Automation in Construction xxx (2013) xxx–xxx

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Automation in Construction

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Flexible optimum design of a bracing system for façade design usingmultiobjective Genetic Algorithms

James N. Richardson a,d,⁎, Guy Nordenson b, Rebecca Laberenne b,Rajan Filomeno Coelho a, Sigrid Adriaenssens c

a BATir — Building, Architecture and Town Planning, Brussels School of Engineering, Université libre de Bruxelles, 50 avenue Fr. D. Roosevelt, CP 194/02, 1050 Brussels, Belgiumb Guy Nordenson and Associates Structural Engineers LLP, 225 Varick Street 6th Floor, New York, NY, USAc Department of Civil and Environmental Engineering, Engineering Quadrangle E332, Princeton University, Princeton, NJ, USAd MEMC — Mechanics of Materials and Construction, Faculty of Engineering Sciences, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium

⁎ Corresponding author at: BATir — Building, ArchitectSchool of Engineering, Université libre de Bruxelles, 50 ave1050 Brussels, Belgium. Tel.: +32 26502169.

E-mail address: jrichard@ulb.ac.be (J.N. Richardson)1 Topology in this context refers to the connectivity of

by means of structural components such as bars and be

0926-5805/$ – see front matter © 2013 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.autcon.2012.12.018

Please cite this article as: J.N. Richardson, etAlgorithms, Automation in Construction (2

a b s t r a c t

a r t i c l e i n f o

Article history:Accepted 13 December 2012Available online xxxx

Keywords:Façade bracing designMultiobjective optimizationTopology optimizationGenetic AlgorithmsStructural optimizationSteel braced systems

X-bracing systems are widely applied in structural design to limit deflections and guarantee stability. Efficientdistribution of bracing over a structure is an important concern and often based on intuition and previous expe-rience. This paper presents a topology optimization procedure for cable bracing of the hanging steel façade of anewmuseum in theUnited States. In this procedure the use of amultiobjectiveGenetic Algorithm allows forflex-ibility during design modifications and accounts for uncertainty of deflection constraint values. The presentedmethod achieves practical solutions to a series of cost minimizing problems, giving the designer a range of opti-mal bracing configurations which can be selected in response to the continuously changing structural and archi-tectural requirements throughout the design process.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

X-bracing systems are some of the most widely applied structuralsystems, particularly in steel construction. Schodek [1] and Taranath [2]describe various bracing systems for frame structures includingX-braced, V-braced, K-braced and Chevron-braced (inverted V) systems,of which X-bracing is the most common. One of the main difficultieswhen designing braced systems is the choice of the location of the brac-ing components. This selection is often based on intuition and previousexperience. While an approach based on experience is invaluable, itdoes not necessarily produce the best results. The cost associated withthese bracing systems lies primarily in the connections. Reducing unnec-essarily large numbers of bracings is thus of great interest to make adesign cost-effective. The effective placement of the necessary amountof bracings constitutes a topology1 optimization problem. This topologyoptimization problem is discrete rather than continuous: the possiblebracing locations are limited to a set of predefined positions (such asthe openings between regularly spaced columns and floors) andpredefined cross-sectional profiles usually from standard catalogs ofsections.

ure and Town Planning, Brusselsnue Fr. D. Roosevelt, CP 194/02,

.nodes in the structural modelams.

rights reserved.

al., Flexible optimum design013), http://dx.doi.org/10.1

Topology optimization methods for discrete structures usuallyrequire metaheuristic approaches such as Simulated Annealing [3], AntColony Optimization [4] or Genetic Algorithms (GAs) [5–7]. GAs for fa-çade design have been used to maximize thermal and lighting perfor-mance of buildings [8]. Only a few researchers have used topologyoptimization for braced frame structures. Mijar et al. [9] developed acontinuum topology optimization formulation with hybrid Voigt–Reussmixing rules for conceptual design of frame bracing layout. Liang et al.[10] and Liang [11] presented a performance based optimization tech-nique with continuum topology optimization to study optimal designsof bracing systems for steel building frameworks. However, thesemethods can only approximate global optima of discrete problems.Academic investigations into discrete bracing topology optimization in-clude Kaveh and Shahrouzi [12] who combined a graph theory approachwith a discrete optimization procedure to optimize braced frame sys-tems. Kaveh and Farhoodi [13] explored an ant system for layout (sizingand topology) optimization for X-bracing of steel frames. Baldock andShea [14] used a genetic programmingmethod for topology optimizationof bracing for steel frames. The methods mentioned above were onlyapplied in a limited academic sense and do not address the complex,ever changing requirements during the design process.

For designers it is often unclear at the initial design stagewhat struc-tural components will be used in the final structural system, what theirdimensions will be, and what the constraint values (such as maximalallowed deflections) will be. However, the largest gains from structuraloptimization are made in the initial design stages, as Luebkeman andShea [15] point out, when the greatest uncertainty exists. Therefore,

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NON-STRUCTURAL SUPPORTFOR CLADDING

Fig. 1. Hanging façade system.

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to be effective, the optimization process needs to account for possibledesign and constraint changes.2 One method of taking these uncer-tainties into account is to simplify the problem's basic geometry andto define the uncertain constraints as additional objective functions.The resulting range of solutions allows the designer to select an appro-priate solution as the design changes over time. This aim can beachieved by a so-called Multiobjective Topology Optimization (MTO)approach.

Population based stochastic methods, such as GAs, are also verywellsuited to multiobjective problems [16], since a number of individuals(population) are considered at any given point in the optimization pro-cedure. This aspect is consistent with the notion of Pareto optimality inwhich a number of non-dominated (i.e. ‘best compromise’) solutionsmake up an optimal set (the Pareto optimal set). The combination ofmultiobjective optimization with GAs has been successfully applied inother fields of structural optimization (an overview can be found in[17]), while few papers [18–20] present MTO of discrete structures;MTO of bracing systems is not present in the literature.

In this paperMTOwill be implemented in the design of a hanging fa-çade system for a new museum in the United States. This work is theresult of a scientific and industrial collaboration between the authorsand represents a novel approach to X-bracing design for steel bracedframes. Several studies have been carried out comparing the perfor-mance of various Multiobjective algorithms in structural optimization[21–23].

The remainder of the paper is structured as follows: following an ex-planation of the structural system for the hanging façade (Section 2),Sections 3 and 4 respectively define the problem and present themultiobjective optimization approach. Next an explanation of the com-putation procedure (Section 5) and the resulting topological designs(Section 6) are presented. The paper concludes with a discussion ofthe implementation and future development of themethod (Section 7).

2. Description of façade structural system

For the design of themuseum, the façade design posed amajor chal-lenge. The structural design of the façade arose from three primaryarchitectural objectives: to create a three-tiered sawtooth profile for

2 Design changes occurring at a later stage can significantly affect the optimal solu-tion. The optimization procedure should take this into account. This observation is par-ticularly valuable for constraints which exclude solutions considered unfeasibleoutside the given bounds, yet may be feasible if these bounds change only slightly.

Please cite this article as: J.N. Richardson, et al., Flexible optimum designAlgorithms, Automation in Construction (2013), http://dx.doi.org/10.1

the façade; to maintain a continuous atrium between the façade andthe interior floors of the building; and to keep this atrium space freeof structure such that the façade is independent and free-spanningbetween the ground and roof.

The structure is a four-sided system consisting of three levels of hor-izontal trusses forming the sawtooth profile which are hung by verticalcables from the fifth floor of the building (Fig. 1). The structure issupported at the base only for lateral loads. A hung system is more effi-cient than a bottom-supported system because it reduces dead load de-flections and places the primary support members in tension ratherthan compression, allowing them to be slender. It also effectively pre-tensions the vertical members such that they are able to support com-pression loads due to wind and seismic effects while remaining slender.

While this system relies on pre-tensioning of the vertical hangers,it is not a cable wall in the sense that it does not rely on the tensionstiffness of the cables to resist lateral loads. Instead, the horizontaltrusses work in combination with the vertical and bracing cables toresist wind pressures on the façade. When wind pressure is appliedto one face of the building, steel mullions transfer wind loads to thethree horizontal trusses (Fig. 2).

These horizontal trusses in turn span across the face of the façade totransfer the wind loads to the orthogonal walls of the façade. The verti-cal cables and X-bracing in the orthogonal walls act as inverted bracedframes to transfer the wind shears up to the fifth floor of the buildingwhere they are taken into the concrete cores of the building. In Fig. 3the corner detail is shown for the façade structure.

3. Façade bracing problem definition

The complex 3D problem can be simplified as a series of four, inde-pendent, two dimensional problems shown in Fig. 4. The main purposeof the X-bracing is to provide in-plane stiffness under wind loading ofthe perpendicular façades as described in the previous section. Thebraced systems are supported along the top edge by pinned supportswhile each vertical hanger is restrained horizontally at the bottom end.

The North façade (Fig. 4(a)) layout is symmetric, as are the possibleX-bracing locations. On the West façade (Fig. 4(b)) two hangers areinterrupted in themiddle tier resulting in an openingwhere no bracingcan be located, and must be braced with two pairs of rigid bars underthis opening. The South and East façades (respectively Fig. 4(c) and(d)) the continuity of several of the hangers is interrupted, creatingopenings. The vertical hangers between the tiers are modeled as com-pressible (cable) truss elements, because they are assumed to be

of a bracing system for façade design using multiobjective Genetic016/j.autcon.2012.12.018

INVERTED BRACE FRAMEON SIDEWALLS

HORIZONTAL TRUSSWIND PRESSURE

Fig. 2. Façade structure wind loading principle.

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pretensioned by the dead load of the façade. The four horizontal trussesare modeled as continuous beams. The number of façade bracings is thegreatest factor influencing the cost of the bracing system, the first objec-tive function considered. The effectiveness of a given number of brac-ings to resist deflections of each tier (the second objective function) isa function of their positions. Together, these two factors can be repre-sented using a single type of variable: the topology variables, and bothobjectives set for the same problem: a multiobjective topology optimi-zation problem.

4. Approach: Multiobjective Topology Optimization using GAs

In order to obtain a range of optimal solutions, a multiobjectiveapproach based on the multiobjective GA (MOGA) introduced byFonseca and Fleming [24] is proposed. The MOGA optimization loop iscoupled to a Finite Element Analysis programwhich generates the nec-essary responses (Fig. 5). The structural domain comprised of a set ofnodes, a set of supports, a set of loads, and the non-variable structuralcomponents. In the three asymmetric problems lateral loading is ap-plied from both directions consecutively. The worst performing of

Fig. 3. Isometry detail of corner of hanging façade system.

Please cite this article as: J.N. Richardson, et al., Flexible optimum designAlgorithms, Automation in Construction (2013), http://dx.doi.org/10.1

these two cases is used to evaluate the fitness of the topology design.A ground structure approach is used to define the upper bound of thetopological search space.

4.1. Multiobjective structural optimization

The optimization problem with k objectives can be formulated asfollows3: given some performance criteria fi(x)(i=1,…,k) which de-scribe the performance of a structure, find the vector x∗=[x1∗,…,xn∗]which minimizes the vector function:

f xð Þ ¼ f 1 xð Þ;…; f k xð Þ½ �

subject to gil≤gi(x)≤gi

u the constraints of the structural behavior. x isthe vector of design variables, where gi

u and gil represent respectively

the upper and lower bounds of the feasible solution space. In discretetopology optimization x is the vector of (discrete) topology design vari-ables. In the MTO sense optimization is most commonly described interms of Pareto optimality.

4.2. Pareto optimality

In single objective optimization one solution can be defined as the‘best’ solution in that it is minimal (or maximal) with respect to allother solutions.Whenmultiple objectives are considered, the definitionof optimal solutions is less intuitive. In order to maintain the vectornature of solutions, the concepts ‘Pareto optimal’ and ‘Pareto front’ areused. Pareto optimality of a set of solutions is defined in terms ofnon-dominance. A design variable vector x* dominates x if, for all i:

f i x�� �≤f i xð Þ ð1Þ

and there exists a j such that (fj(x∗)b fj(x)). The Pareto optimal set Ptrue isthen defined as the set of vectorswhich are non-dominated and the Pare-to front PF is the set of objective function vectors corresponding to Ptrue.

These notions are illustrated in Fig. 6. The concept of Pareto opti-mality is used to define solutions which are equivalent, dependingon the importance the designer gives to each objective function.

4.3. Multiobjective Genetic Algorithm

TheMOGA is a special class of GA, and as such uses a series of biolog-ically inspired processes applied to a population of possible solutions, asillustrated in Fig. 7. As a single objective GA, the MOGA uses a chromo-some representation of the topology variables to define any given de-sign. The chromosome is a string of concatenated entries from the

3 For k=1 the formulation describes a single objective optimization problem. Forthe case k>1 the formulation describes a multiobjective optimization problem.

of a bracing system for façade design using multiobjective Genetic016/j.autcon.2012.12.018

¸(a) North facade ¸(b) West facade

¸(c) South facade ¸(d) East facade

Fig. 4. Simplified models of the four façades: the gray shaded areas represent zones where no bracing can be located. The bracing in the West façade (under the gray zone) arecompulsory design elements.

4 J.N. Richardson et al. / Automation in Construction xxx (2013) xxx–xxx

design variable vector. The initial population (whose size is controlledby the population size parameter) is randomly seeded as a populationof strings containing ones and zeros.

Each ‘1’ in the string indicates the presence of a pair of bracing cablesat a particular location, shown in Fig. 8 as a vector of design variables. Inthis figure the number of design variables (length of the chromosome)

FEM

Optimizer(Multiobjective

Genetic algorithm)

parameters

responses

Fig. 5. Basic algorithm scheme: a multiobjective Genetic Algorithm is coupled to a Fi-nite Element Analysis (FEA) program.

Fig. 6. An illustration of the principals of non-dominance and Pareto front.

Please cite this article as: J.N. Richardson, et al., Flexible optimum designAlgorithms, Automation in Construction (2013), http://dx.doi.org/10.1

corresponds to the number of possible positions (size of the groundstructure) for pairs of bracing cables in the façade. In the design of thestructures, symmetry was a desirable feature. As a result the numberof design variables was reduced where symmetry could be applied. Inthat case each entry in the chromosome refers to the presence or ab-sence of two pairs of cables, located symmetrically about a vertical axis.

The variable values are combined with non-variable model data inthe form of a finite element model. The finite element package FEAP[25] is used for a non-linear structural analysis. Upon evaluating the fit-ness of the systems, a ranking and selection of the current population iscarried out, followed by cross-over and mutation of individuals to pro-duce the next generation in the MOGA. The MOGA used differs from asingle-objective GA in the way it evaluates the relative fitness of solu-tions. The algorithm involves the ranking of individuals at each itera-tion, with non-dominated individuals receiving the highest rank. Acomplete overview of the method can be found in [17]. For these calcu-lations the DAKOTA MOGA method [26] was used. This method per-forms Pareto optimization using a metric tracker to evaluate theconvergence of the algorithm. This tracker evaluates threemetrics asso-ciated with consecutive Pareto fronts, described in detail in the abovereference.

5. Calculation procedure and parameters

In this particular project the designers faced the challenge of limitingthe horizontal deflection relative to the height of each tier, using a brac-ing system. This problem can be cast as an unconstrained, bi-objectiveproblem with the following objective functions:

minx

f xð Þ ¼ f 1; f 2ð Þ

where f 1 ¼Xni¼1

aixi

f2 ¼ maxd1j jh1

;d2−d1j jh2

;d3−d2j jh3

� �

ð2Þ

where f1 is the cost objective function, x is the variable vector of lengthn, ai is a weighting coefficient related to the grouping of componentsbased on symmetry, and xi is the topology variable associatedwith brac-ing(s) i. f2 is the relative tier deflection objective function, hj is theheight of tier j and dj is the measured deflection of tier j from rest posi-tion (Fig. 9).

of a bracing system for façade design using multiobjective Genetic016/j.autcon.2012.12.018

0 1 1 1 0 1 1 1 1 1 ... 1

0 0 1 1 1 0 1 1 1 0 ... 1

1 1 1 0 1 1 0 1 0 1 ... 0

1 1 1 1 0 1

1 0 1 1 ... 01 1 1 1 1 1

0 1 0 0 ... 0

0 1 1 1 0 1 1 1 1 1 ... 1

0 0 1 1 1 0 1 1 1 0 ... 1

1 1 1 0 1 1 0 1 0 1 ... 0

Initiation

Generate/update

Evaluate fitness ofeach individual (FEM)

population

Selection

Mutation

Crossover

Convergence?

End

0 1 1 1 0 1 1 1 1 1 ... 1

1 1 1 0 1 1 0 1 0 1 ... 0

0 1 1 1 0 1

1 1 1 1 ... 11 1 1 0 1 1

0 1 0 1 ... 0

Y

N

Fig. 7. MOGA used in the façade bracing topology optimization. Several biologically inspired processes are used to optimize a population of structures. The binary string manipu-lation is demonstrated next to each operation.

0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1

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The algorithm is judged to have converged once the value of themetric tracker does not change significantly for 10 generations. Thefour façade systems each contain varying numbers of design variablesand varying degrees of symmetry. A summary of the Genetic Algorithmparameters can be found in Table 1. TheMOGA parameters are adjusteduntil, for 10 runs of the problem, a majority of the solutions convergedto the minimum solution. Due to this verification process several itera-tions may be necessary to achieve good results. The population size pa-rameter represents the number of generated designs in the initialpopulation of the MOGA and may vary after the first iteration of theMOGA.

The computational expense of the proposed approach is significant.Depending on the problem size the number of function evaluations canrun into the tens or even hundreds of thousands. Large computer clus-ters are best suited for these computations, since parallel computingoptions can be exploited.

Fig. 8. The relationship between chromosome and the bracing topology.

h1

h2

h3

d1

d2

d3

6. Results: a catalog of optimal solutions

The combined Pareto frontsPFN ,PFW ,PF S andPF E4 are shown in

Fig. 10. Each front corresponds to a set of best compromise solutions fora specific façade. Since the four systems work together in the final de-sign, it is necessary to choose solutions by grouping North, East, SouthandWest façades to comprise one structural design. Grouping of the so-lutions is based on the worst performing front (façade), in this case theNorth façade for most of the solution space. For each solution on theworst performing front, corresponding solutions from the other threefronts can be found, as demonstrated in the inset for region IV inFig. 10. For a given required deflection, if solution A in PFN meets thisconstraint value, it is selected along with B in PFW , C in PF S and D inPF E , since these are the least cost solutionswith atmost as little deflec-tion as A. Similar solutions can be grouped for any of the other regions.For regions V to XI the configuration groupings are shown in Table 2.The total cost of each solution is denoted as f1tot and is simply the sum

4 The subscripts refer respectively to the North, West, South and East façades.

Please cite this article as: J.N. Richardson, et al., Flexible optimum designAlgorithms, Automation in Construction (2013), http://dx.doi.org/10.1

of the costs in the group. The relative deflections are scaled to allowfor easy choice of solutions to meet changing constraint values. Thedominant relative deflection value is denoted as f2tot and has been scaledfrom their original values. The North and West façades where symme-try is enforced contrast the South and East façades, where the solutionsdisplay sole asymmetry. Surprisingly, the asymmetric solutions performbetter than the symmetric solutions: Pareto optimal solutions for theSouth and East façades dominate the Pareto optimal solutions of theirNorth and West counterparts. The reason behind this relates to the bi-nary nature of the variables chosen, and is a remarkable result,explained in a recent paper by the authors [27].

Fig. 9. Deflected shape in plane.

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5e-06

1e-05

1.5e-05

2e-05

2.5e-05

3e-05

10 15 20 25 30

Rel

ativ

e de

flect

ion

(m/m

)

Cost function [-]

North elevation PFNWest elevation PFWSouth elevation PFSEast elevation PFE

I

V

ADC B

IV

III

II

VIVII

XI

IXVIIIX

Fig. 10. Combined Pareto fronts: configuration grouping.

Table 1Genetic algorithm parameters. * indicates built in DAKOTA methods.

Parameter N façade W façade S façade E façade

Number of variables 24 22 27 26Population size 800 800 800 800Cross-over rate 0.9 0.9 0.9 0.9Mutation rate 0.2 0.2 0.2 0.2Fitness type Domination count*Replacement scheme Below limit* of 6Cross-over type Multi-point binary*Mutation type Offset normal*

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Once the catalog of solutions is compiled the designers are able touse it throughout the design process to achieve the most cost-effective bracing layout for a given set of circumstances. For example,consider the hypothetical design adjustments shown in Fig. 11. Assumethe initial design calls for a maximal tier deflection of h

200, which, givensome initial design components, can be met with configuration V.

V

Fig. 11. Hypothetical design evaluation bas

Please cite this article as: J.N. Richardson, et al., Flexible optimum designAlgorithms, Automation in Construction (2013), http://dx.doi.org/10.1

During the initial design solution V,with a cost of 62 units, is used. How-ever, at some point during the design process the tier deflection con-straint is made more stringent so that f tot2 ¼ h

400 is required. Thedesigners are facedwith the option of adding additional bracing, or car-rying out another time-consuming optimization study. With the rangeof solutions given above, however, solution XI f tot1;X ¼ 1

2 ftot1;V

� �can be

chosen directly, the most cost effective (46 extra units) way possibleto add additional bracing and double the stiffness. Further in the designprocess the engineers decide to increase the thickness of the brass clad-ding panels to improve their dynamic behavior. The increased dead-load on the hanging façade reduces lateral deflection so that the lessX-bracing is required to meet the tier deflection limit value. The de-signers are now able to select solution IX, costing 92 units, while stillmeeting the deflection requirement.

7. Conclusions and further work

This paper presents the topology optimization method developedand used for the preliminary design of the bracing system for the hang-ing façade of a new museum in the United States. The approach usesmultiobjective Genetic Algorithms to find a series of best compromise(Pareto optimal) solutions. The value of this method lies in its flexibilityto provide solutions, allowing the designers to select optimal solutionswhen constraints change and modifications to the structural systemoccur. Since themain cost of bracing systems lies in the connections, re-ducing the number of bracings required results in significant costsavings.

The presented method could be extended to allow for greater free-dom in the possible bracing locations. For example, bracing could spanmultiple tiers or across multiple columns. Other bracing typologies,such as V, K or Chevron systems could be included as design variables.These extensions would increase the number of topology variables,but not fundamentally change the problem formulation. This adaptivityis one of the strengths of MOGAs. This method could easily be extendedto more than two criteria, making an even more general approach pos-sible. Furthermore the method is general enough to be applied to a full3Dmodel of the structure, taking all structural components of the hang-ing façade into account. For example in the tall building design of theJohn Hancock Center (Chicago, Illinois, 1969) the diagonal bracing in

XI

IX

ed on changes in design requirements.

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Table 2Several solutions in the ‘catalog’.

7J.N. Richardson et al. / Automation in Construction xxx (2013) xxx–xxx

one façade tends to reduce the shear lag in the perpendicular facades[28]. The solution presented in this paper decouples the structural be-havior of the facades although there is interaction. Bracing in one facade

Please cite this article as: J.N. Richardson, et al., Flexible optimum designAlgorithms, Automation in Construction (2013), http://dx.doi.org/10.1

may have beneficial effects on the perpendicular façades. This phenom-enon has not been investigated in this paper and presents an avenue forfurther research. Other multiobjective methods such as multiobjective

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8 J.N. Richardson et al. / Automation in Construction xxx (2013) xxx–xxx

ant colony optimization [29] andmultiobjective particle swarm [30] (orother classes of solution methods) may be more efficient than theMOGA to solving this problem.While these have not been investigated,a comparative study of the efficiency of various methods may lead to amore practical method for this problem.

Finally the proposed approach should be investigated within thebroader context of multidisciplinary design optimization andmultidisciplinary collaborative design [31], both very promising in-struments for structural design.

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