe solving framework of the QEPF-ATC and Kriging

model combined approach for gear reducer is shown inFigure 8

Creating the Kriging models of the shaft subsystem isintroduced in detail Firstly select 45 sample points byLHS method to create the Kriging models to conduct the

Mathematical Problems in Engineering 9

Table 4 The optimal results of the QEPF-ATC and Kriging model combined approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299453432 26 138 (3504 0698 17 7300 7845 3410 5283)2 299453217 25 124 (3501 0700 17 7310 7763 3353 5286)3 299521246 18 113 (3498 0699 17 7306 7755 3312 5279)4 299564235 10 36 (3499 0702 17 7297 7718 3350 5282)5 299433263 24 121 (3502 0700 17 7298 7720 3347 5287)

Table 5 The optimal results of the QEPF-ATC approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299463432 42 272 (3500 0699 17 7300 7825 3392 5282)2 299551327 38 215 (3500 0703 17 7305 7746 3353 5287)3 299612356 30 198 (3497 0696 17 7300 7713 3326 5275)4 299575363 17 97 (3499 0700 17 7299 7715 3349 5282)5 299521554 52 306 (3500 0700 17 7300 7719 3350 5286)

subsystem analysis and the created models are shown inFigure 9

Then select 20 sample points by random samplingmethod to evaluate the createdKrigingmodels Error analysisfor Kriging models of the shaft subsystem is shown inFigure 10 and it can be seen that the predicted valuesobtained by Kriging models are basically equal to the actualvalues

There are four credibility evaluation standards for approx-imate models that is the average the maximum the rootmean square and the 119877-squared For the average the max-imum and the root mean square the much smaller thevalues are the much better the credibility of approximatemodels is The default threshold of the average and theroot mean square is 02 and the maximum is 03 Forthe 119877-squared the much bigger the value is the muchbetter the credibility of approximate models is and thedefault threshold is 09 Figure 11 describes the values of thefour credibility evaluation standards for the created Krigingmodels of the shaft subsystem and it can be seen that the fourcredibility evaluation standards are all met So the createdKrigingmodels are reliable enough and can be used to replacethe original models with high accuracy in the next solvingprocess

Five initial points are selected and the sequentialquadratic programming (SQP) method is adopted to solvethe gear reducer problem and the results provided by theQEPF-ATC and Kriging model combined approach andthe QEPF-ATC approach are shown in Tables 4 and 5respectively It can be seen that results of the QEPF-ATCand Kriging model combined approach and the QEPF-ATCapproach are very close and the best result provided bythe QEPF-ATC and Kriging model combined approach iscloser to the optimal value of the gear reducer problemcompared to that provided by the QEPF-ATC approachIn terms of the iterations and runtime the QEPF-ATCand Kriging model combined approach saves computingtime by more than one half compared with QEPF-ATCapproach

5 Conclusion

Because ATC approach can flexibly organize the design opti-mization process according to the components of engineeringsystems it has been widely applied in MDOThe QEPF-ATCapproach is proposed to reduce the complicated coordinationprocess and heavy computation cost faced by exiting ATCapproaches where quadratic exterior penalty function isadopted to be the replaced coordination strategy Furtherin order to deal with the design optimization of complexengineering systems the QEPF-ATC and Kriging modelcombined approach is implemented where Kriging modelsare used to replace simulation models with high accuracyin the solving process of MDO problems The QEPF-ATCand Kriging model combined approach can avoid the impactof numerical noise brought by using simulation modelsduring the iterative solving process and reduce computationcost A geometric programming case is given to validatethe usability and effectiveness of the QEPF-ATC approachThe QEPF-ATC and Kriging model combined approach hasbeen successfully applied to solve the speed reducer designproblem

It should be pointed out that usingmetamodels within theQEPF-ATC approach is not without its own set of potentialdrawbacks which are well known and beyond the scope ofthis work Prior to employing the metamodel assisted QEPF-ATC approach the user should become familiar with thestrengths and weaknesses associated with surrogate approxi-mation Variable-fidelity modeling provides a promising wayto flexibly use approximate models in dealing with MDOproblems The combination of the QEPF-ATC approach andvariable-fidelity modeling will be studied to design complexengineering systems in the future

Abbreviations and Acronyms

ALF Augmented Lagrangian functionAM Approximate modelsATC Analytical target cascadingBLISS Bilevel integrated system synthesis

10 Mathematical Problems in Engineering

002minus002minus006minus01minus014minus018minus022minus026minus03

002minus002minus006minus01minus014minus018minus022minus026minus03

8

8 8

8

75

75 75

75

7 7

77

g7 g7

02

01501005

0minus005minus01minus015minus02

02

015010050minus005minus01minus015minus02

8

8 8

8

75

75 75

75

7 7

77

g8g8

x 4

x 4

x5

x5

x 4

x 4

x5

x5

Figure 9 Kriging models of the shaft subsystem

minus006

minus016

minus026minus026 minus016 minus006

Actu

al

Actu

al

Predicted Predicted

005

minus005

minus015minus015 minus005 005

g8g7

Figure 10 Error analysis for Kriging models of the shaft subsystem

Figure 11 The error data of Kriging models of subsystem 2

Mathematical Problems in Engineering 11

CO Collaborative optimizationCSSO Concurrent subspace optimizationDQA Diagonal quadratic approximationLDF Lagrangian dual functionLHS Latin hypercube samplingMDO Multidisciplinary design

optimizationQEPF Quadratic exterior penalty functionQEPF-ATC Quadratic exterior penalty function

based analytical target cascadingSQP Sequential quadratic programming

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by theNational Natural Sci-ence Foundation of China (NSFC) under Grant no 51121002the National Basic Research Program (973 Program) ofChina under Grant no 2014CB046703 and the FundamentalResearch Funds for the Central Universities HUST Grantno 2014TS040 The authors also would like to thank theanonymous referees for their valuable comments

References

[1] J Sobieszczanski-Sobieski ldquoA linear decomposition method forlarge optimization problemsrdquo NASA Technical Memorandum83248 1982

[2] TW Simpson and J R R AMartins ldquoMultidisciplinary designoptimization for complex engineered systems report from anational science foundation workshoprdquo Journal of MechanicalDesign vol 133 no 10 Article ID 101002 2011

[3] M Balesdent N Berend P Depince and A Chriette ldquoA surveyof multidisciplinary design optimization methods in launchvehicle designrdquo Structural and Multidisciplinary Optimizationvol 45 no 5 pp 619ndash642 2012

[4] I Sobieski and I Kroo ldquoAircraft design using collaborativeoptimizationrdquo AIAA Paper 715 1996

[5] H-Z Huang H Yu X Zhang S Zeng and ZWang ldquoCollabo-rative optimization with inverse reliability for multidisciplinarysystems uncertainty analysisrdquoEngineeringOptimization vol 42no 8 pp 763ndash773 2010

[6] R S Sellar S M Batill and J E Renaud ldquoResponse surfacebased concurrent subspace optimization for multidisciplinarysystem designrdquo AIAA Paper 714 AIAA 1996

[7] L S Li J H Liu and S H Liu ldquoAn efficient strategy formultidisciplinary reliability design and optimization based onCSSO and PMA in SORA frameworkrdquo Structural and Multidis-ciplinary Optimization vol 49 no 2 pp 239ndash252 2014

[8] J Sobieszczanski-Sobieski ldquoBi-level integrated system synthesis(BLISS)rdquo AIAA Paper 4916 AIAA 1998

[9] J Sobieszczanski-Sobieski T D Altus M Phillips and RSandusky ldquoBi-level integrated system synthesis (BLISS) forconcurrent and distributed processingrdquo AIAA Paper 54092002

[10] N Michelena H Park and P Y Papalambros ldquoConvergenceproperties of analytical target cascadingrdquo AIAA Journal vol 41no 5 pp 897ndash905 2003

[11] M Gardenghi M M Wiecek and W Wang ldquoBiobjectiveoptimization for analytical target cascading optimality vsachievabilityrdquo Structural and Multidisciplinary Optimizationvol 47 no 1 pp 111ndash133 2013

[12] S Tosserams L F Etman P Y Papalambros and J E RoodaldquoAn augmented Lagrangian relaxation for analytical target cas-cading using the alternating direction method of multipliersrdquoStructural and Multidisciplinary Optimization vol 31 no 3 pp176ndash189 2006

[13] Y Li Z Lu and J J Michalek ldquoDiagonal quadratic approxima-tion for parallelization of analytical target cascadingrdquo Journalof Mechanical Design Transactions of the ASME vol 130 no 5Article ID 051402 2008

[14] H M Kim W Chen and M M Wiecek ldquoLagrangian coor-dination for enhancing the convergence of analytical targetcascadingrdquo AIAA Journal vol 44 no 10 pp 2197ndash2207 2006

[15] D W Kim and J Lee ldquoAn improvement of Kriging basedsequential approximate optimization method via extended useof design of experimentsrdquo Engineering Optimization vol 42 no12 pp 1133ndash1149 2010

[16] P B Backlund D W Shahan and C C Seepersad ldquoA com-parative study of the scalability of alternative metamodellingtechniquesrdquo Engineering Optimization vol 44 no 7 pp 767ndash786 2012

[17] J Zhang S Chowdhury and A Messac ldquoAn adaptive hybridsurrogate modelrdquo Structural and Multidisciplinary Optimiza-tion vol 46 no 2 pp 223ndash238 2012

[18] V Dubourg B Sudret and J-M Bourinet ldquoReliability-baseddesign optimization using kriging surrogates and subset simu-lationrdquo Structural and Multidisciplinary Optimization vol 44no 5 pp 673ndash690 2011

[19] H M Kim N F Michelena P Y Papalambros and T JiangldquoTarget cascading in optimal system designrdquo Transactions of theASMEmdashJournal of Mechanical Design vol 125 no 3 pp 474ndash480 2003

[20] J P Costa L Pronzato and E Thierry ldquoA comparison betweenKriging and radial basis function networks for nonlinearpredictionrdquo in Proceedings of the IEEE-EURASIP Workshop onNonlinear Signal and Image Processing (NSIP rsquo99) pp 726ndash7301999

[21] V Cherkassky D Gehring and F Mulier ldquoComparison ofadaptive methods for function estimation from samplesrdquo IEEETransactions on Neural Networks vol 7 no 4 pp 969ndash9841996

[22] M S Kim H Cho S G Lee J Choi and D Bae ldquoDFSS androbust optimization tool for multibody system with randomvariablesrdquo Journal of System Design and Dynamics vol 1 no 3pp 583ndash592 2007

[23] H M Kim Target cascading in optimal system design [PhDdissertation] University of Michigan 2001

[24] S L Padula N Alexandrov and L L Green ldquoMDO testsuite at NASA Langley Research Centerrdquo in Proceedings of the6th Symposium onMultidisciplinary Analysis and OptimizationAIAA Paper 96-4028 pp 410ndash420 Bellevue Wash USASeptember 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

g1 g2

x1 x2 x3 x4 x5 x6 x7 x11 ge 0

xU3

xU11B

xL3

xL11B

g3 g4

x3 x8 x9 x10 x11 ge 0

x11

xU6

xU11C

xL6

xL11C

g5 g6

x6 x11 x12 x13 x14 ge 0

st

st st

R1 = x1 = r1(x3 x4 x5) = (x23 + xminus24 + x25)12

R2 = x2 = r2(x5 x6 x7) = (x25 + x26 + x27)12

R3 = x3 = r3(x8 x9 x10 x11)= (x28 + xminus29 + xminus210 + x

211)

12

PC minx6x11 x12 x13 x14

f22 =120590

2x6 minus x

U6 )) 2

+ (x11 minus xU11C)2]]

PA minx1x2x3x4x5x6x7x11

f11 = x21 + x22 +

120590

2x3 minus x

L3)

2+ (x6 minus xL6)

2+ (x11 minus xL11B)

2+ (x11 minus xL11C)

2])]

R4 = x6 = r4(x11 x12 x13 x14)= (x211 + x212 + x213 + x214)

12

xPB min

x3x8x9x10f21 =

120590

2x3 minus x

U3 )

2+ (x11 minus xU11B)

2])]11

Figure 5 The solving framework of QEPF-ATC for geometric programming

(7) pass back current results to system level and imple-ment the system level optimization

(8) obtain results if the convergence condition is metotherwise go to step (6) until the convergence con-dition is met

4 Case Studies

41 The Geometric Programming The geometric program-ming [23] that has only the global optimal solution isemployed to verify the usability of QEPF-ATC Its mathemat-ical expression is shown as

min 119891 = 1199092

1+ 1199092

2

st 1198921119909minus2

3+ 1199092

4

1199092

5

le 1 11989221199092

5+ 119909minus2

6

1199092

7

le 1

11989231199092

8+ 1199092

9

1199092

11

le 1

1198924119909minus2

8+ 1199092

10

1199092

11

le 1 11989251199092

11+ 119909minus2

12

1199092

13

le 1

11989261199092

11+ 1199092

12

1199092

14

le 1

ℎ1= 1199092

1minus 1199092

3minus 119909minus2

4minus 1199092

5= 0

ℎ2= 1199092

2minus 1199092

5minus 1199092

6minus 1199092

7= 0

ℎ3= 1199092

3minus 1199092

8minus 119909minus2

9minus 119909minus2

10minus 1199092

11= 0

ℎ4= 1199092

6minus 1199092

11minus 1199092

12minus 1199092

13minus 1199092

14= 0

(19)

Table 1 The optimal results

QEPF-ATC ALF DQA LDF AAO1199091

29049 285 286 287 2841199092

30551 304 310 305 3091199093

24930 248 249 243 2361199094

08805 086 082 079 0761199095

09676 094 095 089 0871199096

27103 272 275 278 2811199097

10357 102 095 103 0941199098

09830 098 098 101 0971199099

07572 078 079 082 08711990910

07104 076 079 081 08011990911

12408 127 126 125 13011990912

08409 086 085 083 08411990913

17088 173 174 171 17611990914

14875 152 149 150 155119891 177721 1736 1778 1753 1759Runtime (s) 289 355 336 368

Figure 5 shows the solution framework of the QEPF-ATCapproach for the problem where 119909

11is the linking variable

(coupling variable) coordinated by the system level 11990911988011119861

119909119880

11119862 1199091198803 and 119909119880

6are the targets of each subproblem passed

from the system level 11990911987111119861

and 11990911987111119862

are the responses of thelinking variable coming from the subproblems 119861 and 119862 119909119871

3

and 1199091198716are the responses of 119861 and 119862

Solve the geometric programming problem according toQEPF-ATC Initialize the design point 1199090 = (1 1 1 1 1 1 11 1 1 1 1 1 1)

119879 and take 120582 = 10 Here the acceptableinconsistency tolerance is 001 for every response variable andlinking variable Table 1 shows the optimal results based on

Mathematical Problems in Engineering 7

Table 2 The optimal results based on 120582

120582 (1199091 1199092 1199093 1199094 1199095 1199096 1199097 11990911) (119909

3 1199098 1199099 11990910 11990911) (119909

6 11990911 11990912 11990913 11990914) 119891 Iterations

14 (27556 30804 22975 0813109171 27718 09840 12678) (23012 09688 08427 07340 12682) (27638 12732 07447 18411 14655) 175992 78

4 (29716 29567 25227 0759808570 26708 09352 12343) (25230 09911 07320 07072 12321) (26791 12298 08409 16923 14686) 175728 27

7 (29662 29757 25012 0720408231 27169 09014 12386) (25011 09863 07492 07115 12386) (27168 12385 08408 17122 14915) 176531 17

10 (29049 30551 24930 0880509676 27103 10357 12408) (24930 09830 07572 07104 12408) (27102 12409 08409 17088 14875) 177721 5

Table 3 Design variables for speed reducer

Variables Range Units1199091 Gear face width 26 le 119909

1le 36 cm

1199092 Teeth module 07 le 119909

2le 08 cm

1199093

Number of teeth of pinion(integer variable) 17 le 119909

3le 28

1199094

Distance between bearings1 73 le 119909

4le 83 cm

1199095

Distance between bearings2 73 le 119909

5le 83 cm

1199096 Diameter of shaft 1 29 le 119909

6le 39 cm

1199097 Diameter of shaft 1 50 le 119909

7le 55 cm

1199091 1199092 and 119909

3are the design variables of the gear subproblem 119909

4and 1199095are

the design variables of the shaft subproblem 1199096and 1199097are linking variables

(coupling variables)

QEPF-ATC three coordination strategies (ALF DQA andLDF) and AAO (All-At-Once)

According to Table 1 the results of the QEPF-ATCapproach are very close to the optimal value of the originalproblem and the runtime is less than the other threemethods120582 affects the convergence efficiency and computational

efficiency of the QEPF-ATC approach In order to obtainmore suitable 120582 to improve the efficiency of the QEPF-ATC approach different values of 120582 are selected to solve theproblem Table 2 shows the results when different values of 120582are selected

In Table 2 the QEPF-ATC approach implement fastconvergence when 120582 is set to different values and the optimalresults are close to the results of the initial problems Withthe increase of 120582 the number of iterations decreases andthe solution error gets bigger (excessive weight) Generally14 le 120582 le 10 is strictly necessary but typically 4 lt

120582 lt 7 is recommended to speed up convergence and reducecomputational cost and get better optimal results

42 The Speed Reducer Design The speed reducer designcase represents the design of a simple gearbox and is amultidisciplinary problem comprising the coupling betweengear design and shaft design This has been used as a testingproblem for MDO method in the literature from NASALangley research center [24] The multidisciplinary systemsand notations of the reducer are shown in Figure 6 and

x1

x2

x3

x4x5

x6

x7

Figure 6 The multidisciplinary systems and notations of reducer

Speed reducer

x6 x7 Shaftx4 x5

Gearx1 x2 x3

Figure 7 The decomposition of speed reducer problem

the modelling is shown in Figure 7 The design objectiveis to minimize the speed reducer weight while satisfying anumber of constraints posed by gear and shaft parts Thereare seven variables in the speed reducer design optimizationTable 3 lists a brief description of the variables and gives theirreference values

According to the decomposition method of ATC wherea system is partitioned by object the speed reducer problemis decomposed into gear subproblem and shaft subproblem(shown in Figure 7) The design optimization problem isdefined as follows

min 119891 = 1198911(119909) + 119891

2(119909)

1198911(119909) = 07854119909

11199092

2

sdot (333331199092

3+ 149334119909

3minus 430934)

minus 150791199091(1199092

6+ 1199092

7)

8 Mathematical Problems in Engineering

The optimizer of system level

The optimizer of subproblem 1 The optimizer of subproblem 2

DOE-LHS DOE-LHS

Subproblem 1

analysis

Kriging model 1

Subproblem 2

analysis

Kriging model 2

g3 g4 g5 g6

xU8 xU7B x

U6B xL8 x

L7B x

L6B xU9 x

U6C x

U7C xL9 x

L6C x

L7C

X1 X1

X1

X1 G1

G1X2 X2

X2

X2 G2

G2

st

st st

R1 = x10 = r1(x3 x4 x5) = x8 + x9

X0 = x6 x7

G1 = g1 g2 g9 g10 g11R2 = x8 = r2(X1) = f1(x)

X1 = x1 x2 x3 x6 x7

G2 = g7 g8R3 = x9 = r3(X2) = f2(x)

X2 = x4 x5 x6 x7

PA min f11 = x10 + x8 minus xL8)

2+ (x9 minus xL9)

2+ (x6 minus xL6B)

2+ (x6 minus xL6C)

2+ (x7 minus xL7B)

2+ (x7 minus xL7C)

2]120590

2)]

PB min f21 = x8 minus xU8 )

2+ (x6 minus xU6B)

2+ (x7 minus xU7B)

2]120590

2)] PC min f22 = x9 minus x

U9 )

2+ (x6 minus xU6C)

2+ (x7 minus xU7C)

2]120590

2)]

Figure 8 The solving framework of the QEPF-ATC and Kriging model combined approach for the gear reducer

1198912(119909) = 7477 (119909

3

6+ 1199093

7) + 07854 (119909

41199092

6+ 11990951199092

7)

st 1198921=

27

(11990911199092

21199093)minus 1 le 0 119892

2=

3975

(11990911199092

21199092

3)minus 1 le 0

1198923=1931199093

4

(119909211990931199094

6)minus 1 le 0 119892

4=1931199093

5

(119909211990931199094

7)minus 1 le 0

1198925= 10119909

minus3

6radic(745119909

minus1

2119909minus1

31199094) + 169 times 107

minus 1100 le 0

1198926= 10119909

minus3

7radic(745119909

minus1

2119909minus1

31199095) + 1575 times 108

minus 850 le 0

1198927=(151199096+ 19)

1199094

minus 1 le 0

1198928=(151199097+ 19)

1199095

minus 1 le 0

1198929= 11990921199093minus 40 le 0 119892

10= 5 minus

1199091

1199092

le 0

11989211=1199091

1199092

minus 12 le 0

(20)

where 1198911is the gear weight and 119891

2is the shaft weight And 119892

1

and 1198922 are respectively gear bending stress and contact stress

constraints 1198923and 119892

4are respectively torsional distortion

constraints of shaft 1 and shaft 2 1198925and 119892

6are stress

constraints of the shaft 1 and shaft 2 1198927and 119892

8are experience

constraints of shaft 1 and shaft 2 and 1198929 11989210 and 119892

11are gear

geometry constraints1199091 1199092 and 119909

3are the design variables of the gear

subproblem 1199094and 119909

5are the design variables of the

shaft subproblem 1199096and 119909

7are linking variables (coupling

variables) According to the QEPF-ATC and Kriging modelcombined approach three variables 119909

8 1199099 and 119909

10are

introduced and meet the following formula

ℎ0= 1199098+ 1199099minus 11990910= 0

ℎ1= 1199098minus 1198911(119909) = 0

ℎ2= 1199099minus 1198912(119909) = 0

(21)

where ℎ0represents the analysis model of system level and

ℎ1and ℎ

2represent the analysis models of subproblem 1 and

subproblem 2 respectivelyThe solving framework of the QEPF-ATC and Kriging

model combined approach for gear reducer is shown inFigure 8

Creating the Kriging models of the shaft subsystem isintroduced in detail Firstly select 45 sample points byLHS method to create the Kriging models to conduct the

Mathematical Problems in Engineering 9

Table 4 The optimal results of the QEPF-ATC and Kriging model combined approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299453432 26 138 (3504 0698 17 7300 7845 3410 5283)2 299453217 25 124 (3501 0700 17 7310 7763 3353 5286)3 299521246 18 113 (3498 0699 17 7306 7755 3312 5279)4 299564235 10 36 (3499 0702 17 7297 7718 3350 5282)5 299433263 24 121 (3502 0700 17 7298 7720 3347 5287)

Table 5 The optimal results of the QEPF-ATC approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299463432 42 272 (3500 0699 17 7300 7825 3392 5282)2 299551327 38 215 (3500 0703 17 7305 7746 3353 5287)3 299612356 30 198 (3497 0696 17 7300 7713 3326 5275)4 299575363 17 97 (3499 0700 17 7299 7715 3349 5282)5 299521554 52 306 (3500 0700 17 7300 7719 3350 5286)

subsystem analysis and the created models are shown inFigure 9

Then select 20 sample points by random samplingmethod to evaluate the createdKrigingmodels Error analysisfor Kriging models of the shaft subsystem is shown inFigure 10 and it can be seen that the predicted valuesobtained by Kriging models are basically equal to the actualvalues

There are four credibility evaluation standards for approx-imate models that is the average the maximum the rootmean square and the 119877-squared For the average the max-imum and the root mean square the much smaller thevalues are the much better the credibility of approximatemodels is The default threshold of the average and theroot mean square is 02 and the maximum is 03 Forthe 119877-squared the much bigger the value is the muchbetter the credibility of approximate models is and thedefault threshold is 09 Figure 11 describes the values of thefour credibility evaluation standards for the created Krigingmodels of the shaft subsystem and it can be seen that the fourcredibility evaluation standards are all met So the createdKrigingmodels are reliable enough and can be used to replacethe original models with high accuracy in the next solvingprocess

Five initial points are selected and the sequentialquadratic programming (SQP) method is adopted to solvethe gear reducer problem and the results provided by theQEPF-ATC and Kriging model combined approach andthe QEPF-ATC approach are shown in Tables 4 and 5respectively It can be seen that results of the QEPF-ATCand Kriging model combined approach and the QEPF-ATCapproach are very close and the best result provided bythe QEPF-ATC and Kriging model combined approach iscloser to the optimal value of the gear reducer problemcompared to that provided by the QEPF-ATC approachIn terms of the iterations and runtime the QEPF-ATCand Kriging model combined approach saves computingtime by more than one half compared with QEPF-ATCapproach

5 Conclusion

Because ATC approach can flexibly organize the design opti-mization process according to the components of engineeringsystems it has been widely applied in MDOThe QEPF-ATCapproach is proposed to reduce the complicated coordinationprocess and heavy computation cost faced by exiting ATCapproaches where quadratic exterior penalty function isadopted to be the replaced coordination strategy Furtherin order to deal with the design optimization of complexengineering systems the QEPF-ATC and Kriging modelcombined approach is implemented where Kriging modelsare used to replace simulation models with high accuracyin the solving process of MDO problems The QEPF-ATCand Kriging model combined approach can avoid the impactof numerical noise brought by using simulation modelsduring the iterative solving process and reduce computationcost A geometric programming case is given to validatethe usability and effectiveness of the QEPF-ATC approachThe QEPF-ATC and Kriging model combined approach hasbeen successfully applied to solve the speed reducer designproblem

It should be pointed out that usingmetamodels within theQEPF-ATC approach is not without its own set of potentialdrawbacks which are well known and beyond the scope ofthis work Prior to employing the metamodel assisted QEPF-ATC approach the user should become familiar with thestrengths and weaknesses associated with surrogate approxi-mation Variable-fidelity modeling provides a promising wayto flexibly use approximate models in dealing with MDOproblems The combination of the QEPF-ATC approach andvariable-fidelity modeling will be studied to design complexengineering systems in the future

Abbreviations and Acronyms

ALF Augmented Lagrangian functionAM Approximate modelsATC Analytical target cascadingBLISS Bilevel integrated system synthesis

10 Mathematical Problems in Engineering

002minus002minus006minus01minus014minus018minus022minus026minus03

002minus002minus006minus01minus014minus018minus022minus026minus03

8

8 8

8

75

75 75

75

7 7

77

g7 g7

02

01501005

0minus005minus01minus015minus02

02

015010050minus005minus01minus015minus02

8

8 8

8

75

75 75

75

7 7

77

g8g8

x 4

x 4

x5

x5

x 4

x 4

x5

x5

Figure 9 Kriging models of the shaft subsystem

minus006

minus016

minus026minus026 minus016 minus006

Actu

al

Actu

al

Predicted Predicted

005

minus005

minus015minus015 minus005 005

g8g7

Figure 10 Error analysis for Kriging models of the shaft subsystem

Figure 11 The error data of Kriging models of subsystem 2

Mathematical Problems in Engineering 11

CO Collaborative optimizationCSSO Concurrent subspace optimizationDQA Diagonal quadratic approximationLDF Lagrangian dual functionLHS Latin hypercube samplingMDO Multidisciplinary design

optimizationQEPF Quadratic exterior penalty functionQEPF-ATC Quadratic exterior penalty function

based analytical target cascadingSQP Sequential quadratic programming

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by theNational Natural Sci-ence Foundation of China (NSFC) under Grant no 51121002the National Basic Research Program (973 Program) ofChina under Grant no 2014CB046703 and the FundamentalResearch Funds for the Central Universities HUST Grantno 2014TS040 The authors also would like to thank theanonymous referees for their valuable comments

References

[1] J Sobieszczanski-Sobieski ldquoA linear decomposition method forlarge optimization problemsrdquo NASA Technical Memorandum83248 1982

[2] TW Simpson and J R R AMartins ldquoMultidisciplinary designoptimization for complex engineered systems report from anational science foundation workshoprdquo Journal of MechanicalDesign vol 133 no 10 Article ID 101002 2011

[3] M Balesdent N Berend P Depince and A Chriette ldquoA surveyof multidisciplinary design optimization methods in launchvehicle designrdquo Structural and Multidisciplinary Optimizationvol 45 no 5 pp 619ndash642 2012

[4] I Sobieski and I Kroo ldquoAircraft design using collaborativeoptimizationrdquo AIAA Paper 715 1996

[5] H-Z Huang H Yu X Zhang S Zeng and ZWang ldquoCollabo-rative optimization with inverse reliability for multidisciplinarysystems uncertainty analysisrdquoEngineeringOptimization vol 42no 8 pp 763ndash773 2010

[6] R S Sellar S M Batill and J E Renaud ldquoResponse surfacebased concurrent subspace optimization for multidisciplinarysystem designrdquo AIAA Paper 714 AIAA 1996

[7] L S Li J H Liu and S H Liu ldquoAn efficient strategy formultidisciplinary reliability design and optimization based onCSSO and PMA in SORA frameworkrdquo Structural and Multidis-ciplinary Optimization vol 49 no 2 pp 239ndash252 2014

[8] J Sobieszczanski-Sobieski ldquoBi-level integrated system synthesis(BLISS)rdquo AIAA Paper 4916 AIAA 1998

[9] J Sobieszczanski-Sobieski T D Altus M Phillips and RSandusky ldquoBi-level integrated system synthesis (BLISS) forconcurrent and distributed processingrdquo AIAA Paper 54092002

[10] N Michelena H Park and P Y Papalambros ldquoConvergenceproperties of analytical target cascadingrdquo AIAA Journal vol 41no 5 pp 897ndash905 2003

[11] M Gardenghi M M Wiecek and W Wang ldquoBiobjectiveoptimization for analytical target cascading optimality vsachievabilityrdquo Structural and Multidisciplinary Optimizationvol 47 no 1 pp 111ndash133 2013

[12] S Tosserams L F Etman P Y Papalambros and J E RoodaldquoAn augmented Lagrangian relaxation for analytical target cas-cading using the alternating direction method of multipliersrdquoStructural and Multidisciplinary Optimization vol 31 no 3 pp176ndash189 2006

[13] Y Li Z Lu and J J Michalek ldquoDiagonal quadratic approxima-tion for parallelization of analytical target cascadingrdquo Journalof Mechanical Design Transactions of the ASME vol 130 no 5Article ID 051402 2008

[14] H M Kim W Chen and M M Wiecek ldquoLagrangian coor-dination for enhancing the convergence of analytical targetcascadingrdquo AIAA Journal vol 44 no 10 pp 2197ndash2207 2006

[15] D W Kim and J Lee ldquoAn improvement of Kriging basedsequential approximate optimization method via extended useof design of experimentsrdquo Engineering Optimization vol 42 no12 pp 1133ndash1149 2010

[16] P B Backlund D W Shahan and C C Seepersad ldquoA com-parative study of the scalability of alternative metamodellingtechniquesrdquo Engineering Optimization vol 44 no 7 pp 767ndash786 2012

[17] J Zhang S Chowdhury and A Messac ldquoAn adaptive hybridsurrogate modelrdquo Structural and Multidisciplinary Optimiza-tion vol 46 no 2 pp 223ndash238 2012

[18] V Dubourg B Sudret and J-M Bourinet ldquoReliability-baseddesign optimization using kriging surrogates and subset simu-lationrdquo Structural and Multidisciplinary Optimization vol 44no 5 pp 673ndash690 2011

[19] H M Kim N F Michelena P Y Papalambros and T JiangldquoTarget cascading in optimal system designrdquo Transactions of theASMEmdashJournal of Mechanical Design vol 125 no 3 pp 474ndash480 2003

[20] J P Costa L Pronzato and E Thierry ldquoA comparison betweenKriging and radial basis function networks for nonlinearpredictionrdquo in Proceedings of the IEEE-EURASIP Workshop onNonlinear Signal and Image Processing (NSIP rsquo99) pp 726ndash7301999

[21] V Cherkassky D Gehring and F Mulier ldquoComparison ofadaptive methods for function estimation from samplesrdquo IEEETransactions on Neural Networks vol 7 no 4 pp 969ndash9841996

[22] M S Kim H Cho S G Lee J Choi and D Bae ldquoDFSS androbust optimization tool for multibody system with randomvariablesrdquo Journal of System Design and Dynamics vol 1 no 3pp 583ndash592 2007

[23] H M Kim Target cascading in optimal system design [PhDdissertation] University of Michigan 2001

[24] S L Padula N Alexandrov and L L Green ldquoMDO testsuite at NASA Langley Research Centerrdquo in Proceedings of the6th Symposium onMultidisciplinary Analysis and OptimizationAIAA Paper 96-4028 pp 410ndash420 Bellevue Wash USASeptember 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 2 The optimal results based on 120582

120582 (1199091 1199092 1199093 1199094 1199095 1199096 1199097 11990911) (119909

3 1199098 1199099 11990910 11990911) (119909

6 11990911 11990912 11990913 11990914) 119891 Iterations

14 (27556 30804 22975 0813109171 27718 09840 12678) (23012 09688 08427 07340 12682) (27638 12732 07447 18411 14655) 175992 78

4 (29716 29567 25227 0759808570 26708 09352 12343) (25230 09911 07320 07072 12321) (26791 12298 08409 16923 14686) 175728 27

7 (29662 29757 25012 0720408231 27169 09014 12386) (25011 09863 07492 07115 12386) (27168 12385 08408 17122 14915) 176531 17

10 (29049 30551 24930 0880509676 27103 10357 12408) (24930 09830 07572 07104 12408) (27102 12409 08409 17088 14875) 177721 5

Table 3 Design variables for speed reducer

Variables Range Units1199091 Gear face width 26 le 119909

1le 36 cm

1199092 Teeth module 07 le 119909

2le 08 cm

1199093

Number of teeth of pinion(integer variable) 17 le 119909

3le 28

1199094

Distance between bearings1 73 le 119909

4le 83 cm

1199095

Distance between bearings2 73 le 119909

5le 83 cm

1199096 Diameter of shaft 1 29 le 119909

6le 39 cm

1199097 Diameter of shaft 1 50 le 119909

7le 55 cm

1199091 1199092 and 119909

3are the design variables of the gear subproblem 119909

4and 1199095are

the design variables of the shaft subproblem 1199096and 1199097are linking variables

(coupling variables)

QEPF-ATC three coordination strategies (ALF DQA andLDF) and AAO (All-At-Once)

According to Table 1 the results of the QEPF-ATCapproach are very close to the optimal value of the originalproblem and the runtime is less than the other threemethods120582 affects the convergence efficiency and computational

efficiency of the QEPF-ATC approach In order to obtainmore suitable 120582 to improve the efficiency of the QEPF-ATC approach different values of 120582 are selected to solve theproblem Table 2 shows the results when different values of 120582are selected

In Table 2 the QEPF-ATC approach implement fastconvergence when 120582 is set to different values and the optimalresults are close to the results of the initial problems Withthe increase of 120582 the number of iterations decreases andthe solution error gets bigger (excessive weight) Generally14 le 120582 le 10 is strictly necessary but typically 4 lt

120582 lt 7 is recommended to speed up convergence and reducecomputational cost and get better optimal results

42 The Speed Reducer Design The speed reducer designcase represents the design of a simple gearbox and is amultidisciplinary problem comprising the coupling betweengear design and shaft design This has been used as a testingproblem for MDO method in the literature from NASALangley research center [24] The multidisciplinary systemsand notations of the reducer are shown in Figure 6 and

x1

x2

x3

x4x5

x6

x7

Figure 6 The multidisciplinary systems and notations of reducer

Speed reducer

x6 x7 Shaftx4 x5

Gearx1 x2 x3

Figure 7 The decomposition of speed reducer problem

the modelling is shown in Figure 7 The design objectiveis to minimize the speed reducer weight while satisfying anumber of constraints posed by gear and shaft parts Thereare seven variables in the speed reducer design optimizationTable 3 lists a brief description of the variables and gives theirreference values

According to the decomposition method of ATC wherea system is partitioned by object the speed reducer problemis decomposed into gear subproblem and shaft subproblem(shown in Figure 7) The design optimization problem isdefined as follows

min 119891 = 1198911(119909) + 119891

2(119909)

1198911(119909) = 07854119909

11199092

2

sdot (333331199092

3+ 149334119909

3minus 430934)

minus 150791199091(1199092

6+ 1199092

7)

8 Mathematical Problems in Engineering

The optimizer of system level

The optimizer of subproblem 1 The optimizer of subproblem 2

DOE-LHS DOE-LHS

Subproblem 1

analysis

Kriging model 1

Subproblem 2

analysis

Kriging model 2

g3 g4 g5 g6

xU8 xU7B x

U6B xL8 x

L7B x

L6B xU9 x

U6C x

U7C xL9 x

L6C x

L7C

X1 X1

X1

X1 G1

G1X2 X2

X2

X2 G2

G2

st

st st

R1 = x10 = r1(x3 x4 x5) = x8 + x9

X0 = x6 x7

G1 = g1 g2 g9 g10 g11R2 = x8 = r2(X1) = f1(x)

X1 = x1 x2 x3 x6 x7

G2 = g7 g8R3 = x9 = r3(X2) = f2(x)

X2 = x4 x5 x6 x7

PA min f11 = x10 + x8 minus xL8)

2+ (x9 minus xL9)

2+ (x6 minus xL6B)

2+ (x6 minus xL6C)

2+ (x7 minus xL7B)

2+ (x7 minus xL7C)

2]120590

2)]

PB min f21 = x8 minus xU8 )

2+ (x6 minus xU6B)

2+ (x7 minus xU7B)

2]120590

2)] PC min f22 = x9 minus x

U9 )

2+ (x6 minus xU6C)

2+ (x7 minus xU7C)

2]120590

2)]

Figure 8 The solving framework of the QEPF-ATC and Kriging model combined approach for the gear reducer

1198912(119909) = 7477 (119909

3

6+ 1199093

7) + 07854 (119909

41199092

6+ 11990951199092

7)

st 1198921=

27

(11990911199092

21199093)minus 1 le 0 119892

2=

3975

(11990911199092

21199092

3)minus 1 le 0

1198923=1931199093

4

(119909211990931199094

6)minus 1 le 0 119892

4=1931199093

5

(119909211990931199094

7)minus 1 le 0

1198925= 10119909

minus3

6radic(745119909

minus1

2119909minus1

31199094) + 169 times 107

minus 1100 le 0

1198926= 10119909

minus3

7radic(745119909

minus1

2119909minus1

31199095) + 1575 times 108

minus 850 le 0

1198927=(151199096+ 19)

1199094

minus 1 le 0

1198928=(151199097+ 19)

1199095

minus 1 le 0

1198929= 11990921199093minus 40 le 0 119892

10= 5 minus

1199091

1199092

le 0

11989211=1199091

1199092

minus 12 le 0

(20)

where 1198911is the gear weight and 119891

2is the shaft weight And 119892

1

and 1198922 are respectively gear bending stress and contact stress

constraints 1198923and 119892

4are respectively torsional distortion

constraints of shaft 1 and shaft 2 1198925and 119892

6are stress

constraints of the shaft 1 and shaft 2 1198927and 119892

8are experience

constraints of shaft 1 and shaft 2 and 1198929 11989210 and 119892

11are gear

geometry constraints1199091 1199092 and 119909

3are the design variables of the gear

subproblem 1199094and 119909

5are the design variables of the

shaft subproblem 1199096and 119909

7are linking variables (coupling

variables) According to the QEPF-ATC and Kriging modelcombined approach three variables 119909

8 1199099 and 119909

10are

introduced and meet the following formula

ℎ0= 1199098+ 1199099minus 11990910= 0

ℎ1= 1199098minus 1198911(119909) = 0

ℎ2= 1199099minus 1198912(119909) = 0

(21)

where ℎ0represents the analysis model of system level and

ℎ1and ℎ

2represent the analysis models of subproblem 1 and

subproblem 2 respectivelyThe solving framework of the QEPF-ATC and Kriging

model combined approach for gear reducer is shown inFigure 8

Creating the Kriging models of the shaft subsystem isintroduced in detail Firstly select 45 sample points byLHS method to create the Kriging models to conduct the

Mathematical Problems in Engineering 9

Table 4 The optimal results of the QEPF-ATC and Kriging model combined approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299453432 26 138 (3504 0698 17 7300 7845 3410 5283)2 299453217 25 124 (3501 0700 17 7310 7763 3353 5286)3 299521246 18 113 (3498 0699 17 7306 7755 3312 5279)4 299564235 10 36 (3499 0702 17 7297 7718 3350 5282)5 299433263 24 121 (3502 0700 17 7298 7720 3347 5287)

Table 5 The optimal results of the QEPF-ATC approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299463432 42 272 (3500 0699 17 7300 7825 3392 5282)2 299551327 38 215 (3500 0703 17 7305 7746 3353 5287)3 299612356 30 198 (3497 0696 17 7300 7713 3326 5275)4 299575363 17 97 (3499 0700 17 7299 7715 3349 5282)5 299521554 52 306 (3500 0700 17 7300 7719 3350 5286)

subsystem analysis and the created models are shown inFigure 9

Then select 20 sample points by random samplingmethod to evaluate the createdKrigingmodels Error analysisfor Kriging models of the shaft subsystem is shown inFigure 10 and it can be seen that the predicted valuesobtained by Kriging models are basically equal to the actualvalues

There are four credibility evaluation standards for approx-imate models that is the average the maximum the rootmean square and the 119877-squared For the average the max-imum and the root mean square the much smaller thevalues are the much better the credibility of approximatemodels is The default threshold of the average and theroot mean square is 02 and the maximum is 03 Forthe 119877-squared the much bigger the value is the muchbetter the credibility of approximate models is and thedefault threshold is 09 Figure 11 describes the values of thefour credibility evaluation standards for the created Krigingmodels of the shaft subsystem and it can be seen that the fourcredibility evaluation standards are all met So the createdKrigingmodels are reliable enough and can be used to replacethe original models with high accuracy in the next solvingprocess

Five initial points are selected and the sequentialquadratic programming (SQP) method is adopted to solvethe gear reducer problem and the results provided by theQEPF-ATC and Kriging model combined approach andthe QEPF-ATC approach are shown in Tables 4 and 5respectively It can be seen that results of the QEPF-ATCand Kriging model combined approach and the QEPF-ATCapproach are very close and the best result provided bythe QEPF-ATC and Kriging model combined approach iscloser to the optimal value of the gear reducer problemcompared to that provided by the QEPF-ATC approachIn terms of the iterations and runtime the QEPF-ATCand Kriging model combined approach saves computingtime by more than one half compared with QEPF-ATCapproach

5 Conclusion

Because ATC approach can flexibly organize the design opti-mization process according to the components of engineeringsystems it has been widely applied in MDOThe QEPF-ATCapproach is proposed to reduce the complicated coordinationprocess and heavy computation cost faced by exiting ATCapproaches where quadratic exterior penalty function isadopted to be the replaced coordination strategy Furtherin order to deal with the design optimization of complexengineering systems the QEPF-ATC and Kriging modelcombined approach is implemented where Kriging modelsare used to replace simulation models with high accuracyin the solving process of MDO problems The QEPF-ATCand Kriging model combined approach can avoid the impactof numerical noise brought by using simulation modelsduring the iterative solving process and reduce computationcost A geometric programming case is given to validatethe usability and effectiveness of the QEPF-ATC approachThe QEPF-ATC and Kriging model combined approach hasbeen successfully applied to solve the speed reducer designproblem

It should be pointed out that usingmetamodels within theQEPF-ATC approach is not without its own set of potentialdrawbacks which are well known and beyond the scope ofthis work Prior to employing the metamodel assisted QEPF-ATC approach the user should become familiar with thestrengths and weaknesses associated with surrogate approxi-mation Variable-fidelity modeling provides a promising wayto flexibly use approximate models in dealing with MDOproblems The combination of the QEPF-ATC approach andvariable-fidelity modeling will be studied to design complexengineering systems in the future

Abbreviations and Acronyms

ALF Augmented Lagrangian functionAM Approximate modelsATC Analytical target cascadingBLISS Bilevel integrated system synthesis

10 Mathematical Problems in Engineering

002minus002minus006minus01minus014minus018minus022minus026minus03

002minus002minus006minus01minus014minus018minus022minus026minus03

8

8 8

8

75

75 75

75

7 7

77

g7 g7

02

01501005

0minus005minus01minus015minus02

02

015010050minus005minus01minus015minus02

8

8 8

8

75

75 75

75

7 7

77

g8g8

x 4

x 4

x5

x5

x 4

x 4

x5

x5

Figure 9 Kriging models of the shaft subsystem

minus006

minus016

minus026minus026 minus016 minus006

Actu

al

Actu

al

Predicted Predicted

005

minus005

minus015minus015 minus005 005

g8g7

Figure 10 Error analysis for Kriging models of the shaft subsystem

Figure 11 The error data of Kriging models of subsystem 2

Mathematical Problems in Engineering 11

CO Collaborative optimizationCSSO Concurrent subspace optimizationDQA Diagonal quadratic approximationLDF Lagrangian dual functionLHS Latin hypercube samplingMDO Multidisciplinary design

optimizationQEPF Quadratic exterior penalty functionQEPF-ATC Quadratic exterior penalty function

based analytical target cascadingSQP Sequential quadratic programming

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by theNational Natural Sci-ence Foundation of China (NSFC) under Grant no 51121002the National Basic Research Program (973 Program) ofChina under Grant no 2014CB046703 and the FundamentalResearch Funds for the Central Universities HUST Grantno 2014TS040 The authors also would like to thank theanonymous referees for their valuable comments

References

[1] J Sobieszczanski-Sobieski ldquoA linear decomposition method forlarge optimization problemsrdquo NASA Technical Memorandum83248 1982

[2] TW Simpson and J R R AMartins ldquoMultidisciplinary designoptimization for complex engineered systems report from anational science foundation workshoprdquo Journal of MechanicalDesign vol 133 no 10 Article ID 101002 2011

[3] M Balesdent N Berend P Depince and A Chriette ldquoA surveyof multidisciplinary design optimization methods in launchvehicle designrdquo Structural and Multidisciplinary Optimizationvol 45 no 5 pp 619ndash642 2012

[4] I Sobieski and I Kroo ldquoAircraft design using collaborativeoptimizationrdquo AIAA Paper 715 1996

[5] H-Z Huang H Yu X Zhang S Zeng and ZWang ldquoCollabo-rative optimization with inverse reliability for multidisciplinarysystems uncertainty analysisrdquoEngineeringOptimization vol 42no 8 pp 763ndash773 2010

[6] R S Sellar S M Batill and J E Renaud ldquoResponse surfacebased concurrent subspace optimization for multidisciplinarysystem designrdquo AIAA Paper 714 AIAA 1996

[7] L S Li J H Liu and S H Liu ldquoAn efficient strategy formultidisciplinary reliability design and optimization based onCSSO and PMA in SORA frameworkrdquo Structural and Multidis-ciplinary Optimization vol 49 no 2 pp 239ndash252 2014

[8] J Sobieszczanski-Sobieski ldquoBi-level integrated system synthesis(BLISS)rdquo AIAA Paper 4916 AIAA 1998

[9] J Sobieszczanski-Sobieski T D Altus M Phillips and RSandusky ldquoBi-level integrated system synthesis (BLISS) forconcurrent and distributed processingrdquo AIAA Paper 54092002

[10] N Michelena H Park and P Y Papalambros ldquoConvergenceproperties of analytical target cascadingrdquo AIAA Journal vol 41no 5 pp 897ndash905 2003

[11] M Gardenghi M M Wiecek and W Wang ldquoBiobjectiveoptimization for analytical target cascading optimality vsachievabilityrdquo Structural and Multidisciplinary Optimizationvol 47 no 1 pp 111ndash133 2013

[12] S Tosserams L F Etman P Y Papalambros and J E RoodaldquoAn augmented Lagrangian relaxation for analytical target cas-cading using the alternating direction method of multipliersrdquoStructural and Multidisciplinary Optimization vol 31 no 3 pp176ndash189 2006

[13] Y Li Z Lu and J J Michalek ldquoDiagonal quadratic approxima-tion for parallelization of analytical target cascadingrdquo Journalof Mechanical Design Transactions of the ASME vol 130 no 5Article ID 051402 2008

[14] H M Kim W Chen and M M Wiecek ldquoLagrangian coor-dination for enhancing the convergence of analytical targetcascadingrdquo AIAA Journal vol 44 no 10 pp 2197ndash2207 2006

[15] D W Kim and J Lee ldquoAn improvement of Kriging basedsequential approximate optimization method via extended useof design of experimentsrdquo Engineering Optimization vol 42 no12 pp 1133ndash1149 2010

[16] P B Backlund D W Shahan and C C Seepersad ldquoA com-parative study of the scalability of alternative metamodellingtechniquesrdquo Engineering Optimization vol 44 no 7 pp 767ndash786 2012

[17] J Zhang S Chowdhury and A Messac ldquoAn adaptive hybridsurrogate modelrdquo Structural and Multidisciplinary Optimiza-tion vol 46 no 2 pp 223ndash238 2012

[18] V Dubourg B Sudret and J-M Bourinet ldquoReliability-baseddesign optimization using kriging surrogates and subset simu-lationrdquo Structural and Multidisciplinary Optimization vol 44no 5 pp 673ndash690 2011

[19] H M Kim N F Michelena P Y Papalambros and T JiangldquoTarget cascading in optimal system designrdquo Transactions of theASMEmdashJournal of Mechanical Design vol 125 no 3 pp 474ndash480 2003

[20] J P Costa L Pronzato and E Thierry ldquoA comparison betweenKriging and radial basis function networks for nonlinearpredictionrdquo in Proceedings of the IEEE-EURASIP Workshop onNonlinear Signal and Image Processing (NSIP rsquo99) pp 726ndash7301999

[21] V Cherkassky D Gehring and F Mulier ldquoComparison ofadaptive methods for function estimation from samplesrdquo IEEETransactions on Neural Networks vol 7 no 4 pp 969ndash9841996

[22] M S Kim H Cho S G Lee J Choi and D Bae ldquoDFSS androbust optimization tool for multibody system with randomvariablesrdquo Journal of System Design and Dynamics vol 1 no 3pp 583ndash592 2007

[23] H M Kim Target cascading in optimal system design [PhDdissertation] University of Michigan 2001

[24] S L Padula N Alexandrov and L L Green ldquoMDO testsuite at NASA Langley Research Centerrdquo in Proceedings of the6th Symposium onMultidisciplinary Analysis and OptimizationAIAA Paper 96-4028 pp 410ndash420 Bellevue Wash USASeptember 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

The optimizer of system level

The optimizer of subproblem 1 The optimizer of subproblem 2

DOE-LHS DOE-LHS

Subproblem 1

analysis

Kriging model 1

Subproblem 2

analysis

Kriging model 2

g3 g4 g5 g6

xU8 xU7B x

U6B xL8 x

L7B x

L6B xU9 x

U6C x

U7C xL9 x

L6C x

L7C

X1 X1

X1

X1 G1

G1X2 X2

X2

X2 G2

G2

st

st st

R1 = x10 = r1(x3 x4 x5) = x8 + x9

X0 = x6 x7

G1 = g1 g2 g9 g10 g11R2 = x8 = r2(X1) = f1(x)

X1 = x1 x2 x3 x6 x7

G2 = g7 g8R3 = x9 = r3(X2) = f2(x)

X2 = x4 x5 x6 x7

PA min f11 = x10 + x8 minus xL8)

2+ (x9 minus xL9)

2+ (x6 minus xL6B)

2+ (x6 minus xL6C)

2+ (x7 minus xL7B)

2+ (x7 minus xL7C)

2]120590

2)]

PB min f21 = x8 minus xU8 )

2+ (x6 minus xU6B)

2+ (x7 minus xU7B)

2]120590

2)] PC min f22 = x9 minus x

U9 )

2+ (x6 minus xU6C)

2+ (x7 minus xU7C)

2]120590

2)]

Figure 8 The solving framework of the QEPF-ATC and Kriging model combined approach for the gear reducer

1198912(119909) = 7477 (119909

3

6+ 1199093

7) + 07854 (119909

41199092

6+ 11990951199092

7)

st 1198921=

27

(11990911199092

21199093)minus 1 le 0 119892

2=

3975

(11990911199092

21199092

3)minus 1 le 0

1198923=1931199093

4

(119909211990931199094

6)minus 1 le 0 119892

4=1931199093

5

(119909211990931199094

7)minus 1 le 0

1198925= 10119909

minus3

6radic(745119909

minus1

2119909minus1

31199094) + 169 times 107

minus 1100 le 0

1198926= 10119909

minus3

7radic(745119909

minus1

2119909minus1

31199095) + 1575 times 108

minus 850 le 0

1198927=(151199096+ 19)

1199094

minus 1 le 0

1198928=(151199097+ 19)

1199095

minus 1 le 0

1198929= 11990921199093minus 40 le 0 119892

10= 5 minus

1199091

1199092

le 0

11989211=1199091

1199092

minus 12 le 0

(20)

where 1198911is the gear weight and 119891

2is the shaft weight And 119892

1

and 1198922 are respectively gear bending stress and contact stress

constraints 1198923and 119892

4are respectively torsional distortion

constraints of shaft 1 and shaft 2 1198925and 119892

6are stress

constraints of the shaft 1 and shaft 2 1198927and 119892

8are experience

constraints of shaft 1 and shaft 2 and 1198929 11989210 and 119892

11are gear

geometry constraints1199091 1199092 and 119909

3are the design variables of the gear

subproblem 1199094and 119909

5are the design variables of the

shaft subproblem 1199096and 119909

7are linking variables (coupling

variables) According to the QEPF-ATC and Kriging modelcombined approach three variables 119909

8 1199099 and 119909

10are

introduced and meet the following formula

ℎ0= 1199098+ 1199099minus 11990910= 0

ℎ1= 1199098minus 1198911(119909) = 0

ℎ2= 1199099minus 1198912(119909) = 0

(21)

where ℎ0represents the analysis model of system level and

ℎ1and ℎ

2represent the analysis models of subproblem 1 and

subproblem 2 respectivelyThe solving framework of the QEPF-ATC and Kriging

model combined approach for gear reducer is shown inFigure 8

Creating the Kriging models of the shaft subsystem isintroduced in detail Firstly select 45 sample points byLHS method to create the Kriging models to conduct the

Mathematical Problems in Engineering 9

Table 4 The optimal results of the QEPF-ATC and Kriging model combined approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299453432 26 138 (3504 0698 17 7300 7845 3410 5283)2 299453217 25 124 (3501 0700 17 7310 7763 3353 5286)3 299521246 18 113 (3498 0699 17 7306 7755 3312 5279)4 299564235 10 36 (3499 0702 17 7297 7718 3350 5282)5 299433263 24 121 (3502 0700 17 7298 7720 3347 5287)

Table 5 The optimal results of the QEPF-ATC approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299463432 42 272 (3500 0699 17 7300 7825 3392 5282)2 299551327 38 215 (3500 0703 17 7305 7746 3353 5287)3 299612356 30 198 (3497 0696 17 7300 7713 3326 5275)4 299575363 17 97 (3499 0700 17 7299 7715 3349 5282)5 299521554 52 306 (3500 0700 17 7300 7719 3350 5286)

subsystem analysis and the created models are shown inFigure 9

Then select 20 sample points by random samplingmethod to evaluate the createdKrigingmodels Error analysisfor Kriging models of the shaft subsystem is shown inFigure 10 and it can be seen that the predicted valuesobtained by Kriging models are basically equal to the actualvalues

There are four credibility evaluation standards for approx-imate models that is the average the maximum the rootmean square and the 119877-squared For the average the max-imum and the root mean square the much smaller thevalues are the much better the credibility of approximatemodels is The default threshold of the average and theroot mean square is 02 and the maximum is 03 Forthe 119877-squared the much bigger the value is the muchbetter the credibility of approximate models is and thedefault threshold is 09 Figure 11 describes the values of thefour credibility evaluation standards for the created Krigingmodels of the shaft subsystem and it can be seen that the fourcredibility evaluation standards are all met So the createdKrigingmodels are reliable enough and can be used to replacethe original models with high accuracy in the next solvingprocess

Five initial points are selected and the sequentialquadratic programming (SQP) method is adopted to solvethe gear reducer problem and the results provided by theQEPF-ATC and Kriging model combined approach andthe QEPF-ATC approach are shown in Tables 4 and 5respectively It can be seen that results of the QEPF-ATCand Kriging model combined approach and the QEPF-ATCapproach are very close and the best result provided bythe QEPF-ATC and Kriging model combined approach iscloser to the optimal value of the gear reducer problemcompared to that provided by the QEPF-ATC approachIn terms of the iterations and runtime the QEPF-ATCand Kriging model combined approach saves computingtime by more than one half compared with QEPF-ATCapproach

5 Conclusion

Because ATC approach can flexibly organize the design opti-mization process according to the components of engineeringsystems it has been widely applied in MDOThe QEPF-ATCapproach is proposed to reduce the complicated coordinationprocess and heavy computation cost faced by exiting ATCapproaches where quadratic exterior penalty function isadopted to be the replaced coordination strategy Furtherin order to deal with the design optimization of complexengineering systems the QEPF-ATC and Kriging modelcombined approach is implemented where Kriging modelsare used to replace simulation models with high accuracyin the solving process of MDO problems The QEPF-ATCand Kriging model combined approach can avoid the impactof numerical noise brought by using simulation modelsduring the iterative solving process and reduce computationcost A geometric programming case is given to validatethe usability and effectiveness of the QEPF-ATC approachThe QEPF-ATC and Kriging model combined approach hasbeen successfully applied to solve the speed reducer designproblem

It should be pointed out that usingmetamodels within theQEPF-ATC approach is not without its own set of potentialdrawbacks which are well known and beyond the scope ofthis work Prior to employing the metamodel assisted QEPF-ATC approach the user should become familiar with thestrengths and weaknesses associated with surrogate approxi-mation Variable-fidelity modeling provides a promising wayto flexibly use approximate models in dealing with MDOproblems The combination of the QEPF-ATC approach andvariable-fidelity modeling will be studied to design complexengineering systems in the future

Abbreviations and Acronyms

ALF Augmented Lagrangian functionAM Approximate modelsATC Analytical target cascadingBLISS Bilevel integrated system synthesis

10 Mathematical Problems in Engineering

002minus002minus006minus01minus014minus018minus022minus026minus03

002minus002minus006minus01minus014minus018minus022minus026minus03

8

8 8

8

75

75 75

75

7 7

77

g7 g7

02

01501005

0minus005minus01minus015minus02

02

015010050minus005minus01minus015minus02

8

8 8

8

75

75 75

75

7 7

77

g8g8

x 4

x 4

x5

x5

x 4

x 4

x5

x5

Figure 9 Kriging models of the shaft subsystem

minus006

minus016

minus026minus026 minus016 minus006

Actu

al

Actu

al

Predicted Predicted

005

minus005

minus015minus015 minus005 005

g8g7

Figure 10 Error analysis for Kriging models of the shaft subsystem

Figure 11 The error data of Kriging models of subsystem 2

Mathematical Problems in Engineering 11

CO Collaborative optimizationCSSO Concurrent subspace optimizationDQA Diagonal quadratic approximationLDF Lagrangian dual functionLHS Latin hypercube samplingMDO Multidisciplinary design

optimizationQEPF Quadratic exterior penalty functionQEPF-ATC Quadratic exterior penalty function

based analytical target cascadingSQP Sequential quadratic programming

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by theNational Natural Sci-ence Foundation of China (NSFC) under Grant no 51121002the National Basic Research Program (973 Program) ofChina under Grant no 2014CB046703 and the FundamentalResearch Funds for the Central Universities HUST Grantno 2014TS040 The authors also would like to thank theanonymous referees for their valuable comments

References

[1] J Sobieszczanski-Sobieski ldquoA linear decomposition method forlarge optimization problemsrdquo NASA Technical Memorandum83248 1982

[2] TW Simpson and J R R AMartins ldquoMultidisciplinary designoptimization for complex engineered systems report from anational science foundation workshoprdquo Journal of MechanicalDesign vol 133 no 10 Article ID 101002 2011

[3] M Balesdent N Berend P Depince and A Chriette ldquoA surveyof multidisciplinary design optimization methods in launchvehicle designrdquo Structural and Multidisciplinary Optimizationvol 45 no 5 pp 619ndash642 2012

[4] I Sobieski and I Kroo ldquoAircraft design using collaborativeoptimizationrdquo AIAA Paper 715 1996

[5] H-Z Huang H Yu X Zhang S Zeng and ZWang ldquoCollabo-rative optimization with inverse reliability for multidisciplinarysystems uncertainty analysisrdquoEngineeringOptimization vol 42no 8 pp 763ndash773 2010

[6] R S Sellar S M Batill and J E Renaud ldquoResponse surfacebased concurrent subspace optimization for multidisciplinarysystem designrdquo AIAA Paper 714 AIAA 1996

[7] L S Li J H Liu and S H Liu ldquoAn efficient strategy formultidisciplinary reliability design and optimization based onCSSO and PMA in SORA frameworkrdquo Structural and Multidis-ciplinary Optimization vol 49 no 2 pp 239ndash252 2014

[8] J Sobieszczanski-Sobieski ldquoBi-level integrated system synthesis(BLISS)rdquo AIAA Paper 4916 AIAA 1998

[9] J Sobieszczanski-Sobieski T D Altus M Phillips and RSandusky ldquoBi-level integrated system synthesis (BLISS) forconcurrent and distributed processingrdquo AIAA Paper 54092002

[10] N Michelena H Park and P Y Papalambros ldquoConvergenceproperties of analytical target cascadingrdquo AIAA Journal vol 41no 5 pp 897ndash905 2003

[11] M Gardenghi M M Wiecek and W Wang ldquoBiobjectiveoptimization for analytical target cascading optimality vsachievabilityrdquo Structural and Multidisciplinary Optimizationvol 47 no 1 pp 111ndash133 2013

[12] S Tosserams L F Etman P Y Papalambros and J E RoodaldquoAn augmented Lagrangian relaxation for analytical target cas-cading using the alternating direction method of multipliersrdquoStructural and Multidisciplinary Optimization vol 31 no 3 pp176ndash189 2006

[13] Y Li Z Lu and J J Michalek ldquoDiagonal quadratic approxima-tion for parallelization of analytical target cascadingrdquo Journalof Mechanical Design Transactions of the ASME vol 130 no 5Article ID 051402 2008

[14] H M Kim W Chen and M M Wiecek ldquoLagrangian coor-dination for enhancing the convergence of analytical targetcascadingrdquo AIAA Journal vol 44 no 10 pp 2197ndash2207 2006

[15] D W Kim and J Lee ldquoAn improvement of Kriging basedsequential approximate optimization method via extended useof design of experimentsrdquo Engineering Optimization vol 42 no12 pp 1133ndash1149 2010

[16] P B Backlund D W Shahan and C C Seepersad ldquoA com-parative study of the scalability of alternative metamodellingtechniquesrdquo Engineering Optimization vol 44 no 7 pp 767ndash786 2012

[17] J Zhang S Chowdhury and A Messac ldquoAn adaptive hybridsurrogate modelrdquo Structural and Multidisciplinary Optimiza-tion vol 46 no 2 pp 223ndash238 2012

[18] V Dubourg B Sudret and J-M Bourinet ldquoReliability-baseddesign optimization using kriging surrogates and subset simu-lationrdquo Structural and Multidisciplinary Optimization vol 44no 5 pp 673ndash690 2011

[19] H M Kim N F Michelena P Y Papalambros and T JiangldquoTarget cascading in optimal system designrdquo Transactions of theASMEmdashJournal of Mechanical Design vol 125 no 3 pp 474ndash480 2003

[20] J P Costa L Pronzato and E Thierry ldquoA comparison betweenKriging and radial basis function networks for nonlinearpredictionrdquo in Proceedings of the IEEE-EURASIP Workshop onNonlinear Signal and Image Processing (NSIP rsquo99) pp 726ndash7301999

[21] V Cherkassky D Gehring and F Mulier ldquoComparison ofadaptive methods for function estimation from samplesrdquo IEEETransactions on Neural Networks vol 7 no 4 pp 969ndash9841996

[22] M S Kim H Cho S G Lee J Choi and D Bae ldquoDFSS androbust optimization tool for multibody system with randomvariablesrdquo Journal of System Design and Dynamics vol 1 no 3pp 583ndash592 2007

[23] H M Kim Target cascading in optimal system design [PhDdissertation] University of Michigan 2001

[24] S L Padula N Alexandrov and L L Green ldquoMDO testsuite at NASA Langley Research Centerrdquo in Proceedings of the6th Symposium onMultidisciplinary Analysis and OptimizationAIAA Paper 96-4028 pp 410ndash420 Bellevue Wash USASeptember 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 4 The optimal results of the QEPF-ATC and Kriging model combined approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299453432 26 138 (3504 0698 17 7300 7845 3410 5283)2 299453217 25 124 (3501 0700 17 7310 7763 3353 5286)3 299521246 18 113 (3498 0699 17 7306 7755 3312 5279)4 299564235 10 36 (3499 0702 17 7297 7718 3350 5282)5 299433263 24 121 (3502 0700 17 7298 7720 3347 5287)

Table 5 The optimal results of the QEPF-ATC approach

Initial point 119891min Iterations Runtime (s) (1199091 1199092 1199093 1199094 1199095 1199096 1199097)

1 299463432 42 272 (3500 0699 17 7300 7825 3392 5282)2 299551327 38 215 (3500 0703 17 7305 7746 3353 5287)3 299612356 30 198 (3497 0696 17 7300 7713 3326 5275)4 299575363 17 97 (3499 0700 17 7299 7715 3349 5282)5 299521554 52 306 (3500 0700 17 7300 7719 3350 5286)

subsystem analysis and the created models are shown inFigure 9

Then select 20 sample points by random samplingmethod to evaluate the createdKrigingmodels Error analysisfor Kriging models of the shaft subsystem is shown inFigure 10 and it can be seen that the predicted valuesobtained by Kriging models are basically equal to the actualvalues

There are four credibility evaluation standards for approx-imate models that is the average the maximum the rootmean square and the 119877-squared For the average the max-imum and the root mean square the much smaller thevalues are the much better the credibility of approximatemodels is The default threshold of the average and theroot mean square is 02 and the maximum is 03 Forthe 119877-squared the much bigger the value is the muchbetter the credibility of approximate models is and thedefault threshold is 09 Figure 11 describes the values of thefour credibility evaluation standards for the created Krigingmodels of the shaft subsystem and it can be seen that the fourcredibility evaluation standards are all met So the createdKrigingmodels are reliable enough and can be used to replacethe original models with high accuracy in the next solvingprocess

Five initial points are selected and the sequentialquadratic programming (SQP) method is adopted to solvethe gear reducer problem and the results provided by theQEPF-ATC and Kriging model combined approach andthe QEPF-ATC approach are shown in Tables 4 and 5respectively It can be seen that results of the QEPF-ATCand Kriging model combined approach and the QEPF-ATCapproach are very close and the best result provided bythe QEPF-ATC and Kriging model combined approach iscloser to the optimal value of the gear reducer problemcompared to that provided by the QEPF-ATC approachIn terms of the iterations and runtime the QEPF-ATCand Kriging model combined approach saves computingtime by more than one half compared with QEPF-ATCapproach

5 Conclusion

Because ATC approach can flexibly organize the design opti-mization process according to the components of engineeringsystems it has been widely applied in MDOThe QEPF-ATCapproach is proposed to reduce the complicated coordinationprocess and heavy computation cost faced by exiting ATCapproaches where quadratic exterior penalty function isadopted to be the replaced coordination strategy Furtherin order to deal with the design optimization of complexengineering systems the QEPF-ATC and Kriging modelcombined approach is implemented where Kriging modelsare used to replace simulation models with high accuracyin the solving process of MDO problems The QEPF-ATCand Kriging model combined approach can avoid the impactof numerical noise brought by using simulation modelsduring the iterative solving process and reduce computationcost A geometric programming case is given to validatethe usability and effectiveness of the QEPF-ATC approachThe QEPF-ATC and Kriging model combined approach hasbeen successfully applied to solve the speed reducer designproblem

It should be pointed out that usingmetamodels within theQEPF-ATC approach is not without its own set of potentialdrawbacks which are well known and beyond the scope ofthis work Prior to employing the metamodel assisted QEPF-ATC approach the user should become familiar with thestrengths and weaknesses associated with surrogate approxi-mation Variable-fidelity modeling provides a promising wayto flexibly use approximate models in dealing with MDOproblems The combination of the QEPF-ATC approach andvariable-fidelity modeling will be studied to design complexengineering systems in the future

Abbreviations and Acronyms

ALF Augmented Lagrangian functionAM Approximate modelsATC Analytical target cascadingBLISS Bilevel integrated system synthesis

10 Mathematical Problems in Engineering

002minus002minus006minus01minus014minus018minus022minus026minus03

002minus002minus006minus01minus014minus018minus022minus026minus03

8

8 8

8

75

75 75

75

7 7

77

g7 g7

02

01501005

0minus005minus01minus015minus02

02

015010050minus005minus01minus015minus02

8

8 8

8

75

75 75

75

7 7

77

g8g8

x 4

x 4

x5

x5

x 4

x 4

x5

x5

Figure 9 Kriging models of the shaft subsystem

minus006

minus016

minus026minus026 minus016 minus006

Actu

al

Actu

al

Predicted Predicted

005

minus005

minus015minus015 minus005 005

g8g7

Figure 10 Error analysis for Kriging models of the shaft subsystem

Figure 11 The error data of Kriging models of subsystem 2

Mathematical Problems in Engineering 11

CO Collaborative optimizationCSSO Concurrent subspace optimizationDQA Diagonal quadratic approximationLDF Lagrangian dual functionLHS Latin hypercube samplingMDO Multidisciplinary design

optimizationQEPF Quadratic exterior penalty functionQEPF-ATC Quadratic exterior penalty function

based analytical target cascadingSQP Sequential quadratic programming

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by theNational Natural Sci-ence Foundation of China (NSFC) under Grant no 51121002the National Basic Research Program (973 Program) ofChina under Grant no 2014CB046703 and the FundamentalResearch Funds for the Central Universities HUST Grantno 2014TS040 The authors also would like to thank theanonymous referees for their valuable comments

References

[1] J Sobieszczanski-Sobieski ldquoA linear decomposition method forlarge optimization problemsrdquo NASA Technical Memorandum83248 1982

[2] TW Simpson and J R R AMartins ldquoMultidisciplinary designoptimization for complex engineered systems report from anational science foundation workshoprdquo Journal of MechanicalDesign vol 133 no 10 Article ID 101002 2011

[3] M Balesdent N Berend P Depince and A Chriette ldquoA surveyof multidisciplinary design optimization methods in launchvehicle designrdquo Structural and Multidisciplinary Optimizationvol 45 no 5 pp 619ndash642 2012

[4] I Sobieski and I Kroo ldquoAircraft design using collaborativeoptimizationrdquo AIAA Paper 715 1996

[5] H-Z Huang H Yu X Zhang S Zeng and ZWang ldquoCollabo-rative optimization with inverse reliability for multidisciplinarysystems uncertainty analysisrdquoEngineeringOptimization vol 42no 8 pp 763ndash773 2010

[6] R S Sellar S M Batill and J E Renaud ldquoResponse surfacebased concurrent subspace optimization for multidisciplinarysystem designrdquo AIAA Paper 714 AIAA 1996

[7] L S Li J H Liu and S H Liu ldquoAn efficient strategy formultidisciplinary reliability design and optimization based onCSSO and PMA in SORA frameworkrdquo Structural and Multidis-ciplinary Optimization vol 49 no 2 pp 239ndash252 2014

[8] J Sobieszczanski-Sobieski ldquoBi-level integrated system synthesis(BLISS)rdquo AIAA Paper 4916 AIAA 1998

[9] J Sobieszczanski-Sobieski T D Altus M Phillips and RSandusky ldquoBi-level integrated system synthesis (BLISS) forconcurrent and distributed processingrdquo AIAA Paper 54092002

[10] N Michelena H Park and P Y Papalambros ldquoConvergenceproperties of analytical target cascadingrdquo AIAA Journal vol 41no 5 pp 897ndash905 2003

[11] M Gardenghi M M Wiecek and W Wang ldquoBiobjectiveoptimization for analytical target cascading optimality vsachievabilityrdquo Structural and Multidisciplinary Optimizationvol 47 no 1 pp 111ndash133 2013

[12] S Tosserams L F Etman P Y Papalambros and J E RoodaldquoAn augmented Lagrangian relaxation for analytical target cas-cading using the alternating direction method of multipliersrdquoStructural and Multidisciplinary Optimization vol 31 no 3 pp176ndash189 2006

[13] Y Li Z Lu and J J Michalek ldquoDiagonal quadratic approxima-tion for parallelization of analytical target cascadingrdquo Journalof Mechanical Design Transactions of the ASME vol 130 no 5Article ID 051402 2008

[14] H M Kim W Chen and M M Wiecek ldquoLagrangian coor-dination for enhancing the convergence of analytical targetcascadingrdquo AIAA Journal vol 44 no 10 pp 2197ndash2207 2006

[15] D W Kim and J Lee ldquoAn improvement of Kriging basedsequential approximate optimization method via extended useof design of experimentsrdquo Engineering Optimization vol 42 no12 pp 1133ndash1149 2010

[16] P B Backlund D W Shahan and C C Seepersad ldquoA com-parative study of the scalability of alternative metamodellingtechniquesrdquo Engineering Optimization vol 44 no 7 pp 767ndash786 2012

[17] J Zhang S Chowdhury and A Messac ldquoAn adaptive hybridsurrogate modelrdquo Structural and Multidisciplinary Optimiza-tion vol 46 no 2 pp 223ndash238 2012

[18] V Dubourg B Sudret and J-M Bourinet ldquoReliability-baseddesign optimization using kriging surrogates and subset simu-lationrdquo Structural and Multidisciplinary Optimization vol 44no 5 pp 673ndash690 2011

[19] H M Kim N F Michelena P Y Papalambros and T JiangldquoTarget cascading in optimal system designrdquo Transactions of theASMEmdashJournal of Mechanical Design vol 125 no 3 pp 474ndash480 2003

[20] J P Costa L Pronzato and E Thierry ldquoA comparison betweenKriging and radial basis function networks for nonlinearpredictionrdquo in Proceedings of the IEEE-EURASIP Workshop onNonlinear Signal and Image Processing (NSIP rsquo99) pp 726ndash7301999

[21] V Cherkassky D Gehring and F Mulier ldquoComparison ofadaptive methods for function estimation from samplesrdquo IEEETransactions on Neural Networks vol 7 no 4 pp 969ndash9841996

[22] M S Kim H Cho S G Lee J Choi and D Bae ldquoDFSS androbust optimization tool for multibody system with randomvariablesrdquo Journal of System Design and Dynamics vol 1 no 3pp 583ndash592 2007

[23] H M Kim Target cascading in optimal system design [PhDdissertation] University of Michigan 2001

[24] S L Padula N Alexandrov and L L Green ldquoMDO testsuite at NASA Langley Research Centerrdquo in Proceedings of the6th Symposium onMultidisciplinary Analysis and OptimizationAIAA Paper 96-4028 pp 410ndash420 Bellevue Wash USASeptember 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

002minus002minus006minus01minus014minus018minus022minus026minus03

002minus002minus006minus01minus014minus018minus022minus026minus03

8

8 8

8

75

75 75

75

7 7

77

g7 g7

02

01501005

0minus005minus01minus015minus02

02

015010050minus005minus01minus015minus02

8

8 8

8

75

75 75

75

7 7

77

g8g8

x 4

x 4

x5

x5

x 4

x 4

x5

x5

Figure 9 Kriging models of the shaft subsystem

minus006

minus016

minus026minus026 minus016 minus006

Actu

al

Actu

al

Predicted Predicted

005

minus005

minus015minus015 minus005 005

g8g7

Figure 10 Error analysis for Kriging models of the shaft subsystem

Figure 11 The error data of Kriging models of subsystem 2

Mathematical Problems in Engineering 11

CO Collaborative optimizationCSSO Concurrent subspace optimizationDQA Diagonal quadratic approximationLDF Lagrangian dual functionLHS Latin hypercube samplingMDO Multidisciplinary design

optimizationQEPF Quadratic exterior penalty functionQEPF-ATC Quadratic exterior penalty function

based analytical target cascadingSQP Sequential quadratic programming

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by theNational Natural Sci-ence Foundation of China (NSFC) under Grant no 51121002the National Basic Research Program (973 Program) ofChina under Grant no 2014CB046703 and the FundamentalResearch Funds for the Central Universities HUST Grantno 2014TS040 The authors also would like to thank theanonymous referees for their valuable comments

References

[1] J Sobieszczanski-Sobieski ldquoA linear decomposition method forlarge optimization problemsrdquo NASA Technical Memorandum83248 1982

[2] TW Simpson and J R R AMartins ldquoMultidisciplinary designoptimization for complex engineered systems report from anational science foundation workshoprdquo Journal of MechanicalDesign vol 133 no 10 Article ID 101002 2011

[3] M Balesdent N Berend P Depince and A Chriette ldquoA surveyof multidisciplinary design optimization methods in launchvehicle designrdquo Structural and Multidisciplinary Optimizationvol 45 no 5 pp 619ndash642 2012

[4] I Sobieski and I Kroo ldquoAircraft design using collaborativeoptimizationrdquo AIAA Paper 715 1996

[5] H-Z Huang H Yu X Zhang S Zeng and ZWang ldquoCollabo-rative optimization with inverse reliability for multidisciplinarysystems uncertainty analysisrdquoEngineeringOptimization vol 42no 8 pp 763ndash773 2010

[6] R S Sellar S M Batill and J E Renaud ldquoResponse surfacebased concurrent subspace optimization for multidisciplinarysystem designrdquo AIAA Paper 714 AIAA 1996

[7] L S Li J H Liu and S H Liu ldquoAn efficient strategy formultidisciplinary reliability design and optimization based onCSSO and PMA in SORA frameworkrdquo Structural and Multidis-ciplinary Optimization vol 49 no 2 pp 239ndash252 2014

[8] J Sobieszczanski-Sobieski ldquoBi-level integrated system synthesis(BLISS)rdquo AIAA Paper 4916 AIAA 1998

[9] J Sobieszczanski-Sobieski T D Altus M Phillips and RSandusky ldquoBi-level integrated system synthesis (BLISS) forconcurrent and distributed processingrdquo AIAA Paper 54092002

[10] N Michelena H Park and P Y Papalambros ldquoConvergenceproperties of analytical target cascadingrdquo AIAA Journal vol 41no 5 pp 897ndash905 2003

[11] M Gardenghi M M Wiecek and W Wang ldquoBiobjectiveoptimization for analytical target cascading optimality vsachievabilityrdquo Structural and Multidisciplinary Optimizationvol 47 no 1 pp 111ndash133 2013

[12] S Tosserams L F Etman P Y Papalambros and J E RoodaldquoAn augmented Lagrangian relaxation for analytical target cas-cading using the alternating direction method of multipliersrdquoStructural and Multidisciplinary Optimization vol 31 no 3 pp176ndash189 2006

[13] Y Li Z Lu and J J Michalek ldquoDiagonal quadratic approxima-tion for parallelization of analytical target cascadingrdquo Journalof Mechanical Design Transactions of the ASME vol 130 no 5Article ID 051402 2008

[14] H M Kim W Chen and M M Wiecek ldquoLagrangian coor-dination for enhancing the convergence of analytical targetcascadingrdquo AIAA Journal vol 44 no 10 pp 2197ndash2207 2006

[15] D W Kim and J Lee ldquoAn improvement of Kriging basedsequential approximate optimization method via extended useof design of experimentsrdquo Engineering Optimization vol 42 no12 pp 1133ndash1149 2010

[16] P B Backlund D W Shahan and C C Seepersad ldquoA com-parative study of the scalability of alternative metamodellingtechniquesrdquo Engineering Optimization vol 44 no 7 pp 767ndash786 2012

[17] J Zhang S Chowdhury and A Messac ldquoAn adaptive hybridsurrogate modelrdquo Structural and Multidisciplinary Optimiza-tion vol 46 no 2 pp 223ndash238 2012

[18] V Dubourg B Sudret and J-M Bourinet ldquoReliability-baseddesign optimization using kriging surrogates and subset simu-lationrdquo Structural and Multidisciplinary Optimization vol 44no 5 pp 673ndash690 2011

[19] H M Kim N F Michelena P Y Papalambros and T JiangldquoTarget cascading in optimal system designrdquo Transactions of theASMEmdashJournal of Mechanical Design vol 125 no 3 pp 474ndash480 2003

[20] J P Costa L Pronzato and E Thierry ldquoA comparison betweenKriging and radial basis function networks for nonlinearpredictionrdquo in Proceedings of the IEEE-EURASIP Workshop onNonlinear Signal and Image Processing (NSIP rsquo99) pp 726ndash7301999

[21] V Cherkassky D Gehring and F Mulier ldquoComparison ofadaptive methods for function estimation from samplesrdquo IEEETransactions on Neural Networks vol 7 no 4 pp 969ndash9841996

[22] M S Kim H Cho S G Lee J Choi and D Bae ldquoDFSS androbust optimization tool for multibody system with randomvariablesrdquo Journal of System Design and Dynamics vol 1 no 3pp 583ndash592 2007

[23] H M Kim Target cascading in optimal system design [PhDdissertation] University of Michigan 2001

[24] S L Padula N Alexandrov and L L Green ldquoMDO testsuite at NASA Langley Research Centerrdquo in Proceedings of the6th Symposium onMultidisciplinary Analysis and OptimizationAIAA Paper 96-4028 pp 410ndash420 Bellevue Wash USASeptember 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

CO Collaborative optimizationCSSO Concurrent subspace optimizationDQA Diagonal quadratic approximationLDF Lagrangian dual functionLHS Latin hypercube samplingMDO Multidisciplinary design

optimizationQEPF Quadratic exterior penalty functionQEPF-ATC Quadratic exterior penalty function

based analytical target cascadingSQP Sequential quadratic programming

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by theNational Natural Sci-ence Foundation of China (NSFC) under Grant no 51121002the National Basic Research Program (973 Program) ofChina under Grant no 2014CB046703 and the FundamentalResearch Funds for the Central Universities HUST Grantno 2014TS040 The authors also would like to thank theanonymous referees for their valuable comments

References

[1] J Sobieszczanski-Sobieski ldquoA linear decomposition method forlarge optimization problemsrdquo NASA Technical Memorandum83248 1982

[2] TW Simpson and J R R AMartins ldquoMultidisciplinary designoptimization for complex engineered systems report from anational science foundation workshoprdquo Journal of MechanicalDesign vol 133 no 10 Article ID 101002 2011

[3] M Balesdent N Berend P Depince and A Chriette ldquoA surveyof multidisciplinary design optimization methods in launchvehicle designrdquo Structural and Multidisciplinary Optimizationvol 45 no 5 pp 619ndash642 2012

[4] I Sobieski and I Kroo ldquoAircraft design using collaborativeoptimizationrdquo AIAA Paper 715 1996

[5] H-Z Huang H Yu X Zhang S Zeng and ZWang ldquoCollabo-rative optimization with inverse reliability for multidisciplinarysystems uncertainty analysisrdquoEngineeringOptimization vol 42no 8 pp 763ndash773 2010

[6] R S Sellar S M Batill and J E Renaud ldquoResponse surfacebased concurrent subspace optimization for multidisciplinarysystem designrdquo AIAA Paper 714 AIAA 1996

[7] L S Li J H Liu and S H Liu ldquoAn efficient strategy formultidisciplinary reliability design and optimization based onCSSO and PMA in SORA frameworkrdquo Structural and Multidis-ciplinary Optimization vol 49 no 2 pp 239ndash252 2014

[8] J Sobieszczanski-Sobieski ldquoBi-level integrated system synthesis(BLISS)rdquo AIAA Paper 4916 AIAA 1998

[9] J Sobieszczanski-Sobieski T D Altus M Phillips and RSandusky ldquoBi-level integrated system synthesis (BLISS) forconcurrent and distributed processingrdquo AIAA Paper 54092002

[10] N Michelena H Park and P Y Papalambros ldquoConvergenceproperties of analytical target cascadingrdquo AIAA Journal vol 41no 5 pp 897ndash905 2003

[11] M Gardenghi M M Wiecek and W Wang ldquoBiobjectiveoptimization for analytical target cascading optimality vsachievabilityrdquo Structural and Multidisciplinary Optimizationvol 47 no 1 pp 111ndash133 2013

[12] S Tosserams L F Etman P Y Papalambros and J E RoodaldquoAn augmented Lagrangian relaxation for analytical target cas-cading using the alternating direction method of multipliersrdquoStructural and Multidisciplinary Optimization vol 31 no 3 pp176ndash189 2006

[13] Y Li Z Lu and J J Michalek ldquoDiagonal quadratic approxima-tion for parallelization of analytical target cascadingrdquo Journalof Mechanical Design Transactions of the ASME vol 130 no 5Article ID 051402 2008

[14] H M Kim W Chen and M M Wiecek ldquoLagrangian coor-dination for enhancing the convergence of analytical targetcascadingrdquo AIAA Journal vol 44 no 10 pp 2197ndash2207 2006

[15] D W Kim and J Lee ldquoAn improvement of Kriging basedsequential approximate optimization method via extended useof design of experimentsrdquo Engineering Optimization vol 42 no12 pp 1133ndash1149 2010

[16] P B Backlund D W Shahan and C C Seepersad ldquoA com-parative study of the scalability of alternative metamodellingtechniquesrdquo Engineering Optimization vol 44 no 7 pp 767ndash786 2012

[17] J Zhang S Chowdhury and A Messac ldquoAn adaptive hybridsurrogate modelrdquo Structural and Multidisciplinary Optimiza-tion vol 46 no 2 pp 223ndash238 2012

[18] V Dubourg B Sudret and J-M Bourinet ldquoReliability-baseddesign optimization using kriging surrogates and subset simu-lationrdquo Structural and Multidisciplinary Optimization vol 44no 5 pp 673ndash690 2011

[19] H M Kim N F Michelena P Y Papalambros and T JiangldquoTarget cascading in optimal system designrdquo Transactions of theASMEmdashJournal of Mechanical Design vol 125 no 3 pp 474ndash480 2003

[20] J P Costa L Pronzato and E Thierry ldquoA comparison betweenKriging and radial basis function networks for nonlinearpredictionrdquo in Proceedings of the IEEE-EURASIP Workshop onNonlinear Signal and Image Processing (NSIP rsquo99) pp 726ndash7301999

[21] V Cherkassky D Gehring and F Mulier ldquoComparison ofadaptive methods for function estimation from samplesrdquo IEEETransactions on Neural Networks vol 7 no 4 pp 969ndash9841996

[22] M S Kim H Cho S G Lee J Choi and D Bae ldquoDFSS androbust optimization tool for multibody system with randomvariablesrdquo Journal of System Design and Dynamics vol 1 no 3pp 583ndash592 2007

[23] H M Kim Target cascading in optimal system design [PhDdissertation] University of Michigan 2001

[24] S L Padula N Alexandrov and L L Green ldquoMDO testsuite at NASA Langley Research Centerrdquo in Proceedings of the6th Symposium onMultidisciplinary Analysis and OptimizationAIAA Paper 96-4028 pp 410ndash420 Bellevue Wash USASeptember 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of