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Generative Design in Architecture:
From Mathematical Optimization to Grammatical
Customization
Pirouz Nourian1, Shervin Azadi2, and Robin Oval3
1University of Twente, Urban Planning and Geo-Information Management, Hallenweg 8, 7522 NH Enschede, the
Netherlands
2Eindhoven University of Technology, Urbanism and Urban Architecture, Het Kranenveld 8, 5612 AZ Eindhoven,
the Netherlands
3Princeton University, Form-Finding Lab, E209A Engineering Quadrangle, Princeton, New Jersey 08544, the
United States of America
Abstract
This chapter provides a methodological overview of generative design in architecture, especially
highlighting the commonalities between three separate lineages of generative approaches in archi-
tectural design, namely the mathematical optimization methods for topology optimization and shape
optimization, generative grammars (shape grammars and graph grammars), and [agent-based] de-
sign games. A comprehensive definition of generative design is provided as an umbrella term re-
ferring to the mathematical, grammatical, or gamified methodologies for systematic synthesis, i.e.
derivation, itemization, or exploration of configurations. Among other points, it is shown that gener-
ative design methods are not necessarily meant to automate design; but rather provide structured
mechanisms to facilitate participatory design or creative mass customization. Effectively, the chapter
provides the theoretical minimum for understanding generative design as a paradigm in computa-
tional design; demystifies the term generative design as a technological hype; shows a precis of
the history of the generative approaches in architectural design; provides a minimalist methodologi-
cal framework summarising lessons from the three lineages of generative design; and deepens the
technological discourse on generative design methods by reflecting on the topological constructs and
techniques required for devising generative systems or design machines, including those equipped
with Artificial Intelligence. Moreover, the notions of discrete design and design for discrete assem-
bly are discussed as precursors to the core concept of design as decision-making in generative
design, thus hinting to avenues of future research in manufacturing-informed combinatorial mass-
customization and discrete architecture in tandem with generative design methods.
1 Introduction
What is Generative Design and why does it matter? What problem can it solve for society? This book
chapter is to answer these questions in addition to providing a technical overview of the most important
generative design processes right ahead of the frontier of the state of the art of generative design tech-
nologies. Here we provide a brief definition of generative design processes given their applications.
As shall be illustrated, contrary to the common belief, generative design processes are not necessarily
meant to automate design processes, albeit except for generative models used for [procedural] content
generation in the games and entertainment industry. One of the main contributions of this piece is to
show the commonalities of three distinct sorts of generative design processes (mathematical deriva-
tion, grammatical itemization, and gamified exploration) in terms of similarities in their methodologies
(algorithms) and representations (data structures). However, the clear distinction between applications
of generative design to design derivation, design customization, and participatory design problems is
key for avoiding the commonplace reduction of computational design to design automation.
This is the author version of the Chapter 1 of the book Computational Design and Digital Manufacturing, edited by P. Kyratsis,
A. Manavis, and J.P. Davim: https://link.springer.com/book/10.1007/978-3-031- 21167-6
1
Topology Optimization
&
Shape Optimization
Generative Grammars
&
Cellular Automata
Decision Support Systems
&
Serious Gaming
Design
Optimization
Design
Customization
Participatory
Design
Unique Solutions
Catalogue of Solutions
Grammatical
Itemization
Gamified
Exploration
Mathematical
Derivation
Convergent Search
Divergent Search
Application
Application
Approach
Approach
Methodology
Methodology
Figure 1: The Spectrum of Archetypal Generative Design Processes
In what follows, we start with a brief introduction to the generative design spectrum; revisit three
types of computational design problems (optimization, diversification, and consensus-building) and
their inherent complexities; take a glance at the key role of discrete design representation in generative
design; revisit some basic definitions and terminological matters essential to understanding the gener-
ative design paradigm, and then present three types of generative design methodologies, illustrated in
Figure 1, in-depth:
• Generative Design in the sense of Form-Finding or Design Discovery through Mathematical
Derivation (Topology Optimization & Shape Optimization);
• Generative Design in the sense of Writing in a Pattern Language or Design Customization
through Grammatical Itemization (Generative Grammars);
• Generative Design in the sense of Co-Creating or Participatory Design through Gamified Explo-
ration (Cellular Automata, Serious Gaming & Multi-Actor Spatial Decision Support Systems).
The theoretical underpinnings of practice in these three lines of work can be traced to the core
concepts of design sciences [1] and the paradigm of design as a matter of ‘rational problem solving’
[2], [3].
The Mathematical Derivation methods can be traced back to Topology Optimization [4] and Al-
gebraic Shape Optimization (e.g. through the Force Density Method introduced by Schek [5]). Gram-
matical Itemization methodologies can be traced back to the linguistic analogy in Space Syntax [6],
Pattern Language [7], and Shape Grammars [8], all of which seem to have been inspired by Chomsky’s
seminal book on Syntax and Formal Grammars [9]. Later, Lindenmayer Systems [10] and Cellular Au-
tomata of Von Neumann, especially those in the style of the [Solitaire] Game of Life of John H. Conway
(see a recent compendium [11]) and Stephen Wolfram [12] as well as the ideas of William J. Mitchel
[13], Jonathan Cagan [14] and Kristina Shea [15], [16] enhanced the grammatical generative design
literature and emboldened it as an approach for promoting combinatorial design creativity. The idea
of Gamified Exploration as a form of generative design can be traced back to multiple ideas, namely
using multi-actor systems in search of a satisfactory equilibrium through group decision-making and
multi-criteria decision analysis introduced by Herbert A. Simon [17], the notion of generative sciences
put forward by Joshua Epstein [18] (i.e. complexity sciences driven by simulations), and the idea of
Markovian Design Machines introduced by Michael Batty [19]. The particular ideas of building “design
games” and “configurators” can be traced back, respectively, to the works of Henry Sanoff [20] and
Yona Friedmann [21].
Generative Design is a paradigm in computational design rooted in the paradigm of design as
rational problem solving based on the ideas of Herbert a. Simon, q.v., the theoretical discussion on
design paradigms by Kees Dorst on this matter [2] [22]. As an adjective, the term “generative” histor-
ically originates from the idea of generative grammars by Chomsky, on the one hand and the notion
of generating (deriving or discovering) a definite spatial design (typically for a mechanical structure)
from given design requirements using topology optimization or shape optimization (a.k.a. form-finding)
on the other hand. While the former sense (the grammatical definition of generative design) has been
well-known in the context of Architecture since at least the 1980’s, the latter (i.e. the derivational defi-
nition of generative design based on gradients or partial differentials a.k.a. sensitivities particularly in
Topology Optimization) has been known in Mechanical and Structural Engineering since the late 1990’s
(while differential/numerical Shape Optimization can be traced back to the 1970’s, e.g. the Force Den-
sity Method by H.J. Schek [5] or even much earlier, actually to the 19th century if we consider physical
2
or geometric shape optimization methods of Gaudi or the Graphic Statics method as formulated by
James Clerk Maxwell). However, the new surge of popularity of the term generative is mainly due
to the more recent popularity of the topology optimization approach since the 2010s for large-scale
architectural/structural applications. The idea of formulating discrete and multiplayer design games
as generative design systems is particularly our proposition as a mathematical formulation of multiple
ideas such as those of Henry Sanoff for Design Games [20] and Yona Friedman for Design Configu-
rators [21]. Additionally, another recent motivation for formulating design games has been the use of
Artificial Intelligence (especially Reinforcement Learning) for playing games (see e.g. the use of RL for
spatial configuration [23]), or more specifically Multi-Agent Systems and Cellular Automata for playing
Simulation Games similar to the MAS+CA system proposed by Ligtenberg et al. [24] and K ¨
onig [25] for
urban planning and design as well as the “Go Design” generative design framework for architectural
design [26].
Here we propose a comprehensive definition that encompasses the three approaches mentioned
above: generative design processes are simulation-driven, feed-forward, gradual, discrete de-
sign processes that can generate a navigable artificial design space of valid, idiomatic, consen-
sual, or optimal design alternatives respectively through design equations, design grammars,
or design games (see Figure 1). All Generative Design methodologies in our definition start from
an explicitly “navigable” representation of the design problems (constraints, moves, objectives, and
the structure of the state-space and/or the design space). What distinguishes generative design ap-
proaches from genetic evolutionary searches, mistakenly referred to as generative design, is this very
explicit nature of the design space exploration and the explainable navigability of the design spaces
in the generative design approaches, except for the “Genetic Programming” approach of Harding and
Shepherd [27] due to its explicit formulation of design spaces based on discrete design variables.
1.1 Navigation and Mapping
Inter alia, the importance of having an explicitly navigable picture of the design space (and its dual
performance space a.k.a. performance landscape) is key to the ability of a designer for explaining and
justifying design decisions.
Two dual problems can be identified in this regard, which can reveal the complexity of design
endeavours from a mathematical stance: the problem of mapping the multiplex associations of many
design choices to their few integral consequences (dubbed here as the primal evaluation problem
or the mapping problem in a manner of speaking) and the more challenging problem of figuring out
what choices must be made to attain a specific multitude of performances or quality levels that can
be regarded as the eventual consequences of the design choices (dubbed here as the dual design
problem or the navigating problem in a manner of speaking) (see Figure 2).
The navigability of the design spaces requires the explicit modelling of the relations between one
design alternative and another. Such relations can be referred to as design operations (such as
moves or updates based on gradient descent), design rules (as in grammatical rules) or design
moves (as in moves of a game) defining how one can get from one design alternative to another. Nav-
igability also implies that we can have a rigorous sense of topology in our design space as to which we
can measure similarity or dissimilarity between designs, depending on the number of operations, rules,
or moves that set them apart. In other words, generative design processes are inherently topological
workflows.
The importance of this explicit notion of similarity can be understood by considering the key role of
two sorts of similarities between designs that would provide for clustering or manifold mapping of design
spaces for simplifying cognitive decision-making for human designers: similarity between designs in
the high-dimensional design [decision] space and similarity between designs in terms of the outcomes
of interest in the performance space (referred to as the vectors of choices and consequences in Figure
2).
1.2 Generative Design Systems
Three archetypical generative design systems are identified here to define the poles of a broad spec-
trum (as introduced in Figure 1):
3
Dual Design Problem
How should we combine the choices to get to the
desired consequences?
Decoding
↬
Primal Evaluation Problem
Navigating Design SpacesMapping Design Spaces
a few integral consequencesthousands of choices
Mapping
of
Choices
to
Consequences
How do the different consequences emerge from
combinations of choices?
Endocding
↫
a few integral consequences thousands of choices
Navigating
from
Consequences
to
Choices
Figure 2: Duality of Mapping and Navigation problems in generative design
•Design Equations: Design Optimization through Mathematical Derivation, in search of optima,
exemplified in Topology Optimization and Shape Optimization processes;
•Design Grammars: Design Customization through Grammatical Itemization, in search of id-
iomata, exemplified in Formal Grammars, L-Systems, and Generative Grammars (for Shape
rewriting or Graph rewriting);
•Design Games: Participatory Design through Gamified Exploration, in search of [Nash] equilibria
or another notion of satisfaction of multi-actor multi-criteria decision analysis problems.
Whilst all of these approaches are feed-forward and gradual as compared to black-box feedback
optimization, it can be noted that the method of reasoning in grammatical design itemization is akin
to forward induction and that the method of reasoning in mathematical design derivation is akin to
backward induction (i.e. working our way backwards from the target objective(s) converging to a par-
ticular vector of decision variables as opposed to the divergent expansion the design catalogue in the
grammatical or rule-based itemization methods).
A similar comparison can be made in terms of the Depth-First-Search and Breadth-First-Search
methods being analogous respectively to the design space navigation methods in mathematical design
derivation and grammatical design itemization. In our proposed framework, gamified design exploration
methods can exploit both types of navigation mechanisms (hereinafter referring to search and induction
processes).
Even though these three types of generative design methods cover the main applications of gen-
erative design, there is a much newer technology that is rapidly developing with excellent potential for
the performance-based generative design that is yet to be explored: generative models, especially
Bayesian Belief Networks, Variational Auto Encoders, Restricted Boltzmann Machines, Flow-Based
Models, and Diffusion Models. Towards the end of the chapter, we shall reflect on the utility of Gener-
ative Models for augmenting performance-based generative design with Artificial Intelligence.
2 Theoretical Background
As stated by Batty [19], if any progress is to be made in systematic design it will require to be based
on an explicit and unambiguous statement of the design framework (states, goals/objectives, moves,
constraints). We argue that it is necessary to have a framework for formulating design problems to
be able to provide for rigorous progress in computational design, systemic change in practice and
pedagogy of computational design for the betterment of methods in terms of explainability of processes
and justifiability of the results in terms of the triad of sustainable development goals (balanced social-
economic equity, social-environmental comfort, and environmental-economic resource efficiency of
solutions).
Furthermore, as has been extensively argued in the design research literature, one can quickly
identify the difficulty of design as a matter of decision-making in that the designer is supposed to
arrive at a detailed conclusion about the definite geometry of a building (colloquially referred to as the
4
Subjective Matters
Objective Matters
Human
Complexity
Physical
Complexity
Multi-Dimensional or Spatial Complexity
Multi-Criteria or Directional Complexity
Multi-Actor or Consensual Complexity
Multi-Value or Normative Complexity
Doubly Complex Design Problems
Figure 3: Duality of Mapping and Navigation problems in generative design
“form” of a building/structure) while they are only given an abstract and often vague description of the
requirements, wishes, and objectives (collectively referred to as the “function” of a building/structure
in the jargon of AEC). The typical characterization of design problems as ill-defined or even wicked
is well-known and rather overstated [3] [2]. Revisiting these difficulties in terms of the complexity of
decision-making, we can identify four types of complexities that make architectural design problems
especially hard (See Figure 3).
The history-old difficulty of transiting from abstract functions to concrete forms has led to a de-
sign culture that is arguably over-reliant on intuition and the so-called tacit knowledge of the design,
which can be characterized by the so-called “logical leap of design”[28]. This leap, refers to the diffi-
cult, unexplainable, and often unjustifiable transition that can be considered as the intuitive or creative
synthesis process that appears to an external observer as jumping to a conclusion or solution without
even explicitly formulating the problem. This gap of reasoning concerning the dichotomy of the form
and function of a design can also be seen in the descriptive Function-Bahviour-Structure (FBS) Design
Framework by John Gero et. al. [29].
This ontological framework identifies some typical representations starting from the most abstract
design requirements (R) to the most concrete eventual design description (D) and the typical actions
leading back and forth to four intermediary representations of a design process: a process of “formu-
lation” of the Function (F) and the expected Behaviour [Performance] (Be) of the system, a process
of “synthesis” leading to the proposed/designed Structure [Form] (S), a process of “analysis” revealing
the actual Behaviour of the structure (Bs), and a process of “evaluation” for comparing the expected
behaviour with the actual behaviour analysed (simulated) on the Structure to assess how appropriate
is the particular Structure proposed in a design iteration, followed by a process of “documentation” de-
tailing the proposed Structure [Form] into a Design Description (D). The description also contains three
“reformulation” processes as to which design processes are sometimes referred to as “co-evolution of
problems and solutions” [30].
Echoing the genuine appeal of Yona Friedman “Towards a Scientific Architecture”, we argue that the
“Sciences of the Artificial”[31] in their plurality, as the decision sciences focused on how to change the
current state of things and environments towards better states, are key to such a scientific transition.
Contrary to the reductionist connotation of the term scientific, the non-reductionist notion of “generative
sciences”[18] takes it for granted that the matter of mapping a myriad of choices to few consequences is
an endeavour in the realm of complexity sciences that can be dealt with utilizing simulations, be it simu-
lation based on first-principles often encapsulated in Partial Differential Equations, stochastic or deter-
ministic simulation of design moves on a decision-tree of grammatical rules (exhaustive-enumerative
search for cataloguing or probabilistic approaches such as the Wave-Function-Collapse [32]), Agent-
Based Simulations, Markov Chains, or Markov Decision Processes, as well as Simulation Games that
can reveal the emergence of “Collective Intelligence” [33].
5
3 Framework
Whilst there is no general way of proving that a design is the best it could possibly be, Generative
Design aims to bridge the logical leap of design by methodically structuring and mapping the design
space and its dual performance space, starting with defining the requirements, defining valid or mean-
ingful design moves (decisions leading to an identifiable change in the discrete state of a design from
one to another), setting explicit quality criteria, and identifying alternatives in the design space through
a systematic navigation process to satisfy the set quality criteria in three distinct problem settings (op-
timization, customization, or participation problems as explained in the Introduction). A design frame-
work must explicitly identify the ‘purpose’ of design or the nature of the design problem, as to which we
have identified three types of purposes, namely: reaching a unique optimum, producing a plethoric cat-
alogue of idiomatic designs, or reaching a fair/equitable equilibrium in a multi-actor participatory design
process. The framework presented here is an extension and elaboration of the GoDesign Framework
for Generative Design [26]. However, one can trace back the roots of this inherently discrete genera-
tive design framework to the configurational studies of pioneers of computational design such as Lionel
March, Philip Steadman and Raymond Matela on discrete design and configurations with Polyominos
and William J. Mitchel’s discrete grammatical approach [34] [35].
3.1 Discrete Design
The most obvious common feature of the three types of generative processes reintroduced here is
their discretized design spaces. While this discreteness is not necessarily carried over to the actual
construction of the designed objects (e.g. in Topology Optimization of small objects), in the relatively
large scale of buildings, the discrete design variables can correspond one to one to the actual build-
ing blocks. Observe that buildings are too large to be made as really monolithic objects and as such
their construction design most probably requires some sort of segmentation (polygonization and poly-
hedralization). In the discrete architecture paradigm [36], discreteness or jaggedness of the results is
not regarded as a limitation but embraced as a digital way of building as an aggregation of discrete
modules. Discretization not only brings about advantages for the construction process, but also, as a
mathematical proviso, provides a discrete notion of an array of design variables that is key to genera-
tive design. As can be seen in the context of the three archetypical generative design processes, such
a discretization of design variables makes the mathematical, computational, and play processes very
straightforward:
1. In topology optimization, the discrete material design variables make a vector of design variables
x∈(0,1]n, as to which the sensitivities (encapsulated in a gradient vector) can be efficiently
computed. Similarly, in shape optimization, a discrete mesh or network model allows for the
straightforward characterization of the decision variables as the coordinates of a number of dis-
tinct vertices, given a constant topology for the mesh or the network, as well as the discretization
of differential operators required for shape optimization (e.g. the gradient, divergence, and Lapla-
cian operators).
2. In grammatical design, the grammar is effectively a system of graph-rewriting or mesh-rewriting
that depends on the identification of distinct topological conditions (antecedent) on the graph
neighbourhoods or mesh cells that should lead to their rewriting (consequent) based on a dis-
crete set of topological-geometrical rules. Similar to mesh subdivision and graph simplification
procedures, the topological (and thus discrete) identification of predicates and functions make
implementing “if this then that” “pattern language” very straightforward.
3. In gamified design, the discretization of the design space and design moves makes designing
as structured, teachable, and score-able as playing board games like Go & Checkers, tile-based
games like Dominoes & Anagrams, or construction sets/toys like LEGO & Lincoln Logs. Discrete
design moves and game pieces lower the participation threshold for non-expert human players
for partaking in decision-making and facilitates the attainment of valid designs and their scores.
Additionally, a discrete state space for the game also provide for teaching machines to play these
games through Reinforcement Learning, Multi-Agent Systems, or Markovian Design Machines,
all of which require the discrete setup of board games or Cellular Automata to work. The remark-
able success of Artificial Intelligence (Reinforcement Learning in particular) in playing retro Atari
6
games [37] and the game Go should provide a convincing motivation in that regard.
In summary, generative design is about discrete decision-making. Note that in mathematics, deci-
sion problems are binary decision problems. When we formulate the design space as a discrete space
we can enumerate design alternatives, and this, inter alia, can allow for measuring the likelihood of one
configuration of decisions (a design) or another, and this, in turn, allows for measuring information con-
tent based on a measure of entropy as in Shannon Entropy, which pertains to the number of questions,
e.g. binary questions, that need to be answered to determine the exact state of a design configuration.
In other words, even though the design space can still contain infinitely many decisions, at least, in this
discrete setting, we have a navigable and countable plethora of designs.
Moreover, the similarity of designs can be rigorously modelled in discrete generative design, pro-
vided the designs are represented with decision-vectors or multiple criteria as performance vectors.
While designs can be compared easily in terms of the similarity of their measurable performance/quality
consequences, describing the similarity of designs in terms of their inner structure, i.e. the similarity of
their combinations of choices, is typically harder and depends on the extent of explication of a design
space. In the mathematical sense of topology optimization, the similarity between decision variables
will be a matter of defining a similarity metric as to an inner product between high-dimensional arrays
of density-like decision variables (typically in the form of x∈(0,1]n)). In the most abstract case, when
the decisions are all completely discrete, i.e. in the grammatical design sense, the notion of similar-
ity between graphs or meshes is purely discrete and pertains to the number of rules that need to be
applied to get from one design to another (the number of design moves or operations in a decision
graph). Describing similarity between design states in tile-based design games is similarly explained
in a [38].
3.2 Gradients for Disaggregated Design Evaluation
Evaluation is the backbone of decision-making, it allows one to compare various options for design
moves and decide based on the projected consequences. In the general sense, computational per-
formance evaluation mechanisms have existed and been applied in mainstream practices since the
1980s or even since the 1950s (if we take into account the history of the Finite Element Method). How-
ever, the specific idea here is to define and formulate evaluations that are disaggregated at the level
of discrete design moves. The kind of evaluations that can inform generative design moves are those
that can be derived as sensitivities or gradients of the aggregate objective functions with respect to the
density of the discrete decision variables determining the relative or absolute existence of some parts
in the aggregate design, i.e. if a single objective function can be formulated as a scalar function of a
vector decision variable in the form of f(x)in which X:= [xi]n×1∈(0,1]n, then its gradient vector
denoted as ∇f(x)can be expected to contain relevant information content at the level of each design
variable revealing the sensitivity or partial differential of the objective function in question to an infinites-
imal change in each one of the design variables corresponding to density (or the relative existence)
of the discrete cells in the design space, i.e. σ:= ∇f(x) = [ ∂f
∂xi]n×1. In the more general case of
the multi-objective design problem setting, i.e. a vector objective function of vector decision variables
f(x)=[fq]o×1, the gradient or the vector of sensitivities is generalized to the Jacobian of the system:
J:= ∇f(x) = [∂ fq
∂xi]o×n= [[∇Tfq]1×n]o×1. Note that also for constrained optimization problems
the method of Lagrange Multipliers requires one to compute the gradient of the explicitly formulated
(equality or inequality) constraints as functions of vector decision variables.
Gradual feed-forward progression in design is the hallmark of generative design that is most ex-
plicitly achieved by utilizing such gradients in topology optimization and shape optimization (either for
making design moves in a manner like the gradient descent or quasi-analytic solutions characterizing
the state of equilibrium which incorporate the gradients in their formulation, such as the Force Density
Method [39]).
However, not all the qualities and constraints are easy to formulate explicitly as analytical functions
of decision variables. The frontier challenges in the art and science of generative design can be directly
bordered by the extent to which one can formulate the objectives and constrains as functions of the
decision variables. In other words, emblematically speaking, the central question of generative design
is how to navigate in the realm of decision variables with a compass informed by the polarities in the
realm of objectives and constraints. In cases where subjective opinions, intangible cultural values, and
7
CONNECTIVITY COLOR
SHAPE
Figure 4: Design space navigation for a mesh: changing shape (form), connectivity (topology), and color (data).
aesthetics, are to be considered, we eventually have to rely on human judgement for the evaluation
process, whether directly by involving humans in the loop or by training neural networks as “function
approximators” using AI.
Architectural design is almost always concerned with multiple validity constraints and quality cri-
teria as to which the alternatives need to be assessed. Often, such criteria are not commensurate
due to their different physical dimensions. Before jumping to the conclusion that architectural design is
about multi-objective optimization, we need to observe that in the mathematical sense, speaking of the
optimum beyond a single objective can be rather absurd in most cases. Instead, in the general case,
the endeavour is directed at finding a satisfactory design that can be characterized either as a hier-
archically dissected chain of multiple optimization problems, a diversification problem concerned with
the enumeration of valid designs for customization (that is not at all about optimization), or satisfac-
tion of decision-making criteria in a multi-actor and/or multi-criteria setting using Multi-Criteria Decision
Analysis (MCDA).
3.3 Design Space Navigation
As stated in our definition, generative design is all about confidently navigating design spaces based
on a possibly partial but explainable map of the associations between design choices and their perfor-
mance consequences. Specifically, however, the navigation strategies differ from one type of gener-
ative design to another, primarily because the nature of the design spaces can be very different from
one type to another. Considering the most general purpose of design processes in Architecture and
Structural Design as forming a particular material distribution or spatial arrangement, we can observe
that discrete design is practically about making changes in some meshes (hyper-graphs containing 0D
vertices, 1D edges, 2D faces, and possibly 3D cells) in the general form of M= (V, E , F )for discrete
2D Manifolds or M= (V, E, F, C )for discrete 3D Manifolds. Without loss of generality, in the most
common cases, the faces and cells are respectively triangular-tetrahedral or quadrilateral-hexahedral.
Generative Design can be described in terms of changes that can be made on such meshes in terms
of the positions of vertices or “shape changes”, attributes of vertices/hyper-edges (edges, faces, cells)
or “colouring changes”, or the very set of hyper-edges and possibly the set of vertices or “topology
changes” (see Figure 4).
Shape Space is the space of possible coordinate positions for the vertices of a mesh V=
[vi]n×3(Shape Optimization or Grammatical Design as in Shape Grammars, Protein Folding Game
FoldIt [40])
Colour Space is the discrete material design domain corresponding to the vector/array of decision
variables x= [xi]n×1, which is either about integer colours or ambient colours indicating relative
density levels.
Topology Space is the space of possible meshes M= (V, E , F, C), possibly attributed (Grammat-
ical Design or Construction Games as a matter of mesh rewriting regulates how one possible design
can be changed to another possible design with a distinct topology).
The so-called state-space of Design Games in which the design moves are defined could be in the
form of the 1st or the 2nd or rarely the 0th type of design spaces defined above, depending on the
application.
Once the nature of the data points in a design space is explicitly specified, one can think of effective
and efficient methods for navigating within a vast design space, considering the nature of informed and
directed design moves that can take one design as a data point in a decision space to another. The
navigation strategies are later introduced in the specific context of the archetypical generative design
8
Generative Design Methods
Data-DrivenModel-Driven Policy-Driven
Topology Optimization
&
Shape Optimization
Multi-Agent Systems
&
Markov Decision Processes
Reinforcement Learning
Statistical Generative Models
&
Bayesian/Markovian
Proabibilistic Graphical Models
Devising Design Games
&
Scoring Mechanisms
Exploratory
Graph/Mesh Rewriting
Grammars
&
Symbolic AI/Expert Systems
Performance-Based (Associative)
Rule-Based
Systematic Data Collection
&
Black-Box Training
Mathematical Derivation
from
Governing Equations
Computational Design
Practice
Black Box ModelsGrey Box ModelsWhite Box Models
Knowledge Data
Formulation of Design Rules
as
Topology Rewriting Methods
Rule-Based Models
Design Rules
A Spectrum of Generative Design Methodologies & Generative Models for Content Generation
Grammatical Analytical/Differential Gamified Statistical-Probabilistic
Mathematics
Graph Theory,
Algebraic Topology,
Cellular Automata
Numerical Methods
Partial Differential Eq.
Multivariae Calculus
Stochastic Games
Markov Processes
Agent Based Models
Diffusion Processes
Markov Chains
Manifold Learning
Figure 5: Spectrum of Design Space Navigation Methodologies and the Essential Mathematics of Generative
Design
methods.
This idea of informing design moves in the case of mathematical derivation of designs can be simply
understood in terms of gradient descent moves in the decision space. In the case of grammatical
design, however, as explained before, the mode of reasoning is forward induction, as to which the
idea of evaluation is present at a meta-level informing the design moves in terms of guaranteeing that
the grammatical rules result in valid designs. Whilst these categories encompass the most common
approaches, there are specific hybrid approaches to design space navigation that do not fit into a
single one of such categories, a notable example of which is the Shape Annealing Method by Kristina
Shea and Johnatan Cagan (Shape Annealing=Shape Grammars+Simulated Annealing by [41][16]).
However, in the general case, the purpose of grammatical design methods is to produce a navigable
catalogue of designs and so, the more explicit use of evaluation is in the feed-forward formulation
of design rules that can be guaranteed to result in valid designs. See a spectrum of design space
navigation approaches in Figure 5.
4 Generative Design Methodologies
Up to this point, we have discussed the common features of generative design processes in terms of
their discreteness, the idea of gradual progression or informed navigation in design spaces, and the
key role of simulations (Solving Partial Differential Equations using Numerical Methods, Deterministic or
Stochastic Simulation of Grammatical Rewriting Logic, Modelling Finite-State Machines or Automata, or
Simulation Games). In what follows, we go into the specifics of each of the mathematical, grammatical,
and gamified generative design methodologies as introduced in the generative design spectrum (see
Figure 1). In the last part of this section, we briefly review the potential of Generative Models in Artificial
Intelligence for performance-aware architectural design applications.
4.1 Mathematical Generative Design
In this section, we explain the foundations of the generative design processes that can be categorised
as being mathematical as to their commonalities in terms of derivation of forms (topologies and ge-
ometries, a.k.a. configurations and shapes).
9
These methodologies are known as topology optimization and shape optimization in the computa-
tional design literature. In this sense, however, the implication of the term optimization goes beyond
fine-tuning a design or haphazardly generating combinations of parameters to create forms in a genetic
or evolutionary fashion. Instead, the search processes characterized as shape and topology optimiza-
tion are directly navigated by the gradients (a.k.a. sensitivities) of the cost (or benefit/utility) functions
for converging to forms that have some measurable and differentiable minimal or maximal properties,
and so it is fair to call such approaches as form-finding approaches. Even though the terms shape
optimization and topology optimization originate from Structural Engineering and refer to notions of
performance related to minimal compliance or minimal materialization of structures under certain load-
cases, we can see that it is possible to generalize their definitions to other domains-specific notions
of performance, e.g. passive climatic design principles (governing equations from geometric optics)
or even environmental psychology (insofar as the governing equations can be formulated in terms of
the decision variables). We shall see in detail that these approaches are most suitable for convergent
search in a design space equipped with a clear notion of performance or quality.
Here we provide two compact and general definitions of these processes:
Shape Optimization Problem Given a network (graph) or a mesh (hyper-graph) with fixed topology,
it is desired to find a valid embedding (i.e. 3D vertex positions) for the network/mesh that would
minimize a cost function describing how costly (uncomfortable) the shape is; possibly subject to multiple
constraints of validity. The general idea is to formulate a so-called [virtual] energy function (typically
explainable as a Dirichlet Energy functional) that measures the extent to which a shape is far from
being relaxed, optimal, or desirable.
Topology Optimisation Problem The core idea of topology optimization is about solving a problem
in this general form to find a [black & white] colouring of a tessellated cell space that would minimize
a cost function. The cell colours are eventually supposed to be integer labels determining whether a
cell is in or out of the ultimate form. The white-coloured cells are effectively removed from the final
configuration. So, the topology of the final configuration effectively alters as compared to the initial
super-graph that is dual to the cellular tessellation of the design domain, hence the name topology
optimisation.
4.1.1 Existing work
The history of structural shape and topology optimization can arguably be traced back some 150 years
(see this literature review [42]) but providing an exact timeline or a comprehensive literature review in
this regard falls out of the scope of this book chapter. In this section, we briefly introduce some critical
references to the existing works on [mostly] structural shape optimization and topology optimization,
specifically those of particular interest in terms of generality and applicability in the AEC. The gener-
alization of structural optimization approaches to architectural design remains a challenge that would
go beyond the scope of this summary. For the sake of brevity, as well as our focus on explainability,
scalability, reproducibility, and justifiability of generative design methods, we shall restrict our attention
to the form-finding or mathematical design derivation approaches that are explicitly and rigorously led
by gradients and the only hint to alternative approaches such as metaheuristic methods or integer
programming, namely topology optimization than shape optimization.
Shape Optimization In the context of discrete generative design, shape optimization refers to meth-
ods for optimizing (typically minimizing) an energy function that takes the shape of a mesh as the input
function. Thus, the objective function is often in the form of a Dirichlet Energy integral. Such an integral
functional, which is typically based on a Partial Differential Equation governing the dynamics of the
movement of the mesh vertices, can be written as a discrete integral or the sum of magnitudes of the
very forces that could be considered as the gradients of the input function (mostly the positions of ver-
tices) over the mesh elements (edges, faces, or cells). This generalized definition points to two types
of approaches for solving the shape optimization problem: those concerned with analytically solving
the governing equations in the strong form (differential form) and thus characterizing the optimality cri-
teria, in which typically the Laplacian (see two prominent introductions by Olga Sorkine[43] and Bruno
10
Levy[44]) or generally the Kirchhoff Stiffness matrix operators (see a thorough mathematical introduc-
tion [45]) are used to discretize the governing equations, the classical example of which is the Force
Density Method by Hans-J¨
org Schek [5]; and those approaches that can be described loosely as being
based on the weak-form (integral form) focused on numerical integration of the energy functional and
updating the discrete differentials alternatively, the classical example of which is the Dynamic Relax-
ation method by Michael R. Barnes [46] [47]. While the former has the elegance and appeal of a purely
mathematical method, the latter has the flexibility of dealing with non-linear optimization problems.
For a more detailed comparison of these two approaches in a broader sense, see a comprehensive
literature review [48] with a thorough mathematical formulation and collation of the methods. Whilst
the Force Density Method already refers to generalizations for possibly non-linear constrained prob-
lems, it is noteworthy that general methods such as the Shape-Up method of Constraint Projections
by Bouaziz et al. [49][50][51] or the Thrust Network Analysis by Philippe Block & John Ochsendorf[52]
have been introduced for interactive and constrained form-finding. Additionally, a class of interactive
methods for confidently navigating the space of optimal shapes (networks) in equilibrium have been
proposed based on Graphic Statics, e.g. an exemplary compendium [53].
Topology Optimization A generalized problem formulation for topology optimization problems can
lead to new frontiers of research in computational design that go much beyond the famous structural
design applications, cf., the formulation of massing and zoning problems in architectural configuration
inspired by structural topology optimization problems in the GoDesign framework [26]. Referring to
the three types of design moves in generative design as mesh editing procedures (for modification of
shape, colour, and topology), it must be noted that contrary to what the name suggests, the most com-
mon and the most scalable approaches to topology optimization do not work with altering the topology
(by using binary decision variables x= [xe]n×1∈ {0,1}n) throughout the optimization process. Even
though the ultimate aim of the topology optimization process is to end with a particular combination of
binary decision variables defining a distinct topology of a cellular material domain, the ambient colours
typically indicate float density-like variables corresponding to the possibility/probability of having a cell
or not. Thus, it is common to consider an array of float density variables corresponding to the cellular
finite elements in the tessellation in the form of x= [xe]n×1∈(0,1]n. This change of decision variable
xpresents two relaxations to the theoretical topology optimization problem. One relaxation concerning
the change of a Boolean decision variable to a float variable brings about straightforward differentiabil-
ity, thus resolving the need to compute a topological derivative, q.v. [54]. Another relaxation is about
changing the infimum from zero to a small non-zero value (e.g. 10E-3) to avoid singularities in the
stiffness matrix (as well as to avoid division by zero in the smoothing/filtering step [55]).
The most crucial step of any analytical topology optimization process is the simulation of the cost
function and its gradient as a function of the decision variable. In the case of structural topology
optimization, this is performed in these steps:[re]computing the Stiffness Matrix with the current density
variables based on the current density distribution; computing a numerical solution to the governing
equations using the Finite Element Method; computing a gradient vector showing the sensitivity of
the cost function to the perturbations of density variables; iteratively seeking a Lagrange multiplier (in
the Optimality Criteria Method) for solving the volume constraint; multiplying the sensitivities by the
Lagrange multiplier; applying some distance based filtering or smoothing on the gradient vector, and
lastly multiplying the next-iteration density vector by the smoothened gradient in a heuristic fashion
assuming that where there is higher sensitivity, there should be more material.
The most common approaches to structural topology optimization, to this date, are either di-
rectly based on or inspired by the Solid Isotropic Microstructure with Penalization (SIMP) methodol-
ogy[56][4][57]. Notably, several pieces of vectorized MATLAB code have been published for Topology
Optimization based on the SIMP methodology including the famous 99 Lines[4] and 88 Lines[58] for
the 2D structural optimization or for 3D structural optimization [55].
Rozvany and Zhou[59] explain in depth why the intuitively appealing and bio-inspired approaches
known as soft-kill, hard-kill, Evolutionary Structural Optimization (ESO), or Bidirectional Evolutionary
Structural Optimization(BESO) that were popularized by Claus Mattheck [60][61] and Yi Min (Mike) Xie
[62][63] are not guaranteed to find optimal solutions. Thus, due to the mathematical orientation of the
proposed approaches in this book chapter, we shall skip these methodologies in favour of those directly
working with the gradients of the objective function.
11
Whilst the use of a numerical simulation is the core of the methodology; it does not have to be based
on the Finite Element Method; there are alternative formulations based on other numerical methods
such as the Discrete Element Method[64].
4.1.2 Navigation Strategies
Here the basic concepts of topology optimization methodologies are explained with a view toward
generalization, from the structure of the design space to the different design space navigation methods.
Tessellation of a Spatial Material Domain A material domain is modelled as a topological space,
technically tessellated into some regular or irregular cells e.g. without loss of generality, some triangles
or quadrilaterals in 2D or tetrahedra or hexahedra in 3D. Together, these cells form an algebraic topol-
ogy (a [combinatorial] cell complex) that can be represented as a graph whose vertices are dual to the
cells of the cell complex.
A Vector Design Variable To every cell of the cell complex, a virtual density-like parameter is at-
tributed, making up an n-vector of [binary] decision variables for ncells in the tessellation. The ultimate
goal can be described as figuring out for every cell if it should exist as a materialized domain in the
final design or not for some minimization or maximization goal to be achieved. However, the most
prominent approaches to solving this problem tend to deviate from defining binary decision variables at
least for the duration of the topology optimization process for multiple reasons that can be summarised
as differentiability and preservation of topology.
Formulating a Cost Function A virtual energy function or a cost function is typically formulated as a
function of the vector decision variable to be minimized (hence the term optimization), which represents
something that conceptually defines the extent to which a design can be assumed to be tense or
unsatisfactory (opposite of relaxed or satisfactory); this is usually a form of the Dirichlet energy, such
as the strain energy of a structure known as the compliance of the structure under a particular load
case given some boundary conditions (supports). More generally, any cost or benefit function that is
explicitly a function of the vector decision variable can be sought to be minimized or maximized in this
regard. Correspondingly, the argmin or argmax will be the optimum design effectively describing a
foggy picture of the minimal or maximal material distribution over the design domain that is typically
snapped to a binary density vector at the end of the process.
Topology Optimization Problem Solving Method From the very description of the design variable,
it must be noted that even in the case of the binary decision variables, the search space expands expo-
nentially in the order of O(2n)and thus, a naive brute-force search quickly becomes intractable. Fur-
thermore, even after introducing the float variables and the common SIMP heuristic (the penalization
scheme) that makes the cost function easily differentiable, the problem is almost always non-convex
and generally difficult to solve with thousands or hundreds of thousands of decision variables. The
approaches for solving Topology Optimization problems bifurcate with respect to at least two major
questions: (a) binary decision variables or float decision variables, and (b) analytical or meta-heuristic.
As explained earlier, the metaheuristic approaches fall out of the scope of this summary. The approach
of using binary decision variables is perhaps interesting or appealing for small educational examples
such as LayOpt[65] but is not scalable to be applicable in high-resolution applications[54]. As for the
analytical approaches, in addition to the most common Optimality Criteria Method introduced briefly
earlier, a notable alternative is a Method of Moving Asymptotes (MMA) [66] and its newer variant, the
Generally Convergent Method of Moving Asymptotes (GCMMA)[67] by Krister Svanberg. However, the
Optimality Criteria method generally remains more accessible and thus more prevalent in mathematical
topology optimization.
Architectural Layout Methods The GoDesign framework[26] explains how the problem of 2D/3D
space layout in Architecture can be broken down into three sub-problems of massing (binary colour-
ing)[68], zoning (multi-colour spatial allocation or space planning), and routing (corridor generation)[69].
12
Figure 6: Structural Topology Optimization for designing a discrete compression-only structure, image credit:
Rick van Dijk[81]; (a) a 2D toy-problem showing the proposed method as compared to the common isotropic
formulation obtained from Ansys Topology Optimization package; (b) a 3D toy-problem showing the discrete
results of the topology optimization process for a compression-only/funicular structure to be built out of discrete
blocks
The problem of Architectural Space Planning, especially the 3D layout is arguably the hardest problem
in computer-aided architectural design especially because it is ill-defined and over-constrained[70][71],
and so to keep the discussion contained we refer the reader to a few literature reviews [72][73][74]
and restrict our attention to explicitly discrete problem formulations such as Space Allocation [75] or
Mesh Colouring [76]; most of which propose mathematical Integer Programming (linear or quadratic
optimization with integer variables using Operations Research solvers) [77][78].
4.1.3 Illustrative example
Applications of the Topology Optimization (TO) methodology in domain areas other than structural
optimization are quite rare; see a range of applications in a recent review by Mike Xie[79]. However,
there is at least one other prominent application of TO other than SO in engineering for designing heat
sinks[55] and at least one application for the design of a complete building structure [80]. Thus, here
we show only two architectural applications of topology optimization which differ significantly from the
mainstream structural applications. The first illustrative example[81], shown in Figure6is a topology
optimization process for finding a compression-only structure with large-scale cells whose self-weights
are not negligible, and thus the example has a few additional constraints of validity on the stress,
self-weights, and the supports. The second illustrative example[82], shown in Figure7is a topology
optimization process for finding an optimal climate-adaptive shape for the envelope of a building with
a mathematical formulation of the cost function and gradient both explicitly formulated in terms of a
discrete opacity (density-like) decision variable. These examples help in illustrating the bigger picture of
topology optimization. in that the general idea of defining a simulation-driven differentiable cost function
as a function of a vector of density-like decision-variables corresponding to a cellular tesselation of a
design space provides for solving a mathematical optimization problem for deriving an optimal bi-
colouring of the lattice graph dual to the cellular tessellation based on its analytical gradients. Such a
problem can be formulated in this compact form:
x:= [xe]n×1∈(0,1]n:decision variable
min c(x) : cost function
s.t.g(x) = 0 : constraint f unction
H(x∗),x∗= arg minxL(x, λ) = c(x)−λg(x) : solution
, where H(x∗)denotes the Heaviside Step function at an arbitrary threshold and L(x, λ), denotes the
Lagrangian functional. In summary, as long as the costs and the constraints are differentiable functions
of a density vector variable corresponding to the cellular elements of a tessellated region, the above
problem can be regarded as a topology optimization problem.
13
Figure 7: Climatic Topology Optimization for deriving a building mass/envelope of optimum solar potential, image
credit: Anastasia Florou [82], (a) a hypothetical binary mass-configuration and the changes in performance from
one configuration to another, (b) the proposed analytical solution to the topology optimization problem for
maximizing the solar potential as a level set in the design space.
4.2 Grammatical Generative Design
Grammatical approaches to generative design can be traced back to the figurative work of Chomsky
on Formal Grammars [9], [83]. Additionally, to the historical example of Shape Grammars initiated by
Stiny [8], [84], there is a congruent framework to discuss modern grammatical approaches to design
based on Graph Grammars or Graph Rewriting Systems from a mathematical and computational point
of view.
Practically, grammatical approaches can be implemented either computationally or as a system of
rules for [pen & paper or physical] games. These processes allow a designer to explore a vast range of
topological designs, especially because they let the designer systematically change the topology of the
design space directly. Such exploration processes are typically not bound to or directed by an explicit
definition of performance.
The introductory example in Figure 8uses a simple two-rule grammar to evolve fractal squares
designs. The marker in pink is positioned with a MOVE rule to choose which square to subdivide with
the SPLIT rule. The SPLIT rule has an optional Boolean parameter: whether the subdivision goes
across all squares. The MOVE rule has two discrete parameters to choose the number of squares to
traverse in the X and Y directions.
SPLIT
CONTINUOUS
MOVE
X+2 & Y+1
SPLITSPLIT
Figure 8: Introducing Grammatical Generative Design: fractal square subdivision of squares by moving a marker
(in pink).
We first present existing work on the development of grammars for generative design, from formal
to shape and finally to graphs. Then we discuss the navigation strategies and illustrate them on an
example.
4.2.1 Existing work
The rule-based design encapsulates different applications with design grammars that can apply to
language, form and structure through different levels of complexity, starting with formal grammars.
Formal grammars Chomsky introduced in the 1950s generative design applied to language using
formal grammars [9], [83]. A finite set of formal grammar rules apply modifications to a finite set of
words, enabling the generation of an infinite set of sentences. The design rules define the language –
or design space – to which these sentences – or designs – belong.
14
L-systems L-systems extend formal grammars for the generation of geometries. In 1968, Linden-
mayer introduces L-systems as a type of formal grammars to algorithmically describe the growth of
plants [10], [85]. A set of rules applies to a string of characters that come from an alphabet. The in-
terpretation of the string generates the corresponding shape in the manner of turtle graphics [86]. The
rules are applied iteratively to generate different levels of growth of the shape. The shape depends
on the starting string axiom and the different parameters in the case of parametric rules. L-systems
can also describe the generation of patterns such as fractals. L-systems can apply to architectural
and structural design as parametrisation strategies for the generation of designs. The combination
of L-systems and genetic algorithms provides a means for topology optimisation for the search for
statics-optimised designs [87], [88].
Shape grammars Shape grammar does not only modify a chain of characters, as formal grammars
do but a geometrical design and found many applications for exploration and design of general geome-
tries. Stiny and Gips introduce in 1971 shape grammars for the generation of shapes in painting and
sculpting [8]. A classification can be found in [89]. Subsequently, shape grammars found a large set of
applications in many fields of design, architecture, and engineering [14], [84], [90].
Architectural grammars Architectural grammars include a variety of applications, such as Palladian
villas [91], Frank Lloyd Wright’s prairie houses [92], Queen Anne Houses [93], Yingazo Fashi Chinese
buildings [94] or Siza’s houses in Malagueira [95]. The grammar decodes each architectural style
or building typology, formalised and structured into a set of rules to generate multiple other designs
that share the same characteristics. Design grammars also apply at other scales than the building.
Urban shape grammars tackle the district, like the Medina of Marrakesh [96], or the city, like Praia
in Cabo Verde [97]. Product shape grammars tackle specific styles of chairs [98], coffeemakers [99]
cars [100] or tableware [101], for instance, allowing mass customisation. An architectural grammar for
housing rehabilitation takes into account the varied information like new usage, among others, instead
of decoding a design style [102].
Functional, structural and force grammars Even though architectural grammars focus on the ar-
rangement of shapes, grammars can include functional aspects other than geometry. Shape grammars
evolve into functional and structural grammars to include non-geometrical data related to structures.
This data can include structural-behaviour and construction-technology requirements. Engineering ap-
plications include houses [13], towers [103], [104], halls [105], bridges, [106], trusses [41] or geodesic
domes [15], [16]. Some applications are specific to a fabrication technology, like CNC machines [107],
[108], instead of a structural system. Force grammars describe the organisation between forces, as
opposed to the one between spaces or elements, which can later materialise into a structure. Force
grammars for graphic statics are introduced by Lee et al. and applied to 2D edge networks [109] and
3D cell decompositions [110].
Mesh and graph grammars The exploration of the topology of patterns and meshes resulted in the
development of several operations, sometimes also framed as grammar rules, which are not specific to
architecture and structures, which abstract a design and generalises the range of applications. Consid-
ering [polygon] meshes as hyper-graphs that can contain higher-dimensional topological entities known
as hyper-edges (more commonly known as faces & cells), Mesh Grammars are also considered Graph
Grammars. The extensive range of applications of meshes sparked the design of very different sets of
rules tuned for specific objectives, mainly for aesthetic purposes. Hansmeyer and Dillenburger [111]
introduce a mesh grammar following a formal grammar approach that modifies the density and the
geometry of meshes to generate highly-detailed ornamental shapes. This grammar focuses on shape.
Indeed the pattern and its singularities are not modified. In computer graphics, quad-mesh grammars
modify the topology to improve the regularity of dense and unstructured meshes [112]–[115]. This
regularity for modelling and representation concerns both geometry and topology. These quad-mesh
grammars consist of a set of local rules that preserve the quad-mesh constraint, unlike other grammars
for triangulated meshes. Another family of quad-mesh grammars does not consist of rules, but differ-
ent patterns [116]–[118]. The patterns feature different singularities to patch N-sided polygons with a
15
prescribed number of subdivisions on each side. Combining these patterns allows for completing the
mesh of different shapes. These rules apply to meshes for the field of computer graphics, animation
and rendering. These rules have little use in the context of architectural and structural design, where
design does not start with a dense unstructured mesh and where many more objectives come into
play. The Conway operators constitute a grammar developed by John H. Conway for the description of
polyhedra [119]. Conway operators can be applied to translate one tessellation into another, already
applied for optimising space frame structures [120], [121]. Indeed, tessellations present different struc-
tural and fabrication properties worth investigating [122]–[124]. The design of a pattern is a trade-off
between topological regularity and irregularity. Oval et al. [125] provide a grammar tailored for the
topological design of vertex singularities in quad meshes. Heisserman [126] presents a grammar on
the boundary of solids represented by meshes for modelling architectural spaces or volumetric objects.
Modifications of the topology result from modifications of the geometry. The topology is updated when
a movement modifies the adjacency or creates an overlap. This grammar explores the topology of the
shape, the underlying mesh being only a representation for computation.
4.2.2 Navigation strategies
Of course, such processes can be coupled with aggregate performance evaluators, but incorporating
disaggregated performance gradients with respect to the discrete design moves (graph-rewriting ac-
tions) would require much more sophisticated mathematics than multi-variate calculus. For the sake of
brevity, we shall restrict our attention to the use of grammatical approaches for exploration (and rewrit-
ing of) design spaces as a matter of itemisation, i.e. itemising a wide range of distinct design options.
We can easily observe that the act of itemisation does not require a notion of performance to guide the
process. It is mainly the definition of the graph-rewriting actions (the proposed graph grammar) that
defines how a design space can change to another while itemising distinct design topologies.
Design space creation: granularity, distance, similarity The grammar defines the design space,
its extent and its granularity. The constraints on the design are directly embedded in the definition of the
grammar rules. These constraints can relate to organisation, feasibility, constructibility, etc. Therefore,
it is crucial to design the grammar rules to provide the desired level of granularity. Low-level rules
apply atomic changes and allow a comprehensive description of design variations at the expense of
applying many rules to obtain designs of significant differences. High-level rules make more substantial
changes in a design, allowing to produce of a large variety of designs with a few rules, though missing
intermediary hybrid designs.
This opposition between low-level and high-level rules may be understood as local search versus
global search. The most relevant ones depend on the application and approach as they have different
pros and cons and can be combined for efficient design space search, including creating intermediary-
level rules.
In the example of the quad-mesh grammar by Oval et al. [125], the grammar consists of two rules
only. These rules ever add or delete a strip of quad faces, which are the fundamental object in quad
meshes. As such, the grammar is constrained to generating quad meshes, not generic polygonal
meshes, and all quad meshes. The design space has the aimed level of granularity: all and only
quad meshes. A grammar adding or deleting edges would generate meshes with any combination of
polygonal faces. A grammar modifying other mesh elements would not guarantee the possibility of
obtaining any quad mesh topology.
Performance search: evaluation, navigation, decision Originally, the grammatical generative de-
sign relies on the user to evaluate the suitability of the design and choose which rules to apply. Indeed,
if the grammar is properly designed, all the generated designs should be pertinent. Every design
generated by Stiny’s Palladian-villa grammar [91] should look like a Palladian villa! The designer can
evaluate qualitatively or quantitatively the design options generated during an open-ended search.
However, this combinatorial space can be daunting or under-explored by a designer if not provided
with search aids, mainly when the grammar consists of numerous rules. Therefore, combinatorial
algorithmic search, with different levels of machine-user interaction, from gamification to automation,
including AI and data-driven search, can contribute to grammatical generative design.
16
4.2.3 Illustrative example
We illustrate Grammatical Generative Design with the example of a truss in Figure 9. The proposed
grammar consists of two types of rules: shape rules that modify the geometry of the truss and connec-
tivity rules that modify its topology. Additional rules could consider the materials and the cross-sections
of the truss elements.
A TRUSS GENERATION GRAMMAR
SHAPE RULES
B | BOX(length, height)
Q | QUADRATIC(length, extremity_height, midspan_height)
CONNECTIVITY RULES
V | VERTICAL(number_subdivisions)
D | DIAGONAL(first_direction, second_direction)
S | SWAP()
SWAP()
B(10.0,3.0) - V(4) - D(0,1) - S(0) - B(10.0,2.0) - V(5) - Q(10.0,1.0,2.5)
QUADRATIC
(10.0,1.0,2.5) VERTICAL(5) BOX(10.0, 2.0)
DIAGONAL(0,1)VERTICAL(4)BOX(10.0,3.0)
Figure 9: Illustrating Grammatical Generative Design: combining shape and connectivity rules to evolve a series
of truss designs.
The shape rules consider two truss profiles: a BOX with a straight bottom and top chords or a
QUADRATIC profile with a curved top chord. They have parameters to control the length and the
heights of the truss directly. More complex shapes could be considered though, probably resulting in
designs that are harder to build with little gain in structural efficiency. The BOX rule can be considered
redundant as such designs are exceptional cases of the QUADRATIC designs. However, the designer
may want to limit design exploration to standard BOX profiles, maybe using automated optimisation of
the height parameter, hence the interest in this rule.
The connectivity rules control the number and type of VERTICAL and DIAGONAL bars between
the two chords. The SWAP rule could also be considered redundant, as the DIAGONAL rule has
parameters allowing to choose any or both directions for the DIAGONAL bars. However, the SWAP
rule enables the designer to change their mind about that aspect of the design without starting over
and further navigating the design space by applying more grammar rules.
The last design in Figure 9results from the following sequence of rules: BOX(10.0,3.0) - VERTI-
CAL(4) - DIAGONAL(0,1) - SWAP(0) - BOX(10.0,2.0) - VERTICAL(5) - QUADRATIC(10.0,1.0,2.5). It
could have been produced in an infinite number of rule combinations. The shortest number of combi-
nations, also called design path, consist of three rules only: QUADRATIC(10.0,1.0,2.5) - VERTICAL(5)
- DIAGONAL(1,0). However, the redundancy provided by this choice of grammar allows the designer
to navigate the tree of possibilities without being constrained by early decisions.
While applying a chain of rules, the designer can, in the first instance, visually and qualitatively
assess the suitability of the design based on their intuition and experience. Additionally, this design
process can be enhanced by informing the designer regarding the performance of the design on one
or multiple metrics after an analysis at each step or at key steps only. Here structural analysis can
provide results about the load path for structural efficiency, for example.
The algorithmic search could aim at finding the combination of rules, including their continuous (e.g.
height), discrete (e.g. vertical subdivisions), and Boolean (e.g. orientation of diagonals) parameters,
that minimize a performance objective, like the total load path for structural efficiency, with respecting
to some constraints, like a fixed length and a maximum height to respect integration in a building or
landscape. Other requirements are by design embedded by the generation grammar, like the limited
shape and connectivity complexity for construction simplicity. The redundancy provided by the gram-
mar can make automated exploration more difficult because of the high number of rules that increase
the combinatorial complexity of the design space without enriching it with new designs. Indeed, several
17
chains of rules yield the same designs, also known as a problem of isomorphism between the descrip-
tion (genotype) and the design (phenotype), making algorithmic learning more complex. Therefore, a
sub-grammar limited to the rules QUADRATIC, VERTICAL, and DIAGONAL, which are independent
of each other and make the rules BOX and SWAP redundant, would potentially be more suitable for
automated optimization.
4.3 Gamified Generative Design
In the middle of the spectrum of generative design methods lies the simulation-driven games where the
interactivity of the grammar itemization meets the robust evaluation and assessment of mathematical
derivation. In a broad sense, the gamified generative design processes are design processes where
a gamified environment allows the players to navigate the design space in a structured way while the
alternatives are being evaluated and the corresponding scores are reported back to the player; hence
emblematically dubbed play & score mechanisms.
The first element, play, refers to serious gaming, which is individual or collective activities to make
decisions with high utility based on conflicting objectives and limiting resources [127]. The second
element, score, pertains to the inherent capability of evaluating different alternatives based on their
integral performance as a design and consequences as a decision vector combining many choices.
Thus we require simulation models that represent the states and dynamic behaviours of real-world
phenomena [128], so we can predict the functional performance and consequences of choices (de-
sign moves). To recapitulate, a Gamified Generative Design process is a serious game that has an
incorporated simulation model of the real world, which interactively informs the players about the con-
sequences of their actions. Such games provide exploration and experimentation environments for
the players to take actions, learn, and design through formal simulations based on logic, rules, first
principles, and statistical co-relations [129]; and role-play in [group decision] simulation (games) based
on experience, negotiation, and intuition [130].
The main challenge of the simulation game designer is to (1) formulate the main objective of the
game, (2) capture the essential dynamics of the real phenomena in the simulation engine, and (3)
provide the proper interaction channels so players can understand decision dynamics. These three
aspects have been named: meaning,reality and play by Harteveled [131].
One of the key characteristics of a game is the level of abstraction which has major consequences
on the ease of reasoning about the system [132] and the focus of the reasoning in the game. For
illustration, board games often have a higher level of abstraction than computer games, as their medium
is more constraining. The degree of abstraction and simplification is essential in the modelling and
design processes; however, in the gamified approach, it is specifically consequential. Games provide
an artificial environment for exploration, assessment, and discussion of decisions. A highly abstracted
game has the risk of omitting the complexities of the original problem, and the lowly abstracted game
has the risk of losing the focus on the main problem. Therefore, the gamified environment needs to
have an adequately abstract representation of the complexity of the real problem.
In Figure 3 we have elaborated on different types of complexities present in a built environment
design problem. These complexities demonstrate numerous inputs and outputs with high levels of
inter-dependencies which can impede human cognition in grasping the details of the problem and
taking action accordingly [133]. Simulation games facilitate the decision makers in understanding,
assessing and exploring these complex problem, hence their utilization can also be understood as an
approach to augmenting human cognition.
4.3.1 Existing work
The earliest use of serious games in urban planning and architectural design goes back to Abt [127] and
Sanoff [20]. During the last decades, the use of games in design and decision-making has increased.
Initially, the main driving force was integrating the simulation in an interactive environment, but later,
designers and planners started to focus on the participatory potential of games as well [134]. In this
section, we provide an overview of different gamification approaches and applications in generative
design.
18
Games for Education One of the main advantages of serious games is that they provide a safe envi-
ronment to explore a model and learn about its dynamics. The majority of the research body in serious
gaming aims to exploit this pedagogical potential of games by educating the players through interaction
in simulation-driven game sessions [135]. For instance, researchers have utilized simulation games to
explore the complex inter-dependencies of sustainable development of the built environment [136] to
allow first-year architectural students to learn by interacting with such systems [137]. The foundations
of how games can contribute have been established in Constructivist learning theories since they high-
light interaction, curiosity, and social negotiation as the main driving force of internalizing knowledge
[138]. Furthermore, it is key to note that other games that do not have a primarily educational goal
exploit this potential for familiarizing the players with the topics and establishing a common language
for discussions and negotiations.
First Principle Simulations Beyond the educational case, other applications of serious games, such
as decision-making and planning support, have received relatively little attention [139]. However, these
applications are more central to the gamified generative design approach, as they support evidence-
based navigation of the design space. The most prominent way to provide evidence for the comparison
of alternatives is first principle simulations and evaluations. In these approaches, a formal model that
is rooted in previously established scientific and theoretical work is used to assess the alternatives.
A classic example of this is the original SimCity game [140], in which the authors used Forrester’s
model to replicate urban dynamics. This helped them predict the population increase or decrease
based on education, unemployment and growth rates [141]. More recently, Sanchez focused on the
ecological aspect of urban development through Block’hood: a simulation-driven voxel placement game
that resembles sand-boxes such as Minecraft and Simcity for neighbourhood development [142].
Specific to urban infrastructure, SimPort-MV2 focuses on land allocation in port-planning [143];
MATRICES in the ProRail games focuses on the scheduling and rail infrastructure to predict the conse-
quences of different scenarios [144]; the Train Fever game (2015) utilizes an integrated transportation
and land-use model to predict population and employment flow based on the rail networks [141]. More
recently, there is increasing interest in games that aim to integrate different domains to achieve a more
holistic representation of the urban planning complexities, such as Water-Energy-Food-Land-Climate
Nexus serious game [145] where the authors embed statistical correlation from different domains to
make an integrated inference engine for assessment of scenarios. Another interactive multiplayer game
is the Sustainable Infrastructure Planning Game (SIPG) which focuses on the inter-dependencies of
different sectors in strategic planning exercises [146].
On the architectural scale, Moloney proposes a serious game for integral sustainable design [137].
They provide a set of spatial quality measures based on the rule of thumb and a peer review mecha-
nism that capture the qualitative views of the players as a collective. Savov proposes a block-based
assembly game for facade design with a setup similar to Jenga. They conduct light and structural
analyses to provide live feedback to the players about the performance of the design [147]. Lin sug-
gests integrating VR and BIM to communicate the exquisite requirements of healthcare buildings to the
players [148].
Agent Based Models and Cellular Automata Spatial systems are integrated with various aspects
of human life, many of which can not be represented formally with first principles. This is especially
true in the case of human behaviour within the built environment. In such cases, Agent-Based Models
(ABM) can be constructive, as they allow the game designers to embed complex relationships between
agents and their environment in the simulation modelSee Figure 12 as an example).
A recent example is the work of Raghothama et al., where the authors utilize the SUMO simulation
model for gamifying the operation of the transport system in two cities of Rome and Haifa [149] within
the ProtoWorld framework [150]. At the architectural scale, crowd simulation ABM has been used to
assess the accessibility of spatial configuration in standard, and emergencies [151].
A particular category of the ABM is Cellular Automata (CA), which consists of spatially fixed agents.
These models are explicitly valuable for design and planning as the agents defined in them have a
fixed topology (relation network) that can encapsulate context-specific spatial relations. In an urban
case of flood mitigation policy development, Khoury et al. embed a CA in a serious game to educate
the participants and reach a consensual policy. They utilize the CADDIES open-source framework,
19
which is 2D CA developed explicitly for urban flood modelling [152]. On an architectural scale, CA
has also been used to represent spatial design’s horizontal and vertical constraints to generate spatial
configurations [153].
Beyond evaluation, ABM can also be used for exploration. Epstein very well establishes the foun-
dations of such approaches as the generative social sciences [18]. A recent example is using artificial
agents for exhausting and mapping all possible scenarios in a serious game environment to achieve a
holistic picture of the design space [145]. This approach has also been applied for design configura-
tions in the GoDesign framework [26] where the authors propose using ABM to allocate the program
of requirement in the predefined envelope (see Figure 12 [154].) Similar to the ABM, artificial agents
can be defined in the simulation environment to explore the design space and learn about dynamics of
it under the Reinforcemont Learning framework, for example for configuring architectural spaces [155].
Grammar Based Games Some games utilize explicit grammar formalism to structure the exploration
of the players. Most of these games are focused on the morphological aspect of the built environment.
In the architectural scale, the exploration of complex spatial configuration can be structured as a voxel-
based grammar in [156], or a shape grammar with structural verification routines in [157]. On the
urban scale, rule-based systems have been used extensively to describe the morphological relations
of the city [158]. These rule-based morphological systems are the basis of the procedural generation
of cityscapes [159]. Furthermore, such procedural methods can be used to optimise the urban con-
figuration by integrating assessments to evaluate the alternatives [160]. In a recent example, authors
integrate a prediction model for population density to evaluate the procedurally generated alternative
and guide the exploration [141].
Games for Consensus Serious games can provide the medium for a common language for a struc-
tured discussion and comparison that can potentially facilitate the process of reaching a consensus.
This potential is mainly the result of the interactive environment of the games that allows an engaging
experience and a transparent relationship between decisions and their consequences. The founda-
tions of this potential of games can be traced to social cognition studies where the dynamics of the
individuals’ opinion concerning their interaction with the group is studied [161].
For example, in [162], the authors propose an online game for mass participation of the neighbour-
hood inhabitants to focus on the ”Not In My Backyard” phenomena. Another example is presented
in [151], where the researchers utilize crowd-sourcing for floor plan generation. They provide an im-
plicit peer review mechanism by allowing players to access the design of other players [151]. Similarly,
Fumarola and Verbraecksuggest suggest constructing a decision tree during the game as the player
interactively makes decisions. Providing additional abilities to navigate this decision tree allows the
player to explore multiple what-if scenarios, understand their relations, and draw a conclusion about
the potential trade-offs [163].
4.3.2 Navigation strategies
The simulation games sit in-between the mathematical derivation and grammatical itemization mainly
as their navigation of the design space is a merger of the two ends of the spectrum. In this segment,
we first elaborate on the similarities and differences of the gamified approach with each of the math-
ematical and grammatical approaches and then explain how they make it a unique methodology in
generative design.
The common thread between grammatical and gamified approaches is twofold. Firstly, they both
limit the navigation to a rule-set. In simulation games, the players’ actions are predefined based on the
game rules. In the grammatical approach, the rules define what the possible design changes that can
be made are. Often, the game rules are more forgiving than grammar structures as they provide for
the players’ exploration. However, in both approaches, decisions/actions are defined based on the rule
sets and the grammar structures possible changes that can be made in the decisions/actions. Thus,
the grammar creates pathways in the design space that make it navigable for human agents. This
brings us to the second similarity between gamified and grammatical approaches: they both rely on
the human agency as the principal controller of the navigation process. This means that the decision
to apply a rule primary lies in the control of the human agents. In some examples, game actions can
20
be delegated to artificial agents or might be based on the system’s recommendation, but the human
player remains the authority in the navigation process.
On the other hand, gamified and mathematical approaches are similar as they both incorporate
evaluation mechanisms that can provide a basis for comparison of them. In simulation games, these
evaluations are generally introduced as indicators that allow the players to have a basis for their com-
parison. Performing the evaluation relies on formal or role-play simulations that essentially project
the consequences of actions and decisions to the identified indicators. The formal simulations can
be embedded in the game as analogue mechanisms in card games or digital simulations similar to
objective functions in the mathematical approach. Role-play simulations are normally introduced to
represent the societal and organizational complexities of the problem. Thus they are closely related to
the consensus-building capabilities of games. In all cases, the overlap of the gamified and mathemati-
cal approach is that given two decisions, it is possible to differentiate them by evaluating indicators. In
the mathematical approach, this differentiation is performed formally in the gradient-based approaches
that also help the process identify the new potential decisions. However, in the gamified approach, the
comparison of the alternatives and the specification of the new decisions are performed purely by the
human agent.
4.3.3 Illustrative example
In this segment, we elaborate on the EquiCity project’s details to dive deeper into the potential of spatial
simulation-driven games as generative design methodologies. EquiCity is a spatial game for planning
the redevelopment of a neighbourhood in Delft, the Netherlands. The existing site was a historical
factory that has been an economic propellant of Delft in the last century. As the factory has stopped
working, the municipality is aiming to redevelop the area with a mixture of functionalities such as res-
idential, commercial, cultural and public spaces. The main objectives of the redevelopment consist of
three main categories: (1) environmental such as light and visibility; (2) social such as accessibility;
and (3) economic concerning the intervention extent. The main planning measures pertain to allo-
cating different functions in various sites within the district. As the game is a multi-player game, the
participants from various backgrounds were involved in deciding about this district.
On the methodological level, the problem has been formulated based on the introduced framework
in Section 3. As indicated in Figure 10, the decisions of the players were structured as the interest
matrix X(t)
n×m×o. Players would specify their decisions individually at the beginning of each round.
Then the game engine would gather all of the decisions, construct the interest matrix, and perform
Opinion Pooling, and Proportional Fitting to achieve a consensual decision on the allocation of functions
V(t)
n×o. As the decisions are on the planning level, the spatial decision on the configuration of each site
is delegated to the agent-based massing process, which produces the prospective massing {k(t)
j}. In
the next step, the game engine evaluates the corresponding massing {k(t)
j}based on the predefined
economic, social, and environmental objective functions. Finally, individual utility values are combined
through Multi-Criteria Decision Analysis (NMCDA) to produce the group score, individual score, and
achievement badges. At this point, one iteration of the game is complete; at this point, players can
explore their scores, previous decisions, and extra analytical information to be able to negotiate for the
next round of decisions.
The gaming platform of EquiCity is a web-based interface that players can individually access to
participate in the game (see Figure 11). Authentications are embedded to ensure that only dedicated
players can access the decision-making interface, while the game-play and the analytical information
are openly accessible for everyone else to follow the flow of the game.
EquiCity is inherently a multiplayer game that aims to structure the negotiations of the stakehold-
ers by providing feedback on the performance of their decisions. The stakeholders in EquiCity are
represented with predefined roles that are described by the corresponding Interest Xand Control C
matrices. The predefined interest X(0) of the stakeholders identifies their individual agenda and how
their individual score is calculated. The control matrix of the stakeholders specifies how influential they
are in each site-function combination. With these two matrices, we can conduct the Opinion Pooling
and Iterative Proportional Fitting process to achieve a consensual decision about the district. However,
this decision might not perform the best for the environmental and social indicators. Therefore, the
stakeholders are motivated to deviate from their initial interest X(0) and produce new decisions X(t)
21
Figure 10: Illustrated Formulation of a the gamification problem in EquiCity game as a combination of a MCDA
problem, a generative massing problem, a consensus building problem, and the creation of extra scores for
encouragement of competitive or cooperative play styles.
to reach a better performance. The translation of this approach in the gameplay is an iterative process
where the stakeholders will negotiate with their peers about their agenda and performance of the area
as a whole and try to form new decisions. At the end of each round, after submitting their decision, they
will get the evaluation of their decision which is the basis for comparison with the previous decision and
is potentially constructive in guiding them toward a new decision.
EquiCity, embodies the principles of gamified generative design as it (1) structures the design space
by defining decision variables (planning measures); establishes a robust evaluation system for differ-
entiating the alternatives based on evidence from the simulations; (3) incorporates the participatory
nature of the problem in the game and facilitates them in reaching consensus; and finally, (4) provides
an interactive environment where the relevant information is accessible to the stakeholder to support
their decision-making process (see Figure 11). Nevertheless, EquiCity is just a detailed example of this
approach. As it has been established in the previous subsections, the gamified generative design can
(1) have various sets of decision variables to address different application areas in the spatial domain;
(2) benefit from various evaluation methods such as first principles, ABM, statistical inference, and
even participatory evaluation such as peer review; (3) incorporate various levels multi-actor complexity
to undertake the societal and organizational challenges of the spatial problem; and finally (4) utilize
various technological infrastructure such as web-based interfaces, game engines, VR, etc. to provide
an interactive environment for the participants.
4.4 Generative Models for Data-Driven Generative
Here we juxtapose the so-called Generative Models in Artificial Intelligence with the three archetypical
generative design processes and reflect on their potential for augmenting generative design processes.
Before proceeding, it is necessary to note that an ambiguous notion of performance can lead to mis-
understandings and unrealistic expectations from such models.
The notion of performance as in the realistic look of generated designs is important in applications
related to the entertainment industries (computer graphics, especially in generative arts, game-level
design and procedural content generation); however, in architectural design, we need to work with a
multi-faceted and much more constrained notion of performance. Producing valid designs that meet
spatial constraints is already a significant challenge in most architectural applications, let alone gener-
ating designs of high-quality w.r.t. multiple quality criteria. Thus, instead of focusing on the generative
capabilities of generative models for generating realistic pictures in the styles learnt from a corpus of
humans (such as the 2D or 3D images typically generated with Generative Adversarial Networks or
DALL.E). Regardless of whether it is a utopian or dystopian future for AI models to generate architec-
22
Figure 11: Screenshots of two iterations of game-play in the EquiCity game, the case of the redevelopment of a
former factory into an urban district featuring a mix of residential, commercial, cultural spaces in addition to
communal/public spaces. The overlaid screenshots show the information on individual and collective scores
(right) and the control, interest, and power difference matrices (left) shown to the players.
Figure 12: Top rows show the application of the Technique for Oder of Preference by Similarity to Ideal Solution
(TOPSIS) for solving the massing configuration problem, the bottom row shows the application of the method of
Multi-Actor Multi-Attribute Gradient Driven Mass Aggregation (MAGMA) Method, a participatory zoning algorithm
based on Fuzzy Aggregation in Configraphics [154], [164]
tural designs, the technological possibility for generating a design is already available.
The more critical question of interest regards the capability of AI for solving hard problems
of performance-based generative design[165] where mapping or navigating the design space is
intractable due to the difficulty of formulating the associations of choices and consequences, see a
comprehensive review of the applications of deep generative models in design[166].
One particular class of models of interest are thus the models that can help make these high-
dimensional “bipartite design spaces” intuitively understandable and tractable for humans; a particularly
relevant example is a one-layer deep Bayesian Belief Network[167] that can be trained to work in both
directions, i.e. both from the design space to the performance space and from the performance space
to the design space. Considering the difficulty of simulating some aspects of performance related to
subjective and cognitive matters such as ergonomics, the need for building function approximators ca-
pable of associating performance levels with spatial configurations can be fulfilled with trainable neural
networks, provided a systematic data collection campaign is conducted; see a connective perspective
on Spatial Computing for Design [168]. In general, however, there are two major types of problems that
can be meaningfully addressed with AI: mapping or associating low-dimensional performance data
23
points with high-dimensional input design variables (encoding spatial configurations) and the inverse
or dual problem of navigating this bipartite space in the opposite direction (see Figure2). Two major
lines of work can be identified as relevant in this regard: one that uses neural networks as function ap-
proximators for regression aimed at Interpolation and Extrapolation between data points; and another
that is concerned with Manifold Learning, Dimensionality Reduction (Encoding), Signal Reconstruction
(Decoding), and Sampling (as in Design of Experiments), see a recent example for making an explain-
able encoder-decoder network architecture [169]. A particular class of Generative Models with special
relevance to the dual problems of mapping and navigation are Probabilistic Graphical Models (PGM),
which are of two general types [170][171]: Bayesian (Directed Acyclic Graphs) and Markovian (Bi-
Directed or Undirected). Examples of such models include Bayesian Belief Networks[empty citation],
Restricted Boltzmann Machines[172], Flow-Based Models, Markov Random Fields, and Diffusion Mod-
els. In short, generative models equipped with AI can augment the three types of generative design
processes where there is a complication related to the difficulty of navigation or mapping in high-
dimensional bipartite design spaces. However, the use of Generative Models for content generation,
no matter how interesting for the entertainment industry, does not introduce a new type of Generative
Design relevant to AEC, unless one adopts a reductionist approach to GD as design automation (au-
tomating the task of a human, not necessarily supposed to produce better results than those made by
humans).
5 Conclusion
Here we look back at the rhetorical questions posed at the beginning and provide succinct answers.
Cedric Price in 1966 expressed the provocation ”Technology is the answer, but what was the ques-
tion?”. When it comes to the application of digital technologies in AEC, sadly, theis rhetorical question
seems to be relevant more than ever before. The AEC industry as a whole has yet to learn from other
Sciences of the Artificial for forming a transparent culture of listing and formulating problems. Before
claiming to have invented yet another solution, one needs to ask rigorously what the problem is. The
most pressing confusion about “the problem of the design problem” seems to be about the notion
of automated design. We hope to have presented compelling reasons to think beyond automation,
optimization, or even problem-solving by presenting the essentially different problem settings where
the purpose of the design process can be the gradual and explainable derivation of design, participa-
tion in design for the sake of ensuring inclusivity and equity, or mass-customization and formation of
rich design languages capable of forming diverse designs, not only for the sake of comprehensively
reaching better performance levels but also for enriching design cultures as integral parts of human
cultures. In this sense, we hope to have adequately shown that it is time to think beyond the cliched
Wicked-Problem mindset and go about formulating design problems rigorously. The following para-
graphs summarise the moral of the paradigm of Generative Design.
Methodological Commonalities As we have illustrated in the three plus one methodologies, Gen-
erative Design is not an agglomeration of isolated and distinct approaches but a coherent spectrum of
methodologies that surround the principles of formulating the problem, structuring the design space,
and navigating the design space based on evidence, knowledge, rules (design principles), or policies
(control or game-playing strategies). Despite the differences in the three approaches, there are com-
monalities in these methodologies, most importantly, they all have explicit discrete decision variables
on which the design space and an explicit formulation of the problem are based. This discreteness
allows the design solutions to be countable, whether finite in the case of mathematical optimization
or infinite in the grammatical approach. The enumerability of the design space is a prerequisite of a
structured navigation strategy.
The three methodologies have an explicit and explainable navigation strategy. In the grammati-
cal approach, the grammar describes the applicability of the rules alternatives, which in total exhibits
pathways through the idomatic expansion of the design space. In the gamified approach, the naviga-
tion pathways are looser to allow for the creative exploration of the players. Nevertheless, the scoring
mechanism facilitates the players to learn about the dynamics of the game and reach a consensus
collectively. In the mathematical approach, navigation is delegated to the gradient-based search to
24
have the utmost objectivity in the design derivation process.
The last common theme is explicit validation and evaluation which appears in all of the methods.
In the mathematical approach, constraints ensure the validation and objective function will be an eval-
uation of the performance of the design that we aim to maximise. In the grammatical approach, the
validation is ensured by the limitations of the grammar and the evaluation is based on the subjective
opinion of the designer. In the gamified approach, the game rules ensure the validity of the design and
the feedback provided by the simulations enables evaluation.
Customization vs Automation More unfortunate than the prevailing false dichotomy of automation
and customization is the preconception that automation is integral to generative design. The map-
ping of the generative design methodologies in this paper clearly illustrates how generative design
ultimately provides for customization of solutions; customization through itemization in the rule-based
grammatical approach; customization through the informed exploration to reach consensus in gami-
fied approaches; and customization for the satisfaction of the specific needs of a particular problem
in mathematical approach. The generative design paradigm aims to formulate the problem, structure
the design space, and devise navigation strategies to facilitate the customization of the design to the
requirements and preferences of the future inhabitants.
Future Research We can envisage two main directions for future research in Generative Design:
development of methodologies and rigorous evaluation of methodologies. The existing methodologies
under the umbrella of generative design are rooted in disparate fields and subjects, thus bringing
incompatible terminologies and notations. As an example, the term decision variable in decision theory
is equivalent to control vector in control theory, planning measures in urban planning, and design
parameters in architectural design. This lack of a common terminology fundamentally impedes the
formation of a rigorous discourse that is necessary for any structured discipline. In addition, much
work is needed for meaningfully mapping mathematical methods and connecting them to appropriate
applications in AEC.
Currently, various methodologies are borrowed and applied based on the individual expertise of
researchers and practitioners. Although enriching, it is unclear how effective each method has been
for the corresponding problem. In other words, it is impossible to benchmark, compare and validate the
application of methods in problem categories. This is because a collective culture of sharply defining,
formulating, and benchmarking problems is practically non-existent in the AEC disciplines. For this, the
generative design paradigm requires a unified way of evaluating the methodologies based on explicitly
mathematical design frameworks. Specifically, we need to evaluate the formulated design space based
on size, evaluate the problem-solving methods based on their informed navigation potential for a human
designer, explainability, reach/coverage, and reproducibility, as well as the justifiability of solutions.
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