Research Article Reliability-Based Robust Design...

Research ArticleReliability-Based Robust Design Optimization ofStructures Considering Uncertainty in Design Variables

Shujuan Wang1 Qiuyang Li1 and Gordon J Savage2

1 Electrical Engineering Harbin Institute of Technology Harbin Heilongjiang 150001 China2 Systems Design Engineering University of Waterloo Waterloo ON Canada N2L 3G1

Correspondence should be addressed to Qiuyang Li 12qiuyligmailcom

Received 5 June 2014 Accepted 4 September 2014

Academic Editor Kang Li

Copyright copy 2015 Shujuan Wang 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

This paper investigates the structural design optimization to cover both the reliability and robustness under uncertainty in designvariables The main objective is to improve the efficiency of the optimization process To address this problem a hybrid reliability-based robust design optimization (RRDO) method is proposed Prior to the design optimization the Sobol sensitivity analysis isused for selecting key design variables and providing response variance as well resulting in significantly reduced computationalcomplexity The single-loop algorithm is employed to guarantee the structural reliability allowing fast optimization process Inthe case of robust design the weighting factor balances the response performance and variance with respect to the uncertainty indesign variables The main contribution of this paper is that the proposed method applies the RRDO strategy with the usage ofglobal approximation and the Sobol sensitivity analysis leading to the reduced computational cost A structural example is givento illustrate the performance of the proposed method

1 Introduction

The deterministic models do not account for parameter vari-ation which is poorly identified and provides an inaccuratepicture of the problem in question In practice a theoreticallyexcellent deterministic solution may be proved catastrophicdue to uncertainty introduced by manufacturing process andenvironmental changes To tackle the uncertainty in designvariables and parameters nondeterministic methods havebeen developed rapidly in the last twenty years [1 2] Theaim of mechanical structure optimization is to find a high-performance systemwhich generally has two criteria robust-ness and reliability Nondeterministic methods thus canbe classified into two approaches namely reliability-baseddesign optimization (RBDO) and robust design optimization(RDO) [3] RBDO concentrates on finding an optimal designwith low probability of failure while RDO aims to reduce thevariability of the systemperformance To obtain a reliable androbust product a hybrid algorithm named reliability-basedrobust design optimization (RRDO) employs both RDO andRBDO techniques to search for robust optima while obeyingreliability type of constraints

In the present study structural sizing RRDO problemsaim to minimize both the weight and the response varianceof the structure A number of RRDO methods have beenreported for structural optimization in the recent years [2 4ndash9] Stochastic design optimization employing Monte Carlosimulation (MCS) is known as the most adequate methodol-ogy which can directly calculate the probability of failure andthe response moments of system [2] Lagaros et al took intoaccount the probabilistic constraints using MCS combinedwith Latin hypercube sampling in the framework of RBDO[5] Then each optimal design was checked if it satisfies theRDO formulation (European design codes for structures) inorder to select the final results However a large numberof sample points need to be generated to ensure accuracywhichmakesMCS time consuming for implementationwhennumerical simulations are involved

Besides MCS reliability level may be estimated throughthe first- or second-order reliability methods (FORM orSORM) in the process of RBDO Reliability index is intro-duced as an optimization criterion instead of the probabilisticconstraints [6] In [7] and later in [8] a combined algorithmwas discussed based on performance measure approach

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 280940 8 pageshttpdxdoiorg1011552015280940

2 Mathematical Problems in Engineering

(PMA) finding the reliable and robust Pareto optimal solu-tions Although reliability methods reduce the sampling setsand simulation cost they make the optimization structuremore complex with double loops (the outer loop is theoptimization loop and reliability analysis is performed in theinner loop) even triple loops (including robustness evalua-tion loop) as presented in [9] Hence a common challenge forthe designers is the computational expense of either RDO orRBDO whose objective is distinctly different Some attemptshave been made to improve the optimization efficiency for aone desired characteristic (robustness or reliability) [10ndash13]and few of the existing works address the development ofefficient techniques for both robustness and reliability

The main purpose of this paper is to improve the optimi-zation efficiency for both the reliability design and the robustdesign under uncertainty To address this problem a hybridRRDOmethod is proposed combining the single-loopRBDOalgorithm and monoobjective RDO formulation to improveefficiency In the case of RBDO the single-loop approachbased on the conjugate gradient is employed to treat the con-straints efficiently The robustness evaluation is fulfilled con-sidering both the mean and the variance of response repre-sented by a weighting factor The proposed RRDOmethod isapplied to a ldquohookrdquo problem with uncertainty in design vari-ables The results are compared with those from othermethods in the literature leading to a reduced computationalcost

2 Design Optimization under Uncertainty

21 Reliability-Based Design Method The objective of RBDOis to find an optimal solution that verifies as a probability offailure lower or equal to the target probability expressed as119875119905

119891 The classical RBDO is expressed in the form

minimizek

119891 (k)

subject to Pr [119892119894(k) lt 0] le 119875119905

119891= Φ(minus120573

119905

119894)

119894 = 1 119898

v119871 le v le v119880

(1)

where v isin R119899 is the design vector v119871 and v119880 are the lowerlimit and upper limit of the design vector respectively 119892

119894(sdot)

is the 119894th limit state function dividing the safety region (119892 gt0) and failure region (119892 lt 0) Φ(sdot) is the standard cumu-lative function for the standard normal distribution and 120573119905

119894

is target reliability index of the 119894th probabilistic constraintrepresenting the reliability level

In the field of RBDO considering uncertainty reliabilityindex approach (RIA) and performance measure approach(PMA) based on the reliability index have been mainlystudied The reliability index approach is a FORM-basedmethod which is a direct method in comparison to otheroptimization approaches The basic formulation of RIA iswritten as

minimizek

119891 (k)

subject to 120573119894(u) ge 120573119905

119894 119894 = 1 119898

k119871 le k le k119880(2a)

The suboptimization problem for the evaluation of reliabilityis defined as

minimize 120573119905119894= u

subject to 119892119894(u) = 0 119894 = 1 119898

(2b)

where u is independent standard normal random variablevectors taking the following form

u = k minus 120583k120590k

(3)

and 119892(u) is the probabilistic constraint defined in u-spaceSimilarly the PMA formulation is expressed as follows

minimizek

119891 (k)

subject to 119866119894(u) ge 0 119894 = 1 119898

k119871 le k le k119880

(4a)

where 119866119894(u) is the minimum of the performance function

among the points that have the target reliability index whichis obtained from a suboptimization problem Consider

minimize 119866119894= 119892119894(u)

subject to u119894= 120573119905

119894 119894 = 1 119898

(4b)

It is necessary for both RIA and PMA to solve the subopti-mization problem in order to estimate the reliability becauseof its double-loop structure

22 Robust Design Method A commonly used definitionstates that a robust design is a design that is insensitive (orless sensitive) to input variations RDO aims to improve thequality of products by minimizing the performance variationwithout eliminating the existing uncertainty [14] To achievethe robustness there is no unifiedmathematical formulationInmost cases designers emphasized obtaining the robustnessof the objective performance rather than satisfying con-straints with more robust limit state functions Thus theconstruction of a robust problem generally is based on themean and standard deviation of objective functions Consider

minimizek

F (k) = [120583119891(k) 120590

119891(k)]

subject to 119892119894(k) ge 0 119894 = 1 119898

k119871 le k le k119880

(5)

where 120583119891(v) and 120590

119891(v) are the mean and standard deviation

of the objective function 119891(v) respectively Within this for-mulation RDO is defined as amultiobjective problem (MOP)that gives a series of solutions known as a set of optimalPareto solutions However MOP suffers from a large degree

Mathematical Problems in Engineering 3

of complexity with regard to numerical implementation andprovides large numbers of candidate solutions for designersto make a decision In this paper the MOP is converted to amonoobjective problem by adding weighting factors beforeeach entry A normalized RDO therefore becomes

minimizek

F (k) = 119908120583119891(k)120583lowast

119891

+ (1 minus 119908)

120590119891(k)120590lowast

119891

0 lt 119908 lt 1

subject to 119892119894(k) ge 0 119894 = 1 119898

k119871 le k le k119880

(6)

where 119908 is the positive weighting factor and 120583lowast119891and 120590lowast

119891

are the values that take the function 119891 as their optimumconsidering only the response mean or standard deviationas objective functions (ie 119908 = 1 and 119908 = 0 resp) Theweighting factor is determined according to the importancebetween minimum performance and robustness

23 Proposed RRDO Method In order to improve the opti-mization efficiency especially for the multicriteria problemthe single-loop approach (SLA) has been used to substitutethe double-loop structure [15] The SLA aims to replace thecharacteristic point with the minimum performance targetpoint integrating systemrsquos reliability The SLA formulation isas follows

minimizek

119891 (k)

subject to 119892119894(z) ge 0 119894 = 1 119898

k119871 le k le k119880

(7a)

such that z = 120583v minus 120590v120573119905120572 with

120572 =120590knablak119892119894 (k)1003817100381710038171003817120590knablak119892119894 (k)

1003817100381710038171003817

(7b)

where z corresponds to the characteristic point connected tothe target level of reliability120573119905 and 120583v and 120590v are themean andstandard deviation of the design variables v respectively

The RRDO problem therefore is formulated combining(5) and (7a) and (7b) in the following form

minimizek

F (k) = 119908120583119891(k)120583lowast

119891

+ (1 minus 119908)

120590119891(k)120590lowast

119891

0 lt 119908 lt 1

subject to 119892119894(z) ge 0 119894 = 1 119898

k119871 le k le k119880

where z = 120583k minus 120590k120573119905 120590knablak119892119894 (k)1003817100381710038171003817120590knablak119892119894 (k)

1003817100381710038171003817

(8)

It can be seen from formulation (8) that two essential ele-ments of the RRDO problem are (i) determining objec-tive function and constraint function and (ii) estimating

the moments (ie mean and standard deviation) of theperformance function To begin with many approximatedmodels have been developed for several decades Detaileddescription ofmeta-modeling can be found in Section 31 Onthe other hand the moments generally can be achieved intwo ways Taylor series expansion and Monte Carlo methodTaylor series expansion method omits higher-order indicesof performance function and becomes difficult as the numberand intersection of variables increase AlthoughMonte Carlosimulation gives more accurate results the computation isexpensive and time consuming In this paper the momentsare calculated through a fully quantitative variance-basedglobal analysis which is also called Sobol method Themoments can be obtained analytically from Sobol functionsfor simple mathematical model or estimated from MonteCarlo calculation when the model is more complex and non-integrable By means of the Sobol method the main sourcesof uncertainty also can be determined This measure will bespecifically discussed in Section 32

In summary the general framework of the proposedRRDO algorithm is depicted in Figure 1 and it can be des-cribed in the following steps

(1) According to the information on the uncertain para-meters determine and generate the sets of randomdesign variables v

(2) Get the studied responses from finite element analy-sis

(3) Construct the approximate models of the objectivefunction and constraint function in (8)

(4) Consider the less significant variables as deterministicwithout variation by the Sobol method and at thesame time obtain mean and standard deviation ofobjective function

(5) Define optimization problem (weighting factor upperand lower limits etc)

(6) Check convergence If there is a feasible result go tothe 7th step otherwise insert more sample sets andgo to the 2nd step

(7) Obtain the optimum design and the correspondingobjective function value

(8) Repeat from the 5th step to the 7th step if anotheroptimization with respect to a different weightingfactor is needed

3 Methodology Used in RRDO

31 Global Approximation To optimize the performance ofcomplex structures the relationship between design variablesand considered functions can be approximated by means of apure mathematical model The global approximation modeltechnology presents an advantage that it can be used for opti-mization through a singlemeta-model rather than a sequenceof fitted local meta-models The parameters of the meta-model are defined on the basis of a limited number ofsimulations such as finite element simulation


Start

Generate sampling points using Latin hypercube method

Get responses from finite element simulation

Construct multivariate approximation

Formulationof f( ) g( )

Variance-based global sensitivity analysis of f( )

Obtain 120583f and 120590f rank the importance of design variables

Calculationof 120583f and 120590f

Specification of weighting factors and reliability index

Evaluate objective and constraints

Converged No

Yes

Single robust and safety optimal solution

Need anothersolution

Stop

Optimization

Changeweightingfactors orreliability

index

Insertmore

samplingpoints of

importantvariables

No

Yes

119959

119959 119959

Figure 1 Flowchart of proposed method

A linear or quadratic polynomial regression model hasbeen used frequently in engineering due to its computationalsimplicityThe used multivariate polynomial expression con-sidering the cross-product term is given as [16]

(119909) = sum11986211989411198942119894119899

119899

prod

119895=1

119909119894119895

119895

119899

sum

119895=1

119894119895le 119875

(9)

where (119909) is the approximated function 119862 are the poly-nomial coefficients 119899 is the number of the polynomialvariables x 119894

119895is the degree of the polynomial variables and

119875 is the polynomial degree The polynomial expression canbe converted to the general form by grouping the fittingparameters C

119886and the corresponding functions of variables

X119879119886as shown

(x) = C119886X119879119886= C119886[1X 119891 (X)]119879 (10)

For the second-order approximation the submatrix of func-tions ofX and119891(X) represent the order and interaction termsof the vector respectively as shown

X119879119886= [1 119909

1sdot sdot sdot 1199091198991199092

1sdot sdot sdot 1199092

11989911990911199092sdot sdot sdot 119909119894119909119895]119879

forall119894 lt 119895

C119886= [1198620 1198621 sdot sdot sdot 119862119899 11986211 sdot sdot sdot 119862119899119899 11986212 sdot sdot sdot 119862119894119895]

forall119894 lt 119895

(11)

To obtain polynomial coefficients the least-square fittingmethod [17] is generally used by keeping aminimumdistance(fitting error e) between the original data and the approxi-mated one Consider

min119862

e = min119862

119899119901

sum

119894=1

[119884119894(k) minus

119894(k)]2

(12)

This method can provide a good fit for the low nonlinearfunctions with a small variable region If the response datahavemultiple local extremes then polynomialmodels cannotbe used to construct an effective meta-model In order toapproximate globally nonlinear functions other algorithmsfor example Kriging interpolationmodel [18 19] radial basisfunction [20] and multilayer perceptron neural networks(MLPNN) [21] have been employed

A percentage error between 119884119894and

119894is considered by

using the 119899119901simulations that constitute the validation set

Percentage error is given as

119864 =119894minus 119884119894

119884119894

(13)


32 Global Sensitivity Analysis A fully quantitative variance-based global sensitivity method namely the Sobol methodcan analyze the impact of the uncertainties of inputs onthe output [22] The Sobol decomposition was originallydeveloped for the analysis with respect to uniformly dis-tributed variables [23] However Sobol method can be usedfor functions of variables with any distribution Here weuse the extended Sobol formulation for normal distributionmentioned in [24]

Let the random design variables X = (1198831 1198832 119883

119899)

with normal distribution functions Φ1(1198831) Φ2(1198832)

Φ119899(119883119899) And then system response 119865(X) in Sobol decompo-

sition format is given by

119865 (X) = 1198650+ sum

1le119894le119899

119865119894(119883119894) + sum

1le119894lt119895le119899

119865119894119895(119883119894 119883119895)

+ sdot sdot sdot + 11986512119899(119883119894 119883

119899)

(14)

In this expansion the terms 119865119894(119883119894) 119865119894119895(119883119894 119883119895) called

the Sobol functions can be calculated by integrating 119865(X)according to

int119877119899

119865 (x) 120593119899(x) dx = 119865

0

int119877119899minus1

119865 (x= 119894 119883119894) 120593119899minus1(x= 119894) dx= 119894

= (1198650+ 119865119894(119883119894)) 1205931(x119894)

int119877119899minus2

119865 (x= 119894119895 119883119894 119883119895) 120593119899minus2(x= 119894119895) dx= 119894119895

= (1198650+ 119891119894(119883119894) + 119865119895(119883119895) + 119865119894119895(119883119894 119883119895)) 1205932(x119894 x119895)

(15)

where120593119899(119909) is the 119899-dimensional probability density function

of normal distribution x= 119894implies a vector of variables

corresponding to all but 119883119894and x

= 119894119895notation implies the

vector of variables corresponding to all but 119883119894and 119883

119895 and

1198650is the mean value of the response function The variance

of 119865(119883) denoted by 119863 can be decomposed as well as thefunction 119865(119883) according to

int119877119899

1198652(x) 120593119899(x) dx minus 1198652

0

= int1198771

1198652

1(1199091) 1205931(1199091) d1199091+ int1198771

1198652

2(1199092) 1205931(1199092) d1199092

+ sdot sdot sdot + int1198772

1198912

12(1199091 1199092) 1205932(1199091 1199092) d11990911199092+ sdot sdot sdot

(16)

thereby the partitioned variance has the form

119863 = sum

1le119894le119899

119863119894+ sum

1le119894lt119895le119899

119863119894119895+ sdot sdot sdot + 119863

12sdotsdotsdot119899 (17)

R1R4 R3

R2

W2

W1

200

100

520

Figure 2 Dimensions of the section of the hook structure

The Sobol index for a group of indices 1198941 1198942 119894

119905 with 1 le

1198941lt sdot sdot sdot lt 119894

119905le 119899 is defined as

1198781198941sdotsdotsdot119894119905

=

1198631198941sdotsdotsdot119894119905

119863 (18)

All the 1198781198941119894119905

are nonnegative and their sum equals 1 Inpractice the sum of first-order Sobol index denoted by119878(1) usually makes up a large part of the whole varianceThus if all or part of the first-order Sobol index satisfies thecondition 1 minus 119878(1) lt 120576 where 120576 is some small numberthe rest of Sobol index is negligible When a relative simplemodel (eg linear or quadratic regression) is insufficient forthe response function the analytic estimating method maybecome difficult to be implemented Hence for engineeringproblems it is convenient to estimate all the Sobol indicesfrom Monte Carlo simulations The specified Monte Carlomethod for calculating Sobol indices is detailed in [23]

4 Case Study

In this section a ldquohookrdquo structure from [25] is selected toexamine the performance of the proposed method as shownin Figure 2 While uncertainty is considered in geometricdimensions of the hook no uncertainty is considered inmaterial properties or load condition It is rational sinceman-ufacturing tolerance is the most likely source of uncertaintyat the conceptual design stage In later design phase variationintroduced by other sources (changes in temperature andfluctuation in load) can be taken into account to search amore thorough optimal design

The hookrsquos load environment is composed of a weldedconstraint around the left hole and even pressure 119875 =50Nmm2 applied to the lower half of the right side Thematerial has the following mechanical properties elasticmodulus 119864 = 71018MPa Poisson ratio ] = 033 and thedensity 120588 = 825 kgm3

The goal of this problem is to find an optimal shape ofhook so that the area of hook should be as small and robustas possible while keeping the maximum von Mises stressless than the admissible constraint 120590ad = 150MPa with aprobability of 9973 (ie the target reliability index 120573

119888= 3)

The shape of hook can be defined using six design variablesthat are modeled by a vector v = 119877

1 1198772 1198773 119877411988211198822

All random design variables are statistically independent anddistributed normally with a standard deviation of 01 The


Table 1 Properties of design variables

Variables 1198771

1198772

1198773

1198774

1198821

1198822

Initial value (mm) 45 20 10 20 75 125Lower limit (mm) 30 6 5 10 65 110Upper limit (mm) 60 30 30 30 85 140

Table 2 Approximation errors (mean 119864 times 100)

QPA RBF MLPNNArea 44 times 10minus11 164 times 10minus8 847 times 10minus8

Stress 72 362 089

initial values and upper and lower limits of design variables(nominal value) are presented in Table 1 The optimizationproblem is therefore expressed in the following form

minimizek

119908120583119860(k)120583lowast

119860

+ (1 minus 119908)120590119860(k)120590lowast

119860

0 lt 119908 lt 1

subject to 120590eq (z) le 120590ad

k119871 le k le k119880

where z = 120583k minus 120590k120573119905120590knablak120590eq (k)10038171003817100381710038171003817120590knablak120590eq (k)

10038171003817100381710038171003817

(19)

In numerical implementation a set of area and stressvalues are obtained through finite element simulations at 200sampling points generated using Latin hypercube methodbased on orthogonal array The minimum number of sim-ulation points is chosen equal to 10 times the number ofvariables as recommended in [26] thus 200 samples areenough for the case of six design variablesThen these sampl-ing points and their corresponding function values are usedto construct the approximation models of the area and themaximum stress Since the complexity of the response behav-ior in the design space is unknown beforehand a familyof approximating algorithms namely quadratic polynomialapproximation (QPA) radial basis functions (RBF) andmultilayer perceptron neural networks (MLPNN) are usedand comparedThe approximation errors (percentage errors)are summarized in Table 2 It is evident that the area indicesare approximated with a sufficient accuracy in the shape ofpolynomial expression Sobol sensitivity analysis thus canbe performed on the polynomial model resulting in theresponse mean 120583

119860and standard deviation 120590

119860as a function of

design variables For the maximum stress data the MLPNNmodel results in the most accurate fit with the maximumerror equal to 089

The global sensitivity analysis is performed by means ofthe simulations based on the approximated model The first-order Sobol indices are computed by considering the designvariables at their initial values reported in Table 1 Each indexis obtained by varying one single variable the other onesbeing constant and equal to those of the reference value Theresults of the variable first-order sensitivity indices for thefunction of area are shown in Figure 3

Table 3 Results of the different solutions

1198771

(mm)1198772

(mm)1198773

(mm)1198774

(mm)1198821

(mm)1198822

(mm)Deterministic 4063 1251 1000 1890 6970 1150RBDO 4149 1264 1000 1933 6983 1150RRDO 4151 1255 1000 1920 7007 1150

Table 4 Comparison of different designs

Deterministic RBDO RRDOObjective (area 119860)120583119860(mm2) 101046 103755 103826

120590119860

18172 18391 18358Constraint (stress 120590

119890119902)

120583120590(MPa) 1500 1457 1458

Pr119865 521 017 017Number of function calls

Objective 157 237 822Constraint 1467 8462 1778

By inspection of Figure 3 it is noticed that the areafunction is not influenced significantly by 119882

1and 119882

2 In

the current case an obvious truncation would include onlythe uncertainty in input variables corresponding to the fourradius values 119877

1 1198772 1198773 and 119877

4 which account in only their

first-order forms for 9803 of the total response varianceThus119882

1and119882

2can be regarded as two deterministic para-

meters which can efficiently reduce the sample size and vari-able dimension in the later optimization process Since gene-tic algorithm (GA) works well in obtaining global optimumit is chosen as the solver in the proposed RRDOoptimizationThe optimal results from the hybrid RRDO (119908 = 05) using(19) deterministic design and RBDO approach are presentedin Table 3

Table 4 compares the results of the different optimaldesigns As shown in Table 4 the deterministic design givesthe smallest values in both the area response and its standarddeviation which is right because the purpose of deterministicoptimization is to obtain the minimal value without regardfor uncertainties Consequentially it has a relatively largeprobability of failure value of 521when themaximum stress120583120590satisfies the constraint condition At the same time both

RRDO and RBDO let the mean values of area increase tomeet the requirements of structural safety To clearly showthe difference between RBDOandRRDO results the range ofarea response can be briefly estimated based on the momentsof area value (mean and standard deviation) By using 3-120590 rule applied for normal distribution the ranges of areafrom RBDO and RRDO are [4858 15893] and [4875 15890]respectively Thus RRDO gives a feasible result with betterrobustness as the range is smaller

As to the constraint of stress three probability densityfunctions (PDFs) of maximum stress are compared with eachother in Figure 4 Note that these three PDFs are expandedacross just for better showing each failure region and itcannot tell the exact number of probability density From


0

01

02

03

04

05

06

30 40 50

R1

S i

60 10 17 24 30

0

01

02

03

04

05

06

S i

R2

10 17 24 30

5 10 15

R3

0

01

02

03

04

05

06

S i

R4 W1

0

01

02

03

04

05

06

S i

65 72 79 85

0

01

02

03

04

05

06

S i

W2

110 120 130 140

0

01

02

03

04

05

06

S i

SR1SR2SR3

SR4SW1

SW2

SR1SR2SR3

SR4SW1

SW2

SR1SR2SR3

SR4SW1

SW2

Figure 3 Nondimensional first-order sensitivity indices

Stre

ss(M

Pa) 150

14581457

521017

017

Failure

RRDO

RBDO

Deterministic

Figure 4 Probability density functions of maximum stress 120590eq

Figure 4 failure occurs over the red line whose value is150MPa and the PDF area belonging to the failure regioncorresponds to the probability of failure It is seen thatboth RBDO and RRDO ensure reliability with the target

probability of failure (ie 017) Considering the numberof function evaluations for all the approaches the proposedmethod shows an economical solutionwithmuch lower com-putational cost over RBDO From the results it is evident thatthe proposed method reduces the computational cost whilemaintaining the accuracy of reliability and robustness ana-lysis

5 Conclusions

In this study a hybrid RRDO approach is presented to finda reliable and robust solution under uncertainty in designvariables This work combined the single-loop approachand monoobjective robust design (weighted sum) methodapplying a RRDO strategy with the usage of global approx-imation and global sensitivity analysisThe central advantageof the proposed method is reducing the computational costwhich is implemented through three ways First using afamily of approximated models to replace the computationalexpensive FE simulations the response model was accurately


constructed Second global sensitivity analysis ensured arelatively small sample size because some insensitive designvariables were considered deterministic without uncertaintyThird the optimization structure was converted into single-loop process by the adoption of single-loop approach Astructural example was tested and the results have beencomparedwith those fromRBDOanddeterministic caseTheaccuracy and efficiency of the proposed method have beendemonstrated

Conflict of Interests

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

References

[1] C Zang M I Friswell and J E Mottershead ldquoA review ofrobust optimal design and its application in dynamicsrdquo Com-puters amp Structures vol 83 no 4-5 pp 315ndash326 2005

[2] G I Schueller and H A Jensen ldquoComputational methods inoptimization considering uncertaintiesmdashan overviewrdquo Com-puter Methods in Applied Mechanics and Engineering vol 198no 1 pp 2ndash13 2008

[3] F Jurecka M Ganser and K-U Bletzinger ldquoUpdate scheme forsequential spatial correlation approximations in robust designoptimisationrdquo Computers amp Structures vol 85 no 10 pp 606ndash614 2007

[4] V Rathod O P Yadav A Rathore and R Jain ldquoOptimizingreliability-based robust design model using multi-objectivegenetic algorithmrdquoComputers amp Industrial Engineering vol 66no 2 pp 301ndash310 2013

[5] N D Lagaros V Plevris and M Papadrakakis ldquoReliabilitybased robust design optimization of steel structuresrdquo Interna-tional Journal for Simulation and Multidisciplinary Design Opti-mization vol 1 pp 19ndash29 2007

[6] X Du and W Chen ldquoTowards a better understanding of mod-eling feasibility robustness in engineering designrdquo Journal ofMechanical Design vol 122 no 4 pp 385ndash394 2000

[7] A F Shahraki and R Noorossana ldquoA combined algorithm forsolving reliability-based robust design optimization problemsrdquoJournal of Mathematics and Computer Science vol 7 pp 54ndash622013

[8] Z Wang H-Z Huang and Y Liu ldquoA unified framework forintegrated optimization under uncertaintyrdquo Journal of Mechan-ical Design vol 132 no 5 Article ID 051008 2010

[9] D Roos K Cremanns and T Jasper ldquoProbability and variance-based stochastic design optimization of a radial compressorconcerning fluid-structure interactionrdquo in Proceedings of the 6thInternational Conference on Adaptive Modeling and Simulation2013

[10] D M Frangopol and K Maute ldquoLife-cycle reliability-basedoptimization of civil and aerospace structuresrdquo Computers andStructures vol 81 no 7 pp 397ndash410 2003

[11] R M Paiva A Carvalho and C Crawford ldquoA robust and relia-bility based design optimization framework for wing designrdquo inProceedings of the 2nd International Conference on EngineeringOptimization Lisbon Portugal 2010

[12] S B Jeong and G J Park ldquoReliability-based robust design opti-mization using the probabilistic robustness index and theenhanced single loop single vector approachrdquo in Proceedings

of the 10th World Congress on Structural and MultidisciplinaryOptimization Orlando Fla USA 2013

[13] X Du A Sudjianto and W Chen ldquoAn integrated frameworkfor optimization under uncertainty using inverse reliability stra-tegyrdquo Journal of Mechanical Design Transactions of the ASMEvol 126 no 4 pp 562ndash570 2004

[14] W Y Fowlkes and C M Creveling Engineering Methods forRobust Product Design Using Taguchi Methods in Technologyand Product Development Addison Wesley Reading MassUSA 1995

[15] S-B Jeong and G-J Park ldquoPreliminary study on a single loopsingle vector approach using the conjugate gradient in relia-bility based design optimizationrdquo in Proceedings of the 14thAIAAISSMOMultidisciplinary Analysis andOptimization Con-ference September 2012

[16] T W Simpson J Peplinski P N Koch and J K Allen ldquoOnthe use of statistics in design and implications for deterministiccomputer experimentsrdquo in Proceedings of the ASME DesignEngineering Technical Conferences (DETC rsquo97) Dallas TexUSA September 1997

[17] D C Montgomery Design and Analysis of Experiments JohnWiley amp Sons Hoboken NJ USA 2005

[18] R T Haftka E P Scott and J R Cruz ldquoOptimization andexperiments a surveyrdquo Applied Mechanics Reviews vol 51 no7 pp 435ndash448 1998

[19] J SacksW JWelch T J Mitchell andH PWynn ldquoDesign andanalysis of computer experimentsrdquo Statistical Science vol 4 no4 pp 409ndash423 1989

[20] S Chen C F N Cowan and P M Grant ldquoOrthogonal leastsquares learning algorithm for radial basis function networksrdquoIEEETransactions onNeuralNetworks vol 2 no 2 pp 302ndash3091991

[21] S Haykin Neural Networks A Comprehensive FoundationPrentice Hall Upper Saddle River NJ USA 2nd edition 1999

[22] I M Sobol1015840 ldquoGlobal sensitivity indices for nonlinear mathe-matical models and their Monte Carlo estimatesrdquoMathematicsand Computers in Simulation vol 55 no 1ndash3 pp 271ndash280 2001

[23] I M Sobol ldquoGlobal sensitivity indices for nonlinear mathemat-ical models and their Monte Carlo estimatesrdquoMathematics andComputers in Simulation vol 55 no 1ndash3 pp 271ndash280 2001

[24] S R Arwade M Moradi and A Louhghalam ldquoVariancedecomposition and global sensitivity for structural systemsrdquoEngineering Structures vol 32 no 1 pp 1ndash10 2010

[25] A El Hami and B Radi Uncertainty and Optimization inStructural Mechanics ISTE Great Britain UK 2013

[26] M Schonlau Computer experiments and global optimization[PhD thesis] University of Waterloo Ontario Canada 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


(PMA) finding the reliable and robust Pareto optimal solu-tions Although reliability methods reduce the sampling setsand simulation cost they make the optimization structuremore complex with double loops (the outer loop is theoptimization loop and reliability analysis is performed in theinner loop) even triple loops (including robustness evalua-tion loop) as presented in [9] Hence a common challenge forthe designers is the computational expense of either RDO orRBDO whose objective is distinctly different Some attemptshave been made to improve the optimization efficiency for aone desired characteristic (robustness or reliability) [10ndash13]and few of the existing works address the development ofefficient techniques for both robustness and reliability

The main purpose of this paper is to improve the optimi-zation efficiency for both the reliability design and the robustdesign under uncertainty To address this problem a hybridRRDOmethod is proposed combining the single-loopRBDOalgorithm and monoobjective RDO formulation to improveefficiency In the case of RBDO the single-loop approachbased on the conjugate gradient is employed to treat the con-straints efficiently The robustness evaluation is fulfilled con-sidering both the mean and the variance of response repre-sented by a weighting factor The proposed RRDOmethod isapplied to a ldquohookrdquo problem with uncertainty in design vari-ables The results are compared with those from othermethods in the literature leading to a reduced computationalcost

2 Design Optimization under Uncertainty

21 Reliability-Based Design Method The objective of RBDOis to find an optimal solution that verifies as a probability offailure lower or equal to the target probability expressed as119875119905

119891 The classical RBDO is expressed in the form

minimizek

119891 (k)

subject to Pr [119892119894(k) lt 0] le 119875119905

119891= Φ(minus120573

119905

119894)

119894 = 1 119898

v119871 le v le v119880

(1)

where v isin R119899 is the design vector v119871 and v119880 are the lowerlimit and upper limit of the design vector respectively 119892

119894(sdot)

is the 119894th limit state function dividing the safety region (119892 gt0) and failure region (119892 lt 0) Φ(sdot) is the standard cumu-lative function for the standard normal distribution and 120573119905

119894

is target reliability index of the 119894th probabilistic constraintrepresenting the reliability level

In the field of RBDO considering uncertainty reliabilityindex approach (RIA) and performance measure approach(PMA) based on the reliability index have been mainlystudied The reliability index approach is a FORM-basedmethod which is a direct method in comparison to otheroptimization approaches The basic formulation of RIA iswritten as

minimizek

119891 (k)

subject to 120573119894(u) ge 120573119905

119894 119894 = 1 119898

k119871 le k le k119880(2a)

The suboptimization problem for the evaluation of reliabilityis defined as

minimize 120573119905119894= u

subject to 119892119894(u) = 0 119894 = 1 119898

(2b)

where u is independent standard normal random variablevectors taking the following form

u = k minus 120583k120590k

(3)

and 119892(u) is the probabilistic constraint defined in u-spaceSimilarly the PMA formulation is expressed as follows

minimizek

119891 (k)

subject to 119866119894(u) ge 0 119894 = 1 119898

k119871 le k le k119880

(4a)

where 119866119894(u) is the minimum of the performance function

among the points that have the target reliability index whichis obtained from a suboptimization problem Consider

minimize 119866119894= 119892119894(u)

subject to u119894= 120573119905

119894 119894 = 1 119898

(4b)

It is necessary for both RIA and PMA to solve the subopti-mization problem in order to estimate the reliability becauseof its double-loop structure

22 Robust Design Method A commonly used definitionstates that a robust design is a design that is insensitive (orless sensitive) to input variations RDO aims to improve thequality of products by minimizing the performance variationwithout eliminating the existing uncertainty [14] To achievethe robustness there is no unifiedmathematical formulationInmost cases designers emphasized obtaining the robustnessof the objective performance rather than satisfying con-straints with more robust limit state functions Thus theconstruction of a robust problem generally is based on themean and standard deviation of objective functions Consider

minimizek

F (k) = [120583119891(k) 120590

119891(k)]

subject to 119892119894(k) ge 0 119894 = 1 119898

k119871 le k le k119880

(5)

where 120583119891(v) and 120590

119891(v) are the mean and standard deviation

of the objective function 119891(v) respectively Within this for-mulation RDO is defined as amultiobjective problem (MOP)that gives a series of solutions known as a set of optimalPareto solutions However MOP suffers from a large degree



minimizek

F (k) = 119908120583119891(k)120583lowast

119891

+ (1 minus 119908)

120590119891(k)120590lowast

119891

0 lt 119908 lt 1

subject to 119892119894(k) ge 0 119894 = 1 119898

k119871 le k le k119880

(6)


119891



minimizek

119891 (k)

subject to 119892119894(z) ge 0 119894 = 1 119898

k119871 le k le k119880

(7a)



1003817100381710038171003817

(7b)



minimizek

F (k) = 119908120583119891(k)120583lowast

119891

+ (1 minus 119908)

120590119891(k)120590lowast

119891

0 lt 119908 lt 1

subject to 119892119894(z) ge 0 119894 = 1 119898

k119871 le k le k119880


1003817100381710038171003817

(8)















Start










Converged No

Yes



Stop

Optimization


index

Insertmore

samplingpoints of

importantvariables

No

Yes

119959

119959 119959



(119909) = sum11986211989411198942119894119899

119899

prod

119895=1

119909119894119895

119895

119899

sum

119895=1

119894119895le 119875

(9)





X119879119886as shown

(x) = C119886X119879119886= C119886[1X 119891 (X)]119879 (10)


X119879119886= [1 119909

1sdot sdot sdot 1199091198991199092


11989911990911199092sdot sdot sdot 119909119894119909119895]119879

forall119894 lt 119895


forall119894 lt 119895

(11)


min119862

e = min119862

119899119901

sum

119894=1

[119884119894(k) minus

119894(k)]2

(12)






119864 =119894minus 119884119894

119884119894

(13)




119899)




119865 (X) = 1198650+ sum

1le119894le119899

119865119894(119883119894) + sum

1le119894lt119895le119899

119865119894119895(119883119894 119883119895)

+ sdot sdot sdot + 11986512119899(119883119894 119883

119899)

(14)



int119877119899

119865 (x) 120593119899(x) dx = 119865

0

int119877119899minus1

119865 (x= 119894 119883119894) 120593119899minus1(x= 119894) dx= 119894

= (1198650+ 119865119894(119883119894)) 1205931(x119894)

int119877119899minus2

119865 (x= 119894119895 119883119894 119883119895) 120593119899minus2(x= 119894119895) dx= 119894119895

= (1198650+ 119891119894(119883119894) + 119865119895(119883119895) + 119865119894119895(119883119894 119883119895)) 1205932(x119894 x119895)

(15)






119895 and



int119877119899

1198652(x) 120593119899(x) dx minus 1198652

0

= int1198771

1198652

1(1199091) 1205931(1199091) d1199091+ int1198771

1198652

2(1199092) 1205931(1199092) d1199092


1198912

12(1199091 1199092) 1205932(1199091 1199092) d11990911199092+ sdot sdot sdot

(16)


119863 = sum

1le119894le119899

119863119894+ sum

1le119894lt119895le119899

119863119894119895+ sdot sdot sdot + 119863


R1R4 R3

R2

W2

W1

200

100

520



119905 with 1 le



1198781198941sdotsdotsdot119894119905

=

1198631198941sdotsdotsdot119894119905

119863 (18)

All the 1198781198941119894119905


4 Case Study




119888= 3)


1 1198772 1198773 119877411988211198822




Variables 1198771

1198772

1198773

1198774

1198821

1198822




Stress 72 362 089


minimizek

119908120583119860(k)120583lowast

119860

+ (1 minus 119908)120590119860(k)120590lowast

119860

0 lt 119908 lt 1


k119871 le k le k119880


10038171003817100381710038171003817

(19)







1198771

(mm)1198772

(mm)1198773

(mm)1198774

(mm)1198821

(mm)1198822




120590119860


119890119902)

120583120590(MPa) 1500 1457 1458




1and 119882

2 In


1 1198772 1198773 and 119877



1and119882







0

01

02

03

04

05

06

30 40 50

R1

S i

60 10 17 24 30

0

01

02

03

04

05

06

S i

R2

10 17 24 30

5 10 15

R3

0

01

02

03

04

05

06

S i

R4 W1

0

01

02

03

04

05

06

S i

65 72 79 85

0

01

02

03

04

05

06

S i

W2

110 120 130 140

0

01

02

03

04

05

06

S i

SR1SR2SR3

SR4SW1

SW2

SR1SR2SR3

SR4SW1

SW2

SR1SR2SR3

SR4SW1

SW2


Stre

ss(M

Pa) 150

14581457

521017

017

Failure

RRDO

RBDO

Deterministic




5 Conclusions






References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minimizek

F (k) = 119908120583119891(k)120583lowast

119891

+ (1 minus 119908)

120590119891(k)120590lowast

119891

0 lt 119908 lt 1

subject to 119892119894(k) ge 0 119894 = 1 119898

k119871 le k le k119880

(6)


119891



minimizek

119891 (k)

subject to 119892119894(z) ge 0 119894 = 1 119898

k119871 le k le k119880

(7a)



1003817100381710038171003817

(7b)



minimizek

F (k) = 119908120583119891(k)120583lowast

119891

+ (1 minus 119908)

120590119891(k)120590lowast

119891

0 lt 119908 lt 1

subject to 119892119894(z) ge 0 119894 = 1 119898

k119871 le k le k119880


1003817100381710038171003817

(8)















Start










Converged No

Yes



Stop

Optimization


index

Insertmore

samplingpoints of

importantvariables

No

Yes

119959

119959 119959



(119909) = sum11986211989411198942119894119899

119899

prod

119895=1

119909119894119895

119895

119899

sum

119895=1

119894119895le 119875

(9)





X119879119886as shown

(x) = C119886X119879119886= C119886[1X 119891 (X)]119879 (10)


X119879119886= [1 119909

1sdot sdot sdot 1199091198991199092


11989911990911199092sdot sdot sdot 119909119894119909119895]119879

forall119894 lt 119895


forall119894 lt 119895

(11)


min119862

e = min119862

119899119901

sum

119894=1

[119884119894(k) minus

119894(k)]2

(12)






119864 =119894minus 119884119894

119884119894

(13)




119899)




119865 (X) = 1198650+ sum

1le119894le119899

119865119894(119883119894) + sum

1le119894lt119895le119899

119865119894119895(119883119894 119883119895)

+ sdot sdot sdot + 11986512119899(119883119894 119883

119899)

(14)



int119877119899

119865 (x) 120593119899(x) dx = 119865

0

int119877119899minus1

119865 (x= 119894 119883119894) 120593119899minus1(x= 119894) dx= 119894

= (1198650+ 119865119894(119883119894)) 1205931(x119894)

int119877119899minus2

119865 (x= 119894119895 119883119894 119883119895) 120593119899minus2(x= 119894119895) dx= 119894119895

= (1198650+ 119891119894(119883119894) + 119865119895(119883119895) + 119865119894119895(119883119894 119883119895)) 1205932(x119894 x119895)

(15)






119895 and



int119877119899

1198652(x) 120593119899(x) dx minus 1198652

0

= int1198771

1198652

1(1199091) 1205931(1199091) d1199091+ int1198771

1198652

2(1199092) 1205931(1199092) d1199092


1198912

12(1199091 1199092) 1205932(1199091 1199092) d11990911199092+ sdot sdot sdot

(16)


119863 = sum

1le119894le119899

119863119894+ sum

1le119894lt119895le119899

119863119894119895+ sdot sdot sdot + 119863


R1R4 R3

R2

W2

W1

200

100

520



119905 with 1 le



1198781198941sdotsdotsdot119894119905

=

1198631198941sdotsdotsdot119894119905

119863 (18)

All the 1198781198941119894119905


4 Case Study




119888= 3)


1 1198772 1198773 119877411988211198822




Variables 1198771

1198772

1198773

1198774

1198821

1198822




Stress 72 362 089


minimizek

119908120583119860(k)120583lowast

119860

+ (1 minus 119908)120590119860(k)120590lowast

119860

0 lt 119908 lt 1


k119871 le k le k119880


10038171003817100381710038171003817

(19)







1198771

(mm)1198772

(mm)1198773

(mm)1198774

(mm)1198821

(mm)1198822




120590119860


119890119902)

120583120590(MPa) 1500 1457 1458




1and 119882

2 In


1 1198772 1198773 and 119877



1and119882







0

01

02

03

04

05

06

30 40 50

R1

S i

60 10 17 24 30

0

01

02

03

04

05

06

S i

R2

10 17 24 30

5 10 15

R3

0

01

02

03

04

05

06

S i

R4 W1

0

01

02

03

04

05

06

S i

65 72 79 85

0

01

02

03

04

05

06

S i

W2

110 120 130 140

0

01

02

03

04

05

06

S i

SR1SR2SR3

SR4SW1

SW2

SR1SR2SR3

SR4SW1

SW2

SR1SR2SR3

SR4SW1

SW2


Stre

ss(M

Pa) 150

14581457

521017

017

Failure

RRDO

RBDO

Deterministic




5 Conclusions






References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start










Converged No

Yes



Stop

Optimization


index

Insertmore

samplingpoints of

importantvariables

No

Yes

119959

119959 119959



(119909) = sum11986211989411198942119894119899

119899

prod

119895=1

119909119894119895

119895

119899

sum

119895=1

119894119895le 119875

(9)





X119879119886as shown

(x) = C119886X119879119886= C119886[1X 119891 (X)]119879 (10)


X119879119886= [1 119909

1sdot sdot sdot 1199091198991199092


11989911990911199092sdot sdot sdot 119909119894119909119895]119879

forall119894 lt 119895


forall119894 lt 119895

(11)


min119862

e = min119862

119899119901

sum

119894=1

[119884119894(k) minus

119894(k)]2

(12)






119864 =119894minus 119884119894

119884119894

(13)




119899)




119865 (X) = 1198650+ sum

1le119894le119899

119865119894(119883119894) + sum

1le119894lt119895le119899

119865119894119895(119883119894 119883119895)

+ sdot sdot sdot + 11986512119899(119883119894 119883

119899)

(14)



int119877119899

119865 (x) 120593119899(x) dx = 119865

0

int119877119899minus1

119865 (x= 119894 119883119894) 120593119899minus1(x= 119894) dx= 119894

= (1198650+ 119865119894(119883119894)) 1205931(x119894)

int119877119899minus2

119865 (x= 119894119895 119883119894 119883119895) 120593119899minus2(x= 119894119895) dx= 119894119895

= (1198650+ 119891119894(119883119894) + 119865119895(119883119895) + 119865119894119895(119883119894 119883119895)) 1205932(x119894 x119895)

(15)






119895 and



int119877119899

1198652(x) 120593119899(x) dx minus 1198652

0

= int1198771

1198652

1(1199091) 1205931(1199091) d1199091+ int1198771

1198652

2(1199092) 1205931(1199092) d1199092


1198912

12(1199091 1199092) 1205932(1199091 1199092) d11990911199092+ sdot sdot sdot

(16)


119863 = sum

1le119894le119899

119863119894+ sum

1le119894lt119895le119899

119863119894119895+ sdot sdot sdot + 119863


R1R4 R3

R2

W2

W1

200

100

520



119905 with 1 le



1198781198941sdotsdotsdot119894119905

=

1198631198941sdotsdotsdot119894119905

119863 (18)

All the 1198781198941119894119905


4 Case Study




119888= 3)


1 1198772 1198773 119877411988211198822




Variables 1198771

1198772

1198773

1198774

1198821

1198822




Stress 72 362 089


minimizek

119908120583119860(k)120583lowast

119860

+ (1 minus 119908)120590119860(k)120590lowast

119860

0 lt 119908 lt 1


k119871 le k le k119880


10038171003817100381710038171003817

(19)







1198771

(mm)1198772

(mm)1198773

(mm)1198774

(mm)1198821

(mm)1198822




120590119860


119890119902)

120583120590(MPa) 1500 1457 1458




1and 119882

2 In


1 1198772 1198773 and 119877



1and119882







0

01

02

03

04

05

06

30 40 50

R1

S i

60 10 17 24 30

0

01

02

03

04

05

06

S i

R2

10 17 24 30

5 10 15

R3

0

01

02

03

04

05

06

S i

R4 W1

0

01

02

03

04

05

06

S i

65 72 79 85

0

01

02

03

04

05

06

S i

W2

110 120 130 140

0

01

02

03

04

05

06

S i

SR1SR2SR3

SR4SW1

SW2

SR1SR2SR3

SR4SW1

SW2

SR1SR2SR3

SR4SW1

SW2


Stre

ss(M

Pa) 150

14581457

521017

017

Failure

RRDO

RBDO

Deterministic




5 Conclusions






References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119899)




119865 (X) = 1198650+ sum

1le119894le119899

119865119894(119883119894) + sum

1le119894lt119895le119899

119865119894119895(119883119894 119883119895)

+ sdot sdot sdot + 11986512119899(119883119894 119883

119899)

(14)



int119877119899

119865 (x) 120593119899(x) dx = 119865

0

int119877119899minus1

119865 (x= 119894 119883119894) 120593119899minus1(x= 119894) dx= 119894

= (1198650+ 119865119894(119883119894)) 1205931(x119894)

int119877119899minus2

119865 (x= 119894119895 119883119894 119883119895) 120593119899minus2(x= 119894119895) dx= 119894119895

= (1198650+ 119891119894(119883119894) + 119865119895(119883119895) + 119865119894119895(119883119894 119883119895)) 1205932(x119894 x119895)

(15)






119895 and



int119877119899

1198652(x) 120593119899(x) dx minus 1198652

0

= int1198771

1198652

1(1199091) 1205931(1199091) d1199091+ int1198771

1198652

2(1199092) 1205931(1199092) d1199092


1198912

12(1199091 1199092) 1205932(1199091 1199092) d11990911199092+ sdot sdot sdot

(16)


119863 = sum

1le119894le119899

119863119894+ sum

1le119894lt119895le119899

119863119894119895+ sdot sdot sdot + 119863


R1R4 R3

R2

W2

W1

200

100

520



119905 with 1 le



1198781198941sdotsdotsdot119894119905

=

1198631198941sdotsdotsdot119894119905

119863 (18)

All the 1198781198941119894119905


4 Case Study




119888= 3)


1 1198772 1198773 119877411988211198822




Variables 1198771

1198772

1198773

1198774

1198821

1198822




Stress 72 362 089


minimizek

119908120583119860(k)120583lowast

119860

+ (1 minus 119908)120590119860(k)120590lowast

119860

0 lt 119908 lt 1


k119871 le k le k119880


10038171003817100381710038171003817

(19)







1198771

(mm)1198772

(mm)1198773

(mm)1198774

(mm)1198821

(mm)1198822




120590119860


119890119902)

120583120590(MPa) 1500 1457 1458




1and 119882

2 In


1 1198772 1198773 and 119877



1and119882







0

01

02

03

04

05

06

30 40 50

R1

S i

60 10 17 24 30

0

01

02

03

04

05

06

S i

R2

10 17 24 30

5 10 15

R3

0

01

02

03

04

05

06

S i

R4 W1

0

01

02

03

04

05

06

S i

65 72 79 85

0

01

02

03

04

05

06

S i

W2

110 120 130 140

0

01

02

03

04

05

06

S i

SR1SR2SR3

SR4SW1

SW2

SR1SR2SR3

SR4SW1

SW2

SR1SR2SR3

SR4SW1

SW2


Stre

ss(M

Pa) 150

14581457

521017

017

Failure

RRDO

RBDO

Deterministic




5 Conclusions






References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Variables 1198771

1198772

1198773

1198774

1198821

1198822




Stress 72 362 089


minimizek

119908120583119860(k)120583lowast

119860

+ (1 minus 119908)120590119860(k)120590lowast

119860

0 lt 119908 lt 1


k119871 le k le k119880


10038171003817100381710038171003817

(19)







1198771

(mm)1198772

(mm)1198773

(mm)1198774

(mm)1198821

(mm)1198822




120590119860


119890119902)

120583120590(MPa) 1500 1457 1458




1and 119882

2 In


1 1198772 1198773 and 119877



1and119882







0

01

02

03

04

05

06

30 40 50

R1

S i

60 10 17 24 30

0

01

02

03

04

05

06

S i

R2

10 17 24 30

5 10 15

R3

0

01

02

03

04

05

06

S i

R4 W1

0

01

02

03

04

05

06

S i

65 72 79 85

0

01

02

03

04

05

06

S i

W2

110 120 130 140

0

01

02

03

04

05

06

S i

SR1SR2SR3

SR4SW1

SW2

SR1SR2SR3

SR4SW1

SW2

SR1SR2SR3

SR4SW1

SW2


Stre

ss(M

Pa) 150

14581457

521017

017

Failure

RRDO

RBDO

Deterministic




5 Conclusions






References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

01

02

03

04

05

06

30 40 50

R1

S i

60 10 17 24 30

0

01

02

03

04

05

06

S i

R2

10 17 24 30

5 10 15

R3

0

01

02

03

04

05

06

S i

R4 W1

0

01

02

03

04

05

06

S i

65 72 79 85

0

01

02

03

04

05

06

S i

W2

110 120 130 140

0

01

02

03

04

05

06

S i

SR1SR2SR3

SR4SW1

SW2

SR1SR2SR3

SR4SW1

SW2

SR1SR2SR3

SR4SW1

SW2


Stre

ss(M

Pa) 150

14581457

521017

017

Failure

RRDO

RBDO

Deterministic




5 Conclusions






References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Reliability-Based Robust Design...

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