An Object-Oriented Evolutionary Algorithm for ... - Semantic Scholar

An Object-OrientedEvolutionary Algorithm for AutomatedAdvancedAnalysisBasedDesign Daniel Schinler

Christopher M. Foley

Marquette University Dept. of Civil & EnvironmentalEngineering Milwaukee. WI53233

Marquefte University Dept. of Civil & EnvironmentalEngineering Milwaukee.WI53233

Abstract

resist lateral loading. krdiyidual 2 contains an eccentricallybracedframe @BF) bay.

Genetic and evolutionary algorithms have long been used for optimized design of building structures. Optimized design using these techniques has most often utilized binary strings to represent individuals. Object oriented programming(OOP) affords convenient encapsulation of building components,which in tum allows alternate methods for representing design variables within EA's. Advanced analvsis based optimizationcombinesinelasticanalysisand constraints in such a manner that design specifications and codes are not needed. Therefore, advanced analysis-baseddesign has many of the elements of performancebased design. The present paper outlines and evaluates an automated design procedure that uses an object-oriented evolutionary algorithm with advancec analysisto desigrrsteel frameworks.

f| I |

f column,shrpa conneclion'tYPe l- bssmshaoe -.lLconnaciion.type I bulldlng,story -{- column.shrpe | Fconnoclion.tyDe F Oe8m.Snaoa---i lconnecllon.typf, | column.Snape

I I tl I | F I I | c I

f column.shape F beamshaDe J*connecllon'tYPo lconnecllon.type | bulldhg story -f column.sh!pe I ----lFconnedlon.tvDe t-bolmsnapa lconnodpn.type |lcdumn.snapo

a

5 e

I

I . l^ Populatim of bullding ob.itrts \y condnuoawith repstitivs hieErchy.

Figure l: Building Object Hierarchy ControlIndividual

INTRODUCTION Binary string representationsof designvariableshas been the de-facto standard means with which to implement design optimization using genetic and evolutionary algorithms. Variations on a themehave been developed(Parmee1995),but there has been little researchrelated to new mechanismsby which structural engineerscan representdesigrrvariables. Object-oriented programming (OOP) has created a very convenient way to accomplishencapsulationin computer programs. The encapsulation(packaging) offered by OOP has resulted in stuchual engineers formulating new ways to represent building structures on the computer (Rivard and Fenves 2000). A building structurecan be visualizedas a hierarchy of objects as shown in Figure 1. Encapsulating building components as objects createsa mechanismby which system optimization can occur. Consider two individuals present in a generational snapshot of the evolution (Figure 2). Individual I (the contol individual) is a hybrid structural steel-concreteframework where a moment resisting frame interactswith concreteshearwalls to

73

Story

Mate Figure2: Object Crossoverin Buildings If the binary string representation was used to accomplish crossover,the allele describing a shear wall would most likely be &astically different than the allele describingan EBF. As a result, crossover of these two elements might not make sense.

However,to the structuralengineer,swappingshear wall and EBF objects betweenindividualsmay resultin rnoreefficient hvbrid frameworks. 2

OBJECT.ORIENTED EVOLUTIONARY ALGORITHM

Control Individual

An OO-EAdiffers from the binarygeneticalgorithm in the mannerwhich crossoverand mutationaffects the individualsin thepopulation(Figure3),

INDryIDUAL B

ll10

xl

xf

x\

x|

xf

Liie-ilqe-'-il !-ql-'-iii-,r tt t2

ODe PolDt at I

Mate

HomologousCrossoverat D.V,2

xt\ INDryIDUAL A

An object-oriented evolutionary algorithm was written to solve the problem. Two designvariable objectswere usedin a very small hierarchy(Figure 4).

Non-HomologousCrossoverat D.V. I ll0l

0001 0011 OEe Polnt at 2

tl0l

l10l

Figure 4: Simple Object Crossover for Variable Optimization Problem.

0010 0011 U3tformwlhMrsk

0110 lr0l

I 0l I

Two

I I I I

0100 0001

Figure 3: Crossover Operation Comparison Assume that the four-bit binary strings represent wide-flange shapes. Traditional one point crossover at point 1 createsa new shapein the offspring which can be considered as rnutation at the design variable level. Variables xz remain intact (but are exchanged). If one-point crossover at point 2 occurs, design variables are exchanged. Uniform cross-over creat€s new genetic material for both design variables. Object crossover acts much like one-point crossover at point 2 in Figure 3. It doesnot createnew genetic material in the offspring, but merely exchangesit. As a result, the crossoverand mutation operations are completely divorced from one another. A quick example can be formulated to illustrate the behavior of an object oriented EA. Consider the optimizationproblem outlined below:

. r,rr*Jr,' Minimize:/(X) = ]-: x,+x, >8 .x,> 0 SubjectTo: x, )0 The optimalsolution,to this problemhas been (1991). as X = | 6 2 lby Jenkins computed

,7^ t:

Two bpes of design variable crossover were implemented (Figure 4). Homologous crossovercan be seen to merely swap genes (ie. design variables), while non-homologous crossover moves the genes along the chromosome.Mutation is a randomchange in value within a predefined range. Research has been undertaken to elucidate the disruotive and beneficial nature of binary crossover and mutation in the evolutionary procesj(Wrr, et.al. 1997). The present object formulation has the possibility to foster the creation and retention of beneficial building blocks. However, the object representation requires crossover and mutation be studiedto ensureexplorationofthe designspace. Numerical simulation was undertaken to empirically explore the object representationand its ability to reliably find the solution that minimizes the previously discussed function. Design variables were restrictedto integer values, r, cll-+i5[ . A general crossover probability, p" wai defin-edas well as subsequentprobability of non-homologous crossover, pn' . The probability of design variable mutation, p., was also defined,. Linear scaling of the fibress values was used and fitness proportional (roulette wheei) selection was employed (Jenkins 1991). Successof the object-orientedevolutionary algorithm is defined as finding the optimal solution. A small emprical study was undertaken to evaluate the effect of population size on the successof the evolutionary algorithm proposed. The results of this study are given in Table 1.

Table1: Effectof PooulationSizeon Success of OO EvolutidnaryAlgorithm: P " = 0 . 6 0 ,P * = o ' 0 . Population Size

Mutation

20 30

0.25 0.25

53 73

0.2s

93

0.30

93

40

Rate

population during the evolution. lf a small population is used, non-homologouscrossovercan improve the evolutionary search. However, the improvement is not as appreciable as mutation. When population sizes are sufficiently large (40 individuals in this case), the non-homologous crossoverhas lessimpact on the successrate.

Success Rate (%)

It wasalsodecidedto evaluatethe effectof mutation rateon the smallestpopulation. This effecton the OO-EA'ssuccess is givenin Table2. Table2: Effectof MutationRateon Success of OO Evolutionary Algorithm: p" =0.60, p* = 0.0, Nro,,= 20. Success Mutation Rate__lB4!91%)_ 53 80 93 87

0.40 0.60 0.80 1.00

Voss and Foley (1999) studied the effect of nonhomologous crossover on a simple cantilever optimization problem. The genetic representation (although hierarchical) in this former study did not utilize objects and therefore, it was decided to study the effect of non-homologous crossover on the OOEA (Table 3). Table 3: Effect of PopulationSize and Mutation Rate on Successof OO Evolutionarv Algorithm:p,=0.60.

40 = P, 0.30

0.40

40

0.60

53

0.80 1.00 0.20 0.40 0.60 0.80 1.00

67 67 93 80 93 87 60

3

ADVANCED ANALYSIS BASED DESIGN OF FR AND PR FRAMES

Advanced analysisof steel frames is related to the plastic design methodologies used for rigid frame steel buildings. The goal of advancedanalysisis to create a design basis capable of omitting specification equations(LRFD 1999). However, a requirement of advanced analysis based design is that the analysis employed should be capable of considering all critical aspects of steel member behavior used to develop the specification design equations(SSRC 1988). Excellent reviews related to the basic assumptions and requirements of advancedanalysisare available @ridge, et.al. 1998; White 1993). An advanced analysis is capable of addressing individual member strengths in light of the redundancy present in the framework. Therefore, performance-based design criteria are easily incorporated into optimized design algorithms that employ advancedanalysis. 3.1

Population Non-Ifomologous Success Size Crossover Rate Rate (7o) 0.20 40

20 = P^ 0 . 2 5

Crossover operations are important to the OO-EA as with binary genetic algorithms. However, one should recognize that the non-homologous operator is important when more desigSrvariables are present and/or the hierarchical representation is "deep" (Voss and Foley 1999). This ensures that the crossover operations are able to move the genetic material around and explore building block formation.

OBJECTTVE FUNCTION

The objective function is based on modified frame weight (Xu, et.al. 1995), ly'd

Na

w=LLrArpr*L ,t'l

The successrate of the OO-EA appearsto be linked to the population size. For small populations, the mutation rate should remain high so that sufficient new genetic material can be introduced in the

JO

n=l

f(u^" \ I L^A_p^L[t rJ-tj (r)

where: Z is the memberlength; I is the crosssectionalarea;and p is the material density. The modiffing factor, (, accountsfor the connectionat the beamends. Partially and fully restrainedconnectionvariations were defined using non-dimensional curyes (Bjorhovde, et.al. 1990). These curves were establishedin such a manner that a full range of connection strength and stiffness were present. Fully resfained connectionswere denotedas Cl

while flexibleconnections weredenotedC5. Three connections existedbetweentheselimits. 3.2 CONSTRAINTS Advancedanalysisbaseddesignallows lowlevel behavioralconstraints to be established. This in-turn allows performance constraints to be easily incorporated into optimization algorithms. Constaintsmust be established at both serviceand ultimateloadlevels. Serviceload level conshaintsusedin the advanced analysisbaseddesignoptimizationwere: 1. appliedloadratio for loadcombinations, 2. connectionrotations, 3. lateral(inter-story)drift, 4. verticaldeflectionof beams, 5. cross-sectionplastification. IJltimate(strength)load level constraintswere: l. appliedloadratio for loadcombinations, 2. connectionrotations, 3. plastichingecurvature, 4. out-of-planecompression buckling, 5. unbraced length for lateral torsional buckling. An additional constaint, designerpreference,was included. This consffaintwas establishedto ensure the colurnnbelow has a nominal depth and weight Iargerthanthe columnabove. 4

EVOLUTIONARYALGORITHM

An evolutionaryalgorithmwas usedto automatethe designof partially and fully restrainedsteel frame configurationswith fixed topology. The small empirical study discussedpreviously was used to initially assignmutationandcrossoverrates.

wherethe plastichinge curvaturelimit is takenfrom Yura,et.al.(1978), gF,db . Kriri, = **r =-E-

All penalties including the designerpreference penaltywerescaledusing(Camp,et.al.1998), p i = 1 . 0+ k ( d r - l ) "

(4)

The exponent,s, was taken to be 1.0 indicating linearscaling.Theparametert wastakento be 5.0. It should be noted that equation (4) is only applicablewhen /, > 1.0. The total penalty for plastic hinge rotation at ultimateload levelsis computedusing, du N,'*

o*=ff fl (r,)."

(s)

where: Nu is the number of ultimate load cases(3 in this study); N.**^ is the number of members. Further details regarding the formulation of the penalties used in this study can be found in Schinler

(2001). 4.3 SELECTION Populationpartitioning(Camp,et.al. 1998;Pezeshh et.al. 2000) is used for selectingindividualsfor mating. This schemeallows the selectionpressure to be controlledand has beenfound to be effective for the building optimization problems studied. Tournamentswerethen utilized to assisnthe matine pool. 4,4

CROSSOVERAND MUTATION Crossover is performed after choosing two individualsfrom the mating pool constructedusing the selectionmechanism.Each individualin this 4.I FITNESS new population is then considered a conftol The fitness statementused for the evolutionarv individual (refer to Figure2). Amate (not the same algorithm is similar to that suggestedby Pezesh( individual) is then chosenfrom the mating pool. et.al.(2000), Crossover results in one new individual being created. (2) In the present sfudy, each column, beam, and i-l connectionobjectin the control individualis chosen where:l/ is definedin (l) and @,aretotalpenalties for possiblecrossover.If a randomnumber satisfies corresponding to the constraintsin the problem. the crossoverprobabiliff, p", then crossoverwill take place with the object in the contol individual 4.2 PENALTIES being exchangedwith a similar object in the mate. A secondrandomnumberis calledandif it satisfies Penalties were written in a form suitable for inclusionin equation(2). For example,the plastic the non-homologouscrossoverrate, p,,n, nonhomologouscrossoveroccurs. If it doesnot satisfy hinge rotation penalty at ultimate load levels is this rate,homologous crossover is performed. wTltten as, Mutation is performed at the object level * (3) corresponding to buildings,stories,beams,columns, 0 *= Kti-i < 7 . 0 and connections. A mutation rate, pa, is

f =wfra,

76

established for eachof theseobjects.Mutationtakes placeafterthecrossover operations. 5

FRAME DESIGNS

The frame chosenfor implementationof the objectorientedevolutionaryalgorithm is the sameas the oneusedby Xu, et.al. (1995). Figure 6 providesan illustrationof the frameworkloading and topology. All beamsand columns are assumedto be Grade ,4'36 steel. The connectionsused in the present studyfollow the non-dimensional connectioncurves previouslydescribed. OL. 0.028 Mn.

Resultsfor the two frame designswith comparison to theresultsobtainedby Xu, et.al.(1995)arefound in Table 6. ConnectionC4 is one step up from flexible(C5), Table6: FrameWeightComparisons (lbs.) Frame Xlu,et.al.(1995) FR 7,03r PR

6,712

Present 7,086

l|xT_8i6,468

Convergencehajectories of the fittest feasible individualaregivenin Figures7 and 8.

LL = 0.083 k/ln.

30,000 E

E 25,000

]

!t o

5 20.000 tt o F

i' rs.ooo 6 o g

= L

io,ooo 5,000

Figure6: SteelFrameUsedin the PresentStudy.

0

The interior columns and exterior columns are groupedwithin eachstory. Furthermore,the beams in any story are assumedto be the same(including connections). The connectionsat each end of the beamsare requiredbe the same. As a result, the presentproblemhasa total of 6 designvariablesfor the fully restrained(FR) caseand 8 designvariables for thepartially restrained(PR)frame.\ The inelastic analysis used to establish an individual's fitness is based on the distributed plasticity model. Zerc lenglh connectionelements are used on all beams. Details of this inelastic analysismodelcanbe foundin Foley and Vinnakota (1999). Otherdetailsregardingthe OO-EA can be foundin Schinler(2001). The evolutionaryalgorithmparametersused in the analysesare as follows. The mutationrate waskept constantthroughouttheevolution. The fixed ratefor beams,columnsand connections(whereapplicable) was 30%. Crossoverwas restrictedto columns, beams and connections. The probability of crossoverwas setat 60%o with the subsequent rate of non-homologouscrossover set at 50%. The partitioning schemeresulted in the mating pool beingdevelopedfrom the top 40% of the population. krdividuals were selectedto enter two-at-a-time toumamentsfrom this upper partition 70yo of the time.

77

5

t0

15 20 25 30 35 lo Ganoratlon

4{t 50 5s

Figure7: FR Frame ConvergenceTrajectoriesfor theFittestFeasibleIndividual.

38,000 E

.E 32,000 o

'

E o

28,000

24,000 E E a o o

= lr

20,000 16,000 12,000 8,000

o

s

to tt

':rt"u"r#n

tt

40 45 50 5s

Figure8: PR Frame ConvergenceTrajectoriesfor theFittestFeasibleIndividual. The evolutionary algorithm developed exhibits stable convergence characteristics.It should be notedthat the fitnessillustratedin the figuresis the modified weight.This weight includesconnection modification factors. As one can see, the same individual doesnot result for all the runs. Further study of the results during the evolution revealed that the algorithmwas making concessionsamong

designvariablesand constraintsand therefore,the sameindividualmightnot be expected. Table6 iliustratesthatthe presentalgorithmandthe optimization problem formulated gives results consistent with thoseof pastresearchers. 6

CONCLUSIONS

An object-orientedevolutionaryalgorithmhasbeen described. The OO representationof design variableshasbeenshownto exhibit slishtlv different overall behavior than genetic algolrithhs using binary strings for design variable representation. Two types of crossoverhave been discussedand applied within the context of the object-oriented representation.A short empirical study illusfrated that object-oriented EA's may require larger populations and higher mutation rates that the corresponding binary GA. An optimizationproblem implementingadvanced analysisbaseddesign assessment was forrnulated. Discussionof the penaltiesneededfor the inelastic analysisbaseddesignprocesswas provided. A two story,three bay steelframe was desigrredusing the object oriented evolutionary algorithm with fully restained andpartially restrainedconnections.The resultsprovidedillustratethat theobjectorientedEA can achievedesignsconsistentwith thoseobtained by pastresearchers. Acknowledgments This researchwas made possiblethrough a grant from the National Science Foundation (CMS 9813216). The authors would also like to acknowledgethe fruitful discussionsrelated to geneticalgorithmswith Mark S.Voss. References Bjorhovde,R., Colson, A., Brozzetti,I. (1990). Classification System for Beam-to-Column Connection s,Journal of StructuralEngineering,116 (11), American Society of Civil Engineers,pp. 3059-3076. Bridge,R.Q.,Clarke,M.J., Osterreider, P., Pi, Y.L., Trahair,N.S.(1998).Designby AdvancedAnalysis, Journalof Constructional SteelResearch, 46 (103), ElsevierSciencePublishers, Paper144. Camp,C., Pezeshk,S., Cao, G. (1998). Optimized Design of Two-DimensionalStructuresUsing a Genetic Algorithm, Journal of Structural Engineering,AmericanSocietyof Civil Engineers, 1 1 9( s ) ,p p .5 5 1 - 5 5 9 . Foley, C.M. and Vinnakota,S. (1999).Inelastic Behaviorof Multistory Partially RestrainedFrames - Part I, Journal of StructuralEngineering,125 (8), pp. 854-861. AmericanSocietyof Civil Engrneers, (6

Jenkins, W.M. (1991). Towards Structural OptimizationVia the GeneticAlgorithm, Computers & Stractures, 40 (5),Pergamon Press,plc, pp, i321t327. LRFD (1999).Load and Resistance Factor Design Specification for Structural Steel Buildings, AmericanInstituteof SteelConstruction,Chicago, IL, December. Parmee,I.C. (1995).DiverseEvolutionarySearch for Preliminary Whole System Design, kr Developments in Neural Networl