Computational Exploration
of the
Structural Design Space
by
Caitlin T. Mueller
B.S. in Art and Design, Department of Architecture
Massachusetts Institute of Technology, 2007
M.S. in Structural Engineering, Department of Civil and Environmental Engineering
Stanford University, 2008
Submitted to the Department of Architecture
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy in Architecture: Building Technology
at the
Massachusetts Institute of Technology
June 2014
© 2014 Caitlin T Mueller. All rights reserved.
The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic
copies of this thesis document in whole or in part in any medium now known or hereafter created.
Signature of Author: ______________________________________________________________
Department of Architecture
May 2, 2014
Certified by: ___________________________________________________________________
John A. Ochsendorf
Professor of Architecture and Civil and Environmental Engineering
Thesis Supervisor
Accepted by: ___________________________________________________________________
Takehiko Nagakura
Professor of Architecture
Chairman, Department Committee on Graduate Studies
Dissertation Committee:
John A. Ochsendorf
Professor of Architecture and Civil and Environmental Engineering
Massachusetts Institute of Technology
Thesis Supervisor
Sigrid Adriaenssens
Assistant Professor of Civil and Environmental Engineering
Princeton University
Thesis Reader
Jerome J. Connor
Professor of Civil and Environmental Engineering
Massachusetts Institute of Technology
Thesis Reader
Terry Knight
Professor of Architecture
Massachusetts Institute of Technology
Thesis Reader
Computational Exploration
of the Structural Design Space
by
Caitlin T. Mueller
Submitted to the Department of Architecture
on May 2, 2014 in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy in Architecture: Building Technology
Abstract
This dissertation focuses on computational strategies for incorporating structural considerations into the
earliest stages of the architectural design process. Because structural behavior is most affected by geometric
form, the greatest potential for structural efficiency and a harmony of design goals occurs when global formal
design decisions are made, in conceptual design. However, most existing computational tools and approaches
lack the features necessary to take advantage of this potential: architectural modeling tools address geometry in
absence of performance, and structural analysis tools require an already determined geometrical form. There is
a need for new computational approaches that allow designers to explore the structural design space, which
links geometric variation and performance, in a free and interactive manner.
The dissertation addresses this need by proposing three new design space strategies. The first strategy, an
interactive evolutionary framework, balances creative navigation of the design space with a focus on
performance. The original contributions of this strategy center on enhanced opportunities for designer
interaction and control. The second strategy introduces structural grammars, which allow for the formulation
of broad and diverse design spaces that span across typologies. This strategy extends existing work in
geometry-based shape grammars by incorporating structural behavior in novel ways. Finally, the third strategy
is a surrogate modeling approach that approximates the design space to enable fast and responsive design
environments. This strategy contributes new ways for non-experts to use this machine-learning-based
methodology in conceptual design.
These three complementary strategies can be applied independently or in combination, and the dissertation
includes a discussion about possibilities and techniques for integrating them. Finally, the dissertation
concludes by reflecting on its potential impact on design in practice, and by outlining important areas for future
work.
Key words: conceptual structural design, design space exploration, structural optimization, interactive
evolutionary algorithm, structural grammar, surrogate modeling, structural design tools
Thesis supervisor: John A. Ochsendorf
Title: Professor of Architecture and Civil and Environmental Engineering
Acknowledgements
This dissertation would not have been possible without the thoughtful feedback and guidance from a variety of
important advisors, colleagues, and friends. First, I am tremendously grateful to my dissertation advisor,
Professor John Ochsendorf, who first introduced me to the joy of structural design in 2006, and who has been a
supportive, creative, and incisive mentor in the years since. In particular, I thank Professor Ochsendorf for his
open-mindedness, for his high expectations, and for his insistence on intellectual clarity.
My additional committee members have also been instrumental in a variety of ways. I thank Professor Jerome
Connor for welcoming me into his weekly research meetings, and for many insightful discussions about the
history and the future of structural engineering and design. I thank Professor Terry Knight for her knowledge
and wisdom in computational creativity, and for her enthusiasm for interdisciplinary work. I thank Professor
Sigrid Adriaenssens for sharing her formidable expertise and experience in innovative structural design, and for
her advice about problem-framing and connecting with practice. As a group, my committee has been helpful,
generous, and harmonious, and I greatly appreciate the productive group dynamic in addition to their
individual contributions.
I acknowledge the help of additional faculty members and informal advisors. The Building Technology faculty
— Professors John Fernández, Leon Glicksman, Les Norford, and Christoph Reinhart — were very helpful as I
shaped my key research questions during the Building Technology Seminar, and in the years since. Faculty
from the Computation for Design and Optimization program, especially Professor Karen Willcox, welcomed me
to the world of computational engineering and provided important feedback. Finally, I thank author and
educator Edward Allen for his groundbreaking books on creativity in structural design, and for his wisdom,
support, and humor in discussions about my research.
I am grateful to my fellow students and friends for inspiring me with broad intellectual curiosity, and for their
support and empathy. In particular, I thank the students of the Building Technology Lab, including Timothy
Cooke, Noel Davis, Teri Hall, Jonathan Krones, Andrea Love, and David Quinn. I also thank my colleagues in
the Structural Design Lab, which has become a flourishing scholarly community that I am honored to be a part
of, especially Rory Clune, Catherine De Wolf, Benjamin Jenett, Samar Malek, and William Plunkett. Finally, I
am indebted to student researchers who have directly helped me with the work in this dissertation: Virginie
Arnaud, Ali Irani, Andrew Sang, Iovana Valdez, and Yuxing (Jocelyn) Wang.
Tireless and incredibly organized members of the MIT staff have played an important role in the success of this
work. Most critically, I thank Kathleen Ross of the Building Technology Program for her never-ending energy
and competence, and for saving the day for me on numerous occasions. I also thank the staff of the Department
of Architecture headquarters, especially Renée Caso, for their clear-headed support and eagerness to help. I
would also like to recognize the knowledge and warmth of the MIT Libraries staff, who have tracked down
many obscure articles for me, and who have brightened my day during my visits to Barker and Rotch. I am also
grateful for technological assistance from CRON, who have responded with patience to many computing
emergencies over the years.
I am grateful to several generous sources of funding during my Ph.D. studies: the MIT Presidential Fellows
program, the MIT Department of Architecture, and the Amar Bose Teaching Fellowship.
Finally, I thank my parents, Mark and Liz Mueller, and my husband, Martijn Stevenson, for their exuberant
support of my academic pursuits. My parents cultivated a love of design and a passion for interdisciplinary
learning in me at a young age, and taught me to aspire to creativity and intellectual courage. Martijn has been a
close and extremely talented collaborator and partner, who continues to share his brilliance and expertise in
software engineering with me, and who has also happily joined me in my explorations of the world of
structures.
Table of Contents
List of Mathematical Symbols ................................................................................................................................... 15
I
INTRODUCTION
17
1. Problem Statement
19
1.1. Conceptual design of architecture and structures..................................................................................... 19
1.1.1. Significance of structural form ......................................................................................................... 20
1.2. Benefits of integrated structural design .................................................................................................... 21
1.2.1. Reduced environmental impact and construction cost .................................................................... 21
1.2.2. Architectural richness and elegance.................................................................................................. 21
1.2.3. Inherent safety and longevity ............................................................................................................22
1.2.4. Counterexamples................................................................................................................................23
1.3. Existing computational design tools ..........................................................................................................24
1.3.1. Geometry-based tools for architects .................................................................................................24
1.3.2. Analysis-based tools for engineers .................................................................................................... 25
1.4. Key structural design tool features ............................................................................................................ 25
1.4.1. Feedback features...............................................................................................................................26
1.4.2. Guidance features...............................................................................................................................26
1.5. Need for guidance-based structural design approach ..............................................................................26
1.5.1. Directed exploration ..........................................................................................................................26
1.5.2. Diversity and surprise ........................................................................................................................26
1.5.3. Rapid and interactive results ............................................................................................................. 27
1.6. Organization of dissertation ....................................................................................................................... 27
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C. T. MUELLER | PH.D. DISSERTATION, 2014
TABLE OF CONTENTS
2. Literature Review
29
2.1. Existing tools for structural design ............................................................................................................29
2.1.1. Graphic statics tools ...........................................................................................................................29
2.1.2. Real-time numerical structural analysis tools ................................................................................. 30
2.1.3. Integrated numerical analysis modules for architectural modeling tools....................................... 33
2.1.4. Critique of feedback-only tools .......................................................................................................... 35
2.1.5. Forming-finding tools for membrane and shell structures .............................................................. 35
2.2. Optimization in structural design .............................................................................................................. 35
2.2.1. Optimization problem formulation .................................................................................................. 38
2.2.2. Gradient-based optimization .............................................................................................................39
2.2.3. Heuristic optimization .......................................................................................................................39
2.2.4. Limitations of optimization in design .............................................................................................. 40
2.3. Promising directions beyond standard optimization ............................................................................... 41
2.3.1. Interactive design space navigation .................................................................................................. 41
2.3.2. Grammar-based design space formulations .....................................................................................43
2.3.3. Design Space approximation through surrogate modeling .............................................................44
2.3.4. Integrated design approach ...............................................................................................................44
2.4. Challenges and opportunities .................................................................................................................... 45
2.4.1. Specific research goals ....................................................................................................................... 45
II DESIGN SPACE STRATEGIES
47
3. Interactive Evolutionary Framework
49
3.1. Background on design space navigation....................................................................................................49
3.1.1. Navigation needs ................................................................................................................................49
3.1.2. Evolutionary algorithms ....................................................................................................................50
3.1.3. Interactive evolutionary algorithms .................................................................................................. 51
3.1.4. Applications in structural design ...................................................................................................... 52
3.1.5. Specific needs ..................................................................................................................................... 52
3.2. Framework overview .................................................................................................................................. 53
3.2.1. Framework and software architecture .............................................................................................. 53
3.2.2. Variables and design models ............................................................................................................. 55
3.2.3. Analysis engines ................................................................................................................................. 56
3.2.4. Population generator ......................................................................................................................... 57
3.2.5. Graphical user interface .....................................................................................................................58
3.2.6. Extensibility ........................................................................................................................................ 59
3.3. Enhanced interactivity and user input ..................................................................................................... 60
3.3.1. Multiple design selection .................................................................................................................. 60
3.3.2. Mutation rate ...................................................................................................................................... 61
3.3.3. Generation size ...................................................................................................................................62
3.4. Design quality and diversity enhancements ..............................................................................................63
3.4.1. Hybrid automatic-interactive functionality ......................................................................................63
3.4.2. Diversity booster ................................................................................................................................64
3.5. Expanded user experience.......................................................................................................................... 65
3.5.1. Model setup ........................................................................................................................................ 65
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C. T. MUELLER | PH.D. DISSERTATION, 2014
TABLE OF CONTENTS
3.5.2. Design refinement .............................................................................................................................. 67
3.6. Design example: cantilevered truss roof .................................................................................................. 68
3.6.1. Design problem formulation ............................................................................................................ 68
3.6.2. Evolution of candidate designs ..........................................................................................................69
3.6.3. Refinement of selected designs .........................................................................................................70
3.7. Additional design examples ....................................................................................................................... 74
3.8. Summary of intellectual contributions ...................................................................................................... 77
4. Trans-typology Structural Grammars
79
4.1. Background on design space formulation ................................................................................................. 79
4.1.1. Trans-typological design ................................................................................................................... 80
4.1.2. Parametric design spaces .................................................................................................................. 83
4.1.3. Rule-based design space ................................................................................................................... 83
4.1.4. Structural grammars ......................................................................................................................... 86
4.1.5. Specific needs ..................................................................................................................................... 87
4.2. Trans-typological design generation ........................................................................................................ 88
4.2.1. General approach .............................................................................................................................. 88
4.2.2. Structural shapes ............................................................................................................................... 88
4.2.3. Recursive rules .................................................................................................................................. 90
4.2.4. Rules and state labels ........................................................................................................................ 90
4.2.5. Parametric and structurally aware rules ........................................................................................... 91
4.2.6. Structural performance evaluation ...................................................................................................92
4.3. Design generation using grammar.............................................................................................................94
4.3.1. Manual rule application ..................................................................................................................... 95
4.3.2. Automatic random computation .......................................................................................................96
4.3.3. Hybrid manual-automatic computation ........................................................................................... 97
4.4. A trans-typology structural grammar for pedestrian bridges ................................................................. 98
4.4.1. Bridge design rules .............................................................................................................................99
4.4.2. Implicit structural information and analysis engine ......................................................................100
4.4.3. Randomly generated pedestrian bridge designs ............................................................................100
4.4.4. Additional possible grammars ......................................................................................................... 103
4.5. Summary of intellectual contributions .................................................................................................... 103
5. Performance-focused Surrogate Modeling
105
5.1. Background on design space approximation .......................................................................................... 105
5.1.1. Need for computation speed ............................................................................................................ 106
5.1.2. Approximation strategies ................................................................................................................108
5.1.3. Surrogate modeling strategies ......................................................................................................... 109
5.1.4. Specific needs ................................................................................................................................... 110
5.2. Ensemble black-box regression models as surrogates ............................................................................ 110
5.2.1. Advantages of black-box and ensemble methods ............................................................................ 111
5.2.2. Ensemble neural networks ............................................................................................................... 111
5.2.3. Random forests ................................................................................................................................ 112
5.3. Performance-focused modeling approach............................................................................................... 113
5.3.1. Weighted sampling plans................................................................................................................. 113
5.3.2. New rank-based error measures ...................................................................................................... 117
5.4. Automatic model building for non-experts ............................................................................................. 122
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TABLE OF CONTENTS
5.4.1. User-specified accuracy ................................................................................................................... 122
5.4.2. User-specified model-building preferences .................................................................................... 122
5.4.3. Automatic parameter setting ........................................................................................................... 124
5.4.4. Graphical testing results .................................................................................................................. 124
5.5. Surrogate modeling case studies .............................................................................................................. 126
5.5.1. Model accuracy ................................................................................................................................. 126
5.5.2. Model building time ......................................................................................................................... 128
5.6. Summary of intellectual contributions .................................................................................................... 128
III INTEGRATION AND CONCLUSIONS
131
6. Integrated Design Approach
133
6.1. Design space strategies applied together................................................................................................. 133
6.1.1. General integration strategy ............................................................................................................ 134
6.2. Evolutionary framework and structural grammars ................................................................................ 134
6.2.1. Design models and variables ........................................................................................................... 135
6.2.2. Crossover and mutation of variables ............................................................................................... 135
6.2.3. Analysis engines ............................................................................................................................... 137
6.2.4. Design problem setup ...................................................................................................................... 138
6.2.5. Design refinement ............................................................................................................................ 139
6.3. Evolutionary framework and surrogate modeling .................................................................................. 139
6.3.1. Automatic surrogate model building .............................................................................................. 139
6.3.2. Model predictions and updates in interactive mode ...................................................................... 140
6.3.3. Use of approximation in refinement mode ..................................................................................... 141
6.4. Structural grammars and surrogate modeling ........................................................................................ 141
6.4.1. Challenge of nonparametric formulation ....................................................................................... 141
6.4.2. Salient and emergent properties ..................................................................................................... 142
6.4.3. Rule counts and parameter values .................................................................................................. 142
6.4.4. Pruning the design vector ................................................................................................................ 143
6.5. Summary of intellectual contributions .................................................................................................... 143
7. Discussion and Conclusions
145
7.1. Need for novel design methodology ........................................................................................................ 145
7.1.1. Beyond guess-and-check ................................................................................................................. 146
7.1.2. Beyond rapid feedback ..................................................................................................................... 146
7.1.3. Beyond standard optimization ........................................................................................................ 146
7.2. Specific contributions ............................................................................................................................... 147
7.2.1. Interactive evolutionary framework ................................................................................................ 148
7.2.2. Trans-typology structural grammars .............................................................................................. 148
7.2.3. Performance-focused surrogate modeling ...................................................................................... 148
7.2.4. Integrated approach ......................................................................................................................... 149
7.3. Applications of proposed strategies ......................................................................................................... 149
7.3.1. In practice ......................................................................................................................................... 150
7.3.2. In the classroom ............................................................................................................................... 150
7.3.3. Historical analysis ............................................................................................................................ 150
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TABLE OF CONTENTS
7.4. Directions for future work .........................................................................................................................151
7.4.1. Practical needs...................................................................................................................................151
7.4.2. Technical needs ................................................................................................................................ 152
7.4.3. Theoretical needs ............................................................................................................................. 152
7.4.4. Cultural needs .................................................................................................................................. 152
7.5. Concluding remarks ................................................................................................................................. 153
IV APPENDICES
155
A. Structural Analysis Code Validation
157
B. Pedestrian Bridge Grammar Details
167
C. Automatic Surrogate Modeling Results
183
D. References
199
13
List of Mathematical Symbols
Symbol
Meaning
Truss element cross-sectional area governed by local buckling
Truss element cross-sectional area governed by axial stress
Cross-sectional area of th truss element
Area of steel
Width
Number of copies used in weighted sampling plans
Cost of assembly
Depth
Minimum allowable distance between designs to ensure diversity
Euclidean distance between th and th designs
Size of the design space based on Euclidean distance
Modulus of elasticity of a material
Axial force in truss element
Objective function
Vector of applied loads
Vector of reactions at supports
Sequence of rules comprising grammatical design representation
Inequality constraint
Equality constraint
Iterations
Required moment of inertia of a cross section
Effective buckling length
Submatrix of global stiffness matrix corresponding to free degrees of freedom
Submatrix of global stiffness matrix corresponding to free and fixed degrees of freedom
Local element stiffness matrix
Submatrix of global stiffness matrix corresponding to free and fixed degrees of freedom
Submatrix of global stiffness matrix corresponding to fixed degrees of freedom
Length of th truss element
Number of top-performing designs already included in group (for diversity check)
Bending moment
Factored design bending moment
Number of elements considered (specifics vary)
Number of top-performing designs used to compute error measures
Performance score
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C. T. MUELLER | PH.D. DISSERTATION, 2014
LIST OF MATHEMATICAL SYMBOLS
Setting of th parameter in a parametric rule
Uniform loading
Cost per connection
Observed rank of th observtion
Cost by volume of material
Mutation rate, ranging from 0 to 1
̂
Predicted rank of th observtion
Thickness
Normalized time
Transformation matrix from global to local coordinate systems
Vector of displacements of free degrees of freedom
Vector of displacements of element degrees of freedom in local coordinate system
Volume of structural material
Scalar weights (randomly generated in evolutionary crossover)
Design vector
th design variable in the design vector
th design variable in the design vector of the th design
Lower bound of
Upper bound of
Design variable setting resulting from crossover
Design variable setting resulting from mutation
Randomly generated design variable setting
Observed value of th observtion
̂
Predicted value of th observtion
Mean of a normal probability distribution
Standard deviation of a normal probability distribution
Variance of a normal probability distribution
Allowable stress of a material
16
PART I:
Introduction
“The loftiest and most difficult problems arise in architecture from the need to realize a synthesis between
opposing sets of factors: harmony of form and the requirements of technology, heat of inspiration and the
coolness of scientific reason, freedom of imagination and the iron laws of economy.”
— Pier Luigi Nervi in Structures, 1956
CHAPTER 1:
Problem Statement
This dissertation presents new computational strategies that encourage creativity in conceptual structural
design. The first chapter motivates this research with a discussion of current design approaches and available
tools, critiquing existing methods and identifying the needs and opportunities that the research in this
dissertation addresses.
1.1
Conceptual design of architecture and structures
In building design disciplines, including architecture and structural engineering, the design process is
conventionally divided into four sequential phases: Conceptual Design, Schematic Design, Design
Development, and Construction Documents (American Institute of Architects, 2007). In practice today, major
decisions regarding the building’s geometry, massing, and overall form are usually made during the first phase,
Conceptual Design (Hsu & Liu, 2000; Wang et al., 2002). This phase is typically carried out by the architecture
team alone, before strong involvement of engineering consultants.
After the project has already taken shape, structural engineers and other consultants typically begin work, with
the task of developing engineering strategies to enable the conceptual design vision, as illustrated in Figure 1.1.
This means that in standard practice, structural considerations are often subservient to architectural goals
(Macdonald, 2001). The design process is necessarily linear and unidirectional, and there are few opportunities
for structural input to inform or improve the initial concept in significant ways (Holgate, 1986).
19
C. T. MUELLER | PH.D. DISSERTATION, 2014
conceptual
design
CHAPTER 1: PROBLEM STATEMENT
schematic
design
design
development
construction
documents
100%
percentage
involvement of
structural engineers
design
freedom
design
knowledge
time into design process
Figure 1.1: Relationship between design freedom and design knowledge in building design projects. The most
opportunity for design impact and creativity occurs during conceptual design, but structural considerations usually enter
the process far later. This limits the ability of structural engineers to contribute impactful ideas in the design process.
After Fabrycky & Blanchard (1991) and Paulson (1976).
The structural engineering team’s tasks during schematic design include structural material and system
selection, preliminary structural member sizing, and the development of structural strategies for unusual
design elements and conditions. However, because much of the overall design geometry has already been set at
this stage, the engineering team rarely provides advice or feedback to the architecture team on form.
1.1.1 Significance of structural form
History, theory, and nature show that for structural performance, overall form matters much more than
material, member sizing, or internal topology (Thompson, 1942; Zalewski et al., 1998; Larsen & Tyas, 2003;
Allen & Zalewski, 2010). The geometry of a building’s structure directly determines the distribution and
magnitude of the forces it must resist (Macdonald, 2001). Uruguayan structural designer Eladio Dieste (1917 –
2000) is quoted in an elegant expression of this point:
The resistant virtues of the structures that we seek depend on their form; it is through their form that
they are stable, not because of an awkward accumulation of material. There is nothing more noble and
elegant from an intellectual viewpoint than this: to resist through form (Anderson, 2004).
As a simple example, Figure 1.2 shows three possible geometries for a long-span arch roof. As noted, the
maximum force in the least efficient form is three times that in the most efficient.
Today, with advances in a broad range of technologies, it is possible to design, analyze, and build forms
regardless of their structural performance (Addis, 1994). In fact, there is a recognized ingenuity in meeting the
challenge of making structurally poor forms work in spite of their inefficiencies (Macdonald, 2001). However,
this does not mean that this is the best way forward. This dissertation argues for an alternate paradigm in
which structural considerations are integrated into form-making in the earliest phase of the design process:
conceptual design.
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C. T. MUELLER | PH.D. DISSERTATION, 2014
(a) Fmax = 1500 kips
CHAPTER 1: PROBLEM STATEMENT
(b) Fmax = 750 kips
(c) Fmax = 500 kips
Figure 1.2: Three possible geometries for a long-span arch roof, with maximum axial force under uniform vertical
loading in the arch noted below. The increased curvature of design (c) reduces the internal forces in the arch by a factor of
three compared to design (a).
1.2
Benefits of integrated structural design
Integrating structure into conceptual design offers a way to harness the power of good structural form. There
are considerable advantages to this approach, as evidenced by both the historical and more recent examples
highlighted in the following sections.
1.2.1 Reduced environmental impact and construction cost
By finding efficient structural forms during conceptual design, considerable amounts of structural material can
be saved. Material savings means consuming fewer resources and spending less on construction. Historically,
structural designers such as Robert Maillart of Switzerland (1872 – 1940) were awarded projects by developing
the most cost effective designs through creative form exploration (Billington, 1983). More recently, the
Luxembourgian structural designer Laurent Ney (b. 1964) has similarly won design competitions with
structurally efficient and visually striking forms (Ney et al., 2010). Finally, examples like the Pines Calyx
conference center in Dover, England are able to significantly reduce embodied energy through a clear and
architecturally integrated structure developed in conceptual design (Ramage, 2007). These examples are
illustrated in Figure 1.3.
1.2.2 Architectural richness and elegance
Many in the architecture and design community have argued that a harmony between the aesthetic and
technical goals in a project imparts crucial value and rigor. For example, in describing the work of Italian
architect-engineer Pier Luigi Nervi (1891 – 1971), the architectural critic Ada Louise Huxtable writes, “His
buildings are most remarkable for the clarity of their engineering. The power and grace of these extraordinary
shapes and patterns stems directly from their structural logic, and are inseparable from it” (1960). Indeed,
Nervi’s approach used structure directly as a form-generating principle to discover new and exciting shapes for
architecture, as shown in Figure 1.4.
Other examples demonstrate the success achievable by simultaneously solving architectural and structural
problems. In the Dulles Airport Terminal by architect Eero Saarinen (1910 – 1961), the evocative swoop of the
hanging roof suggests flight, but also reveals the flow of internal forces through its structural simplicity. In the
San Francisco International Terminal designed by Skidmore, Owings & Merrill (SOM), the three-dimensional
21
C. T. MUELLER | PH.D. DISSERTATION, 2014
CHAPTER 1: PROBLEM STATEMENT
truss design achieves a long, column-free span while also allowing filtered daylight to enter the space. These
examples are illustrated in Figure 1.5.
(a) Shed roof by Robert Maillart in
Chiasso, Switzerland (1924).
Image from Billington (1990).
(b) Footbridge by Ney +
Partners in KnokkeHeist, Belgium
(2007). Image from
Ney et al. (2010).
(c) Pines Calyx Conference Center by
Cameron Taylor Bedford and the MIT
Guastavino Team in Dover, England
(2005). Image from The Bay Trust
(2012).
Figure 1.3: Design examples illustrating materials savings, and thereby reduced cost and environmental impact, through
integrating structure into conceptual design and architectural form selection.
(a) Airplane hangar in Orvieto,
Italy (1935). Image from Nervi
(1957).
(b) Gatti Wool Factory in Rome,
Italy (1951). Image from Nervi
(1956).
(c) Turin Exhibit Hall B
in Turin, Italy
(1949). Image from
Nervi (1957).
Figure 1.4: Projects designed by the Italian architect-engineer Pier Luigi Nervi.
1.2.3 Inherent safety and longevity
Building forms that result from integrated structural design are safe by their nature, rather than through
extreme exertion on the part of structural engineers and the high-strength materials they employ. Lower
internal forces make structures more robust and forgiving of material and construction variation. Examples
that still stand after hundreds of years, such as the masonry cathedrals of Europe and the timber stave churches
of Scandinavia, shown in Figure 1.6, prove that such forms are enduring.
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C. T. MUELLER | PH.D. DISSERTATION, 2014
(a) Washington Dulles Terminal by
Eero Saarinen in Virginia, U.S.
(1962). Image from California
Literary Review.
CHAPTER 1: PROBLEM STATEMENT
(b) San Francisco International Terminal
by SOM (2000). Image by Oleg
Sklyanchuk.
Figure 1.5: Examples of projects that concurrently fulfill architectural and structural goals.
(a) Cathedral in Reims,
France (1275). Image
by Magnus Manske.
(b) Borgund Stave Church in Norway (1180).
Image by Flickr user zoetnet.
Figure 1.6: Historical examples of projects that have endured due to their structural forms.
1.2.4 Counterexamples
In contrast, when architectural concepts are developed in absence of structural influence, results can be
wasteful, expensive, maintenance-intensive, and in the worst cases, unsafe. Architect Frank Gehry’s Walt
Disney Concert Hall in Los Angeles required a complex and materially intensive structure to fit inside and
support its whimsical forms (Naeim et al., 1999). Unlike his later Dulles Airport, the thin-shell roof of Eero
Saarinen’s Kresge Auditorium at MIT was famously designed according to geometric rather than structural
principles (Billington, 1983; Mark, 1990), resulting in unexpected large initial deflections and years of repairs
(Cohen et al., 1985). Finally, Terminal 2E of the Charles de Gaulle airport in Paris was shaped in a way that
induced large internal forces and depended on high-strength materials to stand up. This building collapsed in
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C. T. MUELLER | PH.D. DISSERTATION, 2014
CHAPTER 1: PROBLEM STATEMENT
2004, less than a year after its opening, killing four people and resulting in €130 million in repair and
replacement costs (Clark, 2008). These examples are illustrated in Figure 1.7.
(a) Walt Disney Concert Hall
by Frank Gehry in Los
Angeles , U.S. (2003).
Image by Flickr user
BudCat14/Ross.
(b) Kresge Auditorium by
Eero Saarinen in
Cambridge, U.S. (1955).
Image by Wikipedia user
Dadero.
(c) Collapse of the Charles de Gaulle
Terminal 2E in Paris, France
(2004). Image from the Daily
Mail.
Figure 1.7: Projects with forms not primarily guided by structural behavior.
1.3
Existing computational design tools
Today’s architecture and engineering practices make widespread use of computational tools throughout the
design process, and currently available tools both reflect and enforce existing design strategies (Hsu & Liu,
2000; Wang et al., 2002).
1.3.1 Geometry-based tools for architects
Architecture tools, starting with Computer-Aided Drafting programs in the 1980s, allow users to thoroughly
document, and more recently generate, both conceptual and detailed designs. An increasing interest in
complex geometry has led to powerful 3D modeling software which, coupled with scripting capabilities, enables
the development of impressively intricate forms, as shown in Figure 1.8.
Figure 1.8: Geometries generated using generative algorithms in the program Rhino and the plugin Grasshopper
(Khabazi, 2012).
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1.3.2 Analysis-based tools for engineers
Computational tools for structural analysis mirror architecture tools in their power and capacity for complexity,
and yet also maintain existing design roles. Finite element analysis (FEA) programs are capable of determining
stresses, deflections, and dynamic behavior for highly complicated geometry using sophisticated techniques, as
shown in Figure 1.9. Recent developments focus on increased accuracy and speed under a range of conditions.
However, these tools are of little use in conceptual design; they require that a geometry be provided to be
analyzed, and are incapable of assisting with geometry generation. Again, these tools relegate engineers to the
tasks of verifying the form and sizing the members, thus limiting or eliminating their involvement in conceptual
design.
Figure 1.9: Sample analysis output from SAP2000, a finite element analysis program (Computers and Structures, 2012).
1.4
Key structural design tool features
The emerging research area of conceptual structural design computation seeks to bridge the gap between these
existing computational approaches, enabling a true integration of structural input during conceptual design.
This dissertation identifies two key types of features for such tools, feedback and guidance, as shown in Figure
1.10.
Figure 1.10: Key features for structural design tools that encourage integrated conceptual design.
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1.4.1 Feedback features
A clear remedy for the lack of performance evaluation in geometry-generation tools is to integrate structural
analysis capabilities into such software. It is critical that such analysis be fast, or ideally real-time, to allow for
an interactive user experience. This type of feature shows users how design changes will affect structural
performance according to metrics such as required material volume, structural stiffness, or estimated
construction costs. This has been implemented in a number of applications both in research and practice, but
is limited by the speed of computational structural analysis.
1.4.2 Guidance Features
To shift engineering software from the existing analysis and verification focus, tools for structural design should
include form-guiding capabilities. This type of feature enables the software to suggest new geometries to the
user in order to improve the structural performance of a design concept. While the field of optimization offers
insight into ways to achieve this, there has been little progress in developing guidance-based tools for
conceptual design both in research and practice. To truly encourage integrated conceptual structural design
through modern computational tools, it is critical to develop methodologies that achieve this functionality.
1.5
Need for guidance-based structural design approach
This dissertation addresses the problem of integrating structural guidance into conceptual design through
computational means. To achieve this, there are three specific requirements for which this research offers
solutions through novel intellectual contributions.
1.5.1 Directed exploration
First, guidance-based tools must carefully balance the ability to suggest design changes with freedom of
exploration within the design environment. There is no single correct answer in architectural design, and it is
crucial that such tools allow for a plurality of design options, while nevertheless encouraging the user towards
those with better performance. Chapter 3 offers a new approach to achieve these goals using an interactive
evolutionary algorithm for design space navigation.
1.5.2 Diversity and surprise
For use in conceptual design, a guidance-based methodology should perform like a talented team member in a
brainstorming session, generating a broad range of new and unexpected design ideas. This capability is
important not only to improve structural performance, but also to discover exciting architectural forms. To
accomplish this, the methodology should incorporate a broad and varied design space. Chapter 4 presents a
strategy to formulate broad and diverse design spaces through structural grammars.
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1.5.3 Rapid and interactive results
Another challenge in integrating structure into computational design tools is that structural analysis can be
computationally expensive and slow. A successful computational approach should include rapid performance
prediction strategies to allow for an interactive, real-time user experience. Chapter 5 addresses this issue by
introducing a performance-focused surrogate modeling strategy for design space approximation.
1.6
Organization of dissertation
This dissertation is divided into three parts: Introduction, Design Space Strategies, and Integration and
Conclusions.
The first part, Introduction, includes the problem statement and critical literature review for the research
question considered in this dissertation.
Chapter 2 presents existing work in the field of computational tools and methodologies for conceptual
structural design. This includes a critical review of structural optimization. Additionally, this chapter contains
an overview of existing work relating to structural design tools in the three specific research areas that comprise
the original contributions of this thesis. Further detailed background and literature review for these topics are
provided in the subsequent chapters that present original work. Finally, this chapter summarizes current
challenges and identifies the opportunities that this dissertation responds to.
The second part, Design Space Strategies, describes three new computational strategies for conceptual
structural design, including additional literature review relevant to strategies employed in each chapter.
Chapter 3 introduces an interactive evolutionary framework for structural design that guides users toward high
performing designs while allowing for architectural exploration. This chapter includes extended background
information on interactive evolutionary algorithms, parametric problem formulation, design performance
evaluation metrics, and user controls and experience.
Chapter 4 proposes a new approach for trans-typology structural grammars that generate conceptual design
possibilities across typology boundaries. A detailed review of shape grammars and a prescription for the new
approach of structural grammars is included, as well as a discussion of structural typologies in conceptual
design. A specific new grammar is presented, including a discussion of grammar properties, states, and rules,
and examples of generated designs are also illustrated.
Chapter 5 discusses an approach for design space approximation to enable rapid performance evaluation of
candidate design concepts. It includes a review of surrogate modeling in other optimization applications as well
as a review of nonparametric regression techniques developed in the field of machine learning. The chapter
also discusses new model training procedures and error measures, and introduces a strategy for building
surrogate models in an automated way accessible for non-experts. The approach is exemplified through several
case studies.
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The third part, Integration and Conclusions, shows how the three previously presented methodologies could be
integrated into combined design approaches to solve a range of conceptual design problems, and concludes the
dissertation with a summary of contributions.
Chapter 6 discusses the possibilities and challenges of integrating the strategies in pairwise combinations and
into a single unified approach, including suggestions of techniques for achieving these integrations.
Chapter 7 summarizes the specific intellectual contributions of the thesis and discusses potential impact,
envisioned applications, and important directions for future research.
Additionally, there are three appendices that document detailed results referred to in the Design Space
Strategies chapters. References are also included in Appendix D.
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CHAPTER 2:
Literature Review
This chapter presents existing work in the field of computational tools for conceptual structural design,
including a critical review of existing feedback-based tools and the field of structural optimization.
Additionally, this chapter identifies and discusses specific developments in three key areas, and illustrates the
need for further research that the work of this dissertation addresses.
2.1
Existing tools for conceptual structural design
As noted in Chapter 1, the majority of computational tools used in architecture and engineering in practice are
either geometry-driven or analysis-driven, and reflect the lack of overlap between the two disciplines.
However, some progress has been made in developing tools that bring these functionalities together to assist
with conceptual structural design.
Almost all such tools employ feedback functionality, one of the two key features identified in Chapter 1. This
section will present an overview of these tools, and will argue that new innovations are needed to bring the
second key feature, guidance, to tools available for practitioners.
2.1.1 Graphic statics tools
Graphic statics is a graphical, as opposed to numerical, method of calculating internal forces in axially-loaded
structures such as arches, cables, and trusses. Developed from fundamentals established in the early 1800s, the
method was formalized in 1866 by Culmann in his book Die graphische Statik (1866). A recent book by Allen
and Zalewski (2009) gives an overview of the method and applies the method to conceptual design problems.
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Engineers made widespread use of this technique for both design and analysis until the 1970s, when numerical
methods gained prominence due to the increasing calculation power of computers.
Recently, there has been renewed interest in graphic statics because of its rediscovered simplicity and power.
Several researchers have developed computational implementations that allow users to manipulate structures
in real time and observe the how internal forces change through the force polygon. One pioneering example is
Active Statics, an online tool that contains seven interactive design examples (Greenwold & Allen, 2003). A
screenshot from this tool is shown in Figure 2.1.
Further advancements are evident in eQULIBRIUM, an online interactive tool that illustrates graphic statics
techniques on a wider range of example problems (Van Mele et al., 2009-2012). Additionally, Shearer (2009)
has created RhinoStatics, a plug-in for the 3D modeling software Rhinoceros that performs graphics statics
analysis of structures drawn by users. These are illustrated in Figure 2.3 and Figure 2.2 respectively.
While constituting an important step forward, this class of tools is limited in several ways. First, graphic statics
techniques are restricted to relatively simple problems, generally two-dimensional and statically determinate.
Second, with the exception of Shearer’s work, most currently available graphic statics computational tools work
only on pre-set examples, and are not flexible enough to provide feedback on a design problem presented by the
user. These issues are addressed in the next class of tools, which provide real-time numerical structural
analysis.
2.1.2 Real-time numerical structural analysis tools
Several tools have been developed that employ full numerical structural analysis, or finite element analysis, to
provide real-time or rapid feedback about structural performance, including internal forces, reactions, and
sometimes required material or cost, to users. These tools tend to be structural analysis programs directed at
engineers, with the promise of allowing for a more free exploration of structural forms. The advantage of this
class of tools over traditional structural analysis programs is the speed with which they convey results.
There are numerous examples, both in academic research and commercial use, as shown in Figure 2.4 through
Figure 2.8. One of the first programs of this type is Arcade, a free academic software tool, which uses a physics
engine to simulate the dynamic behavior of two-dimensional structures in real time (Martini, 2006). SAP2000,
a widespread commercial structural analysis program, first introduced the Model-Alive feature, which offers
real-time analysis for small to medium-sized structures, with its Version 12 (Computers and Structures, 2008).
Dr. Frame 3D is a commercial software program that allows for real-time static analysis for a range of threedimensional problems (Dr. Software, 2009). Work by Clune (2010) includes a two-dimensional structural
design environment for truss structures that provides real-time feedback for multiple objectives – weight,
compliance, and cost – and also incorporates optimization functionality. More recently, Autodesk released
Force Effect, a tablet application that allows users to analyze and design structures in real time using a mobile
device (Autodesk, 2011).
While these tools are effective to varying degrees, they are all restricted in the size and complexity of the
structures that they analyze in real time because of computational limitations. Additionally, most tools of this
class exist within the realm of structural engineering software, and are not designed to be used by architects or
designers with less technical backgrounds. This limits their applicability in conceptual design.
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Figure 2.1: Active Statics computational tool to use graphic statics in an interactive environment (Greenwold & Allen,
2003).
Figure 2.2: eQULIBIRUM, an interactive online tool that illustrates graphic statics techniques through a range of
examples (Van Mele et al., 2009-2012).
Figure 2.3: RhinoStatics computational tool to implement graphic statics within a CAD environment as a Rhino plugin
(Shearer, 2009).
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Figure 2.4: A screen shot of Arcade, showing a design process based on rapid feedback from the tool (Martini, 2006).
Figure 2.5: Model-Alive in SAP2000, showing the removal of a member and updated analysis (Computers and
Structures, 2008; 2011);
Figure 2.6: Several screenshots of Dr. Frame 3D, a real-time three-dimensional structural analysis program (Dr.
Software, 2009).
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Figure 2.7: Work by Clune (2010) showing an interactive real-time analysis modeling environment.
Figure 2.8: ForceEffect real-time statics simulation for tablet and phone devices by Autodesk (2011).
2.1.3 Integrated numerical analysis modules for architectural modeling tools
Finite element analysis tools that integrate directly into architectural drawing and modeling programs allow for
a smooth and fluid workflow, without the need to transfer design information and results between software
programs. These tools are conceived as modules or plugins that perform structural analysis directly on
architectural or geometric models. There are a number of examples of such tools, including Geometry Gym for
Rhinoceros (Mirtschin, 2011) and Robot for Revit (Autodesk, 2012) .
While attractive, these tools have several drawbacks. From a practical standpoint, they are tied to the modeling
program into which they are integrated, and are therefore only accessible to designers who use that program.
Due to both the high rate of technology turnover and the breadth of programs in use, this presents a serious
limitation. From a theoretical standpoint, it is generally problematic to treat an architectural geometry model
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directly as a structural model, since many assumptions about structural properties, boundary conditions, and
behavior must be made in translation.
.
Figure 2.9: Finite element analysis of a component within the Rhinoceros modeling environment using Geometry Gym
(Mirtschin, 2011).
Figure 2.10: Integration of Autodesk Revit and Robot, a structural analysis software program (Autodesk, 2012).
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2.1.4 Critique of feedback-only tools
The types of tools reviewed in this section begin to address the gap between geometry tools and analysis tools
by bringing structural analysis to architectural geometry in a rapid way. However, while rapid feedback
improves existing design methods by increasing speed, it does not fundamentally change them. Feedback-only
tools still enforce a geometry-first, analysis-second paradigm that amounts to a guess-and-check approach. To
move beyond this, tools must offer a way to synthesize new geometries using structural principles implicitly.
2.1.5 Form-finding tools for membrane and shell structures
One compelling way for designers to explore this synthesis is with a set of tools that employ form-finding
techniques. These tools use various algorithms to discover equilibrium configurations for spatial structures
that contain little or no bending, and move beyond feedback in important ways. Key examples of such tools
include CADenary, a particle-spring tool for exploring pure-compression and pure-tension structures (Kilian &
Ochsendorf, 2005; Kilian, 2006), RhinoVAULT, a tool for designing compression-only structures using thrust
network analysis (Rippmann et al., 2012), and a web-based numerical form-finding tool from Princeton’s Form
Finding Lab that uses dynamic relaxation for shell design (Adriaenssens et al., 2012; Adriaenssens, 2014).
These tools move beyond feedback to guide designers to high-performing design options. However, they only
work for a narrow range of structural typologies, and are not generally applicable to problems beyond
membrane and shell structures. It is therefore necessary to look for a broader approach that can be used
systematically on a range of problem types. As suggested in Chapter 1, this can be achieved, in theory, by
structural optimization.
2.2
Optimization in structural design
Structural optimization is a promising field with a rich history, but it has nevertheless yet to make a significant
impact on structural design in practice. This section explains the development of structural optimization theory
and discusses the reasons for its disconnect with design.
The history of structural optimization can be traced back to Galileo Galilei (1564 – 1642), who in 1638
determined the minimal-material shape of a cantilevered beam subjected to a point load at its free end
(Timoshenko, 1953; Heyman, 1998). By finding the parabolic profile, as illustrated in Figure 2.11, Galileo
showed that mathematics can be used to find forms that use material as efficiently as possible to support a
given load. For many years since, this has been the goal of structural optimization.
Since Galileo, scholars have solved a steady stream of increasingly complex structural optimization problems
(Wasiutynski & Brandt, 1963). One of the most well-known contributions comes from Anthony G. M. Michell’s
(1870 – 1959) work on another cantilever problem almost three hundred years after Galileo’s original work.
Michell showed how to find an optimal truss solution for the point-loaded cantilever problem (and a few
others) in his seminal 1904 paper, “The Limits of Economy of Material in Frame-structures,” as shown in
Figure 2.12. Like Galileo, Michell was looking for minimal-material analytical solutions for key canonical
problems, rather than offering a general approach for optimization of any structure.
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Figure 2.11: Drawings from Galileo’s Dialogues Concerning Two New Sciences (1638), showing in (a) an incorrect
linearly varying solution for the minimal-material shape of a cantilevered constant-width beam supporting a point load at
its tip, along with (b), the correct parabolically varying solution (Timoshenko, 1953).
Figure 2.12: Illustration from Michell’s 1904 paper which laid the foundations of truss optimization. This figure shows
the optimal form and member distribution of a cantilevered planar truss structure subject to a point load.
A more general approach that resembles methods in use today was developed in the 1960s, with critical work by
Schmit (1960). A cohesive overview of work since is given by Spillers & MacBain (2009). In contrast with the
analytical methods of scholars like Galileo and Michell, the new numerical methods attempted to find the
optimum by iterating through potential solutions in a systematic way (Kirsch, 1981). While iterative
approaches were practically impossible in the days of manual calculation, the newly developed computers
brought rapid calculations for large problems to reality.
Importantly, structural optimization researchers in the 1960s referred to their discipline as structural synthesis
(Schmit, 1981; Vanderplaats, 2010), revealing the early aspirations of the field and evoking ideas of design in its
truest sense: creating something new. However, the work actually dealt with choosing member cross sections
for predetermined geometries and member configurations (Fox & Schmit, 1966). For example, Figure 2.15
shows a three-dimensional truss tower with 25 elements, whose cross sections were selected using a numerical
weight minimization algorithm. This type of problem is referred to as size optimization. While improvements
since the 1960s have broadened the reach of structural optimization strategies, the general disconnect between
the goals and reality of structural optimization persist today. In short, although structural optimization aims to
generate new and exciting forms, most applications are limited to rather narrow problem spaces.
An important step forward in structural optimization was the development of shape optimization, or the
determination of overall structural form as opposed to element sizes (Vanderplaats, 1982; Bennett & Botkin,
1986; Haftka & Grandhi, 1986). Most applications of this early work were in structural design of components in
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the automotive and aerospace industries, where an improved part would be used hundreds or thousands of
times, yielding extensive savings, although there are also examples of shape optimization for trusses,
sometimes called geometry optimization. Because it deals with overall form, shape optimization is more
relevant to conceptual design than size optimization. An illustration of shape optimization for mechanical
design is given in Figure 2.14.
Figure 2.13: 25-bar trussed tower with member cross sectional diameters and wall thicknesses chosen by an
optimization algorithm (Fox & Schmidt, 1966).
Figure 2.14: Shape optimization of a mechanical bracket supporting a rigid axel, with the objective of “minimizing
structural weight while assigning maximum allowable values to the von Mises stress” (Bennett & Botkin, 1986).
The third type of structural optimization used today is topology optimization, or the optimal connective
arrangement of elements in a structure, developed numerically in the late 1980s (Bendsøe & Kikuchi, 1988;
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Rozvany, 2001; Rozvany, 2007). This type of optimization can also be integrated with shape optimization and
size optimization.
Specific methods have been developed to address each of the three classes of structural optimization problems,
but in general they share a common formulation, described in the following subsection.
2.2.1 Optimization problem formulation
Formally, structural optimization is a numerical method of finding the best solution according to
mathematically formulated functional requirements, or objectives, while conforming to mathematically
formulated constraints. The solution is expressed in the form of numerical values for a design vector, , which
represents a list of design decisions to be made – for example, nodal positions, material selections, cross
sections – called design variables.
The objective function, ( ), is often a calculation of the weight or volume of the structure, such that a minimalmaterial structure can be found. However, this function can also consider stiffness, strain energy, deflection,
dynamic behavior, or other quantitative goals, structural or otherwise. Objective functions are standardly given
as functions to minimize, although maximization functions can easily be used, converted to standard form by
minimizing the negative of the function. As indicated, the objective function is computed based on the values of
the design vector. The optimal design vector will yield an objective function with the smallest possible value.
The constraints, ( )
and ( )
, and the variable bounds,
and
, restrict the solutions according
to design or behavioral requirements. More specifically, design constraints can represent geometric or spatial
requirements, constructability or fabrication limitations, or other functional considerations (Kirsch, 1981).
Behavioral constraints set limitations on structural behavior, and include restrictions on performance metrics
like internal stresses, deflections, or buckling capacity (Kirsch, 1981). Like the objective function, constraint
functions are calculated based on the values of the design vector. A feasible design solution must not violate
any of the constraints. It is also possible for a problem to be formulated without constraints; this is referred to
as unconstrained optimization.
Together, the design vector, constraints, variable bounds, and objective function define a design space, or
solution space, for a given problem. The dimension of this space is given as one more than the number of
design variables, to represent the space of possible design vector values and their resulting objective, or
performance, values. Structural design problems often have design spaces that are large and complicated. As
an example, a simple structural optimization problem and its design space are shown in Figure 2.15.
The optimization problem is stated mathematically as follows:
( )
( )
( )
Depending on the nature of the design variables, the objective function, and the constraints, a variety of
optimization algorithms are available to solve this problem. The two main classes of optimization algorithms,
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gradient-based and heuristic, will be presented in brief overviews in the following subsections. As a whole, this
approach to numerically computing the design vector to minimize the objective function according to
constraints will be referred to as standard optimization in this dissertation.
2.2.2 Gradient-based optimization
In the broadest terms, gradient-based optimization works by finding the point in the design space at which the
gradient or derivative of the objective function is zero, or where no improvement can be made without violating
constraints. There are a wide variety of sophisticated algorithms that use this general approach, or at least
make use of gradient information (Bertsekas, 1999; Papalambros & Wilde, 2000). This class of algorithms has
the benefit of extensive theory, including guarantees that computation will converge to an optimal result at
proven rates.
(a)
(b)
(c)
Figure 2.15: A simple 3-bar truss sizing problem (a); the variable and constraint plot (b); and the design space showing
objective function contours (c). (Kirsch, 1981).
However, there are limitations for using gradient-based optimization on so-called messy problems, which are
the types often found in engineering and design. Some of the biggest issues include a lack of convexity,
meaning that multiple local optima exist. Gradient-based approaches are unable to handle this on their own.
Additionally, in engineering problems and structural design, the objective function is usually evaluated in such
a way that derivatives do not exist, such as through black-box simulations. Gradient-based approaches must
then work around this issue by approximating gradients through many expensive function evaluations, which
can be both time-consuming and inaccurate.
2.2.3 Heuristic optimization
Heuristic optimization algorithms, sometimes called stochastic optimization algorithms, address these issues
well. Instead of using gradient information in design space exploration, they incorporate randomness in a
variety of ways. The most well-known method in heuristic optimization is genetic algorithms, a form of
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evolutionary computing that uses Darwinian natural selection theory to grow and evolve populations of designs
(Bentley, 1999).
These approaches are attractive for messy engineering problems, but also have important drawbacks. Unlike
gradient-based methods, heuristic optimization approaches are not guaranteed to find the optimal solution,
and may take a long time. However, they have been shown empirically to work well on the types of problems
found in structural design, which usually have many local optima and undefined gradient information
(Rayward-Smith et al., 1996).
2.2.4 Limitations of optimization in design
Despite the rich academic history of structural optimization, it has had relatively little impact on structural
engineering in practice. (One important counterexample is the work of SOM’s William Baker and his
collaborators, who have worked to apply structural optimization to real design projects in new ways (Stromberg
et al., 2011; Baker et al., 2012). However, their efforts remain exceptional in the broader building engineering
and design industries). Fundamentally, this can be attributed to an inherent difference in goals between
optimization and the design of buildings. While optimization is necessarily a convergent process, or one in
which an iterative and systematic algorithm converges upon a single solution, design is decidedly divergent. In
design, it is recognized that a variety of significantly different yet suitable solutions can be found from a single
starting point.
Moreover, the exercise of mathematically formulating objectives and constraints is difficult or impossible in the
design of buildings. Many important goals and requirements are qualitative, or even subjective, such as visual
impact, spatial experience, contextual fit, and overall architectural value. Since most structural design cannot
occur in the absence of architectural goals, this presents a significant challenge.
In addition, the design process for buildings is often one of discovery: designers do not know all of their
objectives and constraints at the beginning of the process, but develop them as they explore design possibilities.
The designer’s interaction with the process of evaluation and iteration is key. In contrast, standard
optimization is a relatively rigid and automated process in which goals and requirements must be enumerated
completely at the start. Unlike the human design process, optimization on its own cannot handle unformulated
objectives and constraints.
Another limitation of optimization in conceptual structural design is the design vector . Like the objectives
and constraints, this list of design parameters must be fully established at the beginning of the process.
Because the design vector completely defines design possibilities in a narrow way, it effectively predetermines
the final design. This precludes optimization as an explorative approach able to generate design diversity,
which is critical in conceptual design and should be included in a computational guidance-based tool.
From a more practical perspective, structural optimization can be very computationally time-consuming for
realistically sized problems. This contrasts strongly with the rapid-fire brainstorming sessions typical of
conceptual design, and seriously limits the use of optimization in an interactive design tool. Part of this issue is
due to the high-powered structural analysis engines running behind optimization algorithms, which are
arguably too detailed and sophisticated for the lower level of accuracy needed in conceptual design.
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Finally, most structural designers lack intensive training in optimization, and there are few tools or approaches
available that make optimization accessible to non-experts. Furthermore, optimization tools that do exist are
often text-based or severely limited in their graphical displays, and often rely on piecing several pieces of
software together. Human designers are necessarily highly visual, and can process and evaluate information
much more quickly and fully when it is presented graphically. Therefore, in order to be useful for designers in
practice, tools that use optimization should be easy to use, integrated, and strongly graphical.
2.3
Promising directions beyond standard optimization
Given the issues with standard optimization in conceptual architectural and structural design, it is necessary to
look beyond the established approaches to find ways to bring computational design guidance to conceptual
design tools. This section identifies three promising methodological directions that suggest remedies to the
problems noted above, and which constitute the basis for the novel intellectual work of this dissertation.
This section outlines background work and identifies needs for improvement in each area. A further and more
detailed review of relevant theory and development in each area is given in the beginning of the corresponding
chapter that presents novel work.
In addition to developing individual methodologies, this dissertation also addresses the need to combine
disparate methods into integrated approaches for conceptual structural design. This section will also discuss
existing work in the field of unified and integrated design environments for structural design.
2.3.1 Interactive design space navigation
As discussed previously, heuristic optimization is often preferable in real-world engineering problems due to its
robustness and ease of use. Arguably, the use of randomness also approximates human creativity by allowing
extraneous influences to enter the process of design discovery. However, on their own, heuristic optimization
algorithms still have the problem of being too heavy-handed and reliant on a pre-formulated and quantitative
problem setup.
Interactive heuristic optimization addresses this issue in a simple but compelling way: the designer is allowed
to interact with the computer algorithm in deciding which designs to pursue in the iterative optimization
process. The exact mechanics of the interaction depend on the specific heuristic algorithm chosen. In general,
the interactive element allows the user to only partially formulate the design problem in a quantitative way, and
to use unformulated or newly discovered objectives and constraints to make design selections.
A growing body of research in this field suggests that interactive heuristic optimization is a promising way to
move beyond standard optimization for problems with qualitative goals, like architectural and structural
design. Most notably, von Buelow (2008) has shown important results for the design of truss bridges and other
simple structures using interactive evolutionary algorithms. Martini (2011) has also recently presented
important work using a harmony search algorithm to produce multiple design results for a tied arch bridge.
Examples from these contributions are shown in Figure 2.16 and Figure 2.17.
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These results are encouraging, but suggest several challenges. First, existing work is very problem-specific and
not yet generalized to apply a single approach for a range of problems. Second, the literature lacks adequate
discussion on the full user experience of using such an algorithm, including problem setup and parameter
setting. Third, there is little work on the quality and diversity of solutions presented to the user.
These issues are addressed in the original contributions presented in Chapter 3, which also includes a more
thorough background on design space navigation approaches.
Figure 2.16: Truss designs using von Buelow’s interactive evolutionary design tool (von Buelow, 2008).
Figure 2.17: Tied-arch bridge design from Martini’s multi-modal close harmony search algorithm (Martini, 2011).
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2.3.2 Grammar-based design space formulations
As previously noted, classical optimization approaches are limited by the definition of the design variables,
sometimes called design parameters. Designs generated according to a list of parameters must necessarily be
parametric variations of each other, and there is a finite and enumerable list of possible design solutions. In
conceptual structural design, it is preferable to consider as wide a range of options as possible. Truly broad
design space exploration is difficult with parametric approaches.
In contrast, grammars can generate an infinitely wide range of designs through the iterative application of
rules. In computational architecture research, there is a wealth of literature on shape grammars, originally
developed by Stiny and Gips (1972), which illustrates this point. Shape grammars are sets of rules that act on
geometric shapes to generate designs.
In the world of structural design, and more broadly engineering design, there is promising research suggesting
that shape grammars are a way forward. For example, Shea & Cagan (1997; 1998; 1999a; 1999b) have shown
that grammars for truss design can be integrated into structural design tools, shown in Figure 2.18.
Additionally, Byrne et al. (2011) have developed work on shape grammars for bridge design using evolutionary
algorithms, shown in Figure 2.19.
Figure 2.18: Trussed arch designs generated using a structural grammar (Shea & Cagan, 1998).
Figure 2.19: Arched pedestrian bridges designed using a three-dimensional shape grammar (Byrne et al., 2011).
One strong limitation of existing work is its lack of implicit engineering information. Structural designs are
defined by far more than geometry, even at the conceptual stage, as they contain information about materials,
connections, loads, boundary conditions, and general behavior. Structural grammars go beyond shape
grammars to encode this information into generated designs.
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Furthermore, the true power of diverse design generation has yet to be shown through the use of grammars in
the structural design realm. Existing grammars like the one found in Byrne et al. (2011) work within specific
typologies of structures, such as suspension bridges or long-span trusses. A successful structural grammar for
conceptual design should allow users to move between existing typologies, and generate and compare designs
of several types. In other words, such a grammar should be trans-typology.
These issues are addressed in Chapter 4, which also includes a more thorough review of shape grammars and
grammars used in engineering design.
2.3.3 Design space approximation through surrogate modeling
As indicated earlier, the computational approaches used in existing tools, including optimization algorithms,
can be very slow for large design problems. While inconvenient in standard optimization, this becomes
prohibitive in interactive approaches in which the user expects speedy responses from the computer in a design
session.
Optimization scholars have developed a set of strategies to mitigate this issue by substituting a lower fidelity, or
surrogate, model that is faster to analyze for the original high fidelity design model. In other words, the
performance of a design can be predicted by an approximate version of the design in a shorter amount of time.
One specific version of this approach that has been successful involves building regression models of the
problem’s design space as the approximation (Forrester et al., 2008; Quiepo et al., 2005).
While surrogate modeling is an established technique in the field of optimization and some applications, such
as aircraft design, there are no existing tools or approaches that utilize it in structural design for buildings. One
reason for this is that surrogate models take expertise to construct, and while they save analysis time, they can
be time-consuming to create in their own right. Additionally, accuracy of surrogate models for structural
problems can be difficult to attain due to the underlying structural equations, which often yield discontinuous
and highly nonlinear design spaces.
Regression modeling has been used in related fields in which objective functions and constraints can be
formulated with relatively simple equations. For example, researchers in building science have successfully
approximated building energy behavior for quick feedback during conceptual design (Signor et al., 2001).
However, because of the complicated design space in structural design problems, regression approaches that
merely choose coefficients for preset equations, or parametric regression, are not sufficient.
These issues are addressed in Chapter 5, which also includes additional background on surrogate modeling
techniques and machine learning approaches.
2.3.4 Integrated design approach
While each of these techniques has been well developed within its own discipline, there have been very few
examples of adaptation for use in conceptual structural design. Furthermore, there are no existing tools that
offer ways to integrate these techniques into a single design approach.
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Research in software for structural design argues that the underlying code should be written in a modular and
extensible way so that additional model types and analysis techniques can be easily integrated (Clune et al.,
2012). The combination of the particular techniques discussed here presents additional challenges related to
their specific nature. A unified approach should also allow additional modules to be added or replaced by
future users.
Besides the underlying software architecture for a unified tool, serious attention should be given to the user
experience and interface design. Academic software tends to overlook this aspect, but it is critical that the user
interface be well developed and designed for several reasons. First, an interactive design approach depends on
user input, and the success of the interaction is highly linked to the user’s experience. Second, tools that are
difficult or unpleasant to use tend to be quickly overlooked or forgotten, limiting the impact of the underlying
research.
Suggestions for integrated design approaches that addresses these issues, including an implementation through
a computational design environment, are presented in Chapter 6. This chapter also gives more specific
background on combining evolutionary algorithms, structural grammars, and surrogate modeling
approximation.
2.4
Challenges and opportunities
This chapter has shown that in the realm of structural design tools, feedback features are well represented but
not sufficient on their own. Features that provide guidance, or suggestions for high performing designs, are
both important and lacking in tools available to designers.
While optimization is the clear and well-established computational approach for design guidance, its effect on
tools used in practice has been minimal, for the reasons outlined in this chapter. This dissertation proposes
three strategies for moving beyond standard structural optimization to develop tools that provide design
guidance and that are useful and accessible for designers. Fundamentally, these strategies address the issues of
design space navigation, design space formulation, and design space approximation. Additionally, this
dissertation discusses the potential for combining these strategies in integrated computational design
environments.
As shown in the previous section, work in each area is encouraging, but preliminary, with critical needs to be
addressed. This dissertation aims to both make new intellectual contributions in the three specific areas
outlined, as well as to suggest combined approaches for creative conceptual structural design.
2.4.1 Specific research goals
Based on the existing work reviewed in this chapter and the identified challenges, this dissertation has several
specific research goals:

Expand upon work in interactive evolutionary algorithms for structural design to generalize the
approach into a framework, incorporate a more fluid and interactive user experience, and improve the
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quality and diversity of designs presented to the user. Work that addresses this need is presented in
Chapter 3.

Develop a computational approach to generate diverse structural designs using grammars. Work in
this area is presented in Chapter 4.

Apply surrogate modeling techniques to structural design tools, with an emphasis on automatic model
generation and rank-based preferences. Chapter 5 presents research that addresses these needs.

Address key issues in combining the three new strategies into integrated approaches for conceptual
structural design. Chapter 6 presents work in this area.
46
PART II:
Design Space Strategies
“The process of visualizing or conceiving a structure is an art. Basically it is motivated by an inner
experience, by an intuition. It is never the result of mere deductive logical reasoning. Yet, as in all art, there
is the possibility of establishing certain general rules, though it should be well understood that those
enunciated here are not all and may not be even the most important ones.
…
There is no method that enables us automatically to discover the most adequate structure type to fit a specific
problem, as it is faced by the designer.”
— Eduardo Torroja in Philosophy of Structures, 1958
CHAPTER 3:
Interactive Evolutionary Framework
This chapter introduces the first of three design space strategies, an interactive evolutionary framework for
conceptual structural design. This framework is an extensible and generalized approach for using interactive
evolutionary algorithms to navigate the design space of a broad range of structural design problems.
Specifically, three original intellectual developments are presented in this chapter: enhanced approaches for
user interactivity, a new method to promote design quality and diversity, and a broadened user experience.
3.1
Background on design space navigation
Chapter 2 introduced interactive heuristic optimization as a strategy to move beyond the pitfalls of standard
optimization in structural design. This section further develops the case for interactive heuristic methods, and
specifically argues that interactive evolutionary algorithms are a promising approach for conceptual design.
3.1.1 Navigation needs
In what ways can designers navigate the space of possible solutions to a design problem in search of good
alternatives? The most obvious way is through random or educated guessing: think of several possible
solutions and evaluate them. Through rapid feedback, design tools offer users a way to map out small portions
of the design space in this way. Another approach is to be guided to the best solutions by a computer algorithm.
Optimization-based guidance can bring users directly to the point of interest, the optimal design.
However, neither of these approaches is completely satisfying. Ideally, a tool should point users in the
directions of good designs, but should still allow them the freedom to explore. This is well illustrated by a
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simple example. Figure 3.1 shows a planar seven-bar simply supported truss with a point load on its central
node. A small design problem is to determine the horizontal and vertical position of the lower left node, which
is mirrored by the lower right node through bilateral symmetry. The performance of the design can be
computed as the weight of structural material required to support the load, which will vary with the shape of
the truss. In this case, the truss is assumed to be made from steel tubes with a wall thickness of 5% of the outer
radius. Even a very straightforward problem like this has a very complicated design space. Because there are
two design variables, the design space can be visualized in three dimensions, as shown in Figure 3.1.
Figure 3.1: A seven-bar simply supported planar truss with a central point load, and a resulting design space. There are
two design variables, the horizontal and vertical position of the lower left node, as indicated by the arrows. The designs
highlighted on the right are isoperforming alternatives that perform 10%, 20%, and 30% worse than the optimal point,
exhibiting increasing diversity and potential to meet a range of important but non-numerical design goals.
In this example, there are two local optima, one of which is the global optimum. However, it is also important
to note the shallowness of the design space around these optimal designs: there are many designs that perform
almost as well as the two best, while varying significantly in their appearance. An ideal design guidance
approach would lead users toward these high-performing regions, but also expose them to rich design diversity.
A powerful strategy to accomplish this is the evolutionary algorithm.
3.1.2 Evolutionary algorithms
Evolutionary algorithms are a general class of optimization strategies that use the principles of Darwinian
natural selection to grow and evolve populations of designs (Bentley, 1999). Like other heuristic algorithms
discussed in Chapter 2, they have the advantages of being robust and well-suited to complicated engineering
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problems. Because they incorporate randomness, they avoid getting stuck in local optima, and can effectively
hop around the design space in search of better solutions.
Furthermore, because they work with populations of candidate designs, evolutionary algorithms are especially
useful in promoting design diversity. Unlike algorithms that focus on improving single solutions, these
algorithms improve a set of alternative options as they iterate. The general procedure is to randomly initialize a
first generation, evaluate the fitness of each member of the generation, identify the top performers, and use
those to create a subsequent generation by combining and mutating them. In standard evolutionary
algorithms, the process runs automatically until preset criteria are reached, and a single solution is presented as
the optimum. However, it is also possible to take better advantage of the design diversity created by this
approach by incorporating human interaction.
3.1.3 Interactive evolutionary algorithms
On their own, evolutionary algorithms are subject to the same criticisms as other standard optimization
approaches, as detailed in Chapter 2. However, because of their population-based approach and selection
mechanics, evolutionary algorithms lend themselves particularly well to human interaction. Interactive
evolutionary algorithms are a subclass of optimization algorithms that use principles of evolution combined
with human input to drive design space navigation. The general iterative process for this type of algorithm is
illustrated in the diagram in Figure 3.2. The cycle differs from standard evolutionary algorithms at the design
selection step. The algorithm identifies top performers, but solicits input from the user to make final choices
about which designs to proceed with to form the subsequent generation. This key difference allows the
designer to adjust the optimization process based on unformulated goals, such as visual impact or
constructability requirements. Furthermore, the user may adapt goals across generations, based on newly
realized design criteria discovered in the explorative process.
Figure 3.2: General diagram of an interactive evolutionary algorithm, including the interactive step in which the
designer selects offspring (highlighted in blue).
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The first interactive evolutionary algorithms were developed by Sims (1992) for the purpose of finding visually
interesting cellular automata. In this early case, selection was entirely based on user preferences, rather than
on a combination of user preferences with calculated objective functions. The literature includes many
subsequent examples of this strict type of interactive evolutionary algorithm, including for the design of web
pages (Oliver et al., 2002) and coffee blends (Herdy, 1997).
Contributions from Parmee and collaborators led to some of the first interactive evolutionary algorithms that
used both computation and human input to drive selection (Parmee, 1997; Parmee & Bonham, 2000; Parmee,
2001). Unlike the earlier examples, which focused on design problems with highly subjective performance
metrics, this work is in the realm of engineering, which has both quantitative and qualitative goals. An example
of this type of design problem is shown in Figure 3.3. This work laid the foundations for further research in the
applications of interactive evolutionary computation to structural design.
3.1.4 Applications in structural design
More recently, some progress has been made in applying interactive evolutionary computation specifically to
the realm of structural design. Most notably, von Buelow has proposed an interactive genetic design tool for
creative exploration of design spaces, including for the design of trusses (2008) and folded plate structures
(2011), shown in Figure 3.4.
3.1.5 Specific needs
Existing work suggests specific challenges to be addressed by a new interactive evolutionary framework. First,
existing approaches implement interactivity in limited ways. Interactive features should be expanded to allow
more incorporation of requirements and criteria from the designer. These features can also help the designer
direct navigation of the design space in a more precise way, further improving the effectiveness of an interactive
evolutionary approach.
Second, the reviewed literature lacks strategies to filter and promote diversity and quality of designs. Without
such filters, interactive evolutionary algorithms can produce results that are too similar to each other or
otherwise undesirable.
Finally, existing research treats interactive evolutionary algorithms as a stand-alone approach without
considering the broader user design experience. There is a need to incorporate general problem setup
strategies and design refinement functionalities into an expanded approach, along with the evolutionary
approach itself.
The framework presented in this chapter is a new holistic approach that generalizes the use of interactive
evolutionary algorithms in conceptual structural design, and also addresses these specific needs.
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Figure 3.3: Urban furniture, or park benches, designed using an interactive evolutionary algorithm (Machwe & Parmee,
2009).
Figure 3.4: Folded plate designs generated by an interactive genetic design tool (von Buelow, 2011).
3.2
Framework overview
This section introduces a new framework that adapts a generalized interactive evolutionary algorithm for
conceptual structural design, as well as its implementation as a software tool. Detailed descriptions of specific
original features of the framework are discussed more fully in subsequent sections.
3.2.1 Framework and software architecture
The software implementation of this framework reflects its generalized nature. The program is written in
C#/.NET, an object-oriented programming language, and is designed to be modular and extensible. There are
four general types of backend classes: variables, design models, structural analysis engines, and the interactive
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evolutionary algorithm population generator. The population generator connects with a graphical user
interface to allow input from the user. The interaction of these parts is illustrated in Figure 3.5.
Variables
Design Models
• Upper and
lower bounds
• Crossover
implementation
• Mutation
implementation
• Geometry
description
• Loads, boundary
conditions, material
properties
• List of variables
Analysis
Engines
Population
Generator
Graphical User
Interface (GUI)
• Given a design
model, determine
a score based on
structural
behavior
• Create random
populations
• Evaluate, rank,
and identify
top performers
• Show user top
performing
designs
• Allow user to make
design selections
BACKEND
FRONTEND
Figure 3.5: Software architecture diagram for the interactive evolutionary framework, illustrating main class types and
interactions. As shown, each type of design model can have variables of multiple types and be associated with multiple
types of analysis engines. Also, a variable type can apply to different design model types, and a type of analysis engine
can work on multiple types of design models. The population generator uses a particular design model type and a
particular analysis engine type, and communicates its results with the graphical user interface.
This diagram shows the versatile nature of the framework. Variables, design models, and analysis engines are
all designed using interfaces, meaning that each can be implemented as a variety of types. For example,
variable types can be horizontal and vertical nodal positions, as shown in Figure 3.1, but they could also be
material properties, joint fixities, member topologies, or other design decisions. Design models can be truss
structures, again as introduced previously, but they could also be frame structures, continuous solid structures,
or other structural types. A design model type must have one or more analysis engine type that can apply to it.
For example, truss structures are associated with a truss analysis engine, but could also be analyzed by more
sophisticated analysis engine types. Examples of variable types, design model types, and analysis engine types
are presented in the following subsections.
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The population generator works with a particular design model type and a particular associated analysis engine
type. Using the design model and its variables, it creates a generation through crossover and mutation. Using
the analysis engine, it applies a fitness score to each candidate design. It then presents the best designs to the
user through the graphical user interface, which also allows the user to make selections. These selections are
sent back to the population generator, which produces a new generation.
3.2.2 Variables and design models
As discussed in the previous subsection, the interactive evolutionary framework supports multiple variable
types and design model types. To illustrate how these classes work, the example of a truss design model with
variable nodal positions will be used. Figure 3.6 shows a seven-bar truss similar to the one shown in Figure 3.1,
but with three design variables instead of two. The truss model is defined by its nodes and members. Nodes
are defined by degrees of freedom, which have coordinates, loads, and supports. In this two-dimensional case,
nodes have two degrees of freedom. Members are defined by their start and end nodes and their material
properties. Like all design model types, the truss model also has a vector of variables. This is the model’s
design vector, or parametric representation.
In this type of design problem, the coordinate of each degree of freedom can be a variable. Any variable type
must have defined upper and lower bounds. In this case, the upper and lower bounds are the allowable range
for the coordinate, illustrated in Figure 3.6 with the dashed lines for x1, x2, and x3.
Figure 3.6: A planar seven-bar truss design problem with three design variables: the horizontal and vertical positions of
the lower left node (x1 and x2) and the vertical position of the central node (x3). Like the two-variable problem shown in
Figure 3.1, this truss is simply supported, has a central point load, and is bilaterally symmetrical.
Additionally, any variable type must implement analogs of the biological concepts of crossover and mutation.
Conceptually, crossover combines encoded information from more than one parent to create offspring that
have traits from each of them. Mutation then randomly perturbs the newly formed offspring in order to
encourage diversity. For this example, the implementations of mutation and crossover are given in Equations
[3.1] through [3.3], and apply to continuous variables in general beyond the degree of freedom coordinate.
Crossover is accomplished through a weighted average of seed variable values with random weights. Mutation
updates a variable value with a random variable from a normal distribution with a standard deviation related to
the variable’s set mutation rate, rmutation. For discrete or integer variables, these same approaches can be used
with minor modifications.
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Crossover:
CHAPTER 3: INTERACTIVE EVOLUTIONARY FRAMEWORK
∑
∑
seed variables
[3.1]
uniformly distributed random weights
Mutation:
[3.2]
normal probability distribution
where
and
|
[3.3]
|
The framework also supports parametric relationships between variables and non-variables. For example, the
truss design model presented here allows for mirror and offset relationships between degree of freedom
coordinates. The former is illustrated in the problem shown in Figure 3.6, which uses bilateral symmetry to
define the position of the lower right node based on the position of the lower left node.
3.2.3 Analysis engines
Design model types must be associated with at least one analysis engine type, although the framework supports
the use of multiple analysis engines. Any analysis engine must determine a quantitative fitness score for a given
design model, based on structural criteria. For example, in the case of the truss model, a truss analysis engine
can find the required volume of a structure with a given geometry, loading, and support conditions. The engine
calculates this metric as follows: compute the forces in each member using the direct stiffness method, assign
required cross sectional areas to each member based on allowable stress and buckling considerations (using
material properties and cross-section assumptions), and find the sum of the area lengths times their required
areas. These steps are illustrated in Equations [3.4] through [3.11]. An important note is that for statically
indeterminate structures, this particular process is affected by initial member sizes used to compute forces. In
this case, optimal member sizing can be computed through iteration, or an approximate result found through
initial equal member sizing can be accepted.
The code for this truss analysis engine was implemented by the author, using the open-source Math.NET
numerical analysis library for matrix operations (Math.NET Project, 2012), and a validation of this code is given
in Appendix A. However, analysis engines could also make use of commercial structural analysis codes. In
addition to the strength-based approach given above, quantitative evaluation methods could also compute
metrics related to serviceability, such as deflections and frequencies, to material-specific failure modes, such as
over-reinforcement in concrete, and to stability, such as global buckling. Furthermore, depending on the
requirements on the problem, these metrics could be used either as the main design criterion, or as a postprocessing check in combination with a different criterion.
Global stiffness equation:
[ ]
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[
][
]
[3.4]
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Compute global displacements:
]
[
] [ ]
[3.5]
[ ]
[
][
]
[3.6]
[ ][ ][ ]
[3.7]
[
Compute global reactions:
Compute local axial forces for each member:
Compute required area for each member:
{
(
[3.8]
)
[3.9]
⁄
(
Compute total required volume for structure:
)
∑
(
)
[3.10]
[3.11]
Depending on the size and complexity of the design model, and the fidelity of the analysis engine, evaluation of
design fitness can be very time consuming. This issue is compounded by the fact that the population generator
must evaluate the fitness of an entire generation, which can include up to hundreds of designs. A detailed
discussion and systematic solution for this problem are presented in Chapter 5.
3.2.4 Population generator
The population generator in this framework implements a simple and flexible interactive evolutionary
algorithm that can be easily controlled by the user and adapted to a wide range of variable, design model, and
analysis engine types. As explained previously, the interactive evolutionary algorithm is an iterative approach
that can be repeated until the user is satisfied.
The first step of the algorithm is to generate a random population of a preset number of candidate designs. For
the first generation, this is based on random perturbations from an initial structure defined by the user.
Specifically, for each candidate design in the new generation, each design variable is mutated from initial values
from the user-defined initial structure. Mutation is carried out in the manner previously discussed, and
illustrated in Equations [3.2] and [3.3] for the example of continuous variables.
Next, the algorithm uses the analysis engine to assign a fitness score to each candidate design. The algorithm
then sorts the designs according to this score and presents a top-performing subset of designs to the user
through the graphical user interface. The user is then able to visually evaluate the designs and choose those
that best meet the qualitative or otherwise unformulated goals for the design process. The designs that the user
chooses are used as seeds for creating new designs in the iterative process.
The seeds produce a new generation using the previously discussed crossover and mutation functionalities. The
newly formed generation of new candidate designs is then evaluated, sorted, and presented again, and this
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process can continue as long as the user wishes. There are also several ways for the user to interrupt the
process. If the user does not like any of the presented designs, or wishes to make changes to designs previously
selected, the user can return to a previous generation, adjust selections, and rerun the algorithm from that
point. Also, the user can choose to select no designs, and the algorithm will reset and start with the previously
defined initial structure once again.
3.2.5 Graphical user interface
The graphical user interface (GUI) enables the interactive step of the interactive evolutionary algorithm by
showing the user top-performing designs graphically and allowing the user to make selections. The GUI is
implemented using Silverlight, a platform-agnostic technology that supports interactive user applications that
run in a web browser (Microsoft, 2012). There are several advantages to this approach, in comparison with
traditional desktop applications or integration into existing software. First, the program is highly accessible:
anyone with a web browser can use it, regardless of operating system, and there is no need to download or
install it. Second, there is no need for the user to own other commercial software, such as Rhino or AutoCAD,
to run the program, and the program is not tied to software trends, which tend to change relatively quickly in
the architectural computation realm. Finally, the web-based interface lends itself naturally to analysis
calculations on remote servers. While all calculations are currently executed on the client-side, or on the user’s
computer, future use of server-side calculations through remote resources or cloud computing could
significantly improve performance.
The GUI is implemented as a web-based tool called structureFIT (Mueller, 2014), and a screenshot is shown in
Figure 3.7. It is designed to be simple and user-friendly, while still allowing for powerful user control. The
main feature of the interface is the matrix of designs, shown in numbered rows. Each row represents a
generation created by the population generator, and the designs shown are the top ten performers. The
number under each design corresponds to its score, normalized by the score of a base design, which is shown,
along with the initial design, in the upper left corner of the interface. Designs with scores less than 1.00
perform better than the base design, and those with scores higher than 1.00 perform worse. A closer view of
generated designs and their scores is shown in Figure 3.8. After each generation is produced, the user is able to
select zero, one, or more designs by clicking on them, and selected designs are indicated with a gray square.
The user then clicks the main “generate” button to produce a new generation.
The user can return to a previous generation by clicking the “<” button next to the corresponding row. This will
erase the designs generated since, and the user can change the selected designs and rerun the computation.
The user can also adjust the mutation rate and population size for each generation, and can choose to turn on a
hybrid approach that automatically computes several generations in a row. These features are discussed in
more detail in subsequent sections.
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Figure 3.7: Screenshot of the web-based graphical user interface, showing the evolution of solutions for the design
problem presented in Figure 3.6.
Figure 3.8: A closer view of several candidate designs created by the population generator and presented to the user,
with scores normalized by a base design’s score shown underneath each.
3.2.6 Extensibility
The parts of this framework are explained through a specific implementation, but it is important to reiterate
that the framework is designed to be general, flexible, and extensible. As described, it can support a range of
variable types, design model types, and analysis engine types. Specifically, the analysis engines can vary in type
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of analysis performed, but also in type of result reported, and could include information related to stiffness,
deflection, or dynamic behavior in addition to volume. Structural performance metrics, as well as other related
quantitative metrics, can also be combined to arrive at a score that incorporates multiple objectives. Because of
the versatility of this framework, structureFIT and its underlying principles move beyond specific examples and
are applicable to a wide range of problems encountered during conceptual structural design.
3.3
Enhanced interactivity and user input
This framework includes several novel features that allow for enhanced interactivity between the user and the
evolutionary algorithm. As discussed previously, all interactive evolutionary algorithms include user input in
the form of design selection, but this framework includes additional and unprecedented ways for the user to
incorporate design intentions into the computation. Enhanced user involvement enables more design freedom
and less automation, which in turn helps the framework incorporate more qualitative and unformulated but
important design considerations.
3.3.1 Multiple design selection
Standard evolutionary algorithms follow evolution observed in nature, in which offspring are produced by a
recombination of genes from two parents. However, this requirement is arbitrary in the design world, and
indeed, designers may wish to explore combinations of three or more parent designs. Due to the flexible nature
in which the evolutionary algorithm is implemented, specifically the crossover functionality illustrated in
Equation [3.1], this framework allows users to select any number of designs to seed the next generation. An
example of results of crossover from more than two parent designs is shown in Figure 3.9.
a
b
c
Figure 3.9: Designs resulting from multiple selections. The designs in the second row exhibit a combination of traits
from the three selected designs in the first row. Some have lower nodes that are close together, like selected design a, some
are flat, like selected design b, and some have a central node that is lower than the supports, like selected design c. Each of
the resulting designs combines these traits in different ways, due to the random nature of the crossover functionality.
As noted previously, the user may also select only a single design or no designs at all. When a single design is
selected, a generation of offspring is created through mutation alone. In the case of no selected designs, the
algorithm generates a fresh population randomly, as at the beginning of the process.
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3.3.2 Mutation rate
Through a user interface control, users can directly manipulate the mutation rate used in populating the next
generation of designs. This can be changed in each generation to control the type of exploration that the
algorithm produces. Additionally, users can rerun generations with varying mutation rates.
Although the mutation rate is a technical algorithm parameter, non-experts can quickly grasp its meaning
through simple experimentation and familiarity with the general theory of evolution. Small mutation rates
focus the design space search to the area around the selected designs, and lead to less diversity in the results.
This is preferable when the user has found a part of the design space of interest, and wishes to fine-tune the
design by exploring small variations. Large mutation rates increase the likelihood of offspring to jump to
regions of the design space far from their parents. This behavior is useful when the user is looking for a breadth
of ideas. A comparison of results with varying mutation rates is shown in Figure 3.10.
Figure 3.10: Resulting designs from the same parent, the design selected in the first row, using different mutation rates.
The second row shows designs found with a mutation rate of 0.2, the third row with 0.4, and the fourth row with 0.6. As
the mutation rate increases, the design diversity and distance from the original parent also tend to increase.
In standard evolutionary algorithms, the proper range of feasible values for a mutation rate varies depending
on the specifics of the design problem. This framework automatically takes these considerations into account
through its mutation formulation, as given in [3.2]. The user does not need any technical knowledge to set the
mutation rate, and is always presented with a consistently labeled global value, a decimal between o and 1, to
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control. This ensures that the user can interact with the evolutionary algorithm in a powerful, intuitive way,
while avoiding excessive technical jargon and a steep learning curve.
3.3.3 Generation size
As with mutation rate, the user may also directly manipulate the generation size produced by the evolutionary
algorithm. Again, this value can be changed for each generation and helps the user direct the manner in which
the design space is explored. The effects of modifying the generation size can be learned quickly through
experimentation, and can again also be understood through the lens of the natural selection and evolution
metaphor.
Independent of the generation size, the user is shown a fixed number – by default, ten – of top-performing
designs. Therefore, large generations are more likely to yield better results, since the algorithm displays a
smaller percentage of the best quantitative performers. This is ideal behavior when the user is looking for
optimal designs or optimal regions of the design space. However, the user may want to explore a suboptimal
region of the design space that is otherwise interesting for qualitative reasons. In this case, a small generation
size will help the user maintain the general area of exploration without pushing the results away towards higher
performers. A comparison of results with varying generation sizes (and a fixed mutation rate) is shown in
Figure 3.11.
Figure 3.11: Resulting designs from the same parent, the design selected in the first row, using different generation sizes.
The second row shows designs found with a generation size of 20, the third row with 40, and the fourth row with 60. As
the generation size increases, the number of designs performing better than the parent tends to increase, and the variety of
designs tends to decrease.
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In general, design problems of higher dimensions require larger generation sizes to adequately explore
variation within the design space. For example, enumeration of all designs for an -dimensional problem with
settings for each variable would result in
design options. Empirical research in genetic algorithms
suggests that generation sizes be set up to
for binary variables (Alander, 1992). This framework allows the
user to vary the generation size from ten designs up to 200% of this recommended value, converted for
continuous variables. Because the bounds for generation size are computed automatically by the framework,
the user does not require expertise in evolutionary algorithms to get reasonable results.
3.4
Design quality and diversity enhancements
This framework also includes two new techniques to improve the solutions it generates, a hybrid automaticinteractive feature and a diversity booster. The hybrid feature finds better performing designs by automatically
running through a set number of evolutionary iterations. This helps the user find top performing designs in a
desired region of the design space with reduced manual effort. The diversity booster filters out designs that are
too similar to those already under consideration in the group of top performing designs. This increases the
number of significantly different designs that the user can consider, thereby also enhancing the design space
exploration.
3.4.1 Hybrid automatic-interactive functionality
While the interactive aspect of the interactive evolutionary framework is critical for achieving a harmony
between quantitative and qualitative goals, there are times when interaction is not required at every step. In
situations in which the user selects the very top performers of each generation for several cycles in a row, the
user is not truly applying qualitative selection criteria, and is instead seeking the optimal design according to
the analysis engine in a particular region of the design space. To improve the user experience in these cases, the
framework offers the option to automatically compute multiple cycles, selecting the top two performers as seeds
for the subsequent generation. The user can specify how many generations should be automatically computed.
This functionality is important for several reasons. First, in design problems that are highly multi-modal, or
that have a large number of local optima, the hybrid approach allows users to quickly drill down to a local
optimum once an interesting design direction has been discovered. This prevents user fatigue from working
through many generations manually, and improves the resulting design by finding the best possible version of a
particular design “family.” Second, it encourages the user to move through sub-optimal and visually
unappealing portions of the design space that may stand between two high-performing regions. While the user
may be reluctant to move in such a direction, the automatic mode can traverse across the space and find
interesting regions without user input. Third, this functionality effectively allows the user to use a more
traditional optimization-like approach when necessary. The user is able to decide how to balance the tradeoff
between pure optimization and open exploration, allowing an additional high-level form of interaction and user
control. An example of the automatic mode in use is shown in Figure 3.12.
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3.4.2 Diversity booster
Conceptual design demands that designers consider a broad range of alternative solutions, so it is crucial that
this framework encourage design diversity. In standard form, evolutionary algorithms tend to homogenize
populations of designs as evolutions proceed. This means that interactive evolutionary algorithms often show
users top-performing designs that are quite similar to each other. This effectively reduces the user’s choice in
selection, and limits the ability of the user to explore a variety of options.
Figure 3.12: Resulting designs from the same parent, the design selected in the first row, with the automatic generation
option turned off and on. The second row shows designs found without automatic generation computation, and the third
row with the feature enabled for five generations. The automatic feature results in the highest performing designs related
to the parent.
This framework addresses this problem by filtering the presented designs according to diversity criteria.
Specifically, it ensures that each design in the top ten is not unduly close to those ahead of it in the list, using a
measure of quantitative distance to determine design similarity. The distance metric is computed as the
Euclidean distance between two points in the design space, as calculated from their underlying design vectors.
In assembling the group of top performing designs, the diversity boosting filter omits any design too close to
those ahead of it, reaching further down into the population to find those that are significantly different from
each other. The calculations to determine whether a design is diverse enough are given in Equations [3.12]
through [3.15].
If the population does not include enough diverse designs to generate ten designs to present to the user, the
diversity criterion is relaxed and additional designs are included. An example comparing results with and
without the diversity booster is given in Figure 3.13.
Size design space of
variables:
√∑(
64
)
[3.12]
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Compute allowable distance based on design
space size and mutation rate (factor of 0.3
empirically determined):
[3.13]
Compute distance between two designs:
√∑(
{
Decide whether to add design to list of top
performers based on IsDiverse:
[3.14]
)
[
]
[3.15]
Figure 3.13: Resulting designs from the same initial design shown in Figure 3.6. The first row shows designs with the
diversity booster disabled, and the second row has it enabled. The diversity booster results in designs that are less similar
to each other, effectively increasing the user’s choice in design selection and improving the interactive nature of the
framework.
3.5
Expanded user experience
In addition to the interactive evolutionary design experience, this framework contributes original functionality
that can be used before and after. Before evolutionary design exploration, the user can set up the design
problem by drawing in a graphical and intuitive user interface. This makes the framework general beyond
specific examples. After the evolutionary design evaluation, the user can refine an evolved design using realtime performance feedback. These additional features help bring this framework beyond an algorithm and
toward an approach usable for real design problems.
3.5.1 Model setup
The design setup mode allows the user to define a design problem by building a structural model and
identifying variables. The user can draw a structure by clicking and dragging to create nodes and members on a
canvas, or by modifying entries in an adjacent spreadsheet. The user can then assign loads and supports to
defined notes, and define variables, including upper and lower bounds. Finally, the user can define planes of
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symmetry and parametric relationships, including mirror and offset relations. The information entered by the
user is updated dynamically in the graphical view of the structural model. This functionality is illustrated in
Figure 3.14.
Figure 3.14: Screenshot of the model setup mode, in which the user can input a design problem, specified by structural
geometry, loads, materials, boundary conditions, and variable definitions.
The user may also choose to open one of a range of preset design examples that can be run directly, or modified
to adapt to new problems. Additionally, the user can choose to save a custom setup structure that can be
opened again later in the design session, or to save the setup structure as a text file to disk (called a .fit file) that
can be opened in a different session. Once the setup structure has been finalized, the user can click the button
in the upper left of the screen to set it as the initial design for the interactive evolutionary mode. If the structure
is not stable, or contains no loads or variable definitions, the program will identify these issues for the user to
correct.
This setup mode is important because it makes the interactive evolutionary framework both highly flexible and
easy to use. The framework is not tied to any particular example or case study, and can be used by designers for
real design problems. Additionally, the GUI for design input is powerful and user friendly, so that designers
can define problems quickly and move on to exploring solutions in the interactive evolutionary mode.
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3.5.2 Design refinement
Once the user has found an interesting design, it can be studied and refined further in the design refinement
mode. This mode allows the user to graphically adjust variable settings for a selected design to fine-tune its
appearance, while also receiving real-time feedback on the performance implications of the adjustments. In the
case of nodal coordinate variables, the user is able to adjust the nodal positions by clicking and dragging, and
may observe the change in the overall design score. The program also instantly updates the required thickness
of individual members, shown graphically on the members themselves and numerically in a spreadsheet. The
user is able to save particular designs found in this design refinement mode and return to them for comparison.
Once an attractive solution is found, the user can export it for use in more advanced modeling and analysis
software. A screenshot of this design mode is shown in Figure 3.15.
Figure 3.15: Screenshot of the design refinement mode, in which the user can adjust designs found in the interactive
evolutionary exploration with real-time performance feedback in terms of the overall score and individual member sizing.
The members are drawn with required thicknesses shown to scale, with blue indicating tension and red compression.
Like the model setup mode, the design refinement mode adds crucial novel functionality to the interactive
evolutionary framework. By combining a guidance-based approach with a feedback-based post-processing
step, the framework is able to expand design freedom for users.
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3.6
CHAPTER 3: INTERACTIVE EVOLUTIONARY FRAMEWORK
Design example: cantilevered truss roof
Now that the interactive evolutionary framework has been presented and its original features highlighted, this
section will illustrate its use through a realistic conceptual structural design problem: finding the shape of a
cantilevered truss roof. This example illustrates the general use of the framework as well as the value of the
specific intellectual developments presented in this chapter.
3.6.1 Design problem formulation
The cantilevered truss roof is shown in the setup mode in Figure 3.16. The roof has a total length of 90 feet,
comprising a 15-foot cantilever on the left, a 50-foot central span, and a 25-foot cantilever on the right. The
roof is pitched, and is supported at two points by pairs of splayed, pin-based columns that provide lateral
stability, the shorter pair with a height of about 22 feet, and the taller with a height of about 28 feet. The top
chord of the roof truss has joints spaced at 10-foot intervals, and each joint has a downward point load of 10
kips. This load corresponds to a tributary width for the truss of 10 feet, and a total uniform gravity load of 100
psf. The structure is assumed to be made of steel tubes similar to those used in previous examples.
Figure 3.16: Screenshot of the model setup mode for the cantilevered truss roof design example, showing geometry,
boundary conditions, loads, variables, and variable bounds.
While the flat upper chord of the truss is fixed, the nodes along the bottom chord are defined as variables in
both the horizontal and vertical directions. All of these nodes can vary 100 inches up and down, and most can
vary 40 inches left and right, with the exception of the nodes connected to the columns. The intention of these
design variables is to find an elegant and efficient form for the lower profile of the roof truss that performs
better than the initial flat chord. Like most structural design problems, there is an inherent tradeoff between
decreased member forces, achieved through increased truss depth, and decreased member lengths, achieved
through decreased truss depth. In this case, the most benefit from increased depth occurs over the supports,
where the bending moment demand for an equivalent continuous beam is greatest. The goal of the
evolutionary exploration is to find solutions that find a balance of reduced member lengths and forces.
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3.6.2 Evolution of candidate designs
Now that the design model and variables are defined, the user can begin exploring the design space through the
interactive evolutionary mode. Because the process can be interrupted and restarted, the exploration cannot be
described with a single screenshot. The first few generations found by the user are shown in Figure 3.17. As
illustrated, the first generation, found using a generation size of 100 and mutation rate of 0.50, contains a
diverse group of designs, some of which perform better than the initial model, and some which perform worse.
The user selects the best performing design, which requires only 59% of the original volume, but also two other
designs that suggest a more sleek profile. This results in a new generation that shows a balance between these
parents: high performing, but less depth compared to the best-performing predecessor.
Figure 3.17: First three generations of evolutionary exploration for the cantilevered truss roof design problem.
In the next generation (line 1 of Figure 3.17), the user is interested in a design that shows a node moving above
the flat plane of the roof, perhaps suggesting a strategy for a skylight that could bring natural light into the
space. To explore this type of solution more deeply, the user selects only this design and reduces the mutation
rate to 0.15. This allows the user to find a range of higher performing versions of this selected design in the
third generation. The user selects one of these resulting designs to study further in the design refinement
mode, which will be discussed in the next section.
Meanwhile, the user now decides to return to the second generation to change the selected design and explore a
different path through the design space, shown in Figure 3.18. This time, two designs are selected: one which
brings the node connected to the support on the left down, and another which articulates the cantilever on the
right as a thin taper. The user finds a new generation with a mutation rate of 0.50 and a generation size of 100,
leading to considerable diversity and high performance in the results. The user chooses three designs that
include overall depth and elegantly tapered curves, and creates a new generation using the automatic
computation mode with a lower mutation rate of 0.30. As a result, the algorithm computes four generations
automatically, producing a generation of designs in the fourth row that are high performing and of high quality,
meaning in this case that the forms are relatively smooth. The user selects one of the resulting designs, and
increases the mutation rate to 0.60 and reduces the generation size to 30, in an effort to find more variety,
perhaps at the expense of performance. The next generation includes an attractively shaped form that the user
chooses to explore further in the refinement mode.
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Figure 3.18: Five additional generations of evolutionary exploration for the cantilevered truss roof design problem.
The user is then interested in exploring one more family of solutions, returning to the second generation, as
shown in Figure 3.19. This time, a design that shows increased depth in the midspan region is selected. From a
structural perspective, this design could evoke the concept of two cantilevers simply supporting a central span,
as in the Firth of Forth Bridge in Scotland, or perhaps the choice is purely for architectural reasons, suggesting
a sort of belly that hangs above the main space and could be used to house services. The user creates a new
generation using this choice, decreasing the mutation rate to 0.15 and increasing the generation size to 150 to
find better versions of the selected design. The top performing design in the third generation includes a more
pronounced belly, a smoother shape, and a better score, so the user also selects this design to study further.
Figure 3.19: A new third generation found by changing the selected design and evolutionary parameters.
3.6.3 Refinement of selected design
After a design has been selected for further study in the refinement mode, the user may make adjustments to
variable settings to fine-tune the design and note the performance changes through real-time feedback. The
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first design that the user has selected, the version with the node raised above the roof for a possible skylight, is
shown in Figure 3.20. The top image, (a), shows the original design found through the interactive evolutionary
process, along with its score, 0.66, relative to the initial flat-truss design shown in Figure 3.16. Again, this
means that this design requires 34% less material than the flat truss version.
(a)
(b)
(c)
Figure 3.20: The original form selected to be refined, (a), and two new forms found by the user through small
adjustments of nodal positions, (b) and (c). The relative scores associated with each design are shown on the right.
By clicking and dragging the variable nodes, the user is able to adjust this design slightly, evening out the
curvature and realigning the position of the skylight as shown in (b). These small adjustments affect the overall
performance of the structure, mainly by increasing the structural depth near the points of supports in this
instance. This information is communicated to the user by the score and status bar as well as the rendered
thickness of the members. In this case, fine-tuning the variable nodal positions has decreased the required
material by an additional two percent, while also improving the architectural quality of the design.
In the third image, (c), the user has chosen to push this design concept further, exploring raising the height of
the skylight ridge and deepening the truss to maintain a relatively smooth curve. Because of the increased
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structural depth, the score reduces even further, to 0.55. While this design performs better structurally, the
user may decide that the increased bulk is not desirable, and finally settle on the design found in (b).
(a)
(b)
(c)
Figure 3.21: The original form selected to be refined, (a), and two new forms found by the user through small
adjustments of nodal positions, (b) and (c). The relative scores associated with each design are shown on the right.
The next design that the user has chosen to study, the elegantly curved form, is shown in Figure 3.21. The top
image, (a), shows the original form found in the interactive evolutionary mode. In image (b), the user has
evened out the curvature of the cantilevers, slightly improving the structural performance and the architectural
shape. In the third image, (c), the user explores the structural and architectural impacts of emphasizing the
curve at the left support and reducing the curve at the right support, perhaps to bring more light into the space
through fenestration in the right façade. This change slightly increases the required material for the structure,
but a reduction is structural performance may be acceptable in exchange for improvements in other areas.
Because this framework allows users to quantify this tradeoff, conceptual design decisions can be well
informed.
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The final design that the user selected, the version with the mid-span belly, is shown in Figure 3.22. The
originally selected design is shown in the top image, (a). In the second image, (b), the user has straightened the
bottom chord in the exterior cantilevers, and has also imposed linearity on internal elements to more strongly
distinguish the rounded belly form. These changes slightly increased the score, but resulted in a more
compelling overall form. In the third image, (c), the user explores further emphasizing the depth of the central
belly and reducing the depths of the cantilevers as the supports. This further worsens the structural
performance, since in this case, truss depth is better used at the supports compared to the midspan, but
perhaps strengthens the architectural concept. This adjustment may also allow for more ductwork to pass
through the central span, and more light to enter through the sides.
(a)
(b)
(c)
Figure 3.22: The original form selected to be refined, (a), and two new forms found by the user through small
adjustments of nodal positions, (b) and (c). The relative scores associated with each design are shown on the right.
This example has shown that structureFIT can be used to discover a variety of design alternatives that perform
considerably better than the initial design idea. In these cases, about 40% of the initial structural material was
saved through the evolutionary navigation process, and the three resulting designs were able to meet specific
and distinct architectural goals.
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3.7
CHAPTER 3: INTERACTIVE EVOLUTIONARY FRAMEWORK
Additional design examples
To further illustrate the power of this framework, this section presents additional resulting solutions from six
design problems, including the cantilevered truss problem and five new conceptual structural design problems.
The problem setups, including initial geometry, loading, supports, and variable definitions, are shown in Figure
3.23. The first image, (a), shows the familiar cantilevered truss with the variable bottom chord. The additional
design problems are as follows: (b) determine a shape for the bottom chord of a gabled truss, (c) determine the
inner profile for a trussed rigid frame, (d) determine the outer profile and interior node heights for a tower
subject to lateral loading, (e) determine the shape of a trussed arch, and (f), determine the nodal positions of
the top chord for a trussed bridge. A possible range of solutions found for each of these six design problems
using the interactive evolutionary framework is shown in Figure 3.24.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3.23: Additional example design problems to be explored using the interactive evolutionary framework.
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1.00
0.84
0.76
0.71
0.66
0.57
0.56
0.52
0.49
0.47
1.00
0.93
0.89
0.86
0.82
0.76
0.75
0.74
0.64
0.52
1.00
0.88
0.84
0.80
0.78
0.74
0.72
0.72
0.69
0.63
1.00
0.96
0.86
0.82
0.80
0.74
0.73
0.72
0.68
0.59
1.00
0.91
0.90
0.85
0.82
0.79
0.76
0.74
0.72
0.71
1.00
0.81
0.79
0.71
0.68
0.63
0.56
0.54
0.54
0.47
Figure 3.24: Sample design solutions found using the interactive evolutionary framework for six design problems. The score under each design is normalized by the score of the initial design, shown in the leftmost column. The score is a measure of required structural material
volume, so a lower score indicates better performance. These examples illustrate the rich diversity of high performing solutions possible to discover using the new methodology presented in this chapter.
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These examples show that highly efficient solutions can be found as alternatives to standard design ideas.
However, they also illustrate the plurality of options available between the initial idea and the best-performing
solution. This space in between contains meaningful richness and offers designers a range of possibilities that
are not necessarily structurally pure or completely rational, but that nevertheless perform better than standard
ideas.
As a final note, it is important to acknowledge that as structures become increasingly light, like some of the
options shown in Figure 3.24, their performance may no longer be governed by strength-based considerations
and downward gravity loading. For example, in the design of lightweight pedestrian bridges, vibration
concerns require the structure to be stiffer and heavier than what is needed for stresses alone. Lightweight
long-span roofs must handle uplift from wind in addition to standard gravity forces. A key extension of this
work would allow for adaptable structural goals that account for and anticipate these shifting criteria.
3.8
Summary of intellectual contributions
This chapter has presented a new and general framework for using interactive evolutionary optimization in
conceptual structural design. This work is important because it helps enable a guided exploration of structural
design spaces, while still allowing for creativity and freedom, addressing the issues found in standard
optimization identified in Chapter 2.
This framework builds upon existing work in interactive evolutionary algorithms and in structural design tools,
addressing specific issues that remain unresolved in previous literature. The original developments contributed
by this chapter are as follows:

Enhanced capabilities for interaction and user input through multiple-selection and simple but
powerful parameter controls.

Design quality and diversity boosters that significantly improve the designs presented to the user,
improving the effectiveness of the design space exploration.

An expanded user experience that includes generalized problem setup and post-evolution design
refinement through real-time analysis.
Additionally, this chapter has illustrated the use of this framework in a series of conceptual structural design
examples, highlighting the impact of these novel functionalities, and has also shown a variety of additional
design problems and solutions found by using the framework.
The flexibility and extensibility of this framework allow it to be combined with the two additional
methodologies presented in the Chapter 4 and Chapter 5. Chapter 6 outlines how these three strategies could
be integrated.
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CHAPTER 4:
Trans-typology Structural Grammars
This chapter presents the second of three new design space strategies, a grammatical approach to generating
conceptual designs across multiple structural typologies. The approach uses rule applications, as opposed to
more common parameter settings, to formulate broader and more diverse design spaces that offer rich and
often unexpected design possibilities. The specific intellectual contributions of this chapter include a
generalized prescription for successful grammars of this type, as well an example grammar that illustrates the
power and possibilities of this approach.
4.1
Background on design space formulation
In computational design, the design space contains all possible solutions to a problem system. Optimization
methods focus on how to locate the best performer(s) in a given design space, and more nuanced approaches
like the interactive evolutionary approach presented in Chapter 3 allow free yet directed design space
navigation. However, it is also important to consider the design space itself. No matter how well optimization
or navigation approaches work, they are limited by the solutions that can be found in the design space of a
particular problem formulation.
This section motivates the need for ways to define broad and diverse design spaces, and discusses types of
design spaces for conceptual structural design. Based on existing literature, this section makes the case for
rule-based, or grammatical, approaches to design space formulation, and identifies the areas for further
development addressed later in this chapter.
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CHAPTER 4: TRANS-TYPOLOGY STRUCTURAL GRAMMARS
4.1.1 Trans-typological design
The earliest steps in the contemporary conceptual structural design process involve choosing a typology or
system. For instance, in a long-span roof design, should the structural action be carried out with an arch, a
cable, a fan-like scheme, a bending option, or with a truss? The world’s best structural designers are able to
brainstorm a range of creative ideas and can intuitively estimate relative performance of competing concepts.
For example, the German structural engineer Jörg Schlaich (b. 1934) generated sixteen innovative conceptual
design possibilities for a bridge competition, illustrated in Figure 4.1 and Figure 4.2 (The Happy Pontist, 2009).
Other examples are given in British engineer Tony Hunt’s (b. 1932) published “sketchbooks” (1999; 2003),
exemplified in Figure 4.3, and in Heino Engel’s extensive structural catalog (1967).
Currently, in the most successful examples, the generation of these typological ideas and the selection between
them are carried out by expert practitioners with many years of experience and keen intuitions, like Schlaich
and Hunt. In less successful instances, fewer typological ideas are considered, or an ill-fitting typology is
chosen without adequate consideration. There is plenty of room for bias and human error to influence this step
in the process, which is arguably the most important step because it determines many characteristics of the
overall form.
There is a strong and unaddressed need to develop computational methodologies for exploring possibilities
across typological boundaries. While some luminaries in the structural design field excel at doing this by hand,
as shown in the previous examples, the computer can help in several ways. First, given a broad enough design
space formulation, computational techniques can automatically generate a range of solutions to consider,
behaving like a creative brainstorming partner. Second, computation can be used to quantitatively evaluate
design options according to structural behavior. This is standard practice as a way to compare designs within a
set typology, such as trusses of various configurations, but is rarely used to compare designs across typologies.
There is also a more subtle, yet very important, argument for trans-typological explorations on the computer.
The Luxembourgian structural designer Laurent Ney (b. 1964) argues that structural typologies are artificial
constructs developed by 19th and 20th century engineers to categorize successful preceding solutions, but do not
constitute all possibilities:
A typology has a name, and the form and the relationship between the elements is described. The
advantage of this is that it is easy to talk about the structure, but the disadvantage is that how the
structure looks is predetermined… This approach has a perverse effect: the vocabulary freezes the
object, and the objects thus frozen assume a sort of inviolable legitimacy. In order to arrive at new
forms and concepts we have to free ourselves from such pre-defined typologies. (Ney et al., 2010)
Indeed, the physics that governs structural behavior is a continuum, and design possibilities exist between and
beyond the boundaries of traditional typologies. A unified computational approach to exploring multiple
typologies will also include the spaces in between them, enabling designers to generate unexpected possibilities
that have never been discovered before.
A design space of trans-typological breadth is the key to computational explorations of this type. The following
two subsections introduce two types of design space formulation, parametric and rule-based, and argue that the
former is not able to achieve the required breadth for trans-typological exploration on its own.
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Figure 4.1: Conceptual bridge designs by leading German structural engineer Jörg Schlaich of Schlaich Bergerman und
Partner, developed during collaboration on a competition entry with architect Frank Gehry (The Happy Pontist, 2009).
These concepts illustrate the breadth of possible solutions to a design problem, which span across many typologies:
suspension bridges, cable-stayed bridges, arch bridges, beam bridges, and several in between.
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Figure 4.2: Foam models completed by architect Frank Gehry’s office based on the conceptual sketches of Jörg Schlaich
shown in Figure 4.1 (The Happy Pontist, 2009).
Figure 4.3: Tony Hunt’s design concepts for an unbuilt factory requiring long clear spans in 1985 (Hunt, 1999). Like
those shown in Figure 4.1, these designs illustrate a range of structural typologies, including trusses, arches, and cables
in various configurations.
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4.1.2 Parametric design spaces
In both optimization and architectural computation in general, the most obvious type of design space is one
that is parameter-based, sometimes called variable-based. Each parameter or design variable constitutes one
dimension in the space, and a particular point in the
-dimensional space represents a particular parameter
setting for each of the parameters (plus one dimension for the performance metric). All of the examples in
Chapter 3 utilize this type of design space. Parameters can explicitly relate to particular spatial definitions of a
design, or they can more globally control a design’s geometry as a whole, which helps limit the design space
dimension.
In both cases, the designs found in this type of space are parametric variations of each other. Through clever
parameter definition, it is possible to define somewhat broad design spaces that exhibit diversity in possible
solutions, as illustrated by the example in Figure 4.4 (Furuto, 2012). This type of space can be useful in
exploring design decisions once the overall formal strategies and structural systems have been decided upon.
However, it is practically impossible to define a parametric design space that covers the range of possibilities
that one would like to consider during conceptual structural design, such as those shown in Figure 4.1, Figure
4.2, and Figure 4.3. This is related to the fact that one can enumerate a parametric design space – that is, list
every possible design it contains – or at least map it exhaustively at a finite resolution.
Figure 4.4: Parametric variations of a design concept for a mixed-use complex in Tehran by ContemporARchitectURban
Designer’s Group (Furuto, 2012). While these designs exhibit a range of design possibilities, they are clearly part of the
same family and share significant formal and organizational characteristics.
4.1.3 Rule-based design spaces
One effective way to move beyond the limitations of parametric variation is by using rule-based systems, or
grammars, instead of parameter settings to generate designs. Based on Noam Chomsky’s theories of generative
grammars in language (1956), George Stiny and James Gips proposed generative grammars for geometric
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shapes, or shape grammars (1972). As Stiny later explained, “[Chomsky’s] idea was that a grammar had a
limited number of rules that could generate an unlimited number of different things, and that the resulting
language was the set of things the rules produced” (Stiny, 2006). Just as there are an unlimited number of new
and creative sentences that can be uttered in a language, a grammar for shapes can yield an infinite number of
new and creative designs.
Since its introduction in the early 1970s, the theory of shape grammars has been used as a way to analyze
existing design types and styles, and also to generate new ones. For example, Koning and Eizenberg (1981)
presented a compelling grammatical study of Frank Lloyd Wright prairie houses, shown in part in Figure 4.5.
Additionally, they generated several new convincing designs in the same style, as shown in Figure 4.6. Such
examples reveal the power of the rule-based approach, which leads to widely varying yet meaningful results.
Figure 4.5: Illustration of a small subset of possible rule applications in the Frank Lloyd Wright prairie house grammar
(Koning & Eizenberg, 1981). This shows the power of simple rules to generate increasingly diverse and complicated forms
through repeated applications.
Depending on the specifics of the rules, a rule-based design space can be infinite and non-enumerable beyond
the case of bounded continuous variables in parametric design spaces. While in the latter case, there can
technically be an infinite number of possibilities due to the nature of real numbers, rule-based design spaces
can be infinite in the sense of unboundedness. One reason this characteristic arises is because grammatical
rules can be recursive, or can be applied repeatedly without end. Also, rule-based design spaces offer the
possibility of emergence, in which sequences of rule applications lead to results that have not been predefined,
and that can be operated on by subsequent rules in unexpected ways.
Because of the breadth and richness of design spaces defined by grammars and rules, they are a better
candidate for enabling trans-typological explorations than parametric design spaces (Al-kazzaz et al., 2010).
However, the application of geometric shape grammars to the field of conceptual structural design is not trivial.
While the generative power of grammars is great, there is a danger that grammars can be too broad, capable of
generating forms that make little sense in the physical and structural world. It is therefore critical that
grammars used in structural design be sufficiently restrictive and incorporate structural information into rule
definitions.
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Figure 4.6: Three new designs in the style of Frank Lloyd Wright’s prairie houses developed using a shape grammar
(Koning & Eizenberg, 1981). The complexity and variation of the results indicates a power beyond that of parametric
variation.
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4.1.4 Structural grammars
Grammars in architectural and engineering domains that move beyond shapes were first suggested by Mitchell
(1991), who proposed functional grammars with rules that incorporate engineering and fabrication knowledge,
shown in Figure 4.7. Cagan and Mitchell (1993) then combined grammars with performance goals in their
formative paper on shape annealing. Shape annealing is a computational technique to generate optimally
directed grammar-based shapes through the stochastic simulated annealing optimization algorithm. In the
original paper, geometric properties are used as the objective function, but the method has also been applied
using structural criteria as the objective.
Figure 4.7: Sample rules for the first functional grammar proposed by Mitchell (1991). Rules are applied according to
structural requirements, such as lateral stability and spanning roof support.
The shape annealing approach has been further developed and applied extensively to truss structures, most
notably by Shea and Cagan (Shea & Cagan, 1997; Shea et al., 1997; Shea & Cagan, 1998; Shea & Cagan, 1999a;
Shea & Cagan, 1999b), with an example illustrated in Figure 4.8. This approach has been shown to produce a
wide variety of high-performing designs within a relatively narrow problem domain, such as a cantilevered
truss-beam or a domed roof. Because of this, shape annealing is most applicable to post-conceptual design,
once the global structural typology has been selected.
Figure 4.8: Two truss solutions found using a shape grammar coupled with a simulated annealing exploration strategy
(Shea & Cagan, 1999b).
More recently, Geyer (2008) has extended Mitchell’s idea of functional grammars and combined it with
multidisciplinary optimization, and applied his approach to the design of a planar gravity and lateral frame
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system for a large hall. The resulting designs are trans-typological, but not in an unexpected way. Despite the
grammatical formulation, the design space is combinatorial, with all of the resulting designs effectively
predefined through relatively inflexible rules. Because of this, the proposed approach does not take full
advantage of the power of grammars over parametric design definitions; indeed, the proposed problem design
space could be formulated as one that is parametric. The inclusion of optimization makes this approach very
useful for comparing between a predetermined set of options at a stage after conceptual design.
Figure 4.9: A sample of rules and resulting designs from a structural grammatical framework combined with
multidisciplinary optimization (Geyer, 2008).
Similar ideas have also been explored in the computer graphics field of procedural modeling, which uses similar
principles to shape grammars for the automatic generation of digital 3D scenes for video games and other
animations. Specifically, procedural modeling has been used to generate cities and buildings (Parish & Müller,
2001; Müller et al., 2006), and more relevantly, to generate structurally stable masonry structures (Whiting et
al., 2009). Again, because this work is specific to a particular problem type, it does not yet address the need to
consider multiple structural typologies.
4.1.5 Specific needs
This section has shown that there is a need for a computational approach to enable trans-typological
exploration in conceptual structural design. Such an approach would help designers generate a broad range of
design possibilities, quantitatively compare options of varying types, and discover unprecedented solutions
between the boundaries of traditional forms.
Conventional parametric design space definitions are inadequate for trans-typological design generation
because they are limited by an enumerable list of parametric variations. In contrast, rule-based design space
definitions are capable of the breadth and diversity needed for trans-typological exploration. Adapting the
existing shape grammar approach, which is geometry-based, for structural design requires that physical
behavior and properties be incorporated.
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Some existing literature begins to address how grammars can be used in engineering applications, but no work
yet focuses on generative breadth. There is a need to further develop the concept of structural grammars, and
to propose a generalized approach for developing grammars that can define creative trans-typological structural
design spaces for use in generating conceptual design alternatives. Furthermore, it is necessary to develop a
computational implementation of such an approach that can use the grammars to generate design ideas.
This chapter addresses these needs in the subsequent sections, and also illustrates the power of the proposed
approach and implementation through an example grammar that enables computational trans-typology design.
4.2
Trans-typology grammar features
Based on the needs identified in the previous section, this section proposes an original and general grammarbased approach for defining trans-typological design spaces that contain a wide range of diverse, yet
structurally feasible, design options. To help illustrate the proposal, a simple structural grammar that
generates tension and compression funicular forms is introduced and used to explain concepts in the following
subsections. A more sophisticated grammar capable of producing more realistic and complex designs is
presented later in the chapter.
4.2.1 General approach
The trans-typology grammar approach involves three types of computational classes: shapes, grammars, and
analysis engines. A particular type of shape is operated upon by a particular grammar, and analyzed for
structural performance by a particular analysis engine. In the generalized approach presented here, these
classes implement generic computational interfaces: IShape, IGrammar, and IAnalysis. This means that any
shape/grammar/analysis set can be used that follows the same pattern. This section will use the example of a
SimpleShape class, which is associated with a SimpleGrammar class and a SimpleAnalysis class. The
SimpleShape class contains data, or properties, that include geometric information, but also additional internal
organization and hierarchy. The SimpleGrammar class contains a list of rules that can apply to certain
SimpleShape objects by modifying their properties. Similarly, the SimpleAnalysis class can provide a
performance score for certain SimpleShape objects based on their properties. These relationships are
illustrated in the diagram in Figure 4.10.
4.2.2 Structural shapes
Structural shapes like the SimpleShape class are defined by their properties. A SimpleShape object is an
instantiation of the SimpleShape class that has particular property settings. Properties are shape-specific and
include single and group functional designations for geometric elements like lines, points, and areas that dictate
their behavior. Properties also include a state label, which will be discussed later. For example, as illustrated in
Figure 4.11, the SimpleShape class has three lists of lines among its properties: a list of vertical elements, a list
of horizontal elements, and a list of funicular elements. It also contains two designated points: a start and an
end. These designations allow rules and analysis engines to identify and act on certain parts of the structural
shape.
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PROPERTIES
SimpleShape : IShape
SimpleGrammar : IGrammar
SimpleAnalysis : IAnalysis
Rule 01
Rule 02
Rule 03
Score =
SimpleShape
Figure 4.10: Relationship of SimpleShape class with the SimpleGrammar class, which contains rules for a SimpleShape,
and the SimpleAnalysis class, which can structurally evaluate a SimpleShape.
SimpleShape : IShape
PROPERTIES
List<ShapeLine> Verticals
List<ShapeLine> Horizontals
List<ShapeLine> Funiculars
SimpleShapeState State
ShapePoint Start
ShapePoint End
state Start
state SubdivideHor
state AddFunicular
state End
Figure 4.11: Properties of the SimpleShape class, including designated lines and points and a state label.
As discussed in the previous section, structural shapes must include more than pure geometric data. Important
structural information, such as loading, material properties, support conditions, and allowable structural
behavior should be encoded and accessed by rules and analysis engines. This is accomplished by incorporating
non-visual data into the computational representation of the objects within the structural shape, such as lines,
points, and areas. While the graphical depiction shows the geometry, the underlying formulation contains a
richer set of properties. An illustration of this concept is shown in Figure 4.12.
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Applied Distributed Loading = 1 kip/ft
Modulus of Elasticity = 29,000 kip/in2
Allowable Stress = 3 kip/in2
Figure 4.12: Relationship between geometric shape and underlying structural information encoded into the shape’s
computational formulation.
4.2.3 Recursive rules
A trans-typology structural grammar can be described by the list of rules that it contains and an initial
structural shape to begin rule application with. Rules adjust the structural shape through addition, subtraction,
subdivision, and other modifications to geometric or structural properties. A rule has a left-hand side, or LHS
(the structural shape prior to rule application) and a right-hand side, or RHS (the structural shape after
application), and can only apply to a structural shape that matches its left-hand side. A sample rule for a
structural shape is shown in Figure 4.13.
When rules can be applied recursively, there are an infinite number of rule application sequences that
determine unique designs. In the case of the recursive rule in Figure 4.13, the effect of its repeated application
is shown in Figure 4.14. This simple example shows how a one-rule grammar can define an infinite design
space, and the same principle can be used in more complex structural grammars to generate a broad range of
designs.
Figure 4.13: A sample subdividing structural grammar rule with a left-hand side and a right-hand side. This rule can be
applied recursively an infinite number of times. An example of recursive applications of this rule is shown in Figure 4.14.
4.2.4 Rules and state labels
Figure 4.11 shows that the SimpleShape class contains a property specifying a state label. A state label is a way
to control which rules can be applied to structural shapes at various times in the rule application process. In
the trans-typological structural grammar approach presented here, a structural shape is always in a particular
state, and a rule can only apply to structural shapes in one or more specified states. Rules can change the state
of a structural shape, thereby changing the rules that can subsequently apply to it, although they may also
maintain the current state. This can occur along with a geometrical or other substantive modification, or alone.
Examples of rules with state labels are shown in Figure 4.15.
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1 rule application
1 possible design
2 rule applications
2 possible designs
3 rule applications
5 possible designs
4 rule applications
14 possible designs
Figure 4.14: Possible resulting designs for four rule applications of the rule shown in Figure 4.13. While the results are
initially predictable, the number of possibilities grows more quickly with each subsequent rule application.
state
state
state
state
Figure 4.15: A subdividing rule that maintains the state of a SimpleShape object (top) and a rule that does nothing
besides changing the state (bottom).
State-labeled rules are important for a trans-typology grammar because they create a general order in which
rules can be applied. The order is not completely prescribed, however, as that would counteract the benefits of
the grammatical approach. Instead, state labels simply define a general blueprint for generating reasonable
structural forms. Most states have several rules that can apply, and some rules can apply in more than one
state. Furthermore, a structural shape can return to a previous state during the rule application process. The
full power and flexibility of the state label system will be shown in examples in upcoming sections.
4.2.5 Parametric and structurally aware rules
To allow for maximum flexibility in rule applications, trans-typology structural grammars also include
parametric rules. The application of a parametric rule is dependent on one or more parameters that help to
define its behavior. Parameters can be continuous numerical values, integers, binary values, or members of a
discrete set. In all of these cases, the parametric rule must bound the parameter values that are possible, either
by defining upper and lower bounds, or by enumerating the possible discrete values.
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Two examples are given in Figure 4.16, a rule that raises the end of a SimpleShape object, and a rule that adds a
funicular form to a SimpleShape. In the first rule, there are two parameters. The first parameter,
, is a
discrete parameter and can either have the value “Left Support” or “Right Support.” The second parameter, ,
is a continuous parameter that can vary between 0 and 10 feet. In the second rule, there are also two
parameters, and again, one is discrete and one is continuous. The first parameter, , determines whether the
funicular structure acts in tension or compression, or whether it is an arch or a cable. The second parameter,
, identifies the magnitude of the horizontal thrust exerted by the funicular form. A lower value leads to a
deeper form, while a higher value results in less depth.
(a)
or
or
(b)
Figure 4.16: Two parametric rules that can apply to a SimpleShape. The second rule is also an example of structural
awareness; the funicular form is found according to equilibrium requirements.
The second rule is also an example of an important type of rule for structural grammars: those that are
inherently structurally aware. In the case of the funicular form, the chosen shape is an equilibrium solution
with internal forces that are axial only, with no imposed bending moment. This is important because it limits
the results to those that are structurally feasible. In contrast, a rule that chose an arch or cable shapes
arbitrarily would likely yield highly irrational or impossible forms.
In this case, the funicular form is found using graphic statics. The rule first analytically computes the force in
each vertical cable/strut as a reaction for a continuous beam. Using the graphic statics procedure, a force
polygon is the constructed and the horizontal force parameter is used to locate the pole, so that the rest of the
geometry can be developed (Allen & Zalewski, 2010). This is illustrated in Figure 4.17. While this rule is
specific, there are many other ways that rules can be structurally aware. For example, in truss problems, rules
can ensure that stability is maintained through the established relationship of joints, members, and reactions.
In frame problems, rules can govern the number of hinges allowed. In general, structurally aware rules help to
restrict the design space to possible solutions that are feasible, while still maintaining ample design space
breadth within the feasible range.
4.2.6 Structural performance evaluation
While structurally aware rules in trans-typology structural grammars are useful for restriction, it is still usually
necessary to compare among structurally feasible design possibilities via quantitative performance evaluation.
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There is an inherent tradeoff between embedding structural knowledge in rules versus structural evaluation as
a post-processing step. Design spaces that are too restricted by structural rules may become overly narrow and
contain few interesting or unexpected options. However, design spaces that are too open and that rely only on
evaluation may contain many undesirably or infeasible results, making them difficult to navigate. Furthermore,
such design spaces require an evaluation method in order to be useful, while those that embed structural logic
can convey interesting and feasible design options even without evaluation.
The performance evaluation method is necessarily grammar-specific, since different grammars include
different assumptions about structural behavior. In general, the evaluation method should utilize some kind of
analysis engine that produces a numerical score for a given structural shape.
In the case of the SimpleShape and SimpleGrammar, a SimpleAnalysis engine is developed that provides a
score according to required structural material cost. Cost is used instead of volume in this case because the
SimpleShape includes two materials: steel for the vertical and funicular elements, and reinforced concrete for
the horizontal elements. The steel cost is calculated according to required sectional forces of elements based on
axial forces, and the concrete cost is based on sectional properties that resist combined bending and axial stress
in the horizontal elements supported by the verticals. Unit costs for each material are based on current
averages in the construction industry in the United States, and could be replaced by more accurate values in a
more sophisticated analysis. The procedure followed by the SimpleAnalysis engine is given in Equations [4.1]
through [4.7].
Compute moments at each vertical support
of continuous beam deck by solving
simultaneous three-moment equations
(Gere & Timoshenko, 1990):
(
Compute reaction forces in vertical elements
from continuous beam moments (Gere &
Timoshenko, 1990):
(
Compute axial force in funicular elements
using graphic statics:
)
[4.1]
)
[4.2]
(see Figure 4.17)
a
A
B
C
o
b
c
Figure 4.17: Illustration of graphic statics calculation of forces in funicular elements (Allen & Zalewski, 2010).
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Compute required area for each vertical and
funicular steel element:
(
{
)
[4.3]
⁄
(
)
Compute required thickness for horizontal
reinforced concrete elements using
approximate estimate for reinforcement
depth (Alsamsam & Kamara, 2004):
(
(
)
)
(Assume
(
∑
∑
[4.4]
)
(Assume
Compute total volume for steel and
reinforced concrete:
)
for
for
)
steel elements
concrete elements
[4.5]
Compute total material cost (estimate
$3000/ton for steel, $100/cubic yard for
reinforced concrete):
[4.6]
Compute total cost, including a penalty for
connections (estimate $50/connection):
[4.7]
4.3
Design generation using grammar
The SimpleGrammar set of rules discussed above is given formally in Figure 4.18. There are four rules, and
four possible state labels for a SimpleShape. The initial shape for this grammar is shown in the left-hand side
of Rule 1, and rule application stops when the structural shape reaches the
state. The design space for the
SimpleGrammar contains arch and cable solutions, but the variation provided by the subdivision rule also
suggests forms that are more truss-like and forms that are more continuous. This grammar can be used
manually, meaning that the designer chooses which rules to apply and which parameter values to use, or
automatically, meaning that these choices are made randomly by a computer. A hybrid approach between these
two options is also possible. These three modes of design generation are discussed in the following subsections,
illustrated using the SimpleGrammar example.
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state
state
Rule 01
or
state
state
state
state
state
state
Rule 02
Rule A
Rule 03
or
Figure 4.18: The four rules in the SimpleGrammar rule set, exemplifying the various properties of trans-typology
structural grammars: recursion, state labels, parameters, and structural awareness. The naming convention used here
assigns numbers to rules that manipulate shape geometry or structural properties, and letters to rules that only change
the state label.
4.3.1 Manual rule application
The first way a designer can use a trans-typology grammar is by manually applying rules to arrive at different
possible solutions. Examples of two different manual computations are given in Figure 4.19. The designer
starts with an initial shape, and chooses a rule and parameter values to apply. In the SimpleGrammar, Rule 01
must be applied first, since it is the only rule that can be applied to shapes with the initial shape label, . After
that, Rule 02 must be applied once, and then Rule 02 or Rule A may be applied, including repeated applications
of Rule 02. Finally, once Rule A has been applied, Rule 03 is applied to bring the structure to the
state.
Depending on parameter settings and the exact sequence of applied rules, a variety of different resulting
designs can be generated.
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(a)
$16,722
(b)
$14,146
Figure 4.19: Two sample manual computations using the SimpleGrammar, with resulting cost scores indicated. The
values under the arrows indicate parameter settings, where applicable.
4.3.2 Automatic random computation
While the user must choose rules and parameter setting in manual design computation, it is also possible for
the computer to automatically and randomly generate designs through rule applications. This process is
outlined in Figure 4.20. To generate a new design, the algorithm identifies a list of rules that can apply to an
initial shape, randomly chooses one of them, identifies the parameters and bounds associated with the rule,
randomly sets each of them, and finally applies the rule, resulting in a new structural shape. The cycle then
continues, starting with the algorithm identifying possible rules for the shape based on its new state label and
other properties.
Automatic random computation is useful in exploring the scope of a grammar and determining whether it
behaves as designed. Furthermore, random design generation is very useful in conceptual design as a way to
brainstorm design ideas. Additionally, this technique is a necessary first step for more sophisticated design
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space navigation approaches, such as the interactive evolutionary framework presented in Chapter 3. A sample
of designs randomly generated using the SimpleGrammar is given in Figure 4.21.
What rules are
possible to
apply?
initial shape
new shape
Randomly
choose
parameters
Randomly choose
rule
What values of
parameters are
permitted?
Figure 4.20: The automatic random computation process for generating new structural shapes.
Figure 4.21: Six sample designs, with their calculated material costs, generated through automatic random computation
using the SimpleShape grammar.
4.3.3 Hybrid manual-automatic computation
In addition to manual and automatic computation, it is possible to generate structural designs using a hybrid of
the two approaches. This allows the user to decide on particular rules and parameter settings, while allowing
the computer to randomly complete the rest of the computation. In one version of this approach, the user
determines the entire rule derivation, or list of rules in the computation, but lets the computer determine
parameter values. Another version has the user decide on rules and parameter values up to a certain point in
the computation, allowing the computer to randomly finish the design. Similarly, the computer can start a
computation randomly to be finished by the user.
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The hybrid approach is useful in situations where it is desirable to avoid user fatigue, which occurs when the
user is required to make too many mundane decisions. Additionally, the hybrid approach can be employed
when the designer has made progress on a design, and would like to investigate alternatives for a small portion
of the rule derivation.
4.4
A trans-typology structural grammar for pedestrian bridges
The SimpleGrammar shown previously exemplifies the trans-typology structural grammar approach clearly,
but as its name implies, it is rather simple. To demonstrate the power of this approach to generate diverse and
interesting designs, this section introduces a more realistic and complex trans-typology structural grammar
developed to generate designs for short- and medium-span pedestrian bridges. The grammar is inspired by
creative and innovative bridge designs involving a variety of types of cable solutions, such as those shown in
Figure 4.22. This section gives the rules of the grammar, outlines the evaluation method, and illustrates a range
of generated designs.
Figure 4.22: Innovative bridge designs that inspired the pedestrian bridge grammar presented in this chapter. From the
top left, the bridges are the Shin Ohashi in Tokyo (1976, Image by Tommydigital), the Erasmusbrug in Rotterdam by Bert
van Berkel (1996, Image by Massimo Catarinella), the Sunniberg Bridge in Switzerland by Christian Menn (1998, Image
by Christof Sonderegger), the unbuilt Strait of Messina Bridge by Sergio Musmeci (1969, Image from The Happy Pontist),
the Rhine-Main-Danube Channel Bridge in Kelheim, Germany by Schlaich Bergermann und Partner (1987, Image by
SBP), and the First Traversina Bridge in Switzerland by Conzett Bronzini Gartmann (1996).
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4.4.1 Bridge design rules
There are 21 rules in the pedestrian bridge grammar, described in full detail in Appendix B. Like the rules in
the SimpleGrammar, these rules use parameters and state labels, and often incorporate structural logic and
knowledge. A summary of the rules is given in Table 4.1, and a sample rule is given in Figure 4.23.
Rule
Description
LHS State Label
RHS State Label
Parameters
01
Sets the height of the tower
MakeTower
AddBranches
1
02
Rotates a tower element about its base
MakeTower
MakeTower
1
03
Branches a tower element
AddBranches
ModifyTower
3
04
Deletes tower branches
ModifyTower
ModifyTower
1
05
Changes the length of tower branches
ModifyTower
ModifyTower
2
06
Rotates the tower 180 degrees
ModifyTower
MakeDeck
0
07
Adds a horizontal deck
MakeDeck
MakeInfill
3
08
Fills in space between tower branches with
narrow angles
MakeInfill
MakeInfill
2
09
Adds cable outline
MakeInfill
MultipleTowers
0
10
Adds a second tower based on the first
MultipleTowers
Subdivide
2
11
Divides the deck
Subdivide
AddSupports
1
12
Adds support cables at deck subdivision
points
AddSupports
ModifySupports
1
13
Removes cables supporting the deck
ModifySupports
ModifySupports
1
14
Connects each cable to the closest tower top
ConnectSupports
End
0
15
Connects each cable to the tower top
resulting in the steepest slope
ConnectSupports
End
0
16
Connects cables in a parallel configuration
ConnectSupports
End
0
17
Connects support cables to suspension
cables
ConnectSupports
End
1
A
Changes state label
ModifyBranches
MakeDeck
0
B
Changes state label
AddBranches
ModifyTower
0
C
Changes state label
ModifyTower
MakeDeck
0
D
Changes state label
ModifySupports
ConnectSupports
0
Table 4.1: Summary of rules for pedestrian bridge grammar, including the right-hand side (RHS) and left-hand side
(LHS) state labels and the number of parameters.
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Figure 4.23: One of the 21 rules in the pedestrian grammar. The full rule list is given in Appendix B.
4.4.2 Implicit structural information and analysis engine
Like the SimpleGrammar, the pedestrian bridge grammar incorporates structural and other functional
behavior through its rule applications and internal properties. For example, Rule 17 determines a funicular
cable shape for a given loading, similar to Rule 03 in in the SimpleGrammar. Rule 03 branches tower
elements, which, when rotated by Rule 06, allows for a wide stance to improve structural stability. When
branched towers are not inverted, they can improve constructability by providing more space for cable
connections, and can reduce cable forces by allowing slopes that are closer to vertical.
The evaluation method for designs generated using the pedestrian bridge grammar is not implemented, but
could be similar to that used for the SimpleGrammar, extended to account for tower behavior, connections,
and types of supports. The evaluation method could assume that horizontal and vertical connections are
possible at the ends of the deck, meaning that the bridges are not necessarily self-anchored. The towers could
be assumed to be fixed at their base, unless they are branched and can achieve a moment reaction through
multiple pinned connections.
4.4.3 Randomly generated pedestrian bridge designs
Figure 4.24 shows 50 designs generated by the pedestrian bridge grammar using the automatic random
computation method previously discussed. The full computations for the first 25 designs are given in Appendix
B. These designs demonstrate the breadth of the grammatical design space, including both the cable-stayed
bridge typology, the suspension bridge typology, and space in between the two. There are many unexpected
results that emerge from a relatively small set of rules, potentially suggesting innovative and creative solutions
that have yet to be built.
The structures shown incorporate structural principles through an implicit understanding of gravity loading
and considerations of forces in the grammatical rules. Future iterations could incorporate a broader range of
structural logic, especially considerations about stiffness and dynamic behavior, which are often critical in
pedestrian bridge design. In that case, bridges with very shallow structures, such as 11 and 15, or those that are
very sparse in their structural configuration, such as 50, might be modified to better meet realistic design goals.
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1
6
11
16
21
26
31
36
41
46
2
7
12
17
22
27
32
37
42
47
3
8
13
18
23
28
33
38
43
48
4
9
14
19
24
29
34
39
44
49
5
10
15
20
25
30
35
40
45
50
Figure 4.24: 50 pedestrian bridge designs randomly generated using the same structural grammar. Some designs, such as 2 and 14, resemble the standard typologies of cable-stayed and suspension bridges, while others are less expected. Computations for the first 25 designs
are given in Appendix B.
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4.4.4 Additional possible grammars
The pedestrian bridge grammar is just one possible implementation of the trans-typology structural grammar
approach, and many other grammars can be developed. For example, a grammar for long-span roof projects,
such as airport terminals, could be developed to generate designs that include arch, cable, truss, and tree-like
solutions, such as those found in existing examples, as well as new design ideas in between. A threedimensional grammar for the design of sports stadia could explore the design space within and between dome
and membrane typologies. Following the approach described in this chapter, such grammars could generate
new structural design ideas for many different building and project types.
4.5
Summary of intellectual contributions
This chapter has presented a new way to formulate broad, diverse design spaces that can generate unexpected
and innovative design alternatives for conceptual structural design. Through the use of trans-typology
structural grammars, designers can explore concepts that range across traditional typologies in an automated,
computational manner. This is important because both because new forms can be discovered, and because a
broad range of forms can be quickly and quantitatively compared.
The approach presented here extends the geometric shape grammar methodology to problems that incorporate
physical behavior and functionality, building upon existing work in functional and structural grammars. The
specific contributions developed in this chapter are as follows:

A prescription for trans-typology structural grammars that includes parametric rules, state labels,
embedded structural information, structurally aware rules, and an evaluation method

A computational framework for generating structural designs across typologies through manual
computation, random computation, or a hybrid of the two.

A specific trans-typology grammar for generating pedestrian bridge designs, including an illustration
of the wide variety of results
In addition to automatic random computation, the design spaces formulated using trans-typological grammars
can be explored using the interactive evolutionary framework presented in Chapter 3. Discussion of the
integration of these strategies is given in Chapter 6.
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CHAPTER 5:
Performance-Focused Surrogate Modeling
This chapter presents the third of three design space strategies, a surrogate modeling approximation approach
that greatly reduces the computational speed required to evaluate performance in conceptual structural design
tools. Surrogate modeling substitutes a low fidelity, computationally inexpensive model, or surrogate, for an
original high fidelity model. In general, the challenge of this method is to find a surrogate model that is
sufficiently accurate. This chapter proposes an approach that focuses on accuracy in high-performing design
space regions, tunes models automatically, and adapts to fit user preferences.
5.1
Background on design space approximation
Even in conceptual design, mathematical models of structural designs can become unwieldy and difficult to
evaluate in a manner rapid enough for a fast-paced, interactive design tool. This is because performance
evaluation methods for structures, such as finite element analysis, typically involve solving large linear systems.
While the response time for a single analysis run is tolerable in a traditional application, newer design space
exploration approaches, such as the interactive evolutionary framework presented in Chapter 3, require the
evaluation of tens or hundreds of designs at once, and demand increased computational performance.
To facilitate such exploration, an approximation of the design space can be used: the performance of a design
concept at a particular point in the design space is predicted using a data-based response surface. The benefit
of an approximation approach like this is that compared to more accurate analysis-based performance
evaluation, performance prediction takes negligible computation time. Therefore, hundreds or thousands of
design points can be visited and approximately evaluated by the computer nearly instantaneously. The
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drawback of this type of approach is that the performance prediction may be quite inaccurate, so design
decisions made based on the predictions may be ill-informed. The key to design space approximation is
navigating this tradeoff between wait time and accuracy.
5.1.1 Need for computational speed
To motivate the need for increased computational speed, Table 5.1 and Figure 5.2 show the relationship
between design complexity and required evaluation time for a range of planar truss design problems given in
Figure 5.1. Modest increases in the number of nodes, members, and design variables result in order-ofmagnitude increases in evaluation time. This effect is compounded when the number of evaluations of a design
problem reflects the need to sample larger problems more thoroughly—the so-called curse of dimensionality.
(a)
(c)
(b)
(d)
Figure 5.1: Four parametric design problem setups for exploration of nodal positions.
For example, in genetic algorithm literature, best-practice recommendations dictate that the number of
members to be evaluated in a generation should be equal to 4 , where
is the length of the binary string
representing the design vector. An equivalent “4 ” requirement can be found for the real-valued nodal position
variables used in the problems shown here, as given in Equation [5.1]. As shown in the equation, this
recommended value is a function of both the number of design variables and the allowable range in which they
may vary. Larger, more complex design problems tend to have both more design variables and larger allowable
ranges, leading to very high recommended population sizes.
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Compute equivalent binary string
length for real-valued design
vector of length
∑
|
|
[5.1]
:
This effect is also important in applications beyond genetic algorithms; any design space navigation or
optimization strategy requires more performance evaluations for more complex problems. Coupled with the
fact that more complex problems also take longer to evaluate, this leads to prohibitively slow computational
requirements very quickly.
Design Problem
Setup
Number of
Nodes
Number of
Members
Number of
Variables
4
(a)
5
7
3
68
1.0
0.8
(b)
14
25
6
192
9.1
17.4
(c)
23
39
16
456
31.9
147.9
(d)
37
83
17
340
127.6
450.6
Table 5.1: Computational time required for evaluation of the four design problems shown in Figure 5.1, normalized by the
time required to evaluate 100 versions of design (a). Two metrics are provided: the time required to evaluate 100
versions of the design,
, and the time required to evaluate the recommended number of designs for a genetic
algorithm population,
. Calculations used the analysis engine for structureFIT introduced in Chapter 3.
500
100 Evaluations
4n Evaluations
Normalized Time
400
300
200
100
0
(a)
(b)
(c)
(d)
Increasing Design Complexity
Figure 5.2: The normalized evaluation times for the four design problems shown in Figure 5.1 and detailed in Table 5.1.
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5.1.2 Approximation strategies
There are several types of approximation strategies that can be used to remedy the problem of prohibitive
computational runtimes. At a conceptual level, such strategies can be divided into those that account for
physical behavior, and those that are based on data alone. Physical approximations include approaches such as
traditional hierarchical modeling, which introduces incrementally more complicated mathematical models as
the design process progresses (Bucalem & Bathe, 2011). For example, a high-rise steel-framed building can be
modeled first as a cantilevered beam, then as a series of lumped masses connected by springs, then by a linear
frame model, and finally by a detailed solid finite element model, as shown in Figure 5.3. These approaches
have the advantage of relating directly to structural theory and engineering intuition. However, in general,
hierarchical models require engineering expertise to build, and do not lend themselves well to computational
automation because they depend on the specifics of the problem.
Figure 5.3: Hierarchical models of increasing complexity and accuracy for a high-rise steel-framed building.
Another important physics-based approach is basis reduction, in which the number of design variables is
significantly reduced through computational techniques, including establishing dependencies between design
variables. A related approach is the use of adaptive finite element techniques, which attempt to automatically
adjust mesh density to balance speed and accuracy. Again, these techniques are attractive because of their
relation to structural principles, but are hard to employ in an automatic, systematic way on a broad range of
problems. Designers and engineers who use such approaches must apply expert knowledge about the
particular nature of the problem.
In contrast to physics-based strategies, data-based strategies are agnostic about underlying physical behavior,
and operate under the assumption that performance values are derived from design variables via a black box
function. This type of approach is unsatisfying to those who are looking to establish analytical and physicsbased relationships, but works very well if such a requirement can be relaxed. There are several key advantages
to data-based approaches compared to physics-based approaches for use in computational conceptual
structural design: First, they can be developed in a much more generalized and systematic way, due to their
agnostic nature. Second, their effectiveness is independent of the breadth of physical behavior represented by
the design space; candidate designs that behave very differently can all be approximated in the same way, as
long as there is a unified approach for actual evaluation. Finally, data-based models tend to be extremely fast to
use for prediction, regardless of accuracy. In other words, a more accurate model takes longer to build, but not
to use for evaluation.
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Across disciplines and applications, there are several terms that scholars use to describe the data-based
modeling approach for design space approximation. In statistics and machine learning, such an approach is
referred to as “regression modeling”, and is also used to predict the behavior based on non-computational
empirical or experimental data (Hastie et al., 2009). In optimization and engineering, the approach is
sometimes referred to as the “response surface methodology” (Box & Draper, 1987). More recently, the
optimization community has used the term “surrogate modeling” to describe the approximation of physical
behavior based on computational data points for the purpose of navigation and optimization (Forrester et al.,
2008). Since design space navigation is an important goal in conceptual structural design, this dissertation will
also use the latter term.
5.1.3 Surrogate modeling strategies
Figure 5.4 illustrates the basic concept of surrogate modeling: statistical models are built to attempt to fit a
curve (in one dimension) or surface (in multiple dimensions) to a set of data points generated through
computer simulation. This curve or surface is then used to predict the performance of newly generated data
points, avoiding computationally expensive simulation. The curve or surface generally includes some degree of
error, both at the points it is trained on, and the newly generated points it is tested on.
Figure 5.4: Illustration of basic surrogate modeling concept: a surrogate regression model is trained based on a
collection of pre-computed data points, and then used to predict performance for other points in the design space with
some degree of error.
Surrogate modeling approaches vary in the specific data-based model used to approximate the design space.
The simplest surrogate models are polynomial models, which use polynomial functions of the design variables
to predict performance (Box & Draper, 1987). Model training involves choosing weights, or coefficients, for
terms in a predefined polynomial expression. Radial basis function models expand this concept, predicting
performance through a weighted combination of predetermined basis functions, each evaluated at the distance
from a predetermined point (Forrester et al., 2008). A specific type of radial basis function is used in a
surrogate modeling approach called Kriging, which has found to be very effective in certain types of problems
(Quiepo et al., 2005).
These standard approaches all share several advantages. First, they develop an analytical approximation
function that can easily evaluated or used in a gradient-based optimization routine. Second, they have all been
found to perform well on particular problem types. However, there are also common disadvantages to these
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surrogate modeling strategies. They require expertise to apply, since good results require careful selection of
polynomial terms or basis functions prior to building the regression model. Furthermore, any one approach,
especially one with a predetermined function structure, will generally not work well on a wide range of
problems. Because of these drawbacks, it is difficult to integrate existing surrogate modeling strategies in an
automated, widely applicable manner for non-experts.
5.1.4 Specific needs
This section has illustrated that design space approximation is necessary to make conceptual structural design
tools fast and interactive enough for practical use. Without approximation, exploring even modestly complex
design problems requires orders of magnitude more time than simple examples, both because of model
complexity and design space size and dimension. Data-based approximation approaches such as surrogate
modeling have proven effective at significantly improving computational performance, while maintaining a
reasonable level of prediction accuracy.
However, most existing surrogate modeling strategies require careful, problem-specific application. Such
requirements limit the use of surrogate modeling to experts who spend considerable time understanding the
details of their problem to build a model that is sufficiently effective. To use surrogate modeling in a
generalized conceptual structural design tool, it is necessary to find modeling strategies that are more robust
and effective on wide range of problems than those discussed in the previous subsection. Section 5.2 addresses
this need by proposing the use of ensemble black-box regression models as surrogates.
A surrogate model built using an automated approach will likely be less accurate than a carefully custom-built
model, a reasonable and expected tradeoff for increased robustness. However, this issue can be mitigated by
concentrating surrogate model accuracy in regions of the design space of most interest – that is, the highperforming regions. Furthermore, conceptual design applications are often more concerned with relative
performance than absolute performance, focusing on comparing or exploring a set of alternative design
options. There is a need to explore procedures for building and evaluating surrogate models through this lens.
Section 5.3 introduces novel strategies for sampling data points and evaluating accuracy in candidate models
that focus on high performance and relative rank.
Finally, there is a need for a systematic, automated approach for generating surrogate models for designers who
are not experts in statistics or surrogate modeling. Section 5.4 proposes such an approach, which incorporates
the ensemble black-box modeling strategies and novel sampling plans and error measures previously
introduced. The approach also includes simple and intuitive controls for the user to steer the model-building
process, and graphical result visualizations that help the user decide whether the model is sufficient.
5.2
Ensemble black-box regression models as surrogates
Because polynomial and other function-based modeling strategies are difficult to apply successfully and
consistently to a wide range of problems in an automated way, it is necessary to consider other regression
model types as potential surrogates. In machine learning, significant study has been given to off-the-shelf or
black-box methods that work well on many problem types without much tuning (Hastie et al., 2009).
Furthermore, the machine learning technique of bagging, or bootstrap aggregating, has been found to be an
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effective way to increase model robustness by averaging the results of an ensemble of regression models (Hastie
et al., 2009). This section discusses these ideas further, and proposes two machine learning regression
modeling approaches as surrogate models for conceptual structural design applications.
5.2.1 Advantages of black-box and ensemble methods
As indicated earlier, model types that do not use a predetermined underlying analytical expression are much
more flexible and robust in fitting a wide range of problem types. Regression modeling that uses this type of
model is sometimes referred to as nonparametric regression, or black-box regression. The key advantages of
this approach are ease of application, insensitivity to suboptimal tuning, and applicability to many different
design space shapes.
Ensemble methods have been also been found to confer robustness, and to increase predictive power, especially
in combination with black-box models (Hastie et al., 2009). In concept, ensemble methods work by generating
many individual regression models from a single training data set, and performing prediction by aggregating
the results through averaging. Bagging is a particular ensemble technique that uses the statistical
bootstrapping method to generate new data sets from the training set for model fitting. The new sets are
obtained by randomly sampling the original training set uniformly and with replacement, meaning that a point
can be sampled more than once. Each bagged data set therefore contains a subset of the original set, with
points occurring more than once. The effect of bagging is generally reduced noise and bias, as compared to
prediction from an individual model.
The combination of these methods results in regression modeling techniques that work very well as off-theshelf approaches for automatic predictive modeling. Two different modeling strategies of this type, ensemble
neural networks and random forests, have become popular in the machine learning realm for their combination
of robustness and predictive power. Despite their common use in machine learning, they have not been
frequently applied in surrogate modeling applications. It is proposed that they be used instead of standard
surrogate modeling types in cases where systematic, non-expert model building is important, such as in a
conceptual structural design environment.
5.2.2 Ensemble neural networks
Neural networks, sometimes called artificial neural networks, comprise a computational modeling approach
that simulates groups of interconnected biological neurons found in the nervous system. While this modeling
technique was originally developed to study brain activity, it has been found to be an effective predictive
modeling tool for regression problems in general. A diagram of a simple neural network is given in Figure
5.5(a), showing multiple input variables ( ) connected to nodes ( ) in an intermediate layer, called a hidden
layer, which are connected to nodes that represent output values ( ). In the design problems considered here,
the number of input variables is equal to the number of design variables, and there is only one output value, the
predicted performance.
Mathematically, a neural network fits a function of a linear combination of inputs to produce outputs at each
layer. Fitting of a neural network requires choosing optimal weights to apply to each input for each layer. This
is typically done using the standard least squares approach. Based on this description, it is evident that neural
networks do use a predetermined analytical formulation, so they are not truly black-box models. However, they
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are often referred to as black-box or nonparametric modeling methods due to their composite nature involving
multiple nodes and sometimes multiple hidden layers. Indeed, it has been shown that a neural network with
enough internal nodes can perfectly fit any data set (Lee, 2000). Aggregating neural networks in an ensemble
method further removes the modeling approach from a predetermined form. This is what makes the approach
flexible and widely applicable.
Individual neural networks have been proposed for surrogate modeling in comparison with polynomial models
(Carpenter & Barthelemy, 1993), and more recently, they have been considered in ensemble for surrogate
modeling (Goel et al., 2007). However, there is relatively little literature on the use of ensemble neural
networks as compared to the more standard analytical approaches, especially in comparison to their
widespread use in non-surrogate modeling regression applications.
The work presented in this chapter uses an open-source implementation of ensemble neural networks, ALGLIB
(ALGLIB Project, 2012).
(b)
(a)
Figure 5.5: Diagrams of a neural network (a) and a regression tree (b), from Hastie et al. (2009). These are bagged to
create ensembles in ensemble neural network and random forest modeling, respectively.
5.2.3 Random forests
Random forest models are a special kind of ensemble of decision tree model, sometimes called classification
and regression tree (CART) models. A simple illustration of a regression tree model is given in Figure 5.5(b),
showing the actual tree and the resulting predictive surface. The basic approach of a decision tree uses
repeated binary splitting, based on design variable values. To use the tree for prediction, one simply moves
down the branches, following the path based on given design variables. The terminus of a branch gives the
predicted output, or design performance, value. As more branches are “grown,” the predictive surface becomes
more and more refined, allowing it to match the shape of any kind of design space.
Random forests were proposed by Brieman (2001), and use the previously discussed bagging technique with
the modification that each tree only use a randomly selected subset of features, or design variables. The
purpose of this modification is to reduce correlation between individual trees in the ensemble to mitigate bias
(Hastie et al., 2009). The random forest technique is truly nonparametric in the sense that there is no
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predetermined analytical formulation for the trees or the ensemble. It has been found to be a good regression
predictor with very little tuning required (Hastie et al., 2009).
Surprisingly, there are relatively few examples in the literature of using random forests as surrogate models for
design. This may be due to their non-analytical nature, and possibly because their development in machine
learning is still somewhat recent. Additionally, while random forest models tend to perform well overall, they
do not usually produce smooth predictive surfaces, and therefore cannot be used for gradient-based
optimization. One example of their proposed use is as a screen for important design variables, that is, variables
whose setting significantly affects design performance (Serna & Bucher, 2009). However, due to their
demonstrated flexibility, robustness, and ease of tuning, it is proposed here that they be used as true surrogate
models for conceptual structural design.
As with the ensemble neural networks, the random forest work presented here uses the open-source ALGLIB
library for implementation (ALGLIB Project, 2012).
5.3
Performance-focused modeling approach
In addition to the type of surrogate model used, the strategies used to build the model also greatly affect its
performance. Two important considerations of this type are addressed in this section: sampling plans and
error measures. Sampling plans dictate how the data points used in model building are generated. Error
measures are used in deciding between candidate surrogate models, and in evaluating whether the accuracy of a
surrogate model is acceptable. This section reviews existing standards and proposes novel approaches for both
of these strategies, with a focus on conceptual design relevance.
Before discussing these strategies in detail, it is important to review the model-building process for surrogate or
other data-based regression models. It is standardly recommended that three data sets be used: a training data
set, a validation data set, and a testing data set (Hastie et al., 2009). Each data set is a list of observations
generated using computational analysis, with an observation consisting of a list of features, or design variables,
and an output, or performance score. The training set contains the data used to actually fit the model. The
validation set is used in selecting a model among multiple candidates. The candidates can differ in surrogate
model type, or in the tuning of model-specific “nuisance parameters,” so-called because the optimal setting for
a particular problem requires experimentation. In validation, multiple models are fit to the training data using
varying nuisance parameter settings. The model that performs best on the validation set according to an error
measure is selected. Finally, the testing set is used to objectively evaluate the accuracy of the chosen surrogate
model. Since the training set and the validation set were used to fit and choose the model, they are not fair data
points to use to test the model, so a new set must be used.
It is clear that the data points generated for each set and the error measures used to evaluate accuracy greatly
affect the surrogate model building process. The following sections explore these two issues in detail.
5.3.1 Weighted sampling plans
In conventional machine learning and other statistics applications, model builders do not have much control
over the collection of data points; models are often built from pre-existing data that is assumed to be random.
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In contrast, in surrogate modeling, data points are generated, or sampled, deliberately through computer
simulations, and it is possible to explicitly dictate how this should be done. In the literature, the decision of
how to computationally generate data points is called the sampling plan or the design of experiments (DOE). It
has been argued that sampling plans for computer-generated data should be space-filling as opposed to
completely random (Sacks et al., 1989). There is a great deal of literature on sampling plans that are spacefilling; this dissertation will use random Latin hypercube sampling (LHS) as one example.
It is worth considering whether space-filling samples are truly best for building surrogate models for use in
conceptual design exploration. The space-filling criterion suggests that the best surrogate models are built
from data that most completely represents the full design space. However, as seen in previous chapters, design
spaces for structural design problems often feature discontinuities, where performance metrics spike towards
high values, indicating very low performance. These regions of the design space are very difficult to fit with a
surrogate model, and in attempting to do so, model accuracy may be compromised in other high-performing
regions. Furthermore, a conceptual design approach that considers performance will likely involve exploration
in high-performing design space regions. These considerations suggest that weighted or adaptive sampling
plans that include more samples from better performing design space regions are preferable in conceptual
design.
This dissertation proposes two simple weighted sampling plans, as well as a more complex adaptive sampling
plan. All three plans result in samples that are not space-filling, but rather performance-focused. To help
illustrate the sampling plans, a simple two-dimensional design problem similar to problems in previous
chapters is presented in Figure 5.6(a). The two variables are the vertical positions of two of the nodes of the
truss, and the performance metric is the required volume. A contour plot of the design space is given in Figure
5.6(b).
The points resulting from the sampling plans for the example problem are shown in plots in Figure 5.7, and
resulting predicted design space plots using a random forest surrogate model are given in Figure 5.8. First, the
sampling plans are described in detail below:

Weighted Random Uniform Sampling: This plan generates random uniformly distributed data points,
and then discards them, keeps them, or keeps multiple copies of them depending on their performance
score. The copying plan is given in Equation [5.2].
normalized performance score, relative to initial design
copies of generated design added to sampled set
maximum permitted score for sampled set
{
[5.2]

Weighted Latin Hypercube Sampling: This plan generates points using a Latin hypercube sampling
plan, and then discards them, keeps them, or keeps multiple copies of them depending on their
performance score. This strategy also uses the copying plan given in Equation [5.2].

Adaptive Sampling: This plan uses an evolutionary approach to evolve data points in high-performing
regions of the design space. The procedure iterates over a set number of generations, identifying and
mutating high-performing points to obtain additional data. As the generations proceed, the mutation
rate decreases. The mutation procedure used is the same as given in Chapter 3.
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The weighted and adaptive sampling plans have the advantage of focusing the surrogate model to the highperforming design space regions in a simple way. Because the internal model fitting procedures work by
minimizing error over the design space, duplicate points effectively more heavily weight error at their location.
However, if a model disregards poorly performing designs too greatly, there is a danger that it will predict high
performance for poor designs, so poor-performing design space regions should not be completely ignored.
50
20
0
-30
-50
-80
(a)
(b)
Figure 5.6: Simple five-bar truss design problem (a), with two design variables, the vertical positions of n2 and n4, and
the resulting design space (b), where required normalized volume is the performance metric.
The resulting sampled points in Figure 5.7 show that as expected, the Latin hypercube sampling plan results in
the most evenly distributed set of points. The adaptive plan is similar to the random uniform sampling plan, in
that more points are focused in high-performing design space regions. However, while the adaptive plan
sometimes works very well, it may easily miss local minima in the design space, and may entirely disregard
poorly performing regions. The random uniform plan is also susceptible to this issue, to a lesser extent. The
most conservative and consistent approach is the weighted Latin hypercube, which balances even point
distribution with performance-focused weighting through the copying scheme.
(a)
(b)
(c)
Figure 5.7: 100 data points resulting from the weighted random uniform sampling (a), weighted Latin hypercube
sampling (b), and adaptive sampling (c). In plots (a) and (b), the size of the point indicates the number of duplicate copies.
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Random Uniform Sampling
(a) Unweighted
(b) Weighted:
,
(c) Weighted:
,
(f) Weighted:
,
Latin Hypercube Sampling
(d) Unweighted
(e) Weighted:
,
Adaptive Sampling
(g)
(h)
(i)
Figure 5.8: Predicted design space plots generated by random forest surrogate models using different sampling plans.
In general, the weighted and adaptive schemes tend to lead to more accurate models in the high-performing design space
regions.
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The predicted design space plots in Figure 5.8 show that in general, the weighted and adaptive plans result in
better predictions; both of these approaches map out the high-performing design space regions more correctly
than the unweighted random uniform and Latin hypercube approaches. However, as indicated above, the
adaptive plan is inconsistent compared to the weighted plans, due to its random nature. The best results come
from the using the weighted Latin hypercube approach with heavy weighting, and it is therefore suggested that
this plan be used for building surrogate models for conceptual structural design.
5.3.2 New rank-based error measures
Building and selecting surrogate models is a process that involves maximizing model accuracy, or in other
words, minimizing error. This occurs both during the training of the model, during which the structure of the
model is set, and during model validation, during which model tuning parameters are selected. Additionally,
model testing that occurs after the surrogate model is developed seeks to show that error is acceptably low. In
all three of these cases, a formalized numerical error measure must be used. In the case of model training, the
structure of the error measure is critical to the training process. For example, it must be computable and
differentiable at each point in the design space. In contrast, error measures used in validation and testing can
relax these requirements and be treated in a more black-box manner. This section will introduce new error
measures that do not meet the strict requirements of training error measures, but can be used in validation and
testing to select and verify surrogate models that perform better for conceptual design.
Global error for a surrogate model is conventionally given as the mean value of a loss function. Most existing
surrogate modeling schemes measure error using the mean square error (MSE) or root mean square error
(RMSE) functions, given in Equations [5.3] and [5.4]. These measures treat error equally over the entire design
space, and more heavily penalize high inaccuracies in model prediction due to the squaring operation. The
RMSE function is sometimes preferred, since the square root transforms the error measure back to the units of
performance.
total design points
actual performance
MSE:
∑ ̂
̂ predicted performance
RMSE:
√ ∑ ̂
[5.3]
[5.4]
The mean absolute error (MAE) function has been used in cases when it is undesirable to give more weight to
outliers in assessing the overall error of a surrogate mode. This is given in Equation [5.5]. A hybrid of the MSE
and MAE is the error computed with the Huber loss function, a piecewise function that is parabolic in the
region close to 0, and linear beyond this region (Hastie et al., 2009). A comparison of these three loss functions
is shown in Figure 5.9.
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MAE:
∑| ̂
|
[5.5]
Figure 5.9: Three possible loss functions that can be averaged over a design space to measure the accuracy and efficacy
of a predictive model (Hastie et al., 2009).
These existing error measures were developed in the realm of statistical regression modeling, used to create
data-based predictive surfaces from empirical results or findings. However, because the goals of conceptual
design differ considerably from data analysis, it is important to reconsider what error measures mean and
should convey in this new context. There are two key differences in the priorities of predictive modeling for
design compared to analysis. First, the desired accuracy for the surrogate model is not uniform across the
entire design space; more accuracy is needed in high-performing regions of the design space, where most of the
exploration will occur. Second, conceptual design involves alternative competing design concepts, so the ability
of the model to correctly compare the performance of design points is more important than the values it
predicts.
These differences suggest that standard error measures that incorporate value discrepancies across the entire
design space do not lead to the best surrogate models for conceptual design. To address these issues, this
dissertation proposes novel error measures that more heavily weight error in high-performing regions of the
design space, and that use rank instead of value to compute discrepancies. To facilitate comparisons with
existing error measures, these error measures all decrease with improvement, have 0 as the best value, and are
somehow normalized. Four novel error measures of this type are given in Equations [5.6] through [5.9] and
explained below:

Mean Rank Error (MRE): This measure is similar to MAE, but uses rank instead of value, and
normalizes the value by the average rank across the design space. It is given by the absolute difference
between the actual rank and predicted rank, normalized by the average rank, and averaged over the
entire design space.
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total design points
actual rank
∑
̂ predicted rank

| ̂
|
[5.6]
Top Mean Rank Error (TMRE): This measure only accounts for error in top-performing designs, to
account for the fact that many conceptual design applications focus on high-performing options. Its
computation is the same as the MRE, but averaged only over the top
selected by the user. A typical value for
otherwise noted.
is 10 to 20. This chapter will generally use 20, unless
top-performing design points

performing designs, with
∑
| ̂
Top Ratio Error (TRE): This measure computes how many of the top
|
[5.7]
designs have been correctly
ranked as within the top designs by the predictive model, and is given by the difference between
and the correctly identified designs, normalized by the total number of design points considered.
total design points
[5.8]
top-performing design points
correctly identified top-performing designs

Top Factor Error (TFE): This measure computes how far out of the top
has placed the top performers. It is given by the difference between
designs the predictive model
and the worst predicted rank of a
top- design, normalized by the total number of design points considered.
̂
maximum predicted rank of top-
̂
design
[5.9]
To illustrate the computation and visualization of these error measures, three surrogate models and their
predictions for randomly simulated data sets are given in Table 5.2, with error measures summarized in Figure
5.10. It is important to note that none of the models performs best according to every error measure: Model 1
performs best according to the traditional error measures (RMSE and MAE), Model 2 performs best according
to TMRE and TFE, and Model 3 performs best according to TRE and MRE.
These differences can be conceptually understood by examining the highlighted rows in Table 5.2, which
correspond to the predicted locations of the top performing designs. Model 1 distributes the top performers
somewhat evenly across its ranking, only correctly placing one top design in its top five. In contrast, Model 2
places the top designs much closer to the top in its rankings. While it only correctly placed one top design in its
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top five as well, the other top designs follow closely behind, suggesting that this model would perform well in
cases where it is critical to correctly place all top designs near the top. Model 3 performs very well in that it
correctly identifies three of the top five designs. However, it placed its incorrectly identified designs very far
down the list. This behavior may be desirable in situations in which it is important to identify most of the very
top performers.
Since each of these new error measures accounts for different criteria, it is proposed that they be used in
combination with weights that reflect their relative importance in a particular conceptual design application.
Despite the fact that each measure is normalized, they are nevertheless not directly comparable and therefore
cannot be combined using a simple linear combination. Instead, a weighted linear combination of the ranks of
the model according to each error measure can be used instead, as shown in Equation [5.10].
weight indicating importance of th error measure
[5.10]
∑
model rank according to th error measure
Model 1
Model 2
Model 3
RMSE
MAE
MRE
TMRE
TRE
TFE
Figure 5.10: Normalized comparison of error measures for the three example surrogate models shown in Table 5.2.
While Model 1 has the lowest error values according to traditional error measures (RMSE and MAE), it performs fairly
poorly in the other error categories presented here. In general, the choice of the best model depends on the accuracy
criteria most important to the specific application.
To summarize, this subsection has presented four novel error measures that can be used alone or in
combination with existing error measures to select between candidate surrogate models during the model
validation stage, and to evaluate the accuracy of a chosen surrogate model during the testing phase. Since the
novel measures are nondifferentiable and require knowledge of all the sampled points to evaluate error at any
one point, there is no clear way to use them during model training, which may continue to use existing error
measures such as MSE. However, use of the novel error measures in validation helps to find surrogate models
that perform better for conceptual design applications.
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CHAPTER 5: PERFORMANCE-FOCUSED SURROGATE MODELING
Surrogate Model 1
RMSE
MAE
MRE
TMRE
TRE
TFE
Surrogate Model 2
̂
̂
Surrogate Model 3
̂
̂
1
2
0.91
1.04
̂
̂
4
10
1.02
1.04
1
2
9.74
1.02
29
6
0.99
0.99
1
2
4.74
1.07
27
11
0.99
0.99
0.94
3
0.99
3
0.97
6
0.99
3
0.93
5
1.05
3
30.39
30
0.99
4
0.89
3
0.99
4
1.17
16
1.08
4
1.45
22
1.00
5
2.98
25
0.99
5
0.88
2
1.14
5
0.98
5
1.00
6
0.94
5
1.00
6
0.97
7
1.15
6
7
1.00
7
1.21
17
1.25
7
30
1.02
8
1.14
13
1.26
8
4
1.03
9
2.28
23
1.29
9
10
82.57
30
1.29
10
1.12
13
1.00
7
2.57
24
1.00
9
0.99
137.2
4
0.94
1.45
21
1.00
8
3.37
25
1.04
10
1.03
9
1.05
1.07
10
1.04
11
0.82
2
1.06
11
1.15
15
1.29
11
1.09
12
1.45
21
2.19
12
1.10
13
17.16
28
2.69
13
1.07
9
1.04
12
0.78
1
1.11
12
1.06
13
3.28
26
0.92
1
1.06
14
1.43
22
1.10
14
1.39
20
3.16
14
1.05
8
1.08
15
1.08
12
1.10
15
1.3
18
3.27
15
0.92
2
1.09
16
16.86
28
1.10
16
1.05
11
3.95
16
1.10
17
1.14
14
3.98
17
1.09
11
1.09
17
1.13
16
6.94
26
1.10
18
22.33
29
1.10
18
0.95
6
4.02
18
1.11
19
0.89
3
5.34
19
20
3.86
25
6.13
20
7.11
27
1.10
19
1.49
24
1.04
7
1.10
20
0.99
8
1.12
7.56
28
1.10
21
1.11
15
1.12
21
3.11
24
7.59
21
1.14
22
1.04
9
8.31
22
1.19
23
0.98
8
8.33
23
0.97
4
1.10
22
1.36
20
1.28
19
1.10
23
1.03
10
1.2
18
1.10
24
1.10
14
1.19
24
2
22
8.52
24
1.12
14
1.12
25
1.08
13
1.19
25
4.89
27
8.53
25
1.49
23
1.17
26
1.36
21
1.21
26
1.07
12
10.22
26
1.22
27
0.7
1
10.51
27
1.15
15
1.18
27
1.45
23
1.42
20
1.22
28
1.17
18
1.22
28
1.31
19
12.74
28
1.22
29
35.72
29
14.28
29
1.22
30
4.47
26
20.28
30
1.19
17
1.22
29
1.19
19
1.18
16
1.22
30
1.15
17
5.95
2.10
25.35
6.14
16.41
7.16
0.65
0.57
0.54
6.00
2.16
4.00
0.13
0.13
0.07
0.57
0.23
0.73
Table 5.2: Visualization of predictions from three surrogate models, with error measures given. In this case, = 30 and
= 5. It is noteworthy that the first model performs the best according to standard error measures (RMSE and MAE), but
not according to the novel performance-focused error measures proposed here.
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5.4
CHAPTER 5: PERFORMANCE-FOCUSED SURROGATE MODELING
Automatic model building for non-experts
As discussed in the previous section, the surrogate modeling procedure typically involves deciding on a
regression model type and sampling plan, training a set of models using different nuisance parameter settings,
manually selecting the best performing model according to error measures and perhaps other qualitative
considerations, and testing the finalized model to confirm sufficient accuracy. Many of these steps depend on
the intuition, judgment, and expertise of the modeler, which limits the use of surrogate models to a narrow
group of specialists. This section proposes a way to combine automation with input from a non-expert user to
systematically build models that perform well and meet the user’s needs.
5.4.1 User-specified accuracy
One of the key decisions in building surrogate models is the tradeoff between accuracy and the time required to
build the model. More accurate models require more data points for training, validation, and testing, and since
data points result from computationally expensive simulations, they are time-consuming to generate. The
number of samples needed for a reasonably accurate model depends on the number of design variables, the
complexity of the design space, and the tolerance for various types of error.
The approach proposed here makes an initial suggestion for the number of data points to the user, along with
an estimate of the required time to build the model. The recommended number of data points is the minimum
of 100 and the
value given in Equation [5.1]. The user can choose to adjust the number of data points in the
training, validation, and testing sets through a slider control, with the estimated build time reflecting this
adjustment, as shown in Figure 5.11. Once the model is built, the user can decide whether it is acceptable, with
the help of the error measures given in Section 5.3 and the visualization to be discussed shortly. If the accuracy
is insufficient, the user can easily rebuild the model with an adjusted setting for the number of data points.
This simple user control is important because it helps to tailor the surrogate model to the user’s needs. In some
cases, the user may want to quickly mock up design possibilities, and may be willing to tolerate significant error
in exchange for very fast model-building and performance prediction. In other cases, the user may be
interested in a more detailed and refined study of a design problem, and may not mind waiting five or ten
minutes for the program to sample data points and build a higher fidelity model, especially considering that
this will allow for immediate and accurate performance predictions in a design exploration framework. By
explicitly showing estimated wait time, this approach helps the user make an informed decision.
5.4.2 User-specified model-building preferences
In addition to the tradeoff between wait time and accuracy, there are several other model-building preferences
that the user may choose to adjust. It is proposed that the user be able to determine the relative weights, or
importance, of standard error measures and those proposed in Section 5.3. The specified weights are used to
select a model that performs best according to a weighted sum of ranks during the validation step.
Recommended settings are given that more heavily weight rank predictions than value predictions, and that
consider a top-performing subset of designs instead of the entire design space. Based on specific applications,
the user may choose to adjust these using more slider controls, as shown in Figure 5.11.
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(a)
(b)
Figure 5.11: Dashboard for the user to control model-building settings in an intuitive way. The user may choose to accept
all recommended settings (a), and automatically build a surrogate model through a single button click, or to adjust more
settings (b), including the tradeoff between wait time and speed, the error measure weights, the types of models considered,
and the sampling plan.
Additionally, the user may choose to vary the types of models considered. The recommended setting includes
both random forests and ensemble neural networks, since both perform well in general. However, ensemble
neural networks tend to produce approximations that are smoother but take more time to build. To save time,
the user may choose to consider only random forest models. Alternately, it may be desirable to omit random
forest models in applications where smoothness and continuity are important.
Finally, the user may choose to adjust the sampling plan used to generate the training and validation data sets.
The recommended plan is the weighted Latin hypercube approach, suggested in Section 5.3, which leads to well
distributed data sets but focuses the model-fitting on higher performing designs. However, the user may
choose to try a different sampling plan, such as the adaptive approach also suggested in Section 5.3, to try to
improve the performance of the surrogate model for a given problem. Regardless of the sampling plan used to
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build and validate the model, this approach uses random uniform sampling to test the model. This is so that
the test simulates the performance of the model on any random part of the design space that the user may
choose to explore.
5.4.3 Automated parameter-setting
Based on recommended or adjusted weights of error measures, the proposed approach automatically completes
the validation step of the model-building process. In this step, nuisance parameters must be set for both types
of models under consideration. This is accomplished by fitting five models of each type, random forest and
ensemble neural network, with different parameter settings to the training data set. The ten models are then
ranked according to each of the ten error measures, and the model with the best combined weighted rank is
selected.
The nuisance parameter selected for each model type is a sort of regularization parameter, or one that controls
how closely the model fits the training data in tradeoff with model regularity, or smoothness. A higher
regularization parameter setting leads to smoother models that fit the specific training data less well, but may
fit new data points better, avoiding overfitting. The optimal setting for these parameters depends on the
specifics of the design problem, necessitating the validation step.
In the random forest model, the regularization parameter used here is the ratio parameter, a decimal value
between 0 and 1 that determines the percentage of training points used to fit any one decision tree in the forest
ensemble (ALGLIB Project, 2012). The values considered for this parameter are [0.15, 0.30, 0.45, 0.60, 0.75].
In the ensemble neural network model, this approach uses the number of nodes in the hidden layer as the
regularization parameter. Including more nodes leads to a more detailed model, but too many nodes can
overcomplicate the model in some cases. The values considered for the number of nodes are [4, 6, 8, 10, 12].
Once the program has selected the best model and parameter setting according to the given error measure
weights, it automatically tests the selected model, and creates a graphical report of the results, which
communicates the chosen model and parameter setting to the user. The report also includes values for six error
measures, the settings used discussed in the previous subsections that are used to build the model, and the time
taken to build the model. As a separate tab, the results report contains a table with the raw data used in the test
set, including actual performance, actual rank, predicted performance, predicted rank, and design variable
settings. Finally, the results report includes intuitive visualizations of the test error, described in the following
subsection.
5.4.4 Graphical testing results
While the numerical error values suggested in 5.3 are useful for quantifying the accuracy of a surrogate model,
they are not necessarily meaningful to a non-expert user. Therefore, this approach includes graphical
information to help the user visually understand model accuracy, regardless of expertise or background. Two
types of graphics are presented: plots of observed versus predicted ranks and values, and a set of new rankbased error diagrams. An example of observed-predicted plots is given in Figure 5.12. In these types of plots,
model accuracy is indicated by how closely the data points fall to the central diagonal. The example shown in
the figure shows a model that performs well at predicting rank, but less well at predicting performance values,
at least for poorly-performing designs.
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An example of the rank-based error diagrams is given in Figure 5.13. There are two versions of the diagram,
one sorted by actual, or observed, rank, and one sorted by predicted rank. In the actual rank sorting, the
colored bars represent the rank position of the designs predicted to be in the top subset. In the predicted rank
sorting, the colored bars correspond to the position of designs that are actually in the top subset. Accuracy is
shown by how close the colored bars fall to the left of the diagram. A colored bar far to the right in the actual
rank sorting indicates that a design that performs fairly badly is predicted to perform very well. In the
predicted rank sorting, this indicates that the surrogate model has predicted that a very good design performs
badly. Depending on the application, tolerance for both of these types of error may vary.
These error visualizations are an important way for the user to gauge the success of the surrogate model
without needing to process quantitative error information. The visual results also convey more than the
summarizing error measures are able to, in that they visually represent all of the data in the test set. If the user
is dissatisfied with the results, it is possible to rebuild the model, using more data points, a different sampling
plan, or different error weights, and compare the error graphics. Once an acceptable model has been found, the
user may select it for use in a conceptual design application, such as evolutionary navigation.
Figure 5.12: Observed-predicted plots showing actual rank vs. predicted rank and actual value vs. predicted value. A
perfectly predictive model would place each point on the central diagonal line.
Figure 5.13: Rank-based error bar diagrams showing surrogate model accuracy. The colored bars correspond to the
locations of the top-performing designs, according to the predictive model (in the actual rank sorting) or to the actual
results (in the predicted rank sorting). A perfectly predictive model would place all colored bars in the leftmost segment of
each diagram.
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5.5
CHAPTER 5: PERFORMANCE-FOCUSED SURROGATE MODELING
Surrogate modeling case studies
This section illustrates the effectiveness of the surrogate modeling approach presented in this chapter through
the sample design problems introduced in Figure 5.1. For each design problem, surrogate models were built
using the automatic procedure for several different sample sizes. The results are compared with those from
models built using standard techniques, i.e. unweighted Latin hypercube sampling and RMSE as the validation
step error measure. This comparison is summarized in Table 5.3. More detailed results for these case studies
are given in Appendix C.
5.5.1 Model accuracy
In general, the results in Table 5.3 support the argument that this chapter makes for weighted sampling plans
and rank- and performance-focused error measures. The rank bar diagrams show that as more samples are
generated, the models’ predictive power tends improves in all cases. However, for a given sample size,
especially smaller sample sizes, the proposed approach tends to be more accurate in terms of the measures of
mean rank error and top ratio error, as shown in Figure 5.14. This is an important result because it suggests
that in cases where only a small number of data points can be generated, the modifications proposed in this
chapter have significant advantages.
(a) Standard
(b) Proposed
(c) Proposed
(c) Standard
(d) Proposed
(d) Standard
Top Mean Rank Error
(b) Standard
4
0.16
3.5
0.14
3
0.12
2.5
Top Ratio Error
(a) Proposed
2
1.5
0.1
0.08
0.06
1
0.04
0.5
0.02
0
0
100
200
400
800
16n
Number of Samples
100
200
400
800
16n
Number of Samples
Figure 5.14: Test error values for surrogate models built using the standard and proposed approaches for the design
problems introduce in Figure 5.1.
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Design
Problem
Setup
CHAPTER 5: PERFORMANCE-FOCUSED SURROGATE MODELING
Number of
Samples
Model Build
Time [sec]
TMRE
TMRE*
TRE
TRE*
100
5
2.340
3.385
0.109
0.137
200
9
2.236
2.353
0.092
0.111
400
23
1.889
2.361
0.077
0.091
16 = 544
36
1.664
2.151
0.079
0.085
100
12
1.360
1.737
0.061
0.095
200
34
1.150
1.366
0.080
0.085
400
74
0.845
1.304
0.054
0.078
16 = 1536
273
0.744
0.781
0.045
0.050
100
53
1.177
2.075
0.085
0.078
200
112
0.958
1.727
0.080
0.087
400
204
1.325
1.698
0.074
0.075
16 = 3648
1500
0.916
0.979
0.055
0.063
100
197
0.707
0.975
0.055
0.054
200
377
0.841
0.858
0.060
0.064
400
581
1.063
0.791
0.070
0.051
16 = 2720
3403
0.466
0.491
0.035
0.041
Rank Bar Diagram
Rank Bar Diagram*
**
**
**
**
(a)
(b)
(c)
(d)
Table 5.3: Summary of surrogate modeling results for the design problems introduced in Figure 5.1. The column headings marked by * indicate results from surrogate
models built using a standard approach, i.e. Latin hypercube sampling and RMSE, while the other columns include results from the new approach proposed here, i.e.
weighted Latin hypercube sampling and a weighted combination of novel error measures. The ** mark indicates that the rank bar diagram is too fine to be displayed.
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Although the results generally confirm expected behavior, there are some surprises. Problem (a), which is the
simplest both in terms of structural complexity and number of design variables, was the most difficult design
space to approximate, attaining the worst error results for nearly each of the given sample sizes. Problem (d),
which was the most complex structural model with the most variables, was the easiest to approximate. This
may be due to the relatively small bounded regions for each of the variables in (d), compared to the large
variation allowed of the variables in (a). This result suggests that explorations of complex models are a very
worthwhile application for surrogate modeling when the variation under consideration is small, since the actual
evaluation time is large, and the approximation is quite accurate.
5.5.2 Model building time
A plot showing computational time required for model building versus the resulting top mean rank error is
shown in Figure 5.15. This plot shows that while a significant error improvement occurs when increasing the
number of evaluations, and therefore the model building time, for the simpler models, there are diminishing
returns once the build time reaches approximately 100 seconds. This is a somewhat unexpected result, and
may relate to the specific cases considered here more closely than to design problems in general. However, if
generalized, the conclusion is a positive one: reasonably effective surrogate models for conceptual structural
design can be built in a few minutes, and perform nearly as well as those that take an order of magnitude more
time to create.
2.5
(a)
(b)
Top Mean Rank Error
2
(c)
(d)
1.5
1
0.5
0
1
10
100
Model Building Time [sec]
1000
10000
Figure 5.15: Test error values for surrogate models built using the standard and proposed approaches for the design
problems introduced in Figure 5.1.
5.6
Summary of intellectual contributions
This chapter has motivated and addressed the need for a design space approximation approach in conceptual
structural design tools. Because computational approaches for conceptual structural design often require fast
and repeated structural evaluations, such as in real-time analysis or evolutionary exploration, they are currently
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limited to very simple example problems or are very slow. After a review of existing methods in structural
approximation, this chapter has proposed surrogate modeling as the best option for an automated tool.
While surrogate modeling is a well-established approach in the optimization community, this chapter proposes
new modifications to adapt surrogate modeling for use in conceptual structural design applications beyond
optimization. The proposed approach strategically sacrifices model accuracy and analytical representation,
which are not as important in conceptual design as in optimization, in exchange for broad robustness and
applicability. The specific contributions in this area are:

The use of ensemble black-box regression strategies, and particularly ensemble neural networks and
random forests, over the more standard analytical surface-fitting surrogate modeling approaches, due
to their robustness and ease of application.

Weighted sampling plans and novel error measures that shift model accuracy to the best-performing
regions of the design space, and account for comparative performance predictions (i.e. rank) instead of
predicted performance values.

An automated model-building procedure and user interface that gives the designer intuitive control
and results, while removing the need for model-building expertise.
This chapter has illustrated the effectiveness of this approach on design problems of varying complexity,
showing that automated surrogate modeling can lead to reasonably accurate performance predictions that take
negligible time to compute, a key feature for fast and interactive conceptual design tools.
The contributions of this chapter are clearly applicable to the interactive evolutionary framework presented in
Chapter 3, and can also be applied in combination with the trans-typological structural grammars proposed in
Chapter 4. Chapter 6 discusses the integration of these three methodologies in detail.
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PART III:
Integration and Conclusions
“The intention behind our approach became clear: this bridge finally convinced us that there is still a world of
forms to be discovered.”
— Laurent Ney in Shaping Forces, 2010
CHAPTER 6:
Integration of Design Space Strategies
The previous three chapters introduced three computational design space strategies for conceptual structural
design: design space navigation through an interactive evolutionary framework, design space formulation
through trans-typology structural grammars, and design space approximation through performance-focused
surrogate modeling. This chapter outlines possible ways to integrate the three strategies into approaches that
offer creative freedom, design diversity, and fast computational interaction. There are several challenges to
combining the three strategies, and these are addressed specifically in three subsequent sections that discuss
each pairwise combination.
6.1
Design space strategies applied together
This dissertation has shown that effective computational approaches for conceptual structural design should
treat problems in a systematic way through the notion of the design space: navigation, formulation, and
approximation of design spaces are all important to achieve the key goals of creativity, diversity, and
performance. The individual strategies presented in previous chapters comprise a palette of computation tools
available in conceptual structural design. For some design problems, a single strategy used alone makes the
most sense. In other cases, using two or all three of the strategies together offers improved opportunities for
design space exploration. This chapter considers these cases.
With an approach that integrates all three strategies, a designer can navigate a broad and diverse space of
design options with the aid of performance-based guidance, in a manner rapid enough to align with the pace of
an analog conceptual design session.
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The integration of these three ideas into one approach has yet to be presented in the literature. Because the
methodologies developed here have origins in disparate disciplines, it is not trivial to combine them
successfully. Of particular difficulty is the incorporation of grammar-based design spaces, since the majority of
work in optimization, evolutionary algorithms, and machine learning centers around the parametric design
vector. The resolution of these and other integration issues constitutes important and original work, and
possible solutions to these challenges are presented in subsequent sections.
6.1.1 General integration strategy
A pairwise approach is used to consider the integration of the three strategies, individually addressing each of
the three overlapping regions in the Venn diagram in Figure 6.1. Combining the interactive evolutionary
framework with trans-typology structural grammars involves resolving crossover and mutation for a
nonparametric design formulation, integrating a grammar-specific analysis engine, and generalizing the user
experience to support grammar-based design spaces. Integrating the performance-focused surrogate modeling
approach into the evolutionary framework requires a strategy for when to use the approximation, how to
update and adapt it, and how to use it for real-time analysis in design refinement. Finally, combining
performance-based surrogate modeling with the structural grammar approach requires deriving common
variables or features from grammar-based designs that can be used to build regression models. Each of these
issues is addressed in the following sections.
Trans-typology
Structural
Grammars
Interactive
Evolutionary
Framework
3
6
5
4
Performance-focused
Surrogate Modeling
Integrated
Design
Approaches
Figure 6.1: Venn diagram showing overlapping design space strategies integrated into a single approach. The numbers
in the three regions signify corresponding chapters in this dissertation.
6.2
Evolutionary framework and structural grammars
Chapter 4 introduced a novel way to generate grammar-based design spaces for structural design problems that
are far broader and more diverse than parametric design spaces. Clearly, it is important to have a way to
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CHAPTER 6: INTEGRATION OF DESIGN SPACE STRATEGIES
navigate such a design space in a free but guided manner. Chapter 4 includes two simplistic ways to traverse
the design space: random design generation and manual rule application. A more effective design space
navigation technique is interactive evolutionary exploration, as presented in Chapter 3. This approach allows
designers to take full advantage of the powerful grammar-based design space; by exploring it in a rapid and
interactive way through the power of evolution, it is possible to quickly discover new, unexpected, and highperforming designs.
As stated previously, the conceptual challenge of integrating a grammar-based approach into an evolutionary
framework is that evolutionary algorithms, and optimization-based approaches in general, rely on a parametric
design formulation and design space definition. Specifically, the exploration strategies of crossover and
mutation, the combination of parent designs to produce offspring, depends on an underlying design vector of
equal length for each parent design. Other challenges include integration with the additional aspects of the
interactive evolutionary framework: establishing a method for structural performance evaluation, and
modifying the design setup and design refinement modes to support establishing and exploring nonparametric
design spaces.
6.2.1 Design models and variables
The interactive evolutionary framework presented in Chapter 3 depends on design models, which have
variables of several types. The underlying formulation for all of the examples given in that chapter is a
parametric design vector, or a list of each of the variable settings in a given order. In contrast, a grammarbased design has a rule derivation as its equivalent to the design vector. The rule derivation gives the list of
rules, and their parameter settings if applicable, in the order in which they were applied to arrive at the final
design. Due to the non-deterministic nature of the structural grammars presented in Chapter 4, the rule
derivation does not have a predetermined length. The difference between design vectors and rule derivations is
illustrated in Figure 6.2.
Variables are defined in a parametric design formulation as the parameters, or entries in the design vector. In a
rule derivation, each rule application is a variable. The fact that the number of variables is not fixed is what
makes grammar-based approaches so powerful, and design spaces formulated using grammars so broad and
diverse. However, this also makes it more difficult to treat designs in a systematic way.
6.2.2 Crossover and mutation of variables
Chapter 4 addresses the need to randomly generate grammar-based designs using the computer, which is a
requirement of the interactive evolutionary framework. A step beyond random generation involves random
crossover and random mutation. In Chapter 3, these two strategies were implemented on the variable level;
each type of variable was required to have crossover and mutation functionalities defined. An example of such
definitions for continuous variables was given.
The grammar-based approach can fit into this framework if it similarly defines crossover and mutation for its
variable types, which are rule applications. Because of the nonstandard length, crossover cannot be defined in
an element-wise fashion, and must instead apply to an entire design, or rule derivation. This type of crossover
is more closely related to biological crossover, and also more conceptually related to crossover implementations
in standard genetic algorithms: one or more crossover points are identified and portions from each parent are
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CHAPTER 6: INTEGRATION OF DESIGN SPACE STRATEGIES
swapped to generate new designs. Several implementations along these lines have been proposed for use with
grammars (Harper & Blair, 2005; O'Neill et al., 2003; Byrne et al., 2011). In existing literature, a byte encoding
is typically used, and the goal of crossover is to most quickly arrive at optimal designs.
0.02
1.81
-0.67
4.37
6.01
11.26
23.05
19.58
37.52
Rule 01
= 2.43
Rule 01
= 6.78
Rule 01
= 5.19
Rule 02
= true
=4
Rule 02
= false
=6
Rule 02
= false
=3
Rule 02
= true
=3
Rule 04
= 11.34
Rule 02
= true
=4
Rule 03
= 20.45
-0.54
6.94
2.06
3.20
-13.63
2.64
Rule 02
= false
=7
5
8
1
Rule 03
= 36.97
Rule 04
= 14.21
Rule 02
= true
=3
Rule 05
=0
Rule 03
= 14.27
Rule 02
= false
=2
Rule 05
= -3
Rule 04
= 8.83
Rule 05
=1
(a)
(b)
Figure 6.2: A parametric design formulation (a) for a fictitious parametric design space, and a grammar-based design
formulation (b), which uses a fictitious sample grammar. In (a), three design vectors are given; each has the same length,
and the th entry in each corresponds to the same design variable. In contrast, in (b), the rule derivations are of varying
length, and an entry at a particular index in one derivation does not necessarily relate directly to a corresponding entry in
another derivation.
Since the goals of interactive evolutionary exploration differ from optimization, the goals of crossover differ as
well; crossover should conceptually combine traits from two or more parents in an intelligible manner, so that
offspring reflect the selections that the designer has chosen. The method proposed here is simple and generates
offspring designs that are visibly “related” to their parents. It works as follows: first, identify all possible splice
point pairs in parent design derivations. These are points at which crossover is allowed, or in other words,
points after which the next rule can apply to the same state label in both parents. Then, randomly choose a
splice point pair, and create new designs by combining the derivation before the splice point of one parent with
the derivation after the splice point of the other parent, and vice versa. This procedure is illustrated in Figure
6.3. Figure 6.4 gives an example of applying this approach to the pedestrian bridge grammar introduced in
Chapter 4. It is important to note that because the splice points are chosen only at permissible locations, the
structural logic and analyzability of the resulting crossed over designs are maintained.
This procedure can be expanded to create spliced offspring from more than two parents, although it is not
necessarily guaranteed that multiple pairs of splice points will exist. An alternative approach to achieve
multiple design selection, which is a feature of the interactive evolutionary framework, is to randomly choose
two of the selected designs to crossover. Through multiple generations of crossover, the traits from more than
two parents can still be incorporated in this way.
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CHAPTER 6: INTEGRATION OF DESIGN SPACE STRATEGIES
Rule 01
= 2.43
Rule 01
= 6.78
Rule 01
= 2.43
Rule 01
= 6.78
Rule 02
= true
=4
Rule 02
= false
=6
Rule 02
= true
=4
Rule 02
= false
=6
Rule 02
= true
=3
Rule 04
= 11.34
Rule 02
= true
=3
Rule 03
= 36.97
Rule 03
= 20.45
Rule 02
= false
=7
crossover
Rule 04
= 14.21
Rule 03
= 20.45
Rule 05
=0
Rule 05
= -3
Rule 02
= false
=7
Rule 03
= 36.97
Rule 04
= 11.34
Rule 05
= -3
Rule 04
= 14.21
Rule 05
=0
Figure 6.3: Illustration of crossover concept for rule derivations.
Like crossover, mutation must also be implemented, and Chapter 3 does this using an elementwise approach
for parametric design vectors. Unlike crossover, mutation does not depend on a particular vector length, so a
similar elementwise approach can be used for grammar-based designs. To limit the disruptive power of
mutation, it is proposed that only the parameters in the rule derivation be mutated, and not the rule
applications themselves. This approach expands the design space exploration without generating designs that
are too far away from the user’s preferences. The mutation of rule parameters can be implemented in nearly
the exact same way as parametric design vector mutation, with the adjustment that only some of the
parameters be mutated at a time. Like the actual degree of mutation, the number of parameters that are
mutated is tied to the mutation rate set by the user.
6.2.3 Analysis engines
The interactive evolutionary framework presented in Chapter 3 requires at least one analysis engine that can
give numerical scores based on structural behavior for particular types of designs. In Chapter 3, the truss
problems used a direct stiffness method truss analysis engine to generate structural performance scores related
to required volume of material. Because the trans-typology structural grammars presented in Chapter 4 also
require an analysis method for performance evaluation, it is simple to address this requirement. The
performance evaluation method that is specific to a trans-typology grammar can be used as the analysis engine
in the interactive evolutionary framework.
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Parent 1:
CHAPTER 6: INTEGRATION OF DESIGN SPACE STRATEGIES
Parent 2:
Parent 1:
Parent 2:
Figure 6.4: Crossover between pedestrian bridge grammatical designs. For each set of parents, three offspring are
given, with contributions from each parent’s rule derivation highlighted. The resulting offspring display traits from both
parents combined in different ways.
6.2.4 Design problem setup
In addition to the interactive exploration mode, the interactive evolutionary framework also contains additional
modes that comprise an expanded user experience, described in Chapter 3. Before evolutionary exploration,
the user is able to define the design problem by drawing a structure, identifying design variables, and
determining allowable ranges. This is equivalent to defining a design space. To extend this idea to structural
grammars, the setup mode allows the user to define a grammar instead of a single parametric structure. This
can involve defining new rules with state labels and parameters, and defining a performance evaluation
method.
For easier and faster design space definition, the user can work with predefined grammars in the setup mode,
modifying the set of rules included in the grammar, the allowable ranges for the rule parameters, and the states
in which various rules can apply. The setup mode for grammar-based design space definition also allows the
user to establish overall design parameters that are not necessarily rule parameters, such as the overall height
and span of the structure, the value of the applied load, and material properties used in the analysis engine.
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6.2.5 Design refinement
The interactive evolutionary framework also includes a design refinement mode, to be used after interactive
exploration as a way to fine-tune a selected design using real-time analysis. For truss and other parametric
structures, this is achieved by allowing the user to click and drag on nodes in the design, while continuously
updating the performance score.
Expanding this design mode to include grammar-based designs involves allowing the user to adjust the rule
derivation and its parameters. Slider controls are introduced as a way to modify parameter values, with the
resulting design changes visually updated in real-time, along with the resulting performance score. A more
significant adjustment would allow the user to replace actual rules in the derivation list. This is a more difficult
functionality to implement, since changing one rule may invalidate all of the rules further down in the
derivation. The integration approach proposed here therefore allows only rule parameters to be adjusted
during design refinement.
6.3
Evolutionary framework and surrogate modeling
Because of the widespread use of surrogate modeling, it is not difficult to apply the surrogate modeling method
developed in Chapter 5 to the interactive evolutionary framework presented in Chapter 3. There are quite a few
examples of using surrogate modeling as an approximation algorithm for various evolutionary algorithms (Nair
& Keane, 1998). In concept, the surrogate model must be built in an offline approximation mode, after which
the interactive evolutionary exploration can use the surrogate model rather than full structural analysis to sort
through hundreds or thousands of designs in a generation very quickly. This allows users to explore complex
design problems without having to wait for minutes or hours between generations, and without the need to use
supercomputers to perform the analysis. As previously discussed, this is a key requirement for a computational
approach to be practical and useful for designers.
6.3.1 Automatic surrogate model building
To incorporate the offline approximation mode into the evolutionary framework, it must be introduced after the
design setup mode, and before the interactive exploration mode, as shown in Figure 6.5. The automatic modelbuilding procedure introduced in Chapter 5 is systematic and can be applied to any problem setup that the user
generates. The approximation mode makes recommendations for model-building settings, which the user may
accept or modify. Of particular importance is the slider allowing the user to decide on a tradeoff between wait
time and accuracy, which helps the user customize the surrogate model to specific needs.
The user may also elect not to use a surrogate model. This is a good choice in cases where the design problem is
relatively simple and fast to analyze, such as the 7-bar truss example used in Chapter 5. The approximation
mode can inform a user about whether a surrogate model is recommended, based on the computational time
measured to evaluate the design problem once.
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Set up
initial
structure
Build
surrogate
model
User defines starting
design, variables,
and constraints
“Offline” Mode:
Develop surrogate
model based on
randomly generated
data sets
CHAPTER 6: INTEGRATION OF DESIGN SPACE STRATEGIES
Generate
designs
“Online” Mode:
Exploration of design
space using interactive
evolutionary algorithm
Refine
design
Post-processing:
User can fine-tune
design in real-time
analysis environment
Figure 6.5: Illustration of design modes in the computational design approach presented in this dissertation. The
insertion of the surrogate model building mode allows the evolutionary exploration mode to be fast and interactive.
6.3.2 Model predictions and updates in interactive mode
Once the surrogate model is built, it can be used to quickly navigate the design space in place of
computationally expensive structural analysis. The specific focus on rank in the modeling approach proposed
in Chapter 5 is especially helpful in an interactive evolutionary context, in which top-performing designs are
presented to the user. While the score value predictions were often inaccurate in the examples shown in
Chapter 5, the surrogate models predicted relative rank fairly well. This means that the surrogate model will
generally perform reasonably well at identifying top designs.
To address the issue of value inaccuracies, it is proposed that the top designs predicted by the surrogate be
actually evaluated using the structural analysis engine. This ensures that they are presented in the correct
order to the user, and with the correct score. As an additional safety measure, it is proposed that the program
actually evaluate twice the number of designs that will be presented, pushing poor-performing outliers out of
the set that the user is shown.
The designs that are actually evaluated constitute new data points that can be used by the surrogate model to
improve its accuracy by retraining the model with an expanded data set. Furthermore, if multiple copies of the
new data points are introduced to the training set, the surrogate will adjust itself to be more accurate in the
region that the designer has expressed interest in. In this way, the surrogate improves and adapts itself as
evolutionary exploration continues.
Since the same number of designs are actually evaluated in each generation, regardless of generation size, this
approach essentially decouples wait time from the number of designs explored. This is an important outcome,
since large generation sizes are often desirable during evolutionary exploration. It should be noted that both
surrogate training and prediction take negligible time compared to actual evaluation of multiple generations for
most design problems.
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6.3.3 Use of approximation in refinement mode
While surrogate models are standardly used for optimization, they can also be used for feedback-based features
like the one used in the design refinement mode. For complex design problems, the real-time analysis is too
slow to quickly update performance values as the user makes adjustments. The approximate surrogate model
can be used instead as a way to deliver quick feedback.
The challenge here is that while the surrogate modeling strategy proposed in Chapter 5 tends to produce
models with good rank-based results in top-performing regions, automatically built surrogate models do not
perform very well at predicting performance values over the whole design space. With this caveat, the
surrogate model can nevertheless be used as a stand-in analysis approach, to be updated by full structural
analysis once the user stops adjusting the model.
6.4
Structural grammars and surrogate modeling
The most challenging of the three pairwise integrations is the use of surrogate modeling to approximate design
spaces formulated through the use of structural grammars. As discussed previously, surrogate modeling
techniques require design vectors, and are incompatible with variable-length rule derivations as design
formulations. No existing approaches for integrating these two strategies have been found in the literature.
The combination of these methodologies is nevertheless important: grammars lead to complex structures that
often have computationally expensive analysis engines. Without approximation, it is not possible to explore
broad and interesting grammar-based design spaces in a rapid, interactive way.
6.4.1 Challenge of nonparametric formulation
The data-based surrogate modeling strategies used in Chapter 5 build predictive systems that produce an
output, given a vector of inputs. In standard surrogate modeling, the natural candidate for the input vector is
the design vector. When applying surrogate modeling to grammar-generated designs, it is necessary to
generate a reasonable input vector based on the design without directly using its rule derivation. Once the rule
derivation is transformed into a constant-length design vector, it can be used to build surrogate models, and to
predict performance of new designs using the surrogate. This concept is illustrated in Figure 6.6.
How can a rule derivation be transformed into a design vector? There is no obvious answer, and it is a rather
uncommon problem, since the goal of using grammar-based design spaces is to transcend parametric
definition. If a grammar-based design could be represented perfectly by a parametric design vector, then there
is no point in using a grammatical formulation at all.
It is therefore accepted that the design vector transformed from the rule derivation will necessarily be
approximate, and not completely descriptive of the design. Since the surrogate model works with performance,
the design vector will attempt to capture information most related to structural behavior. Several strategies to
do this are described in the following sections.
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surrogate
model
surrogate
model
(a)
(b)
Figure 6.6: Illustration of (a) surrogate model building and (b) prediction through the transformation of grammarbased rule derivations into more standard design vectors.
6.4.2 Salient and emergent properties
For designs generated by a particular grammar, it is possible to identify and calculate common salient
properties that affect design performance, such as dimensions, clear spans, and other geometric information.
As long as these properties do not take long to compute, they are good candidates for entries in a transformed
design vector. Such properties are referred to as emergent when their values are not explicitly encoded in the
design’s formulation, but rather emerge from the combined effects of rule applications.
The challenge with using salient properties is that they must be explicitly defined for any grammar, an
additional requirement beyond those given in Chapter 4. However, in many cases, at least some salient
properties are simple to define, and they often have a large impact on design performance.
6.4.3 Rule counts and parameter values
In addition to salient properties, it is proposed that rule derivations be converted into parameter formulations
more directly, through rule count and parameter value entries in the transformed design vector. For example,
for each rule, an entry is created whose value is equal to the number of times the rule is applied. Additionally,
more entries are created that correspond to the average value of the parameters for each rule.
Rule counts and parameter values affect performance in less direct and obvious ways than salient properties,
but are nevertheless relevant. In most cases, some rules and parameters will be more important in design
performance than others. Therefore, it is suggested that after adding the salient, rule count, and parameter
values to the design vector, it be pruned, or reduced, to eliminate transformed parameters that make little
difference.
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6.4.4 Pruning the design vector
A technique called variable reduction is sometimes used in surrogate modeling applications to minimize the
design vector to only the variables that are the most impactful, using statistically determined variable
importance measures to determine which variables to eliminate. It is proposed that such an approach be used
here, to avoid overly long design vectors which are harder to fit.
6.5
Summary of intellectual contributions
This chapter has outlined the challenges and potential solutions to synthesizing the work presented in the
previous three chapters into unified computational approaches for conceptual design through three pairwise
combinations. A fourth approach that unifies all three strategies is also possible, and could be realized by
addressing the challenges of the each of the three pairwise approaches in unison.
These design approaches have the potential to combine the benefits of the individual strategies previously
introduced: guided creative exploration, diverse and unexpected design generation, and rapid interactivity. The
specific intellectual contributions of combining these three strategies come from pairwise integrations:

Novel crossover and mutation proposals for integrating structural grammars into the interactive
evolutionary framework.

Adaptive surrogate modeling procedure that replaces full analysis and adapts itself during interactive
evolutionary exploration.

An original solution to applying surrogate modeling to grammar-based designs by transforming the
rule derivation into a design vector.
While future work is needed to further develop and fully implement these concepts, this chapter offers the first
step of diagnosing the challenges and suggesting new directions for solutions.
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CHAPTER 7:
Discussion and Conclusions
This dissertation has argued for and presented a set of new computational strategies for integrating structural
principles into the conceptual design of architecture. Previous chapters have motivated this problem, reviewed
background literature, introduced original design space strategies, and discussed the integration of strategies
into unified design approaches. This final chapter summarizes the novel intellectual work of the dissertation,
discusses potential applications, and offers concluding remarks.
7.1
Need for new design approaches
Because of the critical and innate relationship of architectural form and structural behavior, there is a great
potential for elegant, materially efficient, and intellectually rigorous design achieved by conceiving of form and
behavior in concert. Currently, standard approaches for conceptual design in architecture and structural design
lack mechanisms to integrate structural considerations into the form-making process. Instead, engineering
knowledge is usually applied after basic geometry has been established, as a way to enable a formal idea.
This dissertation identifies a lack of integrated computational design tools as a key obstacle for addressing this
well-documented problem. Current computational tools reflect and strengthen the separation of formal design
from structural behavior, with architectural tools focusing on geometry independent of performance, and
structural tools focusing on analysis of an already established geometry. This section reviews the motivation to
move beyond existing strategies, both well-established and in development, to truly harness the power of
computation to improve conceptual structural design.
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CHAPTER 7: DISCUSSION AND CONCLUSIONS
7.1.1 Beyond guess-and-check
Because of the current state of computational tools for formal and structural design, the primary workflow for
reconciling geometrical and structural issues is iterative trial and error, or guess-and-check. This is a slow and
painstaking cycle that can usually only be completed a few times during the schedule of conceptual design in
practice. The issues of moving between different tools are compounded by the fact that the traditional
modeling and analysis of multiple candidate concepts offers little economy of scale; nearly as much time and
effort as invested in the first idea are required for each additional option under consideration.
Since the slow and cumbersome guess-and-check process limits the number of options to be compared, the
designer has a great responsibility to generate and test high-quality ideas, relying on previous experience and
intuition. A small number of practicing structural design leaders excel at this activity, but even the best
designers tend to stick with typologies and canonical approaches seen previously, limiting innovation. In other
cases, designers with less experience consider too narrow a scope, or choose a structural system not well suited
to their problem.
7.1.2 Beyond rapid feedback
A step beyond the guess-and-check approach to structural design introduces speed in the form of rapid
feedback. This approach performs structural analysis in real-time as the designer manipulates design geometry
and other variables through a graphical user interface. As discussed in previous chapters, this approach marks
an important departure from traditional methods in that it integrates geometry and structural performance in a
single environment. Because of the ease of formal manipulation and the fast reporting of structural
performance, this approach allows users to consider many more design options in a quantitative manner.
However, rapid feedback suffers from the same basic issues as guess-and-check: the user must decide which
designs to consider in the first place, usually starting with previously established design typologies. In a
realistic design problem with a large number of variables, manual exploration is still too slow and arbitrary to
ensure that the designer will stumble upon the best design options, or even considerably different design
options. The fundamental issue with guess-and-check and rapid feedback approaches is that they do not help
the user understand how to choose a new candidate design once one has been analyzed. In other words, while
they give feedback on performance, they give no guidance for improvement.
7.1.3 Beyond optimization
The well-known approach for computational design guidance is optimization, a computational algorithm that
finds the best solution to a mathematically formulated design problem, given mathematically formulated
objectives and constraints. While optimization is widely used in practice in some engineering disciplines,
notably mechanical, aerospace, and automotive design, it has yet to become commonplace in structural design
for buildings, bridges, and other civil structures. This is partly due to the difference in performance goals
between mass-produced products and one-off designs: material savings have far more impact when a part or
product is manufactured in large quantities.
However, this dissertation asserts that they key barrier preventing the use of optimization in the design of
architectural structures is the difficulty of properly formulating a highly qualitative design problem in a
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CHAPTER 7: DISCUSSION AND CONCLUSIONS
mathematical way. Aside from structural and other areas of technical performance, most ways to evaluate the
important aspects of architecture – visual impact, occupant experience, contextual suitability, rhythm and
composition – are non-numerical. Furthermore, while defining good architecture is not subjective, there is
certainly more than one valid answer, and often many very different answers, to a given architectural problem.
Finally, architectural goals and assessment tend to evolve during the design process, reflecting newly
discovered ideas and preferences not stated or known at the outset.
A sample comparison of designs explored during conceptual design using these three processes – guess-andcheck, rapid feedback, and optimization – is given in Figure 7.1. It is clear from the diagrams that none of the
approaches is completely satisfactory, and a new approach that incorporates a focus on performance, design
diversity, and interactive speed is needed.
GUESS-AND-CHECK
RAPID FEEDBACK
OPTIMIZATION
*
*
*
Figure 7.1: Design options explored using guess-and-check, rapid feedback, and optimization approaches on a sample
two-variable design problem. The global optimum is indicated with a * symbol. In the first two approaches, only a small
and arbitrary region of the design space is explored. In the third approach, the exact optimum solution is found, but it is
the only solution offered.
7.2
Specific contributions
The new approach presented in this dissertation overcomes these issues with strategies that operate on the
design space, a formal construct that links a mapping of design possibilities with quantitative structural
performance. These strategies are summarized as follows:

Chapter 3 introduced an interactive evolutionary framework that allows for exploration of the design
space, guided by a combination of quantitative performance goals and the designer’s creative
preferences.

In Chapter 4, an approach for formulating broader, more diverse design spaces through trans-typology
structural grammars was presented.

Chapter 5 introduced a methodology for design space approximation using a data-based surrogate
modeling strategy modified to fit the goals of conceptual structural design.
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
CHAPTER 7: DISCUSSION AND CONCLUSIONS
Finally, Chapter 6 addressed the challenges of integrating these design space methodologies into
combined design approaches.
This section reviews these contributions and highlights their impact.
7.2.1 Interactive evolutionary framework
The interactive evolutionary framework presented in Chapter 3 is a design space navigation strategy that
combines the biological analogy of natural selection with designer freedom and creativity. This approach
allows the user to work collaboratively with the computer program to identify top performing designs in a
population and generate new offspring designs that improve upon their parents. The framework introduced in
this dissertation is novel because of its generalized nature, its enhanced interactivity, its mechanisms for
diversity and design quality, and its expanded user experience.
This contribution is important because it improves a strong alternative to the existing design space navigation
strategies previously reviewed. While interactive evolutionary algorithms have previously been shown to be
promising in the field of structural design, the work presented in this dissertation encapsulates the algorithm in
a flexible and user-friendly framework that balances powerful user control with ease of access.
7.2.2 Trans-typology structural grammars
Chapter 4 introduces a methodology to define grammatical design spaces that contain a diverse range of
structural designs of varying typology. This approach is presented as an alternative to conventional parametric
design spaces, which comparatively offer less variety and more predictability due to the limitations of
parametric variation. The grammar-based approach presented in this dissertation builds upon existing work in
the small field of engineering shape grammars, prescribing the necessary features to develop a grammar that
incorporates structural behavior and encompasses designs across typologies.
The impact of this work is the potential for broader, more creative, and more unexpected structural design
exploration than possible with existing approaches. The emphasis on trans-typological design spaces increases
the relevance of computation in conceptual design, during which designers must enumerate and compare as
wide a range of design options as possible.
7.2.3 Performance-focused surrogate modeling
The surrogate modeling approach introduced in Chapter 5 is a strategy to approximate design spaces to
increase computational speed in evaluating design performance. This strategy is critical for making
computational explorations of the design space practical for realistically sized problems, which can otherwise
take a prohibitive amount of time or computational resources to evaluate in large numbers. The approach
presented in this dissertation modifies existing surrogate modeling techniques to improve approximation
performance for conceptual design, and introduces a method to automatically build models and report
graphical and numerical results.
These methodological developments are significant because they expand the accessibility of surrogate modeling
to designers and practitioners who are not experts in statistics or optimization. Approximation through
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CHAPTER 7: DISCUSSION AND CONCLUSIONS
surrogate modeling is important for moving beyond conventional design tools, which often require long
runtimes that prohibit interactive exploration. Because approximation allows designers to consider many more
options due to increased computational speed, designers are able to discover and develop better, higher
performing designs.
7.2.4 Integrated approaches
The three design space strategies discussed above can be used alone, in pairwise combinations, or integrated
into a unified design approach, as suggested in Chapter 6, which addresses the issues of combining them. This
approach incorporates broad, interesting design spaces with an effective means to explore and navigate them at
a fast, interactive pace. The three approaches complement each other and achieve more than the sum of their
parts in their synthesis, which this dissertation is the first to propose.
The integrated design approach represents a new way forward for making use of computation in creative design
pursuits. It moves beyond the pitfalls of existing approaches, allowing for broader design spaces, better
navigation, and faster exploration of more points, as shown in Figure 7.2. Finally, this approach is also
generalized and extensible so that it can be applied to a wide range of design problems.
*
Figure 7.2: An example of design options explored using the approach presented in this dissertation, in contrast with
those shown in Figure 7.1. It is noteworthy that multiple high-performing regions are investigated over a broad space in a
manner rapid enough to allow for many designs to be considered.
7.3
Applications of proposed strategies
The new strategies presented in this dissertation were developed in response to a need for computational tools
that allow for integrated structural design. The strategies can be used by architects, structural engineers, and
designers who straddle both disciplines, either in collaboration or independently. In addition to applicability in
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CHAPTER 7: DISCUSSION AND CONCLUSIONS
practice, these design approaches can also be helpful in academia: in the classroom, they can build intuition
and foster creativity, and in research, they can be used as a lens for historical analysis. This section discusses
each of these applications in more detail.
7.3.1 In practice
Tools that implement the design space strategies introduced in this dissertation could significantly improve
conceptual design exercises in practice, as a way to generate and compare a wide range of design ideas quickly
and easily. An architect with basic structural knowledge could use such a tool alone or as a supplement to
working with a creative structural engineer early in the design process. A structural designer could also use the
tool to develop innovative structural concepts to discuss with the architect for further development. In a more
integrated approach, a team of architects and engineers could use the tool together during conceptual design,
collaboratively developing design alternatives that perform well structurally and achieve architectural design
goals. Finally, the tool could be useful in facilitating discussions between designers and clients, helping clients
understand tradeoffs between options and cost implications of design ideas at the earliest stages.
7.3.2 In the classroom
Possible applications in the classroom mirror those in practice: architecture students could use tools
implementing these strategies for exploring early design options for studio projects, and engineering students
could use such a tool for engineering design projects. However, design tools based on this research also have
additional didactic potential for developing intuition for structural behavior in architecture and engineering
students, a very important and increasingly neglected aspect of education in both disciplines. For engineering
students, such tools could also offer a way to encourage design creativity, another significant but overlooked
area. Furthermore, a tool used by students from both disciplines together would foster collaboration and
improve students’ cross-disciplinary communication skills, which are much needed in practice.
7.3.3 Historical analysis
In addition to discovering design possibilities for new projects, these strategies could also be useful in studying
existing work within the context of a formal design space. Most architectural history research does not include
detailed analyses on structural performance, which can be of value in evaluating success and identifying lessons
to move forward with. The design space strategies presented here allow researchers to consider a historical
work as a point in a space of alternatives of varying structural performance and formal attributes, potentially
gaining insights on design decisions and process. For example, Robert Maillart’s concrete shed roof in Chiasso,
Switzerland, designed in 1924, is shown in Figure 7.3, along with related design alternatives explored using the
approach presented in this dissertation. It is evident that there is a family of solutions of varying performance,
some of which share more in common with Maillart’s design, which achieves a constant force in the gable
elements, and some less. Such a study could provide a new context through which designs could be analyzed,
understood, and revisited in the future.
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CHAPTER 7: DISCUSSION AND CONCLUSIONS
Figure 7.3: Robert Maillart’s 1924 design for a shed roof in Chiasso, compared with designs discovered using the novel
computational approach presented in this dissertation. It is notable that Maillart’s design requires no diagonal elements,
like the design found on the far right, due to the constant force in its gable elements. Other design alternatives suggest
additional possible solutions.
7.4
Directions for future work
To fully enable the applications envisioned in the previous section, especially widespread use in practice, future
work beyond this research is needed in several key areas. This section organizes these needs into three
categories: practical, technical, theoretical, and cultural.
7.4.1 Practical needs
First, expansion of the types of structural models and analysis methods used is needed to broaden possible
design problem applications beyond what has been illustrated in this work. Because the strategies were
developed with the goal of flexibility and generality, it is a practical, rather than technical, challenge to extend
them beyond the examples shown in this dissertation. For example, three-dimensionality, planar and surface
forms, and more complex curvature are all possible and promising directions. Analysis engines can move
beyond the simple truss analysis and graphic statics method presented here to include more robust finite
element analyses, nonlinear considerations, and dynamic performance. In parametric design spaces, variables
can be expanded to include material properties, connection types, and parameters in more complex parametric
relationships. More complicated grammatical design spaces can also easily be included.
It is also important to continue implementation and software development of these strategies. The existing
work in this dissertation presents important first steps: a flexible, object-oriented back end and a web-based,
user-friendly front end. These can be improved and expanded upon in several ways. First, it is important to
continue development of the web-based interface, currently implemented using Microsoft Silverlight. Since
this technology will eventually be discontinued, a new version that makes use of new HTML5 canvas
technology, which is even more browser-integrated and platform-independent, is envisioned. This future
version could be used on smartphones and mobile devices, in addition to traditional computers, for on-the-go
design sessions and classroom learning. Because the back and front ends of the current implementation were
designed to be independent, an upgraded front end would could still reuse most of the existing code developed
for this dissertation.
It would also be valuable to implement these strategies within design environments commonly used in practice,
like Rhino 3D and Autodesk Revit. While the web-based interface was a better choice for this initial research,
connecting the back end to commercial software would allow for more complicated structural systems and
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CHAPTER 7: DISCUSSION AND CONCLUSIONS
geometry. This would also allow these strategies to integrate into the daily workflows of design practitioners
more seamlessly. Again, because the code developed for this research is well separated, the existing back end
could be integrated with commercial software relatively easily.
7.4.2 Technical needs
There are also important technical areas for future development in the realm of computation and software
architecture. As a means for expanding the structural model types, analysis methods, and grammars available
for these strategies, it would be helpful to expose the software platform to the structural design communities in
research and practice. This would enable other developers to use the interactive evolutionary framework,
trans-typology structural grammar method, and surrogate modeling strategy and contribute new modules that
could be used by all. The existing software architecture, which relies heavily on extensible interfaces, makes
this step feasible.
Another important step would be to offload the significant computation of the new strategies, which currently
runs on the user’s computer, to a remote server through cloud computing. Because most of the existing
computation is highly parallelizable, such as the evaluation of a generation of designs in evolutionary design
space navigation, it is well suited to this new approach, which would further improve response times.
7.4.3 Theoretical needs
A key theoretical expansion of the work presented here is to consider more than one quantitative performance
metric, for multi-objective design space exploration. Additional measures of performance could be structural,
such as serviceability or dynamic behavior, related to construction, such as numbers or types of connections, or
linked more broadly to building technology, in areas such as energy consumption, thermal performance, or
daylighting quality. This would more fully reflect the challenges of many design projects, which should
consider architectural quality as well as a variety of quantitative, technical goals in conceptual design. This
expansion would introduce a second multidimensional space to the computational design approach: the
objective space, which has as many dimensions as quantitative goals. Exploring the objective space in addition
to the design space would present new challenges and opportunities for visualization and organization of the
conceptual design process.
7.4.4 Cultural needs
Finally, there are existing cultural barriers that may need to be dismantled for widespread use of this research
in practice. The current divide between the disciplines of architecture and structural engineering presents a
strong obstacle to integrated design. This research attempts to bridge this gap through computation,
developing strategies that overcome the limitations of currently used tools and techniques. Because this work
considers both architectural geometry and structural performance, it aims to encourage architects and
engineers to overlap in their roles. However, it also requires a willingness on behalf of architects to engage in
the technical, and on engineers to engage in the qualitative (and even subjective). These requirements may
violate the comfort zones established by traditional education in both disciplines. Cultural and pedagogical
evolution is needed to broaden architectural and engineering perspectives, and while computational solutions
can help with this, they are not sufficient in themselves.
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7.5
CHAPTER 7: DISCUSSION AND CONCLUSIONS
Concluding remarks
Historical and current masterpieces in structural design show that there are an infinite number of innovative,
exciting, and unexpected design possibilities for addressing structural requirements in architecture. Designers
are limited only by their means to explore and discover them. As the need to use resources responsibly grows,
it becomes increasingly important for designers to direct their exploration toward high-performing solutions.
At the same time, computational power is unprecedented and rapidly increasing, and has already shown its use
in powerful geometric modeling and sophisticated structural analysis for use in later design process stages.
This dissertation shows that there are new ways to take advantage of computation for exceptional creativity and
freedom in conceptual design exploration, with the possibility of discovering new ideas for moving forward.
153
PART IV:
Appendices
“It is admittedly fairly widespread opinion that the dimensions should be unequivocally and finally
determined by calculation. However, in view of the impossibility of taking into account all possible
contingencies, any calculation can be nothing but a guidance to the designer.”
— Robert Maillart via Hans Straub in A History of Civil Engineering, 1952
“In practice, however, the ideal forms defined by abstract principles can only be realized with a high degree of
fidelity if the task is sufficiently narrow in scope. As soon as some complication is introduced, the ideal form
is 'disrupted.' Dealing with such disruptions – an integral part of the engineer's role – means adjusting the
project in line with more subjective assumptions."
— Jürg Conzett in Structure as Space, 2006
APPENDIX A:
Structural Analysis Code Validation
This appendix compares results from the custom structural analysis code written for and used in this
dissertation with results from SAP2000, a commercially available structural analysis software package
(Computers and Structures, 2012). Results are compared for three planar truss problems, and are given in the
form of axial member forces. Each problem uses a single material, A36 steel, which has a modulus of elasticity
of 29,000 kips/in2. The geometry for each problem is shown graphically in a diagram, and is also described
numerically in node and member tables.
Units are given in the table column headings, and in general follow the imperial system: inches for length and
kips for force. Nodal loads use the following sign conventions: positive for rightward horizontal loads, and
positive for upward vertical loads. The convention for axial forces used in these tables is a positive sign for
tension and a negative sign for compression.
The results for each problem consistently show that the custom analysis code and the commercial software
package are in complete agreement on member force magnitudes and direction.
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A.1
APPENDIX A: STRUCTURAL ANALYSIS CODE VALIDATION
Seven-bar truss
TABLE OF NODES
Name
X-Coordinate
[inches]
Y-Coordinate
[inches]
Horizontal
Load [kips]
Vertical Load Horizontal
[kips]
Fixity
Vertical
Fixity
n1
0
0
0
0
reaction
reaction
n2
30
-30
0
0
free
free
n3
90
-30
0
0
free
free
n4
60
0
0
-10
free
free
n5
120
0
0
0
free
reaction
TABLE OF MEMBERS
Name
Start Node
End Node
Cross-Sectional
Area [in2]
Custom Code
Axial Force
[kips]
m1
n1
n4
1.00
-5.000
-5.000
0.00%
m2
n1
n2
1.00
7.071
7.071
0.00%
m3
n2
n3
1.00
10.000
10.000
0.00%
m4
n3
n5
1.00
7.071
7.071
0.00%
m5
n4
n5
1.00
-5.000
-5.000
0.00%
m6
n2
n4
1.00
-7.071
-7.071
0.00%
m7
n4
n3
1.00
-7.071
-7.071
0.00%
158
SAP2000
Axial Force
[kips]
Difference
C. T. MUELLER | PH.D. DISSERTATION, 2014
A.2
APPENDIX A: STRUCTURAL ANALYSIS CODE VALIDATION
Cantilevered truss roof
TABLE OF NODES
Name
X-Coordinate
[inches]
Y-Coordinate
[inches]
Horizontal
Load [kips]
n1
0.0
240.0
0
-10
free
free
n2
60.0
216.0
0
0
free
free
n3
120.0
259.2
0
-10
free
free
n4
180.0
235.2
0
0
free
free
n5
240.0
278.4
0
-10
free
free
n6
300.0
254.4
0
0
free
free
n7
360.0
297.6
0
-10
free
free
n8
420.0
273.6
0
0
free
free
n9
480.0
316.8
0
-10
free
free
n10
540.0
292.8
0
0
free
free
n11
600.0
336.0
0
-10
free
free
n12
660.0
312.0
0
0
free
free
n13
720.0
355.2
0
-10
free
free
n14
780.0
331.2
0
0
free
free
n15
840.0
374.4
0
-10
free
free
n16
900.0
350.4
0
0
free
free
159
Vertical Load Horizontal
[kips]
Fixity
Vertical
Fixity
C. T. MUELLER | PH.D. DISSERTATION, 2014
APPENDIX A: STRUCTURAL ANALYSIS CODE VALIDATION
Name
X-Coordinate
[inches]
Y-Coordinate
[inches]
Horizontal
Load [kips]
Vertical Load Horizontal
[kips]
Fixity
Vertical
Fixity
n17
960.0
393.6
0
-10
free
free
n18
1020.0
369.6
0
0
free
free
n19
1080.0
412.8
0
-10
free
free
n20
156.0
0.0
0
0
reaction
reaction
n21
192.0
0.0
0
0
reaction
reaction
n22
756.0
0.0
0
0
reaction
reaction
n23
792.0
0.0
0
0
reaction
reaction
SAP2000
Axial Force
[kips]
Difference
TABLE OF MEMBERS
Name
Start Node
End Node
Cross-Sectional
Area [in2]
Custom Code
Axial Force
[kips]
m1
n1
n2
1.00
-19.233
-19.233
0.00%
m2
n1
n3
1.00
18.084
18.084
0.00%
m3
n2
n3
1.00
22.004
22.004
0.00%
m4
n2
n4
1.00
-36.169
-36.169
0.00%
m5
n3
n4
1.00
-38.465
-38.465
0.00%
m6
n3
n5
1.00
72.337
72.337
0.00%
m7
n4
n5
1.00
-44.008
-44.008
0.00%
m8
n4
n6
1.00
-35.355
-35.355
0.00%
m9
n5
n6
1.00
19.233
19.233
0.00%
m10
n5
n7
1.00
18.084
18.084
0.00%
m11
n6
n7
1.00
-22.004
-22.004
0.00%
m12
n6
n8
1.00
0.814
0.814
0.00%
m13
n7
n8
1.00
0.000
0.000
0.00%
m14
n7
n9
1.00
0.000
0.000
0.00%
m15
n8
n9
1.00
0.000
0.000
0.00%
m16
n8
n10
1.00
0.814
0.814
0.00%
m17
n9
n10
1.00
-19.233
-19.233
0.00%
m18
n9
n11
1.00
18.084
18.084
0.00%
160
C. T. MUELLER | PH.D. DISSERTATION, 2014
APPENDIX A: STRUCTURAL ANALYSIS CODE VALIDATION
Custom Code
Axial Force
[kips]
SAP2000
Axial Force
[kips]
Difference
Name
Start Node
End Node
Cross-Sectional
Area [in2]
m19
n10
n11
1.00
22.004
22.004
0.00%
m20
n10
n12
1.00
-35.355
-35.355
0.00%
m21
n11
n12
1.00
-38.465
-38.465
0.00%
m22
n11
n13
1.00
72.337
72.337
0.00%
m23
n12
n13
1.00
44.008
44.008
0.00%
m24
n12
n14
1.00
-107.692
-107.692
0.00%
m25
n13
n14
1.00
-57.698
-57.698
0.00%
m26
n13
n15
1.00
162.758
162.758
0.00%
m27
n14
n15
1.00
-66.013
-66.013
0.00%
m28
n14
n16
1.00
-108.506
-108.506
0.00%
m29
n15
n16
1.00
38.465
38.465
0.00%
m30
n15
n17
1.00
72.337
72.337
0.00%
m31
n16
n17
1.00
-44.008
-44.008
0.00%
m32
n16
n18
1.00
-36.169
-36.169
0.00%
m33
n17
n18
1.00
19.233
19.233
0.00%
m34
n17
n19
1.00
18.084
18.084
0.00%
m35
n18
n19
1.00
-22.004
-22.004
0.00%
m36
n4
n20
1.00
-8.081
-8.081
0.00%
m37
n4
n21
1.00
-31.874
-31.874
0.00%
m38
n14
n22
1.00
-27.509
-27.509
0.00%
m39
n14
n23
1.00
-32.713
-32.713
0.00%
161
C. T. MUELLER | PH.D. DISSERTATION, 2014
A.3
APPENDIX A: STRUCTURAL ANALYSIS CODE VALIDATION
Trussed rigid frame
TABLE OF NODES
Name
X-Coordinate
[inches]
Y-Coordinate
[inches]
Horizontal
Load [kips]
Vertical Load Horizontal
[kips]
Fixity
Vertical
Fixity
n1
-144
0
0
0
reaction
reaction
n2
-120
24
0
0
free
free
n3
-144
48
0
0
free
free
n4
-120
72
0
0
free
free
n5
-144
96
0
0
free
free
n6
-120
120
0
0
free
free
n7
-144
144
0
0
free
free
n8
-120
168
0
0
free
free
n9
-144
192
0
0
free
free
n10
-120
216
0
0
free
free
n11
-144
240
0
0
free
free
n12
-120
264
0
0
free
free
n13
-144
288
20
-4
free
free
162
C. T. MUELLER | PH.D. DISSERTATION, 2014
APPENDIX A: STRUCTURAL ANALYSIS CODE VALIDATION
Name
X-Coordinate
[inches]
Y-Coordinate
[inches]
Horizontal
Load [kips]
Vertical Load Horizontal
[kips]
Fixity
Vertical
Fixity
n14
-120
0
0
0
reaction
reaction
n15
144
0
0
0
reaction
reaction
n16
120
24
0
0
free
free
n17
144
48
0
0
free
free
n18
120
72
0
0
free
free
n19
144
96
0
0
free
free
n20
120
120
0
0
free
free
n21
144
144
0
0
free
free
n22
120
168
0
0
free
free
n23
144
192
0
0
free
free
n24
120
216
0
0
free
free
n25
144
240
0
0
free
free
n26
120
264
0
0
free
free
n27
144
288
0
-4
free
free
n28
120
0
0
0
free
free
n29
-96
288
0
-4
free
free
n30
-72
264
0
0
free
free
n31
-48
288
0
-4
free
free
n32
-24
264
0
0
free
free
n33
0
288
0
-4
free
free
n34
24
264
0
0
free
free
n35
48
288
0
-4
free
free
n36
72
264
0
0
free
free
n37
96
288
0
-4
free
free
n24
120
216
0
0
free
free
n25
144
240
0
0
free
free
n26
120
264
0
0
free
free
n27
144
288
0
0
free
free
n28
120
0
0
0
reaction
reaction
n29
-96
288
0
0
free
free
163
C. T. MUELLER | PH.D. DISSERTATION, 2014
APPENDIX A: STRUCTURAL ANALYSIS CODE VALIDATION
Name
X-Coordinate
[inches]
Y-Coordinate
[inches]
Horizontal
Load [kips]
Vertical Load Horizontal
[kips]
Fixity
Vertical
Fixity
n30
-72
264
0
0
free
free
n31
-48
288
0
0
free
free
n32
-24
264
0
0
free
free
n33
0
288
0
0
free
free
n34
24
264
0
0
free
free
n35
48
288
0
0
free
free
n36
72
264
20
-4
free
free
n37
96
288
0
0
free
free
SAP2000
Axial Force
[kips]
Difference
TABLE OF MEMBERS
Name
Start Node
End Node
Cross-Sectional
Area [in2]
Custom Code
Axial Force
[kips]
m1
n1
n2
1.00
11.652
11.652
0.00%
m2
n15
n16
1.00
-16.632
-16.632
0.00%
m3
n2
n3
1.00
-11.652
-11.652
0.00%
m4
n16
n17
1.00
16.632
16.632
0.00%
m5
n3
n4
1.00
11.652
11.652
0.00%
m6
n17
n18
1.00
-16.632
-16.632
0.00%
m7
n4
n5
1.00
-11.652
-11.652
0.00%
m8
n18
n19
1.00
16.632
16.632
0.00%
m9
n5
n6
1.00
11.652
11.652
0.00%
m10
n19
n20
1.00
-16.632
-16.632
0.00%
m11
n6
n7
1.00
-11.652
-11.652
0.00%
m12
n20
n21
1.00
16.632
16.632
0.00%
m13
n7
n8
1.00
11.652
11.652
0.00%
m14
n21
n22
1.00
-16.632
-16.632
0.00%
m15
n8
n9
1.00
-11.652
-11.652
0.00%
m16
n22
n23
1.00
16.632
16.632
0.00%
m17
n9
n10
1.00
11.652
11.652
0.00%
164
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APPENDIX A: STRUCTURAL ANALYSIS CODE VALIDATION
Custom Code
Axial Force
[kips]
SAP2000
Axial Force
[kips]
Difference
Name
Start Node
End Node
Cross-Sectional
Area [in2]
m18
n23
n24
1.00
-16.632
-16.632
0.00%
m19
n10
n11
1.00
-11.652
-11.652
0.00%
m20
n24
n25
1.00
16.632
16.632
0.00%
m21
n11
n12
1.00
11.652
11.652
0.00%
m22
n25
n26
1.00
-16.632
-16.632
0.00%
m23
n12
n13
1.00
35.550
35.550
0.00%
m24
n26
n27
1.00
-64.140
-64.140
0.00%
m25
n1
n3
1.00
53.254
53.254
0.00%
m26
n15
n17
1.00
-76.255
-76.255
0.00%
m27
n2
n4
1.00
-49.966
-49.966
0.00%
m28
n16
n18
1.00
41.445
41.445
0.00%
m29
n3
n5
1.00
36.775
36.775
0.00%
m30
n17
n19
1.00
-52.733
-52.733
0.00%
m31
n4
n6
1.00
-33.487
-33.487
0.00%
m32
n18
n20
1.00
17.923
17.923
0.00%
m33
n5
n7
1.00
20.297
20.297
0.00%
m34
n19
n21
1.00
-29.212
-29.212
0.00%
m35
n6
n8
1.00
-17.009
-17.009
0.00%
m36
n20
n22
1.00
-5.598
-5.598
0.00%
m37
n7
n9
1.00
3.819
3.819
0.00%
m38
n21
n23
1.00
-5.690
-5.690
0.00%
m39
n8
n10
1.00
-0.531
-0.531
0.00%
m40
n22
n24
1.00
-29.120
-29.120
0.00%
m41
n9
n11
1.00
-12.659
-12.659
0.00%
m42
n23
n25
1.00
17.832
17.832
0.00%
m43
n10
n12
1.00
15.948
15.948
0.00%
m44
n24
n26
1.00
-52.642
-52.642
0.00%
m45
n11
n13
1.00
-29.138
-29.138
0.00%
m46
n25
n27
1.00
41.354
41.354
0.00%
m47
n1
n14
1.00
0.000
0.000
0.00%
165
C. T. MUELLER | PH.D. DISSERTATION, 2014
APPENDIX A: STRUCTURAL ANALYSIS CODE VALIDATION
Custom Code
Axial Force
[kips]
SAP2000
Axial Force
[kips]
Difference
Name
Start Node
End Node
Cross-Sectional
Area [in2]
m48
n2
n14
1.00
-66.444
-66.444
0.00%
m49
n15
n28
1.00
0.000
0.000
0.00%
m50
n16
n28
1.00
64.967
64.967
0.00%
m51
n13
n29
1.00
-45.138
-45.138
0.00%
m52
n12
n30
1.00
34.328
34.328
0.00%
m53
n26
n36
1.00
-38.065
-38.065
0.00%
m54
n27
n37
1.00
45.354
45.354
0.00%
m55
n29
n30
1.00
-4.312
-4.312
0.00%
m56
n30
n31
1.00
4.312
4.312
0.00%
m57
n31
n32
1.00
-9.969
-9.969
0.00%
m58
n32
n33
1.00
9.969
9.969
0.00%
m59
n33
n34
1.00
-15.626
-15.626
0.00%
m60
n34
n35
1.00
15.626
15.626
0.00%
m61
n35
n36
1.00
-21.283
-21.283
0.00%
m62
n36
n37
1.00
21.283
21.283
0.00%
m63
n29
n31
1.00
-43.039
-43.039
0.00%
m64
n30
n32
1.00
28.229
28.229
0.00%
m65
n31
n33
1.00
-32.941
-32.941
0.00%
m66
n32
n34
1.00
14.131
14.131
0.00%
m67
n33
n35
1.00
-14.843
-14.843
0.00%
m68
n34
n36
1.00
-7.967
-7.967
0.00%
m69
n35
n37
1.00
11.255
11.255
0.00%
m70
n12
n29
1.00
-1.345
-1.345
0.00%
m71
n26
n37
1.00
-26.940
-26.940
0.00%
166
APPENDIX B:
Pedestrian Bridge Grammar Details
This appendix includes the rules for the pedestrian bridge trans-typology structural grammar introduced and
summarized in Chapter 4, and also gives sample computations for 25 designs generated using the grammar.
B.1
Grammar Rules
There are 21 rules in the pedestrian bridge grammar: 17 numbered rules that modify the design geometry, and 4
lettered rules that only modify the state label. Each rule is shown below, including a verbal description and a
sample graphical depiction. Where applicable, rule parameters are also noted.
167
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APPENDIX B: PEDESTRIAN BRIDGE GRAMMAR DETAILS
168
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APPENDIX B: PEDESTRIAN BRIDGE GRAMMAR DETAILS
169
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APPENDIX B: PEDESTRIAN BRIDGE GRAMMAR DETAILS
170
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APPENDIX B: PEDESTRIAN BRIDGE GRAMMAR DETAILS
171
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APPENDIX B: PEDESTRIAN BRIDGE GRAMMAR DETAILS
172
C. T. MUELLER | PH.D. DISSERTATION, 2014
B.2
APPENDIX B: PEDESTRIAN BRIDGE GRAMMAR DETAILS
Sample Computations
Figure 4.24 in Chapter 4 shows 50 randomly generated designs using the pedestrian bridge grammar given in
B.1. The full computations for the first 25 designs are given below.
Design 1
Design 2
Design 3
173
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APPENDIX B: PEDESTRIAN BRIDGE GRAMMAR DETAILS
Design 4
Design 5
174
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APPENDIX B: PEDESTRIAN BRIDGE GRAMMAR DETAILS
Design 6
Design 7
Design 8
175
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APPENDIX B: PEDESTRIAN BRIDGE GRAMMAR DETAILS
Design 9
Design 10
Design 11
176
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APPENDIX B: PEDESTRIAN BRIDGE GRAMMAR DETAILS
Design 12
Design 13
Design 14
177
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APPENDIX B: PEDESTRIAN BRIDGE GRAMMAR DETAILS
Design 15
Design 16
Design 17
178
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APPENDIX B: PEDESTRIAN BRIDGE GRAMMAR DETAILS
Design 18
Design 19
Design 20
179
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APPENDIX B: PEDESTRIAN BRIDGE GRAMMAR DETAILS
Design 21
Design 22
180
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APPENDIX B: PEDESTRIAN BRIDGE GRAMMAR DETAILS
Design 23
Design 24
Design 25
181
C. T. MUELLER | PH.D. DISSERTATION, 2014
APPENDIX B: PEDESTRIAN BRIDGE GRAMMAR DETAILS
182
APPENDIX C:
Automatic Surrogate Modeling Results
This appendix includes additional results for the case studies presented in Chapter 5. A range of surrogate
models are automatically developed for four different design problems, introduced in Figure 5.1, using both the
strategy proposed in Chapter 5 as well as a more standard approach. The results of these studies are
summarized in Section 5.5. A more detailed report of results is presented here, with a detailed sample of output
given in the first section, and graphical results for all of the case studies provided subsequently.
C.1
Sample of full testing results
This section gives an example of the testing data set randomly generated to evaluate the accuracy of the
surrogate model built using the strategy described in Chapter 5. The following table reviews 101 design
variations for problem (a) of Figure 5.1, sorted by the actual score computed by structural analysis. The
predicted score and predicted rank are also given; these values are estimated by a surrogate model built using
the approach proposed in Chapter 5. Finally, the elements of the three-dimensional design vector, which
defines the design, are also given in each case.
183
C. T. MUELLER | PH.D. DISSERTATION, 2014
Computed
Score
Computed
Rank
APPENDIX C: AUTOMATIC SURROGATE MODELING RESULTS
Predicted
Score
Predicted
Rank
̂
̂
Horizontal
Position of n2
[in]
Vertical
Position of n2
[in]
Vertical
Position of n4
[in]
0.73
0.87
0
1
2.14
0.95
63
0
46.8
43.6
7.0
-26.8
-39.8
20.3
0.89
2
1.05
5
32.2
-25
13.5
0.90
3
1.25
28
45.1
-14.3
27.7
0.90
4
2.17
65
25.9
3.0
-33.6
0.90
5
0.96
1
27.7
-24.8
20.8
0.90
6
1.04
4
37.3
-22.5
31.6
0.91
7
0.96
2
47.6
-20.3
35.7
0.91
8
1.25
30
39.1
-35
22.9
0.92
9
1.15
15
41.7
-43.6
3.1
0.94
10
1.12
10
45.7
-33.7
30.2
0.96
11
1.44
45
44.5
-2.9
38.1
0.96
12
1.40
44
44.4
-15.8
14.5
0.97
13
0.96
3
31.3
-41.6
20.3
0.98
14
1.47
48
18.5
-36.3
2.3
0.98
15
1.33
38
18.5
-11.1
33.2
0.98
16
1.46
46
45.9
-41.7
-3.8
0.99
17
1.13
12
39.4
-55.3
-3.6
0.99
18
1.69
57
41.2
-18.5
9.7
0.99
19
3.89
88
15.4
7.1
-28.9
1.00
20
1.49
51
29.3
-5.3
28.6
1.02
21
1.35
40
23.8
-3.2
33.3
1.02
22
1.64
54
12.3
-15.7
32.7
1.03
23
4.31
93
19.8
0.1
-34.7
1.03
24
2.76
78
35.4
-7.2
21.1
1.04
25
1.29
36
42.9
-61.8
-7.6
1.04
26
1.11
8
36.6
-57.6
9.4
1.05
27
1.35
42
42.3
-63.2
-10.7
1.07
28
1.12
11
43.0
-64.1
-0.2
1.07
29
1.17
19
29.2
-64.4
-17.0
1.07
30
1.24
27
18.4
-59.2
-12.3
1.07
31
1.15
16
37.2
-60.8
-18.1
1.08
32
1.29
35
36.5
-66.5
-5.6
1.08
33
2.42
74
24.1
-28.0
-1.1
1.09
34
1.25
32
28.3
-67.4
-13.3
1.10
35
1.93
58
26.3
-36.3
-8.7
1.10
36
1.11
9
49.9
-64.7
5.5
184
C. T. MUELLER | PH.D. DISSERTATION, 2014
Computed
Score
Computed
Rank
APPENDIX C: AUTOMATIC SURROGATE MODELING RESULTS
Predicted
Score
Predicted
Rank
̂
̂
Horizontal
Position of n2
[in]
Vertical
Position of n2
[in]
Vertical
Position of n4
[in]
1.11
37
1.40
43
11.0
-35.2
-4.3
1.12
38
1.35
41
21.0
-64.0
-1.0
1.12
39
1.66
56
37.9
-2.5
23.7
1.12
40
1.21
24
26.6
-62.8
-24.6
1.13
41
1.25
31
10.5
-58.0
-14.1
1.13
42
1.10
7
43.3
-47.0
35.8
1.13
43
2.31
68
22.0
-56.9
-22.5
1.14
44
1.52
52
36.4
-53.5
-20.5
1.14
45
2.04
60
21.7
-50.8
-19.4
1.15
46
1.33
39
37.9
-64.2
-24.2
1.16
47
1.13
13
11.5
-53.7
10.7
1.16
48
4.13
91
23.5
-1.8
-37.2
1.17
49
1.55
53
17.7
-39.1
-11.7
1.17
50
1.48
49
14.6
-25.6
0.1
1.17
51
1.21
22
30.1
-59.8
-26.1
1.18
52
1.1
6
15.8
-61.3
7.4
1.18
53
4.83
95
22.2
-1.7
-36.4
1.19
54
1.21
25
37.1
-52.6
33.2
1.19
55
1.49
50
48.0
-43.1
-14.5
1.20
56
2.33
69
20.7
-54.8
-24.4
1.20
57
1.17
20
14.8
-39.6
36.3
1.21
58
1.21
23
10.6
-46.3
25.4
1.21
59
1.19
21
25.2
-60.7
18.8
1.23
60
1.14
14
17.3
-68.8
3.5
1.24
61
1.17
18
49.3
-54.7
37.5
1.24
62
2.38
72
39.3
-17.3
2.0
1.26
63
1.46
47
17.0
-21.2
0.8
1.27
64
1.16
17
17.1
-45.6
36.6
1.27
65
1.24
26
32.8
-52.8
37.8
1.28
66
2.14
64
40.3
7.6
39.1
1.28
67
1.64
55
45.1
-14.0
5.9
1.29
68
1.26
33
12.0
-53.9
24.9
1.34
69
1.25
29
41.6
-63.5
33.0
1.34
70
2.1
61
25.1
-37.2
-17
1.35
71
1.32
37
36.5
-68.8
25.6
1.36
72
2.52
76
22.4
-23.9
-5.3
185
C. T. MUELLER | PH.D. DISSERTATION, 2014
Computed
Score
Computed
Rank
APPENDIX C: AUTOMATIC SURROGATE MODELING RESULTS
Predicted
Score
Predicted
Rank
̂
̂
Horizontal
Position of n2
[in]
Vertical
Position of n2
[in]
Vertical
Position of n4
[in]
1.38
73
2.13
62
47.0
-4.4
-28.2
1.38
74
2.58
77
39.3
-40.6
-21.3
1.42
75
3.18
84
38.7
-5.5
-31.1
1.43
76
3.56
85
46.2
-34.9
-16.6
1.51
77
3.79
87
35.0
-38.7
-22.5
1.52
78
1.28
34
21.9
-69.2
30.5
1.64
79
2.79
79
39.4
-9.6
4.8
1.67
80
2.97
83
47.5
-21.1
-6.7
1.78
81
4.86
97
22.3
-44.7
-30.1
1.80
82
2.03
59
18.3
5.5
-7.8
1.89
83
7.46
100
35.3
-10.2
-35.7
2.05
84
3.98
89
42.0
-18.4
-8.3
2.38
85
2.94
81
29.2
-43.3
-34.4
2.49
86
3.65
86
48.0
5.0
-5.2
2.50
87
2.35
70
14.0
7.3
-2.0
2.57
88
2.44
75
27.6
-1.3
8.8
3.73
89
2.39
73
10.3
5.8
-0.6
4.14
90
4.33
94
41.1
-23.0
-19.5
4.43
91
2.97
82
45.6
-41.4
-38.5
6.08
92
4.85
96
38.7
0.7
-3.5
6.80
93
2.79
80
13
-4.3
-9.7
12.60
94
2.37
71
33.2
7.6
10.1
12.61
95
5.23
98
41.3
-13.1
-14.4
14.94
96
4.21
92
30
-3.6
-9.1
21.73
97
2.23
66
25.1
3.2
5.8
24.87
98
2.27
67
39.7
6.5
11
32.14
99
6.11
99
22.6
-27.9
-27.4
35.30
100
4.07
90
38.5
1.5
1
C.2
Graphical testing results
This section provides graphical results for each of the 32 surrogate models summarized in Table 5.3. For each
of the four case study problems, two surrogate models, one using proposed techniques and one using standard
techniques, were built at four different sample sizes. For a description of the plots, see Section 5.4.4.
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APPENDIX C: AUTOMATIC SURROGATE MODELING RESULTS
Problem (a) / 100 samples / proposed approach
Problem (a) / 100 samples / standard approach
Problem (a) / 200 samples / proposed approach
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Problem (a) / 200 samples / standard approach
Problem (a) / 400 samples / proposed approach
Problem (a) / 400 samples / standard approach
188
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APPENDIX C: AUTOMATIC SURROGATE MODELING RESULTS
Problem (a) / 16n = 544 samples / proposed approach
Problem (a) / 16n = 544 samples / standard approach
Problem (b) / 100 samples / proposed approach
189
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APPENDIX C: AUTOMATIC SURROGATE MODELING RESULTS
Problem (b) / 100 samples / standard approach
Problem (b) / 200 samples / proposed approach
Problem (b) / 200 samples / standard approach
190
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APPENDIX C: AUTOMATIC SURROGATE MODELING RESULTS
Problem (b) / 400 samples / proposed approach
Problem (b) / 400 samples / standard approach
Problem (b) / 16n = 1536 samples / proposed approach
191
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APPENDIX C: AUTOMATIC SURROGATE MODELING RESULTS
Problem (b) / 16n = 1536 samples / standard approach
Problem (c) / 100 samples / proposed approach
Problem (c) / 100 samples / standard approach
192
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APPENDIX C: AUTOMATIC SURROGATE MODELING RESULTS
Problem (c) /200 samples / proposed approach
Problem (c) / 200 samples / standard approach
Problem (c) / 400 samples / proposed approach
193
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APPENDIX C: AUTOMATIC SURROGATE MODELING RESULTS
Problem (c) / 400 samples / standard approach
Problem (c) / 16n = 3648 samples / proposed approach
Problem (c) / 16n = 3648 samples / standard approach
194
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APPENDIX C: AUTOMATIC SURROGATE MODELING RESULTS
Problem (d) / 100 samples / proposed approach
Problem (d) / 100 samples / standard approach
Problem (d) / 200 samples / proposed approach
195
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APPENDIX C: AUTOMATIC SURROGATE MODELING RESULTS
Problem (d) / 200 samples / standard approach
Problem (d) / 400 samples / proposed approach
Problem (d) / 400 samples / standard approach
196
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APPENDIX C: AUTOMATIC SURROGATE MODELING RESULTS
Problem (d) / 16n = 2720 samples / proposed approach
Problem (d) 16n = 2720 samples / standard approach
197
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