OptiGAN: Topological Optimization in Design Form-Finding With Conditional GANs

Abstract

With the rapid development of computers and technology in the 20th century, the topological optimisation (TO) method has spread worldwide in various fields. This novel structural optimisation approach has been applied in many disciplines, including architectural form-finding. Especially Bi-directional Evolutionary Structural Optimisation (BESO), which was proposed in the 1990s, is widely used by thousands of engineers and architects worldwide to design innovative and iconic buildings. To integrate topological optimisation with artificial intelligence (AI) algorithms and to leverage its power to improve the diversity and efficiency of the BESO topological optimisation method, this research explores a non-iterative approach to accelerate the topology optimisation process of structures in architectural form-finding via conditional generative adversarial networks (GANs), which is named as OptiGAN. Trained with topological optimisation results generated through Ameba software, OptiGAN is able to predict a wide range of optimised architectural and structural designs under defined conditions.

Full text

OPTIGAN: TOPOLOGICAL OPTIMISATION IN DESIGN FORM-
FINDING WITH CONDITIONAL GANS
XUYOU YANG1, DING WEN BAO2,* , XIN YAN 3 and YUCHENG
ZHAO4
11DesignLab
2,*School of Architecture and Urban Design & Centre for Innovative
Structures and Materials, RMIT University
3Tsinghua University
4ByteDance
1xuyou.yang.92@gmail.com, 0000-0002-6294-378X
2, *nic.bao@rmit.edu.au, 0000-0003-1395-8747
3yanxin13@mails.ucas.ac.cn, 0000-0002-5033-3597
4zhaoyucheng.joe@bytedance.com, 0000-0002-9291-2554
Abstract. With the rapid development of computers and technology
in the 20th century, the topological optimisation (TO) method has
spread worldwide in various fields. This novel structural optimisation
approach has been applied in many disciplines, including architectural
form-finding. Especially Bi-directional Evolutionary Structural
Optimisation (BESO), which was proposed in the 1990s, is widely used
by thousands of engineers and architects worldwide to design
innovative and iconic buildings. To integrate topological optimisation
with artificial intelligence (AI) algorithms and to leverage its power to
improve the diversity and efficiency of the BESO topological
optimisation method, this research explores a non-iterative approach to
accelerate the topology optimisation process of structures in
architectural form-finding via conditional generative adversarial
networks (GANs), which is named as OptiGAN. Trained with
topological optimisation results generated through Ameba software,
OptiGAN is able to predict a wide range of optimised architectural and
structural designs under defined conditions.
Keywords. BESO (Bi-directional Evolutionary Structural
Optimisation); Artificial Intelligence; Deep Learning; Topological
Optimisation; Form-Finding; GAN (Generative Adversarial
Networks); SDG 12; SDG 9.
1. Introduction
Structural optimisation, including topology optimisation, plays a significant role in
architectural design. It can increase the performance of structures and the efficiency of
material use and thus reduce material waste and carbon impact in the fabrication and
construction process. By integrating topology optimisation with artificial intelligence
POST-CARBON, Proceedings of the 27th International Conference of the Association for Computer-
Aided Architectural Design Research in Asia (CAADRIA) 2022, Volume 1, 121-130. © 2022 and
published by the Association for Computer-Aided Architectural Design Research in Asia (CAADRIA),
Hong Kong.

X. YANG ET AL.
(AI) for more efficient use of materials in the industry, it seeks to help achieve the
United Nations Sustainable Development Goal 12: Ensure sustainable consumption
and production patterns (United Nations, 2015).
1.1. STRUCTURAL OPTIMISATION
Structural optimisations aim to achieve the best structural performance while meeting
the requirement of various constraints. Over the past three decades, high-speed
computers and rapid improvements in algorithms have been used to develop better
structural optimisation solutions by a number of engineering researchers.
1.1.1. Topology Optimisation
Topology optimisation is one of the most popular optimal structural design methods
for discrete structures, such as trusses and frames. It is developed to search for the
optimal spatial order and connectivity of the bars. Topology optimisation of continuum
structures is to find optimal designs by determining cavities' best locations and
geometries in the design domains.
Topology optimisation can be readily used to perform shape optimisation by simply
restricting the structural modification to the existing boundaries (Huang & Xie, 2010).
In the field of topology optimisation, there are several notable methods based on finite
element analysis (FEA) developed, such as the homogenisation method (Bendsøe &
Kikuchi, 1988), the solid isotropic material with penalisation (SIMP) method (Bendsøe
& Sigmund, 1999), the evolutionary structural optimisation (ESO) (Xie & Steven,
1993), the bi-directional evolutionary structural optimisation (BESO) (Huang & Xie,
2010; Huang et al., 2007) and the level-set method (LSM) (Wang et al., 2003). In this
paper, bi-directional evolutionary structural optimisation (BESO) proposed by Huang
and Xie (2010) is adopted to develop a new integrated topology optimisation algorithm
(Figure 1).
Figure 1 Bi-directional evolutionary structural optimisation (BESO) result
1.1.2. Bi-directional Evolutionary Structural Optimisation (BESO)
Bi-directional evolutionary structural optimisation (BESO) is the emerging technology
that is an extension of evolutionary structural optimisation (ESO) developed by Xie
and Steven in 1992 (Xie & Steven, 1993). Both ESO and BESO algorithms are based
on finite element analysis (FEA) for topology optimisation of continuum structures.
BESO algorithm aims to find the solution with the highest structural performance
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FORM-FINDING WITH CONDITIONAL GANS
under certain material limitations by removing or adding material elements step by step
(Bao et al., 2020). The ESO method also inspires the Extended ESO method, widely
used in architecture design projects, such as the Akutagawa River Side Project in Japan
by Ohmori and Qatar National Convention Centre by Arata Isozaki, to generate an
optimised model with not only high structural performance but also some different
characteristics to meet more functional requirements or aesthetic preferences. In the
past few years, Mike Xie and his team have modified many detailed control strategies
for topology optimisation in architectural design and development during the process
(Yan et al., 2021).
1.1.3. Ameba Software
Because of the benefit of form-finding through topology optimisation and Bi-
directional evolutionary structural optimisation (BESO), more and more designers and
architects seek to use topology optimisation methods to design buildings and furniture.
However, due to the complexity and slow speed to directly use the algorithm for
architectural design, a new Rhinoceros plug-in named Ameba, a topology optimisation
tool based on the BESO method and FEniCS open-source computing platform (Zhou
et al., 2018), has been developed. More and more architects and designers have gained
opportunities to use this intelligent method to work with computers interactively to
create innovative, efficient, and organic architectural forms using Ameba. In this work,
the authors use it as the topology optimisation tool to form the dataset for training
generative adversarial networks to assist and investigate the research.
1.2. GENERATIVE ADVERSARIAL NETWORK AND ITS APPLICATION
IN TOPOLOGY OPTIMISATION
Allowed by the development of deep learning algorithms and fast-growing
computational power, artificial neural networks, including generative adversarial
networks (GANs), have been increasingly used in architectural and structural
explorations such as topology optimisation in the design process.
1.2.1. Generative Adversarial Networks
A generative adversarial network (GAN) is a particular artificial neural network that
learns from a collection of examples and their probability distribution. It is then able to
generate more examples from the estimated probability distribution (Goodfellow et al.,
2020). A typical GAN often consists of a generator that defines a prior probability
distribution P(z) based on a vector z as input and a discriminator which examines
whether data x is real (sampled from the training examples) or fake (sampled from the
output of the generator).
GANs can further be extended to conditional models (cGANs) where both the
generator and discriminator are conditioned on extra information y as input (Mirza &
Osindero, 2014). Besides examining whether x is real or fake, the discriminator of a
cGAN also evaluates whether it matches the condition y. For example, when using
cGANs to solve topology optimisation problems, x can be the expected optimisation
results, given the corresponding boundary and load conditions of y.
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1.2.2. Topology Optimisation via Deep Learning
In recent years, there has already been some research into solving topology
optimisation problems through artificial neural networks, especially GANs. For
example, the TopologyGAN (Nie et al., 2021) is developed on a cGAN, whose
generator combines a U-Net (Ronneberger et al., 2015) and ResNet (He et al., 2016).
It takes displacement, load boundary conditions and target volume fraction augmented
with dense initial fields computed over the unoptimised domain as the input to the
model. By doing so, this method vastly improves the accuracy of predicting topology
optimisation results compared to some baseline models. Another research by Yu et al.
(2019) transforms the boundary conditions into multi-channel images as input to a
convolutional-neural-network-based encoder to generate low-resolution topology
optimisation results. It inputs the low-resolution results into a GAN to produce final
results in high resolution. Differently, the proposed method utilises cGANs directly. It
requires very brief input to keep the models easy to operate and thus broaden the
potential range of users with or without professional structural knowledge.
1.3. PROPOSED METHOD
This research suggests an approach to accelerate the topology optimisation process of
structures in architectural form-finding by replacing iterative calculation procedures
with an end-to-end algorithm via conditional generative adversarial networks. This
method is named OptiGAN by the authors as the ultimate goal is to generate topology
optimisation results efficiently and accurately. Trained with a small number of
topological optimisation results generated with Ameba software, the proposed method
is able to predict a wide range of optimised two-dimensional structural forms under
defined conditions.
2. Methods
To achieve the research target, a coarse-to-fine network of cGANs is developed and
trained with a dataset collected by the authors.
2.1. DATA COLLECTION AND PRE-PROCESSING
To train OptiGAN, an original dataset of topology optimisation results is collected with
Ameba software. In detail, the material is kept as steel during the data generating
process, and the volume fraction is set to 0.5 consistently. The parameters that can vary
from case to case are design domain, fixing edge and load conditions, which are the
very parameters to use as input parameters of OptiGAN. In practice, in the Ameba
script, the fixing edge of a square design domain is always set to the left edge as fixing
at other edges are seen as the same as rotating left-edge-fixing conditions in the later
data augmentation process when training the cGAN models. In the first stage of the
research, a total number of 1385 optimisation results are included in the dataset.
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FORM-FINDING WITH CONDITIONAL GANS
During the pre-processing, the input parameters are translated into a three-channel
input image at 256 * 256 pixels in size. By doing so, the model is provided with two-
dimensional spatial clues. Thus, the difficulty for training the model to predict two-
dimensional results can be reduced compared to using merely numerical inputs directly
without spatial suggestions. Specifically, as Figure 2 demonstrates, the design domain
and fixing edge are expressed in all three channels by assigning the value of 0 to the
corresponding pixels representing the geometries. Load locations and load unit vectors
projected onto the X and Y axis are documented in the first and second channel
respectively in values remapped into a range between 0 and 255. Values of the other
pixels are assigned 255 in all three channels by default.
Figure 2 Translated input image of 1000*1000 mm design domain, fixing at the left edge, load one
at (900, 1000) location in 45° angle, load two at (1000, 700) location in 180° angle. The values of
the rest pixels are all 255.
2.2. OPTIGAN ARCHITECTURE
Unlike the previous researches mentioned in section 1.2.2, there are no dense initial
fields, low-resolution results, or any other inputs than design domain, fixing edge and
load conditions for the OptiGAN generator. Keeping the inputs simple can make it
potentially as easy as adjusting a few number sliders for the OptiGAN users to operate.
At the same time, it dramatically increases the difficulty for the generator to speculate
the results by providing less input information. To respond to the conflict, OptiGAN
adopts a coarse-to-fine network architecture.
Specifically, as Figure 3 demonstrates, there are two generators and discriminators
in the network. The initial input of conditions in the form of translated input images is
first fed to the coarse generator, which then predicts a rough output examined by the
coarse discriminator. Then the rough output together with the initial conditions are
input to the fine generator, which outputs the final results. Although compared to
conventional cGANs, the input of OptiGAN consists only the conditions y alone
without vector z, the networks can also learn from only the conditions. This choice of
input is also being suggested in Pix2Pix (Isola et al., 2017), one of the most successful
image-to-image translation models.
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Figure 3 The architecture of OptiGAN
2.2.1. Generators and Discriminators
Both the coarse and fine generators used in OptiGAN are U-Net (Ronneberger et al.,
2015), which has a mirrored encoder-decoder network architecture with skip
connections between symmetrical layers. It can work effectively with very few training
data. Meanwhile, both discriminators are PatchGAN (Isola et al., 2017), which focuses
on 156 * 156 patches as through testing, such patch size provides the best outcome in
the tasks.
2.2.2. Objective
The objective that OptiGAN tries to optimise can be expressed as equation (1). It
consists of two parts, the cGAN loss (equation (2)) and L1 loss (equation (3)). Besides
generating images that look real, as the other goal is to eventually achieve results as
close to the BESO optimisation outcomes as possible, L1 loss is added to the total loss
with a considerable weight of λ to force the output to be close to the ground truth. L1
loss is chosen out of L2 loss because it encourages less blurring effect of images
compared to L2 loss. In the experiments, the weight is set to 125 (λ = 125) to emphasise
the importance of L1 loss in this particular task.
=arg min
max
ℒ (,)+ ℒ1 () (1)
ℒcGAN(,) = ,[log (,)] +[log(1− (,())] (2)
ℒ1()=,[‖ − ()‖1] (3)
2.3. TRAINING OPTIGAN
OptiGAN is trained in two steps: the coarse generator and discriminator are trained
first, after which the fine generator and discriminator are trained. Both parts of the
network are trained for 200 epochs to achieve the demonstrated results. Following the
conventions, instead of minimising log(1− (,()), the trainings maximise
log((,()) to avoid saturating log(1− (,()) in the early training stage.
The initial learning rate is 0.0005 decaying to 0 in the last 100 epochs and the Adam
optimiser is used. Figure 4 shows the history of the cGAN loss and L1 loss during the
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FORM-FINDING WITH CONDITIONAL GANS
two training procedures.
Figure 4 The cGAN loss and L1 loss during the coarse training and fine training
The coarse generator and fine generator take different types of input and are paired
with different discriminators as introduced in section 2.1.1, so it is not considered
practical to compare the absolute value of the corresponding cGAN losses of the coarse
and fine models. Even though, the cGAN loss improves from the beginning to the last
epoch in both cases, as shown in Figure 4. In contrast, for both generators, the L1 loss
is calculated according to the same ground truth, and it can be discovered that the fine
generator further decreases the L1 loss based on the coarse generator, which indicates
that the fine generator is able to further improve the accuracy of predicted topology
optimisation results by OptiGAN.
3. Results
During the training process, the L1 loss of OptiGAN reduced from over 80 to less than
20. More importantly and precisely, the pixel-wise accuracy is used to evaluate the
performance of the models. It is equal to the percentage of accurate pixels in a
prediction out of the total pixels of that image. Tested with 150 randomly selected
pieces of data different from the training set, the average pixel-wise accuracy of
OptiGAN is able to achieve 83.15%. Figure 5 demonstrates some of the testing results
in different load conditions with pixel-wise differences between the predictions and
ground truth visualised for each case.
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Figure 5 Some prediction results of OptiGAN with ground truth and pixel-wise difference
4. Discussion
OptiGAN demonstrates the ability of a novel approach and its application in
architectural and structural form-finding. It is the extension of the SwarmBESO (multi-
agent-based topology optimisation) method proposed by Bao & Yan in 2020 (Bao et
al., 2021) to improve the diversity of the topological optimisation generative method.
It has the potential to significantly help architects and engineers save material and
produce more efficient structural layouts and building envelopes. It is valuable to
integrate two intelligent computational design methods, deep learning and topology
optimisation, for designers in the conceptual design phases.
However, the research is in a rudimentary phase and is temporarily constrained in
a range of two-dimensional solutions. Although set as an input variable, the design
domain in the current dataset includes only square geometries, despite that it can
perform well in this geometric range, as demonstrated in the testing results. To truly
diversify the spectrum of results and further increase the accuracy, it is very crucial that
OptiGAN must be trained with much more data of various design domains and load
conditions. Future line of research also includes further equipping the model with the
ability to solve three-dimensional topology optimisation problems.
5. Conclusion
This research develops OptiGAN, a non-iterative method to accelerate the topology
optimisation process of structures in architectural form-finding via conditional
generative adversarial networks with high accuracy. It demonstrates the process of
integrating topology optimisation and generative adversarial networks to establish an
artificial intelligence (AI) based structural optimisation technique. This new
methodology holds great potential for practical application in architecture and
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engineering fields. It increases the diversity of outcome of the topology optimisation
generative design such as the application of shell (Figure 6).
Figure 6 Diverse BESO results of shell optimisation
Acknowledgements
We thank Nanjing Ameba Engineering Structure Optimization Research Institute for
providing educational version of Ameba software to support our research. (For Ameba,
see Ameba.xieym.com)
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