A Hybrid Method for Truss Mass Minimization considering...

Research ArticleA Hybrid Method for Truss Mass Minimizationconsidering Uncertainties

Herbert M Gomes1 and Leandro L Corso2

1Federal University of Rio Grande do Sul Av Sarmento Leite 425 Sala 202 2∘ Andar 90050-170 Porto Alegre RS Brazil2University of Caxias do Sul R Francisco Getulio Vargas 1130 95070-560 Caxias do Sul RS Brazil

Correspondence should be addressed to Herbert M Gomes herbertmecanicaufrgsbr

Received 10 November 2016 Revised 2 March 2017 Accepted 16 May 2017 Published 13 June 2017

Academic Editor David Greiner

Copyright copy 2017 Herbert M Gomes and Leandro L CorsoThis is an open access article distributed under theCreativeCommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In real-world structural problems a number of factors may cause geometric imperfections load variability or even uncertainties inmaterial properties Therefore a deterministic optimization procedure may fail to account such uncertainties present in the actualsystem leading to optimum designs that are not reliable the designed systemmay show excessive safety or sometimes not sufficientreliability to carry applied load due to uncertainties In this paper we introduce a hybrid reliability-based design optimization(RBDO) algorithm based on the genetic operations of Genetic Algorithm the position and velocity update of the Particle SwarmAlgorithm (for global exploration) and the sequential quadratic programming for local searchThe First-Order ReliabilityMethodis used to account uncertainty in design and parameter variables and to evaluate the associated reliability The hybrid method isanalyzed based onRBDObenchmark examples that range from simple to complex truss parametric sizing optimizations with stressdisplacements and frequency deterministic and probabilistic constraints The proposed final problem which cannot be handledby single loop RBDO algorithms highlights the importance of the proposed approach in cases where the discrete design variablesare also random variables

1 Introduction

In the engineering science the cost reduction in manufac-turing is pursued in order to obtain efficiency and save timein production Although the use of optimization methods isincreasing in practical terms the design parameters andordesign variables used in deterministic optimization proce-dures may present uncertainties It means that analyzing theresponses obtained by deterministic optimization proceduresin terms of failure probability nonacceptable levels maybe present in the engineering point of view Accordingto [1] uncertainties in real-world optimization problemsinclude factors such as data incompleteness mathematicalmodel inaccuracies and environmental condition variationjust to name a few This directly affects the optimizationproblem since in a structural engineering design economyand safety are competing goals [2] One way to link suchuncertainties in an optimization problem is using a reliabilityindex (120573) a measure of the degree of reliability in the design

according to [1] This procedure is referred to as reliability-based design optimization (RBDO) in this case the relia-bility index becomes an extra constraint to the optimizationproblem According to [3] in the last decade the RBDOsignificance and conceptual and mathematical complexityhave been intensively studied The optimization proceduremay require a high computational cost [4] For this reasonseveral authors have studied methods that use single loopRBDO [3 5ndash7] These authors have applied reliability-baseddesign optimization strategies like SAP (Sequential Approx-imate Programming) and SORA (Sequential Optimizationand Reliability Assessment) which are capable of solving themajority of practical cases when multiple modes of failureare present but for linear and smooth functions Accordingto [8] the use of approximation procedures in the sequentialoptimization and the evaluation of the reliability index ina single step may result in spurious optimal points Thatis evident since these methods are deterministic and theyshow a tendency of convergence to local minima points

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 2324316 14 pageshttpsdoiorg10115520172324316

2 Mathematical Problems in Engineering

due to possible nonlinear function behavior in probabilisticproblems In order to increase robustness authors [9] pro-posed to construct a response surface that is the product ofindividual performance functions and be sued as surrogateto get optimal design solutions Despite the large amount ofliterature in the theme a good introduction to the RBDOsubject can be found in [10]

In this article a hybrid RBDO methodology is proposedand applied to a range of spatial trusses in order to findthe optimum parameters for minimum mass consideringmaximum allowable deflections and limited stress The First-Order Reliability Method (FORM) is used to evaluate thereliability index (120573) along the optimization steps Due tothe high computational cost of the traditional algorithms tosolve nonconvex optimization problems we propose a hybridoptimizationmethod Comparisons about the computationalcost and quality of the solution are performed using asbaseline standard structural problems The proposed hybridglobal optimization method based on existent global search(metaheuristic) and deterministic algorithms is detailedThisnew method is presented and discussed by using reliability-based optimization on structural truss examples

2 Reliability Index Evaluation

As stated by [11] reliability analysis can be applied to manyengineering fields such as in aeronautical mechanical andcivil problems By definition the reliability is the complementof the probability of failure that is the likelihood of failureof a specific event or set of events from a complex system Alimit state function that relates the failure event (violation ofspecific set of constraints) as function of several variables isstated by the mathematical expression

119892 (1198831 119883119899) = 0 (1)

where 119892 means the limit state function that defines a con-straint Some design variables may present random compo-nents In case 119883119894 is a set of 119899 random variables that affectsthat constraint this limit state function becomes also randomand some probability of violation is implicit [11 12] 119892(sdot) le 0means that the system is in the failure domain 119863 and 119892(sdot) gt0 means that the system is in the safe domain (constraintwas not violated) The probability of failure can be evaluatedby the integration of the joint probability density function119891119883(1198831 119883119899) as indicated in [11]

119875119891 = int sdot sdot sdot int119863119891119883 (1198831 119883119899) 1198891198831 sdot sdot sdot 119889119883119899 cong Φ (minus120573) (2)

where the failure domain is defined by 119863 means that 119892(sdot) le0 Φ is the cumulative standard distribution function and120573 is a safety index metric Equation (2) presents a closesolution in some particular cases where 119891119883(sdot) is Gaussianand for linear and quadratic 119892(sdot) However it is difficultto be handled in case of several random variables 119899 Inthis case Monte Carlo (MC) Simulation can be used to getapproximate asymptotic solutions Moreover statistic valuesfor function119891119883(119883) are not known a priori and the number ofMC samples frequently is not enough to ensure confidence

fX(X)

g(X) lt 0 g

g

g

Pf

g(X)

Figure 1 Concept of probability of failure and the reliability index 120573for the first-order approximation

So a very simple but not so robust way to get and estimatefor the reliability index (that is related to the probabilityof survival) 120573 is using the probability density function firstand second moments (mean and variance) for the limit statefunction 119892(1198831 119883119899) When the limit state function 119892(119883)is linear and the random variables are normally distributedand uncorrelated the reliability index 120573 can be approximatedby the following (see Figure 1)

120573 = 120583119892120590119892 (3)

where 120583119892 and 120590119892 represent respectively the mean value andthe standard deviation (square root of variance) for function119892(119883)

Let 120590119894 be the mechanical stress that can be measured ina loaded component 119894 assuming a failure situation wherethis value exceeds the imposed material strength limit value(120590lim) Equation (1) can be rewritten for the limit statefunction as follows

119892119894 (120590119894) = 1 minus 120590119894120590lim (4)

Linearization of function 119892(119883) by a Taylor expansion upto linear terms can be used to get approximated values for120583119892 and 120590119892 when the limit state functions are nonlinear Thepoint around the linearization performed affects 120583119892 and 120590119892values A method to obtain the reliability index 120573 that isindependent of the limit state function formulation is knownas AFOSM (Advanced First-Order SecondMoment) and wasfirst proposed by [13] For uncorrelated Gaussian randomvariables 119883119894 they are transformed into normalized ones 119880119894by the transformation

119880119894 = Φminus1 [119865119883119894 (119909119894)] = 119879 (119883119894) (5)

where 119865119883119894(119909119894) and Φminus1(sdot) are the cumulative distributionfunction of the random variable 119883119894 and the inverse of thecumulative standard Gaussian distribution respectively Inthis way the limit state function in the real space 119883 istransformed to the uncorrelated normalized space 119880 so

ℎ (119880) cong 119892 (119883) (6)

The linearization of the limit state function ℎ(119880) isperformed at the 119880lowast point that presents the shorter distanceto the origin of the uncorrelated space 119880 and that ensures

Mathematical Problems in Engineering 3

ℎ(119880) = 0 The 119880lowast point is called the design point and thereliability index 120573 is evaluated as mentioned previously as

120573 = min (119880)subject to ℎ (119880) le 0 (7)

21 Rackwitz-Fiessler Algorithm In order to solve (7) theefficient algorithm proposed by Rackwitz and Fiessler [13] isused It can be described in the following steps

Step 1 Define the limit state function for the problem 119892(X) =0Step 2 Assume initial values for the design point in the realspace Xlowast = (1198831 119883119899)119879 and evaluate the correspondingvalues for the limit state function 119892(X) (eg assume an initialdesign point as the mean values of random variables)

Step 3 Evaluate the equivalent Gaussian mean value andstandard deviation for the random variables

120590119873119883119894 = 120601 (Φminus1 [119865119883119894 (119909119894)])119891119883119894 (119909119894) 120583119873119883119894 = 119883119894 minus 120590119873119883119894Φminus1 [119865119883119894 (119909119894)]

(8)

Step 4 Transform random variables from real space X tonormal uncorrelated U The design variables values at thedesign point will be evaluated as follows

119880119894 = 119883119894 minus 120583119873119883119894120590119873119883119894 (9)

Step 5 Evaluate the sensitivities 120597119892(119883)120597119883119894 at the designpoint Xlowast

Step 6 Evaluate the partial derivatives 120597119892(119883)120597119880119894 in thenormal uncorrelated space using the chain rule

120597119892 (119883)120597119880119894 = 120597119892 (119883)120597119883119894 120597119883119894120597119880119894 (10)

Step 7 Evaluate the new value for the design point 119880lowast119894 in thenormal uncorrelated space using the recurrent equation

119880lowast119894119896+1 = [nabla119892 (119880lowast119894119896)119879119880lowast119894119896 minus 119892 (119880lowast119894119896)] nabla119892 (119880lowast119894119896)10038161003816100381610038161003816nabla119892 (119880lowast119894119896)10038161003816100381610038161003816 (11)

Step 8 Evaluate the distance from the origin to this newpointand estimate the new reliability index

120573 = 119880 = radic 119899sum119894=1

(119880lowast119894 )2 (12)

Step 9 Check convergence of 120573 along iterations using apredefined tolerance

Step 10 Evaluate the random variables at the new designpoint using

119883119894 = 120583119873119883119894 + 120590119873119883119894119880lowast119894 (13)

Step 11 Evaluate119892(X) value for the new randomvariables andverify a final convergence criterion for instance Δ119892(X) lttolerance and ΔX lt tolerance

Step 12 If both criteria are met stop iterating otherwiserepeat Steps 3ndash11

This algorithm assumes all the random variables asnoncorrelated in the original actual space If there exists cor-relation between random variables using Cholesky decom-position of the covariancematrix the correlated variables canbe transformed to uncorrelated ones and the algorithm is stillvalid [8 11 14] This transformation is presented in

Z = Lminus1 (X minus 120583) with cov (XX) = C = LL119879 (14)

3 Reliability-Based DesignOptimization (RBDO)

In the RBDO the objective function should satisfy predefinedprobabilistic constraints which are set as new constraints tothe problem Probability failure analyses are performed alongthe optimization process in order to check if the probabilisticconstraints are met This is used to guide the optimization tothe minimum weight that complies with the target reliabilitylevels The easier formulation for RBDO implements thealgorithm with a double loop where the optimization is splitinto two stages (a) on a first stage the objective functionoptimization is performed focusing on the design variables(b) on a second stage the optimization is performed focusingon the random variables starting from the design variablesfrom the outer loop More details can be found in [15] Adeterministic model for the minimization can be definedgenerally as follows [16]

Minimize 119891 (V119889 p)Subject to ℎ119894 (V119889 p) = 0 119894 = 1 119898

119892119895 (V119889 p) lt 0 119895 = 1 119899V119889119897 le V119889 le V119889119906

(15)

where V119889 is the vector of design variables p is the vectorof parameters for the optimization problem ℎ119894(sdot) = 0 is119894th modelrsquos equality constraint from a total of 119898 equalityconstraints 119892119895 are the 119899 inequality constraints and V119889119906 andV119889119897 are the vectors that contains upper and lower values forthe design variables However a deterministic optimizationdoes not consider the uncertainties in the design variablesnor fixed design parameters In RBDO the probabilistic con-straints are added increasing the set of constraint equationsSince the reliability index can be defined in terms of the


accumulated probability function for the limit state function(and vice versa) the following holds

119875119891 (V119889 p) = Φ (minus120573)or 120573 = minusΦminus1 [119875119891 (V119889 p)] (16)

where Φ is the cumulative standard distribution function Inthis article the reliability constraint is formulated as follows

119892119895 (V119889 p) = 1 minus 120573120573targetlt 0 119895 = 119899119903 + 1 119899119901 (17)

where 119892119895(sdot) is the probabilistic constraint defined by thedimensionless ratio between the evaluated reliability index 120573and the target reliability index 120573target This means that if theevaluated reliability index 120573 during the optimization is largerthan the reliability target index 120573target then 119892119895(sdot) le 0 and theprobabilistic criterion is met Otherwise a penalization forthe objective function will take place Numerically duringthe optimization process the failure function needs to beevaluated a number of times so one can find the probabilityof failure value (or conversely the reliability index) In aRBDO both parameter vector p and design variables V119889 canbe random variables

Figure 2 shows a geometric interpretation for the dif-ference between a deterministic and a reliability-based opti-mization The implementation of the optimization may beperformed using two different approaches RIA (ReliabilityIndex Approach) or PMA (PerformanceMeasure Approach)

31 Reliability Index Approach (RIA) This approximationfor the reliability constraint is treated as an extra constraintthat is formulated by the reliability index 120573 (for the sake ofsimplicity in the uncorrelated design space) So the followingcan be written

Minimize 119891 (u p)Subject to p119871 lt p lt p119880

119892119894 (119864 [u p]) lt 0 119894 = 1 119898ℎ119895 (119864 [u p]) = 0 119895 = 1 119899120573target minus 120573 (119891119896 (u p) = 0) lt 0

119896 = 1 119901120573target minus 120573 (119891119897 (u p) = 0) = 0

119897 = 1 119902

(18)

where u is the standard uncorrelated random variables and119892119894 and ℎ119895 are the 119898 inequality and 119899 equality deterministicconstraints with corresponding 119901 probabilistic equalitiesand 119902 probabilistic inequalities constraints 119864[sdot] means theexpected values In order to find 120573 the reliability problemis defined as 120573 = min(119880) subject to 119892(119880) le 0

32 Performance Measure Approach (PMA) This formula-tion is performed with the inverse of the previous analysis byRIA in such a way that the following can be written


119892119894 (119864 [u p]) lt 0 119894 = 1 119898ℎ119895 (119864 [u p]) = 0 119895 = 1 119899119892119896 (u = 120573target p) minus 119892119896 (u p) lt 0

119896 = 1 119901ℎ119897 (u = 120573target p) minus ℎ119897 (u p) = 0

119897 = 1 119902

(19)

where u is the vector the normalized uncorrelated randomvariables 119892119894 and ℎ119895 are the 119898 and 119899 inequality and equalityconstraints and 119892119896 and ℎ119897 are the 119901 and 119902 probabilisticinequalities constraints respectively So differently fromRIA for a fixed reliability index u = 120573target the equalityand inequality constraints are ensured beforehand so a linesearch for the point where hypersphere u = 120573target cutsthe constraints should be performed The advantages anddisadvantages in using this or the previous PMA formulationcan be found in [17]

4 Proposed Hybrid Optimization Method

Evolutionary algorithms are widely considered powerful androbust techniques for global optimization and can be usedto solve full-scale problems which contain several localoptima [18] Although the implementation of such methodsis easy they can demand a high computational cost dueto high number of function calls Therefore they are notrecommended for problems in which the objective functionpresents high computational cost [19] Besides it is importantthat the algorithmparameters be tuned adequately in order toavoid premature convergence of the algorithm

Pioneering works with hybrid algorithms for optimiza-tion have been performed by [20] with the so-called Aug-mented Lagrangian Genetic Algorithm that could deal withconstraints by penalization functions The algorithm hasan outer loop to update penalization parameter based onconstraints values and an inner loop for traditional GAusing the penalized fitness function The algorithm canalso avoid trial and error selection and manually increasepenalty constants in the optimization problem In a differentway the hybrid method proposed here is based on the useof genetic operators of GA like mutation crossover andelitism associated with a gradient search by SQP algorithmperformed on best individual of the GA population Besides20 of the best individuals will pass by a hybrid PSOoperation which allows position update based on velocityand global and local cognitive coefficients These operatorsare inherited from the traditional PSO algorithm [21] Using


Failure domain

Failure domain

Failure domainSafety domain

Optimum deterministicdesign

Optimum reliability-based design

Safet

y

Safet

y

Failu

re

Failu

re

g1(X) = 0

g2(X) = 0

x2

x1

Figure 2 Geometric definition for the reliability-based design optimization

1

m

Evaluation andranking of m individuals

Generations from 1 to n

PSO operators(i) 20 of bestindividuals selected(ii) Position and velocity updated

SQP

Genetic operators(i) Selection(ii) One point crossover(blended)(iii) Adaptive mutation

2

Latin hypercube sampling (LHS) generation of m

individuals

1

m

GA

Figure 3 Flowchart schematic for the HGPS

elitism the best individual of the group passes to a costfunction enhancement by sequential quadratic programming(SQP) and then is sent to the next generation This methodis hereafter called HGPS The main goal of hybrid methodis to accelerate the convergence rate of a global search whilemaintaining the exploration capability The SQP method isjustified by the intrinsic speed in finding local optima whileGA and PSO preserve the diversity in the exploration of newregions of the search space

Although the convergence rate can be accelerated gettingstuck in local optima still remains possible For this reasonthe adaptive mutation is inherited from GA PSO operators(like momentum) are also used in order to help the escapefrom local minima Initial tests carried out by [22 23]showed that loss in diversity may still occur resulting inpremature convergence thus adaptive mutation operationis recommended mainly in cases where the problem hasa complex solution (eg in nonsmooth or discontinuousfunctions) Their use is also based on the good results foundin preliminary tests in literature [24]

A simple sketch illustrating the idea of mixing bestfeatures from several algorithms is shown in Figure 3 consid-ering 119898 individuals and 119899 generations Algorithm 1 presentsthe corresponding pseudocode with the steps followed bythe proposed algorithm It is important to notice that the

codification for the design variables (in a GA sense) is thereal one that is each individual 119894 for the generation 119905 isrepresented by b119894119905 = (1199091 1199092 119909119899119888) where 119899119888 is the numberof genes (design variables)

In the pseudocode in Algorithm 1 119909119896119894119895 is the current valueof the design variable 119895 of the particle 119894 of the generation119896 of the GA The V119896+1119894119895 is the updated velocity of the designvariable 119895 of the particle 119894 of the generation 119896 of the GA The119909119897119887119890119904119905119896119894119895 is the best design variable 119895 from the selected 20of best individuals and 119909119892119887119890119904119905119896119895 is the best design variable 119895of generation 119896 Related to the stopping criteria a criterionis chosen that is based on the coefficient of variation (120590120583) ofthe objective function of the elite individuals within a definednumber of generations For the following problems it hasbeen considered that within five consecutive generationsif the change in the coefficient of variation of the objectivefunction of the elite individuals is lower than a definedtolerance the convergence criterion is met

This hybridized methodology (HGPS) was proposedaiming at convergence acceleration and maintaining thediversity of individuals to avoid premature convergenceTherefore this approach takes advantage from the use ofadaptive mutation PSO operator performed on individualsfrom consecutive generations and SQP improvements on the


(1) Initialize generations Initialize Population size ldquo119898rdquo mutation probability ldquo119875119898rdquo crossover probability ldquo119875119888rdquonumber of genes per individual ldquo119899119888rdquo upper and lower bounds values for each gene ldquo119909max(119899119888)rdquo ldquo119909min(119899119888)rdquo

Main Loop (while stopping criteria are not met)(2) Evaluate and rank the population of individuals by the objective function vector(3) Generation of new population(31) SQP methodThe best individual of the population is the starting point of search SQP method which

replaces the same point in the next generation(32) PSO method 20 of the best individuals suffers PSO operations constrained to 119909max and 119909min

According toV119896+1119894119895 = 120594[120603V119896119894 + 11988811199031(119909119897119887119890119904119905119896119894119895 minus 119909119896119894119895) + 11988821199032(119909119892119887119890119904119905119896119895 minus 119909119896119894119895)]119909119896+1119894119895 = 119909119896119894119895 + V119896+1119894119895This work assumes 1198881 = 1198882 = 15 and 120603 = 04(1 +min lowast (cov 06) according to [25])

(33) GAMethod (genetic operators)(331) Crossover It is assumed 119875119888 = 08 according to [26]ldquoOne point RecombinationrdquoLoop 119894 = 1 to119898 minus 1 Step 2

If random(0 1) lt 119875119888 do120572 = 05Δ = max[119887119894119905(119896) 119887119894+1119905(119896)] minusmin[119887119894119905(119896) 119887119894+1119905(119896)]119887119894119905(119896) = random(min[119887119894119905(119896) 119887119894+1119905(119896)] minus 120572Δmax[119887119894119905(119896) 119887119894+1119905(119896)] + 120572Δ)End If

End Loop 119894(332) Adaptive Mutation [22]

Loop 119894 = 1 to119898If 119891119894 gt 119891min then 119875119898 = 02Else

119875119898 = 05(119891119894 minus 119891min)(119891 minus 119891min)End If

End Loop 119894If random(0 1) lt 119875119898 do119896 = random(0 1) lowast 119899119888119887119894119905(119896) = random(119909max(119896) 119909min(119896))End If

End of the Main Loop

Algorithm 1 Pseudocode for the HGPS

solution of the best individual found so far The probabilityof mutation is not a constant value but varies according tothe fitness function used by the GA that is the mean ofindividual objective function value 119891 According to [22] thevalue of 119875119898 should depend not only on 119891 minus 119891min (a measureof convergence) but also on the fitness function value 119891119894 ofthe solution The closer 119891119894 is to 119891min the smaller 119875119898 shouldbe assuming zero at 119891119894 = 119891min To prevent the overshootingof 119875119898 beyond 10 the constraint for 119875119898 is achieved by setting119875119898 = 02 whenever 119891119894 gt 119891min (since such solutions need tobe completely disrupted) otherwise it is set to 119875119898 = 05(119891119894 minus119891min)(119891 minus 119891min)5 Results

The comparisons are performed using only simple optimiza-tion algorithms like GA (Genetic Algorithm) DE (Differen-tial Evolution) PSO (Particle Swarm Optimization) FMA(Firefly Metaheuristic Algorithm) and the proposed HGPSalgorithmWhen possible the deterministic SQP (sequential

quadratic programming) algorithm will be used for com-parisons Specifically in example 3 where discrete designvariables are present this comparison will not be possibleSince there is no theorem that gives the best parameters valuesfor the algorithms that may serve to any problem basedon literature recommendations and some trial and errorsome effort was made in finding those parameters for eachmetaheuristic algorithm In all studied cases in order to havefair comparisons the same set of optimization parameterswas applied that were previously tuned for best performancein each algorithm Table 1 shows such parameters for thetested algorithms

51 Case 1 10 Bar Truss Problem This classical model waspresented by [6] that performed the minimization of thetruss mass with stress and displacement constraints takinginto account reliability index for stress and displacementsThe analyzed truss is presented in Figure 4 The bar length119886 = 9144m(360 in)The ten bar cross sections are the designvariables The lower and upper limit for the design variable


Table1Summaryof

parametersfor

each

optim

izationmetho

d

GA

DE

PSO

FMA

HGPS

Num

bero

find

ividuals

200

Num

bero

findividu

als

50Num

bero

fparticles

50Num

bero

fparticles

50Num

bero

findividu

als

50

Crossover

one

point

Weightin

gratio

15So

cialconstant

201

Beta

10So

cial

constantcognitiv

econstant

201201

Prob

abilityof

crossover

80

Crossoverp

rob

08

Cognitiv

econ

stant

201

Beta

min

01

Prob

abilityof

crossover

80

Prob

abilityof

mutation

1Search

Startin

gpo

int

(119909 max+119909 m

in)2

Inertia

mom

ent

08

Gam

ma

15Prob

abilityof

mutation

1

Tolerance(successiv

eObj

Funcdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

bleo

rmeshsiz

e)10minus6

Tolerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6Maxim

umnu

mbero

fgeneratio

ns200

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800


2

3

4

5

6

11 2

3 4

5 6

7

8

9

10

a

P1 P2

a a

Figure 4 10 bar truss geometry nodes and connectivity

are 645 times 10minus6m2 (001 in2) and 161 times 10minus2m2 (25 in2)respectively The memberrsquos elastic modulus are 119864 = 6895 times1010 Pa (107 Psi) and assumed deterministic The appliedloads are assumed random and uncorrelated following alognormal distribution with mean 120583 = 4448 times 105N (10 times105 lbf) and standard deviation 120590 = 2224 times 104N (50 times103 lbf) Material strength is assumed as Gaussian randomvariable with mean and standard deviation being equal to1724times108 Pa (25times104 Psi) and 1724times107 Pa (25times103 Psi)respectively The vertical displacement has a constraint ofplusmn01143m (plusmn45 in)

The deterministic constraints for stress and displace-ments are treated as limit state functions from the probabilis-tic point of viewThe reliability indexes for the stress values inany member (tension or compression) and for the maximumvertical displacement at any node are set to 30Therefore themathematical RBDO problem can be stated as follows

Minimize 119891 (A) = 120588 119899sum119894=1

119860 119894119871 119894Subject to 10038161003816100381610038161205901198941003816100381610038161003816 le 1724 times 108 Pa 119894 = 1 1198991003816100381610038161003816100381611988911989510038161003816100381610038161003816 le 01143m 119895 = 1 119899119899

120573119904 (up) ge 120573119871119904120573119889 (up) ge 120573119871119889p119871 le p le p119880

(20)

where p is the vector of design variable (member area)120573119871119889 120573119871119904 are the target reliability indexes and 120573119904(u p) and120573119889(u p) are the reliability indexes for stress and displace-ment respectively that are function of vector of designvariables p and vector of random variables u (two loads andthe material strength) The parameters p119871 and p119880 representlower and upper values for design variables respectively 119899119899is the total number of nodes

Table 2 shows the obtained results with the optimizationmethods averaged for 20 independent runs In the sametable the efficiency parameter (ratio between total numberof objective function (deterministic) and limit state function(probabilistic) evaluations to the corresponding value forthe best valid solution found so far ie less weight and

not constrained) and the quality parameter (best objectivefunction values ratio) for each result are presented

Results from GA FMA and SQP show to be heavierthen that reported by [6] using SORASQPmethod but bothPSO and HGPS (particularly HGPS) achieved a lighter trusssolution The author [8] reported that the methods basedon SORA may fall into local optima points that is localoptimum originated mainly due to the used gradient-basedmethods involved in optimization of the objective functionand constraints So the best results are attributed to the use ofmetaheuristic capabilities in the proposed HGPS algorithm

In addition it is possible to note that the GA and FMAsolutions were not better (objective function) than SORAsolution meaning that the reported configuration seemsnot to be the best for this problem Apparently GA andFMA get stuck into local minima It was also observed thatthe PSO SQP and HGPS methods found slightly differentdesign variable solutions even with reliability index con-straints being satisfied and having similar objective functionvalues This seems to represent a flat design space near theoptimum solution It is also possible to notice the not sogood HGPS efficiency when compared to PSO algorithm orSPQ (the most efficient) although HGPS have a superiorsolution (quality) It should the emphasized that any of thedeterministic and probabilistic constraints are violated forthe HGPS solution For the 20 independent runs the HGPSpresented the best mass value of 125231 kg mean valueof 126215 kg (median 126497 kg) the worst mass value of126968 kg and a coefficient of variation of 0012

52 Case 2 37 Bar Truss Problem In this example a Pratttype truss with 37 members is analyzed The goal of thedeterministic optimization problem is to minimize the massThere are nonstructural masses of 119898 = 10 kg attached toeach of the bottom node see Figure 5 The lower chordis modelled as bar elements with fixed cross-sectional area119860 = 4 times 10minus3m2 The remaining symmetric bars are designvariables also modelled as bar elements with initial crosssection of 1198600 = 1 times 10minus4m2 This problem is considereda truss optimization on size and geometry since all nodesof the upper chord are allowed to vary along the 119910-axis ina symmetric way and all the diagonal and upper bars areallowed to vary the cross-sectional area within upper andlower values Figure 5 describes the structure In this examplethe design variables are (1199103 1199105 1199107 1199109 11991011 1198601 1198602 11986014)The lower limit for the design variables are (0 0 0 0 0 1 times10minus4 1 times 10minus4 1 times 10minus4) in m and m2 respectively Theupper limit is (5 5 5 5 5 5 times 10minus4 5 times 10minus4 5 times 10minus4) inm and m2 A deterministic constraint for the 3 first naturalfrequencies is imposed to the original problem see Table 3The optimum mass found for the deterministic problem is36056 kgwith the design variables being (093911 13327 1521116656 17584 29941 times 10minus4 10019 times10minus4 10033 times 10minus4 24837times 10minus4 11804 times 10minus4 12643 times 10minus4 25903 times 10minus4 15983 times 10minus415209 times 10minus4 25021 times 10minus4 12443 times 10minus4 13579 times 10minus4 23928times 10minus4 10014 times 10minus4) m and m2 and the natural frequenciesconstraints are satisfied (1198911 ge 20 1198912 ge 40 and 1198913 ge 60)


Table 2 Results obtained using the optimization methods (10 bar truss)

Parameters GA FMA PSO SQP SORASQP [6] HGPSBest mass value (kg) 144704 166775 125375 125610 125391 125231120573119904 3004 3181 3004 3000 30lowastlowast 3000120573119889 3001 3000 3000 3000 30lowastlowast 3000lowastMean number of objective function evaluations 9031 10005 2012 3207 4030 2454lowastMean number of LSFlowast evaluations 15630 16028 12121 4580 1775 29640Efficiency 424 448 243 134 100 553Quality 1155 1332 1001 1003 1001 10001198601 (1198982) 72529 times 10minus3 83642 times 10minus3 74611 times 10minus3 74563 times 10minus3 74580 times 10minus3 74782 times 10minus31198602 (1198982) 51958 times 10minus3 15359 times 10minus3 49060 times 10minus3 55108 times 10minus3 49032 times 10minus3 48697 times 10minus31198603 (1198982) 11237 times 10minus2 10675 times 10minus2 99456 times 10minus3 93988 times 10minus3 99483 times 10minus3 10026 times 10minus21198604 (1198982) 12833 times 10minus3 50408 times 10minus3 78759 times 10minus6 74939 times 10minus6 64516 times 10minus6 69707 times 10minus61198605 (1198982) 35329 times 10minus3 37096 times 10minus3 65592 times 10minus6 20610 times 10minus5 64516 times 10minus6 64516 times 10minus61198606 (1198982) 30688 times 10minus3 85805 times 10minus3 64516 times 10minus6 76501 times 10minus6 64516 times 10minus6 69707 times 10minus61198607 (1198982) 61501 times 10minus3 56797 times 10minus3 69485 times 10minus3 66513 times 10minus3 69548 times 10minus3 69358 times 10minus31198608 (1198982) 45216 times 10minus3 59744 times 10minus3 53252 times 10minus3 52952 times 10minus3 53354 times 10minus3 52899 times 10minus31198609 (1198982) 98911 times 10minus4 59404 times 10minus3 64516 times 10minus6 78343 times 10minus6 64516 times 10minus6 69707 times 10minus611986010 (1198982) 64405 times 10minus3 21937 times 10minus3 69428 times 10minus3 72971 times 10minus3 69419 times 10minus3 69180 times 10minus3lowastMean number of LSF evaluations the mean number of times the probabilistic constraints are evaluated in order to get the reliability index (20 independentruns)lowastlowastIt is assumed that the constraints become active The original paper did not mention this value

10Ｇ

x

2 4 6 8 10 12 13 16 181

3 5 7 9 11 13 15 17 19

1 2 3

4

5 6

7 10 13 16 19 22 25

27262423212018171514121198

y

FF F F F F F F F F

Added mass

1Ｇ

Figure 5 37 bar truss problem

Table 3 Structural properties and constraints for the 37 bar Pratttruss

Parameter Value Unit119864 (modulus of elasticity) 21 times 1011 Pa119865 (applied load) 50 N120588 (material density) 7800 kgm3

Natural frequencyconstraints

1198911 ge 20 1198912 ge 40 and1198913 ge 60 Hz

Material strength 30 times 108 PaDisplacement limit 50 times 10minus4 m

The probabilistic problem is then adapted from thisdeterministic example described in [27 28] following thesame geometry and deterministic frequency constraint Dis-placement and stress constraints as well as concentrated loadsare added to the original problem in order to result in extraprobabilistic constraints for stress and displacements

In this problem the first three natural frequencies(1198911 1198912 and 1198913) are of interest and a reliability constraint isconsidered in the probabilistic problem Therefore the goalof the optimization problem is to minimize the mass of thetruss taking into account constraints for stress displacementand natural frequency reliability indexes All remaining sym-metric member areas are considered random variables withlognormal distribution with coefficient of variation of 001Therefore in this problem the mean value of cross-sectionalareas is either of the design variables for the optimizationproblem as random variables for the probabilistic problemas well The modulus of elasticity is considered following anormal distribution withmean 21 times 1011 Pa and coefficient ofvariation of 005 (as reported in [29]) The areas of the lowerchord bars are equal to 4times 10minus3m2 and assumeddeterministicand they are not design variables The physical propertiesand the constraints are shown in Table 3 The violation ofany of these constraints will represent a failure mode In thisparticular model as previously described three reliability


Table 4 Final objective function constraints and parameters by optimization method (objective function values do not take into accountadded masses)

Parameters GA FMA DE PSO HGPSBest mass value (kg) 44885 41329 39475 39710 374187120573119891 3084 3803 365 3038 3275120573119904 1000 1000 100 1000 1000120573119889 1000 1000 100 1000 1000Mean number of objective function evaluations 1982 3084 10007 4003 1839lowastMean number of LSF evaluations 767715 392793 1443497 591936 805459Efficiency 194 10 367 151 200Quality 120 110 105 106 100Maximum absolute member stress (Pa) 779 times 105 126 times 106 111 times 106 211 times 106 113 times 106Maximum absolute node displacement (m) 609 times 10minus5 632 times 10minus5 537 times 10minus5 679 times 10minus5 658 times 10minus51198911 (Hz) 2176 2101 2327 2178 21851198912 (Hz) 5364 4653 4433 4884 43821198913 (Hz) 7069 6699 6630 6493 6563lowastMean number of LSF evaluations the mean number of times the probabilistic constraints are evaluated in order to get the reliability index (20 independentruns)

indexes are considered as extra constraints for the problem120573119891 120573119904 and 120573119889 They represent the reliability for frequencystress and displacement constraints

Equation (21) in this case can represent the RBDOmathematical model where 120573119871119891 = 30 120573119871119904 = 20 and 120573119871119889 =20 represent the target reliability indexes for frequencystress and displacement constraints

Minimize 119891 (A) = 120588 119899sum119894=1

119860 119894119871 119894Subject to 1198911 ge 20Hz

1198912 ge 40Hz1198913 ge 60Hz10038161003816100381610038161205901198941003816100381610038161003816 le 30 times 108 Pa 119894 = 1 1198991003816100381610038161003816100381611988911989510038161003816100381610038161003816 le 50 times 10minus4m 119895 = 1 119899119899120573119891 (u p) ge 120573119871119891120573119904 (u p) ge 120573119871119904120573119889 (u p) ge 120573119871119889p119871 le p le p119880

(21)

where p represents the vector of design variables (five 119910coordinates and 14 member areas) and u represents therandom variable vector (14 member areas and the Youngmodulus) 119899119899 is the total number of nodes

Table 4 presents a summary of the results and Table 5shows the obtained values for each design variable for thetested optimization algorithms For the 20 independent runsthe HGPS presented the best mass value of 374187 kg meanvalue of 396680 kg (median 39813 kg) worst mass value of405097 kg and a coefficient of variation of 0037

0 1 2 3 4 5 6 7 8 9 10x

y

005

115

225

335

4

InitialGAFMA

DEPSOHGPS

Figure 6 Superimposed geometries obtained with the optimizationmethods

Figure 6 presents the resulting superimposed geometricconfigurations obtained with the optimization methodsHGPS in this case give the best mass value (quality) althoughwith not so good efficiency as the PSO algorithm Thereliability for displacements and stress constraints resulted inhigh values (120573 = 10 means negligible failure of probability)indicating that the frequency constraint in this case is adominant mode of failure that prevails over the two otherconstraints As expected the absolute value for displacementstress and frequencies in the optimum designs are far belowthe corresponding limits

53 Case 3 25 Bar Space Truss Problem In this example themass minimization of a 25 bar truss is performed subject toconstraints in the reliability indexes for stress displacementand first natural frequency The deterministic optimizationproblem was previously presented by [30] where the massminimization was performed not taking into account uncer-tainties see Figure 7 The problem is described as a discrete


Table 5 Design variables for each optimization method

Variable GA FMA DE PSO HGPS1198843 11988419 (m) 216 075 108 210 0911198845 11988417 (m) 225 148 126 269 1181198847 11988415 (m) 237 170 160 264 1481198849 11988413 (m) 211 233 180 275 16011988411 (m) 224 229 185 275 1731198601 11986027 (m2) 334 times 10minus4 308 times 10minus4 420 times 10minus4 278 times 10minus4 334 times 10minus41198602 11986026 (m2) 424 times 10minus4 298 times 10minus4 100 times 10minus4 284 times 10minus4 130 times 10minus41198603 11986024 (m2) 433 times 10minus4 152 times 10minus4 166 times 10minus4 123 times 10minus4 118 times 10minus41198604 11986025 (m2) 317 times 10minus4 337 times 10minus4 288 times 10minus4 277 times 10minus4 383 times 10minus41198605 11986023 (m2) 345 times 10minus4 257 times 10minus4 100 times 10minus4 207 times 10minus4 144 times 10minus41198606 11986021 (m2) 139 times 10minus4 110 times 10minus4 100 times 10minus4 238 times 10minus4 124 times 10minus41198607 11986022 (m2) 344 times 10minus4 366 times 10minus4 500 times 10minus4 275 times 10minus4 325 times 10minus41198608 11986020 (m2) 184 times 10minus4 309 times 10minus4 436 times 10minus4 289 times 10minus4 175 times 10minus41198609 11986018 (m2) 394 times 10minus4 401 times 10minus4 299 times 10minus4 270 times 10minus4 188 times 10minus411986010 11986019 (m2) 476 times 10minus4 296 times 10minus4 392 times 10minus4 295 times 10minus4 456 times 10minus411986011 11986017 (m2) 441 times 10minus4 448 times 10minus4 100 times 10minus4 241 times 10minus4 211 times 10minus411986012 11986015 (m2) 271 times 10minus4 357 times 10minus4 380 times 10minus4 259 times 10minus4 158 times 10minus411986013 11986016 (m2) 408 times 10minus4 216 times 10minus4 500 times 10minus4 278 times 10minus4 363 times 10minus411986014 (m2) 264 times 10minus4 420 times 10minus4 102 times 10minus4 263 times 10minus4 100 times 10minus4

1

23

45

67

89

10

11

12

13

14 15

16

17

18

19

2021

22

23

24

25

3a

3a

3a

8a8a

4a

4a

Figure 7 25 bar space truss Dimensions numbering and boundary conditions

optimization problem since the member areas can varyfollowing a table of discrete values that ranges from 64516 times10minus5m2 (01 in2) to 2258 times 10minus3m2 (35 in2) by incrementsof 64516 times 10minus5m2 (01 in2) There are stress constraints inall members such that a strength limit of plusmn27579 times 108 Pa

(plusmn40 ksi) should be verified A displacement constraint ofplusmn889 times 10minus3m (plusmn035 in) is imposed to nodes 1 and 2 in119909 and 119910 direction The fundamental frequency should beconstrained to 1198911 ge 55Hz Figure 7 shows the dimensionsof the 25 bar spatial truss example The physical properties of


the material upper and lower limits for the design variablesand the deterministic constraints are listed in Table 6

In this example eight sets of grouped bars are assumedresulting in eight design variables namely set 1 nodes 1-2 set2 nodes 1ndash4 2-3 1ndash5 and 2ndash6 set 3 nodes 2ndash5 2ndash4 1ndash3 and1ndash6 set 4 nodes 3ndash6 and 4-5 set 5 nodes 3-4 and 5-6 set 6nodes 3ndash10 6-7 4ndash9 and 5ndash8 set 7 nodes 3ndash8 4ndash7 6ndash9 and5ndash10 set 8 nodes 3ndash7 4ndash8 5ndash9 and 6ndash10 Table 7 presents theload direction and magnitudeThe spatial truss is fixed at thelower nodes and the loads are applied as indicated in Figure 7(where dimension 119886 = 9144m) The obtained final mass bythe deterministic optimization is 21760 kg (47972 lbm) withthe optimumdesign variable vector (discrete areas) as (64516times 10minus5 19354 times 10minus4 22580 times 10minus3 64516 times 10minus5 10322 times 10minus358064 times 10minus4 32258 times 10minus4 22580 times 10minus3)m2

For the RBDO problem the areas are either designvariables (deterministic) or random (probabilistic) They areassumed uncorrelated following a lognormal distributionwith coefficient of variation of 002 the elastic modulusfollows a Gaussian distribution with coefficient of variationof 005 Thus there are in total nine random variables Thereliability index levels for each constraint (frequency stressand displacement) are set to 120573119871119891 ge 15 120573119871119904 ge 20 and 120573119871119889 ge15 Therefore the reliability-based optimization problem iswritten as indicated by

Minimize 119891 (A) = 120588 119899sum119894=1

119860 119894119871 119894Subject to 1198911 ge 55Hz10038161003816100381610038161205901198941003816100381610038161003816 le 27579 times 108 Pa 119894 = 1 1198991003816100381610038161003816100381611988911989510038161003816100381610038161003816 le 889 times 10minus3m

119895 = 1 2 in 119909 and 119910 direction

120573119891 (u p) ge 120573119871119891120573119904 (u p) ge 120573119871119904120573119889 (u p) ge 120573119871119889p119871 le p le p119880

(22)

where p represents the vector of design variables (eightdiscrete member areas) and u represents the random variablevector (8 discrete member areas and the elastic modulus) p119871and p119880 are the lower and upper limits for the design variables119899119899 is the total number of nodes

Table 8 presents a summary of the results and Table 9shows the obtained design variables for each optimizationmethod For the 20 independent runs the HGPS presentedthe best mass value of 25380 kg mean value of 25576 kg(median 25401 kg) the worst mass value of 26573 kg and acoefficient of variation of 0014 Particularly in this examplePSO and HGPS found the same close results (best qualityindexes) althoughHGPS resulted in a better efficiency indexAll reliability constraints were satisfied and the displacementconstraint presented as the most important and dominant

Table 6 Structural properties and constraints for the 25 bar spacetruss

Properties Values119864 (Youngrsquos modulus) 6895 times 1010 Pa (1 times 104 ksi)120588 (density) 276799 kgm3 (01 lbmin3)Area lower limit 64516 times 10minus5m2 (01 in2)Area upper limit 2258 times 10minus3m2 (35 in2)Material Strength 27579 times 108 Pa (40 ksi)Displacement limit 889 times 10minus3m (035 in)1st natural frequency constraint 1198911 ge 55Hz

Table 7 Applied loads

Node Load component119865119909 119865119910 1198651199111 444822N

(1000 lbf)minus4448222N(minus10000 lbf) minus4448222N

(minus10000 lbf)2 00N minus4448222N

(minus10000 lbf) minus4448222N(minus10000 lbf)

3 222411 N(500 lbf) 00N 00N

6 266893N(600 lbf) 00N 00N

mode of failure Stress and frequency constraints resulted inhigh reliability indexes (negligible probability of failure)

6 Final Remarks and Conclusions

In this article a hybridized method for RBDO is pro-posed It takes advantage of the main features presentedin some heuristic algorithms like PSO GA and gradient-based methods like SQP Simple to gradually more complextruss examples are analyzed using the proposedmethodologycomparedwith literature examplesThe best result is obtainedwith the hybrid method HGPS in the analyzed examplesalbeit in the last example PSO and HGPS presented thesame result Moreover it is relevant to mention that due tocomplexity of the optimization problems with uncertaintiesin most of situations the heuristic methods solely werenot able to find the optimal design although they resultedin objective function with same order of magnitude Thissuggests that a different convergence criterion may be usedfor each algorithm in order to avoid premature convergence

In all problems the resulting final mass obtained usingRBDO is larger than that obtained by deterministic optimiza-tion however the corresponding reliability levels in stressdisplacements or frequency have been met These reliabilitylevels are not attained with the original deterministic opti-mization

Even with a large number of design variables in theoptimization analysis all presented global search methodswere able to find feasible solutionsTheoptimizationmethodsfound different optimum design variables for similar objec-tive function It can be argued that those design points do notrepresent the best quality converged design points and may


Table 8 Results for the optimization methods in 25 bar truss example

Parameters GA FMA DE PSO HGPSBest mass value (kg) 27269 26138 25445 25380 25380120573119891 10000 9031 9014 9803 9803120573119904 10000 10000 10000 10000 10000120573119889 1973 1996 2021 1963 1963lowastMean number of objective function evaluations 3452 2002 5012 3002 1002lowastMean number of LSF evaluations 499001 281583 185877 333634 74143Efficiency 669 377 254 448 100Quality 1074 1029 1002 1000 1000Maximum absolute stress (Pa) 52752 times 107 3504 times 107 41136 times 107 3904 times 107 3904 times 107

Maximum absolute displacement (m) 7992 times 10minus3 7998 times 10minus3 8882 times 10minus3 7998 times 10minus3 7998 times 10minus31198911 (Hz) 7542 7738 5968 7381 7381lowastMean number of LSF evaluations the mean number of times the probabilistic constraints are evaluated in order to get the reliability index (20 independentruns)

Table 9 Best design variables for tested optimization methods

Design variable GA FMA DE PSO HGPS1198601 (m2) 645 times 10minus5 193 times 10minus4 645 times 10minus5 645 times 10minus5 645 times 10minus51198602 to 5 (m2) 129 times 10minus3 774 times 10minus4 645 times 10minus4 516 times 10minus4 516 times 10minus41198606 to 9 (m2) 180 times 10minus3 206 times 10minus3 225 times 10minus3 225 times 10minus3 225 times 10minus311986010 and 11 (m2) 645 times 10minus4 129 times 10minus4 645 times 10minus5 645 times 10minus5 645 times 10minus511986012 and 13 (m2) 258 times 10minus4 387 times 10minus4 774 times 10minus4 116 times 10minus3 116 times 10minus311986014 and 17 (m2) 903 times 10minus4 709 times 10minus4 709 times 10minus4 709 times 10minus4 709 times 10minus411986018 and 21 (m2) 645 times 10minus4 903 times 10minus4 645 times 10minus4 580 times 10minus4 580 times 10minus411986022 and 25 (m2) 219 times 10minus3 219 times 10minus3 225 times 10minus3 226 times 10minus3 226 times 10minus3

represent a local minimum however for nonexplicit limitstate function problems this can only be guaranteed checkingthe results with different algorithms

The hybrid method HGPS when compared to the GAin the conventional form presented a lower mass value(quality in result) with a low number of function evaluations(efficiency in the optimization process) The better efficiencywas attained with the gradient- basedmethods at the expenseof worse quality

The proposed final problem which cannot be handledby single loop RBDO algorithms (design variables are alsorandomvariables) highlights the importance of the proposedapproach in cases where the discrete design variables arealso random variables Some new improvements in theHGPS method are being planned in future research like theparallelization of the code More challenging problems areforeseen to test the proposed method

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors acknowledge Brazilian Agencies CNPq andCAPES for the partial financial support for this research

References

[1] H Agarwal Reliability Based Design Optimization Formulationand Methodologies Aerospace and Mechanical EngineeringUniversity of Notre Dame formulation and methodologies2004

[2] A T Beck and C C Verzenhassi ldquoRisk optimization of a steelframe communications tower subject to tornado windsrdquo LatinAmerican Journal of Solids and Structures vol 5 no 3 pp 187ndash203 2008

[3] S Shan and G G Wang ldquoReliable design space and completesingle-loop reliability-based design optimizationrdquo ReliabilityEngineering System Safety vol 93 no 8 pp 1218ndash1230 2008

[4] F Li T Wu A Badiru M Hu and S Soni ldquoA single-loopdeterministic method for reliability-based design optimiza-tionrdquoEngineeringOptimization vol 45 no 4 pp 435ndash458 2013

[5] X Du and W Chen ldquoSequential optimization and reliabilityassessment method for efficient probabilistic designrdquo Journal ofMechanical Design vol 126 no 2 pp 225ndash233 2004

[6] 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

[7] R H Lopez and A T Beck ldquoReliability-based design opti-mization strategies based on FORM a reviewrdquo Journal of theBrazilian Society ofMechanical Sciences and Engineering vol 34no 4 pp 506ndash514 2012


[8] H M Gomes Reliability analysis of reinforced concrete struc-tures [PhD Thesis] Brazil Porto Alegre 2001

[9] X Zhuang R Pan and X Du ldquoEnhancing product robustnessin reliability-based design optimizationrdquo Reliability Engineeringamp System Safety vol 138 pp 145ndash153 2015

[10] H Bae R V Grandhi and R A Canfield ldquoStructural designoptimization based on reliability analysis using evidence the-oryrdquo SAE Technical Paper 2003-01-0877 2007

[11] D M Frangopol ldquoProbability concepts in engineering empha-sis on applications to civil and environmental engineeringrdquoStructure and Infrastructure Engineering vol 4 no 5 pp 413-414 2008

[12] I Gondres Torne R Baez Prieto S Lajes Choy and A delCastillo Serpa ldquoDeterminacion de la confiabilidad en interrup-tores de potencia caso de estudiordquo Ingeniare Revista Chilena deIngenierıa vol 21 no 2 pp 271ndash278 2013

[13] A M Hasofer and D Lind ldquoAn exact and invariant first-orderreliability formatrdquo Journal of Engineering Mechanics ASCE vol100 pp 111ndash121 1974

[14] A Haldar and S Mahadevan ldquoProbability reliability and statis-tical methods in engineering designrdquo in Engineering Design JW Sons Ed vol 77 p 379 2000

[15] R E Melchers Structural Reliability Analysis and Prediction CJohnWiley amp Sons Ltd New York NY USA 2nd edition 1999

[16] AD Belegundu andTRChandrupatlaOptimizationConceptsand Applications in Engineering Cambridge University PressUK 2nd edition 1999

[17] J Tu K K Choi and Y H Park ldquoA new study on reliability-based design optimizationrdquo Journal of Mechanical Design vol121 no 4 pp 557ndash564 1999

[18] V Kelner F Capitanescu O Leonard and L Wehenkel ldquoAhybrid optimization technique coupling an evolutionary and alocal search algorithmrdquo Journal of Computational and AppliedMathematics vol 215 no 2 pp 448ndash456 2008

[19] I E Grossmann and L T Biegler ldquoPart II Future perspectiveon optimizationrdquo Computers and Chemical Engineering vol 28no 8 pp 1193ndash1218 2004

[20] H Adeli and N-T Cheng ldquoAugmented lagrangian geneticalgorithm for structural optimizationrdquo Journal of AerospaceEngineering vol 7 no 1 pp 104ndash118 1994

[21] F van den Bergh and A P Engelbrecht ldquoA study of particleswarm optimization particle trajectoriesrdquo Information Sciencesvol 176 no 8 pp 937ndash971 2006

[22] M Srinivas and L M Patnaik ldquoAdaptive probabilities ofcrossover and mutation in genetic algorithmsrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 24 no 4 pp 656ndash667 1994

[23] S Marsili Libelli and P Alba ldquoAdaptive mutation in geneticalgorithmsrdquo Soft Computing vol 4 no 2 pp 76ndash80 2000

[24] R Hassan B Cohanim OWeck and G Venter ldquoA comparisonof particle swarm optimization and the genetic algorithmrdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics Material Conference Austin Tex USA2005

[25] L J Li Z B Huang F Liu and Q H Wu ldquoA heuristic particleswarm optimizer for optimization of pin connected structuresrdquoComputers and Structures vol 85 no 7-8 pp 340ndash349 2007

[26] R Riolo U-M OrsquoReilly and T McConaghy ldquogenetic pro-gramming theory and practice VIIrdquo in Genetic amp EvolutionaryComputation Series Ed Springer 2010

[27] D Wang W H Zhang and J S Jiang ldquoTruss optimization onshape and sizingwith frequency constraintsrdquoAIAA Journal vol42 no 3 pp 622ndash630 2004

[28] W Lingyun Z Mei W Guangming and M Guang ldquoTrussoptimization on shape and sizing with frequency constraintsbased on genetic algorithmrdquo Computational Mechanics vol 35no 5 pp 361ndash368 2005

[29] G D Cheng L Xu and L Jiang ldquoA sequential approximateprogramming strategy for reliability-based structural optimiza-tionrdquo Computers and Structures vol 84 no 21 pp 1353ndash13672006

[30] W H Tong and G R Liu ldquoOptimization procedure fortruss structures with discrete design variables and dynamicconstraintsrdquo Computers and Structures vol 79 no 2 pp 155ndash162 2001

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


due to possible nonlinear function behavior in probabilisticproblems In order to increase robustness authors [9] pro-posed to construct a response surface that is the product ofindividual performance functions and be sued as surrogateto get optimal design solutions Despite the large amount ofliterature in the theme a good introduction to the RBDOsubject can be found in [10]

In this article a hybrid RBDO methodology is proposedand applied to a range of spatial trusses in order to findthe optimum parameters for minimum mass consideringmaximum allowable deflections and limited stress The First-Order Reliability Method (FORM) is used to evaluate thereliability index (120573) along the optimization steps Due tothe high computational cost of the traditional algorithms tosolve nonconvex optimization problems we propose a hybridoptimizationmethod Comparisons about the computationalcost and quality of the solution are performed using asbaseline standard structural problems The proposed hybridglobal optimization method based on existent global search(metaheuristic) and deterministic algorithms is detailedThisnew method is presented and discussed by using reliability-based optimization on structural truss examples

2 Reliability Index Evaluation

As stated by [11] reliability analysis can be applied to manyengineering fields such as in aeronautical mechanical andcivil problems By definition the reliability is the complementof the probability of failure that is the likelihood of failureof a specific event or set of events from a complex system Alimit state function that relates the failure event (violation ofspecific set of constraints) as function of several variables isstated by the mathematical expression

119892 (1198831 119883119899) = 0 (1)

where 119892 means the limit state function that defines a con-straint Some design variables may present random compo-nents In case 119883119894 is a set of 119899 random variables that affectsthat constraint this limit state function becomes also randomand some probability of violation is implicit [11 12] 119892(sdot) le 0means that the system is in the failure domain 119863 and 119892(sdot) gt0 means that the system is in the safe domain (constraintwas not violated) The probability of failure can be evaluatedby the integration of the joint probability density function119891119883(1198831 119883119899) as indicated in [11]

119875119891 = int sdot sdot sdot int119863119891119883 (1198831 119883119899) 1198891198831 sdot sdot sdot 119889119883119899 cong Φ (minus120573) (2)

where the failure domain is defined by 119863 means that 119892(sdot) le0 Φ is the cumulative standard distribution function and120573 is a safety index metric Equation (2) presents a closesolution in some particular cases where 119891119883(sdot) is Gaussianand for linear and quadratic 119892(sdot) However it is difficultto be handled in case of several random variables 119899 Inthis case Monte Carlo (MC) Simulation can be used to getapproximate asymptotic solutions Moreover statistic valuesfor function119891119883(119883) are not known a priori and the number ofMC samples frequently is not enough to ensure confidence

fX(X)

g(X) lt 0 g

g

g

Pf

g(X)

Figure 1 Concept of probability of failure and the reliability index 120573for the first-order approximation

So a very simple but not so robust way to get and estimatefor the reliability index (that is related to the probabilityof survival) 120573 is using the probability density function firstand second moments (mean and variance) for the limit statefunction 119892(1198831 119883119899) When the limit state function 119892(119883)is linear and the random variables are normally distributedand uncorrelated the reliability index 120573 can be approximatedby the following (see Figure 1)

120573 = 120583119892120590119892 (3)

where 120583119892 and 120590119892 represent respectively the mean value andthe standard deviation (square root of variance) for function119892(119883)

Let 120590119894 be the mechanical stress that can be measured ina loaded component 119894 assuming a failure situation wherethis value exceeds the imposed material strength limit value(120590lim) Equation (1) can be rewritten for the limit statefunction as follows

119892119894 (120590119894) = 1 minus 120590119894120590lim (4)

Linearization of function 119892(119883) by a Taylor expansion upto linear terms can be used to get approximated values for120583119892 and 120590119892 when the limit state functions are nonlinear Thepoint around the linearization performed affects 120583119892 and 120590119892values A method to obtain the reliability index 120573 that isindependent of the limit state function formulation is knownas AFOSM (Advanced First-Order SecondMoment) and wasfirst proposed by [13] For uncorrelated Gaussian randomvariables 119883119894 they are transformed into normalized ones 119880119894by the transformation

119880119894 = Φminus1 [119865119883119894 (119909119894)] = 119879 (119883119894) (5)

where 119865119883119894(119909119894) and Φminus1(sdot) are the cumulative distributionfunction of the random variable 119883119894 and the inverse of thecumulative standard Gaussian distribution respectively Inthis way the limit state function in the real space 119883 istransformed to the uncorrelated normalized space 119880 so

ℎ (119880) cong 119892 (119883) (6)

The linearization of the limit state function ℎ(119880) isperformed at the 119880lowast point that presents the shorter distanceto the origin of the uncorrelated space 119880 and that ensures








(8)


119880119894 = 119883119894 minus 120583119873119883119894120590119873119883119894 (9)



120597119892 (119883)120597119880119894 = 120597119892 (119883)120597119883119894 120597119883119894120597119880119894 (10)




120573 = 119880 = radic 119899sum119894=1

(119880lowast119894 )2 (12)



119883119894 = 120583119873119883119894 + 120590119873119883119894119880lowast119894 (13)








119892119895 (V119889 p) lt 0 119895 = 1 119899V119889119897 le V119889 le V119889119906

(15)












119896 = 1 119901120573target minus 120573 (119891119897 (u p) = 0) = 0

119897 = 1 119902

(18)






119897 = 1 119902

(19)






Failure domain

Failure domain




Safet

y

Safet

y

Failu

re

Failu

re

g1(X) = 0

g2(X) = 0

x2

x1


1

m




SQP


2


individuals

1

m

GA


























Table1Summaryof

parametersfor

each

optim

izationmetho

d

GA

DE

PSO

FMA

HGPS

Num

bero

find

ividuals

200

Num

bero

findividu

als

50Num

bero

fparticles

50Num

bero

fparticles

50Num

bero

findividu

als

50

Crossover

one

point

Weightin

gratio

15So

cialconstant

201

Beta

10So

cial

constantcognitiv

econstant

201201

Prob

abilityof

crossover

80

Crossoverp

rob

08

Cognitiv

econ

stant

201

Beta

min

01

Prob

abilityof

crossover

80

Prob

abilityof

mutation

1Search

Startin

gpo

int

(119909 max+119909 m

in)2

Inertia

mom

ent

08

Gam

ma

15Prob

abilityof

mutation

1

Tolerance(successiv

eObj

Funcdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

bleo

rmeshsiz

e)10minus6

Tolerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6Maxim

umnu

mbero

fgeneratio

ns200

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800


2

3

4

5

6

11 2

3 4

5 6

7

8

9

10

a

P1 P2

a a




Minimize 119891 (A) = 120588 119899sum119894=1



(20)










10Ｇ

x

2 4 6 8 10 12 13 16 181

3 5 7 9 11 13 15 17 19

1 2 3

4

5 6

7 10 13 16 19 22 25

27262423212018171514121198

y

FF F F F F F F F F

Added mass

1Ｇ





1198911 ge 20 1198912 ge 40 and1198913 ge 60 Hz









Minimize 119891 (A) = 120588 119899sum119894=1

119860 119894119871 119894Subject to 1198911 ge 20Hz


(21)



0 1 2 3 4 5 6 7 8 9 10x

y

005

115

225

335

4

InitialGAFMA

DEPSOHGPS







1

23

45

67

89

10

11

12

13

14 15

16

17

18

19

2021

22

23

24

25

3a

3a

3a

8a8a

4a

4a








Minimize 119891 (A) = 120588 119899sum119894=1


119895 = 1 2 in 119909 and 119910 direction


(22)










3 222411 N(500 lbf) 00N 00N

6 266893N(600 lbf) 00N 00N

















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











(8)


119880119894 = 119883119894 minus 120583119873119883119894120590119873119883119894 (9)



120597119892 (119883)120597119880119894 = 120597119892 (119883)120597119883119894 120597119883119894120597119880119894 (10)




120573 = 119880 = radic 119899sum119894=1

(119880lowast119894 )2 (12)



119883119894 = 120583119873119883119894 + 120590119873119883119894119880lowast119894 (13)








119892119895 (V119889 p) lt 0 119895 = 1 119899V119889119897 le V119889 le V119889119906

(15)












119896 = 1 119901120573target minus 120573 (119891119897 (u p) = 0) = 0

119897 = 1 119902

(18)






119897 = 1 119902

(19)






Failure domain

Failure domain




Safet

y

Safet

y

Failu

re

Failu

re

g1(X) = 0

g2(X) = 0

x2

x1


1

m




SQP


2


individuals

1

m

GA


























Table1Summaryof

parametersfor

each

optim

izationmetho

d

GA

DE

PSO

FMA

HGPS

Num

bero

find

ividuals

200

Num

bero

findividu

als

50Num

bero

fparticles

50Num

bero

fparticles

50Num

bero

findividu

als

50

Crossover

one

point

Weightin

gratio

15So

cialconstant

201

Beta

10So

cial

constantcognitiv

econstant

201201

Prob

abilityof

crossover

80

Crossoverp

rob

08

Cognitiv

econ

stant

201

Beta

min

01

Prob

abilityof

crossover

80

Prob

abilityof

mutation

1Search

Startin

gpo

int

(119909 max+119909 m

in)2

Inertia

mom

ent

08

Gam

ma

15Prob

abilityof

mutation

1

Tolerance(successiv

eObj

Funcdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

bleo

rmeshsiz

e)10minus6

Tolerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6Maxim

umnu

mbero

fgeneratio

ns200

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800


2

3

4

5

6

11 2

3 4

5 6

7

8

9

10

a

P1 P2

a a




Minimize 119891 (A) = 120588 119899sum119894=1



(20)










10Ｇ

x

2 4 6 8 10 12 13 16 181

3 5 7 9 11 13 15 17 19

1 2 3

4

5 6

7 10 13 16 19 22 25

27262423212018171514121198

y

FF F F F F F F F F

Added mass

1Ｇ





1198911 ge 20 1198912 ge 40 and1198913 ge 60 Hz









Minimize 119891 (A) = 120588 119899sum119894=1

119860 119894119871 119894Subject to 1198911 ge 20Hz


(21)



0 1 2 3 4 5 6 7 8 9 10x

y

005

115

225

335

4

InitialGAFMA

DEPSOHGPS







1

23

45

67

89

10

11

12

13

14 15

16

17

18

19

2021

22

23

24

25

3a

3a

3a

8a8a

4a

4a








Minimize 119891 (A) = 120588 119899sum119894=1


119895 = 1 2 in 119909 and 119910 direction


(22)










3 222411 N(500 lbf) 00N 00N

6 266893N(600 lbf) 00N 00N

















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of














119896 = 1 119901120573target minus 120573 (119891119897 (u p) = 0) = 0

119897 = 1 119902

(18)






119897 = 1 119902

(19)






Failure domain

Failure domain




Safet

y

Safet

y

Failu

re

Failu

re

g1(X) = 0

g2(X) = 0

x2

x1


1

m




SQP


2


individuals

1

m

GA


























Table1Summaryof

parametersfor

each

optim

izationmetho

d

GA

DE

PSO

FMA

HGPS

Num

bero

find

ividuals

200

Num

bero

findividu

als

50Num

bero

fparticles

50Num

bero

fparticles

50Num

bero

findividu

als

50

Crossover

one

point

Weightin

gratio

15So

cialconstant

201

Beta

10So

cial

constantcognitiv

econstant

201201

Prob

abilityof

crossover

80

Crossoverp

rob

08

Cognitiv

econ

stant

201

Beta

min

01

Prob

abilityof

crossover

80

Prob

abilityof

mutation

1Search

Startin

gpo

int

(119909 max+119909 m

in)2

Inertia

mom

ent

08

Gam

ma

15Prob

abilityof

mutation

1

Tolerance(successiv

eObj

Funcdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

bleo

rmeshsiz

e)10minus6

Tolerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6Maxim

umnu

mbero

fgeneratio

ns200

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800


2

3

4

5

6

11 2

3 4

5 6

7

8

9

10

a

P1 P2

a a




Minimize 119891 (A) = 120588 119899sum119894=1



(20)










10Ｇ

x

2 4 6 8 10 12 13 16 181

3 5 7 9 11 13 15 17 19

1 2 3

4

5 6

7 10 13 16 19 22 25

27262423212018171514121198

y

FF F F F F F F F F

Added mass

1Ｇ





1198911 ge 20 1198912 ge 40 and1198913 ge 60 Hz









Minimize 119891 (A) = 120588 119899sum119894=1

119860 119894119871 119894Subject to 1198911 ge 20Hz


(21)



0 1 2 3 4 5 6 7 8 9 10x

y

005

115

225

335

4

InitialGAFMA

DEPSOHGPS







1

23

45

67

89

10

11

12

13

14 15

16

17

18

19

2021

22

23

24

25

3a

3a

3a

8a8a

4a

4a








Minimize 119891 (A) = 120588 119899sum119894=1


119895 = 1 2 in 119909 and 119910 direction


(22)










3 222411 N(500 lbf) 00N 00N

6 266893N(600 lbf) 00N 00N

















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Failure domain

Failure domain




Safet

y

Safet

y

Failu

re

Failu

re

g1(X) = 0

g2(X) = 0

x2

x1


1

m




SQP


2


individuals

1

m

GA


























Table1Summaryof

parametersfor

each

optim

izationmetho

d

GA

DE

PSO

FMA

HGPS

Num

bero

find

ividuals

200

Num

bero

findividu

als

50Num

bero

fparticles

50Num

bero

fparticles

50Num

bero

findividu

als

50

Crossover

one

point

Weightin

gratio

15So

cialconstant

201

Beta

10So

cial

constantcognitiv

econstant

201201

Prob

abilityof

crossover

80

Crossoverp

rob

08

Cognitiv

econ

stant

201

Beta

min

01

Prob

abilityof

crossover

80

Prob

abilityof

mutation

1Search

Startin

gpo

int

(119909 max+119909 m

in)2

Inertia

mom

ent

08

Gam

ma

15Prob

abilityof

mutation

1

Tolerance(successiv

eObj

Funcdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

bleo

rmeshsiz

e)10minus6

Tolerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6Maxim

umnu

mbero

fgeneratio

ns200

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800


2

3

4

5

6

11 2

3 4

5 6

7

8

9

10

a

P1 P2

a a




Minimize 119891 (A) = 120588 119899sum119894=1



(20)










10Ｇ

x

2 4 6 8 10 12 13 16 181

3 5 7 9 11 13 15 17 19

1 2 3

4

5 6

7 10 13 16 19 22 25

27262423212018171514121198

y

FF F F F F F F F F

Added mass

1Ｇ





1198911 ge 20 1198912 ge 40 and1198913 ge 60 Hz









Minimize 119891 (A) = 120588 119899sum119894=1

119860 119894119871 119894Subject to 1198911 ge 20Hz


(21)



0 1 2 3 4 5 6 7 8 9 10x

y

005

115

225

335

4

InitialGAFMA

DEPSOHGPS







1

23

45

67

89

10

11

12

13

14 15

16

17

18

19

2021

22

23

24

25

3a

3a

3a

8a8a

4a

4a








Minimize 119891 (A) = 120588 119899sum119894=1


119895 = 1 2 in 119909 and 119910 direction


(22)










3 222411 N(500 lbf) 00N 00N

6 266893N(600 lbf) 00N 00N

















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






















Table1Summaryof

parametersfor

each

optim

izationmetho

d

GA

DE

PSO

FMA

HGPS

Num

bero

find

ividuals

200

Num

bero

findividu

als

50Num

bero

fparticles

50Num

bero

fparticles

50Num

bero

findividu

als

50

Crossover

one

point

Weightin

gratio

15So

cialconstant

201

Beta

10So

cial

constantcognitiv

econstant

201201

Prob

abilityof

crossover

80

Crossoverp

rob

08

Cognitiv

econ

stant

201

Beta

min

01

Prob

abilityof

crossover

80

Prob

abilityof

mutation

1Search

Startin

gpo

int

(119909 max+119909 m

in)2

Inertia

mom

ent

08

Gam

ma

15Prob

abilityof

mutation

1

Tolerance(successiv

eObj

Funcdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

bleo

rmeshsiz

e)10minus6

Tolerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6To

lerance(successiv

eObjFun

cdesig

nvaria

ble)

10minus6Maxim

umnu

mbero

fgeneratio

ns200

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800

Maxim

umnu

mbero

fiteratio

ns800


2

3

4

5

6

11 2

3 4

5 6

7

8

9

10

a

P1 P2

a a




Minimize 119891 (A) = 120588 119899sum119894=1



(20)










10Ｇ

x

2 4 6 8 10 12 13 16 181

3 5 7 9 11 13 15 17 19

1 2 3

4

5 6

7 10 13 16 19 22 25

27262423212018171514121198

y

FF F F F F F F F F

Added mass

1Ｇ





1198911 ge 20 1198912 ge 40 and1198913 ge 60 Hz









Minimize 119891 (A) = 120588 119899sum119894=1

119860 119894119871 119894Subject to 1198911 ge 20Hz


(21)



0 1 2 3 4 5 6 7 8 9 10x

y

005

115

225

335

4

InitialGAFMA

DEPSOHGPS







1

23

45

67

89

10

11

12

13

14 15

16

17

18

19

2021

22

23

24

25

3a

3a

3a

8a8a

4a

4a








Minimize 119891 (A) = 120588 119899sum119894=1


119895 = 1 2 in 119909 and 119910 direction


(22)










3 222411 N(500 lbf) 00N 00N

6 266893N(600 lbf) 00N 00N

















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





2

3

4

5

6

11 2

3 4

5 6

7

8

9

10

a

P1 P2

a a




Minimize 119891 (A) = 120588 119899sum119894=1



(20)










10Ｇ

x

2 4 6 8 10 12 13 16 181

3 5 7 9 11 13 15 17 19

1 2 3

4

5 6

7 10 13 16 19 22 25

27262423212018171514121198

y

FF F F F F F F F F

Added mass

1Ｇ





1198911 ge 20 1198912 ge 40 and1198913 ge 60 Hz









Minimize 119891 (A) = 120588 119899sum119894=1

119860 119894119871 119894Subject to 1198911 ge 20Hz


(21)



0 1 2 3 4 5 6 7 8 9 10x

y

005

115

225

335

4

InitialGAFMA

DEPSOHGPS







1

23

45

67

89

10

11

12

13

14 15

16

17

18

19

2021

22

23

24

25

3a

3a

3a

8a8a

4a

4a








Minimize 119891 (A) = 120588 119899sum119894=1


119895 = 1 2 in 119909 and 119910 direction


(22)










3 222411 N(500 lbf) 00N 00N

6 266893N(600 lbf) 00N 00N

















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







10Ｇ

x

2 4 6 8 10 12 13 16 181

3 5 7 9 11 13 15 17 19

1 2 3

4

5 6

7 10 13 16 19 22 25

27262423212018171514121198

y

FF F F F F F F F F

Added mass

1Ｇ





1198911 ge 20 1198912 ge 40 and1198913 ge 60 Hz









Minimize 119891 (A) = 120588 119899sum119894=1

119860 119894119871 119894Subject to 1198911 ge 20Hz


(21)



0 1 2 3 4 5 6 7 8 9 10x

y

005

115

225

335

4

InitialGAFMA

DEPSOHGPS







1

23

45

67

89

10

11

12

13

14 15

16

17

18

19

2021

22

23

24

25

3a

3a

3a

8a8a

4a

4a








Minimize 119891 (A) = 120588 119899sum119894=1


119895 = 1 2 in 119909 and 119910 direction


(22)










3 222411 N(500 lbf) 00N 00N

6 266893N(600 lbf) 00N 00N

















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









Minimize 119891 (A) = 120588 119899sum119894=1

119860 119894119871 119894Subject to 1198911 ge 20Hz


(21)



0 1 2 3 4 5 6 7 8 9 10x

y

005

115

225

335

4

InitialGAFMA

DEPSOHGPS







1

23

45

67

89

10

11

12

13

14 15

16

17

18

19

2021

22

23

24

25

3a

3a

3a

8a8a

4a

4a








Minimize 119891 (A) = 120588 119899sum119894=1


119895 = 1 2 in 119909 and 119910 direction


(22)










3 222411 N(500 lbf) 00N 00N

6 266893N(600 lbf) 00N 00N

















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







1

23

45

67

89

10

11

12

13

14 15

16

17

18

19

2021

22

23

24

25

3a

3a

3a

8a8a

4a

4a








Minimize 119891 (A) = 120588 119899sum119894=1


119895 = 1 2 in 119909 and 119910 direction


(22)










3 222411 N(500 lbf) 00N 00N

6 266893N(600 lbf) 00N 00N

















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of








Minimize 119891 (A) = 120588 119899sum119894=1


119895 = 1 2 in 119909 and 119910 direction


(22)










3 222411 N(500 lbf) 00N 00N

6 266893N(600 lbf) 00N 00N

















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




A Hybrid Method for Truss Mass Minimization considering...

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