Performance Based Exploration Of Generative Design Solutions Using Formex Algebra Anahita Khodadadi and Peter von Buelow
international journal of architectural computing
issue 3, volume 12
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Performance Based Exploration Of Generative Design Solutions Using Formex Algebra Anahita Khodadadi and Peter von Buelow
This paper illustrates the use of a computational formfinding 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.
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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.
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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
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Parameters Span of the bridge Width of the bridge Frequency of geometrical modules along the length of the bridge Total rise of the bridge at the top part Total rise of the bridge at the bottom part The rise of the arch at the top truss The rise of the arch at the bottom truss The top rise in non-arched part of the truss The bottom rise in non-arched part of the truss Curvature sign of the top truss
Sign L W m
Constant/Variable Constant Constant variable
HX1 HX2 X1 X2 H1 H2 Sign1
Variable Variable Variable Variable H1 = HX1 - X1 H2 = HX2 - X2 Variable
Curvature sign of the bottom truss
Sign1
Variable
Pattern of top truss
Top
Variable
Pattern of bottom truss
Bottom
Variable
Table 1: Geometrical parameters on which the configuration processing of truss bridges are based.
Acceptable interval/ value 48 m 8m m should be an even integer. [4 to 20] is recommended. [0.5 to 16] m [0.5 to 16] m [0.01 to 16] m [0.01 to 16] m [0.49 to 16) m [0.49 to 16) m 1= positive curvature -1 = negative curvature 0 = none 1= positive curvature -1 = negative curvature 0 = none D1, D3, D5, D7, D9, D11, D13, D15, D17 (see figure 2) D2, D4, D6, D8, D10, D12, D14, D16, D18 (see figure 2)
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
Figure 2:The truss patterns that are used for the top and the bottom parts of the bridges
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Figure 3: A part of a curved bridge configuration and the related parameters within a 3direcional reference system
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.
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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.
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 NonDestructive 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.
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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.
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 suboptimal solutions in the exploration process [9]. Furthermore, in case of any modification of criteria, the poor performing solutions can also be reconsidered and re-used in the breeding process to form new solutions.
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Figure 5: Schematic of the ParaGen cyclic method
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.
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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
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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. 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. Table 2: Filtering properties for a series of solutions in the ParaGen interface which compare to the Foster Bridge
Properties Number of panels Max deflection Total weight Number of joints
Desired value = 8 ≤3 ≤ 400 ≤ 85
Figure 7: Number of joints vs. maximum deflection.The highlighted solutions are depicted in figure 6
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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.
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,
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Figure 10: A plot of load capacity to weight ratio vs. number of joints within the 219 bridge models.
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 11:The display of highlighted models in figure 10 chosen as probable desirable solutions
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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.
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Figure 13: Pareto optimal set from the graph in figure 12.
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.
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.
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•
•
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.
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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, Toronto: 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.
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Anahita Khodadadi and Peter von Buelow University of Michigan Taubman College of Architecture and Urban Planning 2000 Bonisteel Blvd. Ann Arbor, MI48109
[email protected] [email protected]
