Truss topology optimization for mass and reliability considerations—co-evolutionary multiobjective formulations

Struct Multidisc Optim (2012) 45:589–613DOI 10.1007/s00158-011-0709-9

RESEARCH PAPER

Truss topology optimization for mass and reliabilityconsiderations—co-evolutionary multiobjective formulations

David Greiner · Prabhat Hajela

Received: 28 September 2010 / Revised: 5 July 2011 / Accepted: 19 August 2011 / Published online: 6 October 2011c© Springer-Verlag 2011

Abstract The paper presents an approach for simultaneousoptimization of structural mass and reliability in discretetruss structures. In addition to member sizing, the selectionof an optimal topology from a pre-specified ground struc-ture is a feature of the proposed methodology. To allow for aglobal search, optimization is performed using a multiobjec-tive evolutionary algorithm. System reliability is based on arecently developed computational approach that is efficientand could be integrated within the framework of an evo-lutionary optimization process. The presence of multipleallowable topologies in the optimization process was han-dled through co-evolution in competing subpopulations. Aunique feature of the algorithm is an automatic reunificationof these populations using hypervolume measure-basedindicator as reunification criterion to attain greater searchefficiency. Numerical experiments demonstrate the compu-tational advantages of the proposed method. These advan-tages become more pronounced for large-scale optimizationproblems, where the standard evolutionary approach fails toproduce the desired results.

Keywords Structural topology optimization ·Multiobjective optimization · Trusses · Reliability analysis ·Evolutionary algorithms

D. Greiner (B)Institute of Intelligent Systems and Numerical Applications inEngineering (SIANI), Universidad de Las Palmas de Gran Canaria,Las Palmas de Gran Canaria, 35017, Spaine-mail: [email protected]

P. HajelaDepartment of Mechanical, Aerospace and Nuclear Engineering(MANE), Rensselaer Polytechnic Institute, Troy, 12180, USAe-mail: [email protected]

1 Introduction

The use of evolutionary algorithms in structural optimiza-tion was pioneered in the mid 80’s (Goldberg and Samtani1986). A focus on their use as global optimizers requiringno gradient information was presented in a 1987 publica-tion (Hajela 1990) and since that time, these algorithms havebeen applied in a number of design problems in the fields ofscience and engineering. Early applications of evolutionaryalgorithms in optimal topology design of discrete structuressuch as trusses are available in Shankar and Hajela (1991),Hajela et al. (1993), Grierson and Pak (1993a) and Hajelaand Lee (1995). Application to frame structures may befound in Grierson and Pak (1993b). A state of the art reviewof topology optimization using evolutionary algorithms isavailable in Hajela and Vittal (2000).

Multiobjective evolutionary optimization has extendedthe capabilities of evolutionary optimization algorithms toaccount for more than a single criterion function. Since late1990’s, the performance of these algorithms has improvednotably through the inclusion of the elitism operator asdescribed in Coello Coello (2006). A compilation of severalapplications of evolutionary multiobjective optimization isavailable in Coello Coello et al. (2007). An early applicationof multiobjective optimization with evolutionary algorithmsin structural design was presented in Hajela and Lin (1992).Early works in multiobjective optimum design of trussstructures with Pareto-based evolutionary algorithms arerepresented by Cheng and Li (1997), and of elitist Pareto-based evolutionary algorithms (Greiner et al. 2000, 2001).A multicriteria topology optimization of trusses approach ispresented in Ruy (2001), where simultaneous minimizationof weight and displacement at certain nodes was performed,and a non-dominated set of designs with different topologies

590 D. Greiner, P. Hajela

was determined (no reliability considerations were takeninto account). Recent work related to reliability-based opti-mization of truss structures with multiobjective evolutionaryalgorithms is documented in Tsompanakis et al. (2008) andOkasah and Frangopol (2009).

There has been considerable interest in including reliabil-ity considerations into topology optimization of structuralsystems. A specific area of focus for this research is in thedomain of topology optimization related to a structural con-tinuum and is documented in several recent publications(Kharmanda and Olhoff 2002; Maute and Frangopol 2003;Kharmanda et al. 2004). Mozumder et al. (2006) has pre-sented some improvements to the original method approachproposed by Kharmanda. Examples of other work in thisfield include Chen et al. (2010), where level set basedrobust shape and topology optimization is introduced, andSilva et al. (2010), where practical applications of topologyoptimization with reliability considerations are presented inthe context of an automotive control arm and a 3D kneestructure.

Research on integrating reliability with topology opti-mization in discrete bar structures is documented inThampan and Krishnamoorthy (2001), where a geneticalgorithm was used to perform a single objective opti-mization including both reliability indices (incorporated asconstraints values and handled using a penalty functionapproach) and variations of configurations (fixed number ofbars but with variable geometry of the nodes); Chen et al.(1999) present an approach for topology optimization thatincludes reliability constraints and uses an improved sim-plex method as a solution strategy; Mogami et al. (2006)model the structure as a series system for reliability con-siderations and include both stiffness and eigenfrequency inthe design process; Lagaros et al. (2008) describe a singleobjective reliability based optimization using evolutionaryalgorithms applied to 3D frame structures with topologyvariations consisting of varying the spacing among the gridof columns in the horizontal plane.

An example of research involving multiobjective evolu-tionary algorithm applications to truss topology optimiza-tion with reliability considerations is available in Mathakariand Gardoni (2007). The authors used multiobjective evo-lutionary algorithms to simultaneously minimize the struc-tural weight and maximize the reliability index of electricaltower trusses. This problem included a total of nine discretevariables, and categorized failure of the structural systembased on the failure of its first element; importance sam-pling in a MonteCarlo simulation was used in the reliabilitycalculation.

The present paper describes an approach for simultane-ous optimization of the mass (minimization) and reliability(maximization) in structural truss structures, where exter-

nal applied loads and material properties were consideredas random variables. Given that the two objectives foroptimization are in conflict, a multiobjective optimizationapproach provides for a clear emergence of a tradeoff pat-tern among the competing criteria. The approach helpsin promoting an understanding of the influence of cross-sectional sizing, the degree of structural redundancy, anddifferent structural layouts, on weight and system reliability(Fig. 1). The approach is not limited to inclusion or dele-tion of bar elements that increases or decreases the structureredundancy, but also allows for different structural layouts.The ultimate output of the optimization process at the endof a single run is a set of structural designs, each with thelower weight for each reliability level. The paper shows thata standard multiobjective evolutionary approach does notperform well for large number of design variables and inclu-sion of multiple layouts. A new evolutionary algorithm isdeveloped and implemented in this paper that successfullyidentifies the complete non-dominated solution front evenwhen considering different competing layout topologies inthe search process. The approach, referred to as CoSAR(Competing Subpopulations with Automatic Reunification)is based on allowing co-evolution among competing sub-populations and ultimately combining these subpopulationsas the optimal solutions are identified. The approach isintegrated with a novel and recently proposed reliabilityindex calculation for a structural system (Park et al. 2004).This computation yields quick estimates of reliability indexthat is well suited for use in a computationally demandingevolutionary optimization environment.

Subsequent sections of this paper summarize the reliabil-ity computation approach (Section 2), the development andimplementation of the CoSAR approach (Section 3), andapplication of the algorithm to test problems (Section 4).A discussion of the results and major conclusions aresummarized in Sections 5 and 6, respectively.

Fig. 1 Relationships among factors in minimum weight trussreliability

Truss topology optimization for mass and reliability considerations—co-evolutionary multiobjective formulations 591

2 Reliability method for evolutionary optimization

In evolutionary optimization, where multiple function eval-uations are required due to the population-based nature ofthe search, simple and fast function evaluations greatlyenhance the numerical efficiency of the search. Reliabilityestimation is generally a nested optimization process thatwould add to the computational burden in an evolutionaryoptimization process; improving the efficiency of reliabil-ity index calculation greatly helps with the computationalexpediency of the optimization. Park et al. (2004) have pro-posed a computationally efficient method for evaluating theprobability of failure of discrete structures (trusses as wellas frames). For purposes of this paper, this approach willbe referenced as the PCSS method (initial of its authors’surnames). Bar structures are considered as hybrid systems(neither parallel nor series systems), where statistical cor-relation exists among elements and failure modes. Underthe assumption of completely correlation of each and everymode of failure, an estimate of an upper bound of theprobability of failure of each parallel subsytem is easilycomputed. In addition, the sequence of failure of the com-ponents of a given mode is ignored, which implies that thenumber of failure modes is drastically reduced along witha corresponding reduction in the number of required struc-tural analyses. With that, the rule of the PCSS method maybe interpreted as determining the lower bound of the upperbound of failure mode probabilities:

If the failure probabilities of the elements are rear-ranged in order of descending magnitude a1 > a2 >

a3 > ... > aN, and if the redundancy of the structure (orsubstructure) is s, an estimate of the failure probabilityof the system is given by Psystem = as.

A more detailed description and validation of this rule fortrusses and frames can be found in the original paper. Theglobal failure mode is directly related to the redundancy ofthe whole structure. Similarly, the failure of a substructure isdirectly related to the redundancy of this substructure (localfailure modes). Each failure mode is a parallel system, andthe whole structure is a series of parallel subsystems, whoseprobability of failure (or reliability index) is computed fromthe probabilities of failure of their elements.

The reliability of the elements is evaluated using the first-order second-moment method (FOSM) (Cornell 1969), alsoreferred to as the mean value first-order second-momentmethod MVFOSM, that is equivalent to the Hasofer-Lindmethod (Hasofer and Lind 1974) in case of normal randomvariables and linear limit states (Haldar and Mahadevan2000; Choi et al. 2007). For these conditions, the safety orreliability index, β, represents the minimum distance of the

limit state line to the origin in transformed space (larger β

implies a higher reliability). The point on the limit state cor-responding to the minimum distance from the origin to limitsurface constitutes the most probable point of failure.

The limit state function (M) is represented by:

G (R, S) = M = R − S ≤ 0 (1)

The mean μ and the standard deviation σ of M are writtenas follows.

μM = μR − μS (2)

σM =√

σ 2R + σ 2

S (3)

Then, the reliability index, β, is defined as,

β = μM

σM= μR − μS√

σ 2R + σ 2

S

(4)

and is related to the probability of failure as follows.

pf = P (R − S ≤ 0) = �(−β) (5)

In the above, � is the standard normal distribution function(zero mean and unit variance). A value of reliability index of3.0 corresponds roughly to a probability of failure of 10−3.Similarly, a reliability index of 7.0 would correspond to aprobability of failure of 10−12.

Here, following Park et al. (2004), the basic randomvariables are the yield stress (σ Y ) of the material and theexternal applied load (Fexternal). Therefore, for each bar ele-ment, the limit state function in terms of stress units (Pa) iswritten as follows,

G (R, S) = R − S = σY − σelement = σY − k1 Fexternal (6)

where,

σelement = Felement

Aelement= k Fexternal

Aelement= k1 Fexternal (7)

In (7), Felement depends on the external applied loadand k1 is evaluated through a finite element calculation(

k1 = kAelement

; k = FelementFexternal

).

Then, the reliability index can be written as follows.

β = μM

σM= μR − μS√

σ 2R + σ 2

S

= μSy − k1μFex√σ 2

Sy + k21σ 2

Fex

(8)


In case of having n different external forces each of whichconstitutes an independent random variable, the limit statefunction assumes the following form.

G (R, S) = R − S = StressY − Stresselement

= StressY −n∑

i=1

ki Fi = SY −n∑

i=1

ki Fi (9)

And the reliability index results in:

β = μM

σM= μR − μS√

σ 2R + σ 2

S

=μSy −

n∑i=1

kiμFi

√σ 2

Sy +n∑

i=1(kiμFi )

2

(10)

In addition to maximizing the reliability index β above,minimization of the structural weight is the second crite-rion in the design problem. The structural weight is simplydependent on the lengths of the truss structural elements,cross sectional areas, and the material density.

The design formulation of the optimum design problemhandled in this paper is as follows:

max βsystem (s) (11)

min Mstructure (s) =Nbars∑

i=1

Ai Liρi (12)

subject to si ∈ Rd

where βsystem is the reliability index of the structural system(which is evaluated using the PCSS method and depends onthe reliability indexes of the bar elements), Mstructure is thestructural mass (with Ai, the cross-section area of each bar,Li the length of each bar and ρi the material density of eachbar), and s is a vector of the design variables, which can takevalues only from a discrete given set Rd.

The vector containing the design variables defines theevolutionary chromosome, which contains the variableinformation defining each solution. In the most generalcase: Layout size of the structure, Reliability redundancyof each bay, and Cross-section type of each bar.

The minimization of the structural mass is of interestbecause it reduces the raw material cost, which is highlycorrelated with the whole structural cost. The maximizationof the reliability index is of interest because it equals quanti-tatively to reduce the failure probability of the structure. Thereliability index is a measure of structural reliability oftenused in international codes (e.g.: EN 1990 Eurocode – Basisof structural design. 2002; ISO 2394 General principles onreliability for structures, ISO 1998). A multiobjective opti-mization is dealt when there are two or more objectives in

conflict, as is here the case of the maximization of the reli-ability index and the minimization of the structural weight.This simultaneous optimization of various objectives leadsnot to a single optimum, but to a set of equally optimumsolutions which are non-dominant in the Pareto optimalitysense. Evolutionary multiobjective algorithms allow us tosearch for obtaining the complete Pareto optimal set in onesingle execution. Therefore, this set of equally optimal solu-tions can be observed in the problem handled in this paperas obtaining the structural designs that constitute either: 1.The minimum weight solution that corresponds to each reli-ability index value; or either: 2. The maximum reliabilityindex that corresponds to each structural weight value. TheCoSAR algorithm proposed in this paper can realize thisobjective considering different layout sizes, redundanciesand cross-section sizing of the bar structure. After obtain-ing that set of optimum designs (a representative subset isshown for example in Table 14 corresponding to the appli-cation case of Section 4), from the engineering point ofview, there are solutions that are of more interest than oth-ers and corresponds to the engineer or decision maker tochoose these design(s) that adapts more to its requirements.For example, according to the ISO 2394 international code,structural reliability target values vary considering both:(a) failure consequences and (b) safety measure associatedcosts from 0 (low and high, respectively) to 4.3 (serious andlow, respectively); other codes have even greater values ofstructural reliability targets (e.g. in Argentina is above 5).

As a first step, the validation of the PCSS reliabilityindex calculation was performed for a simplistic redundanttruss structure. This was a six-bar truss structure shown inFig. 2 and taken from an example in Murotsu et al. (1984).The ultimate limit state of the truss based on resistance(stresses) was taken into account. Uncertainties in externalload and yield stress were represented as Gaussian normaldistributions.

Numerical data for the truss test case was as follows.Truss height h = 0.9144 m., width w = 1.219 m., Youngmodulus E = 2.06 × 1011Pa., cross-sectional area of bars

Fig. 2 Structural truss first test case (with bar numbering)


Table 1 Reliability index calculation of bar elements in first truss testcase

Reliability index β Present computation Reference

(Park et al. 2004)

Bar 1 3.375 3.375

Bar 2 4.970 4.969

Bar 3 2.688 2.689

Bar 4 2.688 2.689

Bar 5 4.970 4.969

Bar 6 3.375 3.375

Whole structure 2.688 2.689

1, 2, 5, 6 = 1.33 cm2, cross-sectional area of bars 3, 4 =1.49 cm2, mean value of yield stress μy = 2.76 × 108Pa.,standard deviation of yield stress σy = 2.76×107Pa., meanvalue of P, μP = 44.5 × 102N., standard deviation of P,σP = 4.45 × 102N, density of steel, ρ = 7,850 kg/m3.

Table 1 shows the values of the reliability index obtainedusing the proposed approach. The table also shows a com-parison with results presented in the reference publicationand illustrates very good agreement. From the reliabilityindex β, the probability of failure is computed as Pf =�(−β) = 1−�(β), where � is the cumulative distributionfunction of the standard normal variate.

Having validated the process of reliability estimation,this evaluation was integrated into an evolutionary mul-tiobjective optimization procedure with two conflictingcriteria—minimization of the structural weight and the max-imization of the total structural reliability index (which is

equivalent to the minimization of the structural probabil-ity of failure). The final result of each optimization runfrom the design process was a non-dominated front withthe minimum weight structure corresponding to each struc-tural safety index. The NSGA-II algorithm (Deb et al. 2000,2002) was used in the optimization, and discrete values ofcross-sectional areas were considered. To facilitate com-parison with the reference publications, 16 different crosssection areas (four bits per bar) were taken in the range of0.85 cm2 and 2.05 cm2 so that the values of 1.33 cm2 and1.49 cm2 considered in the reference were part of the admis-sible values of the design variables listed as 0.85, 0.93, 1.01,1.09, 1.17, 1.25, 1.33, 1.41, 1.49, 1.57, 1.65, 1.73, 1.81,1.89, 1.97, 2.05 (cm2). Figure 3 shows results of the nondominated front (upper left corner corresponding to the opti-mization direction), coming from a single execution of thealgorithm (104 fitness function evaluations), with a popula-tion size of 100, and a uniform mutation rate of 3%. Thedesign corresponding to the test case reference original datais also included in the figure (single star point), as are fourother designs.

The cross sectional areas of these designs are presentedin Table 2. For the sake of clarity the areas are substitutedby the ordering in the 16 cross-section type database (DB-ord), where the lower the order, the lower the area; also theprobability of failure corresponding to each structural relia-bility index has been included as a percentage. In addition,Table 2 shows the probability of failure results obtained byan independent Monte Carlo simulation using 3 × 107 eval-uations; these results are in very good agreement with thevalues obtained for the optimized designs using the PCSSreliability calculation method (third column).

Fig. 3 Optimization ofreference test case drawn fromthe PCSS paper


Table 2 Detailed non-dominated structural designs of first truss test case

Weight, Reliab. Prob. Monte Carlo Bar 1, Bar 2, Bar 3, Bar 4, Bar 5, Bar 6,

kg index β Fail. % Prob. Fail. % DB-ord DB-ord DB-ord DB-ord DB-ord DB-ord

Op Des 1 5.244 0.003 49.8803 49.9071 2 1 4 1 1 1

Op Des 2 7.216 2.725 0.3215 0.3209 7 3 7 11 1 4

Op Des 3 7.905 3.233 0.0612 0.0613 9 4 9 13 4 3

Op Des 4 10.814 4.945 4 × 10−5 4.3 × 10−5 16 16 16 16 6 11

Test Case 8.019 2.688 0.3594 0.3599 7 7 9 9 7 7

This simple example illustrates the use of the reliabilityindex calculation drawn from PCSS, and its integration ina multiobjective optimization based on evolutionary com-putation. The approach allows a simultaneous optimizationwherein the minimum weight structure corresponding toeach value of the reliability index β is evaluated.

3 Truss structure optimization including redundanciesand layout variations

A significant focus of the present research was to look atmethods that allow for varying redundancies and layoutgeometries in structural trusses, while searching for opti-mal discrete designs that minimize structural weight andmaximize the reliability. In this approach, external loadsand yield stress were considered as random variables withGaussian distribution. In this context, two layout topologies(small and medium) were considered. These have the sameexternal dimensions and are subjected to identical externalloading. The possible structural configurations for whichsizing optimization was also included are shown in Figs. 4and 5 (small layout: denoted as cR01 to cR03, and mediumlayout: denoted as cR11 to cR19).

The dimensions in height and width are constant in eachof the cases and are based on the well-known 10-bar opti-

Fig. 4 Small layout structures

mization test case in (Rajeev and Krishnamoorthy 1992):The height of the truss was 18.288 m, its width was 9.144 m,and loads P = 454 KN were applied at the same node loca-tions in all configurations. Both the admissible stress andload P, were assumed as normally distributed variables, withmean values as previously indicated, and with coefficientsof variation Cv = 0.1. The structural material was alu-minum, with Young modulus 68.9 × 109 Pa, admissiblestress 172 × 106 Pa and density 2,770 kg/m3.

A discrete optimization was performed, where sixteendifferent cross-section types (CrS-n) were considered foreach bar, and were selected from the AISC standard cross-sections as indicated in Table 3. Each bar was encoded as a4 bit gene in the chromosome.

To compare the quality of the entire Pareto front duringthe evolutionary search process, the S-metric or hyper-volume was evaluated. Originally proposed in Zitzler andThiele (1998), the hypervolume is a distinguished unaryquality measure for solution sets in Pareto optimization(Knowles et al. 2006), taking into account both coverageand approximation to the optimum Pareto front. The ref-erence point considered was (5000, −10). As discussed inFonseca et al. (2006)1 the higher the value of the S-metric,the better was the quality of the non-dominated solution set.

3.1 Standard optimization

A standard panmictic multiobjective evolutionary optimiza-tion procedure (the whole population is dealt as a single poolof individuals, see e.g. Alba and Tomassini 2002) was usedin this section. The complexity of the design problems wasincreased in steps as described in subsequent sections of thepaper. For ease of reference, these cases are identified inTable 4.

The objective is to allow the simultaneous optimiza-tion of the mass and reliability of truss structures takinginto account layout sizes, redundancies and discrete cross-section type sizing. In order to have reference results for

1Source code available at: http://sbe.napier.ac.uk/∼manuel/hypervolume.

http://www.sbe.napier.ac.uk/~manuel/hypervolume

http://www.sbe.napier.ac.uk/~manuel/hypervolume


Fig. 5 Medium layoutstructures

comparing the optimization outcomes, also optimizationof only discrete cross-section type sizing of each possiblestructure is performed in case A. A gradual increase in theproblem difficulty implies including redundancies jointlywith cross-section type sizing (without integrating layoutsizes in the optimization), which is performed in case B.Therefore independent results corresponding to each lay-out size are obtained, having good agreement with caseA reference outcomes. The whole optimization with thethree features (layout sizes, redundancies and discrete cross-section type sizing) starts with a standard panmictic algo-rithm (case C). Their results are not capable of achieving theproposed goal, because the design solutions belonging to thesmallest layout size dominate the population. In Section 3,different evolutionary multiobjective algorithm proceduresare progressively shown in order to achieve this objective.Therefore, different proposals with enhanced diversity char-acteristics (from cases D to H) related with the layout size

are progressively tested with the purpose of reaching theresults until showing the CoSAR strategy (case H) proposedin this paper and which succeeds efficiently in obtainingoptimum designs.

3.1.1 Optimization of separate layouts

Case A To establish results for baseline comparison, mul-tiobjective optimization for weight minimization and max-imization of reliability index of the small and mediumlayouts was first performed. For the small layout, thismeant independent optimization of the three different sta-ble configurations of the case; two non-redundant and onefirst-degree redundant cases, represented in Fig. 4 andlabelled as cR01, cR02 and cR03, respectively. Ten inde-pendent executions of each case, with population sizes of100 and 200 individuals, 1.5% uniform mutation rate, uni-form crossover and standard binary reflected gray code were


Table 3 DetailedNon-dominated structuraldesigns of first truss test case

Order Cross-section Area (cm2) Radius of gyration (cm)

CrS-01 L2.5 × 2 × 3/16 10.45 2.014

CrS-02 L3 × 2 × 1/4 15.35 2.423

CrS-03 L2.5 × 2 × 3/8 19.94 1.951

CrS-04 L4 × 3.5 × 5/16 28.97 3.200

CrS-05 L6 × 3.5 × 5/16 37.03 4.953

CrS-06 L6 × 4 × 3/8 46.58 4.902

CrS-07 L7 × 4 × 3/8 51.42 5.766

CrS-08 L8 × 4 × 1/2 74.19 6.579

CrS-09 L8 × 6 × 1/2 87.1 6.502

CrS-10 L8 × 8 × 1/2 100 6.350

CrS-11 L6 × 6 × 3/4 109.03 4.648

CrS-12 L8 × 6 × 3/4 128.39 6.426

CrS-13 L8 × 8 × 3/4 147.74 6.274

CrS-14 L8 × 8 × 7/8 170.97 6.223

CrS-15 L8 × 8 × 1 193.55 6.198

CrS-16 L8 × 8 × 11/8 216.13 6.147

used in the optimization process. The accumulated non-dominated fronts after 20200 function evaluations with apopulation size of 100 individuals are shown in the Fig. 6.

These results clearly show that the structure cR01 (diag-onal crosses in Fig. 6) that has a diagonal bar at the nodewhere the horizontal load is applied dominates the structurecR02 (diamonds in Fig. 6) completely. From an engineer-ing structural design point of view this can be interpreted tomean that cR01 is always preferable to cR02 designs. The

redundant configuration cR03 (crosses in Fig. 6) seems tobe better than the non-redundant, except in a few designsthat are in the lower left part of the graph (correspond-ing to the lowest weight, but also lowest reliability index,and not of interest as useful solutions). Some representativenon-dominated solutions are selected and shown in Table 5.

This exercise was repeated for the medium layout, con-sidering as independent cases the nine different stableconfigurations of this topological layout. These included

Table 4 Optimization methodscompared in this paper Paper section Case ID Optimization method Optimized features

Section 3.1.1 Case A Separate optimization of all possible Cross-section type sizing

structures for small and medium layout optimization

Section 3.1.1 Case B Separate optimization of small and medium Cross-section type sizing

layouts—optimization across all allowable and redundancy optimization

structures within a layout

Section 3.1.2 Case C Combined optimization across allowable Cross-section type sizing,

structures of both layouts (panmictic) redundancy, and layout

size optimization

Section 3.2.1 Case D Optimization with constraints ensuring a

minimal number of members from

each layout (panmictic)

Section 3.2.2 Case E Optimization with co-evolution

in subpopulations

Section 3.2.3 Case F Subpopulations with pre-specified

reunification

Section 3.2.4 Case G Subpopulations with automatic reunification

Section 3.2.5 Case H Competing subpopulations with automatic

reunification—CoSAR


Fig. 6 Accumulatednon-dominated front (smalllayout)

four cases of statically determinate structures (cR11 tocR14), four with first-degree redundancies (cR15 to cR18),and one with second-degree redundancy (cR19), eachshown in Fig. 5.

Ten independent optimization runs, each of populationsize 100, involving 104 generations of evolution and witha 1.5% uniform mutation rate were performed. The non-dominated fronts obtained in these runs are shown in Fig. 7.In this figure, final structural designs are identified by thedegree of redundancy of the structure to which they belong.

These figures show how an increase in the degree ofredundancy is beneficial, as these structures are dominanton the Pareto front everywhere except in lower left portionof the front (optimal solutions with lower reliability andweight). This is clear from Figs. 6 and 7, and is true forboth small and medium layouts. The non-dominated frontsof both layouts are shown together in Fig. 8.

In the context of the hypervolume metric, the populationof the reliability optimized structures of test cases cR01-03(small layout) and cR11-19 (medium layout) configurationswere analyzed and the metrics are summarized in Tables 6and 7.

The three structures with best hypervolume are, in ordercR19, cR03 and cR15. These results simply confirm whatis visually evident from the figures showing the final non-dominated fronts. In the small layout, the ordered values ofhypervolume metric are cR03 (redundant structure degreeone) > cR01 > cR02 (non-redundant structures). Similarly,in the medium layout, the ordered values of hypervol-ume metric are cR19 (redundant structure degree two) >

cR15, cR17 (redundant structures degree one) > cR11 (non-redundant structure). When comparing small and mediumlayouts, the accumulated hypervolume of the medium lay-out (78069) is greater than the corresponding value of smalllayout (76904).

Case B Next, the optimization approach was applied soas to include the possible redundancy combinations in eachlayout separately. In addition to the four bits associatedwith each bar to code its cross-section type, each bay wasassigned two additional bits. The first bit simply defined theredundancy (zero and one referring to a non redundant orredundant bay, respectively). The second bit identified thetype of non redundant structure—a zero implied a diagonal

Table 5 Detailednon-dominated structuraldesigns of small layout (barnumbering as in Fig. 4)

Weight (kg) Reliability Bar 1 Bar 2 Bar 3 Bar 4 Bar 5 Bar 6 Struct.

index β

712.57 1.09 CrS-01 CrS-01 CrS-06 CrS-03 CrS-05 CrS-03 cR03








Fig. 7 Accumulatednon-dominated front (mediumlayout)

element from top left to bottom right, as in the cR01 design;similarly, a value of one was used to denote a diagonalelement from top right to bottom left, as in cR02 design.

The combined optimization of the small layout (includ-ing each of the cR01, cR02 and cR03 structures) wasnext considered. The problem parameters for this numeri-cal experiment were identical to those used in the case ofthe independent structures. The results measured in termsof hypervolume are shown in Table 8, and demonstratethat the combined optimization of small layout outperformsany of the individual structures optimizations. It is alsoable to achieve non-dominated designs equivalent to thoseobtained in the optimization of the fixed structures and withsimilar values of the fitness function (with the advantageof only requiring a single optimization run). The accumu-lated hypervolume for the final front resulting from adding

the independent structure optimizations (76904, Table 6) issimilar to that of combined optimization (76908, Table 8).

This optimization of size and redundancy was next per-formed for the medium layout, where nine different struc-tures were possible (cR11 to cR19). Using identical problemparameters to those of previous cases, the optimizationresults measured in terms of hypervolume are shown inTable 9. These results qualitatively follow the same trendsas those for the small layout. The accumulated hypervolumefor the final front resulting from adding the independentstructure optimizations (78069, Table 7) was similar to thatof combined optimization (78068, Table 9).

Results of case B show that it is advantageous to includeredundancies into the optimization procedure in additionto cross-section type sizing, allowing the same quality ofsolutions, but with much lower optimizations. While case

Fig. 8 Accumulatednon-dominated fronts (small andmedium layouts)


Table 6 Hypervolumeconsidering ten independentruns (small layout)

After 20.200 evaluations Hypervolume Hypervolume best Hypervolume Hypervolume in

average (over all run) standard accumulated

deviation final front

cR-01 71891.37 71891.37 0.00 71891.37

cR-02 63239.60 63239.60 0.00 63239.60

cR-03 76520.93 76624.07 78.96 76731.83

cR01 + 02 + 03 – – – 76904.00

Table 7 Hypervolumeconsidering ten independentruns (medium layout)

After 1.000.000 Hypervolume Hypervolume best Hypervolume Hypervolume in

evaluations average (over all run) standard accumulated


cR-11 71799.11 72147.68 1045.71 72147.68

cR-12 68019.07 68019.07 0.00 68019.07

cR-13 67728.08 69763.5 1332.49 69763.5

cR-14 62695.45 64754.21 2058.76 64754.21

cR-15 75836.18 75966.91 139.69 76006.87

cR-16 71256.85 71273.76 4.89 71282.89

cR-17 73054.51 73064.48 2.43 73067.84

cR-18 68638.73 68638.73 0.00 68638.73

cR-19 77553.09 77627.04 41.89 77802.28

cR11+...+cR19 – – – 78069.32

Table 8 Hypervolumesconsidering ten independentruns, combined cR01-03optimization

Combined cR01-03 Hypervolume Hypervolume best Hypervolume Hypervolume in



After 20.200 evaluations 76679.52 76809.08 97.44326 76891.48

After 25.200 evaluations 76730.36 76844.08 101.7664 76908

Table 9 Hypervolumesconsidering ten independentruns, combined cR11-19optimization

Combined cR11-19 Hypervolume Hypervolume best Hypervolume Hypervolume in

average (over all run) standard deviation accumulated final

front (over all run)

After 1.000.100 77791.93 77872.06 70.84 78068.03

evaluations

Table 10 Additionalhypervolume reference values inten independent runs (computedfrom the independent layoutsize optimization)

After 1.000.000 evaluations Hypervolume Hypervolume best Hypervolume Hypervolume in



Combined cR01-03 (Pop. Size 200) 76887.36 76913.54 47.67 76915.63

Combined cR11-19 (Pop. Size 200) 78047.38 78069.54 10.73 78117.93

Combined Small+Medium Layouts – – – 78223.02

(Pop. Size 100 + 100)

Combined Small+Medium Layouts – – – 78272.55

(Pop. Size 200 + 200)


Fig. 9 Independent small andmedium layouts: average

A needs one algorithm execution per individual structure(that is three cases in small layout size and nine cases inmedium layout size) versus only one algorithm execution asa whole for each layout size in case B, meaning a 33% ofcomputational cost in case of the small layout and 11% ofcomputational cost in the case of the medium layout size.

3.1.2 Layout optimization includingboth layout geometries

To search for the best structural design independent of thelayout to which it belongs, an optimization was performedthat included all structural possibilities from the small andmedium layout in a single run. Results obtained from theindependent layout size optimization methodology (Case B)were used as a reference basis for comparison. In particular,the total non-dominated front and its hypervolume metricwere quantities of interest. As the hypervolume depends

also on the population size, results for both population sizesof 100 and 200 are included. The final numerical values areshown in Table 10.

The evolution of the average values of the hypervolumewith separate layouts (small and medium) and both popula-tion sizes (100 and 200) of this independent layout valuesare shown in Fig. 9. The population size of 200 typicallyhas a higher value of hypervolume than the population sizeof 100 (as the hypervolume depends on the number of solu-tions located on the front). This plot is replicated againin Figs. 10, 11, 12 and 13, where it serves as referencesolution.

Case C In addition to the sizing and redundancy, opti-mization was also performed using both layouts (smalland medium) simultaneously in the optimization process.One additional bit distinguished between the layouts (zerodenotes the small layout while one is indicative of themedium layout). These results were compared against those

Fig. 10 Combinedsmall–medium layouts: average






obtained from the independent optimization of separatelayouts; identical problem parameters were used in thesenumerical experiments. Figure 10 shows the hypervolumeaverage evolution, using a standard panmictic optimization.The performance was rather poor, with results that were noteven close to the best results from the medium layout.

A closer examination of the numerical results shows thatalthough both layouts are equally distributed at the startof the evolution, the population is rapidly saturated withmembers belonging to the small layout, and members ofthe medium layout disappear. Only two out of ten casesfor a population size of 100, and three out of ten cases forthe population size of 200, had solutions belonging to themedium layout in the final population; in all other cases,they disappeared in the very early stages of the evolution.In evolutionary optimization, the lower the number of vari-ables in a problem, the faster the algorithm converges thesevariables to a solution: in the combined optimization, thealgorithm tends to obtain faster improvement of the non-dominated front for the small layout (5 or 6 bars), and theyvery quickly dominate the population, eliminating membersof medium layout (9 to 11 bars), thereby precluding thediscovery of better solutions.

Results of case C show that the joint optimization of lay-out sizes, redundancies and cross-section type sizing usinga standard panmictic evolutionary multiobjective algorithmlead to premature convergence due to the population domi-nance of suboptima belonging to the smallest size problem.The next section describes a strategy aimed at overcomingthis deficiency in the optimization process.

3.2 Strategy to ensure population diversity

This section describes an optimization approach whereinexplicit steps were taken to ensure the presence of a minimalnumber of designs belonging to a particular layout (smalland medium).

3.2.1 Optimization with constraints on minimal populationsize for each layout

Case D Balling et al. (2006) proposed an operator referredto as topology fitness that was used effectively in trussweight and topology design. For a population of designs,this operator limits the number of chromosomes corre-sponding to a particular topology. In the present problemthis constraint operator was not applied to structural topolo-gies individually but rather to the family of the structurallayout (small or medium size).

The operator ensured that at least M-members of eachlayout were present in a population, independently of theirfitness value (here M = −1 + N/2 in a population size ofN). With this operator, the algorithm was forced to main-tain at each stage of its evolution, a minimum number ofindividuals belonging to each layout. This allowed for thediscovery of some good solutions of the medium layoutin the Pareto front obtained from the combined optimiza-tion. These results are summarized in terms of average inFig. 11, obtained using the same parameters as those usedin previous cases. The black dotted lines show an improve-ment when compared to the standard optimization approach

Table 11 Hypervolumestatistics; combinedsmall+medium layoutsoutcomes (cases labels inTable 4)

Combined small+medium Hypervolume Hypervolume best Hypervolume Hypervolume in

after 1.000.000 evaluations average (over all run) standard accumulated


Case C

Pop. Size 100 76872.4 77527.76 247.94 77607.31

Pop. Size 200 77103.19 77658.75 348.51 77690.78

Case D

Pop. Size 100 77508.71 77717.52 243.99 78090.07

Pop. Size 200 77818.95 77879.41 37.50 77998.74

Case E

Pop. Size 100: 50 + 50 77352.67 77515.39 71.1 77960.38

Pop. Size 100 + 100 77874.15 78048.89 218.3 78213.72

Case F

Pop. Size 50 + 50 77885.95 77974.15 67.2 78189.12

Pop. Size 100 + 100 78123.64 78207.18 92.87 78269.15

Case G

Pop. Size 100 + 100 78150.68 78218.82 81.55 78273.77

Case H (CoSAR)

Pop. Size 100 + 100 78169.52 78213.45 41.83 78273.04


Fig. 14 Combinedsmall–medium layouts: 200population; average

Fig. 15 Combinedsmall–medium layouts: 200population; best

Fig. 16 Combinedsmall–medium layouts: 200population; standard devistion


Fig. 17 Combinedsmall–medium layouts: 200population; average

(shown in Fig. 10), but are still relatively poor when com-pared against results obtained from the independent mediumlayout optimization case. Final values of hypervolume arepresented in Table 11.

3.2.2 Optimization with subpopulations

Co-evolution of subpopulations has received considerableattention in the literature. Benefits of parallelizing an evolu-tionary search have been realized in many fields and recentadvanced applications of parallel multiobjective evolution-ary algorithms in engineering design can be found in Lee etal. (2008) and Herrero et al. (2009). A tutorial on parallelevolutionary algorithms is available in Alba and Tomassini(2002).

Case E In the current problem, a simplistic implemen-tation of co-evolution was implemented, with separated

subpopulations for each layout size. This approach explic-itly enforces diversity in layout geometries and focuses thesearch of each chromosome set independently. With thisstrategy, results of accumulated hypervolume belonging totwo subpopulations of size 100 each (small and medium lay-outs)(Fig. 12) are compared against those obtained in theapproach of Case D with a population size of 200 (Fig. 11).The subpopulation approach yields results that are betterin terms of both best and average values of hypervolume,while those of the two subpopulations of size 50 each (smalland medium layouts) are worse when compared to the con-strained layout approach with 100 individuals. Final valuesof hypervolume are shown in Table 11.

3.2.3 Optimization with subpopulations with reunif ication

Case F Combining both subpopulations after they havepartially converged, allows the benefits of drawing upon

Fig. 18 Combinedsmall–medium layouts: 200population; best


Fig. 19 Combinedsmall–medium layouts: 200population; standard deviation

genetic information learned during separate evolution, andto discriminate those designs that are dominated by others.As a first step in understanding the benefits of subpopu-lation unification, a fixed reunification was established at600,000 fitness evaluations.

The plot in Fig. 13 shows how this reunification improvesthe result in comparison to those from independent layoutoptimization for two different population sizes. The evolu-tion after reunification of the subpopulations achieves finalvalues of the hypervolume that are higher than those ofthe medium layout size and are near the combined lay-outs values. Final values of hypervolume are presented inTable 11.

3.2.4 Optimization with subpopulations and automaticreunif ication

Case G The determination of the convergence of a popu-lation is often required in evolutionary optimization to notonly establish a stopping criterion but also to determinewhen certain operators could be invoked. Examples of suchoperators in single criterion structural optimization are therebirth operator in Galante (1996), the auto-adaptive rebirthoperator in Greiner et al. (2004), or a restart operator in Tanget al. (2005). In multiobjective optimization, such appli-cations are less widely used. The hypervolume indicatoris used e.g. in Durillo et al. (2010) as a stopping crite-rion to measure quality outcomes in non-dominated fronts.This criterion is used in the present work as a measure todetect the convergence of each subpopulation and there-fore, to define an automatic reunification strategy in lieuof the predetermined reunification point (600,000 functionevaluations) used in Section 3.2.3.

The trends in the convergence pattern of hypervolumeshave been shown in various figures in preceding sections

of this paper. They follow the characteristic logarithmictrend as the evolutionary proceeds. After a steep initialascent, the plot tends towards an asymptote with slope δ

that approaches zero; minor oscillations in the convergencetrend are seen as a natural result of perturbations intro-duced in the evolutionary search process. Saving a history ofhypervolumes during the algorithm evolution allows one tocompute a discrete approximation of this slope δ, and to useit as a convergence indicator that automatically initiates thereunification process. One approach is to let δ be the relativechange in the value of the hypervolume between the currentvalue and the hypervolume at a fixed number of generationsbefore the current one (percentage variation was used in thepresent work). Then, denoting K as the number of timesδ was equal or less than zero, a reunification criterion canbe prescribed as the point where K exceeds a prespecifiedlimit value in all the subpopulations. Figures 14, 15, 16show the results for cases where K exceeds 100. Referenceplots in these figures are the trends of hypervolume evolu-tion determined in the fixed-point reunification strategy ofthe previous section. The automatic reunification strategy

Table 12 Number of function evaluations required to achieve cer-tain Hypervolume values (combined small+medium layouts), totalpopulation size 200 (cases labels in Table 4)

Values of hypervolume 77500 78000 78100

metric average

Case C – – –

Case D 213887 – –

Case E 109098 – –

Case F 113761 646221 871411

Case G 104368 389297 569168

Case H (CoSAR) 70326 294000 452294


Fig. 20 Optimum designstructures belonging to the finalnon-dominated front

allows a faster convergence without losing accuracy. Finalvalues of hypervolume are shown in Table 11.

3.2.5 Optimization with Competing Subpopulationsand Automatic Reunif ication (CoSAR)

Case H As different subpopulations handle different sizeproblems, evolution in these subpopulations tend to exhibitdifferent convergence trends. To account for this behavioura further modification to the algorithm was studied, wherethose subpopulations with higher percentage variation inthe hypervolume (δ) were favored in the evolutionary pro-cess. This approach is referred to here as Competing Sub-populations and Automatic Reunification (CoSAR). Sub-populations compete among themselves to be selected tocontinue their next evolutionary generation based on thevalue of δ for the subpopulation. The higher the value ofδ the greater is the probability of selection. This translatessimply into higher probability of evolution to that subpopu-lation for which the hypervolume is improving at the fastestrate.

A description of the CoSAR algorithm is included withdetailed steps:

1. One subpopulation per layout size is created.2. Co-evolution of the subpopulations are performed: each

subpopulation searches its non-dominated solutions,considering cross-section type sizing and redundancies.

Competing subpopulations: Only one subpopulationper generation among subpopulations is allowed toevolve based on a competing criterion δi, where δi isthe hypervolume percentage variation of subpopulationi after g generations. At generation t, δi is evaluatedaccording to the following equation:

δi =(H ypervolume(t−g) − H ypervolumet

)

H ypervolumet· 100

δi values are ranked and subpopulations are assignedan increasing probability Psubi of being chosen for the

Fig. 21 Non-dominatedsolutions of combined small,medium and big layouts: allsolutions


Table 13 Detailed structuraldesigns from results depicted inFig. 21; small layout size

Structure Weight(kg) Reliability Bar 1 Bar 2 Bar 3 Bar 4 Bar 5 Bar 6

index

cR01 217.998 −8.103 CrS-01 CrS-01 CrS-01 CrS-01 CrS-01 –

271.746 −6.273 CrS-01 CrS-01 CrS-01 CrS-01 CrS-03 –

cR03 302.006 −6.129 CrS-01 CrS-02 CrS-01 CrS-01 CrS-01 CrS-01

449.573 −2.658 CrS-01 CrS-04 CrS-02 CrS-01 CrS-03 CrS-01

643.673 0.637 CrS-01 CrS-06 CrS-02 CrS-01 CrS-04 CrS-03


next evolving generation according to it (the greater δi

the greater Psubi)

3. Reunification of the subpopulations occurs automati-cally when the δi values are K times equal or less thanzero.

4. Finally, evolution of a single panmictic population con-sidering cross-section type sizing, redundancies andlayout size is performed until final stopping criterionis met.

A study of the influence of the probability of subpopulationselection was conducted with values of 67%, 80% and 95%assigned to the subpopulation with higher δ; the remaining

subpopulations were assigned equal probabilities (e.g. fora 80% probability assigned to best performing subpopula-tion out of three, the remaining two would be assigned aprobability of evolution of 10% each). It was observed thatthe highest value, 95%, yielded the best results. Figures 17,18, 19 show results from the CoSAR strategy comparedagainst those from Case G (Section 3.2.4). Final values ofhypervolume are presented in Table 11.

Comparing the hypervolume values in accumulated finalfront of the different proposed strategies in Table 11, versusthe equivalent values obtained from the independent opti-mization of layouts in Table 10, we see how both strategiesof Cases G and H were able (with a total population of

Table 14 Detailed structuraldesigns from results depicted inFig. 21; medium layout size

Structure Weight(kg) Reliability Bar 1 Bar 2 Bar 3 Bar 4 Bar 5 Bar 6

index Bar 7 Bar 8 Bar 9 Bar 10 Bar 11 –


CrS-01 CrS-01 CrS-01 CrS-02 – –

373.946 −4.109 CrS-01 CrS-02 CrS-01 CrS-01 CrS-02 CrS-01

CrS-02 CrS-01 CrS-01 CrS-03 – –


CrS-03 CrS-01 CrS-01 CrS-01 CrS-01 –









1183 4.994 CrS-02 CrS-07 CrS-07 CrS-06 CrS-08 CrS-03











Table 15 Detailed structuraldesigns from results depicted inFig. 21; Big Layout size

Structure Weight(kg) Reliability Bar 1 Bar 2 Bar 3 Bar 4 Bar 5 Bar 6 Bar 7

index Bar 8 Bar 9 Bar 10 Bar 11 Bar12 Bar13 Bar14

Bar15 Bar16 Bar 17 Bar 18 Bar19 Bar20 Bar21

jG81 4528.66 8.68 CrS-05 CrS-11 CrS-16 CrS-08 CrS-09 CrS-06 CrS-01

CrS-09 CrS-06 CrS-11 CrS-08 CrS-12 CrS-04 CrS-16


5509.99 8.881 CrS-10 CrS-16 CrS-16 CrS-07 CrS-08 CrS-08 CrS-01



6325.44 8.983 CrS-12 CrS-16 CrS-16 CrS-05 CrS-08 CrS-08 CrS-04



200 individuals) to achieve comparable accumulated hyper-volumes to those generated from the final non-dominatedfronts obtained from the addition of the two populationsof small and medium layouts of 200 individuals each. Thecomputational savings resulting from working with half thepopulation size in Cases G and H are self-evident. Theyneed only half the fitness evaluation functions to achieveresults of equivalent quality in terms of hypervolume metricof the non-dominated final optimum designs.

Table 12 shows the number of fitness function evalua-tions required on average in the ten independent executionsof each case, to achieve certain values of the hypervolumemetric average with a total population size of 200 individ-uals (in case of subpopulation division it pertains to 100members in each group before reunification). These valueswere obtained from linear interpolation of the results showngraphically in Figs. 10, 11, 12, 13, 14 and 17. This providesa measure of the speed with which an algorithm attains acertain quality of solution for the final non-dominated frontof optimal designs. When the algorithm was not capable ofreaching the desired value, a blank entry is shown in the

table. For each of the three values considered in this anal-ysis, the outcome of algorithm performance is the same:CoSAR outperforms the approach of subpopulations withautomatic reunification that in turn, is better than just the useof subpopulations without any reunification. Each of thesealgorithms performs better than using a simple constraintoperator for maintaining diversity of layouts in the popula-tion. A standard optimization is the poorest performer andis not recommended for these types of problems.

CoSAR is the algorithm with higher average and lowerstandard deviation in the final values of the hypervolume(Table 11), as well as the fastest algorithm during theevolution (Table 12).

4 Application case

A bigger layout of the previous test case was also consid-ered in these numerical experiments. This application casehas been proposed because it gathers the features which

Fig. 22 Combinedsmall–medium–big layouts:(400 / 100 + 100 + 200)populations; average


Fig. 23 Combinedsmall–medium–big layouts:(400 / 100 + 100 + 200)populations; best

increase the complexity of the application in order to testthe proposed CoSAR method:

1. Increase in the number of levels of layout size: The pro-posed application case has three levels of layout sizes(instead of two in Section 3 cases);

2. Increase in the search space: The search space of theproposed application case has increased notably: Con-sidering the sizing problem of the big layout size whichhas 21 bars, each of them considered as an optimiza-tion variable: 221×4 = 284 ∼ 1, 9 × 1025 (expressedin kg, this mass is more than three times the mass ofplanet earth; expressed in meters, this distance is aroundtwenty times smaller than the radius of the observableuniverse); compared versus the sizing problem of themedium layout size shown in Section 3 which has 11bars: 211×4 = 244 ∼ 1, 8 × 1013 (expressed in kg, thismass is of the order of the mass of all the cars in theworld; expressed in meters, this distance is travelled by

light in less than 17 h). Also the redundancies combina-tions of this third layout size leads to a greater numberof possible structures (81).

In addition, it allows having the previous results of Section 3(where two layout sizes have been shown) as reference,which permits to compare in a fair context the CoSARmethodology in a much more complex optimization envi-ronment. We have chosen to emphasize the increment inoptimization complexity in this proposed application casewithout losing the possibility of having the previous resultsas comparison.

The new layout was obtained by subdividing each of thetwo bays of the medium layout size in half, resulting ina four bay layout while maintaining the external dimen-sions of the truss. The most redundant structure is denotedas jG81 in Fig. 20. An additional bit was included in thechromosome to allow this new layout to be represented inthe population of designs. The possible combinations (three

Fig. 24 Combinedsmall–medium–big layouts:(400 / 100 + 100 + 200)populations; Std. Dev


Table 16 Number of cases withsolutions of layouts (out of 10) Case C Case H (CoSAR) Case H (CoSAR)

small+medium+big small+medium+big small+medium

Small layout 10 10 10

Medium layout 0 10 10

Big layout 0 7 –

different possibilities for each bay: two diagonals, one diag-onal from bottom left to upper right, or one diagonal fromupper left to bottom right) result in a total of 81 differentstructures. Between 18 and 21 bars must be sized in theseconfigurations.

Two different solution strategies (CoSAR and standardpanmictic optimization) were explored in the context of thistest problem. Additionally, the CoSAR strategy applied tosmall and medium layouts only is included in the compari-son. A maximum number of 4 million function evaluationswere allowed in a simulation with a population size of400 and a K parameter set to 200. Since the optimal non-dominated front for this problem was large, a referencepoint for the hypervolume metric was set as (7500, −10).The large layout size contributes to greater search difficulty,and the subpopulation size was accordingly set to ensurehigher diversity. Subpopulation sizes of 200 for the largelayout and 100 for each of the small and medium lay-outs were considered: this results in a total population of400. The remaining parameters were kept at values as inSection 3.

The non-dominated solutions obtained by accumulatingthe best solutions of the ten independent runs are repre-sented in Fig. 21. The consolidated front with the layouttypes clearly distinguished is shown in this figure.

From a total of 93 structures (3, 9 and 81 belonging to thesmall, medium and large layouts, respectively), only five arepresent in the non-dominated front (two belonging to smalllayout, two belonging to medium layout and one belongingto the large layout), and are shown in Fig. 20. They rep-resent the optimal solutions that yield the lowest weight foreach reliability index value, according to the non-dominated

sets shown in Fig. 21. The extremes solutions of each struc-ture, as well as other representative designs, are detailedincluding cross-section types in Tables 13, 14 and 15, corre-sponding to small, medium and large layouts, respectively.Figure 20 shows also the numbering of the bars representedin these tables.

Figures 22, 23, 24 show the hypervolume average, bestand standard deviations from 10 independent numericalsimulations. Table 16 presents the number of appearancesof each layout size for the different strategies, and Table 17show the numerical values of hypervolume at the end ofevolution.

The standard panmictic evolutionary algorithm fails toyield acceptable results. None of the ten independent runswas able to locate solutions corresponding to the larger lay-out. Additionally, this approach also failed to locate themedium layout structure (Table 16). Only solutions of thesmall layout size appear in the final population: even fromthe early stage of the algorithm evolution, these topologiesdominate the population. This is also the reason for the lowvalues of the average, best, and standard deviation hyper-volume measures for this strategy shown in Figs. 22, 23, 24,and documented as final values in Table 17.

The CoSAR strategy locates solutions of the three layoutsizes, and was able to correctly identify solutions belongingto the small and medium layouts in all the runs, and the largelayout in seven out of ten of the outcomes (Table 16). Whencomparing the results obtained using the CoSAR strategywith only small and medium layouts (using identical prob-lem parameters) against those from the CoSAR strategywith also the biggest layout size, all the gain in hypervol-ume average (Fig. 22), hypervolume best (Fig. 23), and

Table 17 Hypervolumestatistics; combined small +medium + big layouts outcomes

After 4.000.000 Hypervolume Hypervolume best Hypervolume Hypervolume

evaluations Average (over all run) standard in accumulated


Combined small+medium+big 122916.6 122951.1 103.3 122951.1

Case C (100 + 100 + 200)

Combined small+medium+big 125103.6 125495.1 372.6 125655.17

Case H (CoSAR) (100 + 100 + 200)

Combined small+medium 124975.6 124978.8 1.5 124985.9

Case H (CoSAR) (200 + 200)


in hypervolume standard deviation (Fig. 24) is due to thecontributions from the large layout size problem. A sim-ilar conclusion applies to their final values, as well as tothe hypervolume in the accumulated final front (Table 17).Given that a larger combinatorial problem is included in theanalysis, the evolution of the algorithm is slowed, as shownin the results (Figs. 22 and 24).

5 Discussion

From the results documented in Sections 3 and 4, somegeneral conclusions about layout size, redundancy and dis-crete cross-section type sizing, and their relations with theminimum weight truss reliability can be drawn.

The layout size is the factor that delineates the three non-dominated front zones, each belonging to one layout size.The small layout zone defines the lower weight solutions,while the medium and large layouts define the increasingweight solutions. This is to be expected from a qualitativepoint of view as well; an increase of the layout size resultingin an increase in weight of the optimal designs. Never-theless, the procedure based on the postulates describedin Section 2 and in the method described in Section 3,allows us to define it from the quantitative point of view,and which is ultimately the objective of engineering prac-tice. For example, an overlap in the non-dominated frontsolutions of the small and medium layout structures, andnon-overlap between the medium and large layouts (as inFig. 21), is neither intuitive nor is it obvious—numericalsimulations alone yield those results more precisely.

With regards to redundancy, all optimal designs withdifferent layout sizes share a common attribute—the mostredundant solutions of each layout size dominate the finalfront (with scarcely a few exceptions in the small andmedium layout sizes, both belonging to the least reliablesolutions of each layout size). Under the assumptions givenin the PCSS method (Section 2), the increase in redundancyseems to be beneficial to the minimum weight truss reliabil-ity: despite the increase in length of the bars, these structuresare simultaneously the lightest and most reliable.

Finally, the cross-section types define the best designswithin the choice of layout size and redundancy.

The proposed CoSAR strategy has been shown to beeffective in solving large combinatorial design optimizationproblems. Multiobjective evolutionary algorithms are pow-erful tools that enable solutions to complex problems, butare sensitive to the selection of different problem parameterssuch as mutation rate and the size of the population. In theapplications studied in this work, (particularly Section 4),the convergence characteristics are governed by the mostchallenging big layout geometry. A suitable population sizeto ensure population diversity, and a reunification parameter

that does not impede subpopulation evolution are required.Without this consideration, subpopulations (representinglayout sizes) may not correctly converge to their optimalfronts. If premature reunification is allowed, the subpopula-tion that is the most converged could dominate the solutionand suppress the discovery of the true optimal front.

The proposed evolutionary optimization algorithm hasall the advantages and disadvantages associated with thesekind of methods. When considering the scalability of theaffordable problems, the CoSAR strategy is limited by theproblem of the biggest layout size, which defines the biggesthandled search space. If the fitness function evaluationdepends on numerically heavy design analysis tools (highcomputational cost), then the CoSAR strategy is compatiblewith associated additional resources designed to facilitatethe problem resolution in such cases: parallelization of thesearch and/or use of solver surrogate methods (e.g.: neuralnetworks, radial basis functions, support vector machines,etc).

In this problem a S-metric (or hypervolume) based mea-sure has been shown an effective procedure for determin-ing the subpopulation convergence (stopping criteria) inevolutionary multiobjective algorithms.

6 Conclusions

An ensemble of best solutions for structural bar trusses interms of Pareto non-dominance criterion was obtained usinga new co-evolution strategy (CoSAR). For each reliabilityindex it was possible to define the structural designs thatgives the lowest weight (or also it could be interpreted asfor each structural weight it is possible to define the designthat gives the higher reliability) considering as factors ofthe analysis: layout size variations, redundancies, and dis-crete cross-section type sizing. Layout size has been shownas a major factor governing the main influence zones ofthe Pareto non-dominance optimum designs. Redundancyhas been shown as beneficial to the minimum weight trussreliability, as the non-dominated structural designs iden-tified corresponded to the most redundant structures in eachlayout size.

The CoSAR strategy should be explored in other scienceand engineering related design problems where the popula-tion diversity enhancement is a key issue for success. Ananalysis of its performance in other fields could providemore light about the practical use and generalization of thismultiobjective approach.

Acknowledgments This research was supported by a mobility grantof the Agencia Canaria de Investigación, Innovación y Sociedad dela Información – ACIISI (boc-a-2010-119-3494), cofunded in 85% by


the European Social Fund (ESF). Also computational resources aresupported by the project UNLP08-3E-2010 of the Secretaría de Estadode Universidades e Investigación, Ministerio de Ciencia e Innovación(Spain) and FEDER. Dr. Varun Sakalkar is gratefully acknowledgedfor valuable advice and discussion.

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Truss topology optimization for mass and reliability considerations—co-evolutionary multiobjective formulations