ToposGAN: a deep-learning model for optimising structural topologies

Journal of Engineering Design

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Title A deep-learning model for optimising structural topologies Authors (1)

Efthimis Kapsalis, (1,2) Ioannis Daganis

(1)

: AP Engineering, (2): Aristotle University

Abstract This paper presents a novel methodology aimed at accelerating topology optimization in architectural and industrial design form-finding by employing generative adversarial networks (GANs) in place of traditional iterative calculation procedures. The proposed technique efficiently and accurately optimizes two- and three-dimensional structural configurations under specific conditions, despite being trained on a limited dataset generated by a third-party software. By integrating deep learning and topology optimization, the methodology offers substantial benefits to architects and engineers in material conservation and the development of more effective structural arrangements and building enclosures, particularly during the early stages of the design process. The study's findings indicate that the proposed model has considerable potential for real-world implementation in the field of structural engineering, as it demonstrates the ability to produce diverse outcomes related to the enhancement of structural topologies, including trusses. This research paves the way for further exploration and refinement of computational design techniques in the pursuit of more efficient and sustainable design solutions.

Keywords: topology optimisation, generative adversarial networks, form-finding, structural engineering, architectural engineering

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1. Introduction “

Structural optimization, encompassing topology optimization, holds significant

importance in the fields of architectural and industrial design. This procedure enhances the structural performance and material utilisation efficiency, leading to a reduction in material waste and carbon footprint during the manufacturing and construction stages. The integration of topology optimisation and artificial intelligence (AI) facilitates the implementation of resource-efficient practises in the industrial sector, thereby promoting sustainable consumption and production patterns.

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1.1. Structural optimisation “

The objective of structural optimisation is to attain the most favourable structural

efficiency while accommodating a variety of constraints. Over the last three decades, the swift advancement of high-speed computing technology and algorithmic improvements has facilitated engineering researchers in formulating more sophisticated structural optimisation solutions.

Topology optimisation is a widely used technique for achieving optimal structural design of discrete structures, including trusses and frames. The present approach endeavours to ascertain the optimal spatial configuration and interlinkage of bars.

Topology optimisation is a methodology that seeks to identify optimal designs for continuum structures by determining the optimal locations and shapes of voids within the design domains. Topology optimisation can be utilised to attain shape optimisation by limiting structural modifications to pre-existing boundaries (Huang & Xie, 2010). Several prominent techniques based on finite element analysis (FEA) have been developed in the field of topology optimisation. These include 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). The present study introduces a novel integrated topology optimisation algorithm that incorporates Huang and Xie's (2010) bidirectional evolutionary structural optimisation (BESO) technique (Figure 1).

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Figure 1: An example result from BESO. Source: Huang & Xie (2010). “

The Bi-directional evolutionary structural optimisation (BESO) technology is an

advanced extension of the evolutionary structural optimisation (ESO) theory developed by Xie and Steven in 1992 (Xie & Steven, 1993). The ESO and BESO algorithms are based on the principles of finite element analysis (FEA) for the purpose of topology optimisation of continuum structures. The primary aim of the BESO algorithm is to identify optimal solutions that exhibit superior structural performance while adhering to specified material constraints. This is achieved by iteratively adding or removing material components (Bao et al., 2020). The ESO method has served as a source of inspiration for the development of the Extended ESO method, which has been widely employed in architectural design endeavors, including the London Aquatics Centre by Zaha Hadid Architects and the Taichung Metropolitan Opera House by Toyo Ito. This method is utilised to produce optimised models that not only demonstrate superior structural performance but also possess unique features that cater to supplementary functional requirements or aesthetic preferences. Xie and his colleagues have developed various precise control strategies for topology optimisation within the realm of architectural design and advancement in contemporary times (Yan et al., 2021).

1.2. Generative Adversarial Networks for topology optimisation The utilisation of artificial neural networks has become more prevalent in architectural and structural investigations due to the progress in deep learning algorithms and the rapid expansion of computational capabilities. These networks have been applied to tasks such as topology optimisation during the design process.

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A Generative Adversarial Network (GAN) is a type of artificial neural network that

learns from a given set of examples and their corresponding probability distributions.

As a result, it has the capability to produce supplementary instances predicated on the approximated probability distribution (Goodfellow et al., 2020). In the context of GANs, a typical setup involves a generator that defines a prior probability distribution P(s) based on an input vector s, and a discriminator that is tasked with discerning the authenticity of data x, which may be either genuine (drawn from training examples) or synthetic (generated by the generator). Conditional Generative Adversarial Networks (cGANs) are an extension of GANs that incorporate supplementary information y as input for both the generator and discriminator (Mirsa & Osindero, 2014). The cGAN discriminator performs an evaluation of both the authenticity of x and its congruence with condition y. In the context of topology optimisation problems, x may denote the expected optimisation outcomes based on the corresponding boundary and load conditions of y.

In recent times, scholars have undertaken investigations pertaining to the resolution of topology optimisation predicaments through the utilisation of artificial neural networks, with a specific emphasis on Generative Adversarial Networks (GANs). An instance of this is the TopologyGAN (Nie et al., 2021), which utilises a conditional generative adversarial network (cGAN) and incorporates a U-Net (Ronneberger et al., 2015) and ResNet (He et al., 2016) in its generator. The present model utilises displacement, load boundary conditions, and target volume fraction, along with dense initial fields computed over the unoptimised domain, as its input. As a result, this methodology significantly improves the precision of topology optimisation predictions when compared to specific baseline models. Yu et al. (2019) conducted a study wherein they transformed boundary conditions into multi-channel images to serve as input for an encoder based on convolutional neural network. The outcome of this process was low-resolution topology optimisation results. Subsequently, the low-resolution outputs are inputted into a Generative Adversarial Network (GAN) to produce high-resolution ultimate outcomes. The proposed approach utilises cGANs directly and requires minimal input to ensure ease of use and simplicity, thereby expanding its usability to a broader range of users, regardless of their level of structural expertise.

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1.3. Purpose The current research introduces a methodology aimed at expediting the process of topology optimisation for structures in the fields of architectural and industrial design form-finding. This is achieved by replacing iterative calculation procedures with an end-to-end algorithm that utilises conditional generative adversarial networks (cGANs). We introduce a technique named ToposGAN, which aims to produce topology optimisation results with high efficiency and accuracy. The proposed methodology has the ability to predict a wide range of optimised two- and threedimensional structural configurations under specific conditions, despite being trained on a limited number of topological optimisation results generated by a third-party software.

2. Methods used 2.1. Data collection “

A distinctive dataset of topology optimisation results is compiled through the

utilisation of a Grasshopper plugin named Topos, for the purpose of training ToposGAN. The material of choice for data generation is consistently steel, with a uniform volume fraction of 0.5. The input parameters for ToposGAN are determined by several factors that may vary across different cases, including the design domain, fixing edge, and load conditions. In the context of Topos script, it is customary to position the fixing edge of a square design domain on the left side. This is because securing the other edges is deemed equivalent to rotating the left-edge-fixing conditions during the data augmentation phase that follows when training the cGAN models. At the outset of the investigation, a dataset comprising 1745 optimisation outcomes was compiled.

During the pre-processing phase, the input parameters undergo a transformation process resulting in the creation of a three-channel input image with a resolution of 256 x 256 pixels. The incorporation of two-dimensional or three-dimensional spatial cues in this conversion process serves to alleviate the challenge of training the model to forecast spatially-specific outcomes, as opposed to relying solely on numerical inputs that lack spatial indications. Figure 2 depicts an example of

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allocating the 0 value to the pixels outlining 2D geometries, thereby representing the

design domain and fixing edge across all three channels. The first and second channels of the data capture the locations and unit vectors projected onto the X and Y axes, respectively. These values have been remapped to a range of 0 to 255. As a default setting, a value of 255 is allocated to all remaining pixels across the three channels.

Figure 2: Translation of an input picture with a design domain of 1000x1000 mm, fixing at the left edge, with the first load at (900, 1000) in a 45° angle and the second load at (1000, 700) in a 180° angle. The remaining pixels all have 255 values.

2.2. System architecture In contrast to the previous research expounded upon in section 1.2, the ToposGAN generator does not necessitate dense initial fields, low-resolution results, or any inputs beyond the design domain, fixing edge, and load conditions. The potential facilitation of ToposGAN usage can be achieved through the maintenance of simplicity in inputs, which enables users to easily adjust a few numerical sliders.

Concurrently, the task of inferring outcomes for the generator is considerably heightened as a result of the restricted input data. The conflict at hand is resolved through the utilisation of a network architecture that follows a coarse-to-fine approach, as implemented by ToposGAN. The network is composed of a pair of generators and discriminators, as depicted in Figure 3. The preliminary set of conditions, represented as ground truth images, is initially fed into the coarse generator. This generator then generates a crude output, which is evaluated by the coarse discriminator. The crude output, combined with the initial conditions, are subsequently inputted into the fine generator, which produces the final outcome. The aforementioned input selection is suggested in Pix2Pix, a

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highly effective model for image-to-image translation, as documented by Isola et al.

(2017).

Figure 3: Architecture of ToposGAN “

The ToposGAN model utilises U-Net architecture, as proposed by Ronneberger et

al. (2015), in both the coarse and fine generators. This architecture is characterised by a mirrored encoder-decoder network, which incorporates skip connections between symmetrical layers. The U-Net architecture has been shown to be effective even in scenarios where the amount of available training data is limited.

Simultaneously, the two discriminators are instantiated as PatchGAN, a technique introduced by Isola et al. (2017) that focuses on analysing 156 x 156 patches. This specific patch size has been determined to produce the most favourable outcomes in the current research endeavours.

The primary goal of ToposGAN is to optimise a specific objective, which can be mathematically represented in equation (1). This equation is composed of two distinct components, namely the cGAN loss (equation (2)) and the L1 loss (equation (3)). The primary objective is to attain outcomes that closely approximate the BESO optimisation results, while also producing real images. As a result, the total loss function integrates L1 loss with a substantial weight of λ, which enforces the output to closely approximate the ground truth. The selection of L1 loss over L2 loss is based on its ability to minimise image blurring to a greater extent than L2 loss. The experiments involve assigning a weight of 125 (λ = 125) to the weight variable, in order to underscore the importance of L1 loss in the given task.

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2.3. Model training “

The ToposGAN model undergoes a two-stage training process, wherein the initial

stage involves the training of the coarse generator and discriminator, followed by the training of the fine generator and discriminator. The two segments of the network were subjected to 200 epochs of training in order to achieve the observed results.

Consistent with conventional approaches, the training procedure seeks to optimise log(𝐷𝐷𝐷𝐷(𝑥𝑥𝑥𝑥,𝐺𝐺𝐺𝐺(𝑥𝑥𝑥𝑥)) as opposed to minimising log(1 − 𝐷𝐷𝐷𝐷(𝑥𝑥𝑥𝑥,𝐺𝐺𝐺𝐺(𝑥𝑥𝑥𝑥)) in order to

prevent the saturation of log(1 − 𝐷𝐷𝐷𝐷(𝑥𝑥𝑥𝑥,𝐺𝐺𝐺𝐺(𝑥𝑥𝑥𝑥)) in the initial stages of training. The learning rate is initialised to 0.0005 and gradually decreases to 0 over the last 100

epochs, utilising the Adam optimisation algorithm. The history of the cGAN loss and L1 loss throughout the two training phases are depicted in Figure 4.

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Figure 4: The cGAN loss and L1 loss during the coarse training (upper set) and fine training (lower set).

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As explicated in Section 2.1., the coarse generator and fine generator have

disparate input types and are coupled with dissimilar discriminators. Consequently, it is deemed unfeasible to compare the absolute values of the corresponding cGAN losses for the coarse and fine models. However, it can be observed from Figure 4 that there is an enhancement in cGAN loss from the initial epoch to the final epoch in both cases. In contrast, the L1 loss is computed using identical ground truth for both generators. The findings suggest that the utilisation of the fine generator results in a decrease in L1 loss in comparison to the coarse generator. This implies that the fine generator has the potential to improve the precision of the anticipated topology optimisation outcomes produced by ToposGAN. 3. Results During the course of the training regimen, the ToposGAN's L1 loss exhibited a notable decrease from an initial value exceeding 80 to a final value below 20. The evaluation of the models' performance is primarily based on pixel-wise accuracy, which is considered to be of utmost importance and specificity. The aforementioned metric denotes the ratio of correctly predicted pixels to the overall number of pixels present in an image. Upon conducting a test with a sample size of 150 data points selected at random, which were distinct from the training set, it was observed that the ToposGAN algorithm achieved an average pixel-wise accuracy of 89.67%. The provided examples pertain to various design alternatives for (a) a bridge truss section, (b) a bridge truss structure, and (c) a space truss. The results of multiple tests conducted under different load conditions are depicted in Figure 5-7 repsectively. The discrepancies between the predictions and ground truth are illustrated for each instance on a pixel-wise basis.

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Figure 5: Topology optimisation of a bridge truss section on ToposGAN.

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Figure 6: Topology optimisation of a bridge truss structure on ToposGAN

Figure 7: Topology optimisation of a space truss on ToposGAN

4. Discussion “

The ToposGAN methodology represents a new approach and its application in the

field of structural form-finding. A prior approach, i.e., SwarmBESO, has been extended to further improve the diversity of topological optimisation through a generative method. This extension was proposed by Bao & Yan (2021). The proposed model possesses the capability to substantially assist architects and engineers in the preservation of materials and the creation of more effective structural arrangements and building enclosures. The integration of two sophisticated computational design techniques, namely deep learning and topology optimization, has the potential to offer significant benefits to designers in the early stages of the design process.

However, it should be noted that the current research is still in its early stages and is presently confined to a limited scope of structures. Despite the fact that the design domain has been designated as an input variable, the current dataset exclusively consists of square geometries. Although the model has exhibited satisfactory performance within the geometric scope, as evidenced by the testing results, it is imperative that ToposGAN undergoes training with a more heterogeneous set of

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design domains and load conditions to effectively expand the spectrum of outcomes and augment precision. The findings of our study suggest that the proposed model holds significant potential for real-world implementation in the field of structural engineering. Our results demonstrate that the model produces diverse outcomes related to the enhancement of structural topologies, including trusses. In light of these observations, we can conclude that the model is a promising tool for optimising structural designs.

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