Alternative means of navigating parameter spaces

Proceedings of the IASS Annual Symposium 2016 “Spatial Structures in the 21st Century” 26–30 September, 2016, Tokyo, Japan

K. Kawaguchi, M. Ohsaki, T. Takeuchi (eds.)

Copyright © 2016 by Max Marschall, Saqib Aziz and Christoph Gengnagel Published by the International Association for Shell and Spatial Structures (IASS) with permission.

Alternative means of navigating parameter spaces Max MARSCHALL*, Saqib AZIZ, Christoph GENGNAGEL

*Department for Structural Design and Technology Berlin University of the Arts (UdK)

Hardenbergstraße 33, 10623 Berlin, GERMANY [email protected]

Abstract Generative design processes are characterized by modifications of a design model's parameters with the aim of improving its quality. The entirety of possible parameter combinations in a given design scenario is referred to as parameter space, in which each parameter represents a dimension. The efficacy of a design process could be measured by the quality of the final design found in the vastness of a multidimensional parameter space, as well as by the time needed to arrive at this solution. In this paper we propose a novel method aimed at increasing this efficacy. Our approach of self-organized fitness landscapes (SOFL) facilitates the convergence towards optimized design trade-offs between numeric and subjective criteria. This is achieved by visualizing subdomains of the parameter space comprehensibly, while still leaving architectural quality assessment to the designer. This paper describes our method's applicability in architecture based on a case study in stadium design.

Keywords: parameter space, self-organizing map, optimization, large-scale design, virtual reality, interactive evolutionary computation, digital tooling, stadium.

1. Introduction Architectural design tasks have numeric as well as subjective requirements. For instance, the design brief for a soccer stadium may require sheltered stands with unobstructed viewing conditions, as well as an aesthetically appealing design that the city and its soccer club can identify with. Any design process seeks to find the solution with the best possible tradeoff between all the – often contradictory – requirements, be they based on a client’s brief or the personal convictions of the architect.

Figure 1: “Morphological space” (Raup [2]). The axes denote parameters describing the geometry of shells.

Proceedings of the IASS Annual Symposium 2016 Spatial Structures in the 21st Century

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The entirety of possible solutions to choose from is sometimes referred to as “morphospace” [1] (Figure 1) or “phase space”. Since this paper is directed at architects, it uses the term “parameter space” due to its correlation with the concept of “parametric models”, which most architects are familiar with. The use of parametric models has enabled us to quickly produce a multitude of design variations, thus in effect gaining handy access to large parameter spaces. Small parameter spaces can be visualized by brute force, representing every possible design solution. In any realistic design situation, however, this method is too time-consuming. This has created a new need: an efficient way of parsing parameter spaces for optimum design solutions in the myriad possible outcomes which are now so readily at our disposal. Numerous approaches for this already exist, each with their respective strengths and weaknesses. The main idea of the work presented here is to combine existing methods of problem-solving in order to create a suitable workflow to aid in the complex and ambivalent design tasks of architectural practice. The following is a description of the concepts that our workflow proposal draws upon.

2. Methodologie

2.1. Self-Organizing Map The self-organizing map (SOM) is a type of artificial neural network (ANN) which can be used to display multidimensional data. Its functional principle is based on the neurobiological insight that while input signals - such as visual stimuli - are multidimensional, the corresponding brain regions usually exhibit linear or planar topologies. In addition to that, similar inputs are mapped in close proximity to one another in the nervous system. Designers can use the SOM algorithm to redistribute multidimensional parameter spaces onto a plane. This is done by inputting samples of variations that lie within the parameter space that is to be displayed (Figure 1). Using a neighborhood function, a two-dimensional mapping is created which incorporates the samples, as well as fills in the intermediary space in between. Its benefit over other flat parameter space representations is that similar variations appear close to one another in the SOM. [2]

Figure 2: Concept of SOMs. a) Multidimensional parameter space. b) Volume to be displayed. c) Input samples.

d) SOM.

2.2. Fitness Landscape The arduous task of optimizing a design for numeric criteria can often be facilitated through the use of optimization algorithms such as evolutionary solvers. These require the designer to define a parameter space and a so-called fitness function by which the design solutions are to be assessed. Parameter space and fitness function together constitute the so-called fitness landscape. This can be imagined by picturing a possibly high-dimensional parameter space being flattened out onto a two-dimensional plane, and then evaluating each variation for numeric criteria and assigning it a height over the plane


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accordingly. The result is an undulating terrain that forms peaks in areas of high numeric performance and valleys with variations of poor valuation by the fitness function (Figure 2). Evolutionary and other generic solvers aim to identify the peaks in the landscape [3]. While they are efficient in doing so, their shortcoming is precisely that only numeric criteria can be evaluated. It is hard to imagine a mathematical fitness function that could evaluate subjective factors like aesthetic quality or symbolic gestures in the way a human designer could.

Figure 3: Fitness landscape example by David Rutten. a) Parameter space as two-dimensional plane. b) Fitness

landscape. c) High quality solution. d) Local optimum. e) Low quality solution.

2.3. Interactive Evoltionary Computation Contrary to evolutionary computation, the fitness function in interactive evolutionary computation (IEC) is the designer’s personal, subjective quality assessment. He or she is presented with an array of design variations to manually choose from. The chosen variation is then mutated a number of times and acts as the parent for a new generation of variations with similar properties, which are again shown to the designer for selection (Figure 4). One of the assumed benefits of IEC is its application in early design stages as an inspirational source, due to its ability to produce unexpected results. It has so far found limited utilization in the architectural industry, possibly because apart from choosing from the computer-generated instances, the designer has little influence on the process. Another setback is precisely that numeric criteria are not taken into account.

Figure 4: Example of a generation of mutated pictures for selection. Source: picbreeder.org

Our novel contribution is the proposition of a computational design workflow that enables a realistic application in the field of architecture by implementing the following features:

• Evaluation of numeric (Section 5.2) as well as subjective (Section 5.3) criteria

• Comprehensible visualization of parameter spaces (Section 5.4)

• Accounting for the non-linear nature of design processes by documenting the succession of design decisions and enabling the branching out of different design options (Section 5.5)


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3. Self-Organized Fitness Landscapes (SOFL) As mentioned in Section 2, the proposed method combines different concepts of problem-solving with the aim of making it viable for use in the multifaceted process of architectural design. It is part of a larger workflow concept described below (Figure 5). The method is based on the prerequisite that the design begins with some rudimentary initial form of geometry, which is successively edited by algorithmic definitions that are successively created in an iterative process.

Figure 5: Overlying workflow diagram with the concept of self-organized fitness landscapes at its core.

Design processes are characterized by the creation of design variations commonly represented by drawings or physical models. Traditionally, these variations are made visible side by side, enabling a direct comparison. One arguable disadvantage of most parametric models, which are varied by modifying their numeric parameters, is that the variations are viewed in succession. Our method acknowledges the cognitive benefit of simultaneous representation, and therefore implements a method of visualizing parameter spaces. Since its dimensionality is dependent on the number of explicit parameters of the current algorithmic definition, it is necessary to supply a method capable of displaying spaces of higher dimensions. In this case, the method of the self-organizing map is adopted for this purpose. The reasons for doing so are the following:

• Similar variations are located in close proximity, resulting in of gradual changes occurring between variations (Figure 6). Such an ordered representation is visually beneficial in comparison with other representations of multidimensional spaces, which often have a chaotic or fractal nature

• Instead of inefficiently visualizing an entire parameter space, the SOM makes it possible to restrict the confines of the space by basing its training algorithm on samples selected by the user. Therefore, the user is able to experiment for some time with the parameters of his or her parametric definition and preselect a number of favorable variations as inputs for the map creation

Figure 6: Example of an SOM by John Harding, using his SOM implementation for Grasshopper. a)

Multidimensional input. b) Node location of input in the SOM.


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It is intended that the designer choose numeric criteria by which to evaluate the design variations found in the SOM, in order to visualize the map as a fitness landscape. Instead of visualizing it diagrammatically as a curved surface (Figure 3), our method makes a point of displaying the actual geometric representations at their corresponding locations on the landscape. In doing so, the designer is aided in making informed decisions on which variations to choose. Numeric as well as subjective criteria can be taken into account, and acceptable trade-offs between the two can be found more easily. The chosen “elite”, consisting of a single or several preferred variations, can then either be accepted as the final design or fed back into the next iteration. A valuable concept in design and research is the documentation of data. These processes are characterized by testing various paths of inquiry and upon hitting dead ends, back-pedaling in one’s decision chain to pursue new paths. Hence, the workflow furthermore promotes documenting the iterative cycle in family tree logic, giving the designer an overview of his decision chain and enabling him to make upstream modifications or retrogress from dead-ends in the design process. All data relevant to reinstate past states of work is stored and serves as a form of mind map to aid decision making.

4. Related Work An important contribution to interactive evolutionary computation in the field of architecture is Robert Vierlinger’s evolutionary optimization framework “Octopus” for Grasshopper. Not only does the user-friendly application provide explicit access to various optimization parameters and allow for multi-objective searches, it also grants various interactive features that give the designer more control over the numeric optimization process. The user interface offers an optional depiction of design instances and other helpful visual feedback that aids in the selection of design options. Apart from being able to change objectives or select preferred solutions during a search, it also records the search history, giving the user access to optimized solutions from past iterations and enables data export to text files. The workflow proposed in this paper is an attempt to build on these concepts to make them more accessible to a larger range of architects. In Robert Vierlinger’s latest research, he investigates the applicability of artificial intelligence into the generative design processes. His studies evolve ANNs through aesthetic selection processes by the designer, and aim to create large ANNs capable of developing higher level concepts. [4]

5. Case Study Stadium design is a specialized architectural discipline. It demands a broad range of both numeric and subjective requirements, but generally has a clearly defined usage program as a typology. It is characterized by rationalizable planning and construction processes and is therefore suitable for a high degree of parameterization. Common numeric requirements include unobstructed sightlines and favorable viewing angles from the stands, as well as environmental considerations or economic reflections concerning spectator numbers and construction costs. Many stadia have iconic characteristics and serve as a carrier of identity, often exposed at an international scale. Therefore, great demands are also placed on the subjectively aesthetic or even symbolic perception of the design. Advances in CAD software and the integration of algorithmic modeling processes enable novel research opportunities to inform design decisions through in-depth investigation of geometric interdependencies and analytical models. The case study described hereafter explores these potentials for the design of stadium hull shapes in early design stages. Stadium hulls consist of a roof and a façade, which in many modern stadia form an entity. Their counterpart is the stadium “bowl”, which refers chiefly to the spectator stands.


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5.1. Modifying Definitions The iterative workflow proposal described above allows for the successive modification of geometry through definitions created by the designer in the course of the design process. This method has been shown to produce diverse results (Figure 12). Figure 7 shows three possible model instances generated by such a definition. The dimensionality of each SOM mirrors the number of parameters used by the corresponding algorithmic definitions. In this case, three parameters are involved:

• The first divides the base curve, which is an offset of the perimeter from a predefined stadium bowl not visible in the picture.

• The second parameter causes a rotational divergence of these division points from the original bowl shape.

• A polyline is generated from the new points; its corners are then filleted with a radius described by the third and last parameter.

The sliders underneath the example geometries qualitatively indicate the corresponding vectors, that is, the parameter settings for each geometry. For example, the third slider of the first stadium is set to its least value, causing the stadium to have a minimum fillet radii and therefore sharp corners; the last stadium is round due to the higher value on its last parameter.

Figure 7: Example of the possible outputs of a modifying definition. a) Parameter settings. b) Corresponding

design instances.

5.2. Numeric Evaluation The development of algorithmic models for the generation and analysis of stadium geometry was crucial to enable a feasible research workflow. The exemplary metrics used to numerically evaluate hull designs in this case study were:

• Exposure of spectators to rain

• Exposure of spectators to sunlight

• Volume of enclosed space

• Square measure of the hull


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Figure 8: Analysis of rain exposure.

Environmental factors were evaluated in regard to viewer comfort. The perspective representation in Figure 8 illustrates which seats in a stadium are exposed to rain by color-coding them in cyan. The dashed lines indicate rain vectors. A pie chart reveals which percentage of seats is exposed. Both the top view diagram and the perspective suggest a geometric adjustment to the stadium hull in order to provide full rain shelter; the increase in roof area by doing so is also displayed. This analysis was achieved using the application “Ladybug” for Grasshopper to access EnergyPlus Weather Data for a specific location and time. Grasshopper is an algorithmic modeling add-on for the CAD software Rhinoceros. Figure 9 shows another example of numeric evaluation, in which the spectators’ exposure to sunlight is displayed for various stadium geometries, over a defined time frame at a specified date and location. The output from this analysis is the sum of time each spectator in a stadium spends in the sun over the course of a soccer game.

Figure 9: Analysis of sunlight exposure.

5.3. Subjective Evaluation Examples of possible subjective criteria for evaluating stadium designs are the articulation of symbolic gestures, the visibility of sound architectural concepts or pure aesthetic preference. For obvious reasons, these factors cannot be evaluated by mathematical functions. The subjective evaluation in this case study was conducted with the authors’ personal assessment. The factors influencing subjective judgment are dependent on the parameters of the algorithmic definitions that were generated to modify input geometries (Section 5.1). The geometries for the case study were simple, volumetric stadium hull models. As is common in early architectural design stages, these displayed spatial concepts without detailing structural systems. Geometric properties that influenced the subjective evaluation included:

• Volume


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• Footprint

• Proximity to stands

• Correlation to geometry of stands

• Planarity vs. bumpiness

• Polygonization and segmentation

• Roof opening size and shape

• Height and inclination

5.4. Application of Self-organizing Fitness Landscape As described in Section 2.1, SOMs are used to sort multidimensional data in order to display it in a planar arrangement. The learning algorithm is trained using samples of data vectors. In our case study, a vector is a list of values that describe the parameter settings of our modifying definitions (Section 5.1). The number of parameters is equivalent to the dimensionality of the vector and thus of the parameter space that is to be displayed. Figure 10 shows a four-dimensional SOFL trained with seven samples. The map vectors describe the offset from the stands, the segmentation of the offset curve, the rotation of its control points and a fillet radius. The input vectors were hand-picked after some manual experimentation with the definition. These inputs are present in the map and form centers of clusters with similar individuals that gradually morph towards other clusters. The fitness criteria used to evaluate these variations was the hull footprint. Assuming in this case that a higher utilization of a construction site might be beneficial, the larger individuals are therefore located at a higher position in the map. For better readability, the height is furthermore indicated with a color gradient. Visualizing geometries and displaying their numeric performance at the same time in this way, the designer is flexible in screening the map to seek out aesthetically preferred variations and get feedback on where they rank in performance. If the performance is insufficient, then moving up the nearest hill in the landscape might lead to better performance while only slightly changing the preferred appearance.

Figure 10: An example of our self-organized fitness landscapes (SOFL). A fitness landscape is created from a

multidimensional parameter space sorted by a self-organizing map (SOM). For better comparability in the design process, the resulting geometric variations are displayed at their actual location on the map.


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5.4. Application of Self-organizing Fitness Landscape Architects are often reluctant to introduce computational methods into their workflows. One reason for this might be the assumption that they disrupt existing, more traditional procedures. The overarching workflow described in Section 3 therefore introduces the documentation of data as a way of creating more flexibility in the use of the SOFL method. The variations in the SOFL are labeled and their vectors stored. This enables the designer to reinstate a previous SOFL and choose a new elite. The input vectors are also stored in case an SOFL has to be recalculated with more informed samples. The workflow enables a closer adherence to traditional design methods by allowing for the successive revision of design instances and facilitating branching trains of thought in order to simultaneously pursue several design options. By choosing new modifying definitions with each iterative loop, the parameter space is observed bit by bit, only visualizing the portions that are of interest at a given moment in the design process. Figure 11 is a diagrammatic example of such a design record, in which SOFLs are depicted in family tree logic, revealing parallel lines of investigation. For simplicity, the node vectors are applied to the parameters of simple boxes, rather than instantiating all the actual stadium geometries. Each train of thought is comprised of a chronological succession of research steps. This enables the designer to document varying design concepts. The diagram on the right gives an overview of the hierarchy of dependency within the tree structure. Modifications in an SOFL or revisions of elite selections cause a ripple of updates downstream indicated by the arrows. As the shading of the arrows show, SOFLs that were created early on have a broader field of influence on successive steps.

Figure 11: Design record display.

A different method of representation is shown in Figure 12. Here, only the elite choices are displayed, but again in family tree logic. Omitting the SOFLs renders a more concise overview of the design process. Each design instantiation is annotated for a better comprehension of design choices retrospectively.


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Figure 12: Family tree display.

6. Conclusion: submission of contributions The proposed method of SOFL attempts to merge various concepts of problem-solving, thereby creating a framework that is applicable to a larger degree in architectural practice than current optimization concepts allow for. The case study in Section 5 demonstrates how a workflow involving SOFL can be beneficial as a generative method in early design stages. The proposition advocates the need to efficiently visualize parameter spaces for more in-depth comparability of design variations, in order to harness the full potential of the parameterized modeling paradigm. As suggested in related research, combining numeric and subjective evaluation methods may produce higher cognitive gains, which can lead to more informed decision-making. Recording the design history is a conceivably useful concept deployed as well by other applications; the proposed method to accommodate this aims to offer visualization methods that are readable and therefore useful to generic design processes, but would benefit from more advanced data management techniques. Its applicability has been showcased on the basis of a theoretical case study; its usability in architectural practice has yet to be confirmed. In a broader sense, our research reflects on a development within the architectural field - the increasing application of computational methods in the creative design field - and postulates their use as catalyzers for design processes.

Acknowledgements Thanks to Robert Vierlinger for introducing us to machine learning concepts in architectural practice. The self-organizing maps were created with the support of John Harding and the utilization of his Grasshopper implementation of Teuvo Kohonen’s SOM algorithm. Environmental analysis was conducted with Mostapha Sadegh’s plug-in „Ladybug“ for Grasshopper.

References [1] Kohonen T., Essentials of the self-organizing map. Neural Networks, Volume 37, January 2013,

52-65. [2] Raup D., Geometric Analysis of Shell Coiling: General Problems, Journal of Paleontology 40,

1966, 1178-1190. [3] Rutten D., Navigating multidimensional landscapes in foggy weather as an analogy for generic

problem solving. Paper presented at the 16th International Conference on Geometry and Graphics, Innsbruck, Austria, 2014

[4] Vierlinger R., Towards AI drawing agents. Paper presented at the Design Modelling Symposium, Copenhagen, Denmark, 2015

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Alternative means of navigating parameter spaces