Research ArticleAn Enhanced Reliability Index Method and Its Application inReliability-Based Collaborative Design and Optimization
Debiao Meng 1 Yan Li1 Shun-Peng Zhu 1 Gang Lv2
Joseacute Correia 3 and Ab-lio de Jesus 3
1School of Mechanical and Electrical Engineering University of Electronic Science and Technology of China Chengdu China2Zigong Innovation Center of Zhejiang University Zigong China3INEGI Faculty of Engineering University of Porto Porto 4200-465 Portugal
Correspondence should be addressed to Debiao Meng dbmenguestceducn and Jose Correia jacorreiainegiuppt
Received 19 December 2018 Accepted 6 March 2019 Published 25 March 2019
Academic Editor Elena Zaitseva
Copyright copy 2019 Debiao Meng 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
When designing complex mechanical equipment uncertainties should be considered to enhance the reliability of performanceThe Reliability Index Method (RIM) is a powerful tool which has been widely utilized in engineering design under uncertaintiesTo reduce computational cost in RIM first or second order Taylor approximation is introduced to convert nonlinear probabilityconstraint to the equivalent linear constraint during optimization process Generally this approximation process is performed atMost Probable Point (MPP) to reduce the loss of reliability analysis accuracy However it is difficult for the original RIM to beutilized in the situation that MPP is collinear and RIM has the same direction with the gradient of performance function at MPPTo tackle the above challenges an EnhancedRIM (ERIM) is proposed in this studyTheCollaborativeOptimization (CO) strategy iscombined with ERIMThe formula of CO using ERIM is given to solve reliability-based multidisciplinary design and optimizationproblems A design problem of the speed reducer is utilized in this study to show the effectiveness of the proposed method
1 Introduction
TheReliability-Based Multidisciplinary Design Optimization(RBMDO) has obtained more and more attention for thehigh reliability and safety of complex engineering systems[1ndash10] Generally the RBMDO process involves a three-leveloptimization loop [11ndash16] The inner loop deals with theinteractions between coupled variables and the outer loopexplores design space to obtain optimal design solutionsBetween the inner and outer loop is uncertainty analysis loopIf RBMDO is performed directly the heavy computationalcost will affect the whole optimization process significantly[17ndash20] To deal with the three-level optimization loop struc-ture many sophisticated optimization strategies have beendeveloped [21ndash25] According to the integration strategiesof optimization and uncertainty analysis these methods canbe roughly categorized into single-loop decouple-loop anddouble-loop approaches [26ndash33]
The single-loop approaches are suitable for design prob-lems with moderate nonlinear performance function The
Karush-Kuhn-Tucker conditions are adopted by single-loopapproaches to replace the uncertainty analysis loop in an opti-mization process while the decouple-loop approaches per-formdeterministic design optimization and uncertainty anal-ysis sequentially When new design solutions are obtained bydeterministic optimization uncertainty analysis will be con-ducted to find Most Probable Point (MPP) The acquisitionof MPP is important for constructing shifting vectors Theshifting vectors are utilized to move limit state constraintsinto the safer feasible region Compared with the single-loopand the decouple-loop approaches the strategy of double-loop approaches is simple and robust Many strategies havebeen introduced into double-loop approaches to reducethe computational cost In general these strategies includemodifying the formulation of probability constraint [34ndash36] and enhancing efficiencies of optimization algorithms inreliability analysis [31 37ndash39]
The Reliability Index Method (RIM) is an effective toolwhich can modify the formulation of probability constraintin RBMDO [40 41] In RIM the first or second order Taylor
HindawiMathematical Problems in EngineeringVolume 2019 Article ID 4536906 10 pageshttpsdoiorg10115520194536906
2 Mathematical Problems in Engineering
Optimizer
Analysis at subsystem level
Design at system level
(a)
Optimizer
Design at subsystem level
Analysis at system level
(b)
Optimizer
Design at subsystem level
Design at system level
Optimizer
(c)
Figure 1 The single-level methods and multilevel methods for MDO
approximation expansion is introduced to convert nonlinearprobability constraint to the equivalent linear constraintMPP is the expansion point at which the accuracy loss ofreliability analysis due to approximation can be minimizedHowever in somepractical optimization processesMPPmaybe collinear and RIMhas the same direction with the gradientof performance function at MPP It will result in the lowefficiency of the original RIM To tackle the above challengesan Enhanced RIM (ERIM) is proposed hereThe formulationof CO using ERIM (CO-ERIM) is also proposed to solveRBMDO problems
The rest of this study will be given as follows In Section 2the general formulation of RBMDO is given Also theuncertainty information in practical engineering is discussedand the strategy of RBMDO is briefly reviewedThe RIM andthe Performance Measure Approach (PMA) are introducedin detail in Section 3The proposed ERIM is also discussed inthis part The CO-ERIM is proposed in Section 4 includingthe formulation and procedure An RBMDO problem of thespeed reducer is given in Section 5 to show the efficiencyand accuracy of CO-ERIM The conclusions are given inSection 6
2 Brief Review of RBMDO
When dealing with the design problems of complex engi-neering there are two aspects of challenges which shouldbe taken into consideration [43ndash46] One is the complex-ity of multidisciplinary system analysis The other is theinformation exchange of coupling disciplines involved Themultidisciplinary system analysis process is based on itera-tive calculations between coupled disciplines This processrequires the high computational cost How to reduce thecomputational cost and improve the efficiency of systemanalysis is important for the application of MultidisciplinaryDesign and Optimization (MDO) in practical engineeringFurthermore the interaction effect between coupling dis-ciplines complicates the exchange of information in com-plex engineering systems How to organize and managethe interaction information transmission between couplingdisciplines effectively is another important problem whichshould be taken into consideration in practical engineering
MDO is a methodology which can deal with the designproblems of complex coupling engineering systems effec-tively [42 47ndash50] The main motivation of MDO is to drive
Robustness
Reliability ReliabilityProb
abili
ty d
ensit
y fu
nctio
n
Mean Value
Figure 2 Two categories of uncertainty-based design [42]
the performance of an engineering system not only by eachdiscipline but also by their coupling interactions
By introducing MDO strategies into the early stage ofengineering design designers can enhance the performanceof design solutions effectively Also the design cost can bereduced simultaneously Considering these coupling inter-actions in an MDO problem requires sound mathematicalstrategies and their corresponding formulations In generalthe mathematical strategies of MDO can be classified intosingle-level methods and multilevel methods [51 52] Asshown in Figures 1(a) and 1(b) single-level methods have asingle optimizer They utilize the nonhierarchical structuredirectly Compared with single-level methods multilevelmethods introduce a hierarchical structure instead of anonhierarchical structure which is shown in Figure 1(c)There are optimizers at each level of the hierarchical structureBecause a specific MDO method cannot be suitable to all ofthe practical problems universally appropriateMDOmethodshould be chosen to satisfy the industrial requirements
Furthermore to achieve the high reliability and safetyof complex industry systems uncertainties in practical engi-neering should be considered Uncertainties have differenttaxonomies in practical engineering Correspondingly thereare two types of uncertainty-based design reliability-basedand robust-based respectively [42] The connection anddifference between them are illustrated in Figure 2 Thereliability-based design deals with extreme events whichhappen at tails of a probability density function such asfailure of performance catastrophe and so on The robust-based design mainly considers fluctuations of performance
Mathematical Problems in Engineering 3
Cost optimization
Constraint feasibilityRBMDO
Discipline 1
Discipline i
Discipline n
d d X X
Yn∙
Yi∙
Y1∙Y1i
YniYin
Y1n
Yi1
Yn1
s s11
d d X Xs sii
d d X Xs snn
Figure 3 The design for a complex system based on RBMDOstrategy
around the mean value such as degradation deteriorationquality loss
In this study reliability-based design problems are pri-marily considered The general formulation of RBMDO canbe given as
min(d119894ds 120583X119894 120583Xs )
119891 (d119894 ds120583X119894 120583Xs)
st Pr119894 [119892119894 (d119894 dsX119894XsP) le 0] le Φ (minus120573119905)120583Y119894∙ = Y119894∙ (d119894 ds120583X119894 120583Xs
120583Y∙119894)d119871119894 le d119894 le d119880119894 d119871s le ds le d119880s X119871119894 le 120583X119894 le X119880119894 X119871s le 120583Xs
le X119880s Y119871 le 120583Y le Y119880119894 = 1 2 119899
(1)
where 119891(∙) is a cost-type objective function 119892(∙) le 0and Pr[∙] le Φ(minus120573119905) are the inequality constraint andits corresponding probability constraint respectively 120573119905 isthe safety reliability index Φ(minus120573119905) is the allowable failureprobability for 119892(∙) le 0 Φ(∙) is the Cumulative DistributionFunction (CDF) of the standard normal random variablethe subscript ldquosrdquo denotes the sharing design information toall disciplines the subscript ldquo119894rdquo denotes the 119894th disciplinein a complex system d is the input design informationwhich is not accompanied with with uncertainties X andP are the input design information and parameters whichare accompanied with uncertainties 120583 is the mean value ofuncertainty information Y is the coupling information Y∙119894is the input coupling information for the 119894th discipline fromother disciplines whileY119894∙ is the output coupling informationfor the 119894th discipline to other disciplines the superscripts ldquo119871rdquoand ldquo119880rdquo denote the lower and upper bounds of input designinformation respectively 119899 is the number of disciplines
As shown in Figure 3 uncertainties will be propagatedamong coupled disciplines in multidisciplinary systems Ifthe RBMDO problem in (1) is solved directly a triple-loopstrategy will be utilized which is shown in Figure 4 Theouter loop performs the optimization for objective function
to obtain design point the intermediate loop performsthe reliability analysis on the design point the inner loopperforms the multidisciplinary analysis (MDA) between thesubdisciplines
Using the triple-loop strategy the multidisciplinary sys-tem optimization problem requires reliability analysis ineach iterative operation Meanwhile each reliability analysisoperation involves MDA Both of them result in a high com-putational burden To solve this problem ERIM and PMAare introduced in this study The corresponding formulationof CO-ERIM is also proposed to improve the efficiency ofRBMDO
3 The Performance Measure Approach (PMA)Using RIM in Sequential Optimization andReliability Assessment
Because of the existences of probability constraints the PMAusing RIM (PMA-RIM) strategy has been utilized widely inRBMDO to reduce computational cost Researches on thisstrategy mainly include two aspects the modifying formu-lation of probability constraint and the enhanced efficienciesof reliability analysis and optimization algorithms [34]
In this study the limitations of the original PMA-RIM arediscussedThen a PMAbased on Enhanced RI (PMA-ERIM)is proposed here which is on the condition of accepting theapproximate accuracy of the First Order Reliability Method(FORM)
31 The Strategy of RIM In (1) the probability constraint canbe reexpressed using Φminus1
120573119904119894 = minusΦminus1 (119865119892119894 (0)) ge 120573119905 (2)
where 120573119904119894 is the safety reliability index of the 119894th probabilityconstraint The input design information with uncertaintiesin (1) is treated as random variables in this study Thenin RIM the first order safety reliability index 120573119904FORM canbe obtained using FORM This process mainly includes twosteps
First all random variables 119909 of the set of X (119909 isin X)in the X-space are transformed into the standard normaldistribution variable 119906 in the U-space using the Rosenblatttransformation U is a set of the standard normal distributionvariable 119906 119906 isin U The standard normal variable can bedenoted as
119906 = Φminus1 (119865119909 (119909)) (3)
where 119865119909(∙) is the CDF of a random variable 119909 Then theperformance function 119892119894(X) is transformed into the U-spaceas 119892119894(U)
Second an optimization problem formulated as follows issolved
min U2st 119892 (U) = 0 (4)
4 Mathematical Problems in Engineering
Multidisciplinary analysis
Reliability analysis
Optimization loop
Initial design Optimal designOptimization
Design variables Reliability constraints
MDA loop
Reliability analysis loop
Figure 4 The triple-loop strategy of RBMDO [11]
where ∙ 2 is the magnitude of a vector The optimumsolution of (4) on the failure surface (119892(U) = 0) is called MPPulowast119892(U)=0 Also 120573119904FORM = ulowast119892(U)=02 [34 35]32 The Strategy of PMA-RIM Here the probability con-straint in (1) is converted into its equivalent form by 119865minus1119892119894
119866119901119894 = 119865minus1119892119894 [Φ (minus120573119905)] ge 0 (5)
where119866119901119894 is the probabilistic performance measure of the 119894thprobability constraint
If the value 119892119901 ge 0 then Pr(119892(X) le 0) le Φ(minus120573119905) if 119892119901 lt0 Pr(119892(X) le 0) gt Φ(minus120573119905) 119892119901 corresponds to the Φ(minus120573119905)percentile of the CDF of performance function
At first all uncertainty inputs are converted into thestandard normal random inputs in the U-space The firstorder probabilistic performance measure 119866119901FORM can beobtained by solving
min 119866 (U)st U2 = 120573119905 (6)
where the optimal point on the surface U2 = 120573119905 is identifiedas MPP ulowast120573=120573119905 Furthermore 119866119901FORM = 119866(ulowast120573=120573119905)33 The Strategy of PMA-ERIM To improve the efficiencyand accuracy of the original RIM the ERIM is discussed inthis section Recall the statistic description of the failure of aperformance function 119892
Pr (119892 (X) le 0) = 119865119892 (0) = int119892(X)le0
119891X (x) 119889x (7)
Step 1 Using the Rosenblatt transformation (7) is equivalentto
Pr (119892 (U) le 0) = int sdot sdot sdot int119892(11990611199062sdotsdotsdot 119906119899)le0
Step 2 To evaluate the integration more easily the integrandboundary 119892(U) = 0 is approximated FORM utilizes the firstorder Taylor expansion as
119892 (U) asymp 119871119892 (U) = 119892 (ulowast) + [nabla119892 (ulowast)]119879 (U minus ulowast) (9)
where119871119892(U) is the linearized performance function ulowast is theexpansion point 119879 denotes transpose nabla119892(ulowast) is the gradientof 119892 at ulowast
To minimize the accuracy loss the performance functionshould be expanded at MPP MPP can be obtained by (4)
Because at MPP 119892(ulowast) = 0 the performance function islinearized as
119892 (U) asymp 119871119892 (U) = [nabla119892 (ulowast)]119879 (U minus ulowast)= minus [nabla119892 (ulowast)]119879 ulowast + [nabla119892 (ulowast)]119879U (11)
If the gradient nabla119892(ulowast) at ulowast is equal to zero then theperformance function is linearized as 119892(U) asymp 119871119892(U) = 0 Inthis case the linear approximation of the integrand boundarywill cause a large error about the integration because ofthe highly nonlinear character of performance functionTherefore if the above case appears the FORM is not suitableto deal with the problem In the following the case of thegradient of performance function at MPP unequal to zero isdiscussed
Since 119871119892(U) is a linear function of standard normalvariables 119871119892(U) is normally distributed Thus based on (11)the mean value and standard deviation of 119871119892(U) are 120583119871119892 =minus[nabla119892(ulowast)]119879ulowast and 120590119871119892 = nabla119892(ulowast)2 which can be utilized inthe derivation process in (12)
Therefore the probability of failure is calculated as
Pr (119892 (U) le 0) asymp Pr (119871119892 (U) le 0) = Φ(0 minus 120583119871119892120590119871119892 )
At the optimal point ulowast = nabla119892(ulowast)nabla119892(ulowast)2 sdot ulowast2 orulowast = minusnabla119892(ulowast)nabla119892(ulowast)2 sdot ulowast2 Hence the probability offailure is
Pr (119892 (U) le 0)
asymp
Φ(1003817100381710038171003817ulowast10038171003817100381710038172) if ulowast = nabla119892 (ulowast)1003817100381710038171003817nabla119892 (ulowast)10038171003817100381710038172 sdot1003817100381710038171003817ulowast10038171003817100381710038172
Φ(minus 1003817100381710038171003817ulowast10038171003817100381710038172) if ulowast = minus nabla119892 (ulowast)1003817100381710038171003817nabla119892 (ulowast)10038171003817100381710038172 sdot1003817100381710038171003817ulowast10038171003817100381710038172
(13)
From (12) and (13) the difference of judgment on thesatisfaction of probability constraint between RIM and thepractical situation is caused when ulowast = nabla119892(ulowast)nabla119892(ulowast)2 sdotulowast2 Utilizing (11) the probabilities of failure of the perfor-mance functions in (1) can be recalculated
To satisfy the probability constraint from (12) the prob-ability of failure should satisfy
As a well-known multilevel method for MDO the strategy ofCO is suitable for large-scale distributed engineering systemsThe CO algorithm decomposes coordinates and optimizescomplex engineering problems Every discipline in a systemcan enjoy good autonomy regardless of the influence of otherdisciplinesThe consistency betweendisciplines is guaranteedby compatibility constraints attached to the system-leveloptimization problem While the value of compatibility con-straint is obtained through subject-level optimization prob-lem The objectives of subject-level optimization problems
are to minimize the inconsistency between disciplines whilesatisfying the constraints of discipline design
Using CO the RBMDO problem in (1) is converted intothe system-level and subject-level optimization problemsThe formulation of the optimization problem in system levelis
st 119869119894= (d1015840119894 minus d119894)2 + (d1015840s minus ds)2+ (1205831015840X119894 minus 120583X119894)2 + (1205831015840Xs
minus 120583Xs)2
+ (1205831015840Y∙119894 minus 120583Y∙119894)2 le 120576119894 = 1 2 119899
(19)
where 119869119894 is the compatibility constraints d1015840119894 d1015840s1205831015840X119894 1205831015840Xs
and 1205831015840Y∙119894 are the design variables at the system level Theformulation of the optimization problem in subject-leveloptimization problems is
119869119894= (d1015840119894 minus d119894)2 + (d1015840s minus ds)2+ (1205831015840X119894 minus 120583X119894)2 + (1205831015840Xs
minus 120583Xs)2
+ (1205831015840Y∙119894 minus 120583Y∙119894)2st Pr119894 [119892119894 (d119894 dsX119894XsP) le 0]
le Φ (minus120573119905)120583Y119894∙ = Y119894∙ (d119894 ds120583X119894 120583Xs
120583Y∙119894)d119871119894 le d119894 le d119880119894 d119871s le ds le d119880s X119871119894 le 120583X119894 le X119880119894 X119871s le 120583Xs
le X119880s Y119871 le 120583Y le Y119880119894 = 1 2 119899
(20)
The corresponding bilevel strategy of CO is also shown inFigure 5
Considering uncertainties in practical engineering alldiscipline probability reliability constraints in (20) are con-verted into the corresponding RIM-based reliability con-straints These RIM-based reliability constraints only con-sider the performance reliability at MPP which can reducethe cost of reliability analysis and improve the computational
6 Mathematical Problems in Engineering
Table 1 The design information of speed reducer
Variables Lower and upper bound Distribution Mean Standard deviationgear face width 1199091 (cm) [26 36] - - -teeth module 1199092 (cm) [03 10] - - -number of teeth of pinion 1199093 [17 28] - - -distance between bearings 1 1199094 (cm) [73 83] Normal 1205831199094 0011205831199094distance between bearings 2 1199095 (cm) [73 83] Normal 1205831199095 0011205831199095diameter of shaft 1 1199096 (cm) [29 39] Normal 1205831199096 0011205831199096diameter of shaft 2 1199097 (cm) [5 55] Normal 1205831199097 0011205831199097
System optimization problem in Eq (19)
The 1st discipline optimizationproblem in Eq (20)
The 1st discipline analysis
Theproblem in Eq (20)
thi
XＭX
1
Y∙1Y
1∙
XＭX1
Y∙1Y1∙
XＭX
i
Y∙iY
i∙
XＭXi
Y∙iYi∙
XＭX1Y∙1 Y1∙XＭXi Y∙i Yi∙
discipline optimization
The thi discipline analysis
Figure 5 The bilevel strategy of CO
efficiency of RBMDO The formulation of the subject-leveloptimization problem in the U-space using RIM-based relia-bility constraints can be denoted as
transformation of Xd119871119894 le d119894 le d119880119894 d119871s le ds le d119880s X119871119894 le 120583X119894 le X119880119894 X119871s le 120583Xs
le X119880s
Y119871 le 120583Y le Y119880119894 = 1 2 119899
(21)
The detail information of CO-ERIM is as follows
Step 1 Input the original design information the cyclenumber 119896 = 0Step 2 Solve the system-level optimization problem in (19)During this process d1015840119894 d
1015840s 1205831015840X119894 1205831015840Xs and 1205831015840Y∙119894 are treated as the
design parameters
Step 3 Transform random variables in X-space into randomvariables in U-space using the Rosenblatt transformation
Step 4 Solve the subject-level optimization problems in (21)Then send the design solutions to the system level
Step 5 Obtain the value of 119869119894 119894 = 1 2 119899 If 119869119894 le 120576 andthe difference between the objective function values of twoconsecutive iterations is not more than a small number in theoptimization iteration process carry out Step 6 Otherwise119896 = 119896 + 1 and carry out Step 2
Step 6 Stop and output the design solutions
The flowchart of CO-ERIM is shown in Figure 6
5 Example
Speed reducers are generally used in low-speed high-torquetransmission equipment In this study a speed reducerRBMDO problem is introduced to illustrate the utilization ofthe proposedmethodThere are seven design variables in thisexample which is listed in Table 1 Twenty-five constraints areintroduced to ensure that the design solutions can satisfy thestrength stiffness and space requirements The optimizationobject is to minimize the overall weight Further informationcan be obtained in [14 43]
There are three disciplines in this RBMDO problemBearing-Shaft 1 Bearing-Shaft 2 and Gears which is shownin Figure 7The CO-ERIM strategy for this problem is shownin Figure 8 where 120573119905 = 207 Φ(minus120573119905) = 002 and 120576 = 0001
To illustrate the accuracy of design solutions the MonteCarlo Simulation (MCS) method is also introduced here as
Mathematical Problems in Engineering 7
Solve the optimization problemin Eq (19) at system level
Solve the discipline optimizationproblems in Eq (21) at subsystem level
Start
No
Yes
End
The ith discipline analysis
k = 0
Rosenblatt transformation
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1) ９
i∙(kminus1)
８Ｍ(kminus1) ８i(kminus1) ９∙i(kminus1) ９i∙(kminus1)
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1)
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1) ９i∙(kminus1)
k = k + 1
８Ｍ(kminus1) ８i(kminus1)
９∙i(kminus1)
＜ＮＣＨ ＮＢ ＰＦＯ Ｉ
Ji i = 1 2 n
８Ｍ(kminus1) ８i(kminus1) ９∙i(kminus1) ９i∙(kminus1)
(Ｃ = 1simn)
Ji le
Figure 6 The flowchart of CO-ERIM
Table 2 Optimization results of the reducer design
the reference The software Isight is utilized in the compu-tations of optimization The solutions from CO-ERIM arecompared with the ones from original PMA-RIM based COand MCS based CO which is listed in Table 2 From thecomparison of solutions the design results from ERIM arecloser to the design results from MCS Furthermore the
calculation time of ERIM is 17min23s and the calculationtime of RIM is 25min17s which means the proposed methodenjoys higher computational efficiency
6 Conclusions
In this study the efficiency problem of RBMDO is stud-ied The RIM strategy is reviewed and the correspondingalgorithm of ERIM is discussed in detail Furthermore theCO-ERIM strategy is proposed including its formulationand procedure In CO-ERIM the concurrent design idea isadapting to the development of modern engineering systemsCompatibility constraints are introduced into subsystem-level and system-level optimization problems respectivelyThe consistency between different disciplines can be guaran-teed when the RBMDO solutions are obtainedThe introduc-tion of ERIM reduces the computational burden of reliabilityanalysis during optimization iteration process Under thecondition that the first order method is acceptable thereliability analysis accuracy of the proposedmethod is similar
Figure 8 The MDO strategy for the speed reducer problem
to the accuracy ofMCSThe speed reducer example illustratesthe effectiveness of the proposed method
Data Availability
All data used to support the findings of this study are includedwithin the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thesupport from theNational Natural Science Foundation ofChina (Grant No 51605080 and No 11672070) the SichuanScience and Technology Program (Grant No 2019YFG0350and No 2019YFG0348) the China Postdoctoral ScienceFoundation (Grant No 2015M580780 and 2017T100685) andthe National Defense Pre-Research Foundation of China
(Grant No 6140244010216DZ02001) is gratefully acknowl-edged The authors also acknowledge the Portuguese Sci-ence Foundation (FCT) for the financial support throughthe postdoctoral grant SFRHBPD1078252015 as well asthe funding of Projects POCI-01-0145-FEDER-007457 andUIDECI047082019 - CONSTRUCT - Institute of RampDIn Structures and Construction funded by FEDER fundsthrough COMPETE2020 (POCI) and by national funds(PIDDAC) through Portuguese Science Foundation (FCT)
References
[1] L Li H Wan W Gao F Tong and H Li ldquoReliabilitybased multidisciplinary design optimization of cooling turbineblade considering uncertainty data statisticsrdquo Structural andMultidisciplinary Optimization vol 59 no 2 pp 1ndash15 2018
[2] Z L Huang Y S Zhou C Jiang J Zheng and X HanldquoReliability-based multidisciplinary design optimization usingincremental shifting vector strategy and its application inelectronic product designrdquo Acta Mechanica Sinica vol 34 no2 pp 285ndash302 2018
[3] H Xu X WangW Li M Li S Zhang and C Hu ldquoReliability-based multidisciplinary design optimization under correlated
Mathematical Problems in Engineering 9
uncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 7360615 12 pages 2017
[4] D Meng S Yang Y Zhang and S-P Zhu ldquoStructural relia-bility analysis and uncertainties-based collaborative design andoptimization of turbine blades using surrogate modelrdquo Fatigueamp Fracture of Engineering Materials amp Structures pp 1ndash9 2018httpsdoiorg101111ffe12906
[5] F Yang Z Yue L Li and D Guan ldquoHybrid reliability-based multidisciplinary design optimization with random andinterval variablesrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 232 no 1pp 52ndash64 2018
[6] L Wang C Xiong and Y Yang ldquoA novel methodology ofreliability-based multidisciplinary design optimization underhybrid interval and fuzzy uncertaintiesrdquo Computer MethodsApplied Mechanics and Engineering vol 337 pp 439ndash457 2018
[7] L Wang C Xiong J Hu X Wang and Z Qiu ldquoSequentialmultidisciplinary design optimization and reliability analysisunder interval uncertaintyrdquo Aerospace Science and Technologyvol 80 pp 508ndash519 2018
[8] J Zhang M Xiao L Gao H Qiu and Z Yang ldquoAn improvedtwo-stage framework of evidence-based design optimizationrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1673ndash1693 2018
[9] H Xu W Li L Xing and S-P Zhu ldquoMultidisciplinary designoptimization under correlated uncertaintiesrdquo Concurrent Engi-neering Research and Applications vol 25 no 3 pp 262ndash2752017
[10] X Wang R Wang L Wang X Chen and X Geng ldquoAnefficient single-loop strategy for reliability-based multidisci-plinarydesign optimizationunder non-probabilistic set theoryrdquoAerospace Science and Technology vol 73 pp 148ndash163 2018
[11] X P Du J Guo and H Beeram ldquoSequential optimizationand reliability assessment for multidisciplinary systems designrdquoStructural andMultidisciplinary Optimization vol 35 no 2 pp117ndash130 2008
[12] D Meng Y-F Li H-Z Huang Z Wang and Y LiuldquoReliability-based multidisciplinary design optimization usingsubset simulation analysis and its application in the hydraulictransmission mechanism designrdquo Journal of Mechanical Designvol 137 no 5 Article ID 051402 2015
[13] S Yu Z Wang and D Meng ldquoTime-variant reliability assess-ment for multiple failure modes and temporal parametersrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1705ndash1717 2018
[14] D Meng H-Z Huang Z Wang N-C Xiao and X-L ZhangldquoMean-value first-order saddlepoint approximation based col-laborative optimization for multidisciplinary problems underaleatory uncertaintyrdquo Journal of Mechanical Science and Tech-nology vol 28 no 10 pp 3925ndash3935 2014
[15] S-P Zhu Q Liu Q Lei and Q Wang ldquoProbabilistic fatiguelife prediction and reliability assessment of a high pressureturbine disc considering load variationsrdquo International Journalof Damage Mechanics vol 27 no 10 pp 1569ndash1588 2018
[16] D Liao S-P Zhu J A F O Correia A M P De Jesus andR Calcada ldquoComputational framework for multiaxial fatiguelife prediction of compressor discs considering notch effectsrdquoEngineering Fracture Mechanics vol 202 pp 423ndash435 2018
[17] D Meng M Liu S Yang H Zhang and R Ding ldquoAfluid-structure analysis approach and its application in theuncertainty-based multidisciplinary design and optimization
for bladesrdquo Advances in Mechanical Engineering vol 10 no 6pp 1ndash7 2018
[18] Z Meng G Li B P Wang and P Hao ldquoA hybrid chaos controlapproach of the performance measure functions for reliability-based design optimizationrdquo Computers Structures vol 146 pp32ndash43 2015
[19] X Li H Qiu Z Chen L Gao and X Shao ldquoA local Krigingapproximation method using MPP for reliability-based designoptimizationrdquo Computers amp Structures vol 162 pp 102ndash1152016
[20] S-P Zhu Q Liu W Peng and X-C Zhang ldquoComputational-experimental approaches for fatigue reliability assessment ofturbine bladed disksrdquo International Journal of Mechanical Sci-ences vol 142-143 pp 502ndash517 2018
[21] Z Chen H Qiu L Gao and P Li ldquoAn optimal shiftingvector approach for efficient probabilistic designrdquo Structuraland Multidisciplinary Optimization vol 47 no 6 pp 905ndash9202013
[22] W Deng X Lu and Y Deng ldquoEvidential model validationunder epistemic uncertaintyrdquo Mathematical Problems in Engi-neering vol 2018 Article ID 6789635 11 pages 2018
[23] D Meng H Zhang and T Huang ldquoA concurrent reliabilityoptimization procedure in the earlier design phases of complexengineering systems under epistemic uncertaintiesrdquo Advancesin Mechanical Engineering vol 8 no 10 pp 1ndash8 2016
[24] G Li Z Meng and H Hu ldquoAn adaptive hybrid approach forreliability-based design optimizationrdquo Structural and Multidis-ciplinary Optimization vol 51 no 5 pp 1051ndash1065 2015
[25] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018
[26] M A Valdebenito and G I Schueller ldquoA survey on approachesfor reliability-based optimizationrdquo Structural and Multidisci-plinary Optimization vol 42 no 5 pp 645ndash663 2010
[27] Y Aoues and A Chateauneuf ldquoBenchmark study of numericalmethods for reliability-based design optimizationrdquo Structuraland Multidisciplinary Optimization vol 41 no 2 pp 277ndash2942010
[28] J Ching and W-C Hsu ldquoTransforming reliability limit-stateconstraints into deterministic limit-state constraintsrdquo StructuralSafety vol 30 no 1 pp 11ndash33 2008
[29] TM Cho and B C Lee ldquoReliability-based design optimizationusing convex linearization and sequential optimization andreliability assessment methodrdquo Structural Safety vol 33 no 1pp 42ndash50 2011
[30] J Liang Z P Mourelatos and J Tu ldquoA single-loop method forreliability-based design optimisationrdquo International Journal ofProduct Development vol 5 no 1-2 pp 76ndash92 2008
[31] S Shan and G G Wang ldquoReliable design space and completesingle-loop reliability-based design optimizationrdquo ReliabilityEngineering amp System Safety vol 93 no 8 pp 1218ndash1230 2008
[32] P Yi G Cheng and L Jiang ldquoA sequential approximateprogramming strategy for performance-measure-based proba-bilistic structural design optimizationrdquo Structural Safety vol 30no 2 pp 91ndash109 2008
[33] Z M Yaseen and B Keshtegar ldquoLimited descent-based meanvalue method for inverse reliability analysisrdquo Engineering withComputers pp 1ndash13 2018
[34] B D Youn K K Choi and Y H Park ldquoHybrid analysis methodfor reliability-based design optimizationrdquo Journal of MechanicalDesign vol 125 no 2 pp 221ndash232 2003
10 Mathematical Problems in Engineering
[35] B D Youn K K Choi and L Du ldquoEnriched performancemeasure approach for reliability-based design optimizationrdquoAIAA Journal vol 43 no 4 pp 874ndash884 2005
[36] B D Youn and K K Choi ldquoSelecting probabilistic approachesfor realiability-based design optimizationrdquo AIAA Journal vol42 no 1 pp 124ndash131 2004
[37] S P Zhu Q Liu J Zhou and Z Y Yu ldquoFatigue reliabilityassessment of turbine discs under multi-source uncertaintiesrdquoFatigue Fracture of Engineering Materials Structures vol 41 no6 pp 1291ndash1305 2018
[38] H Li R Li H Li and R Yuan ldquoReliability modeling of mul-tiple performance based on degradation values distributionrdquoAdvances in Mechanical Engineering vol 8 no 10 Article ID168781401667375 2016
[39] B D Youn K K Choi and L Du ldquoAdaptive probability analysisusing an enhanced hybrid mean value methodrdquo Structural andMultidisciplinary Optimization vol 29 no 2 pp 134ndash148 2005
[40] M Ito N H Kim and N Kogiso ldquoConservative reliabilityindex for epistemic uncertainty in reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 5 pp 1919ndash1935 2018
[41] Z Meng D Yang H Zhou and B P Wang ldquoConvergencecontrol of single loop approach for reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 3 pp 1079ndash1091 2018
[42] W Yao X Q Chen W C Luo M van Tooren and J GuoldquoReview of uncertainty-based multidisciplinary design opti-mization methods for aerospace vehiclesrdquo Progress in AerospaceSciences vol 47 no 6 pp 450ndash479 2011
[43] R Yuan D Meng and H Li ldquoMultidisciplinary reliabilitydesign optimizationusing an enhanced saddlepoint approxima-tion in the framework of sequential optimization and reliabilityanalysisrdquo Proceedings of the Institution of Mechanical EngineersPart O Journal of Risk and Reliability vol 230 no 6 pp 570ndash578 2016
[44] D Meng X Zhang Y Yang H Xu and H Huang ldquoInteractionbalance optimization in multidisciplinary design optimizationproblemsrdquo Concurrent Engineering Research and Applicationsvol 24 no 1 pp 48ndash57 2016
[45] R Yuan and H Li ldquoA multidisciplinary coupling relationshipcoordination algorithm using the hierarchical control methodsof complex systems and its application in multidisciplinarydesign optimizationrdquo Advances in Mechanical Engineering vol9 no 1 pp 1ndash11 2016
[46] D Meng X Zhang H-Z Huang Z Wang and H Xu ldquoInter-action prediction optimization in multidisciplinary designoptimization problemsrdquoThe Scientific World Journal vol 2014Article ID 698453 7 pages 2014
[47] J R R A Martins and A B Lambe ldquoMultidisciplinary designoptimization a survey of architecturesrdquo AIAA Journal vol 51no 9 pp 2049ndash2075 2013
[48] Y Li and Y Deng ldquoGeneralized ordered propositions fusionbased on belief entropyrdquo International Journal of ComputersCommunications amp Control vol 13 no 5 pp 792ndash807 2018
[49] X Su S Mahadevan W Han and Y Deng ldquoCombiningdependent bodies of evidencerdquo Applied Intelligence vol 44 no3 pp 634ndash644 2016
[50] S-P Zhu H-Z Huang W Peng H-K Wang and S Mahade-van ldquoProbabilistic Physics of Failure-based framework forfatigue life prediction of aircraft gas turbine discs under uncer-taintyrdquoReliability Engineering amp System Safety vol 146 pp 1ndash122016
[51] S I Yi J K Shin andG J Park ldquoComparison ofMDOmethodswith mathematical examplesrdquo Structural and MultidisciplinaryOptimization vol 35 no 5 pp 391ndash402 2008
[52] H XuW Li M Li C Hu S Zhang and XWang ldquoMultidisci-plinary robust design optimization based on time-varying sen-sitivity analysisrdquo Journal of Mechanical Science and Technologyvol 32 no 3 pp 1195ndash1207 2018
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2 Mathematical Problems in Engineering
Optimizer
Analysis at subsystem level
Design at system level
(a)
Optimizer
Design at subsystem level
Analysis at system level
(b)
Optimizer
Design at subsystem level
Design at system level
Optimizer
(c)
Figure 1 The single-level methods and multilevel methods for MDO
approximation expansion is introduced to convert nonlinearprobability constraint to the equivalent linear constraintMPP is the expansion point at which the accuracy loss ofreliability analysis due to approximation can be minimizedHowever in somepractical optimization processesMPPmaybe collinear and RIMhas the same direction with the gradientof performance function at MPP It will result in the lowefficiency of the original RIM To tackle the above challengesan Enhanced RIM (ERIM) is proposed hereThe formulationof CO using ERIM (CO-ERIM) is also proposed to solveRBMDO problems
The rest of this study will be given as follows In Section 2the general formulation of RBMDO is given Also theuncertainty information in practical engineering is discussedand the strategy of RBMDO is briefly reviewedThe RIM andthe Performance Measure Approach (PMA) are introducedin detail in Section 3The proposed ERIM is also discussed inthis part The CO-ERIM is proposed in Section 4 includingthe formulation and procedure An RBMDO problem of thespeed reducer is given in Section 5 to show the efficiencyand accuracy of CO-ERIM The conclusions are given inSection 6
2 Brief Review of RBMDO
When dealing with the design problems of complex engi-neering there are two aspects of challenges which shouldbe taken into consideration [43ndash46] One is the complex-ity of multidisciplinary system analysis The other is theinformation exchange of coupling disciplines involved Themultidisciplinary system analysis process is based on itera-tive calculations between coupled disciplines This processrequires the high computational cost How to reduce thecomputational cost and improve the efficiency of systemanalysis is important for the application of MultidisciplinaryDesign and Optimization (MDO) in practical engineeringFurthermore the interaction effect between coupling dis-ciplines complicates the exchange of information in com-plex engineering systems How to organize and managethe interaction information transmission between couplingdisciplines effectively is another important problem whichshould be taken into consideration in practical engineering
MDO is a methodology which can deal with the designproblems of complex coupling engineering systems effec-tively [42 47ndash50] The main motivation of MDO is to drive
Robustness
Reliability ReliabilityProb
abili
ty d
ensit
y fu
nctio
n
Mean Value
Figure 2 Two categories of uncertainty-based design [42]
the performance of an engineering system not only by eachdiscipline but also by their coupling interactions
By introducing MDO strategies into the early stage ofengineering design designers can enhance the performanceof design solutions effectively Also the design cost can bereduced simultaneously Considering these coupling inter-actions in an MDO problem requires sound mathematicalstrategies and their corresponding formulations In generalthe mathematical strategies of MDO can be classified intosingle-level methods and multilevel methods [51 52] Asshown in Figures 1(a) and 1(b) single-level methods have asingle optimizer They utilize the nonhierarchical structuredirectly Compared with single-level methods multilevelmethods introduce a hierarchical structure instead of anonhierarchical structure which is shown in Figure 1(c)There are optimizers at each level of the hierarchical structureBecause a specific MDO method cannot be suitable to all ofthe practical problems universally appropriateMDOmethodshould be chosen to satisfy the industrial requirements
Furthermore to achieve the high reliability and safetyof complex industry systems uncertainties in practical engi-neering should be considered Uncertainties have differenttaxonomies in practical engineering Correspondingly thereare two types of uncertainty-based design reliability-basedand robust-based respectively [42] The connection anddifference between them are illustrated in Figure 2 Thereliability-based design deals with extreme events whichhappen at tails of a probability density function such asfailure of performance catastrophe and so on The robust-based design mainly considers fluctuations of performance
Mathematical Problems in Engineering 3
Cost optimization
Constraint feasibilityRBMDO
Discipline 1
Discipline i
Discipline n
d d X X
Yn∙
Yi∙
Y1∙Y1i
YniYin
Y1n
Yi1
Yn1
s s11
d d X Xs sii
d d X Xs snn
Figure 3 The design for a complex system based on RBMDOstrategy
around the mean value such as degradation deteriorationquality loss
In this study reliability-based design problems are pri-marily considered The general formulation of RBMDO canbe given as
min(d119894ds 120583X119894 120583Xs )
119891 (d119894 ds120583X119894 120583Xs)
st Pr119894 [119892119894 (d119894 dsX119894XsP) le 0] le Φ (minus120573119905)120583Y119894∙ = Y119894∙ (d119894 ds120583X119894 120583Xs
120583Y∙119894)d119871119894 le d119894 le d119880119894 d119871s le ds le d119880s X119871119894 le 120583X119894 le X119880119894 X119871s le 120583Xs
le X119880s Y119871 le 120583Y le Y119880119894 = 1 2 119899
(1)
where 119891(∙) is a cost-type objective function 119892(∙) le 0and Pr[∙] le Φ(minus120573119905) are the inequality constraint andits corresponding probability constraint respectively 120573119905 isthe safety reliability index Φ(minus120573119905) is the allowable failureprobability for 119892(∙) le 0 Φ(∙) is the Cumulative DistributionFunction (CDF) of the standard normal random variablethe subscript ldquosrdquo denotes the sharing design information toall disciplines the subscript ldquo119894rdquo denotes the 119894th disciplinein a complex system d is the input design informationwhich is not accompanied with with uncertainties X andP are the input design information and parameters whichare accompanied with uncertainties 120583 is the mean value ofuncertainty information Y is the coupling information Y∙119894is the input coupling information for the 119894th discipline fromother disciplines whileY119894∙ is the output coupling informationfor the 119894th discipline to other disciplines the superscripts ldquo119871rdquoand ldquo119880rdquo denote the lower and upper bounds of input designinformation respectively 119899 is the number of disciplines
As shown in Figure 3 uncertainties will be propagatedamong coupled disciplines in multidisciplinary systems Ifthe RBMDO problem in (1) is solved directly a triple-loopstrategy will be utilized which is shown in Figure 4 Theouter loop performs the optimization for objective function
to obtain design point the intermediate loop performsthe reliability analysis on the design point the inner loopperforms the multidisciplinary analysis (MDA) between thesubdisciplines
Using the triple-loop strategy the multidisciplinary sys-tem optimization problem requires reliability analysis ineach iterative operation Meanwhile each reliability analysisoperation involves MDA Both of them result in a high com-putational burden To solve this problem ERIM and PMAare introduced in this study The corresponding formulationof CO-ERIM is also proposed to improve the efficiency ofRBMDO
3 The Performance Measure Approach (PMA)Using RIM in Sequential Optimization andReliability Assessment
Because of the existences of probability constraints the PMAusing RIM (PMA-RIM) strategy has been utilized widely inRBMDO to reduce computational cost Researches on thisstrategy mainly include two aspects the modifying formu-lation of probability constraint and the enhanced efficienciesof reliability analysis and optimization algorithms [34]
In this study the limitations of the original PMA-RIM arediscussedThen a PMAbased on Enhanced RI (PMA-ERIM)is proposed here which is on the condition of accepting theapproximate accuracy of the First Order Reliability Method(FORM)
31 The Strategy of RIM In (1) the probability constraint canbe reexpressed using Φminus1
120573119904119894 = minusΦminus1 (119865119892119894 (0)) ge 120573119905 (2)
where 120573119904119894 is the safety reliability index of the 119894th probabilityconstraint The input design information with uncertaintiesin (1) is treated as random variables in this study Thenin RIM the first order safety reliability index 120573119904FORM canbe obtained using FORM This process mainly includes twosteps
First all random variables 119909 of the set of X (119909 isin X)in the X-space are transformed into the standard normaldistribution variable 119906 in the U-space using the Rosenblatttransformation U is a set of the standard normal distributionvariable 119906 119906 isin U The standard normal variable can bedenoted as
119906 = Φminus1 (119865119909 (119909)) (3)
where 119865119909(∙) is the CDF of a random variable 119909 Then theperformance function 119892119894(X) is transformed into the U-spaceas 119892119894(U)
Second an optimization problem formulated as follows issolved
min U2st 119892 (U) = 0 (4)
4 Mathematical Problems in Engineering
Multidisciplinary analysis
Reliability analysis
Optimization loop
Initial design Optimal designOptimization
Design variables Reliability constraints
MDA loop
Reliability analysis loop
Figure 4 The triple-loop strategy of RBMDO [11]
where ∙ 2 is the magnitude of a vector The optimumsolution of (4) on the failure surface (119892(U) = 0) is called MPPulowast119892(U)=0 Also 120573119904FORM = ulowast119892(U)=02 [34 35]32 The Strategy of PMA-RIM Here the probability con-straint in (1) is converted into its equivalent form by 119865minus1119892119894
119866119901119894 = 119865minus1119892119894 [Φ (minus120573119905)] ge 0 (5)
where119866119901119894 is the probabilistic performance measure of the 119894thprobability constraint
If the value 119892119901 ge 0 then Pr(119892(X) le 0) le Φ(minus120573119905) if 119892119901 lt0 Pr(119892(X) le 0) gt Φ(minus120573119905) 119892119901 corresponds to the Φ(minus120573119905)percentile of the CDF of performance function
At first all uncertainty inputs are converted into thestandard normal random inputs in the U-space The firstorder probabilistic performance measure 119866119901FORM can beobtained by solving
min 119866 (U)st U2 = 120573119905 (6)
where the optimal point on the surface U2 = 120573119905 is identifiedas MPP ulowast120573=120573119905 Furthermore 119866119901FORM = 119866(ulowast120573=120573119905)33 The Strategy of PMA-ERIM To improve the efficiencyand accuracy of the original RIM the ERIM is discussed inthis section Recall the statistic description of the failure of aperformance function 119892
Pr (119892 (X) le 0) = 119865119892 (0) = int119892(X)le0
119891X (x) 119889x (7)
Step 1 Using the Rosenblatt transformation (7) is equivalentto
Pr (119892 (U) le 0) = int sdot sdot sdot int119892(11990611199062sdotsdotsdot 119906119899)le0
Step 2 To evaluate the integration more easily the integrandboundary 119892(U) = 0 is approximated FORM utilizes the firstorder Taylor expansion as
119892 (U) asymp 119871119892 (U) = 119892 (ulowast) + [nabla119892 (ulowast)]119879 (U minus ulowast) (9)
where119871119892(U) is the linearized performance function ulowast is theexpansion point 119879 denotes transpose nabla119892(ulowast) is the gradientof 119892 at ulowast
To minimize the accuracy loss the performance functionshould be expanded at MPP MPP can be obtained by (4)
Because at MPP 119892(ulowast) = 0 the performance function islinearized as
119892 (U) asymp 119871119892 (U) = [nabla119892 (ulowast)]119879 (U minus ulowast)= minus [nabla119892 (ulowast)]119879 ulowast + [nabla119892 (ulowast)]119879U (11)
If the gradient nabla119892(ulowast) at ulowast is equal to zero then theperformance function is linearized as 119892(U) asymp 119871119892(U) = 0 Inthis case the linear approximation of the integrand boundarywill cause a large error about the integration because ofthe highly nonlinear character of performance functionTherefore if the above case appears the FORM is not suitableto deal with the problem In the following the case of thegradient of performance function at MPP unequal to zero isdiscussed
Since 119871119892(U) is a linear function of standard normalvariables 119871119892(U) is normally distributed Thus based on (11)the mean value and standard deviation of 119871119892(U) are 120583119871119892 =minus[nabla119892(ulowast)]119879ulowast and 120590119871119892 = nabla119892(ulowast)2 which can be utilized inthe derivation process in (12)
Therefore the probability of failure is calculated as
Pr (119892 (U) le 0) asymp Pr (119871119892 (U) le 0) = Φ(0 minus 120583119871119892120590119871119892 )
At the optimal point ulowast = nabla119892(ulowast)nabla119892(ulowast)2 sdot ulowast2 orulowast = minusnabla119892(ulowast)nabla119892(ulowast)2 sdot ulowast2 Hence the probability offailure is
Pr (119892 (U) le 0)
asymp
Φ(1003817100381710038171003817ulowast10038171003817100381710038172) if ulowast = nabla119892 (ulowast)1003817100381710038171003817nabla119892 (ulowast)10038171003817100381710038172 sdot1003817100381710038171003817ulowast10038171003817100381710038172
Φ(minus 1003817100381710038171003817ulowast10038171003817100381710038172) if ulowast = minus nabla119892 (ulowast)1003817100381710038171003817nabla119892 (ulowast)10038171003817100381710038172 sdot1003817100381710038171003817ulowast10038171003817100381710038172
(13)
From (12) and (13) the difference of judgment on thesatisfaction of probability constraint between RIM and thepractical situation is caused when ulowast = nabla119892(ulowast)nabla119892(ulowast)2 sdotulowast2 Utilizing (11) the probabilities of failure of the perfor-mance functions in (1) can be recalculated
To satisfy the probability constraint from (12) the prob-ability of failure should satisfy
As a well-known multilevel method for MDO the strategy ofCO is suitable for large-scale distributed engineering systemsThe CO algorithm decomposes coordinates and optimizescomplex engineering problems Every discipline in a systemcan enjoy good autonomy regardless of the influence of otherdisciplinesThe consistency betweendisciplines is guaranteedby compatibility constraints attached to the system-leveloptimization problem While the value of compatibility con-straint is obtained through subject-level optimization prob-lem The objectives of subject-level optimization problems
are to minimize the inconsistency between disciplines whilesatisfying the constraints of discipline design
Using CO the RBMDO problem in (1) is converted intothe system-level and subject-level optimization problemsThe formulation of the optimization problem in system levelis
st 119869119894= (d1015840119894 minus d119894)2 + (d1015840s minus ds)2+ (1205831015840X119894 minus 120583X119894)2 + (1205831015840Xs
minus 120583Xs)2
+ (1205831015840Y∙119894 minus 120583Y∙119894)2 le 120576119894 = 1 2 119899
(19)
where 119869119894 is the compatibility constraints d1015840119894 d1015840s1205831015840X119894 1205831015840Xs
and 1205831015840Y∙119894 are the design variables at the system level Theformulation of the optimization problem in subject-leveloptimization problems is
119869119894= (d1015840119894 minus d119894)2 + (d1015840s minus ds)2+ (1205831015840X119894 minus 120583X119894)2 + (1205831015840Xs
minus 120583Xs)2
+ (1205831015840Y∙119894 minus 120583Y∙119894)2st Pr119894 [119892119894 (d119894 dsX119894XsP) le 0]
le Φ (minus120573119905)120583Y119894∙ = Y119894∙ (d119894 ds120583X119894 120583Xs
120583Y∙119894)d119871119894 le d119894 le d119880119894 d119871s le ds le d119880s X119871119894 le 120583X119894 le X119880119894 X119871s le 120583Xs
le X119880s Y119871 le 120583Y le Y119880119894 = 1 2 119899
(20)
The corresponding bilevel strategy of CO is also shown inFigure 5
Considering uncertainties in practical engineering alldiscipline probability reliability constraints in (20) are con-verted into the corresponding RIM-based reliability con-straints These RIM-based reliability constraints only con-sider the performance reliability at MPP which can reducethe cost of reliability analysis and improve the computational
6 Mathematical Problems in Engineering
Table 1 The design information of speed reducer
Variables Lower and upper bound Distribution Mean Standard deviationgear face width 1199091 (cm) [26 36] - - -teeth module 1199092 (cm) [03 10] - - -number of teeth of pinion 1199093 [17 28] - - -distance between bearings 1 1199094 (cm) [73 83] Normal 1205831199094 0011205831199094distance between bearings 2 1199095 (cm) [73 83] Normal 1205831199095 0011205831199095diameter of shaft 1 1199096 (cm) [29 39] Normal 1205831199096 0011205831199096diameter of shaft 2 1199097 (cm) [5 55] Normal 1205831199097 0011205831199097
System optimization problem in Eq (19)
The 1st discipline optimizationproblem in Eq (20)
The 1st discipline analysis
Theproblem in Eq (20)
thi
XＭX
1
Y∙1Y
1∙
XＭX1
Y∙1Y1∙
XＭX
i
Y∙iY
i∙
XＭXi
Y∙iYi∙
XＭX1Y∙1 Y1∙XＭXi Y∙i Yi∙
discipline optimization
The thi discipline analysis
Figure 5 The bilevel strategy of CO
efficiency of RBMDO The formulation of the subject-leveloptimization problem in the U-space using RIM-based relia-bility constraints can be denoted as
transformation of Xd119871119894 le d119894 le d119880119894 d119871s le ds le d119880s X119871119894 le 120583X119894 le X119880119894 X119871s le 120583Xs
le X119880s
Y119871 le 120583Y le Y119880119894 = 1 2 119899
(21)
The detail information of CO-ERIM is as follows
Step 1 Input the original design information the cyclenumber 119896 = 0Step 2 Solve the system-level optimization problem in (19)During this process d1015840119894 d
1015840s 1205831015840X119894 1205831015840Xs and 1205831015840Y∙119894 are treated as the
design parameters
Step 3 Transform random variables in X-space into randomvariables in U-space using the Rosenblatt transformation
Step 4 Solve the subject-level optimization problems in (21)Then send the design solutions to the system level
Step 5 Obtain the value of 119869119894 119894 = 1 2 119899 If 119869119894 le 120576 andthe difference between the objective function values of twoconsecutive iterations is not more than a small number in theoptimization iteration process carry out Step 6 Otherwise119896 = 119896 + 1 and carry out Step 2
Step 6 Stop and output the design solutions
The flowchart of CO-ERIM is shown in Figure 6
5 Example
Speed reducers are generally used in low-speed high-torquetransmission equipment In this study a speed reducerRBMDO problem is introduced to illustrate the utilization ofthe proposedmethodThere are seven design variables in thisexample which is listed in Table 1 Twenty-five constraints areintroduced to ensure that the design solutions can satisfy thestrength stiffness and space requirements The optimizationobject is to minimize the overall weight Further informationcan be obtained in [14 43]
There are three disciplines in this RBMDO problemBearing-Shaft 1 Bearing-Shaft 2 and Gears which is shownin Figure 7The CO-ERIM strategy for this problem is shownin Figure 8 where 120573119905 = 207 Φ(minus120573119905) = 002 and 120576 = 0001
To illustrate the accuracy of design solutions the MonteCarlo Simulation (MCS) method is also introduced here as
Mathematical Problems in Engineering 7
Solve the optimization problemin Eq (19) at system level
Solve the discipline optimizationproblems in Eq (21) at subsystem level
Start
No
Yes
End
The ith discipline analysis
k = 0
Rosenblatt transformation
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1) ９
i∙(kminus1)
８Ｍ(kminus1) ８i(kminus1) ９∙i(kminus1) ９i∙(kminus1)
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1)
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1) ９i∙(kminus1)
k = k + 1
８Ｍ(kminus1) ８i(kminus1)
９∙i(kminus1)
＜ＮＣＨ ＮＢ ＰＦＯ Ｉ
Ji i = 1 2 n
８Ｍ(kminus1) ８i(kminus1) ９∙i(kminus1) ９i∙(kminus1)
(Ｃ = 1simn)
Ji le
Figure 6 The flowchart of CO-ERIM
Table 2 Optimization results of the reducer design
the reference The software Isight is utilized in the compu-tations of optimization The solutions from CO-ERIM arecompared with the ones from original PMA-RIM based COand MCS based CO which is listed in Table 2 From thecomparison of solutions the design results from ERIM arecloser to the design results from MCS Furthermore the
calculation time of ERIM is 17min23s and the calculationtime of RIM is 25min17s which means the proposed methodenjoys higher computational efficiency
6 Conclusions
In this study the efficiency problem of RBMDO is stud-ied The RIM strategy is reviewed and the correspondingalgorithm of ERIM is discussed in detail Furthermore theCO-ERIM strategy is proposed including its formulationand procedure In CO-ERIM the concurrent design idea isadapting to the development of modern engineering systemsCompatibility constraints are introduced into subsystem-level and system-level optimization problems respectivelyThe consistency between different disciplines can be guaran-teed when the RBMDO solutions are obtainedThe introduc-tion of ERIM reduces the computational burden of reliabilityanalysis during optimization iteration process Under thecondition that the first order method is acceptable thereliability analysis accuracy of the proposedmethod is similar
Figure 8 The MDO strategy for the speed reducer problem
to the accuracy ofMCSThe speed reducer example illustratesthe effectiveness of the proposed method
Data Availability
All data used to support the findings of this study are includedwithin the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thesupport from theNational Natural Science Foundation ofChina (Grant No 51605080 and No 11672070) the SichuanScience and Technology Program (Grant No 2019YFG0350and No 2019YFG0348) the China Postdoctoral ScienceFoundation (Grant No 2015M580780 and 2017T100685) andthe National Defense Pre-Research Foundation of China
(Grant No 6140244010216DZ02001) is gratefully acknowl-edged The authors also acknowledge the Portuguese Sci-ence Foundation (FCT) for the financial support throughthe postdoctoral grant SFRHBPD1078252015 as well asthe funding of Projects POCI-01-0145-FEDER-007457 andUIDECI047082019 - CONSTRUCT - Institute of RampDIn Structures and Construction funded by FEDER fundsthrough COMPETE2020 (POCI) and by national funds(PIDDAC) through Portuguese Science Foundation (FCT)
References
[1] L Li H Wan W Gao F Tong and H Li ldquoReliabilitybased multidisciplinary design optimization of cooling turbineblade considering uncertainty data statisticsrdquo Structural andMultidisciplinary Optimization vol 59 no 2 pp 1ndash15 2018
[2] Z L Huang Y S Zhou C Jiang J Zheng and X HanldquoReliability-based multidisciplinary design optimization usingincremental shifting vector strategy and its application inelectronic product designrdquo Acta Mechanica Sinica vol 34 no2 pp 285ndash302 2018
[3] H Xu X WangW Li M Li S Zhang and C Hu ldquoReliability-based multidisciplinary design optimization under correlated
Mathematical Problems in Engineering 9
uncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 7360615 12 pages 2017
[4] D Meng S Yang Y Zhang and S-P Zhu ldquoStructural relia-bility analysis and uncertainties-based collaborative design andoptimization of turbine blades using surrogate modelrdquo Fatigueamp Fracture of Engineering Materials amp Structures pp 1ndash9 2018httpsdoiorg101111ffe12906
[5] F Yang Z Yue L Li and D Guan ldquoHybrid reliability-based multidisciplinary design optimization with random andinterval variablesrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 232 no 1pp 52ndash64 2018
[6] L Wang C Xiong and Y Yang ldquoA novel methodology ofreliability-based multidisciplinary design optimization underhybrid interval and fuzzy uncertaintiesrdquo Computer MethodsApplied Mechanics and Engineering vol 337 pp 439ndash457 2018
[7] L Wang C Xiong J Hu X Wang and Z Qiu ldquoSequentialmultidisciplinary design optimization and reliability analysisunder interval uncertaintyrdquo Aerospace Science and Technologyvol 80 pp 508ndash519 2018
[8] J Zhang M Xiao L Gao H Qiu and Z Yang ldquoAn improvedtwo-stage framework of evidence-based design optimizationrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1673ndash1693 2018
[9] H Xu W Li L Xing and S-P Zhu ldquoMultidisciplinary designoptimization under correlated uncertaintiesrdquo Concurrent Engi-neering Research and Applications vol 25 no 3 pp 262ndash2752017
[10] X Wang R Wang L Wang X Chen and X Geng ldquoAnefficient single-loop strategy for reliability-based multidisci-plinarydesign optimizationunder non-probabilistic set theoryrdquoAerospace Science and Technology vol 73 pp 148ndash163 2018
[11] X P Du J Guo and H Beeram ldquoSequential optimizationand reliability assessment for multidisciplinary systems designrdquoStructural andMultidisciplinary Optimization vol 35 no 2 pp117ndash130 2008
[12] D Meng Y-F Li H-Z Huang Z Wang and Y LiuldquoReliability-based multidisciplinary design optimization usingsubset simulation analysis and its application in the hydraulictransmission mechanism designrdquo Journal of Mechanical Designvol 137 no 5 Article ID 051402 2015
[13] S Yu Z Wang and D Meng ldquoTime-variant reliability assess-ment for multiple failure modes and temporal parametersrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1705ndash1717 2018
[14] D Meng H-Z Huang Z Wang N-C Xiao and X-L ZhangldquoMean-value first-order saddlepoint approximation based col-laborative optimization for multidisciplinary problems underaleatory uncertaintyrdquo Journal of Mechanical Science and Tech-nology vol 28 no 10 pp 3925ndash3935 2014
[15] S-P Zhu Q Liu Q Lei and Q Wang ldquoProbabilistic fatiguelife prediction and reliability assessment of a high pressureturbine disc considering load variationsrdquo International Journalof Damage Mechanics vol 27 no 10 pp 1569ndash1588 2018
[16] D Liao S-P Zhu J A F O Correia A M P De Jesus andR Calcada ldquoComputational framework for multiaxial fatiguelife prediction of compressor discs considering notch effectsrdquoEngineering Fracture Mechanics vol 202 pp 423ndash435 2018
[17] D Meng M Liu S Yang H Zhang and R Ding ldquoAfluid-structure analysis approach and its application in theuncertainty-based multidisciplinary design and optimization
for bladesrdquo Advances in Mechanical Engineering vol 10 no 6pp 1ndash7 2018
[18] Z Meng G Li B P Wang and P Hao ldquoA hybrid chaos controlapproach of the performance measure functions for reliability-based design optimizationrdquo Computers Structures vol 146 pp32ndash43 2015
[19] X Li H Qiu Z Chen L Gao and X Shao ldquoA local Krigingapproximation method using MPP for reliability-based designoptimizationrdquo Computers amp Structures vol 162 pp 102ndash1152016
[20] S-P Zhu Q Liu W Peng and X-C Zhang ldquoComputational-experimental approaches for fatigue reliability assessment ofturbine bladed disksrdquo International Journal of Mechanical Sci-ences vol 142-143 pp 502ndash517 2018
[21] Z Chen H Qiu L Gao and P Li ldquoAn optimal shiftingvector approach for efficient probabilistic designrdquo Structuraland Multidisciplinary Optimization vol 47 no 6 pp 905ndash9202013
[22] W Deng X Lu and Y Deng ldquoEvidential model validationunder epistemic uncertaintyrdquo Mathematical Problems in Engi-neering vol 2018 Article ID 6789635 11 pages 2018
[23] D Meng H Zhang and T Huang ldquoA concurrent reliabilityoptimization procedure in the earlier design phases of complexengineering systems under epistemic uncertaintiesrdquo Advancesin Mechanical Engineering vol 8 no 10 pp 1ndash8 2016
[24] G Li Z Meng and H Hu ldquoAn adaptive hybrid approach forreliability-based design optimizationrdquo Structural and Multidis-ciplinary Optimization vol 51 no 5 pp 1051ndash1065 2015
[25] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018
[26] M A Valdebenito and G I Schueller ldquoA survey on approachesfor reliability-based optimizationrdquo Structural and Multidisci-plinary Optimization vol 42 no 5 pp 645ndash663 2010
[27] Y Aoues and A Chateauneuf ldquoBenchmark study of numericalmethods for reliability-based design optimizationrdquo Structuraland Multidisciplinary Optimization vol 41 no 2 pp 277ndash2942010
[28] J Ching and W-C Hsu ldquoTransforming reliability limit-stateconstraints into deterministic limit-state constraintsrdquo StructuralSafety vol 30 no 1 pp 11ndash33 2008
[29] TM Cho and B C Lee ldquoReliability-based design optimizationusing convex linearization and sequential optimization andreliability assessment methodrdquo Structural Safety vol 33 no 1pp 42ndash50 2011
[30] J Liang Z P Mourelatos and J Tu ldquoA single-loop method forreliability-based design optimisationrdquo International Journal ofProduct Development vol 5 no 1-2 pp 76ndash92 2008
[31] S Shan and G G Wang ldquoReliable design space and completesingle-loop reliability-based design optimizationrdquo ReliabilityEngineering amp System Safety vol 93 no 8 pp 1218ndash1230 2008
[32] P Yi G Cheng and L Jiang ldquoA sequential approximateprogramming strategy for performance-measure-based proba-bilistic structural design optimizationrdquo Structural Safety vol 30no 2 pp 91ndash109 2008
[33] Z M Yaseen and B Keshtegar ldquoLimited descent-based meanvalue method for inverse reliability analysisrdquo Engineering withComputers pp 1ndash13 2018
[34] B D Youn K K Choi and Y H Park ldquoHybrid analysis methodfor reliability-based design optimizationrdquo Journal of MechanicalDesign vol 125 no 2 pp 221ndash232 2003
10 Mathematical Problems in Engineering
[35] B D Youn K K Choi and L Du ldquoEnriched performancemeasure approach for reliability-based design optimizationrdquoAIAA Journal vol 43 no 4 pp 874ndash884 2005
[36] B D Youn and K K Choi ldquoSelecting probabilistic approachesfor realiability-based design optimizationrdquo AIAA Journal vol42 no 1 pp 124ndash131 2004
[37] S P Zhu Q Liu J Zhou and Z Y Yu ldquoFatigue reliabilityassessment of turbine discs under multi-source uncertaintiesrdquoFatigue Fracture of Engineering Materials Structures vol 41 no6 pp 1291ndash1305 2018
[38] H Li R Li H Li and R Yuan ldquoReliability modeling of mul-tiple performance based on degradation values distributionrdquoAdvances in Mechanical Engineering vol 8 no 10 Article ID168781401667375 2016
[39] B D Youn K K Choi and L Du ldquoAdaptive probability analysisusing an enhanced hybrid mean value methodrdquo Structural andMultidisciplinary Optimization vol 29 no 2 pp 134ndash148 2005
[40] M Ito N H Kim and N Kogiso ldquoConservative reliabilityindex for epistemic uncertainty in reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 5 pp 1919ndash1935 2018
[41] Z Meng D Yang H Zhou and B P Wang ldquoConvergencecontrol of single loop approach for reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 3 pp 1079ndash1091 2018
[42] W Yao X Q Chen W C Luo M van Tooren and J GuoldquoReview of uncertainty-based multidisciplinary design opti-mization methods for aerospace vehiclesrdquo Progress in AerospaceSciences vol 47 no 6 pp 450ndash479 2011
[43] R Yuan D Meng and H Li ldquoMultidisciplinary reliabilitydesign optimizationusing an enhanced saddlepoint approxima-tion in the framework of sequential optimization and reliabilityanalysisrdquo Proceedings of the Institution of Mechanical EngineersPart O Journal of Risk and Reliability vol 230 no 6 pp 570ndash578 2016
[44] D Meng X Zhang Y Yang H Xu and H Huang ldquoInteractionbalance optimization in multidisciplinary design optimizationproblemsrdquo Concurrent Engineering Research and Applicationsvol 24 no 1 pp 48ndash57 2016
[45] R Yuan and H Li ldquoA multidisciplinary coupling relationshipcoordination algorithm using the hierarchical control methodsof complex systems and its application in multidisciplinarydesign optimizationrdquo Advances in Mechanical Engineering vol9 no 1 pp 1ndash11 2016
[46] D Meng X Zhang H-Z Huang Z Wang and H Xu ldquoInter-action prediction optimization in multidisciplinary designoptimization problemsrdquoThe Scientific World Journal vol 2014Article ID 698453 7 pages 2014
[47] J R R A Martins and A B Lambe ldquoMultidisciplinary designoptimization a survey of architecturesrdquo AIAA Journal vol 51no 9 pp 2049ndash2075 2013
[48] Y Li and Y Deng ldquoGeneralized ordered propositions fusionbased on belief entropyrdquo International Journal of ComputersCommunications amp Control vol 13 no 5 pp 792ndash807 2018
[49] X Su S Mahadevan W Han and Y Deng ldquoCombiningdependent bodies of evidencerdquo Applied Intelligence vol 44 no3 pp 634ndash644 2016
[50] S-P Zhu H-Z Huang W Peng H-K Wang and S Mahade-van ldquoProbabilistic Physics of Failure-based framework forfatigue life prediction of aircraft gas turbine discs under uncer-taintyrdquoReliability Engineering amp System Safety vol 146 pp 1ndash122016
[51] S I Yi J K Shin andG J Park ldquoComparison ofMDOmethodswith mathematical examplesrdquo Structural and MultidisciplinaryOptimization vol 35 no 5 pp 391ndash402 2008
[52] H XuW Li M Li C Hu S Zhang and XWang ldquoMultidisci-plinary robust design optimization based on time-varying sen-sitivity analysisrdquo Journal of Mechanical Science and Technologyvol 32 no 3 pp 1195ndash1207 2018
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Mathematical Problems in Engineering 3
Cost optimization
Constraint feasibilityRBMDO
Discipline 1
Discipline i
Discipline n
d d X X
Yn∙
Yi∙
Y1∙Y1i
YniYin
Y1n
Yi1
Yn1
s s11
d d X Xs sii
d d X Xs snn
Figure 3 The design for a complex system based on RBMDOstrategy
around the mean value such as degradation deteriorationquality loss
In this study reliability-based design problems are pri-marily considered The general formulation of RBMDO canbe given as
min(d119894ds 120583X119894 120583Xs )
119891 (d119894 ds120583X119894 120583Xs)
st Pr119894 [119892119894 (d119894 dsX119894XsP) le 0] le Φ (minus120573119905)120583Y119894∙ = Y119894∙ (d119894 ds120583X119894 120583Xs
120583Y∙119894)d119871119894 le d119894 le d119880119894 d119871s le ds le d119880s X119871119894 le 120583X119894 le X119880119894 X119871s le 120583Xs
le X119880s Y119871 le 120583Y le Y119880119894 = 1 2 119899
(1)
where 119891(∙) is a cost-type objective function 119892(∙) le 0and Pr[∙] le Φ(minus120573119905) are the inequality constraint andits corresponding probability constraint respectively 120573119905 isthe safety reliability index Φ(minus120573119905) is the allowable failureprobability for 119892(∙) le 0 Φ(∙) is the Cumulative DistributionFunction (CDF) of the standard normal random variablethe subscript ldquosrdquo denotes the sharing design information toall disciplines the subscript ldquo119894rdquo denotes the 119894th disciplinein a complex system d is the input design informationwhich is not accompanied with with uncertainties X andP are the input design information and parameters whichare accompanied with uncertainties 120583 is the mean value ofuncertainty information Y is the coupling information Y∙119894is the input coupling information for the 119894th discipline fromother disciplines whileY119894∙ is the output coupling informationfor the 119894th discipline to other disciplines the superscripts ldquo119871rdquoand ldquo119880rdquo denote the lower and upper bounds of input designinformation respectively 119899 is the number of disciplines
As shown in Figure 3 uncertainties will be propagatedamong coupled disciplines in multidisciplinary systems Ifthe RBMDO problem in (1) is solved directly a triple-loopstrategy will be utilized which is shown in Figure 4 Theouter loop performs the optimization for objective function
to obtain design point the intermediate loop performsthe reliability analysis on the design point the inner loopperforms the multidisciplinary analysis (MDA) between thesubdisciplines
Using the triple-loop strategy the multidisciplinary sys-tem optimization problem requires reliability analysis ineach iterative operation Meanwhile each reliability analysisoperation involves MDA Both of them result in a high com-putational burden To solve this problem ERIM and PMAare introduced in this study The corresponding formulationof CO-ERIM is also proposed to improve the efficiency ofRBMDO
3 The Performance Measure Approach (PMA)Using RIM in Sequential Optimization andReliability Assessment
Because of the existences of probability constraints the PMAusing RIM (PMA-RIM) strategy has been utilized widely inRBMDO to reduce computational cost Researches on thisstrategy mainly include two aspects the modifying formu-lation of probability constraint and the enhanced efficienciesof reliability analysis and optimization algorithms [34]
In this study the limitations of the original PMA-RIM arediscussedThen a PMAbased on Enhanced RI (PMA-ERIM)is proposed here which is on the condition of accepting theapproximate accuracy of the First Order Reliability Method(FORM)
31 The Strategy of RIM In (1) the probability constraint canbe reexpressed using Φminus1
120573119904119894 = minusΦminus1 (119865119892119894 (0)) ge 120573119905 (2)
where 120573119904119894 is the safety reliability index of the 119894th probabilityconstraint The input design information with uncertaintiesin (1) is treated as random variables in this study Thenin RIM the first order safety reliability index 120573119904FORM canbe obtained using FORM This process mainly includes twosteps
First all random variables 119909 of the set of X (119909 isin X)in the X-space are transformed into the standard normaldistribution variable 119906 in the U-space using the Rosenblatttransformation U is a set of the standard normal distributionvariable 119906 119906 isin U The standard normal variable can bedenoted as
119906 = Φminus1 (119865119909 (119909)) (3)
where 119865119909(∙) is the CDF of a random variable 119909 Then theperformance function 119892119894(X) is transformed into the U-spaceas 119892119894(U)
Second an optimization problem formulated as follows issolved
min U2st 119892 (U) = 0 (4)
4 Mathematical Problems in Engineering
Multidisciplinary analysis
Reliability analysis
Optimization loop
Initial design Optimal designOptimization
Design variables Reliability constraints
MDA loop
Reliability analysis loop
Figure 4 The triple-loop strategy of RBMDO [11]
where ∙ 2 is the magnitude of a vector The optimumsolution of (4) on the failure surface (119892(U) = 0) is called MPPulowast119892(U)=0 Also 120573119904FORM = ulowast119892(U)=02 [34 35]32 The Strategy of PMA-RIM Here the probability con-straint in (1) is converted into its equivalent form by 119865minus1119892119894
119866119901119894 = 119865minus1119892119894 [Φ (minus120573119905)] ge 0 (5)
where119866119901119894 is the probabilistic performance measure of the 119894thprobability constraint
If the value 119892119901 ge 0 then Pr(119892(X) le 0) le Φ(minus120573119905) if 119892119901 lt0 Pr(119892(X) le 0) gt Φ(minus120573119905) 119892119901 corresponds to the Φ(minus120573119905)percentile of the CDF of performance function
At first all uncertainty inputs are converted into thestandard normal random inputs in the U-space The firstorder probabilistic performance measure 119866119901FORM can beobtained by solving
min 119866 (U)st U2 = 120573119905 (6)
where the optimal point on the surface U2 = 120573119905 is identifiedas MPP ulowast120573=120573119905 Furthermore 119866119901FORM = 119866(ulowast120573=120573119905)33 The Strategy of PMA-ERIM To improve the efficiencyand accuracy of the original RIM the ERIM is discussed inthis section Recall the statistic description of the failure of aperformance function 119892
Pr (119892 (X) le 0) = 119865119892 (0) = int119892(X)le0
119891X (x) 119889x (7)
Step 1 Using the Rosenblatt transformation (7) is equivalentto
Pr (119892 (U) le 0) = int sdot sdot sdot int119892(11990611199062sdotsdotsdot 119906119899)le0
Step 2 To evaluate the integration more easily the integrandboundary 119892(U) = 0 is approximated FORM utilizes the firstorder Taylor expansion as
119892 (U) asymp 119871119892 (U) = 119892 (ulowast) + [nabla119892 (ulowast)]119879 (U minus ulowast) (9)
where119871119892(U) is the linearized performance function ulowast is theexpansion point 119879 denotes transpose nabla119892(ulowast) is the gradientof 119892 at ulowast
To minimize the accuracy loss the performance functionshould be expanded at MPP MPP can be obtained by (4)
Because at MPP 119892(ulowast) = 0 the performance function islinearized as
119892 (U) asymp 119871119892 (U) = [nabla119892 (ulowast)]119879 (U minus ulowast)= minus [nabla119892 (ulowast)]119879 ulowast + [nabla119892 (ulowast)]119879U (11)
If the gradient nabla119892(ulowast) at ulowast is equal to zero then theperformance function is linearized as 119892(U) asymp 119871119892(U) = 0 Inthis case the linear approximation of the integrand boundarywill cause a large error about the integration because ofthe highly nonlinear character of performance functionTherefore if the above case appears the FORM is not suitableto deal with the problem In the following the case of thegradient of performance function at MPP unequal to zero isdiscussed
Since 119871119892(U) is a linear function of standard normalvariables 119871119892(U) is normally distributed Thus based on (11)the mean value and standard deviation of 119871119892(U) are 120583119871119892 =minus[nabla119892(ulowast)]119879ulowast and 120590119871119892 = nabla119892(ulowast)2 which can be utilized inthe derivation process in (12)
Therefore the probability of failure is calculated as
Pr (119892 (U) le 0) asymp Pr (119871119892 (U) le 0) = Φ(0 minus 120583119871119892120590119871119892 )
At the optimal point ulowast = nabla119892(ulowast)nabla119892(ulowast)2 sdot ulowast2 orulowast = minusnabla119892(ulowast)nabla119892(ulowast)2 sdot ulowast2 Hence the probability offailure is
Pr (119892 (U) le 0)
asymp
Φ(1003817100381710038171003817ulowast10038171003817100381710038172) if ulowast = nabla119892 (ulowast)1003817100381710038171003817nabla119892 (ulowast)10038171003817100381710038172 sdot1003817100381710038171003817ulowast10038171003817100381710038172
Φ(minus 1003817100381710038171003817ulowast10038171003817100381710038172) if ulowast = minus nabla119892 (ulowast)1003817100381710038171003817nabla119892 (ulowast)10038171003817100381710038172 sdot1003817100381710038171003817ulowast10038171003817100381710038172
(13)
From (12) and (13) the difference of judgment on thesatisfaction of probability constraint between RIM and thepractical situation is caused when ulowast = nabla119892(ulowast)nabla119892(ulowast)2 sdotulowast2 Utilizing (11) the probabilities of failure of the perfor-mance functions in (1) can be recalculated
To satisfy the probability constraint from (12) the prob-ability of failure should satisfy
As a well-known multilevel method for MDO the strategy ofCO is suitable for large-scale distributed engineering systemsThe CO algorithm decomposes coordinates and optimizescomplex engineering problems Every discipline in a systemcan enjoy good autonomy regardless of the influence of otherdisciplinesThe consistency betweendisciplines is guaranteedby compatibility constraints attached to the system-leveloptimization problem While the value of compatibility con-straint is obtained through subject-level optimization prob-lem The objectives of subject-level optimization problems
are to minimize the inconsistency between disciplines whilesatisfying the constraints of discipline design
Using CO the RBMDO problem in (1) is converted intothe system-level and subject-level optimization problemsThe formulation of the optimization problem in system levelis
st 119869119894= (d1015840119894 minus d119894)2 + (d1015840s minus ds)2+ (1205831015840X119894 minus 120583X119894)2 + (1205831015840Xs
minus 120583Xs)2
+ (1205831015840Y∙119894 minus 120583Y∙119894)2 le 120576119894 = 1 2 119899
(19)
where 119869119894 is the compatibility constraints d1015840119894 d1015840s1205831015840X119894 1205831015840Xs
and 1205831015840Y∙119894 are the design variables at the system level Theformulation of the optimization problem in subject-leveloptimization problems is
119869119894= (d1015840119894 minus d119894)2 + (d1015840s minus ds)2+ (1205831015840X119894 minus 120583X119894)2 + (1205831015840Xs
minus 120583Xs)2
+ (1205831015840Y∙119894 minus 120583Y∙119894)2st Pr119894 [119892119894 (d119894 dsX119894XsP) le 0]
le Φ (minus120573119905)120583Y119894∙ = Y119894∙ (d119894 ds120583X119894 120583Xs
120583Y∙119894)d119871119894 le d119894 le d119880119894 d119871s le ds le d119880s X119871119894 le 120583X119894 le X119880119894 X119871s le 120583Xs
le X119880s Y119871 le 120583Y le Y119880119894 = 1 2 119899
(20)
The corresponding bilevel strategy of CO is also shown inFigure 5
Considering uncertainties in practical engineering alldiscipline probability reliability constraints in (20) are con-verted into the corresponding RIM-based reliability con-straints These RIM-based reliability constraints only con-sider the performance reliability at MPP which can reducethe cost of reliability analysis and improve the computational
6 Mathematical Problems in Engineering
Table 1 The design information of speed reducer
Variables Lower and upper bound Distribution Mean Standard deviationgear face width 1199091 (cm) [26 36] - - -teeth module 1199092 (cm) [03 10] - - -number of teeth of pinion 1199093 [17 28] - - -distance between bearings 1 1199094 (cm) [73 83] Normal 1205831199094 0011205831199094distance between bearings 2 1199095 (cm) [73 83] Normal 1205831199095 0011205831199095diameter of shaft 1 1199096 (cm) [29 39] Normal 1205831199096 0011205831199096diameter of shaft 2 1199097 (cm) [5 55] Normal 1205831199097 0011205831199097
System optimization problem in Eq (19)
The 1st discipline optimizationproblem in Eq (20)
The 1st discipline analysis
Theproblem in Eq (20)
thi
XＭX
1
Y∙1Y
1∙
XＭX1
Y∙1Y1∙
XＭX
i
Y∙iY
i∙
XＭXi
Y∙iYi∙
XＭX1Y∙1 Y1∙XＭXi Y∙i Yi∙
discipline optimization
The thi discipline analysis
Figure 5 The bilevel strategy of CO
efficiency of RBMDO The formulation of the subject-leveloptimization problem in the U-space using RIM-based relia-bility constraints can be denoted as
transformation of Xd119871119894 le d119894 le d119880119894 d119871s le ds le d119880s X119871119894 le 120583X119894 le X119880119894 X119871s le 120583Xs
le X119880s
Y119871 le 120583Y le Y119880119894 = 1 2 119899
(21)
The detail information of CO-ERIM is as follows
Step 1 Input the original design information the cyclenumber 119896 = 0Step 2 Solve the system-level optimization problem in (19)During this process d1015840119894 d
1015840s 1205831015840X119894 1205831015840Xs and 1205831015840Y∙119894 are treated as the
design parameters
Step 3 Transform random variables in X-space into randomvariables in U-space using the Rosenblatt transformation
Step 4 Solve the subject-level optimization problems in (21)Then send the design solutions to the system level
Step 5 Obtain the value of 119869119894 119894 = 1 2 119899 If 119869119894 le 120576 andthe difference between the objective function values of twoconsecutive iterations is not more than a small number in theoptimization iteration process carry out Step 6 Otherwise119896 = 119896 + 1 and carry out Step 2
Step 6 Stop and output the design solutions
The flowchart of CO-ERIM is shown in Figure 6
5 Example
Speed reducers are generally used in low-speed high-torquetransmission equipment In this study a speed reducerRBMDO problem is introduced to illustrate the utilization ofthe proposedmethodThere are seven design variables in thisexample which is listed in Table 1 Twenty-five constraints areintroduced to ensure that the design solutions can satisfy thestrength stiffness and space requirements The optimizationobject is to minimize the overall weight Further informationcan be obtained in [14 43]
There are three disciplines in this RBMDO problemBearing-Shaft 1 Bearing-Shaft 2 and Gears which is shownin Figure 7The CO-ERIM strategy for this problem is shownin Figure 8 where 120573119905 = 207 Φ(minus120573119905) = 002 and 120576 = 0001
To illustrate the accuracy of design solutions the MonteCarlo Simulation (MCS) method is also introduced here as
Mathematical Problems in Engineering 7
Solve the optimization problemin Eq (19) at system level
Solve the discipline optimizationproblems in Eq (21) at subsystem level
Start
No
Yes
End
The ith discipline analysis
k = 0
Rosenblatt transformation
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1) ９
i∙(kminus1)
８Ｍ(kminus1) ８i(kminus1) ９∙i(kminus1) ９i∙(kminus1)
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1)
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1) ９i∙(kminus1)
k = k + 1
８Ｍ(kminus1) ８i(kminus1)
９∙i(kminus1)
＜ＮＣＨ ＮＢ ＰＦＯ Ｉ
Ji i = 1 2 n
８Ｍ(kminus1) ８i(kminus1) ９∙i(kminus1) ９i∙(kminus1)
(Ｃ = 1simn)
Ji le
Figure 6 The flowchart of CO-ERIM
Table 2 Optimization results of the reducer design
the reference The software Isight is utilized in the compu-tations of optimization The solutions from CO-ERIM arecompared with the ones from original PMA-RIM based COand MCS based CO which is listed in Table 2 From thecomparison of solutions the design results from ERIM arecloser to the design results from MCS Furthermore the
calculation time of ERIM is 17min23s and the calculationtime of RIM is 25min17s which means the proposed methodenjoys higher computational efficiency
6 Conclusions
In this study the efficiency problem of RBMDO is stud-ied The RIM strategy is reviewed and the correspondingalgorithm of ERIM is discussed in detail Furthermore theCO-ERIM strategy is proposed including its formulationand procedure In CO-ERIM the concurrent design idea isadapting to the development of modern engineering systemsCompatibility constraints are introduced into subsystem-level and system-level optimization problems respectivelyThe consistency between different disciplines can be guaran-teed when the RBMDO solutions are obtainedThe introduc-tion of ERIM reduces the computational burden of reliabilityanalysis during optimization iteration process Under thecondition that the first order method is acceptable thereliability analysis accuracy of the proposedmethod is similar
Figure 8 The MDO strategy for the speed reducer problem
to the accuracy ofMCSThe speed reducer example illustratesthe effectiveness of the proposed method
Data Availability
All data used to support the findings of this study are includedwithin the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thesupport from theNational Natural Science Foundation ofChina (Grant No 51605080 and No 11672070) the SichuanScience and Technology Program (Grant No 2019YFG0350and No 2019YFG0348) the China Postdoctoral ScienceFoundation (Grant No 2015M580780 and 2017T100685) andthe National Defense Pre-Research Foundation of China
(Grant No 6140244010216DZ02001) is gratefully acknowl-edged The authors also acknowledge the Portuguese Sci-ence Foundation (FCT) for the financial support throughthe postdoctoral grant SFRHBPD1078252015 as well asthe funding of Projects POCI-01-0145-FEDER-007457 andUIDECI047082019 - CONSTRUCT - Institute of RampDIn Structures and Construction funded by FEDER fundsthrough COMPETE2020 (POCI) and by national funds(PIDDAC) through Portuguese Science Foundation (FCT)
References
[1] L Li H Wan W Gao F Tong and H Li ldquoReliabilitybased multidisciplinary design optimization of cooling turbineblade considering uncertainty data statisticsrdquo Structural andMultidisciplinary Optimization vol 59 no 2 pp 1ndash15 2018
[2] Z L Huang Y S Zhou C Jiang J Zheng and X HanldquoReliability-based multidisciplinary design optimization usingincremental shifting vector strategy and its application inelectronic product designrdquo Acta Mechanica Sinica vol 34 no2 pp 285ndash302 2018
[3] H Xu X WangW Li M Li S Zhang and C Hu ldquoReliability-based multidisciplinary design optimization under correlated
Mathematical Problems in Engineering 9
uncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 7360615 12 pages 2017
[4] D Meng S Yang Y Zhang and S-P Zhu ldquoStructural relia-bility analysis and uncertainties-based collaborative design andoptimization of turbine blades using surrogate modelrdquo Fatigueamp Fracture of Engineering Materials amp Structures pp 1ndash9 2018httpsdoiorg101111ffe12906
[5] F Yang Z Yue L Li and D Guan ldquoHybrid reliability-based multidisciplinary design optimization with random andinterval variablesrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 232 no 1pp 52ndash64 2018
[6] L Wang C Xiong and Y Yang ldquoA novel methodology ofreliability-based multidisciplinary design optimization underhybrid interval and fuzzy uncertaintiesrdquo Computer MethodsApplied Mechanics and Engineering vol 337 pp 439ndash457 2018
[7] L Wang C Xiong J Hu X Wang and Z Qiu ldquoSequentialmultidisciplinary design optimization and reliability analysisunder interval uncertaintyrdquo Aerospace Science and Technologyvol 80 pp 508ndash519 2018
[8] J Zhang M Xiao L Gao H Qiu and Z Yang ldquoAn improvedtwo-stage framework of evidence-based design optimizationrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1673ndash1693 2018
[9] H Xu W Li L Xing and S-P Zhu ldquoMultidisciplinary designoptimization under correlated uncertaintiesrdquo Concurrent Engi-neering Research and Applications vol 25 no 3 pp 262ndash2752017
[10] X Wang R Wang L Wang X Chen and X Geng ldquoAnefficient single-loop strategy for reliability-based multidisci-plinarydesign optimizationunder non-probabilistic set theoryrdquoAerospace Science and Technology vol 73 pp 148ndash163 2018
[11] X P Du J Guo and H Beeram ldquoSequential optimizationand reliability assessment for multidisciplinary systems designrdquoStructural andMultidisciplinary Optimization vol 35 no 2 pp117ndash130 2008
[12] D Meng Y-F Li H-Z Huang Z Wang and Y LiuldquoReliability-based multidisciplinary design optimization usingsubset simulation analysis and its application in the hydraulictransmission mechanism designrdquo Journal of Mechanical Designvol 137 no 5 Article ID 051402 2015
[13] S Yu Z Wang and D Meng ldquoTime-variant reliability assess-ment for multiple failure modes and temporal parametersrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1705ndash1717 2018
[14] D Meng H-Z Huang Z Wang N-C Xiao and X-L ZhangldquoMean-value first-order saddlepoint approximation based col-laborative optimization for multidisciplinary problems underaleatory uncertaintyrdquo Journal of Mechanical Science and Tech-nology vol 28 no 10 pp 3925ndash3935 2014
[15] S-P Zhu Q Liu Q Lei and Q Wang ldquoProbabilistic fatiguelife prediction and reliability assessment of a high pressureturbine disc considering load variationsrdquo International Journalof Damage Mechanics vol 27 no 10 pp 1569ndash1588 2018
[16] D Liao S-P Zhu J A F O Correia A M P De Jesus andR Calcada ldquoComputational framework for multiaxial fatiguelife prediction of compressor discs considering notch effectsrdquoEngineering Fracture Mechanics vol 202 pp 423ndash435 2018
[17] D Meng M Liu S Yang H Zhang and R Ding ldquoAfluid-structure analysis approach and its application in theuncertainty-based multidisciplinary design and optimization
for bladesrdquo Advances in Mechanical Engineering vol 10 no 6pp 1ndash7 2018
[18] Z Meng G Li B P Wang and P Hao ldquoA hybrid chaos controlapproach of the performance measure functions for reliability-based design optimizationrdquo Computers Structures vol 146 pp32ndash43 2015
[19] X Li H Qiu Z Chen L Gao and X Shao ldquoA local Krigingapproximation method using MPP for reliability-based designoptimizationrdquo Computers amp Structures vol 162 pp 102ndash1152016
[20] S-P Zhu Q Liu W Peng and X-C Zhang ldquoComputational-experimental approaches for fatigue reliability assessment ofturbine bladed disksrdquo International Journal of Mechanical Sci-ences vol 142-143 pp 502ndash517 2018
[21] Z Chen H Qiu L Gao and P Li ldquoAn optimal shiftingvector approach for efficient probabilistic designrdquo Structuraland Multidisciplinary Optimization vol 47 no 6 pp 905ndash9202013
[22] W Deng X Lu and Y Deng ldquoEvidential model validationunder epistemic uncertaintyrdquo Mathematical Problems in Engi-neering vol 2018 Article ID 6789635 11 pages 2018
[23] D Meng H Zhang and T Huang ldquoA concurrent reliabilityoptimization procedure in the earlier design phases of complexengineering systems under epistemic uncertaintiesrdquo Advancesin Mechanical Engineering vol 8 no 10 pp 1ndash8 2016
[24] G Li Z Meng and H Hu ldquoAn adaptive hybrid approach forreliability-based design optimizationrdquo Structural and Multidis-ciplinary Optimization vol 51 no 5 pp 1051ndash1065 2015
[25] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018
[26] M A Valdebenito and G I Schueller ldquoA survey on approachesfor reliability-based optimizationrdquo Structural and Multidisci-plinary Optimization vol 42 no 5 pp 645ndash663 2010
[27] Y Aoues and A Chateauneuf ldquoBenchmark study of numericalmethods for reliability-based design optimizationrdquo Structuraland Multidisciplinary Optimization vol 41 no 2 pp 277ndash2942010
[28] J Ching and W-C Hsu ldquoTransforming reliability limit-stateconstraints into deterministic limit-state constraintsrdquo StructuralSafety vol 30 no 1 pp 11ndash33 2008
[29] TM Cho and B C Lee ldquoReliability-based design optimizationusing convex linearization and sequential optimization andreliability assessment methodrdquo Structural Safety vol 33 no 1pp 42ndash50 2011
[30] J Liang Z P Mourelatos and J Tu ldquoA single-loop method forreliability-based design optimisationrdquo International Journal ofProduct Development vol 5 no 1-2 pp 76ndash92 2008
[31] S Shan and G G Wang ldquoReliable design space and completesingle-loop reliability-based design optimizationrdquo ReliabilityEngineering amp System Safety vol 93 no 8 pp 1218ndash1230 2008
[32] P Yi G Cheng and L Jiang ldquoA sequential approximateprogramming strategy for performance-measure-based proba-bilistic structural design optimizationrdquo Structural Safety vol 30no 2 pp 91ndash109 2008
[33] Z M Yaseen and B Keshtegar ldquoLimited descent-based meanvalue method for inverse reliability analysisrdquo Engineering withComputers pp 1ndash13 2018
[34] B D Youn K K Choi and Y H Park ldquoHybrid analysis methodfor reliability-based design optimizationrdquo Journal of MechanicalDesign vol 125 no 2 pp 221ndash232 2003
10 Mathematical Problems in Engineering
[35] B D Youn K K Choi and L Du ldquoEnriched performancemeasure approach for reliability-based design optimizationrdquoAIAA Journal vol 43 no 4 pp 874ndash884 2005
[36] B D Youn and K K Choi ldquoSelecting probabilistic approachesfor realiability-based design optimizationrdquo AIAA Journal vol42 no 1 pp 124ndash131 2004
[37] S P Zhu Q Liu J Zhou and Z Y Yu ldquoFatigue reliabilityassessment of turbine discs under multi-source uncertaintiesrdquoFatigue Fracture of Engineering Materials Structures vol 41 no6 pp 1291ndash1305 2018
[38] H Li R Li H Li and R Yuan ldquoReliability modeling of mul-tiple performance based on degradation values distributionrdquoAdvances in Mechanical Engineering vol 8 no 10 Article ID168781401667375 2016
[39] B D Youn K K Choi and L Du ldquoAdaptive probability analysisusing an enhanced hybrid mean value methodrdquo Structural andMultidisciplinary Optimization vol 29 no 2 pp 134ndash148 2005
[40] M Ito N H Kim and N Kogiso ldquoConservative reliabilityindex for epistemic uncertainty in reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 5 pp 1919ndash1935 2018
[41] Z Meng D Yang H Zhou and B P Wang ldquoConvergencecontrol of single loop approach for reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 3 pp 1079ndash1091 2018
[42] W Yao X Q Chen W C Luo M van Tooren and J GuoldquoReview of uncertainty-based multidisciplinary design opti-mization methods for aerospace vehiclesrdquo Progress in AerospaceSciences vol 47 no 6 pp 450ndash479 2011
[43] R Yuan D Meng and H Li ldquoMultidisciplinary reliabilitydesign optimizationusing an enhanced saddlepoint approxima-tion in the framework of sequential optimization and reliabilityanalysisrdquo Proceedings of the Institution of Mechanical EngineersPart O Journal of Risk and Reliability vol 230 no 6 pp 570ndash578 2016
[44] D Meng X Zhang Y Yang H Xu and H Huang ldquoInteractionbalance optimization in multidisciplinary design optimizationproblemsrdquo Concurrent Engineering Research and Applicationsvol 24 no 1 pp 48ndash57 2016
[45] R Yuan and H Li ldquoA multidisciplinary coupling relationshipcoordination algorithm using the hierarchical control methodsof complex systems and its application in multidisciplinarydesign optimizationrdquo Advances in Mechanical Engineering vol9 no 1 pp 1ndash11 2016
[46] D Meng X Zhang H-Z Huang Z Wang and H Xu ldquoInter-action prediction optimization in multidisciplinary designoptimization problemsrdquoThe Scientific World Journal vol 2014Article ID 698453 7 pages 2014
[47] J R R A Martins and A B Lambe ldquoMultidisciplinary designoptimization a survey of architecturesrdquo AIAA Journal vol 51no 9 pp 2049ndash2075 2013
[48] Y Li and Y Deng ldquoGeneralized ordered propositions fusionbased on belief entropyrdquo International Journal of ComputersCommunications amp Control vol 13 no 5 pp 792ndash807 2018
[49] X Su S Mahadevan W Han and Y Deng ldquoCombiningdependent bodies of evidencerdquo Applied Intelligence vol 44 no3 pp 634ndash644 2016
[50] S-P Zhu H-Z Huang W Peng H-K Wang and S Mahade-van ldquoProbabilistic Physics of Failure-based framework forfatigue life prediction of aircraft gas turbine discs under uncer-taintyrdquoReliability Engineering amp System Safety vol 146 pp 1ndash122016
[51] S I Yi J K Shin andG J Park ldquoComparison ofMDOmethodswith mathematical examplesrdquo Structural and MultidisciplinaryOptimization vol 35 no 5 pp 391ndash402 2008
[52] H XuW Li M Li C Hu S Zhang and XWang ldquoMultidisci-plinary robust design optimization based on time-varying sen-sitivity analysisrdquo Journal of Mechanical Science and Technologyvol 32 no 3 pp 1195ndash1207 2018
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4 Mathematical Problems in Engineering
Multidisciplinary analysis
Reliability analysis
Optimization loop
Initial design Optimal designOptimization
Design variables Reliability constraints
MDA loop
Reliability analysis loop
Figure 4 The triple-loop strategy of RBMDO [11]
where ∙ 2 is the magnitude of a vector The optimumsolution of (4) on the failure surface (119892(U) = 0) is called MPPulowast119892(U)=0 Also 120573119904FORM = ulowast119892(U)=02 [34 35]32 The Strategy of PMA-RIM Here the probability con-straint in (1) is converted into its equivalent form by 119865minus1119892119894
119866119901119894 = 119865minus1119892119894 [Φ (minus120573119905)] ge 0 (5)
where119866119901119894 is the probabilistic performance measure of the 119894thprobability constraint
If the value 119892119901 ge 0 then Pr(119892(X) le 0) le Φ(minus120573119905) if 119892119901 lt0 Pr(119892(X) le 0) gt Φ(minus120573119905) 119892119901 corresponds to the Φ(minus120573119905)percentile of the CDF of performance function
At first all uncertainty inputs are converted into thestandard normal random inputs in the U-space The firstorder probabilistic performance measure 119866119901FORM can beobtained by solving
min 119866 (U)st U2 = 120573119905 (6)
where the optimal point on the surface U2 = 120573119905 is identifiedas MPP ulowast120573=120573119905 Furthermore 119866119901FORM = 119866(ulowast120573=120573119905)33 The Strategy of PMA-ERIM To improve the efficiencyand accuracy of the original RIM the ERIM is discussed inthis section Recall the statistic description of the failure of aperformance function 119892
Pr (119892 (X) le 0) = 119865119892 (0) = int119892(X)le0
119891X (x) 119889x (7)
Step 1 Using the Rosenblatt transformation (7) is equivalentto
Pr (119892 (U) le 0) = int sdot sdot sdot int119892(11990611199062sdotsdotsdot 119906119899)le0
Step 2 To evaluate the integration more easily the integrandboundary 119892(U) = 0 is approximated FORM utilizes the firstorder Taylor expansion as
119892 (U) asymp 119871119892 (U) = 119892 (ulowast) + [nabla119892 (ulowast)]119879 (U minus ulowast) (9)
where119871119892(U) is the linearized performance function ulowast is theexpansion point 119879 denotes transpose nabla119892(ulowast) is the gradientof 119892 at ulowast
To minimize the accuracy loss the performance functionshould be expanded at MPP MPP can be obtained by (4)
Because at MPP 119892(ulowast) = 0 the performance function islinearized as
119892 (U) asymp 119871119892 (U) = [nabla119892 (ulowast)]119879 (U minus ulowast)= minus [nabla119892 (ulowast)]119879 ulowast + [nabla119892 (ulowast)]119879U (11)
If the gradient nabla119892(ulowast) at ulowast is equal to zero then theperformance function is linearized as 119892(U) asymp 119871119892(U) = 0 Inthis case the linear approximation of the integrand boundarywill cause a large error about the integration because ofthe highly nonlinear character of performance functionTherefore if the above case appears the FORM is not suitableto deal with the problem In the following the case of thegradient of performance function at MPP unequal to zero isdiscussed
Since 119871119892(U) is a linear function of standard normalvariables 119871119892(U) is normally distributed Thus based on (11)the mean value and standard deviation of 119871119892(U) are 120583119871119892 =minus[nabla119892(ulowast)]119879ulowast and 120590119871119892 = nabla119892(ulowast)2 which can be utilized inthe derivation process in (12)
Therefore the probability of failure is calculated as
Pr (119892 (U) le 0) asymp Pr (119871119892 (U) le 0) = Φ(0 minus 120583119871119892120590119871119892 )
At the optimal point ulowast = nabla119892(ulowast)nabla119892(ulowast)2 sdot ulowast2 orulowast = minusnabla119892(ulowast)nabla119892(ulowast)2 sdot ulowast2 Hence the probability offailure is
Pr (119892 (U) le 0)
asymp
Φ(1003817100381710038171003817ulowast10038171003817100381710038172) if ulowast = nabla119892 (ulowast)1003817100381710038171003817nabla119892 (ulowast)10038171003817100381710038172 sdot1003817100381710038171003817ulowast10038171003817100381710038172
Φ(minus 1003817100381710038171003817ulowast10038171003817100381710038172) if ulowast = minus nabla119892 (ulowast)1003817100381710038171003817nabla119892 (ulowast)10038171003817100381710038172 sdot1003817100381710038171003817ulowast10038171003817100381710038172
(13)
From (12) and (13) the difference of judgment on thesatisfaction of probability constraint between RIM and thepractical situation is caused when ulowast = nabla119892(ulowast)nabla119892(ulowast)2 sdotulowast2 Utilizing (11) the probabilities of failure of the perfor-mance functions in (1) can be recalculated
To satisfy the probability constraint from (12) the prob-ability of failure should satisfy
As a well-known multilevel method for MDO the strategy ofCO is suitable for large-scale distributed engineering systemsThe CO algorithm decomposes coordinates and optimizescomplex engineering problems Every discipline in a systemcan enjoy good autonomy regardless of the influence of otherdisciplinesThe consistency betweendisciplines is guaranteedby compatibility constraints attached to the system-leveloptimization problem While the value of compatibility con-straint is obtained through subject-level optimization prob-lem The objectives of subject-level optimization problems
are to minimize the inconsistency between disciplines whilesatisfying the constraints of discipline design
Using CO the RBMDO problem in (1) is converted intothe system-level and subject-level optimization problemsThe formulation of the optimization problem in system levelis
st 119869119894= (d1015840119894 minus d119894)2 + (d1015840s minus ds)2+ (1205831015840X119894 minus 120583X119894)2 + (1205831015840Xs
minus 120583Xs)2
+ (1205831015840Y∙119894 minus 120583Y∙119894)2 le 120576119894 = 1 2 119899
(19)
where 119869119894 is the compatibility constraints d1015840119894 d1015840s1205831015840X119894 1205831015840Xs
and 1205831015840Y∙119894 are the design variables at the system level Theformulation of the optimization problem in subject-leveloptimization problems is
119869119894= (d1015840119894 minus d119894)2 + (d1015840s minus ds)2+ (1205831015840X119894 minus 120583X119894)2 + (1205831015840Xs
minus 120583Xs)2
+ (1205831015840Y∙119894 minus 120583Y∙119894)2st Pr119894 [119892119894 (d119894 dsX119894XsP) le 0]
le Φ (minus120573119905)120583Y119894∙ = Y119894∙ (d119894 ds120583X119894 120583Xs
120583Y∙119894)d119871119894 le d119894 le d119880119894 d119871s le ds le d119880s X119871119894 le 120583X119894 le X119880119894 X119871s le 120583Xs
le X119880s Y119871 le 120583Y le Y119880119894 = 1 2 119899
(20)
The corresponding bilevel strategy of CO is also shown inFigure 5
Considering uncertainties in practical engineering alldiscipline probability reliability constraints in (20) are con-verted into the corresponding RIM-based reliability con-straints These RIM-based reliability constraints only con-sider the performance reliability at MPP which can reducethe cost of reliability analysis and improve the computational
6 Mathematical Problems in Engineering
Table 1 The design information of speed reducer
Variables Lower and upper bound Distribution Mean Standard deviationgear face width 1199091 (cm) [26 36] - - -teeth module 1199092 (cm) [03 10] - - -number of teeth of pinion 1199093 [17 28] - - -distance between bearings 1 1199094 (cm) [73 83] Normal 1205831199094 0011205831199094distance between bearings 2 1199095 (cm) [73 83] Normal 1205831199095 0011205831199095diameter of shaft 1 1199096 (cm) [29 39] Normal 1205831199096 0011205831199096diameter of shaft 2 1199097 (cm) [5 55] Normal 1205831199097 0011205831199097
System optimization problem in Eq (19)
The 1st discipline optimizationproblem in Eq (20)
The 1st discipline analysis
Theproblem in Eq (20)
thi
XＭX
1
Y∙1Y
1∙
XＭX1
Y∙1Y1∙
XＭX
i
Y∙iY
i∙
XＭXi
Y∙iYi∙
XＭX1Y∙1 Y1∙XＭXi Y∙i Yi∙
discipline optimization
The thi discipline analysis
Figure 5 The bilevel strategy of CO
efficiency of RBMDO The formulation of the subject-leveloptimization problem in the U-space using RIM-based relia-bility constraints can be denoted as
transformation of Xd119871119894 le d119894 le d119880119894 d119871s le ds le d119880s X119871119894 le 120583X119894 le X119880119894 X119871s le 120583Xs
le X119880s
Y119871 le 120583Y le Y119880119894 = 1 2 119899
(21)
The detail information of CO-ERIM is as follows
Step 1 Input the original design information the cyclenumber 119896 = 0Step 2 Solve the system-level optimization problem in (19)During this process d1015840119894 d
1015840s 1205831015840X119894 1205831015840Xs and 1205831015840Y∙119894 are treated as the
design parameters
Step 3 Transform random variables in X-space into randomvariables in U-space using the Rosenblatt transformation
Step 4 Solve the subject-level optimization problems in (21)Then send the design solutions to the system level
Step 5 Obtain the value of 119869119894 119894 = 1 2 119899 If 119869119894 le 120576 andthe difference between the objective function values of twoconsecutive iterations is not more than a small number in theoptimization iteration process carry out Step 6 Otherwise119896 = 119896 + 1 and carry out Step 2
Step 6 Stop and output the design solutions
The flowchart of CO-ERIM is shown in Figure 6
5 Example
Speed reducers are generally used in low-speed high-torquetransmission equipment In this study a speed reducerRBMDO problem is introduced to illustrate the utilization ofthe proposedmethodThere are seven design variables in thisexample which is listed in Table 1 Twenty-five constraints areintroduced to ensure that the design solutions can satisfy thestrength stiffness and space requirements The optimizationobject is to minimize the overall weight Further informationcan be obtained in [14 43]
There are three disciplines in this RBMDO problemBearing-Shaft 1 Bearing-Shaft 2 and Gears which is shownin Figure 7The CO-ERIM strategy for this problem is shownin Figure 8 where 120573119905 = 207 Φ(minus120573119905) = 002 and 120576 = 0001
To illustrate the accuracy of design solutions the MonteCarlo Simulation (MCS) method is also introduced here as
Mathematical Problems in Engineering 7
Solve the optimization problemin Eq (19) at system level
Solve the discipline optimizationproblems in Eq (21) at subsystem level
Start
No
Yes
End
The ith discipline analysis
k = 0
Rosenblatt transformation
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1) ９
i∙(kminus1)
８Ｍ(kminus1) ８i(kminus1) ９∙i(kminus1) ９i∙(kminus1)
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1)
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1) ９i∙(kminus1)
k = k + 1
８Ｍ(kminus1) ８i(kminus1)
９∙i(kminus1)
＜ＮＣＨ ＮＢ ＰＦＯ Ｉ
Ji i = 1 2 n
８Ｍ(kminus1) ８i(kminus1) ９∙i(kminus1) ９i∙(kminus1)
(Ｃ = 1simn)
Ji le
Figure 6 The flowchart of CO-ERIM
Table 2 Optimization results of the reducer design
the reference The software Isight is utilized in the compu-tations of optimization The solutions from CO-ERIM arecompared with the ones from original PMA-RIM based COand MCS based CO which is listed in Table 2 From thecomparison of solutions the design results from ERIM arecloser to the design results from MCS Furthermore the
calculation time of ERIM is 17min23s and the calculationtime of RIM is 25min17s which means the proposed methodenjoys higher computational efficiency
6 Conclusions
In this study the efficiency problem of RBMDO is stud-ied The RIM strategy is reviewed and the correspondingalgorithm of ERIM is discussed in detail Furthermore theCO-ERIM strategy is proposed including its formulationand procedure In CO-ERIM the concurrent design idea isadapting to the development of modern engineering systemsCompatibility constraints are introduced into subsystem-level and system-level optimization problems respectivelyThe consistency between different disciplines can be guaran-teed when the RBMDO solutions are obtainedThe introduc-tion of ERIM reduces the computational burden of reliabilityanalysis during optimization iteration process Under thecondition that the first order method is acceptable thereliability analysis accuracy of the proposedmethod is similar
Figure 8 The MDO strategy for the speed reducer problem
to the accuracy ofMCSThe speed reducer example illustratesthe effectiveness of the proposed method
Data Availability
All data used to support the findings of this study are includedwithin the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thesupport from theNational Natural Science Foundation ofChina (Grant No 51605080 and No 11672070) the SichuanScience and Technology Program (Grant No 2019YFG0350and No 2019YFG0348) the China Postdoctoral ScienceFoundation (Grant No 2015M580780 and 2017T100685) andthe National Defense Pre-Research Foundation of China
(Grant No 6140244010216DZ02001) is gratefully acknowl-edged The authors also acknowledge the Portuguese Sci-ence Foundation (FCT) for the financial support throughthe postdoctoral grant SFRHBPD1078252015 as well asthe funding of Projects POCI-01-0145-FEDER-007457 andUIDECI047082019 - CONSTRUCT - Institute of RampDIn Structures and Construction funded by FEDER fundsthrough COMPETE2020 (POCI) and by national funds(PIDDAC) through Portuguese Science Foundation (FCT)
References
[1] L Li H Wan W Gao F Tong and H Li ldquoReliabilitybased multidisciplinary design optimization of cooling turbineblade considering uncertainty data statisticsrdquo Structural andMultidisciplinary Optimization vol 59 no 2 pp 1ndash15 2018
[2] Z L Huang Y S Zhou C Jiang J Zheng and X HanldquoReliability-based multidisciplinary design optimization usingincremental shifting vector strategy and its application inelectronic product designrdquo Acta Mechanica Sinica vol 34 no2 pp 285ndash302 2018
[3] H Xu X WangW Li M Li S Zhang and C Hu ldquoReliability-based multidisciplinary design optimization under correlated
Mathematical Problems in Engineering 9
uncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 7360615 12 pages 2017
[4] D Meng S Yang Y Zhang and S-P Zhu ldquoStructural relia-bility analysis and uncertainties-based collaborative design andoptimization of turbine blades using surrogate modelrdquo Fatigueamp Fracture of Engineering Materials amp Structures pp 1ndash9 2018httpsdoiorg101111ffe12906
[5] F Yang Z Yue L Li and D Guan ldquoHybrid reliability-based multidisciplinary design optimization with random andinterval variablesrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 232 no 1pp 52ndash64 2018
[6] L Wang C Xiong and Y Yang ldquoA novel methodology ofreliability-based multidisciplinary design optimization underhybrid interval and fuzzy uncertaintiesrdquo Computer MethodsApplied Mechanics and Engineering vol 337 pp 439ndash457 2018
[7] L Wang C Xiong J Hu X Wang and Z Qiu ldquoSequentialmultidisciplinary design optimization and reliability analysisunder interval uncertaintyrdquo Aerospace Science and Technologyvol 80 pp 508ndash519 2018
[8] J Zhang M Xiao L Gao H Qiu and Z Yang ldquoAn improvedtwo-stage framework of evidence-based design optimizationrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1673ndash1693 2018
[9] H Xu W Li L Xing and S-P Zhu ldquoMultidisciplinary designoptimization under correlated uncertaintiesrdquo Concurrent Engi-neering Research and Applications vol 25 no 3 pp 262ndash2752017
[10] X Wang R Wang L Wang X Chen and X Geng ldquoAnefficient single-loop strategy for reliability-based multidisci-plinarydesign optimizationunder non-probabilistic set theoryrdquoAerospace Science and Technology vol 73 pp 148ndash163 2018
[11] X P Du J Guo and H Beeram ldquoSequential optimizationand reliability assessment for multidisciplinary systems designrdquoStructural andMultidisciplinary Optimization vol 35 no 2 pp117ndash130 2008
[12] D Meng Y-F Li H-Z Huang Z Wang and Y LiuldquoReliability-based multidisciplinary design optimization usingsubset simulation analysis and its application in the hydraulictransmission mechanism designrdquo Journal of Mechanical Designvol 137 no 5 Article ID 051402 2015
[13] S Yu Z Wang and D Meng ldquoTime-variant reliability assess-ment for multiple failure modes and temporal parametersrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1705ndash1717 2018
[14] D Meng H-Z Huang Z Wang N-C Xiao and X-L ZhangldquoMean-value first-order saddlepoint approximation based col-laborative optimization for multidisciplinary problems underaleatory uncertaintyrdquo Journal of Mechanical Science and Tech-nology vol 28 no 10 pp 3925ndash3935 2014
[15] S-P Zhu Q Liu Q Lei and Q Wang ldquoProbabilistic fatiguelife prediction and reliability assessment of a high pressureturbine disc considering load variationsrdquo International Journalof Damage Mechanics vol 27 no 10 pp 1569ndash1588 2018
[16] D Liao S-P Zhu J A F O Correia A M P De Jesus andR Calcada ldquoComputational framework for multiaxial fatiguelife prediction of compressor discs considering notch effectsrdquoEngineering Fracture Mechanics vol 202 pp 423ndash435 2018
[17] D Meng M Liu S Yang H Zhang and R Ding ldquoAfluid-structure analysis approach and its application in theuncertainty-based multidisciplinary design and optimization
for bladesrdquo Advances in Mechanical Engineering vol 10 no 6pp 1ndash7 2018
[18] Z Meng G Li B P Wang and P Hao ldquoA hybrid chaos controlapproach of the performance measure functions for reliability-based design optimizationrdquo Computers Structures vol 146 pp32ndash43 2015
[19] X Li H Qiu Z Chen L Gao and X Shao ldquoA local Krigingapproximation method using MPP for reliability-based designoptimizationrdquo Computers amp Structures vol 162 pp 102ndash1152016
[20] S-P Zhu Q Liu W Peng and X-C Zhang ldquoComputational-experimental approaches for fatigue reliability assessment ofturbine bladed disksrdquo International Journal of Mechanical Sci-ences vol 142-143 pp 502ndash517 2018
[21] Z Chen H Qiu L Gao and P Li ldquoAn optimal shiftingvector approach for efficient probabilistic designrdquo Structuraland Multidisciplinary Optimization vol 47 no 6 pp 905ndash9202013
[22] W Deng X Lu and Y Deng ldquoEvidential model validationunder epistemic uncertaintyrdquo Mathematical Problems in Engi-neering vol 2018 Article ID 6789635 11 pages 2018
[23] D Meng H Zhang and T Huang ldquoA concurrent reliabilityoptimization procedure in the earlier design phases of complexengineering systems under epistemic uncertaintiesrdquo Advancesin Mechanical Engineering vol 8 no 10 pp 1ndash8 2016
[24] G Li Z Meng and H Hu ldquoAn adaptive hybrid approach forreliability-based design optimizationrdquo Structural and Multidis-ciplinary Optimization vol 51 no 5 pp 1051ndash1065 2015
[25] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018
[26] M A Valdebenito and G I Schueller ldquoA survey on approachesfor reliability-based optimizationrdquo Structural and Multidisci-plinary Optimization vol 42 no 5 pp 645ndash663 2010
[27] Y Aoues and A Chateauneuf ldquoBenchmark study of numericalmethods for reliability-based design optimizationrdquo Structuraland Multidisciplinary Optimization vol 41 no 2 pp 277ndash2942010
[28] J Ching and W-C Hsu ldquoTransforming reliability limit-stateconstraints into deterministic limit-state constraintsrdquo StructuralSafety vol 30 no 1 pp 11ndash33 2008
[29] TM Cho and B C Lee ldquoReliability-based design optimizationusing convex linearization and sequential optimization andreliability assessment methodrdquo Structural Safety vol 33 no 1pp 42ndash50 2011
[30] J Liang Z P Mourelatos and J Tu ldquoA single-loop method forreliability-based design optimisationrdquo International Journal ofProduct Development vol 5 no 1-2 pp 76ndash92 2008
[31] S Shan and G G Wang ldquoReliable design space and completesingle-loop reliability-based design optimizationrdquo ReliabilityEngineering amp System Safety vol 93 no 8 pp 1218ndash1230 2008
[32] P Yi G Cheng and L Jiang ldquoA sequential approximateprogramming strategy for performance-measure-based proba-bilistic structural design optimizationrdquo Structural Safety vol 30no 2 pp 91ndash109 2008
[33] Z M Yaseen and B Keshtegar ldquoLimited descent-based meanvalue method for inverse reliability analysisrdquo Engineering withComputers pp 1ndash13 2018
[34] B D Youn K K Choi and Y H Park ldquoHybrid analysis methodfor reliability-based design optimizationrdquo Journal of MechanicalDesign vol 125 no 2 pp 221ndash232 2003
10 Mathematical Problems in Engineering
[35] B D Youn K K Choi and L Du ldquoEnriched performancemeasure approach for reliability-based design optimizationrdquoAIAA Journal vol 43 no 4 pp 874ndash884 2005
[36] B D Youn and K K Choi ldquoSelecting probabilistic approachesfor realiability-based design optimizationrdquo AIAA Journal vol42 no 1 pp 124ndash131 2004
[37] S P Zhu Q Liu J Zhou and Z Y Yu ldquoFatigue reliabilityassessment of turbine discs under multi-source uncertaintiesrdquoFatigue Fracture of Engineering Materials Structures vol 41 no6 pp 1291ndash1305 2018
[38] H Li R Li H Li and R Yuan ldquoReliability modeling of mul-tiple performance based on degradation values distributionrdquoAdvances in Mechanical Engineering vol 8 no 10 Article ID168781401667375 2016
[39] B D Youn K K Choi and L Du ldquoAdaptive probability analysisusing an enhanced hybrid mean value methodrdquo Structural andMultidisciplinary Optimization vol 29 no 2 pp 134ndash148 2005
[40] M Ito N H Kim and N Kogiso ldquoConservative reliabilityindex for epistemic uncertainty in reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 5 pp 1919ndash1935 2018
[41] Z Meng D Yang H Zhou and B P Wang ldquoConvergencecontrol of single loop approach for reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 3 pp 1079ndash1091 2018
[42] W Yao X Q Chen W C Luo M van Tooren and J GuoldquoReview of uncertainty-based multidisciplinary design opti-mization methods for aerospace vehiclesrdquo Progress in AerospaceSciences vol 47 no 6 pp 450ndash479 2011
[43] R Yuan D Meng and H Li ldquoMultidisciplinary reliabilitydesign optimizationusing an enhanced saddlepoint approxima-tion in the framework of sequential optimization and reliabilityanalysisrdquo Proceedings of the Institution of Mechanical EngineersPart O Journal of Risk and Reliability vol 230 no 6 pp 570ndash578 2016
[44] D Meng X Zhang Y Yang H Xu and H Huang ldquoInteractionbalance optimization in multidisciplinary design optimizationproblemsrdquo Concurrent Engineering Research and Applicationsvol 24 no 1 pp 48ndash57 2016
[45] R Yuan and H Li ldquoA multidisciplinary coupling relationshipcoordination algorithm using the hierarchical control methodsof complex systems and its application in multidisciplinarydesign optimizationrdquo Advances in Mechanical Engineering vol9 no 1 pp 1ndash11 2016
[46] D Meng X Zhang H-Z Huang Z Wang and H Xu ldquoInter-action prediction optimization in multidisciplinary designoptimization problemsrdquoThe Scientific World Journal vol 2014Article ID 698453 7 pages 2014
[47] J R R A Martins and A B Lambe ldquoMultidisciplinary designoptimization a survey of architecturesrdquo AIAA Journal vol 51no 9 pp 2049ndash2075 2013
[48] Y Li and Y Deng ldquoGeneralized ordered propositions fusionbased on belief entropyrdquo International Journal of ComputersCommunications amp Control vol 13 no 5 pp 792ndash807 2018
[49] X Su S Mahadevan W Han and Y Deng ldquoCombiningdependent bodies of evidencerdquo Applied Intelligence vol 44 no3 pp 634ndash644 2016
[50] S-P Zhu H-Z Huang W Peng H-K Wang and S Mahade-van ldquoProbabilistic Physics of Failure-based framework forfatigue life prediction of aircraft gas turbine discs under uncer-taintyrdquoReliability Engineering amp System Safety vol 146 pp 1ndash122016
[51] S I Yi J K Shin andG J Park ldquoComparison ofMDOmethodswith mathematical examplesrdquo Structural and MultidisciplinaryOptimization vol 35 no 5 pp 391ndash402 2008
[52] H XuW Li M Li C Hu S Zhang and XWang ldquoMultidisci-plinary robust design optimization based on time-varying sen-sitivity analysisrdquo Journal of Mechanical Science and Technologyvol 32 no 3 pp 1195ndash1207 2018
Hindawiwwwhindawicom Volume 2018
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Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
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Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 5
At the optimal point ulowast = nabla119892(ulowast)nabla119892(ulowast)2 sdot ulowast2 orulowast = minusnabla119892(ulowast)nabla119892(ulowast)2 sdot ulowast2 Hence the probability offailure is
Pr (119892 (U) le 0)
asymp
Φ(1003817100381710038171003817ulowast10038171003817100381710038172) if ulowast = nabla119892 (ulowast)1003817100381710038171003817nabla119892 (ulowast)10038171003817100381710038172 sdot1003817100381710038171003817ulowast10038171003817100381710038172
Φ(minus 1003817100381710038171003817ulowast10038171003817100381710038172) if ulowast = minus nabla119892 (ulowast)1003817100381710038171003817nabla119892 (ulowast)10038171003817100381710038172 sdot1003817100381710038171003817ulowast10038171003817100381710038172
(13)
From (12) and (13) the difference of judgment on thesatisfaction of probability constraint between RIM and thepractical situation is caused when ulowast = nabla119892(ulowast)nabla119892(ulowast)2 sdotulowast2 Utilizing (11) the probabilities of failure of the perfor-mance functions in (1) can be recalculated
To satisfy the probability constraint from (12) the prob-ability of failure should satisfy
As a well-known multilevel method for MDO the strategy ofCO is suitable for large-scale distributed engineering systemsThe CO algorithm decomposes coordinates and optimizescomplex engineering problems Every discipline in a systemcan enjoy good autonomy regardless of the influence of otherdisciplinesThe consistency betweendisciplines is guaranteedby compatibility constraints attached to the system-leveloptimization problem While the value of compatibility con-straint is obtained through subject-level optimization prob-lem The objectives of subject-level optimization problems
are to minimize the inconsistency between disciplines whilesatisfying the constraints of discipline design
Using CO the RBMDO problem in (1) is converted intothe system-level and subject-level optimization problemsThe formulation of the optimization problem in system levelis
st 119869119894= (d1015840119894 minus d119894)2 + (d1015840s minus ds)2+ (1205831015840X119894 minus 120583X119894)2 + (1205831015840Xs
minus 120583Xs)2
+ (1205831015840Y∙119894 minus 120583Y∙119894)2 le 120576119894 = 1 2 119899
(19)
where 119869119894 is the compatibility constraints d1015840119894 d1015840s1205831015840X119894 1205831015840Xs
and 1205831015840Y∙119894 are the design variables at the system level Theformulation of the optimization problem in subject-leveloptimization problems is
119869119894= (d1015840119894 minus d119894)2 + (d1015840s minus ds)2+ (1205831015840X119894 minus 120583X119894)2 + (1205831015840Xs
minus 120583Xs)2
+ (1205831015840Y∙119894 minus 120583Y∙119894)2st Pr119894 [119892119894 (d119894 dsX119894XsP) le 0]
le Φ (minus120573119905)120583Y119894∙ = Y119894∙ (d119894 ds120583X119894 120583Xs
120583Y∙119894)d119871119894 le d119894 le d119880119894 d119871s le ds le d119880s X119871119894 le 120583X119894 le X119880119894 X119871s le 120583Xs
le X119880s Y119871 le 120583Y le Y119880119894 = 1 2 119899
(20)
The corresponding bilevel strategy of CO is also shown inFigure 5
Considering uncertainties in practical engineering alldiscipline probability reliability constraints in (20) are con-verted into the corresponding RIM-based reliability con-straints These RIM-based reliability constraints only con-sider the performance reliability at MPP which can reducethe cost of reliability analysis and improve the computational
6 Mathematical Problems in Engineering
Table 1 The design information of speed reducer
Variables Lower and upper bound Distribution Mean Standard deviationgear face width 1199091 (cm) [26 36] - - -teeth module 1199092 (cm) [03 10] - - -number of teeth of pinion 1199093 [17 28] - - -distance between bearings 1 1199094 (cm) [73 83] Normal 1205831199094 0011205831199094distance between bearings 2 1199095 (cm) [73 83] Normal 1205831199095 0011205831199095diameter of shaft 1 1199096 (cm) [29 39] Normal 1205831199096 0011205831199096diameter of shaft 2 1199097 (cm) [5 55] Normal 1205831199097 0011205831199097
System optimization problem in Eq (19)
The 1st discipline optimizationproblem in Eq (20)
The 1st discipline analysis
Theproblem in Eq (20)
thi
XＭX
1
Y∙1Y
1∙
XＭX1
Y∙1Y1∙
XＭX
i
Y∙iY
i∙
XＭXi
Y∙iYi∙
XＭX1Y∙1 Y1∙XＭXi Y∙i Yi∙
discipline optimization
The thi discipline analysis
Figure 5 The bilevel strategy of CO
efficiency of RBMDO The formulation of the subject-leveloptimization problem in the U-space using RIM-based relia-bility constraints can be denoted as
transformation of Xd119871119894 le d119894 le d119880119894 d119871s le ds le d119880s X119871119894 le 120583X119894 le X119880119894 X119871s le 120583Xs
le X119880s
Y119871 le 120583Y le Y119880119894 = 1 2 119899
(21)
The detail information of CO-ERIM is as follows
Step 1 Input the original design information the cyclenumber 119896 = 0Step 2 Solve the system-level optimization problem in (19)During this process d1015840119894 d
1015840s 1205831015840X119894 1205831015840Xs and 1205831015840Y∙119894 are treated as the
design parameters
Step 3 Transform random variables in X-space into randomvariables in U-space using the Rosenblatt transformation
Step 4 Solve the subject-level optimization problems in (21)Then send the design solutions to the system level
Step 5 Obtain the value of 119869119894 119894 = 1 2 119899 If 119869119894 le 120576 andthe difference between the objective function values of twoconsecutive iterations is not more than a small number in theoptimization iteration process carry out Step 6 Otherwise119896 = 119896 + 1 and carry out Step 2
Step 6 Stop and output the design solutions
The flowchart of CO-ERIM is shown in Figure 6
5 Example
Speed reducers are generally used in low-speed high-torquetransmission equipment In this study a speed reducerRBMDO problem is introduced to illustrate the utilization ofthe proposedmethodThere are seven design variables in thisexample which is listed in Table 1 Twenty-five constraints areintroduced to ensure that the design solutions can satisfy thestrength stiffness and space requirements The optimizationobject is to minimize the overall weight Further informationcan be obtained in [14 43]
There are three disciplines in this RBMDO problemBearing-Shaft 1 Bearing-Shaft 2 and Gears which is shownin Figure 7The CO-ERIM strategy for this problem is shownin Figure 8 where 120573119905 = 207 Φ(minus120573119905) = 002 and 120576 = 0001
To illustrate the accuracy of design solutions the MonteCarlo Simulation (MCS) method is also introduced here as
Mathematical Problems in Engineering 7
Solve the optimization problemin Eq (19) at system level
Solve the discipline optimizationproblems in Eq (21) at subsystem level
Start
No
Yes
End
The ith discipline analysis
k = 0
Rosenblatt transformation
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1) ９
i∙(kminus1)
８Ｍ(kminus1) ８i(kminus1) ９∙i(kminus1) ９i∙(kminus1)
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1)
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1) ９i∙(kminus1)
k = k + 1
８Ｍ(kminus1) ８i(kminus1)
９∙i(kminus1)
＜ＮＣＨ ＮＢ ＰＦＯ Ｉ
Ji i = 1 2 n
８Ｍ(kminus1) ８i(kminus1) ９∙i(kminus1) ９i∙(kminus1)
(Ｃ = 1simn)
Ji le
Figure 6 The flowchart of CO-ERIM
Table 2 Optimization results of the reducer design
the reference The software Isight is utilized in the compu-tations of optimization The solutions from CO-ERIM arecompared with the ones from original PMA-RIM based COand MCS based CO which is listed in Table 2 From thecomparison of solutions the design results from ERIM arecloser to the design results from MCS Furthermore the
calculation time of ERIM is 17min23s and the calculationtime of RIM is 25min17s which means the proposed methodenjoys higher computational efficiency
6 Conclusions
In this study the efficiency problem of RBMDO is stud-ied The RIM strategy is reviewed and the correspondingalgorithm of ERIM is discussed in detail Furthermore theCO-ERIM strategy is proposed including its formulationand procedure In CO-ERIM the concurrent design idea isadapting to the development of modern engineering systemsCompatibility constraints are introduced into subsystem-level and system-level optimization problems respectivelyThe consistency between different disciplines can be guaran-teed when the RBMDO solutions are obtainedThe introduc-tion of ERIM reduces the computational burden of reliabilityanalysis during optimization iteration process Under thecondition that the first order method is acceptable thereliability analysis accuracy of the proposedmethod is similar
Figure 8 The MDO strategy for the speed reducer problem
to the accuracy ofMCSThe speed reducer example illustratesthe effectiveness of the proposed method
Data Availability
All data used to support the findings of this study are includedwithin the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thesupport from theNational Natural Science Foundation ofChina (Grant No 51605080 and No 11672070) the SichuanScience and Technology Program (Grant No 2019YFG0350and No 2019YFG0348) the China Postdoctoral ScienceFoundation (Grant No 2015M580780 and 2017T100685) andthe National Defense Pre-Research Foundation of China
(Grant No 6140244010216DZ02001) is gratefully acknowl-edged The authors also acknowledge the Portuguese Sci-ence Foundation (FCT) for the financial support throughthe postdoctoral grant SFRHBPD1078252015 as well asthe funding of Projects POCI-01-0145-FEDER-007457 andUIDECI047082019 - CONSTRUCT - Institute of RampDIn Structures and Construction funded by FEDER fundsthrough COMPETE2020 (POCI) and by national funds(PIDDAC) through Portuguese Science Foundation (FCT)
References
[1] L Li H Wan W Gao F Tong and H Li ldquoReliabilitybased multidisciplinary design optimization of cooling turbineblade considering uncertainty data statisticsrdquo Structural andMultidisciplinary Optimization vol 59 no 2 pp 1ndash15 2018
[2] Z L Huang Y S Zhou C Jiang J Zheng and X HanldquoReliability-based multidisciplinary design optimization usingincremental shifting vector strategy and its application inelectronic product designrdquo Acta Mechanica Sinica vol 34 no2 pp 285ndash302 2018
[3] H Xu X WangW Li M Li S Zhang and C Hu ldquoReliability-based multidisciplinary design optimization under correlated
Mathematical Problems in Engineering 9
uncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 7360615 12 pages 2017
[4] D Meng S Yang Y Zhang and S-P Zhu ldquoStructural relia-bility analysis and uncertainties-based collaborative design andoptimization of turbine blades using surrogate modelrdquo Fatigueamp Fracture of Engineering Materials amp Structures pp 1ndash9 2018httpsdoiorg101111ffe12906
[5] F Yang Z Yue L Li and D Guan ldquoHybrid reliability-based multidisciplinary design optimization with random andinterval variablesrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 232 no 1pp 52ndash64 2018
[6] L Wang C Xiong and Y Yang ldquoA novel methodology ofreliability-based multidisciplinary design optimization underhybrid interval and fuzzy uncertaintiesrdquo Computer MethodsApplied Mechanics and Engineering vol 337 pp 439ndash457 2018
[7] L Wang C Xiong J Hu X Wang and Z Qiu ldquoSequentialmultidisciplinary design optimization and reliability analysisunder interval uncertaintyrdquo Aerospace Science and Technologyvol 80 pp 508ndash519 2018
[8] J Zhang M Xiao L Gao H Qiu and Z Yang ldquoAn improvedtwo-stage framework of evidence-based design optimizationrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1673ndash1693 2018
[9] H Xu W Li L Xing and S-P Zhu ldquoMultidisciplinary designoptimization under correlated uncertaintiesrdquo Concurrent Engi-neering Research and Applications vol 25 no 3 pp 262ndash2752017
[10] X Wang R Wang L Wang X Chen and X Geng ldquoAnefficient single-loop strategy for reliability-based multidisci-plinarydesign optimizationunder non-probabilistic set theoryrdquoAerospace Science and Technology vol 73 pp 148ndash163 2018
[11] X P Du J Guo and H Beeram ldquoSequential optimizationand reliability assessment for multidisciplinary systems designrdquoStructural andMultidisciplinary Optimization vol 35 no 2 pp117ndash130 2008
[12] D Meng Y-F Li H-Z Huang Z Wang and Y LiuldquoReliability-based multidisciplinary design optimization usingsubset simulation analysis and its application in the hydraulictransmission mechanism designrdquo Journal of Mechanical Designvol 137 no 5 Article ID 051402 2015
[13] S Yu Z Wang and D Meng ldquoTime-variant reliability assess-ment for multiple failure modes and temporal parametersrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1705ndash1717 2018
[14] D Meng H-Z Huang Z Wang N-C Xiao and X-L ZhangldquoMean-value first-order saddlepoint approximation based col-laborative optimization for multidisciplinary problems underaleatory uncertaintyrdquo Journal of Mechanical Science and Tech-nology vol 28 no 10 pp 3925ndash3935 2014
[15] S-P Zhu Q Liu Q Lei and Q Wang ldquoProbabilistic fatiguelife prediction and reliability assessment of a high pressureturbine disc considering load variationsrdquo International Journalof Damage Mechanics vol 27 no 10 pp 1569ndash1588 2018
[16] D Liao S-P Zhu J A F O Correia A M P De Jesus andR Calcada ldquoComputational framework for multiaxial fatiguelife prediction of compressor discs considering notch effectsrdquoEngineering Fracture Mechanics vol 202 pp 423ndash435 2018
[17] D Meng M Liu S Yang H Zhang and R Ding ldquoAfluid-structure analysis approach and its application in theuncertainty-based multidisciplinary design and optimization
for bladesrdquo Advances in Mechanical Engineering vol 10 no 6pp 1ndash7 2018
[18] Z Meng G Li B P Wang and P Hao ldquoA hybrid chaos controlapproach of the performance measure functions for reliability-based design optimizationrdquo Computers Structures vol 146 pp32ndash43 2015
[19] X Li H Qiu Z Chen L Gao and X Shao ldquoA local Krigingapproximation method using MPP for reliability-based designoptimizationrdquo Computers amp Structures vol 162 pp 102ndash1152016
[20] S-P Zhu Q Liu W Peng and X-C Zhang ldquoComputational-experimental approaches for fatigue reliability assessment ofturbine bladed disksrdquo International Journal of Mechanical Sci-ences vol 142-143 pp 502ndash517 2018
[21] Z Chen H Qiu L Gao and P Li ldquoAn optimal shiftingvector approach for efficient probabilistic designrdquo Structuraland Multidisciplinary Optimization vol 47 no 6 pp 905ndash9202013
[22] W Deng X Lu and Y Deng ldquoEvidential model validationunder epistemic uncertaintyrdquo Mathematical Problems in Engi-neering vol 2018 Article ID 6789635 11 pages 2018
[23] D Meng H Zhang and T Huang ldquoA concurrent reliabilityoptimization procedure in the earlier design phases of complexengineering systems under epistemic uncertaintiesrdquo Advancesin Mechanical Engineering vol 8 no 10 pp 1ndash8 2016
[24] G Li Z Meng and H Hu ldquoAn adaptive hybrid approach forreliability-based design optimizationrdquo Structural and Multidis-ciplinary Optimization vol 51 no 5 pp 1051ndash1065 2015
[25] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018
[26] M A Valdebenito and G I Schueller ldquoA survey on approachesfor reliability-based optimizationrdquo Structural and Multidisci-plinary Optimization vol 42 no 5 pp 645ndash663 2010
[27] Y Aoues and A Chateauneuf ldquoBenchmark study of numericalmethods for reliability-based design optimizationrdquo Structuraland Multidisciplinary Optimization vol 41 no 2 pp 277ndash2942010
[28] J Ching and W-C Hsu ldquoTransforming reliability limit-stateconstraints into deterministic limit-state constraintsrdquo StructuralSafety vol 30 no 1 pp 11ndash33 2008
[29] TM Cho and B C Lee ldquoReliability-based design optimizationusing convex linearization and sequential optimization andreliability assessment methodrdquo Structural Safety vol 33 no 1pp 42ndash50 2011
[30] J Liang Z P Mourelatos and J Tu ldquoA single-loop method forreliability-based design optimisationrdquo International Journal ofProduct Development vol 5 no 1-2 pp 76ndash92 2008
[31] S Shan and G G Wang ldquoReliable design space and completesingle-loop reliability-based design optimizationrdquo ReliabilityEngineering amp System Safety vol 93 no 8 pp 1218ndash1230 2008
[32] P Yi G Cheng and L Jiang ldquoA sequential approximateprogramming strategy for performance-measure-based proba-bilistic structural design optimizationrdquo Structural Safety vol 30no 2 pp 91ndash109 2008
[33] Z M Yaseen and B Keshtegar ldquoLimited descent-based meanvalue method for inverse reliability analysisrdquo Engineering withComputers pp 1ndash13 2018
[34] B D Youn K K Choi and Y H Park ldquoHybrid analysis methodfor reliability-based design optimizationrdquo Journal of MechanicalDesign vol 125 no 2 pp 221ndash232 2003
10 Mathematical Problems in Engineering
[35] B D Youn K K Choi and L Du ldquoEnriched performancemeasure approach for reliability-based design optimizationrdquoAIAA Journal vol 43 no 4 pp 874ndash884 2005
[36] B D Youn and K K Choi ldquoSelecting probabilistic approachesfor realiability-based design optimizationrdquo AIAA Journal vol42 no 1 pp 124ndash131 2004
[37] S P Zhu Q Liu J Zhou and Z Y Yu ldquoFatigue reliabilityassessment of turbine discs under multi-source uncertaintiesrdquoFatigue Fracture of Engineering Materials Structures vol 41 no6 pp 1291ndash1305 2018
[38] H Li R Li H Li and R Yuan ldquoReliability modeling of mul-tiple performance based on degradation values distributionrdquoAdvances in Mechanical Engineering vol 8 no 10 Article ID168781401667375 2016
[39] B D Youn K K Choi and L Du ldquoAdaptive probability analysisusing an enhanced hybrid mean value methodrdquo Structural andMultidisciplinary Optimization vol 29 no 2 pp 134ndash148 2005
[40] M Ito N H Kim and N Kogiso ldquoConservative reliabilityindex for epistemic uncertainty in reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 5 pp 1919ndash1935 2018
[41] Z Meng D Yang H Zhou and B P Wang ldquoConvergencecontrol of single loop approach for reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 3 pp 1079ndash1091 2018
[42] W Yao X Q Chen W C Luo M van Tooren and J GuoldquoReview of uncertainty-based multidisciplinary design opti-mization methods for aerospace vehiclesrdquo Progress in AerospaceSciences vol 47 no 6 pp 450ndash479 2011
[43] R Yuan D Meng and H Li ldquoMultidisciplinary reliabilitydesign optimizationusing an enhanced saddlepoint approxima-tion in the framework of sequential optimization and reliabilityanalysisrdquo Proceedings of the Institution of Mechanical EngineersPart O Journal of Risk and Reliability vol 230 no 6 pp 570ndash578 2016
[44] D Meng X Zhang Y Yang H Xu and H Huang ldquoInteractionbalance optimization in multidisciplinary design optimizationproblemsrdquo Concurrent Engineering Research and Applicationsvol 24 no 1 pp 48ndash57 2016
[45] R Yuan and H Li ldquoA multidisciplinary coupling relationshipcoordination algorithm using the hierarchical control methodsof complex systems and its application in multidisciplinarydesign optimizationrdquo Advances in Mechanical Engineering vol9 no 1 pp 1ndash11 2016
[46] D Meng X Zhang H-Z Huang Z Wang and H Xu ldquoInter-action prediction optimization in multidisciplinary designoptimization problemsrdquoThe Scientific World Journal vol 2014Article ID 698453 7 pages 2014
[47] J R R A Martins and A B Lambe ldquoMultidisciplinary designoptimization a survey of architecturesrdquo AIAA Journal vol 51no 9 pp 2049ndash2075 2013
[48] Y Li and Y Deng ldquoGeneralized ordered propositions fusionbased on belief entropyrdquo International Journal of ComputersCommunications amp Control vol 13 no 5 pp 792ndash807 2018
[49] X Su S Mahadevan W Han and Y Deng ldquoCombiningdependent bodies of evidencerdquo Applied Intelligence vol 44 no3 pp 634ndash644 2016
[50] S-P Zhu H-Z Huang W Peng H-K Wang and S Mahade-van ldquoProbabilistic Physics of Failure-based framework forfatigue life prediction of aircraft gas turbine discs under uncer-taintyrdquoReliability Engineering amp System Safety vol 146 pp 1ndash122016
[51] S I Yi J K Shin andG J Park ldquoComparison ofMDOmethodswith mathematical examplesrdquo Structural and MultidisciplinaryOptimization vol 35 no 5 pp 391ndash402 2008
[52] H XuW Li M Li C Hu S Zhang and XWang ldquoMultidisci-plinary robust design optimization based on time-varying sen-sitivity analysisrdquo Journal of Mechanical Science and Technologyvol 32 no 3 pp 1195ndash1207 2018
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
transformation of Xd119871119894 le d119894 le d119880119894 d119871s le ds le d119880s X119871119894 le 120583X119894 le X119880119894 X119871s le 120583Xs
le X119880s
Y119871 le 120583Y le Y119880119894 = 1 2 119899
(21)
The detail information of CO-ERIM is as follows
Step 1 Input the original design information the cyclenumber 119896 = 0Step 2 Solve the system-level optimization problem in (19)During this process d1015840119894 d
1015840s 1205831015840X119894 1205831015840Xs and 1205831015840Y∙119894 are treated as the
design parameters
Step 3 Transform random variables in X-space into randomvariables in U-space using the Rosenblatt transformation
Step 4 Solve the subject-level optimization problems in (21)Then send the design solutions to the system level
Step 5 Obtain the value of 119869119894 119894 = 1 2 119899 If 119869119894 le 120576 andthe difference between the objective function values of twoconsecutive iterations is not more than a small number in theoptimization iteration process carry out Step 6 Otherwise119896 = 119896 + 1 and carry out Step 2
Step 6 Stop and output the design solutions
The flowchart of CO-ERIM is shown in Figure 6
5 Example
Speed reducers are generally used in low-speed high-torquetransmission equipment In this study a speed reducerRBMDO problem is introduced to illustrate the utilization ofthe proposedmethodThere are seven design variables in thisexample which is listed in Table 1 Twenty-five constraints areintroduced to ensure that the design solutions can satisfy thestrength stiffness and space requirements The optimizationobject is to minimize the overall weight Further informationcan be obtained in [14 43]
There are three disciplines in this RBMDO problemBearing-Shaft 1 Bearing-Shaft 2 and Gears which is shownin Figure 7The CO-ERIM strategy for this problem is shownin Figure 8 where 120573119905 = 207 Φ(minus120573119905) = 002 and 120576 = 0001
To illustrate the accuracy of design solutions the MonteCarlo Simulation (MCS) method is also introduced here as
Mathematical Problems in Engineering 7
Solve the optimization problemin Eq (19) at system level
Solve the discipline optimizationproblems in Eq (21) at subsystem level
Start
No
Yes
End
The ith discipline analysis
k = 0
Rosenblatt transformation
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1) ９
i∙(kminus1)
８Ｍ(kminus1) ８i(kminus1) ９∙i(kminus1) ９i∙(kminus1)
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1)
８Ｍ(kminus1) ８
i(kminus1) ９
∙i(kminus1) ９i∙(kminus1)
k = k + 1
８Ｍ(kminus1) ８i(kminus1)
９∙i(kminus1)
＜ＮＣＨ ＮＢ ＰＦＯ Ｉ
Ji i = 1 2 n
８Ｍ(kminus1) ８i(kminus1) ９∙i(kminus1) ９i∙(kminus1)
(Ｃ = 1simn)
Ji le
Figure 6 The flowchart of CO-ERIM
Table 2 Optimization results of the reducer design
the reference The software Isight is utilized in the compu-tations of optimization The solutions from CO-ERIM arecompared with the ones from original PMA-RIM based COand MCS based CO which is listed in Table 2 From thecomparison of solutions the design results from ERIM arecloser to the design results from MCS Furthermore the
calculation time of ERIM is 17min23s and the calculationtime of RIM is 25min17s which means the proposed methodenjoys higher computational efficiency
6 Conclusions
In this study the efficiency problem of RBMDO is stud-ied The RIM strategy is reviewed and the correspondingalgorithm of ERIM is discussed in detail Furthermore theCO-ERIM strategy is proposed including its formulationand procedure In CO-ERIM the concurrent design idea isadapting to the development of modern engineering systemsCompatibility constraints are introduced into subsystem-level and system-level optimization problems respectivelyThe consistency between different disciplines can be guaran-teed when the RBMDO solutions are obtainedThe introduc-tion of ERIM reduces the computational burden of reliabilityanalysis during optimization iteration process Under thecondition that the first order method is acceptable thereliability analysis accuracy of the proposedmethod is similar
Figure 8 The MDO strategy for the speed reducer problem
to the accuracy ofMCSThe speed reducer example illustratesthe effectiveness of the proposed method
Data Availability
All data used to support the findings of this study are includedwithin the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thesupport from theNational Natural Science Foundation ofChina (Grant No 51605080 and No 11672070) the SichuanScience and Technology Program (Grant No 2019YFG0350and No 2019YFG0348) the China Postdoctoral ScienceFoundation (Grant No 2015M580780 and 2017T100685) andthe National Defense Pre-Research Foundation of China
(Grant No 6140244010216DZ02001) is gratefully acknowl-edged The authors also acknowledge the Portuguese Sci-ence Foundation (FCT) for the financial support throughthe postdoctoral grant SFRHBPD1078252015 as well asthe funding of Projects POCI-01-0145-FEDER-007457 andUIDECI047082019 - CONSTRUCT - Institute of RampDIn Structures and Construction funded by FEDER fundsthrough COMPETE2020 (POCI) and by national funds(PIDDAC) through Portuguese Science Foundation (FCT)
References
[1] L Li H Wan W Gao F Tong and H Li ldquoReliabilitybased multidisciplinary design optimization of cooling turbineblade considering uncertainty data statisticsrdquo Structural andMultidisciplinary Optimization vol 59 no 2 pp 1ndash15 2018
[2] Z L Huang Y S Zhou C Jiang J Zheng and X HanldquoReliability-based multidisciplinary design optimization usingincremental shifting vector strategy and its application inelectronic product designrdquo Acta Mechanica Sinica vol 34 no2 pp 285ndash302 2018
[3] H Xu X WangW Li M Li S Zhang and C Hu ldquoReliability-based multidisciplinary design optimization under correlated
Mathematical Problems in Engineering 9
uncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 7360615 12 pages 2017
[4] D Meng S Yang Y Zhang and S-P Zhu ldquoStructural relia-bility analysis and uncertainties-based collaborative design andoptimization of turbine blades using surrogate modelrdquo Fatigueamp Fracture of Engineering Materials amp Structures pp 1ndash9 2018httpsdoiorg101111ffe12906
[5] F Yang Z Yue L Li and D Guan ldquoHybrid reliability-based multidisciplinary design optimization with random andinterval variablesrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 232 no 1pp 52ndash64 2018
[6] L Wang C Xiong and Y Yang ldquoA novel methodology ofreliability-based multidisciplinary design optimization underhybrid interval and fuzzy uncertaintiesrdquo Computer MethodsApplied Mechanics and Engineering vol 337 pp 439ndash457 2018
[7] L Wang C Xiong J Hu X Wang and Z Qiu ldquoSequentialmultidisciplinary design optimization and reliability analysisunder interval uncertaintyrdquo Aerospace Science and Technologyvol 80 pp 508ndash519 2018
[8] J Zhang M Xiao L Gao H Qiu and Z Yang ldquoAn improvedtwo-stage framework of evidence-based design optimizationrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1673ndash1693 2018
[9] H Xu W Li L Xing and S-P Zhu ldquoMultidisciplinary designoptimization under correlated uncertaintiesrdquo Concurrent Engi-neering Research and Applications vol 25 no 3 pp 262ndash2752017
[10] X Wang R Wang L Wang X Chen and X Geng ldquoAnefficient single-loop strategy for reliability-based multidisci-plinarydesign optimizationunder non-probabilistic set theoryrdquoAerospace Science and Technology vol 73 pp 148ndash163 2018
[11] X P Du J Guo and H Beeram ldquoSequential optimizationand reliability assessment for multidisciplinary systems designrdquoStructural andMultidisciplinary Optimization vol 35 no 2 pp117ndash130 2008
[12] D Meng Y-F Li H-Z Huang Z Wang and Y LiuldquoReliability-based multidisciplinary design optimization usingsubset simulation analysis and its application in the hydraulictransmission mechanism designrdquo Journal of Mechanical Designvol 137 no 5 Article ID 051402 2015
[13] S Yu Z Wang and D Meng ldquoTime-variant reliability assess-ment for multiple failure modes and temporal parametersrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1705ndash1717 2018
[14] D Meng H-Z Huang Z Wang N-C Xiao and X-L ZhangldquoMean-value first-order saddlepoint approximation based col-laborative optimization for multidisciplinary problems underaleatory uncertaintyrdquo Journal of Mechanical Science and Tech-nology vol 28 no 10 pp 3925ndash3935 2014
[15] S-P Zhu Q Liu Q Lei and Q Wang ldquoProbabilistic fatiguelife prediction and reliability assessment of a high pressureturbine disc considering load variationsrdquo International Journalof Damage Mechanics vol 27 no 10 pp 1569ndash1588 2018
[16] D Liao S-P Zhu J A F O Correia A M P De Jesus andR Calcada ldquoComputational framework for multiaxial fatiguelife prediction of compressor discs considering notch effectsrdquoEngineering Fracture Mechanics vol 202 pp 423ndash435 2018
[17] D Meng M Liu S Yang H Zhang and R Ding ldquoAfluid-structure analysis approach and its application in theuncertainty-based multidisciplinary design and optimization
for bladesrdquo Advances in Mechanical Engineering vol 10 no 6pp 1ndash7 2018
[18] Z Meng G Li B P Wang and P Hao ldquoA hybrid chaos controlapproach of the performance measure functions for reliability-based design optimizationrdquo Computers Structures vol 146 pp32ndash43 2015
[19] X Li H Qiu Z Chen L Gao and X Shao ldquoA local Krigingapproximation method using MPP for reliability-based designoptimizationrdquo Computers amp Structures vol 162 pp 102ndash1152016
[20] S-P Zhu Q Liu W Peng and X-C Zhang ldquoComputational-experimental approaches for fatigue reliability assessment ofturbine bladed disksrdquo International Journal of Mechanical Sci-ences vol 142-143 pp 502ndash517 2018
[21] Z Chen H Qiu L Gao and P Li ldquoAn optimal shiftingvector approach for efficient probabilistic designrdquo Structuraland Multidisciplinary Optimization vol 47 no 6 pp 905ndash9202013
[22] W Deng X Lu and Y Deng ldquoEvidential model validationunder epistemic uncertaintyrdquo Mathematical Problems in Engi-neering vol 2018 Article ID 6789635 11 pages 2018
[23] D Meng H Zhang and T Huang ldquoA concurrent reliabilityoptimization procedure in the earlier design phases of complexengineering systems under epistemic uncertaintiesrdquo Advancesin Mechanical Engineering vol 8 no 10 pp 1ndash8 2016
[24] G Li Z Meng and H Hu ldquoAn adaptive hybrid approach forreliability-based design optimizationrdquo Structural and Multidis-ciplinary Optimization vol 51 no 5 pp 1051ndash1065 2015
[25] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018
[26] M A Valdebenito and G I Schueller ldquoA survey on approachesfor reliability-based optimizationrdquo Structural and Multidisci-plinary Optimization vol 42 no 5 pp 645ndash663 2010
[27] Y Aoues and A Chateauneuf ldquoBenchmark study of numericalmethods for reliability-based design optimizationrdquo Structuraland Multidisciplinary Optimization vol 41 no 2 pp 277ndash2942010
[28] J Ching and W-C Hsu ldquoTransforming reliability limit-stateconstraints into deterministic limit-state constraintsrdquo StructuralSafety vol 30 no 1 pp 11ndash33 2008
[29] TM Cho and B C Lee ldquoReliability-based design optimizationusing convex linearization and sequential optimization andreliability assessment methodrdquo Structural Safety vol 33 no 1pp 42ndash50 2011
[30] J Liang Z P Mourelatos and J Tu ldquoA single-loop method forreliability-based design optimisationrdquo International Journal ofProduct Development vol 5 no 1-2 pp 76ndash92 2008
[31] S Shan and G G Wang ldquoReliable design space and completesingle-loop reliability-based design optimizationrdquo ReliabilityEngineering amp System Safety vol 93 no 8 pp 1218ndash1230 2008
[32] P Yi G Cheng and L Jiang ldquoA sequential approximateprogramming strategy for performance-measure-based proba-bilistic structural design optimizationrdquo Structural Safety vol 30no 2 pp 91ndash109 2008
[33] Z M Yaseen and B Keshtegar ldquoLimited descent-based meanvalue method for inverse reliability analysisrdquo Engineering withComputers pp 1ndash13 2018
[34] B D Youn K K Choi and Y H Park ldquoHybrid analysis methodfor reliability-based design optimizationrdquo Journal of MechanicalDesign vol 125 no 2 pp 221ndash232 2003
10 Mathematical Problems in Engineering
[35] B D Youn K K Choi and L Du ldquoEnriched performancemeasure approach for reliability-based design optimizationrdquoAIAA Journal vol 43 no 4 pp 874ndash884 2005
[36] B D Youn and K K Choi ldquoSelecting probabilistic approachesfor realiability-based design optimizationrdquo AIAA Journal vol42 no 1 pp 124ndash131 2004
[37] S P Zhu Q Liu J Zhou and Z Y Yu ldquoFatigue reliabilityassessment of turbine discs under multi-source uncertaintiesrdquoFatigue Fracture of Engineering Materials Structures vol 41 no6 pp 1291ndash1305 2018
[38] H Li R Li H Li and R Yuan ldquoReliability modeling of mul-tiple performance based on degradation values distributionrdquoAdvances in Mechanical Engineering vol 8 no 10 Article ID168781401667375 2016
[39] B D Youn K K Choi and L Du ldquoAdaptive probability analysisusing an enhanced hybrid mean value methodrdquo Structural andMultidisciplinary Optimization vol 29 no 2 pp 134ndash148 2005
[40] M Ito N H Kim and N Kogiso ldquoConservative reliabilityindex for epistemic uncertainty in reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 5 pp 1919ndash1935 2018
[41] Z Meng D Yang H Zhou and B P Wang ldquoConvergencecontrol of single loop approach for reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 3 pp 1079ndash1091 2018
[42] W Yao X Q Chen W C Luo M van Tooren and J GuoldquoReview of uncertainty-based multidisciplinary design opti-mization methods for aerospace vehiclesrdquo Progress in AerospaceSciences vol 47 no 6 pp 450ndash479 2011
[43] R Yuan D Meng and H Li ldquoMultidisciplinary reliabilitydesign optimizationusing an enhanced saddlepoint approxima-tion in the framework of sequential optimization and reliabilityanalysisrdquo Proceedings of the Institution of Mechanical EngineersPart O Journal of Risk and Reliability vol 230 no 6 pp 570ndash578 2016
[44] D Meng X Zhang Y Yang H Xu and H Huang ldquoInteractionbalance optimization in multidisciplinary design optimizationproblemsrdquo Concurrent Engineering Research and Applicationsvol 24 no 1 pp 48ndash57 2016
[45] R Yuan and H Li ldquoA multidisciplinary coupling relationshipcoordination algorithm using the hierarchical control methodsof complex systems and its application in multidisciplinarydesign optimizationrdquo Advances in Mechanical Engineering vol9 no 1 pp 1ndash11 2016
[46] D Meng X Zhang H-Z Huang Z Wang and H Xu ldquoInter-action prediction optimization in multidisciplinary designoptimization problemsrdquoThe Scientific World Journal vol 2014Article ID 698453 7 pages 2014
[47] J R R A Martins and A B Lambe ldquoMultidisciplinary designoptimization a survey of architecturesrdquo AIAA Journal vol 51no 9 pp 2049ndash2075 2013
[48] Y Li and Y Deng ldquoGeneralized ordered propositions fusionbased on belief entropyrdquo International Journal of ComputersCommunications amp Control vol 13 no 5 pp 792ndash807 2018
[49] X Su S Mahadevan W Han and Y Deng ldquoCombiningdependent bodies of evidencerdquo Applied Intelligence vol 44 no3 pp 634ndash644 2016
[50] S-P Zhu H-Z Huang W Peng H-K Wang and S Mahade-van ldquoProbabilistic Physics of Failure-based framework forfatigue life prediction of aircraft gas turbine discs under uncer-taintyrdquoReliability Engineering amp System Safety vol 146 pp 1ndash122016
[51] S I Yi J K Shin andG J Park ldquoComparison ofMDOmethodswith mathematical examplesrdquo Structural and MultidisciplinaryOptimization vol 35 no 5 pp 391ndash402 2008
[52] H XuW Li M Li C Hu S Zhang and XWang ldquoMultidisci-plinary robust design optimization based on time-varying sen-sitivity analysisrdquo Journal of Mechanical Science and Technologyvol 32 no 3 pp 1195ndash1207 2018
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
the reference The software Isight is utilized in the compu-tations of optimization The solutions from CO-ERIM arecompared with the ones from original PMA-RIM based COand MCS based CO which is listed in Table 2 From thecomparison of solutions the design results from ERIM arecloser to the design results from MCS Furthermore the
calculation time of ERIM is 17min23s and the calculationtime of RIM is 25min17s which means the proposed methodenjoys higher computational efficiency
6 Conclusions
In this study the efficiency problem of RBMDO is stud-ied The RIM strategy is reviewed and the correspondingalgorithm of ERIM is discussed in detail Furthermore theCO-ERIM strategy is proposed including its formulationand procedure In CO-ERIM the concurrent design idea isadapting to the development of modern engineering systemsCompatibility constraints are introduced into subsystem-level and system-level optimization problems respectivelyThe consistency between different disciplines can be guaran-teed when the RBMDO solutions are obtainedThe introduc-tion of ERIM reduces the computational burden of reliabilityanalysis during optimization iteration process Under thecondition that the first order method is acceptable thereliability analysis accuracy of the proposedmethod is similar
Figure 8 The MDO strategy for the speed reducer problem
to the accuracy ofMCSThe speed reducer example illustratesthe effectiveness of the proposed method
Data Availability
All data used to support the findings of this study are includedwithin the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thesupport from theNational Natural Science Foundation ofChina (Grant No 51605080 and No 11672070) the SichuanScience and Technology Program (Grant No 2019YFG0350and No 2019YFG0348) the China Postdoctoral ScienceFoundation (Grant No 2015M580780 and 2017T100685) andthe National Defense Pre-Research Foundation of China
(Grant No 6140244010216DZ02001) is gratefully acknowl-edged The authors also acknowledge the Portuguese Sci-ence Foundation (FCT) for the financial support throughthe postdoctoral grant SFRHBPD1078252015 as well asthe funding of Projects POCI-01-0145-FEDER-007457 andUIDECI047082019 - CONSTRUCT - Institute of RampDIn Structures and Construction funded by FEDER fundsthrough COMPETE2020 (POCI) and by national funds(PIDDAC) through Portuguese Science Foundation (FCT)
References
[1] L Li H Wan W Gao F Tong and H Li ldquoReliabilitybased multidisciplinary design optimization of cooling turbineblade considering uncertainty data statisticsrdquo Structural andMultidisciplinary Optimization vol 59 no 2 pp 1ndash15 2018
[2] Z L Huang Y S Zhou C Jiang J Zheng and X HanldquoReliability-based multidisciplinary design optimization usingincremental shifting vector strategy and its application inelectronic product designrdquo Acta Mechanica Sinica vol 34 no2 pp 285ndash302 2018
[3] H Xu X WangW Li M Li S Zhang and C Hu ldquoReliability-based multidisciplinary design optimization under correlated
Mathematical Problems in Engineering 9
uncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 7360615 12 pages 2017
[4] D Meng S Yang Y Zhang and S-P Zhu ldquoStructural relia-bility analysis and uncertainties-based collaborative design andoptimization of turbine blades using surrogate modelrdquo Fatigueamp Fracture of Engineering Materials amp Structures pp 1ndash9 2018httpsdoiorg101111ffe12906
[5] F Yang Z Yue L Li and D Guan ldquoHybrid reliability-based multidisciplinary design optimization with random andinterval variablesrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 232 no 1pp 52ndash64 2018
[6] L Wang C Xiong and Y Yang ldquoA novel methodology ofreliability-based multidisciplinary design optimization underhybrid interval and fuzzy uncertaintiesrdquo Computer MethodsApplied Mechanics and Engineering vol 337 pp 439ndash457 2018
[7] L Wang C Xiong J Hu X Wang and Z Qiu ldquoSequentialmultidisciplinary design optimization and reliability analysisunder interval uncertaintyrdquo Aerospace Science and Technologyvol 80 pp 508ndash519 2018
[8] J Zhang M Xiao L Gao H Qiu and Z Yang ldquoAn improvedtwo-stage framework of evidence-based design optimizationrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1673ndash1693 2018
[9] H Xu W Li L Xing and S-P Zhu ldquoMultidisciplinary designoptimization under correlated uncertaintiesrdquo Concurrent Engi-neering Research and Applications vol 25 no 3 pp 262ndash2752017
[10] X Wang R Wang L Wang X Chen and X Geng ldquoAnefficient single-loop strategy for reliability-based multidisci-plinarydesign optimizationunder non-probabilistic set theoryrdquoAerospace Science and Technology vol 73 pp 148ndash163 2018
[11] X P Du J Guo and H Beeram ldquoSequential optimizationand reliability assessment for multidisciplinary systems designrdquoStructural andMultidisciplinary Optimization vol 35 no 2 pp117ndash130 2008
[12] D Meng Y-F Li H-Z Huang Z Wang and Y LiuldquoReliability-based multidisciplinary design optimization usingsubset simulation analysis and its application in the hydraulictransmission mechanism designrdquo Journal of Mechanical Designvol 137 no 5 Article ID 051402 2015
[13] S Yu Z Wang and D Meng ldquoTime-variant reliability assess-ment for multiple failure modes and temporal parametersrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1705ndash1717 2018
[14] D Meng H-Z Huang Z Wang N-C Xiao and X-L ZhangldquoMean-value first-order saddlepoint approximation based col-laborative optimization for multidisciplinary problems underaleatory uncertaintyrdquo Journal of Mechanical Science and Tech-nology vol 28 no 10 pp 3925ndash3935 2014
[15] S-P Zhu Q Liu Q Lei and Q Wang ldquoProbabilistic fatiguelife prediction and reliability assessment of a high pressureturbine disc considering load variationsrdquo International Journalof Damage Mechanics vol 27 no 10 pp 1569ndash1588 2018
[16] D Liao S-P Zhu J A F O Correia A M P De Jesus andR Calcada ldquoComputational framework for multiaxial fatiguelife prediction of compressor discs considering notch effectsrdquoEngineering Fracture Mechanics vol 202 pp 423ndash435 2018
[17] D Meng M Liu S Yang H Zhang and R Ding ldquoAfluid-structure analysis approach and its application in theuncertainty-based multidisciplinary design and optimization
for bladesrdquo Advances in Mechanical Engineering vol 10 no 6pp 1ndash7 2018
[18] Z Meng G Li B P Wang and P Hao ldquoA hybrid chaos controlapproach of the performance measure functions for reliability-based design optimizationrdquo Computers Structures vol 146 pp32ndash43 2015
[19] X Li H Qiu Z Chen L Gao and X Shao ldquoA local Krigingapproximation method using MPP for reliability-based designoptimizationrdquo Computers amp Structures vol 162 pp 102ndash1152016
[20] S-P Zhu Q Liu W Peng and X-C Zhang ldquoComputational-experimental approaches for fatigue reliability assessment ofturbine bladed disksrdquo International Journal of Mechanical Sci-ences vol 142-143 pp 502ndash517 2018
[21] Z Chen H Qiu L Gao and P Li ldquoAn optimal shiftingvector approach for efficient probabilistic designrdquo Structuraland Multidisciplinary Optimization vol 47 no 6 pp 905ndash9202013
[22] W Deng X Lu and Y Deng ldquoEvidential model validationunder epistemic uncertaintyrdquo Mathematical Problems in Engi-neering vol 2018 Article ID 6789635 11 pages 2018
[23] D Meng H Zhang and T Huang ldquoA concurrent reliabilityoptimization procedure in the earlier design phases of complexengineering systems under epistemic uncertaintiesrdquo Advancesin Mechanical Engineering vol 8 no 10 pp 1ndash8 2016
[24] G Li Z Meng and H Hu ldquoAn adaptive hybrid approach forreliability-based design optimizationrdquo Structural and Multidis-ciplinary Optimization vol 51 no 5 pp 1051ndash1065 2015
[25] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018
[26] M A Valdebenito and G I Schueller ldquoA survey on approachesfor reliability-based optimizationrdquo Structural and Multidisci-plinary Optimization vol 42 no 5 pp 645ndash663 2010
[27] Y Aoues and A Chateauneuf ldquoBenchmark study of numericalmethods for reliability-based design optimizationrdquo Structuraland Multidisciplinary Optimization vol 41 no 2 pp 277ndash2942010
[28] J Ching and W-C Hsu ldquoTransforming reliability limit-stateconstraints into deterministic limit-state constraintsrdquo StructuralSafety vol 30 no 1 pp 11ndash33 2008
[29] TM Cho and B C Lee ldquoReliability-based design optimizationusing convex linearization and sequential optimization andreliability assessment methodrdquo Structural Safety vol 33 no 1pp 42ndash50 2011
[30] J Liang Z P Mourelatos and J Tu ldquoA single-loop method forreliability-based design optimisationrdquo International Journal ofProduct Development vol 5 no 1-2 pp 76ndash92 2008
[31] S Shan and G G Wang ldquoReliable design space and completesingle-loop reliability-based design optimizationrdquo ReliabilityEngineering amp System Safety vol 93 no 8 pp 1218ndash1230 2008
[32] P Yi G Cheng and L Jiang ldquoA sequential approximateprogramming strategy for performance-measure-based proba-bilistic structural design optimizationrdquo Structural Safety vol 30no 2 pp 91ndash109 2008
[33] Z M Yaseen and B Keshtegar ldquoLimited descent-based meanvalue method for inverse reliability analysisrdquo Engineering withComputers pp 1ndash13 2018
[34] B D Youn K K Choi and Y H Park ldquoHybrid analysis methodfor reliability-based design optimizationrdquo Journal of MechanicalDesign vol 125 no 2 pp 221ndash232 2003
10 Mathematical Problems in Engineering
[35] B D Youn K K Choi and L Du ldquoEnriched performancemeasure approach for reliability-based design optimizationrdquoAIAA Journal vol 43 no 4 pp 874ndash884 2005
[36] B D Youn and K K Choi ldquoSelecting probabilistic approachesfor realiability-based design optimizationrdquo AIAA Journal vol42 no 1 pp 124ndash131 2004
[37] S P Zhu Q Liu J Zhou and Z Y Yu ldquoFatigue reliabilityassessment of turbine discs under multi-source uncertaintiesrdquoFatigue Fracture of Engineering Materials Structures vol 41 no6 pp 1291ndash1305 2018
[38] H Li R Li H Li and R Yuan ldquoReliability modeling of mul-tiple performance based on degradation values distributionrdquoAdvances in Mechanical Engineering vol 8 no 10 Article ID168781401667375 2016
[39] B D Youn K K Choi and L Du ldquoAdaptive probability analysisusing an enhanced hybrid mean value methodrdquo Structural andMultidisciplinary Optimization vol 29 no 2 pp 134ndash148 2005
[40] M Ito N H Kim and N Kogiso ldquoConservative reliabilityindex for epistemic uncertainty in reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 5 pp 1919ndash1935 2018
[41] Z Meng D Yang H Zhou and B P Wang ldquoConvergencecontrol of single loop approach for reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 3 pp 1079ndash1091 2018
[42] W Yao X Q Chen W C Luo M van Tooren and J GuoldquoReview of uncertainty-based multidisciplinary design opti-mization methods for aerospace vehiclesrdquo Progress in AerospaceSciences vol 47 no 6 pp 450ndash479 2011
[43] R Yuan D Meng and H Li ldquoMultidisciplinary reliabilitydesign optimizationusing an enhanced saddlepoint approxima-tion in the framework of sequential optimization and reliabilityanalysisrdquo Proceedings of the Institution of Mechanical EngineersPart O Journal of Risk and Reliability vol 230 no 6 pp 570ndash578 2016
[44] D Meng X Zhang Y Yang H Xu and H Huang ldquoInteractionbalance optimization in multidisciplinary design optimizationproblemsrdquo Concurrent Engineering Research and Applicationsvol 24 no 1 pp 48ndash57 2016
[45] R Yuan and H Li ldquoA multidisciplinary coupling relationshipcoordination algorithm using the hierarchical control methodsof complex systems and its application in multidisciplinarydesign optimizationrdquo Advances in Mechanical Engineering vol9 no 1 pp 1ndash11 2016
[46] D Meng X Zhang H-Z Huang Z Wang and H Xu ldquoInter-action prediction optimization in multidisciplinary designoptimization problemsrdquoThe Scientific World Journal vol 2014Article ID 698453 7 pages 2014
[47] J R R A Martins and A B Lambe ldquoMultidisciplinary designoptimization a survey of architecturesrdquo AIAA Journal vol 51no 9 pp 2049ndash2075 2013
[48] Y Li and Y Deng ldquoGeneralized ordered propositions fusionbased on belief entropyrdquo International Journal of ComputersCommunications amp Control vol 13 no 5 pp 792ndash807 2018
[49] X Su S Mahadevan W Han and Y Deng ldquoCombiningdependent bodies of evidencerdquo Applied Intelligence vol 44 no3 pp 634ndash644 2016
[50] S-P Zhu H-Z Huang W Peng H-K Wang and S Mahade-van ldquoProbabilistic Physics of Failure-based framework forfatigue life prediction of aircraft gas turbine discs under uncer-taintyrdquoReliability Engineering amp System Safety vol 146 pp 1ndash122016
[51] S I Yi J K Shin andG J Park ldquoComparison ofMDOmethodswith mathematical examplesrdquo Structural and MultidisciplinaryOptimization vol 35 no 5 pp 391ndash402 2008
[52] H XuW Li M Li C Hu S Zhang and XWang ldquoMultidisci-plinary robust design optimization based on time-varying sen-sitivity analysisrdquo Journal of Mechanical Science and Technologyvol 32 no 3 pp 1195ndash1207 2018
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Figure 8 The MDO strategy for the speed reducer problem
to the accuracy ofMCSThe speed reducer example illustratesthe effectiveness of the proposed method
Data Availability
All data used to support the findings of this study are includedwithin the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thesupport from theNational Natural Science Foundation ofChina (Grant No 51605080 and No 11672070) the SichuanScience and Technology Program (Grant No 2019YFG0350and No 2019YFG0348) the China Postdoctoral ScienceFoundation (Grant No 2015M580780 and 2017T100685) andthe National Defense Pre-Research Foundation of China
(Grant No 6140244010216DZ02001) is gratefully acknowl-edged The authors also acknowledge the Portuguese Sci-ence Foundation (FCT) for the financial support throughthe postdoctoral grant SFRHBPD1078252015 as well asthe funding of Projects POCI-01-0145-FEDER-007457 andUIDECI047082019 - CONSTRUCT - Institute of RampDIn Structures and Construction funded by FEDER fundsthrough COMPETE2020 (POCI) and by national funds(PIDDAC) through Portuguese Science Foundation (FCT)
References
[1] L Li H Wan W Gao F Tong and H Li ldquoReliabilitybased multidisciplinary design optimization of cooling turbineblade considering uncertainty data statisticsrdquo Structural andMultidisciplinary Optimization vol 59 no 2 pp 1ndash15 2018
[2] Z L Huang Y S Zhou C Jiang J Zheng and X HanldquoReliability-based multidisciplinary design optimization usingincremental shifting vector strategy and its application inelectronic product designrdquo Acta Mechanica Sinica vol 34 no2 pp 285ndash302 2018
[3] H Xu X WangW Li M Li S Zhang and C Hu ldquoReliability-based multidisciplinary design optimization under correlated
Mathematical Problems in Engineering 9
uncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 7360615 12 pages 2017
[4] D Meng S Yang Y Zhang and S-P Zhu ldquoStructural relia-bility analysis and uncertainties-based collaborative design andoptimization of turbine blades using surrogate modelrdquo Fatigueamp Fracture of Engineering Materials amp Structures pp 1ndash9 2018httpsdoiorg101111ffe12906
[5] F Yang Z Yue L Li and D Guan ldquoHybrid reliability-based multidisciplinary design optimization with random andinterval variablesrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 232 no 1pp 52ndash64 2018
[6] L Wang C Xiong and Y Yang ldquoA novel methodology ofreliability-based multidisciplinary design optimization underhybrid interval and fuzzy uncertaintiesrdquo Computer MethodsApplied Mechanics and Engineering vol 337 pp 439ndash457 2018
[7] L Wang C Xiong J Hu X Wang and Z Qiu ldquoSequentialmultidisciplinary design optimization and reliability analysisunder interval uncertaintyrdquo Aerospace Science and Technologyvol 80 pp 508ndash519 2018
[8] J Zhang M Xiao L Gao H Qiu and Z Yang ldquoAn improvedtwo-stage framework of evidence-based design optimizationrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1673ndash1693 2018
[9] H Xu W Li L Xing and S-P Zhu ldquoMultidisciplinary designoptimization under correlated uncertaintiesrdquo Concurrent Engi-neering Research and Applications vol 25 no 3 pp 262ndash2752017
[10] X Wang R Wang L Wang X Chen and X Geng ldquoAnefficient single-loop strategy for reliability-based multidisci-plinarydesign optimizationunder non-probabilistic set theoryrdquoAerospace Science and Technology vol 73 pp 148ndash163 2018
[11] X P Du J Guo and H Beeram ldquoSequential optimizationand reliability assessment for multidisciplinary systems designrdquoStructural andMultidisciplinary Optimization vol 35 no 2 pp117ndash130 2008
[12] D Meng Y-F Li H-Z Huang Z Wang and Y LiuldquoReliability-based multidisciplinary design optimization usingsubset simulation analysis and its application in the hydraulictransmission mechanism designrdquo Journal of Mechanical Designvol 137 no 5 Article ID 051402 2015
[13] S Yu Z Wang and D Meng ldquoTime-variant reliability assess-ment for multiple failure modes and temporal parametersrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1705ndash1717 2018
[14] D Meng H-Z Huang Z Wang N-C Xiao and X-L ZhangldquoMean-value first-order saddlepoint approximation based col-laborative optimization for multidisciplinary problems underaleatory uncertaintyrdquo Journal of Mechanical Science and Tech-nology vol 28 no 10 pp 3925ndash3935 2014
[15] S-P Zhu Q Liu Q Lei and Q Wang ldquoProbabilistic fatiguelife prediction and reliability assessment of a high pressureturbine disc considering load variationsrdquo International Journalof Damage Mechanics vol 27 no 10 pp 1569ndash1588 2018
[16] D Liao S-P Zhu J A F O Correia A M P De Jesus andR Calcada ldquoComputational framework for multiaxial fatiguelife prediction of compressor discs considering notch effectsrdquoEngineering Fracture Mechanics vol 202 pp 423ndash435 2018
[17] D Meng M Liu S Yang H Zhang and R Ding ldquoAfluid-structure analysis approach and its application in theuncertainty-based multidisciplinary design and optimization
for bladesrdquo Advances in Mechanical Engineering vol 10 no 6pp 1ndash7 2018
[18] Z Meng G Li B P Wang and P Hao ldquoA hybrid chaos controlapproach of the performance measure functions for reliability-based design optimizationrdquo Computers Structures vol 146 pp32ndash43 2015
[19] X Li H Qiu Z Chen L Gao and X Shao ldquoA local Krigingapproximation method using MPP for reliability-based designoptimizationrdquo Computers amp Structures vol 162 pp 102ndash1152016
[20] S-P Zhu Q Liu W Peng and X-C Zhang ldquoComputational-experimental approaches for fatigue reliability assessment ofturbine bladed disksrdquo International Journal of Mechanical Sci-ences vol 142-143 pp 502ndash517 2018
[21] Z Chen H Qiu L Gao and P Li ldquoAn optimal shiftingvector approach for efficient probabilistic designrdquo Structuraland Multidisciplinary Optimization vol 47 no 6 pp 905ndash9202013
[22] W Deng X Lu and Y Deng ldquoEvidential model validationunder epistemic uncertaintyrdquo Mathematical Problems in Engi-neering vol 2018 Article ID 6789635 11 pages 2018
[23] D Meng H Zhang and T Huang ldquoA concurrent reliabilityoptimization procedure in the earlier design phases of complexengineering systems under epistemic uncertaintiesrdquo Advancesin Mechanical Engineering vol 8 no 10 pp 1ndash8 2016
[24] G Li Z Meng and H Hu ldquoAn adaptive hybrid approach forreliability-based design optimizationrdquo Structural and Multidis-ciplinary Optimization vol 51 no 5 pp 1051ndash1065 2015
[25] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018
[26] M A Valdebenito and G I Schueller ldquoA survey on approachesfor reliability-based optimizationrdquo Structural and Multidisci-plinary Optimization vol 42 no 5 pp 645ndash663 2010
[27] Y Aoues and A Chateauneuf ldquoBenchmark study of numericalmethods for reliability-based design optimizationrdquo Structuraland Multidisciplinary Optimization vol 41 no 2 pp 277ndash2942010
[28] J Ching and W-C Hsu ldquoTransforming reliability limit-stateconstraints into deterministic limit-state constraintsrdquo StructuralSafety vol 30 no 1 pp 11ndash33 2008
[29] TM Cho and B C Lee ldquoReliability-based design optimizationusing convex linearization and sequential optimization andreliability assessment methodrdquo Structural Safety vol 33 no 1pp 42ndash50 2011
[30] J Liang Z P Mourelatos and J Tu ldquoA single-loop method forreliability-based design optimisationrdquo International Journal ofProduct Development vol 5 no 1-2 pp 76ndash92 2008
[31] S Shan and G G Wang ldquoReliable design space and completesingle-loop reliability-based design optimizationrdquo ReliabilityEngineering amp System Safety vol 93 no 8 pp 1218ndash1230 2008
[32] P Yi G Cheng and L Jiang ldquoA sequential approximateprogramming strategy for performance-measure-based proba-bilistic structural design optimizationrdquo Structural Safety vol 30no 2 pp 91ndash109 2008
[33] Z M Yaseen and B Keshtegar ldquoLimited descent-based meanvalue method for inverse reliability analysisrdquo Engineering withComputers pp 1ndash13 2018
[34] B D Youn K K Choi and Y H Park ldquoHybrid analysis methodfor reliability-based design optimizationrdquo Journal of MechanicalDesign vol 125 no 2 pp 221ndash232 2003
10 Mathematical Problems in Engineering
[35] B D Youn K K Choi and L Du ldquoEnriched performancemeasure approach for reliability-based design optimizationrdquoAIAA Journal vol 43 no 4 pp 874ndash884 2005
[36] B D Youn and K K Choi ldquoSelecting probabilistic approachesfor realiability-based design optimizationrdquo AIAA Journal vol42 no 1 pp 124ndash131 2004
[37] S P Zhu Q Liu J Zhou and Z Y Yu ldquoFatigue reliabilityassessment of turbine discs under multi-source uncertaintiesrdquoFatigue Fracture of Engineering Materials Structures vol 41 no6 pp 1291ndash1305 2018
[38] H Li R Li H Li and R Yuan ldquoReliability modeling of mul-tiple performance based on degradation values distributionrdquoAdvances in Mechanical Engineering vol 8 no 10 Article ID168781401667375 2016
[39] B D Youn K K Choi and L Du ldquoAdaptive probability analysisusing an enhanced hybrid mean value methodrdquo Structural andMultidisciplinary Optimization vol 29 no 2 pp 134ndash148 2005
[40] M Ito N H Kim and N Kogiso ldquoConservative reliabilityindex for epistemic uncertainty in reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 5 pp 1919ndash1935 2018
[41] Z Meng D Yang H Zhou and B P Wang ldquoConvergencecontrol of single loop approach for reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 3 pp 1079ndash1091 2018
[42] W Yao X Q Chen W C Luo M van Tooren and J GuoldquoReview of uncertainty-based multidisciplinary design opti-mization methods for aerospace vehiclesrdquo Progress in AerospaceSciences vol 47 no 6 pp 450ndash479 2011
[43] R Yuan D Meng and H Li ldquoMultidisciplinary reliabilitydesign optimizationusing an enhanced saddlepoint approxima-tion in the framework of sequential optimization and reliabilityanalysisrdquo Proceedings of the Institution of Mechanical EngineersPart O Journal of Risk and Reliability vol 230 no 6 pp 570ndash578 2016
[44] D Meng X Zhang Y Yang H Xu and H Huang ldquoInteractionbalance optimization in multidisciplinary design optimizationproblemsrdquo Concurrent Engineering Research and Applicationsvol 24 no 1 pp 48ndash57 2016
[45] R Yuan and H Li ldquoA multidisciplinary coupling relationshipcoordination algorithm using the hierarchical control methodsof complex systems and its application in multidisciplinarydesign optimizationrdquo Advances in Mechanical Engineering vol9 no 1 pp 1ndash11 2016
[46] D Meng X Zhang H-Z Huang Z Wang and H Xu ldquoInter-action prediction optimization in multidisciplinary designoptimization problemsrdquoThe Scientific World Journal vol 2014Article ID 698453 7 pages 2014
[47] J R R A Martins and A B Lambe ldquoMultidisciplinary designoptimization a survey of architecturesrdquo AIAA Journal vol 51no 9 pp 2049ndash2075 2013
[48] Y Li and Y Deng ldquoGeneralized ordered propositions fusionbased on belief entropyrdquo International Journal of ComputersCommunications amp Control vol 13 no 5 pp 792ndash807 2018
[49] X Su S Mahadevan W Han and Y Deng ldquoCombiningdependent bodies of evidencerdquo Applied Intelligence vol 44 no3 pp 634ndash644 2016
[50] S-P Zhu H-Z Huang W Peng H-K Wang and S Mahade-van ldquoProbabilistic Physics of Failure-based framework forfatigue life prediction of aircraft gas turbine discs under uncer-taintyrdquoReliability Engineering amp System Safety vol 146 pp 1ndash122016
[51] S I Yi J K Shin andG J Park ldquoComparison ofMDOmethodswith mathematical examplesrdquo Structural and MultidisciplinaryOptimization vol 35 no 5 pp 391ndash402 2008
[52] H XuW Li M Li C Hu S Zhang and XWang ldquoMultidisci-plinary robust design optimization based on time-varying sen-sitivity analysisrdquo Journal of Mechanical Science and Technologyvol 32 no 3 pp 1195ndash1207 2018
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 9
uncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 7360615 12 pages 2017
[4] D Meng S Yang Y Zhang and S-P Zhu ldquoStructural relia-bility analysis and uncertainties-based collaborative design andoptimization of turbine blades using surrogate modelrdquo Fatigueamp Fracture of Engineering Materials amp Structures pp 1ndash9 2018httpsdoiorg101111ffe12906
[5] F Yang Z Yue L Li and D Guan ldquoHybrid reliability-based multidisciplinary design optimization with random andinterval variablesrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 232 no 1pp 52ndash64 2018
[6] L Wang C Xiong and Y Yang ldquoA novel methodology ofreliability-based multidisciplinary design optimization underhybrid interval and fuzzy uncertaintiesrdquo Computer MethodsApplied Mechanics and Engineering vol 337 pp 439ndash457 2018
[7] L Wang C Xiong J Hu X Wang and Z Qiu ldquoSequentialmultidisciplinary design optimization and reliability analysisunder interval uncertaintyrdquo Aerospace Science and Technologyvol 80 pp 508ndash519 2018
[8] J Zhang M Xiao L Gao H Qiu and Z Yang ldquoAn improvedtwo-stage framework of evidence-based design optimizationrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1673ndash1693 2018
[9] H Xu W Li L Xing and S-P Zhu ldquoMultidisciplinary designoptimization under correlated uncertaintiesrdquo Concurrent Engi-neering Research and Applications vol 25 no 3 pp 262ndash2752017
[10] X Wang R Wang L Wang X Chen and X Geng ldquoAnefficient single-loop strategy for reliability-based multidisci-plinarydesign optimizationunder non-probabilistic set theoryrdquoAerospace Science and Technology vol 73 pp 148ndash163 2018
[11] X P Du J Guo and H Beeram ldquoSequential optimizationand reliability assessment for multidisciplinary systems designrdquoStructural andMultidisciplinary Optimization vol 35 no 2 pp117ndash130 2008
[12] D Meng Y-F Li H-Z Huang Z Wang and Y LiuldquoReliability-based multidisciplinary design optimization usingsubset simulation analysis and its application in the hydraulictransmission mechanism designrdquo Journal of Mechanical Designvol 137 no 5 Article ID 051402 2015
[13] S Yu Z Wang and D Meng ldquoTime-variant reliability assess-ment for multiple failure modes and temporal parametersrdquoStructural andMultidisciplinary Optimization vol 58 no 4 pp1705ndash1717 2018
[14] D Meng H-Z Huang Z Wang N-C Xiao and X-L ZhangldquoMean-value first-order saddlepoint approximation based col-laborative optimization for multidisciplinary problems underaleatory uncertaintyrdquo Journal of Mechanical Science and Tech-nology vol 28 no 10 pp 3925ndash3935 2014
[15] S-P Zhu Q Liu Q Lei and Q Wang ldquoProbabilistic fatiguelife prediction and reliability assessment of a high pressureturbine disc considering load variationsrdquo International Journalof Damage Mechanics vol 27 no 10 pp 1569ndash1588 2018
[16] D Liao S-P Zhu J A F O Correia A M P De Jesus andR Calcada ldquoComputational framework for multiaxial fatiguelife prediction of compressor discs considering notch effectsrdquoEngineering Fracture Mechanics vol 202 pp 423ndash435 2018
[17] D Meng M Liu S Yang H Zhang and R Ding ldquoAfluid-structure analysis approach and its application in theuncertainty-based multidisciplinary design and optimization
for bladesrdquo Advances in Mechanical Engineering vol 10 no 6pp 1ndash7 2018
[18] Z Meng G Li B P Wang and P Hao ldquoA hybrid chaos controlapproach of the performance measure functions for reliability-based design optimizationrdquo Computers Structures vol 146 pp32ndash43 2015
[19] X Li H Qiu Z Chen L Gao and X Shao ldquoA local Krigingapproximation method using MPP for reliability-based designoptimizationrdquo Computers amp Structures vol 162 pp 102ndash1152016
[20] S-P Zhu Q Liu W Peng and X-C Zhang ldquoComputational-experimental approaches for fatigue reliability assessment ofturbine bladed disksrdquo International Journal of Mechanical Sci-ences vol 142-143 pp 502ndash517 2018
[21] Z Chen H Qiu L Gao and P Li ldquoAn optimal shiftingvector approach for efficient probabilistic designrdquo Structuraland Multidisciplinary Optimization vol 47 no 6 pp 905ndash9202013
[22] W Deng X Lu and Y Deng ldquoEvidential model validationunder epistemic uncertaintyrdquo Mathematical Problems in Engi-neering vol 2018 Article ID 6789635 11 pages 2018
[23] D Meng H Zhang and T Huang ldquoA concurrent reliabilityoptimization procedure in the earlier design phases of complexengineering systems under epistemic uncertaintiesrdquo Advancesin Mechanical Engineering vol 8 no 10 pp 1ndash8 2016
[24] G Li Z Meng and H Hu ldquoAn adaptive hybrid approach forreliability-based design optimizationrdquo Structural and Multidis-ciplinary Optimization vol 51 no 5 pp 1051ndash1065 2015
[25] Y Gong X Su H Qian and N Yang ldquoResearch on faultdiagnosis methods for the reactor coolant system of nuclearpower plant based on D-S evidence theoryrdquo Annals of NuclearEnergy vol 112 pp 395ndash399 2018
[26] M A Valdebenito and G I Schueller ldquoA survey on approachesfor reliability-based optimizationrdquo Structural and Multidisci-plinary Optimization vol 42 no 5 pp 645ndash663 2010
[27] Y Aoues and A Chateauneuf ldquoBenchmark study of numericalmethods for reliability-based design optimizationrdquo Structuraland Multidisciplinary Optimization vol 41 no 2 pp 277ndash2942010
[28] J Ching and W-C Hsu ldquoTransforming reliability limit-stateconstraints into deterministic limit-state constraintsrdquo StructuralSafety vol 30 no 1 pp 11ndash33 2008
[29] TM Cho and B C Lee ldquoReliability-based design optimizationusing convex linearization and sequential optimization andreliability assessment methodrdquo Structural Safety vol 33 no 1pp 42ndash50 2011
[30] J Liang Z P Mourelatos and J Tu ldquoA single-loop method forreliability-based design optimisationrdquo International Journal ofProduct Development vol 5 no 1-2 pp 76ndash92 2008
[31] S Shan and G G Wang ldquoReliable design space and completesingle-loop reliability-based design optimizationrdquo ReliabilityEngineering amp System Safety vol 93 no 8 pp 1218ndash1230 2008
[32] P Yi G Cheng and L Jiang ldquoA sequential approximateprogramming strategy for performance-measure-based proba-bilistic structural design optimizationrdquo Structural Safety vol 30no 2 pp 91ndash109 2008
[33] Z M Yaseen and B Keshtegar ldquoLimited descent-based meanvalue method for inverse reliability analysisrdquo Engineering withComputers pp 1ndash13 2018
[34] B D Youn K K Choi and Y H Park ldquoHybrid analysis methodfor reliability-based design optimizationrdquo Journal of MechanicalDesign vol 125 no 2 pp 221ndash232 2003
10 Mathematical Problems in Engineering
[35] B D Youn K K Choi and L Du ldquoEnriched performancemeasure approach for reliability-based design optimizationrdquoAIAA Journal vol 43 no 4 pp 874ndash884 2005
[36] B D Youn and K K Choi ldquoSelecting probabilistic approachesfor realiability-based design optimizationrdquo AIAA Journal vol42 no 1 pp 124ndash131 2004
[37] S P Zhu Q Liu J Zhou and Z Y Yu ldquoFatigue reliabilityassessment of turbine discs under multi-source uncertaintiesrdquoFatigue Fracture of Engineering Materials Structures vol 41 no6 pp 1291ndash1305 2018
[38] H Li R Li H Li and R Yuan ldquoReliability modeling of mul-tiple performance based on degradation values distributionrdquoAdvances in Mechanical Engineering vol 8 no 10 Article ID168781401667375 2016
[39] B D Youn K K Choi and L Du ldquoAdaptive probability analysisusing an enhanced hybrid mean value methodrdquo Structural andMultidisciplinary Optimization vol 29 no 2 pp 134ndash148 2005
[40] M Ito N H Kim and N Kogiso ldquoConservative reliabilityindex for epistemic uncertainty in reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 5 pp 1919ndash1935 2018
[41] Z Meng D Yang H Zhou and B P Wang ldquoConvergencecontrol of single loop approach for reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 3 pp 1079ndash1091 2018
[42] W Yao X Q Chen W C Luo M van Tooren and J GuoldquoReview of uncertainty-based multidisciplinary design opti-mization methods for aerospace vehiclesrdquo Progress in AerospaceSciences vol 47 no 6 pp 450ndash479 2011
[43] R Yuan D Meng and H Li ldquoMultidisciplinary reliabilitydesign optimizationusing an enhanced saddlepoint approxima-tion in the framework of sequential optimization and reliabilityanalysisrdquo Proceedings of the Institution of Mechanical EngineersPart O Journal of Risk and Reliability vol 230 no 6 pp 570ndash578 2016
[44] D Meng X Zhang Y Yang H Xu and H Huang ldquoInteractionbalance optimization in multidisciplinary design optimizationproblemsrdquo Concurrent Engineering Research and Applicationsvol 24 no 1 pp 48ndash57 2016
[45] R Yuan and H Li ldquoA multidisciplinary coupling relationshipcoordination algorithm using the hierarchical control methodsof complex systems and its application in multidisciplinarydesign optimizationrdquo Advances in Mechanical Engineering vol9 no 1 pp 1ndash11 2016
[46] D Meng X Zhang H-Z Huang Z Wang and H Xu ldquoInter-action prediction optimization in multidisciplinary designoptimization problemsrdquoThe Scientific World Journal vol 2014Article ID 698453 7 pages 2014
[47] J R R A Martins and A B Lambe ldquoMultidisciplinary designoptimization a survey of architecturesrdquo AIAA Journal vol 51no 9 pp 2049ndash2075 2013
[48] Y Li and Y Deng ldquoGeneralized ordered propositions fusionbased on belief entropyrdquo International Journal of ComputersCommunications amp Control vol 13 no 5 pp 792ndash807 2018
[49] X Su S Mahadevan W Han and Y Deng ldquoCombiningdependent bodies of evidencerdquo Applied Intelligence vol 44 no3 pp 634ndash644 2016
[50] S-P Zhu H-Z Huang W Peng H-K Wang and S Mahade-van ldquoProbabilistic Physics of Failure-based framework forfatigue life prediction of aircraft gas turbine discs under uncer-taintyrdquoReliability Engineering amp System Safety vol 146 pp 1ndash122016
[51] S I Yi J K Shin andG J Park ldquoComparison ofMDOmethodswith mathematical examplesrdquo Structural and MultidisciplinaryOptimization vol 35 no 5 pp 391ndash402 2008
[52] H XuW Li M Li C Hu S Zhang and XWang ldquoMultidisci-plinary robust design optimization based on time-varying sen-sitivity analysisrdquo Journal of Mechanical Science and Technologyvol 32 no 3 pp 1195ndash1207 2018
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
10 Mathematical Problems in Engineering
[35] B D Youn K K Choi and L Du ldquoEnriched performancemeasure approach for reliability-based design optimizationrdquoAIAA Journal vol 43 no 4 pp 874ndash884 2005
[36] B D Youn and K K Choi ldquoSelecting probabilistic approachesfor realiability-based design optimizationrdquo AIAA Journal vol42 no 1 pp 124ndash131 2004
[37] S P Zhu Q Liu J Zhou and Z Y Yu ldquoFatigue reliabilityassessment of turbine discs under multi-source uncertaintiesrdquoFatigue Fracture of Engineering Materials Structures vol 41 no6 pp 1291ndash1305 2018
[38] H Li R Li H Li and R Yuan ldquoReliability modeling of mul-tiple performance based on degradation values distributionrdquoAdvances in Mechanical Engineering vol 8 no 10 Article ID168781401667375 2016
[39] B D Youn K K Choi and L Du ldquoAdaptive probability analysisusing an enhanced hybrid mean value methodrdquo Structural andMultidisciplinary Optimization vol 29 no 2 pp 134ndash148 2005
[40] M Ito N H Kim and N Kogiso ldquoConservative reliabilityindex for epistemic uncertainty in reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 5 pp 1919ndash1935 2018
[41] Z Meng D Yang H Zhou and B P Wang ldquoConvergencecontrol of single loop approach for reliability-based designoptimizationrdquo Structural and Multidisciplinary Optimizationvol 57 no 3 pp 1079ndash1091 2018
[42] W Yao X Q Chen W C Luo M van Tooren and J GuoldquoReview of uncertainty-based multidisciplinary design opti-mization methods for aerospace vehiclesrdquo Progress in AerospaceSciences vol 47 no 6 pp 450ndash479 2011
[43] R Yuan D Meng and H Li ldquoMultidisciplinary reliabilitydesign optimizationusing an enhanced saddlepoint approxima-tion in the framework of sequential optimization and reliabilityanalysisrdquo Proceedings of the Institution of Mechanical EngineersPart O Journal of Risk and Reliability vol 230 no 6 pp 570ndash578 2016
[44] D Meng X Zhang Y Yang H Xu and H Huang ldquoInteractionbalance optimization in multidisciplinary design optimizationproblemsrdquo Concurrent Engineering Research and Applicationsvol 24 no 1 pp 48ndash57 2016
[45] R Yuan and H Li ldquoA multidisciplinary coupling relationshipcoordination algorithm using the hierarchical control methodsof complex systems and its application in multidisciplinarydesign optimizationrdquo Advances in Mechanical Engineering vol9 no 1 pp 1ndash11 2016
[46] D Meng X Zhang H-Z Huang Z Wang and H Xu ldquoInter-action prediction optimization in multidisciplinary designoptimization problemsrdquoThe Scientific World Journal vol 2014Article ID 698453 7 pages 2014
[47] J R R A Martins and A B Lambe ldquoMultidisciplinary designoptimization a survey of architecturesrdquo AIAA Journal vol 51no 9 pp 2049ndash2075 2013
[48] Y Li and Y Deng ldquoGeneralized ordered propositions fusionbased on belief entropyrdquo International Journal of ComputersCommunications amp Control vol 13 no 5 pp 792ndash807 2018
[49] X Su S Mahadevan W Han and Y Deng ldquoCombiningdependent bodies of evidencerdquo Applied Intelligence vol 44 no3 pp 634ndash644 2016
[50] S-P Zhu H-Z Huang W Peng H-K Wang and S Mahade-van ldquoProbabilistic Physics of Failure-based framework forfatigue life prediction of aircraft gas turbine discs under uncer-taintyrdquoReliability Engineering amp System Safety vol 146 pp 1ndash122016
[51] S I Yi J K Shin andG J Park ldquoComparison ofMDOmethodswith mathematical examplesrdquo Structural and MultidisciplinaryOptimization vol 35 no 5 pp 391ndash402 2008
[52] H XuW Li M Li C Hu S Zhang and XWang ldquoMultidisci-plinary robust design optimization based on time-varying sen-sitivity analysisrdquo Journal of Mechanical Science and Technologyvol 32 no 3 pp 1195ndash1207 2018
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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
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