Content uploaded by Peter Von Buelow

Author content

All content in this area was uploaded by Peter Von Buelow on Feb 01, 2016

Content may be subject to copyright.

321

Performance Based

Exploration Of

Generative Design

Solutions Using Formex

Algebra

Anahita Khodadadi and Peter von Buelow

issue 3, volume 12international journal of architectural computing

322

Performance Based Exploration Of Generative

Design Solutions Using Formex Algebra

Anahita Khodadadi and Peter von Buelow

This paper illustrates the use of a computational form-

finding method called ParaGen which aids the designer

in the exploration of arrays of good solutions.Although

the method is guided by a multi-objective optimization

program, the goal is to promote the exploration of the

solution space based on designer selected

combinations of performance objectives.The digital

form generation of the bridges is carried out using

Formian, a program which uses Formex algebra to

describe a wide array of geometric configurations.This

form generation is linked to structural simulation and

design software (STAAD.Pro) to determine

performance values. Finally, ParaGen is used to build a

database of all solutions and guide the exploration

based on performance values. Using this database, both

visual and numeric characteristics are explored.

1. INTRODUCTION

Designing a form is broadly considered as a creative procedure within

which certain goals are purposefully sought.The designer usually sets some

parameters such as topology or geometry of forms, material properties,

architectural functions and load cases.Then, the design goals are generally

defined by means of some criteria and the extent to which solution(s) can

meet them. In order to expand the designer’s perspective, facilitate

modifications of the forms and explore more possibilities of appropriate

solutions in the early stages of design, the couple of parametric form

generation and evolutionary optimization techniques can be applied [1].

In this research work two methods of form exploration are compared

to study the relation between the geometry and structural performance of

specific types of spatial structures.The first is based on traditional methods

of design exploration including physical modeling and the second is derived

from the coupling of a parametric formulation of the design and the

application of a genetic search process.To demonstrate the application of

computational method in professional context, the design of a trussed

roadway bridge is used, and a comparison is made between solutions found

using the ParaGen exploration method and a successful built bridge.

Furthermore, the solutions are compared to results achieved by design

students through model studies to show the contribution of this method in

an academic context.

A case study of a two-lane truss bridge is opted for in this paper, since it

allows one to consider structural performance as well as other design

concerns like visual characteristics, compatibility with the surrounding

environment and constructability.The truss bridge is based on an actual

design (shown in figure 1) with a width of 8 m and span of 48 m, and is

expected to be composed of two 2D trusses on the two sides that are

braced together laterally.The parametric variables allow for the trusses to

be situated either above or below or both above and below the deck.The

trusses 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 appropriate

solutions regarding the structural performance, visual appearance and a

designer’s individual preference, and how can such a set of suitable solutions

with different geometrical and structural features be found.These questions

involve the subjectivity of the design process which has been the main

concern of recent works considering interactive evolutionary methods of

form exploration in the conceptual design phase [2].

The parametric configuration of the bridge is first processed using

Formex algebra and Formian programming software [3].The geometrical

concepts used within this step are described in the following section. Next,

having produced a geometry, a DXF file is sent to STAAD.Pro for structural

analyses and evaluation.The obtained information including graphic

depictions is stored in a database which is linked to a visual exploration

interface. In the next step pairs of bridges are selected based on the design

objectives to breed further bridges. In selecting a breeding pair, a fitness

function is used to dynamically create a population of the desirable

solutions out of the entire pool of generated forms.The newly generated

truss bridges emerge through the iteration of this cycle to yield an array

increasingly more suitable solutions.This process is described in detail in

section 3 (see figure 5). Finally, a comparison with the Foster Bridge, a

similar, successful constructed bridge over the Huron River in Michigan, and

also the results of form exploration through physical modeling of the truss

bridges, accompany the computational study at the end.

2. FORMEX CONFIGURATION PROCESSING

The configuration processing of truss bridges is described through

geometrical concepts of Formex algebra [3]. Formex algebra is a

mathematical system that allows a designer to define the geometrical

formulation 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 by

using this mathematical system and its associated programming language

Formian.

In the first step, constant parameters, variables and also their acceptable

intervals, described in table 1, are set. Additionally, sketches of truss

patterns, illustrated in figure 2, are provided to assist the designer choosing

the desired pattern more conveniently.

The configuration of trusses can be processed using a 3-directional

reference system similar to the global Cartesian system.Truss elements and

deck members of the bridge are defined in terms of lines, and surfaces

respectively. Figure 3 shows some parts of a simple truss bridge given in an

appropriate reference system and also the related parameters, used to

define 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 the

elements 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

325Performance Based Exploration Of Generative Design Solutions Using Formex Algebra

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 curvature

0 = none

Curvature sign of the bottom truss Sign1 Variable 1= positive curvature

-1 = negative curvature

0 = 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 to

create different forms. In this study,“circular barrel pellevation” is used, see

Hofmann, 1999 [5]. The configurations generated by Formian can be

exported in DXF format and used in the next stage of study which includes

a 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 best

solution and exploration seeks a set of significantly different good solutions.

Exploration can be used at the early stages of design to study a wider range

of possibilities for reasonably good or even unexpected solutions. Form

exploration usually can be accomplished efficiently using parametric tools.

Evolutionary computation methods provide means to search a range of

generated 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 place

recursively or all at once.Topology is the highest level at which forms can

be explored.An instance of a topology will have a specific geometry which

can either be explored in a range under a single topology or linked to

differing topologies. Solutions can be sorted in a database which can be

inspected visually in a relevantly easy way. Specific solutions of a certain

topology and geometry are further composed of members which can be

optimized as well, but with which designers usually have less direct

interaction in terms of form finding. In this research work, form exploration

is 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

326 Anahita Khodadadi and Peter von Buelow

A genetic algorithm, originally described by John Holland, is a search method

that progresses through iterating cycles to find solutions that meet certain

goals [7]. Using mechanisms like recombination and mutation, good

solutions may be found which are not anticipated by designers.The

solutions which are inherently parametric, are described in terms of a list of

variables which are analogous to genes on chromosomes.These

chromosomes are bred to form children that inherit characteristics through

the genes of their parents.

ParaGen uses a non-conventional genetic algorithm called a Non-

Destructive Dynamic Population GA (NDDP GA). It incorporates HUX as

described above in the breeding step and incorporates a database to

maintain a general pool of all solutions found. ParaGen incorporates the

following steps:

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

chromosome.

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 linked

to 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.

327Performance Based Exploration Of Generative Design Solutions Using Formex Algebra

Steps 3 through 8 form an iterative cycle which continues until satisficing

solutions are found.

ParaGen is designed to run using a parallel cluster of PCs linked to a

web server through the internet.The solution database and design interface

are placed on the web server (the interface is a www site), and steps 3-5

run on that web server. Each child is downloaded to a PC linked to the

server simply by connecting to the ParaGen website.

Figure 5: Schematic of the ParaGen

cyclic method

328 Anahita Khodadadi and Peter von Buelow

In a traditional GA approach, any defective or poor performing solutions

are usually removed from the breeding population (killed off). However, in

the NDDP GA all solutions, both well performing and poorly performing

solutions are stored in the database. ParaGen simply stores all performance

values and defers any ranking of these values to the moment of breeding

(thus “dynamic” selection). By retaining data on all solutions, the designer is

able to learn from ill solutions and increase the knowledge of what would

make a good solutions [8]. Recent studies have shown the value of sub-

optimal solutions in the exploration process [9]. Furthermore, in case of any

modification 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 the

designers, concentrates the generation of new solutions in the area of the

solution space defined by the fitness function of the GA, and the focus of

the exploration becomes more defined.Within a productive exploration,

thousands of solutions are evaluated by the program.The designer uses the

ParaGen web interface to filter and sort these solution based on any

combination of geometry or performance data. In this way, a well

performing set of solutions is defined as a manageable quantity, which is

reasonable for the designer to visually inspect and possibly make manual

selections for further breeding.The web interface provides interaction

between the designer and the form exploration system which eventually

leads to choosing the final desirable solutions.The designer’s interaction can

be also based on a totally aesthetic preference.

The incorporation of a relational database into a genetic algorithm gives

ParaGen several advantages over traditional population bases generational

methods [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 beyond

the range of traditional search and optimization methods when dealing with

design exploration.

4. FORM EXPLORATION AND OPTIMIZATION

PROCESS

This section describes in more detail the 8 step process enumerated in

section 3 as the ParaGen method. In step 1. Formian was used to describe a

wide range of truss bridge types and geometries. In the parametric

formulation, 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 for

Formian are listed in table 1.

For this trial, Span was set to 48 m and Width set to 8 m.These 13

variables constituted the “chromosome” description of a solution which is

eventually downloaded to Formian for processing into a geometry (step 6).

This “chromosome” is bred from two parents in step 5 using a GA

crossover technique called HUX [10].The two parents are randomly

selected from a dynamic population (step 4), which is gleaned from the full

database using a SQL query (step 3).This SQL query formulates the search

objective or fitness function for the GA.

329Performance Based Exploration Of Generative Design Solutions Using Formex Algebra

Once the input variables (the “chromosome”) are passed to Formian,

Formian processes the variables to produce one instance of the parametric

bridge geometry. Formian generates a visual perspective view as well as a

DXF file which can be read by other simulation software. In this example a

finite element analysis (FEA) was carried out using STAAD.Pro (step 7).The

process makes use of scripts written in each software used plus a general

Windows interface script (AutoHotkey) to automate the process. Steps 6

and 7 are performed in parallel using a cluster of PCs.At the completion of

the analysis, the original input data (the “chromosome”) plus all of the

associated performance data collected through the simulation software, plus

any number of descriptive images and files, are all uploaded to the server

through the web site interface (step 8). On the server all numeric data is

placed in a database and tagged to the images.The ParaGen web site then

offers a graphic window into this database by displaying the images and

associated performance values.The ParaGen web interface also allows the

designer to sort and filter the displayed images and data in a variety of ways

to 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.

330 Anahita Khodadadi and Peter von Buelow

boxes that allow the designer to display different ranges of the solutions. In

this example, the solutions are filtered in a way that provides a suitable

comparison with the Foster Bridge (table 2, figure 9).This demonstrates

how the website can display a set of solutions for any particular list of

requirements and preferences. In this example, the better final solutions

have fewer joints that contribute to better constructablity or they may be

lighter or have less deflection, indicating better structural performance.All

in all, the final decision will be made regarding the designer’s informed

intuition and preferences.

In addition to solution displays, the ParaGen website provides graphs in

which the designer can compare two specific properties and choose the

most desirable solution. Figures 7 and 8, show graphs in which solutions are

distributed regarding the maximum deflection or total weight versus

number of joints. For example the designer may look for a solution which

has the least maximum deflection and least number of joints at the same

time. Or a designer may opt for a solution with greater weight but fewer

joints which provides better constructability. It is possible to click on the

specific chosen solution and find out more detail regarding the other

geometrical 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

331Performance Based Exploration Of Generative Design Solutions Using Formex Algebra

5. PHYSICAL MODELING

The physical modeling and testing of truss bridges was carried out in the

context of a structures course at University of Michigan, School of

Architecture. Groups of students were asked to make a 1/64 scale model of

a bridge using balsa wood, and to the best of their knowledge choose a

suitable 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.

332 Anahita Khodadadi and Peter von Buelow

the models were loaded until the breakage point and the structural

performance was presented in terms of held load to weight ratio.The ratio

and geometrical features were added to a data base and with a similar

fashion to computational study, the web site allows to sort and filter the

results and offers a graphic window with all the images of the models. In

figure 10 a graph presents the load capacity to weight ratio of the bridges

versus 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

333Performance Based Exploration Of Generative Design Solutions Using Formex Algebra

plotted.The highlighted models in this figure show the best tradeoff

combinations of these two factors and may be considered as desirable

solutions (Pareto optimal).The web site also provides a graphic window that

shows the images and associated performance values of the models.

6. DISCUSSION

This paper has looked at the results of two different forms of design

exploration. One which can be considered “traditional” in which solutions

were found by designers (students) using physical modeling techniques; and

a second, computational approach based on what has been described as the

“ParaGen method”.Although both studies were based on the same

geometric limits (two lane, 48 m span), the loading was approached

differently. In the physical models although a target was set at the same level

as the computational model (AASHTO HL93) in testing the load was

applied until attaining ultimate failure, which was usually much higher. So,

whereas the computational database contained solutions with a safe

capacity 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 by

the level of loading.

Figure 12: Graph of number of

joints vs. total weight from the

computational exploration.

334 Anahita Khodadadi and Peter von Buelow

Figure 12 shows the computational counterpart to figure 10 which used

the physical models.The lighter load level combined with the stronger

material (steel vs wood) results in solutions with much less depth. In the

computational version, the top bracing was removed at any height below 3

m so as not to interfere with traffic.Thus the lightest solutions found in the

graph shown in figure 12 are simple pony truss bridges using only the deck

for lateral bracing. Figure 13 shows solutions on the Pareto frontier of this

graph.These solutions are certainly good solutions, but the designer may

also explore the forms of nearby solutions (near optimal) by simply clicking

on the graphed dots.

Figure 13: Pareto optimal set from

the graph in figure 12.

335Performance Based Exploration Of Generative Design Solutions Using Formex Algebra

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

observed:

•The solutions with best load to weight ratio were lenticular

(curved top and bottom) but tended to have more joints. Flat

chord trusses tended to have fewer joints, but lower load to

weight ratios.

•The quality of construction undoubtedly played a role in the

strength, which was not present in the computational models.

•Because of the longer time required per solution, the traditional

method with physical modeling was more limited in the number of

solutions which could be considered.

•Using physical modeling makes the form finding process more

tangible and comprehensible. It is therefore arguably better in

developing an initial intuition, and sometimes it helps the designer

to gain a firsthand sense of structural performance.

7. CONCLUSION

Coupling the Formex configuration processing and evolutionary

optimization based on the ParaGen method can be applied in early stages of

design process like the conceptual design phase.The method assists the

designer exploring forms and at the same time considering multiple design

objectives. It can expand the designer’s perspective by providing a number

of suitable solutions with different topology, geometrical and structural

properties.The results are not limited to a single best solution, but generate

arrays of good solutions in dynamic response to given criteria.

The application domain of the ParaGen method can be extended to

different simulation works such as structural performance, energy efficiency,

lighting or acoustic analysis.Therefore, any kind of simulation software or

tool which allows scripting and parametric design can be used within the

iterating cycle of form exploration.This method is mainly applicable in

design projects in which multi-objective criteria are considered. However, it

can certainly also contribute in academic projects to expand the designers’

perspective and facilitate exploring the possible solutions.The Foster Bridge

and also the student models indicate that the provided solutions using the

ParaGen method are reasonable and close to what designers intuitively

expect.This verifies the applicability of the method in both professional and

academic fields.

Compared to similar computational form exploration methods, the one

that is described in this paper has the following characteristics:

•Once the solution database is created, the exploration of different

sets of well performing solutions based on different sets of criteria

is very quick and certainly interactive.

•Because performance data about not only a single best solution,

but in fact all solutions is provided, exploration of suboptimal

solutions is also possible.

•The visual display of solutions allows the designer to interact with

the solutions and make selections with regards to visual criteria

•Because all solutions are held in the SQL database, the designer

can use queries and sorts to define solution sets which respond to

a variety of multi-objective criteria.

On the other hand using Formex configuration processing in conjunction

with the ParaGen method requires certain knowledge, skills and facilities.

336 Anahita Khodadadi and Peter von Buelow

Running the form exploration cycle requires certain knowledge in

programming and working with the specific simulation software. Each

software also has its own limitations in terms of programming and

application. Furthermore, license requirements depend on the simulation

software being used. Finally, because of the time required to run the digital

simulations, hardware requirements can be significant. In this example a

cluster of 10 dual core PCs was used.

All in all, the examples of actual constructed bridges as well as the

examples of student models, verify that the pool of suitable solutions found

by the design method is reasonable. Furthermore, in cases of professional

projects where multiple objectives of both a quantitative and qualitative

nature are considered, this form exploration technique should provide

significant help.

REFERENCES

1. 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 for

Creative Conceptual Structural Design, International Association of Shell and Spatial

Structures (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 of

Polyhedric Configurations., In J. F. Gabriel (Ed.), Beyond the Cube:The Architecture of

Space Frames and Polyhedra,New York, Chichester,Weinheim, Brisbane, Singapore,

To r onto: John Wiley & Sons, Inc, 1997, pp. 343-384.

5. S. Hofmann, I., The Concept of Pellevation for Shaping of Structural Forms,University

of Surrey, Department of Civil Engineering, Guilford: Space Structures Research

Centre, 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 Evolutionary

Architectural Design, In G. Goos, J. Hartmanis, & J. van Leeuwen (Ed.), Application

of Evolutionary Optimization, Springer, 2011, pp. 204-213.

7. Holland, J. H.,Adaptation in Natural and Artificial Systems,Ann Arbor:The

University of Michigan Press, 1975.

8. von Buelow, P.,Techniques for more Productive Genetic Design: Exploration with

GAs 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 Model

of Creative Ideation , R Stouffs, P Janssen, S Roudavski and B Tuncer (eds), Open

Systems:CAADRIA 2013,National University of Singapore, 2013, pp. 853-862.

10. Syswerda, G., Uniform Crossover in Genetic Algorithms, Morgan Kaufmann, Third

International Conference of Genetic Algorithms, 1989, pp. 2-9.

337Performance Based Exploration Of Generative Design Solutions Using Formex Algebra

Anahita Khodadadi and Peter von Buelow

University of Michigan

Taubman College of Architecture and Urban Planning

2000 Bonisteel Blvd.

Ann Arbor, MI48109

anahitak@umich.edu

pvbuelow@umich.edu