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Performance BasedExploration OfGenerative DesignSolutions Using FormexAlgebraAnahita Khodadadi and Peter von Buelow

issue 3, volume 12international journal of architectural computing

322

Performance Based Exploration Of GenerativeDesign Solutions Using Formex AlgebraAnahita Khodadadi and Peter von Buelow

This paper illustrates the use of a computational form-finding method called ParaGen which aids the designerin the exploration of arrays of good solutions.Althoughthe method is guided by a multi-objective optimizationprogram, the goal is to promote the exploration of thesolution space based on designer selectedcombinations of performance objectives.The digitalform generation of the bridges is carried out usingFormian, a program which uses Formex algebra todescribe a wide array of geometric configurations.Thisform generation is linked to structural simulation anddesign software (STAAD.Pro) to determineperformance values. Finally, ParaGen is used to build adatabase of all solutions and guide the explorationbased on performance values. Using this database, bothvisual and numeric characteristics are explored.

1. INTRODUCTION

Designing a form is broadly considered as a creative procedure withinwhich certain goals are purposefully sought.The designer usually sets someparameters such as topology or geometry of forms, material properties,architectural functions and load cases.Then, the design goals are generallydefined by means of some criteria and the extent to which solution(s) canmeet them. In order to expand the designer’s perspective, facilitatemodifications of the forms and explore more possibilities of appropriatesolutions in the early stages of design, the couple of parametric formgeneration and evolutionary optimization techniques can be applied [1].

In this research work two methods of form exploration are comparedto study the relation between the geometry and structural performance ofspecific types of spatial structures.The first is based on traditional methodsof design exploration including physical modeling and the second is derivedfrom the coupling of a parametric formulation of the design and theapplication of a genetic search process.To demonstrate the application ofcomputational method in professional context, the design of a trussedroadway bridge is used, and a comparison is made between solutions foundusing the ParaGen exploration method and a successful built bridge.Furthermore, the solutions are compared to results achieved by designstudents through model studies to show the contribution of this method inan academic context.

A case study of a two-lane truss bridge is opted for in this paper, since itallows one to consider structural performance as well as other designconcerns like visual characteristics, compatibility with the surroundingenvironment and constructability.The truss bridge is based on an actualdesign (shown in figure 1) with a width of 8 m and span of 48 m, and isexpected to be composed of two 2D trusses on the two sides that arebraced together laterally.The parametric variables allow for the trusses tobe situated either above or below or both above and below the deck.Thetrusses can also have positive, negative or zero curvature (figure 1).

� Figure 1:A typical

form of the truss

bridges that are to be

explored.

323Performance Based Exploration Of Generative Design Solutions Using Formex Algebra

The questions are raised at the beginning of this study: what are appropriatesolutions regarding the structural performance, visual appearance and adesigner’s individual preference, and how can such a set of suitable solutionswith different geometrical and structural features be found.These questionsinvolve the subjectivity of the design process which has been the mainconcern of recent works considering interactive evolutionary methods ofform exploration in the conceptual design phase [2].

The parametric configuration of the bridge is first processed usingFormex algebra and Formian programming software [3].The geometricalconcepts used within this step are described in the following section. Next,having produced a geometry, a DXF file is sent to STAAD.Pro for structuralanalyses and evaluation.The obtained information including graphicdepictions is stored in a database which is linked to a visual explorationinterface. In the next step pairs of bridges are selected based on the designobjectives to breed further bridges. In selecting a breeding pair, a fitnessfunction is used to dynamically create a population of the desirablesolutions out of the entire pool of generated forms.The newly generatedtruss bridges emerge through the iteration of this cycle to yield an arrayincreasingly more suitable solutions.This process is described in detail insection 3 (see figure 5). Finally, a comparison with the Foster Bridge, asimilar, successful constructed bridge over the Huron River in Michigan, andalso the results of form exploration through physical modeling of the trussbridges, accompany the computational study at the end.

2. FORMEX CONFIGURATION PROCESSING

The configuration processing of truss bridges is described throughgeometrical concepts of Formex algebra [3]. Formex algebra is amathematical system that allows a designer to define the geometricalformulation of forms through concepts that effect movement, propagation,deformation and curtailment [4].

The creation of any type of spatial structure, such as space trusses,domes, vaults, hypar shells, polyhedric and free forms, can be carried out byusing this mathematical system and its associated programming languageFormian.

In the first step, constant parameters, variables and also their acceptableintervals, described in table 1, are set.Additionally, sketches of trusspatterns, illustrated in figure 2, are provided to assist the designer choosingthe desired pattern more conveniently.

The configuration of trusses can be processed using a 3-directionalreference system similar to the global Cartesian system.Truss elements anddeck members of the bridge are defined in terms of lines, and surfacesrespectively. Figure 3 shows some parts of a simple truss bridge given in anappropriate reference system and also the related parameters, used todefine such bridges.Trusses are expected to have curvature at the top and

324 Anahita Khodadadi and Peter von Buelow

bottom chords. Curved trusses can be created using the “pellevation”function which starts with a non-curved truss and deforms it by pushing theelements up or down, along the z axis, to the prescribed shape.The concept

� Table 1: Geometrical parameters on

which the configuration processing of

truss bridges are based.

� Figure 2:The truss patterns that are

used for the top and the bottom parts

of the bridges


Parameters Sign Constant/Variable Acceptable interval/ value

Span of the bridge L Constant 48 m

Width of the bridge W Constant 8 m

Frequency of geometrical modules along the length m variable m should be an even integer.of the bridge [4 to 20] is recommended.

Total rise of the bridge at the top part HX1 Variable [0.5 to 16] m

Total rise of the bridge at the bottom part HX2 Variable [0.5 to 16] m

The rise of the arch at the top truss X1 Variable [0.01 to 16] m

The rise of the arch at the bottom truss X2 Variable [0.01 to 16] m

The top rise in non-arched part of the truss H1 H1 = HX1 - X1 [0.49 to 16) m

The bottom rise in non-arched part of the truss H2 H2 = HX2 - X2 [0.49 to 16) m

Curvature sign of the top truss Sign1 Variable 1= positive curvature-1 = negative curvature0 = none

Curvature sign of the bottom truss Sign1 Variable 1= positive curvature-1 = negative curvature0 = none

Pattern of top truss Top Variable D1, D3, D5, D7, D9, D11, D13, D15, D17(see figure 2)

Pattern of bottom truss Bottom Variable D2, D4, D6, D8, D10, D12, D14, D16, D18(see figure 2)

of pellevation has been presented in a variety of ways that can be applied tocreate different forms. In this study,“circular barrel pellevation” is used, seeHofmann, 1999 [5].The configurations generated by Formian can beexported in DXF format and used in the next stage of study which includesa full structural analysis and design of members.

3. GENETIC ALGORITHM

Form exploration and traditional optimization can both be used in form finding,however, optimization generally is carried out to search for one single bestsolution and exploration seeks a set of significantly different good solutions.Exploration can be used at the early stages of design to study a wider rangeof possibilities for reasonably good or even unexpected solutions. Formexploration usually can be accomplished efficiently using parametric tools.Evolutionary computation methods provide means to search a range ofgenerated solutions in a directed way [1] [6].

Form optimization can be accomplished in different stages.Topology,geometry and determination of member sections.This can take placerecursively or all at once.Topology is the highest level at which forms canbe explored.An instance of a topology will have a specific geometry whichcan either be explored in a range under a single topology or linked todiffering topologies. Solutions can be sorted in a database which can beinspected visually in a relevantly easy way. Specific solutions of a certaintopology and geometry are further composed of members which can beoptimized as well, but with which designers usually have less directinteraction in terms of form finding. In this research work, form explorationis accomplished at topology and geometry levels.

� Figure 3:A part of a curved

bridge configuration and the

related parameters within a 3-

direcional reference system


A genetic algorithm, originally described by John Holland, is a search methodthat progresses through iterating cycles to find solutions that meet certaingoals [7]. Using mechanisms like recombination and mutation, goodsolutions may be found which are not anticipated by designers.Thesolutions which are inherently parametric, are described in terms of a list ofvariables which are analogous to genes on chromosomes.Thesechromosomes are bred to form children that inherit characteristics throughthe genes of their parents.

ParaGen uses a non-conventional genetic algorithm called a Non-Destructive Dynamic Population GA (NDDP GA). It incorporates HUX asdescribed above in the breeding step and incorporates a database tomaintain a general pool of all solutions found. ParaGen incorporates thefollowing steps:

1. The problem is described in terms of parametric variables: achromosome.

2. An initial pool of solutions is generated and stored in a database.3. A population of parents is dynamically pulled from the full solution

pool based on given criteria (the fitness function).4. Two parents are randomly selected from the dynamic population.5. A child is bred (HUX) from the selected parents.6. The chromosome (parametric values) of the child is translated into

a geometric solution.7. The performance of the solution is evaluated based various

simulation software.8. The resulting performance values along with the parametric values

are uploaded to the database. Images are also included and linkedto the solution.

� Figure 4: Half Uniform Crossover

(HUX):The characteristics of parents

1 and 2 are described in terms of

shapes.The child may inherit some

exact characteristics of parent 1 or

exactly those of parent 2 or a

combination of both through a

“breeding” of the genes.


Steps 3 through 8 form an iterative cycle which continues until satisficingsolutions are found.

ParaGen is designed to run using a parallel cluster of PCs linked to aweb server through the internet.The solution database and design interfaceare placed on the web server (the interface is a www site), and steps 3-5run on that web server. Each child is downloaded to a PC linked to theserver simply by connecting to the ParaGen website.

� Figure 5: Schematic of the ParaGen

cyclic method


In a traditional GA approach, any defective or poor performing solutionsare usually removed from the breeding population (killed off). However, inthe NDDP GA all solutions, both well performing and poorly performingsolutions are stored in the database. ParaGen simply stores all performancevalues and defers any ranking of these values to the moment of breeding(thus “dynamic” selection). By retaining data on all solutions, the designer isable to learn from ill solutions and increase the knowledge of what wouldmake a good solutions [8]. Recent studies have shown the value of sub-optimal solutions in the exploration process [9]. Furthermore, in case of anymodification of criteria, the poor performing solutions can also be re-considered and re-used in the breeding process to form new solutions.

ParaGen guided by the multi-objective performance criteria set by thedesigners, concentrates the generation of new solutions in the area of thesolution space defined by the fitness function of the GA, and the focus ofthe exploration becomes more defined.Within a productive exploration,thousands of solutions are evaluated by the program.The designer uses theParaGen web interface to filter and sort these solution based on anycombination of geometry or performance data. In this way, a wellperforming set of solutions is defined as a manageable quantity, which isreasonable for the designer to visually inspect and possibly make manualselections for further breeding.The web interface provides interactionbetween the designer and the form exploration system which eventuallyleads to choosing the final desirable solutions.The designer’s interaction canbe also based on a totally aesthetic preference.

The incorporation of a relational database into a genetic algorithm givesParaGen several advantages over traditional population bases generationalmethods [8].These include being able to:

• Store all solutions without duplicates;• Use multi-objective fitness functions with SQL queries;• Create dynamic parent populations;• Change search direction instantaneously;• Explore solution space with interactive search;• Graph parameters to determine Pareto trade-off sets;• Utilize parallel hardware for efficient computation.

The above points all enhance to the ability of ParaGen to function beyondthe range of traditional search and optimization methods when dealing withdesign exploration.

4. FORM EXPLORATION AND OPTIMIZATIONPROCESS

This section describes in more detail the 8 step process enumerated insection 3 as the ParaGen method. In step 1. Formian was used to describe awide range of truss bridge types and geometries. In the parametricformulation, 11 independent variables plus 2 additional values were used -(Total Top Height and Total Bottom Height) which are depended on others,but useful in formulating constraints.These 13 parametric input variables forFormian are listed in table 1.

For this trial, Span was set to 48 m and Width set to 8 m.These 13variables constituted the “chromosome” description of a solution which iseventually downloaded to Formian for processing into a geometry (step 6).This “chromosome” is bred from two parents in step 5 using a GAcrossover technique called HUX [10].The two parents are randomlyselected from a dynamic population (step 4), which is gleaned from the fulldatabase using a SQL query (step 3).This SQL query formulates the searchobjective or fitness function for the GA.


Once the input variables (the “chromosome”) are passed to Formian,Formian processes the variables to produce one instance of the parametricbridge geometry. Formian generates a visual perspective view as well as aDXF file which can be read by other simulation software. In this example afinite element analysis (FEA) was carried out using STAAD.Pro (step 7).Theprocess makes use of scripts written in each software used plus a generalWindows interface script (AutoHotkey) to automate the process. Steps 6and 7 are performed in parallel using a cluster of PCs.At the completion ofthe analysis, the original input data (the “chromosome”) plus all of theassociated performance data collected through the simulation software, plusany number of descriptive images and files, are all uploaded to the serverthrough the web site interface (step 8). On the server all numeric data isplaced in a database and tagged to the images.The ParaGen web site thenoffers a graphic window into this database by displaying the images andassociated performance values.The ParaGen web interface also allows thedesigner to sort and filter the displayed images and data in a variety of waysto enhance the exploration of the solution space. Figure 6 shows the query

� Figure 6: Display of solutions

comparable with the properties of

Foster Bridge.They show the variety

of possible solutions if the bridge was

to be design as the Foster Bridge.The

solutions are sorted according to their

weight from lighter to heavier and are

also represented in figures 7 and 8

where structural performance and

number of joints are compared.


boxes that allow the designer to display different ranges of the solutions. Inthis example, the solutions are filtered in a way that provides a suitablecomparison with the Foster Bridge (table 2, figure 9).This demonstrateshow the website can display a set of solutions for any particular list ofrequirements and preferences. In this example, the better final solutionshave fewer joints that contribute to better constructablity or they may belighter or have less deflection, indicating better structural performance.Allin all, the final decision will be made regarding the designer’s informedintuition and preferences.

In addition to solution displays, the ParaGen website provides graphs inwhich the designer can compare two specific properties and choose themost desirable solution. Figures 7 and 8, show graphs in which solutions aredistributed regarding the maximum deflection or total weight versusnumber of joints. For example the designer may look for a solution whichhas the least maximum deflection and least number of joints at the sametime. Or a designer may opt for a solution with greater weight but fewerjoints which provides better constructability. It is possible to click on thespecific chosen solution and find out more detail regarding the othergeometrical and structural properties or the diagram of the modal shape.

Properties Desired value

Number of panels = 8

Max deflection ≤ 3

Total weight ≤ 400

Number of joints ≤ 85

� Table 2: Filtering properties for a

series of solutions in the ParaGen

interface which compare to the Foster

Bridge

� Figure 7: Number of joints vs.

maximum deflection.The highlighted

solutions are depicted in figure 6


5. PHYSICAL MODELING

The physical modeling and testing of truss bridges was carried out in thecontext of a structures course at University of Michigan, School ofArchitecture. Groups of students were asked to make a 1/64 scale model ofa bridge using balsa wood, and to the best of their knowledge choose asuitable geometrical pattern to reach a high strength-to-weight ratio.Then,

� Figure 8: Number of joints vs. total

weight.The highlighted solutions are

depicted in figure 6

� Figure 9:The Foster Bridge in Ann

Arbor, Michigan. Build in 1876 by the

Wrought Iron Bridge Co.


the models were loaded until the breakage point and the structuralperformance was presented in terms of held load to weight ratio.The ratioand geometrical features were added to a data base and with a similarfashion to computational study, the web site allows to sort and filter theresults and offers a graphic window with all the images of the models. Infigure 10 a graph presents the load capacity to weight ratio of the bridgesversus number of joints. In this graph the capacity to self-weight ratio(structural performance) versus the number of joints (constructability) is

� Figure 10:A plot of load capacity to

weight ratio vs. number of joints

within the 219 bridge models.

� Figure 11:The display of

highlighted models in figure

10 chosen as probable

desirable solutions


plotted.The highlighted models in this figure show the best tradeoffcombinations of these two factors and may be considered as desirablesolutions (Pareto optimal).The web site also provides a graphic window thatshows the images and associated performance values of the models.

6. DISCUSSION

This paper has looked at the results of two different forms of designexploration. One which can be considered “traditional” in which solutionswere found by designers (students) using physical modeling techniques; anda second, computational approach based on what has been described as the“ParaGen method”.Although both studies were based on the samegeometric limits (two lane, 48 m span), the loading was approacheddifferently. In the physical models although a target was set at the same levelas the computational model (AASHTO HL93) in testing the load wasapplied until attaining ultimate failure, which was usually much higher. So,whereas the computational database contained solutions with a safecapacity of HL93, the physical models tended to range at much higher loads.Looking at the results it is not surprising to find that the form is effected bythe level of loading.

� Figure 12: Graph of number of

joints vs. total weight from the

computational exploration.


Figure 12 shows the computational counterpart to figure 10 which usedthe physical models.The lighter load level combined with the strongermaterial (steel vs wood) results in solutions with much less depth. In thecomputational version, the top bracing was removed at any height below 3m so as not to interfere with traffic.Thus the lightest solutions found in thegraph shown in figure 12 are simple pony truss bridges using only the deckfor lateral bracing. Figure 13 shows solutions on the Pareto frontier of thisgraph.These solutions are certainly good solutions, but the designer mayalso explore the forms of nearby solutions (near optimal) by simply clickingon the graphed dots.

� Figure 13: Pareto optimal set from

the graph in figure 12.


In studying the results of the physical tests a few other points can beobserved:

• The solutions with best load to weight ratio were lenticular(curved top and bottom) but tended to have more joints. Flatchord trusses tended to have fewer joints, but lower load toweight ratios.

• The quality of construction undoubtedly played a role in thestrength, which was not present in the computational models.

• Because of the longer time required per solution, the traditionalmethod with physical modeling was more limited in the number ofsolutions which could be considered.

• Using physical modeling makes the form finding process moretangible and comprehensible. It is therefore arguably better indeveloping an initial intuition, and sometimes it helps the designerto gain a firsthand sense of structural performance.

7. CONCLUSION

Coupling the Formex configuration processing and evolutionaryoptimization based on the ParaGen method can be applied in early stages ofdesign process like the conceptual design phase.The method assists thedesigner exploring forms and at the same time considering multiple designobjectives. It can expand the designer’s perspective by providing a numberof suitable solutions with different topology, geometrical and structuralproperties.The results are not limited to a single best solution, but generatearrays of good solutions in dynamic response to given criteria.

The application domain of the ParaGen method can be extended todifferent simulation works such as structural performance, energy efficiency,lighting or acoustic analysis.Therefore, any kind of simulation software ortool which allows scripting and parametric design can be used within theiterating cycle of form exploration.This method is mainly applicable indesign projects in which multi-objective criteria are considered. However, itcan certainly also contribute in academic projects to expand the designers’perspective and facilitate exploring the possible solutions.The Foster Bridgeand also the student models indicate that the provided solutions using theParaGen method are reasonable and close to what designers intuitivelyexpect.This verifies the applicability of the method in both professional andacademic fields.

Compared to similar computational form exploration methods, the onethat is described in this paper has the following characteristics:

• Once the solution database is created, the exploration of differentsets of well performing solutions based on different sets of criteriais very quick and certainly interactive.

• Because performance data about not only a single best solution,but in fact all solutions is provided, exploration of suboptimalsolutions is also possible.

• The visual display of solutions allows the designer to interact withthe solutions and make selections with regards to visual criteria

• Because all solutions are held in the SQL database, the designercan use queries and sorts to define solution sets which respond toa variety of multi-objective criteria.

On the other hand using Formex configuration processing in conjunctionwith the ParaGen method requires certain knowledge, skills and facilities.


Running the form exploration cycle requires certain knowledge inprogramming and working with the specific simulation software. Eachsoftware also has its own limitations in terms of programming andapplication. Furthermore, license requirements depend on the simulationsoftware being used. Finally, because of the time required to run the digitalsimulations, hardware requirements can be significant. In this example acluster of 10 dual core PCs was used.

All in all, the examples of actual constructed bridges as well as theexamples of student models, verify that the pool of suitable solutions foundby the design method is reasonable. Furthermore, in cases of professionalprojects where multiple objectives of both a quantitative and qualitativenature are considered, this form exploration technique should providesignificant help.

REFERENCES1. von Buelow, P., ParaGen: Performative Exploration of Generative Systems,

International Association for Shell and Spatial Structures, 2012, December.53 (4).

2. Mueller, C., and Ochsendorf, J.,An Integrated Computational Approach forCreative Conceptual Structural Design, International Association of Shell and SpatialStructures (IASS) Symposium 2013 “Beyond the Limits of Man”, Poland, 2013.

3. Nooshin, H., and Disney, P., Formex Configuration Processing I., (H. Nooshin, &Z. S. Makowski, Eds.), International Journal of Space Structures, 2000, 15(1), pp. 1-52.

4. Nooshin, H., Disney, P. L., and Champion, O. C., Computer-Aided Processing ofPolyhedric Configurations., In J. F. Gabriel (Ed.), Beyond the Cube:The Architecture ofSpace Frames and Polyhedra, New York, Chichester,Weinheim, Brisbane, Singapore,Toronto: John Wiley & Sons, Inc, 1997, pp. 343-384.

5. S. Hofmann, I., The Concept of Pellevation for Shaping of Structural Forms, Universityof Surrey, Department of Civil Engineering, Guilford: Space Structures ResearchCentre, 1999.

6. Byrne, J., Fenton, M., Hemberg, E., McDermott, J., O’Neill, M., Shotton, E., & Nally,C., Combining Structural Analysis and Multi-Objective Criteria for EvolutionaryArchitectural Design, In G. Goos, J. Hartmanis, & J. van Leeuwen (Ed.), Applicationof Evolutionary Optimization, Springer, 2011, pp. 204-213.

7. Holland, J. H.,Adaptation in Natural and Artificial Systems,Ann Arbor:TheUniversity of Michigan Press, 1975.

8. von Buelow, P.,Techniques for more Productive Genetic Design: Exploration withGAs using Non-Destructive Dynamic Populations, In K. S. Beesley (Ed.),Proceedings of the Assoc. for Computer-Aided Design in Architecture (ACADIA),Cambridge, Ontario, Canada, 2013, pp. 24-26.

9. Sosa, R and Gero, JS., The Creative Value of Bad Ideas – A Computational Modelof Creative Ideation , R Stouffs, P Janssen, S Roudavski and B Tuncer (eds), OpenSystems: CAADRIA 2013, National University of Singapore, 2013, pp. 853-862.

10. Syswerda, G., Uniform Crossover in Genetic Algorithms, Morgan Kaufmann, ThirdInternational Conference of Genetic Algorithms, 1989, pp. 2-9.


Anahita Khodadadi and Peter von Buelow

University of MichiganTaubman College of Architecture and Urban Planning2000 Bonisteel Blvd.Ann Arbor, MI48109

[email protected]@umich.edu

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