ORIGINAL ARTICLE

Knowledge-enhanced graph neural networks for construction material quantity estimation of reinforced concrete buildings

Yifan Fei1 | Wenjie Liao1 | Xinzheng Lu1 | Hong Guan2

1Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Department of Civil Engineering, Tsinghua University, Beijing 100084, China

2School of Engineering and Built Environment, Griffith University, Gold Coast Campus, Queensland 4222, Australia

Correspondence
Xinzheng Lu, Department of Civil Engineering, Tsinghua University, Beijing, China
Email: luxz@tsinghua.edu.cn

Hong Guan, School of Engineering and Built Environment, Griffith University, Gold Coast Campus, Queensland, Australia
Email: h.guan@griffith.edu.au

Funding information

ABSTRACT

The construction material quantity (CMQ) is widely concerned in structural design of reinforced concrete buildings, and is often included among the objective functions of computer-aided optimization design techniques. To minimize construction cost and carbon emissions, an accurate and efficient CMQ estimation method is timely required. In this study, a novel graph neural network (GNN) is proposed, whose architecture and loss function are specifically designed for CMQ estimation. With a heterogeneous feature fusion mechanism, the GNN can automatically extract features from all CMQ-related information, in contrast to the existing data-driven methods that rely heavily on manually selected features. By further incorporating a prior knowledge inclusion strategy, the GNN can avoid fundamental errors that might be encountered by purely data-driven methods. To enrich the diversity of the CMQ dataset, a data augmentation method is proposed incorporating generative adversarial networks and parametric modeling. Numerical experiments and case studies show that the proposed CMQ estimation method is superior to the existing data-driven methods in terms of accuracy and is 500 times faster than typical commercial structural design software. This study is anticipated to benefit the objective evaluation of computer-aided design, thereby facilitating the promotion of low-cost and low-carbon-emission building designs.

Computer-Aided Civil and Infrastructure Engineering, 2023

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1        INTRODUCTION

Material consumption, as a critical component of the overall construction cost and carbon emissions, has been widely concerned in the design of building structures (Elfaki et al., 2014; Gan et al., 2020; Sharma et al., 2021). Contractors typically consider construction material quantity (CMQ) as an important aspect in selecting a structural design scheme that can achieve the minimum estimated overall construction cost. Furthermore, CMQ is directly related to the carbon emissions of the construction industry, which must be reduced to realize carbon neutrality and achieve the United Nations Sustainability Development Goals (SDGs) (UNEP, 2022). Although CMQ only affects a portion of the building budget and lifetime carbon footprint, it is the primary component that can be optimized through structural design. Hence, the design objectives of computer-aided structural design methods typically include CMQ minimization (Afzal et al., 2020; M芍laga-Chuquitaype, 2022).

Reinforced concrete (RC) buildings are among the most widely used structural types being built with concrete and reinforcing steel. Steel consumption, which is relatively difficult to estimate, is a major concern for engineers in practical projects due to its significant impact on the construction cost. Concrete consumption is also gaining increasing attention nowadays after the SDGs become an urgent call for action by all countries. Currently, the mainstream CMQ estimation methods for RC buildings can be classified into three categories.

Category 1: Simplified formulas can be used to estimate CMQ (Sharafi et al., 2012; Zhang & Mueller, 2017; Zhou et al., 2022). In this category, the methods are highly efficient; however, they typically either display low accuracy or are only applicable to simple structural systems, such as frame structures.

Category 2: Structural design software, which is used to conduct detailed design, can also estimate the corresponding concrete and steel consumptions (Kim et al., 2013; El-Sokkary & Galal, 2020; Zhang et al., 2021). This method is the most accurate; however, it highly relies on a complicated physics-based analysis model and is therefore time-consuming and inefficient, normally requiring 1每10 minutes of calculation time for a typical RC building. Among computer-aided optimization design techniques, heuristic methods are usually recognized as effective and practical (Aldwaik & Adeli, 2014). The possibility of finding the optimal solution by heuristic methods is positively related to the number of evaluated solutions in the design space. Typically, heuristic methods require hundreds of evaluations during iterations before finding a rational solution (Adeli & Sarma, 2006; Boscardin et al., 2019; Zhang et al., 2023). Another kind of emerging design technique is artificial intelligence (AI)-based methods (M芍laga-Chuquitaype, 2022). During the training of AI models, the prediction results must be evaluated to guide parameter updating. Due to the nature of big data, at least thousands of evaluations are needed for training. Consequently, with the current level of efficiency of structural design software in obtaining the CMQs of RC buildings, design and optimization requirements are difficult to meet. To reduce the computational cost, surrogate models have been proposed, as discussed in the next category.

Category 3: The CMQ can be estimated using AI-based or data-driven methods (Idowu & Lam, 2020; Antoniou et al., 2023; Kovačević et al., 2023), which allows a good balance to be achieved between accuracy and efficiency. They have been widely used as surrogate models to evaluate the objective functions of evolutionary algorithms (Jin et al., 2019; Chugh et al., 2019). The foregoing is the focus of this study as elaborated in the following sections.

The fundamental paradigm of data-driven CMQ estimation methods lies in three aspects (Singh, 1991; Yeh, 1998; Son et al., 2013; Fragkakis et al., 2015; Garc赤a de Soto et al., 2014; Dimitriou et al., 2018; Beljkaš et al., 2020; Idowu & Lam, 2020; Kovačević et al., 2023; Antoniou et al., 2023). (1) Real-world engineering cases are collected, and CMQs are used as label data. (2) Key features are manually selected as input data. (3) A data-driven model is selected to establish the mapping relationship between inputs and labels. However, the existing research on this paradigm has the following limitations: (a) The number of engineering cases collected is typically limited; therefore, the generalization of the model requires further exploration. (b) CMQ is affected by heterogeneous data such as component layout (topology), component section size (geometry), and design conditions (text). However, due to the limitations of their model architecture, existing methods cannot directly handle this data and require manual feature selection, which is inevitably restricted by the knowledge and experience of the developers. (c) A data-driven model is a ※black box§, which is not guided by prior knowledge; consequently, it is prone to fundamental errors.

In recent years, intelligent structural design methods based on generative AI have developed rapidly (Liao et al., 2021; Fu et al., 2023). This technology can quickly generate numerous structural designs and provide valuable data that can be applied to real-world engineering cases. Deep learning models can automatically mine high-dimensional hidden features from data, thereby avoiding the subjectivity of manually defined input features. In addition, in previous research, the authors have successfully integrated engineering knowledge into deep learning, effectively improving the reliability of data-driven models (Fei et al., 2022a; Lu et al., 2022). These previous studies have established a solid foundation for the present study.

Specifically, this study proposes to establish the following strategies to solve the problems that have been encountered by the existing studies. (1) A data augmentation method is proposed, incorporating generative adversarial networks (GANs) and parametric modeling. A considerable amount of synthetic data is generated, and therefore dataset diversity is enriched. (2) A graph neural network (GNN) architecture is proposed herein, which enables direct mapping from all related information to CMQs through a heterogeneous feature fusion mechanism, thereby curtailing the limitations of manually selected features. (3) A prior knowledge inclusion strategy is proposed, which avoids fundamental errors by modifying the output layer and the loss function of the GNNs.

To validate the effectiveness of the proposed methods, RC shear wall building structures, typically used in high-rise residential buildings, are considered as practical examples. The CMQs of these buildings are difficult to estimate because of the presence of shear walls, but are a key factor that must be considered in structural design. Accordingly, the use of a data-driven method to accurately and efficiently estimate the CMQs of RC shear wall building structures is desirable.

The remainder of this paper is organized as follows. Section 2 reviews related studies. Section 3 presents the outline of a novel CMQ estimation method for RC buildings. Section 4 introduces the knowledge-enhanced GNN specifically designed for CMQ estimations. Section 5 describes the data augmentation of the CMQ dataset based on GANs and parametric modeling. Section 6 introduces three other data-driven methods for comparison. Section 7 presents the numerical experiments. Section 8 discusses the case study of several shear wall structures of residential buildings. Finally, the conclusions of this study are presented in Section 9.

2        REVIEW OF CURRENT LITERATURE

2.1  Data-driven CMQ estimation

The research on CMQ estimation using data-driven methods can be traced back to the 1990s (Singh, 1991), and continues to advance recently (Antoniou et al., 2023). The common methods used in relevant research include artificial neural network (ANN) (Adeli & Wu, 1998; Dimitriou et al., 2018; Beljkaš et al., 2020), support vector machine (Idowu & Lam, 2020), and regression analysis (Fragkakis et al., 2015; Son et al., 2013). The research targets include buildings (Garc赤a de Soto et al., 2014; Idowu & Lam, 2020), bridges (Beljkaš et al., 2020; Kovačević et al., 2023), and civil infrastructures (Fragkakis et al., 2015; Antoniou et al., 2023).

Similarly, a significant amount of research is dedicated to estimating construction costs (Adeli & Wu, 1998; Rafiei & Adeli, 2018). These estimation methods are widely used in the cost optimization of concrete structures (Ahmadkhanlou & Adeli, 2005; Aldwaik & Adeli, 2016), steel structures (Sarma & Adeli, 2000a; Sarma & Adeli, 2000b), and composite structures (Kim & Adeli, 2001; Adeli & Kim, 2001). This study mainly focuses on the CMQ estimation because the material cost can be easily calculated based on CMQ if the material price is provided.

Table 1 summarizes some related studies on CMQ estimation of RC structures, including the target, method, number of variables, data size, and prediction error measured by the mean absolute percentage error (MAPE) (if applicable). These studies support the decision-making of stakeholders of construction projects by estimating CMQs. These estimations can be used in design optimization as long as the design variables of RC structures are included as the input variables of the data-driven model. The ground truth of the CMQs is obtained from the design documents, especially the bill of quantities. The accuracy of existing methods (2% ≒ MAPE ≒ 30%) is influenced by many factors and is in need of further improvement. These factors include: (1) only a small number of manually selected variables (three to ten) are considered and the methods may have overlooked useful information that developers are not aware of; (2) data-driven models are prone to fundamental errors if no prior knowledge is incorporated; (3) the data size (or the number of different RC structure samples) is small (6 to 400) compared to those of nowadays* big data tasks (Sun et al., 2017), and the sample diversity is limited. The above-mentioned factors are further discussed in Sections 2.2, 2.3, and 2.4, respectively.


TABLE 1. Related studies on CMQ estimation of RC structures

Reference

Target

Method

Number of variables

Data size

Concrete MAPE (%)

Steel MAPE (%)

Antoniou et al., 2023

Metro station

RA/ANN

5

6

2.13

14.77

Kovacevic et al., 2023

Bridge

GPR

6

181

11.65

N/A

Idowu & Lam, 2020

Building

SVM

10

80

12.19~28.35

12.59~29.73

Beljkas et al., 2020

Bridge

ANN

6

101

8.56

17.31

Dimitriou et al., 2017

Bridge

ANN

5

68

37

31

Garc赤a de Soto et al., 2015

Building

RA/ANN

7

58

3.48

7.44

Fragkakis et al., 2015

Culvert

RA

3

104

13.78

19.79

Yeh, 1998

Building

ANN

10

400

N/A

5.5~14.7

Note: RA=Regression Analysis; ANN=Artificial Neural Networks; GPR=Gaussian Process Regression; SVM=Support Vector Machine; MAPE=Mean Absolute Percentage Error; N/A=Not Applicable.


2.2   Graph neural networks

With the development of deep learning technology, deep neural networks can automatically extract key features from massive amounts of information. Deep learning approaches are gradually replacing traditional feature engineering methods in the fields of computer vision and natural language processing (Russakovsky et al., 2015; Bubeck et al., 2023).

GNN is a type of deep learning approach that operates on graphs, a data structure that represents a set of objects (nodes) and their relationships (edges) (Zhou et al., 2020). In recent years, variants of GNNs such as graph convolutional networks (GCN) (Kipf & Welling, 2016) and Graph-SAGE (Hamilton et al., 2018) have exhibited excellent performances on many deep learning tasks. Nowadays, the applications of GNN have been extended to various tasks with graph-like data structures, such as building structures (Song et al., 2022; Zhao et al., 2023a), water distribution networks (Fan et al., 2023), package-based constraint management (Wu et al., 2023), autonomous vehicle networks (Chen et al., 2021), neural network architectures (Xue et al., 2021), neural signals (Feng et al., 2021), and electroencephalogram signals (Zhao et al., 2021; Che et al., 2022; Lian & Xu, 2022).

However, existing GNNs cannot be directly applied to CMQ estimation because of the heterogeneous input data of this task, including component layout (topology), component section sizes (geometry), and design conditions (text). Hence, this study proposes a novel GNN architecture with a heterogeneous feature fusion mechanism, so that all related information can be directly input into the model without specifying any features artificially.

2.3  Knowledge-enhanced neural networks

Two principal rules must be satisfied to achieve the CMQ estimation of RC buildings based on prior knowledge in structural engineering practice. (1) The total concrete consumption of a building structure is less than the sum of the calculated volumes of all structural components. This is due to the common duplications in calculating the volumes of components. For example, doubling the concrete volume of overlapped beam每column joints must be avoided. (2) The amount of steel reinforcement for any structural component must satisfy the minimum requirements specified in the design codes (MOHURD, 2010, 2015, 2016). However, to the best knowledge of the authors, such prior knowledge has not been considered in the existing data-driven CMQ estimation methods. Hence, unexpected major and fundamental errors may occur during the data-driven model estimation process considering the statistical nature of such models.

Generally, for neural networks, prior knowledge is introduced by modifying the following three elements of the network (Dash et al., 2022): (1) input data (Yin et al., 2019; Marino et al., 2021), (2) loss function (Lu et al., 2022; Fei et al., 2022a; Du et al., 2022), and (3) network architecture (Sourek et al., 2018; Du et al., 2020; Li et al., 2021). The second element, viz., modifying the loss function, is adopted in this study. The output layer of the neural networks is also modified to ensure 100% satisfaction of the prior knowledge.

2.4  Data augmentation of structural design

Table 1 indicates that the data sizes used in the existing studies are no larger than 400, which is much smaller than those of typical deep learning tasks (Sun et al., 2017). This may result in a limited generalization of trained models and difficulty in applying them to practical projects. Since collecting large-scale real-world engineering project data is costly or even unattainable under certain circumstances (D*Amico et al., 2019), the use of synthetic data generated by generative AI to supplement real-world data is desirable.

Structural design methods based on AI have emerged rapidly in recent years and have received extensive attention (Adeli, 2020; Sun et al., 2021; M芍laga-Chuquitaype, 2022). The AI technologies used in relevant research projects include GANs (Zhang et al., 2021; Khan et al., 2023), GNNs (Wang & Zhang, 2021; Chang & Cheng, 2020), convolutional neural networks (CNNs) (Yu et al., 2019; Ampanavos et al., 2022), and reinforcement learning (Hayashi & Ohsaki, 2021; Jeong & Jo, 2021). In particular, a series of research studies have focused on GAN-based design methods for building structures, providing a good foundation for this study (Pizarro et al., 2021; Liao et al., 2021; Fu et al., 2023).

The GAN approach is also a well-recognized data augmentation method (Shorten & Khoshgoftaar, 2019) that can generate realistic synthetic data. However, it has yet to be applied to CMQ estimation tasks because of the difficulty in automatically obtaining the CMQ of synthetic structural design. Therefore, this study proposes a data augmentation method, combining parametric modeling with the GAN-based design method, thereby increasing the CMQ data size by an order of magnitude.

3        Outline of the proposed CMQ estimation method

This study aims to resolve the problems encountered in the CMQ estimation of RC buildings. Consider an RC shear wall building structure as an example. Given the actual structural design (including the spatial layout and geometric dimensions of shear walls and beams) and design conditions (including seismic, site, and height conditions), the concrete and steel consumptions (of shear walls, beams, and slabs) are estimated. The CMQ estimation method proposed in this study, including three modules, is shown in Figure 1.

(1) Model application module: The structural design and design conditions to be considered are represented as a graph (Section 4.1). The concrete and steel consumptions are estimated by inputting the graph into a trained GNN model (Section 4.2). The GNN parameters are generated from Module (2).

(2) Model training module: The GNN model is trained based on the CMQ dataset. The model can extract and fuse Global, Edge, and Node features, and is thereby named as Graph-GEN. It can also embed prior knowledge to ensure that the CMQ estimation satisfies the principal rules of not exceeding the maximum concrete consumption and not below the minimum steel consumption (Section 4.3). The CMQ dataset, including input and label data, is generated from Module (3).

(3) Data augmentation module: A GAN model is trained using a dataset collected from real-world engineering projects. It is then used to generate a large number of synthetic structural designs corresponding to synthetic design conditions. After that, refined finite element models are established using parametric modeling, and their CMQs are determined. This process expands the size of the CMQ dataset by an order of magnitude, thus facilitating the effective training of Graph-GEN (Section 5).

4        Knowledge-enhanced GNNs

A novel GNN architecture called Graph-GEN is proposed that incorporates a heterogeneous feature fusion mechanism and a prior knowledge inclusion strategy. It is capable of extracting and fusing high-dimensional features in various forms, including component layout (topology), component section sizes (geometry), and design conditions (text), which are subsequently mapped into CMQs. Innovatively, prior knowledge is introduced into Graph-GEN by modifying the loss function and output layer.


Figure 1. The proposed CMQ estimation method


4.1    Feature representation 每 model application module

To use a GNN for CMQ estimation, the structural design of an RC building must be represented as a graph. A straightforward approach is adopted to represent the topological relationship of structural components. It is applicable to walls, columns, beams, braces, and so forth, where the structural components are represented by graph edges and the intersections of the structural components are represented by graph nodes (Zhao et al., 2023b). Among all the floors of a building structure, a standard floor can best represent the planar layout of structural components. Therefore, a standard floor of a shear wall building structure is considered herein as an example. The planar layout of the shear walls and beams is represented as a graph, as shown in Figure 2, in which the coordinates of the intersections are used as node features, and the geometric dimensions of the structural components are used as edge features. Thus, all the structural design information can be represented as a graph. For this standard floor of a shear wall building structure, the node features are two-dimensional vectors containing the X and Y coordinates, and the edge features are three-dimensional vectors comprising the length, width, and height of the structural components, as described in Table 2.

Design conditions also significantly affect the CMQ estimation of a building structure (MOHURD, 2010). For example, if a building structure is subjected to a large horizontal seismic action, the amount of reinforcement of the vertical components on the ground floor must increase accordingly. For shear wall building structures, three design conditions are selected for illustration, including seismic, site, and height design conditions, which are represented by the design seismic acceleration ( ), the site characteristic period ( ), and the number of floors ( ), respectively (MOHURD, 2016). Further details are provided in Appendix A. The foregoing design conditions are also represented as a three-dimensional vector known as a global feature because they affect the CMQs of all structural components, as shown in Table 2. The global feature is highly scalable, and its size can be adjusted to consider various design conditions.

The CMQs to be predicted are represented as six-dimensional or two-dimensional vectors comprising the concrete and steel consumptions of shear walls, beams, and slabs (separated or combined), as given in Table 2. These two representations will be discussed in Section 7.2.


TABLE 2. Input and output variables of Graph-GEN

Type

Variable

Representation form

Input

Graph

Node set  and edge set

Node features { }

, where x and y are the coordinates of the intersections of structural components

Edge features { }

, where l, w, and h are length, width, and height of the structural components, respectively

Global features

, where  is the design seismic acceleration,  is the site characteristic period, and  is the number of floors

Output

Vector representations

 or , which represents the concrete and steel consumptions of shear walls, beams, and slabs (separated or combined)

FIGURE 2. Graph representation of shear wall building structure: (a) planar layout of shear walls and beams; (b) graph nodes and edges

Figure 3. Graph-GEN architecture


4.2    Model architecture - Graph-GEN

Given the fact that both the structural design (represented by a graph) and design conditions (represented by a vector) significantly affect the CMQs, this study therefore proposes a novel GNN architecture, named Graph-GEN, which extracts and fuses Global, Edge, and Node features, as demostrated in Figure 3.

Firstly, each node representation is generated by aggregating the representations of its adjacent nodes and the features of its incident edges, the intuitive meaning of which is the CMQ requirements of structural joints (Algorithm 1 in Figure 3). Secondly, each edge prediction is generated using the representations of itself and its end nodes, with an intuitive meaning being the CMQ in each structural component. All edge predictions are then summed up into a graph representation, intuitively standing for an initial CMQ estimation (Algorithm 2 in Figure 3). Thirdly, the processed global features (design conditions) and the graph representation are concatenated and mapped into the final CMQ estimation (Algorithm 3 in Figure 3). The core algorithms with matching steps are introduced below, and the input and output features have been described in Section 4.1.

Step 1: Input the graph , node features , and edge features  into the GNN layers to generate the node representations . The GNN layer is constructed upon Graph-SAGE (Hamilton et al., 2018) and improved to consider the edge features (geometric dimensions of the structural components) (Zhao et al., 2023a).

Algorithm 1 describes the feature aggregation process of Graph-GEN (Figure 3). For the k-th GNN layer, each node first concatenates the current node representations , the edge features , and the neighbor node representations . Then, the concatenated vector is fed into a fully connected layer with non-linear activation and mapped into a temporary vector. After that, all temporary vectors in the immediate neighborhood are summed up into a single vector . This vector is concatenated with the node*s current representation . Note that for the first GNN layer, the node representation  is the input node feature. The concatenated vector is again fed into a fully connected layer with non-linear activation and mapped into an updated node representation  to be used in the next GNN layer.

Algorithm 1. Feature aggregation of Graph-GEN

Input:

Graph ; input node features ; input edge features ; graph depth K; weight matrices for message function ; weight matrices for update function ; non-linear activation function ; tensor concatenation function ; neighborhood function

Output:

Vector representation  for all

 

for  do

for  do

 

 

end for

end for

 

Step 2: Based on the updated node representations  and non-updated edge features , the edge predictions and the graph representation  can be obtained (Figure 3). Specifically, this is accomplished by concatenating the edge feature and end node representations of each edge, and passing the concatenated vector to fully connected layers with non-linear activation to obtain the edge prediction  (Zhao et al., 2023a). Since this operation is performed on the edge level, the size of the weight matrices is irrelevant to the size of the input graph. Subsequently, all edge predictions are summed into one vector and passed onto the fully connected layers with non-linear activation to obtain the graph representation y. Summing all edge predictions ensures that the size of the graph representation y is consistent and irrelevant to the size of the input graph. This approach is intuitive. First, the CMQs of each structural component are estimated by simultaneously considering the structural component itself (edge feature) and the joints at both of its ends (end node representations). Then, the preliminary CMQs of the entire building structure are obtained by summing those of all structural components. Algorithm 2 below provides further details. Note that Algorithm 2 is different from the existing methods in its unique way to obtain the graph representation, considering all edge and node features.

Algorithm 2. Global representation of Graph-GEN

Input:

Graph ; node representations ; input edge features ; fully connected layer depth M; weight matrices for fully connected layer ; non-linear activation function ; tensor concatenation function

Output:

Vector representation  for the graph

for  do

 

for  do

 

end for

 

end for

 

Step 3: The CMQs can be estimated based on the graph representation  and the global feature . Specifically, the global feature is inputted into the fully connected layers with non-linear activation, and the output vector is concatenated with the graph representation. The concatenated vector is then inputted into the fully connected layers with non-linear activation and mapped as the CMQ representation  (Figure 3). The intuitive attribute of this approach is to use the global features (design conditions) to adjust the graph representations (preliminary CMQs). Further details are provided in Algorithm 3 below. Note that the novelty of Algorithm 3 lies in its separate processing of the global feature and effective fusion of heterogeneous features.

Details of Graph-GEN architecture, including the specific setting of each neural network layer, can be found in Appendix B. Three architecture settings are presented with different graph depth K, fully connected layer depths M and N, and are later discussed in Section 7.1.

Algorithm 3. Feature fusion of Graph-GEN

Input:

Graph representation ; input global features ; fully connected layer depth N; weight matrices for fully connected layer ; non-linear activation function ; tensor concatenation function

Output:

Vector representation  for the construction material quantity

 

for  do

 

end for

 

4.3    Knowledge inclusion - model training module

In the CMQ estimation task for RC buildings, the following prior knowledge described in Section 2.3 is elaborated herein.

(1) The total concrete consumption of the building structure is less than the sum of the calculated volumes of all structural components. This is apparent because an inevitable duplication exists in calculating the volumes of structural components. The concrete volume of the overlapped regions (e.g., beam每column and beam每wall joints) must not be doubly counted.

(2) The steel consumption of any structural component must satisfy the minimum reinforcement requirements as specified in the design codes (MOHURD, 2010, 2015, 2016). Evidently, this prior knowledge must be determined according to the local codes. For the RC shear wall building structures constructed in China, the minimum reinforcement requirements for shear walls, beams, and slabs are presented in Appendix C.

This study introduces the foregoing prior knowledge into data-driven models by modifying the loss function and output layer. Specifically, to ensure that the prior knowledge is 100% satisfied, knowledge transformations are performed at the output layer, and a knowledge inclusion loss function ( ) is derived based on these transformations (Dash et al., 2022). This approach is applicable not only to the proposed Graph-GEN but also to other neural network-based models. The loss function of such knowledge-enhanced neural networks is expressed by Equations (1)每(3):

 

(1)

 

(2)

 

(3)

where  is the loss function of knowledge-enhanced neural networks;  is the data-driven loss function;  is the knowledge inclusion loss function;  is the mean square error loss function;  returns the zero tensor of the same shape as the input tensor; and  is the weight of knowledge loss ( );  and  are the labels of concrete and steel consumptions, respectively.

 and  are the concrete and steel consumptions after the transformations, as given by Equations (4) and (5), respectively.

 

(4)

 

(5)

where  is the weight of knowledge transformation ( );  and  are the concrete and steel consumptions directly outputted by the neural networks, respectively;  and  are the portions of concrete and steel consumptions that do not conform to the prior knowledge, as given by Equations (6) and (7), respectively.

 

(6)

 

(7)

where  is the rectified linear unit activation function;  and  are the adjusting factors obtained from datasets, as given by Equations (8) and (9).

 

(8)

 

(9)

where  and  are the labels for concrete and steel consumptions of the i-th sample, respectively;  and  are the maximum concrete and minimum steel consumptions of the i-th sample, respectively; N is the number of samples in the dataset; Min{﹞} and Max{﹞} return the minimum and maximum elements in the list, respectively.

 and  are the maximum concrete and minimum steel consumptions calculated according to prior knowledge, as given by Equations (10) and (11) (for RC shear wall building structures), respectively.

 

(10)

 

(11)

where , , and  are the calculated volumes of a single shear wall, beam, and slab, respectively (the steel volume is neglected); , , and  are the minimum steel consumptions of a single shear wall, beam, and slab, respectively, calculated according to the minimum reinforcement requirements given in Appendix C.

Note that the values of  and  have significant impacts on the model performance, especially in extreme cases when  or ; this is discussed in Section 7.3.

5        Data augmentation module

The performance and generalization of data-driven models are closely related to their data size. As mentioned in Section 2.1, collecting a number of real-world engineering design cases much larger than 400 for CMQ estimation is rather challenging. Therefore, this study proposes a data augmentation method based on GAN and parametric modeling to expand the size of the CMQ dataset by an order of magnitude. First, the GAN is trained based on real-world design data. Subsequently, a large amount of synthetic design data is generated using GAN. Finally, the CMQs corresponding to the synthetic design data are obtained using parametric modeling. To provide a specific explanation of the foregoing, consider an RC shear wall building structure again as an example.

5.1    Structural design dataset

To train the GAN, a structural design dataset containing 430 input data-output data pairs is collected from real-world engineering projects involving shear wall structures of residential buildings, as illustrated in Figure 4. Note that the data size (430) will be increased following data augmentation.

Each data pair includes input and label data. The input data refer to the architectural design (planar layout of partition walls, doors, and windows) and design conditions ( , , and , as described in Section 4.1). The label data denote the structural design (planar layout of shear walls, which is critical in the schematic design of shear wall building structures). The architectural and structural designs are represented as pixel images, and the design conditions are represented as text (Liao et al., 2022). The real-world dataset is split into two parts: 80% of the dataset is used for training and validation, and 20% is set aside for testing.


Figure 4. Structural design dataset

Figure 5. StructGAN architecture


5.2    StructGAN

A well-established GAN model, called StructGAN (Liao et al., 2022), is adopted in this study, as shown in Figure 5. StructGAN consists of a generator and a discriminator, both being deep neural networks. Through a zero-sum game between the generator and discriminator, StructGAN is able to reproduce the probability distribution of the label data based on the input data, thereby establishing a mapping relationship between the input and output. StructGAN is specially proposed for shear wall structure design. Its input is the architectural design being represented by a pixel image and design conditions being represented by texts. Its output is the structural design also being represented by a pixel image. For a given input architectural design, StructGAN can generate different structural designs in response to changes in design conditions.

StructGAN is trained on the structural design dataset described in Section 5.1 (only the training set is used, which is 80% of the 430 data pairs). After training, StructGAN is used to generate synthetic structural designs. For each actual architectural design as input, 10 synthetic design condition combinations are randomly selected. The possible values of , , and  are as follows: , , and . By doing so, 10 synthetic structural designs are obtained by StructGAN for each actual architectural design. In consequence, the size of the structural design dataset is increased by 10 times.

Note that synthetic structural designs are automatically converted from pixel images into vector data using computer vision techniques previously proposed by our team (Fei et al., 2022b).

5.3    Parametric modeling

As described in Section 5.2, the structural design dataset has already been enlarged by 10 times. Note that for this dataset, the input data refer to design conditions and architectural design, and the label data refer to structural design (see Figure 4). However, for the CMQ dataset required in this study, the input data denote design conditions and structural design, and the label data denote CMQs (see Table 2). Therefore, the CMQs corresponding to the structural design should be further obtained.

To facilitate the automatic acquison of CMQs, a parametric modeling procedure for RC shear wall building structures is proposed herein, as shown in Figure 6. The proposed procedure includes five phases, namely parameter interpretation, structural modeling, load definition, analysis & design, and result output.

The material grade significantly affects the physical performance and CMQ of the building structures. To reduce problem complexity, commonly used material grades are generally selected and consistently used in the optimization design of the structural schemes (Sharafi et al., 2012; Zhang & Mueller, 2017; Jeong & Jo, 2021). Consequently, the focus is set on optimizing the structural layout and component dimensions. Accordingly, this study adopts C35 concrete (standard cubic compressive strength: 35 MPa) and HRB400 steel (standard yield strength: 400 MPa), which are the most commonly used materials in building construction projects in China. These materials generally satisfy the requirements of the CMQ evaluation of structural schemes in the related optimization design tasks. When other material grades are used in the design, adjustments must be implemented during parametric modeling.

The parameter interpretation phase is used to read the synthetic structural design generated by StructGAN (i.e., the vector coordinates of shear walls) and the corresponding design conditions. The structural modeling phase is used to create the finite element model of the shear wall building structure. This modeling phase (1) creates the shear walls according to the coordinates and (2) creates the beams and slabs based on hard-coded rules (Fei et al., 2022b). Moreover, it (3) defines the component section sizes according to the empirical formulas (Lu et al., 2022; Zhao et al., 2022) and (4) defines materials as C35 concrete and HRB400 steel. (5) Finally, it assembles the structural components into a structural model. The shear walls and slabs are modeled using shell elements, and the beams are modeled using beam elements. The load definition phase is used to apply loads, including gravity and seismic loads, to the finite element model according to the design codes (MOHURD, 2012, 2016). The design & analysis phase is utilized to perform finite element analysis and reinforcement design to determine the consumption of reinforcing steel of all structural components. The result output phase is employed to count the concrete and steel consumptions of the entire building structure and then output the results as label data. It also converts the structural layout and section sizes into a graph, according to the method presented in Section 4.1, and then saves the graphs as input data.

The foregoing parametric modeling procedure can automatically obtain the CMQs corresponding to the synthetic structural design, thereby establishing the CMQ dataset. Its input data are synthetic structural design (represented as graphs) and design conditions (represented as vectors), and its label data are the CMQs. The augmented CMQ dataset is 10 times larger than the collected structural design dataset, thereby effectively facilitating the training of the CMQ estimation model.


Figure 6. Parametric modeling procedure


6        Alternative CMQ estimation modeling options

Three existing neural network architectures are utilized to establish the CMQ estimation models. Their capacities have been validated in either CMQ estimation or shear wall building structure-related design tasks, and are therefore served as benchmarks. These neural network estimation models are then compared with the proposed Graph-GEN in Section 7 to demonstrate the superiority of the latter.

6.1    ANN

One of the most widely used models for CMQ estimation tasks is ANN (Table 1). This model, also referred to as a multilayer perceptron (MLP), includes several fully connected layers and adopts dropout and batch normalization techniques. Based on the characteristics of an RC shear wall building structure, nine input features are selected: total length of the shear walls, total length of the beams, average thickness of the shear walls, average height of the beams, floor area, floor height, number of floors ( ), design seismic acceleration ( ), and site characteristic period ( ).

6.2    CNN

CNN is a widely used architecture in computer vision. To fit the input form of the CNN, the structural design (shear wall and beam layout shown in Figure 2(a)) and the design conditions are represented as a six-channel pixel image (i.e., a 512 ℅ 256 ℅ 6 matrix). For each channel (i.e., a 512 ℅ 256 matrix), a nonzero element indicates that a structural component exists at this position, whereas a zero element indicates otherwise (Fei et al., 2022a). The values of the nonzero elements in the six channels represent the component width, component height, number of floors ( ), design seismic acceleration ( ), site characteristic period ( ), and scale factor (to scale the structural design into a 512 ℅ 256 matrix). A classical CNN model, ResNet (He et al., 2016), is selected to establish the mapping relationship between the input matrix and CMQ vector. ResNet has been proven successful in its application to the seismic response prediction of RC shear wall building structures, indicating its potential in investigating structural design-related problems (Lu et al., 2022).

6.3    GNN

An existing GNN model, Graph-SF, is also used for comparison (Zhao et al., 2023a). This model considers edge features in the feature aggregation process. Compared with the classical GCN (Kipf & Welling, 2016) and Graph-SAGE (Hamilton et al., 2018), Graph-SF is more suitable for tasks focusing on edges and has been successfully applied to shear wall layout tasks (Zhao et al., 2023b).

The input of Graph-SF only includes a graph and the corresponding node and edge features. Therefore, the global features are considered as part of the edge features (that is, the edge features are six-dimensional vectors (component length, component width, component height, , , and )). The output of Graph-SF is a vector representation of each edge. The output layer is modified such that it is applicable to CMQ estimation. Similar to Graph-GEN, the vector representations of all the edges are summed and mapped into one vector through fully connected layers (Algorithm 2). This vector represents the whole graph and is therefore considered the target CMQ estimation.

As demonstrated above, Graph-SF is less suitable for CMQ estimation than Graph-GEN. The deficiency in Graph-SF architecture results in mixed global and edge features, which causes information redundancy during feature aggregation in GNN layers. Additionally, Graph-SF fails to take prior knowledge into consideration.

7        Numerical experiments and discussions

Numerical experiments are performed on a personal computer with an NVIDIA GeForce RTX 3090 GPU and an Intel Core i7-10700K CPU @ 3.80 GHz. Both StructGAN and Graph-GEN are implemented in the PyTorch deep learning framework (PyTorch, 2023) and trained on a GPU. Graphs are created using the Deep Graph Library (DGL, 2023). Through a secondary development, the parametric modeling procedure of the RC shear wall building structure is established using the structural design software YJK-GAMA (YJK, 2023) and executed on the CPU. YJK is one of the most widely used structural design programs in China. Its reinforcement design and material consumption results are those typically accepted by structural engineers.

Data augmentation is performed, as described in Section 5. The training strategy and hyperparameter selection for StructGAN have been described in Liao et al. (2022). The trained StructGAN and established parametric modeling procedure are applied to a structural design dataset with 430 cases, as described in Section 5.1. Note that the dataset is split into training (80%) and testing (20%) sets before data augmentation. To train the CMQ estimation models, a five-fold cross-validation is adopted. The optimizer used is Adam (Kingma & Ba, 2017), which had a learning rate of 0.001 and a weight decay of 0.0001. The batch size and number of epochs are 8 and 100, respectively. The prediction results of the five CMQ estimation models (from the five-fold cross-validation) are averaged to improve the performance stability.

The widely used MAPE and standard deviation of absolute percentage error (SDAPE) are used as evaluation metrics for the CMQ estimation models (Garc赤a de Soto et al., 2015; Wang et al., 2016). They can be calculated using Equations (12)每(14):

 

(12)

 

(13)

 

(14)

where  is the averaged CMQ estimated by the five estimation models;  is the accurate CMQ; and  is the number of samples.

The hyperparameters of data-driven models are selected based on the MAPE and SDAPE in the following sections.

7.1    Discussion on data-driven modeling options

Several data-driven models〞MLP, ResNet, Graph-SF, and Graph-GEN〞are trained and tested on the CMQ dataset. Each model had three variants each having a different number of layers ( ), as shown in Table 3. For GNN models, K=M= , where K is the graph depth in Algorithm 1 and M is the number of fully connected layers in Algorithm 2; N= , where N is the number of fully connected layers in Algorithm 3;  is selected as the nonlinear activation function  in Algorithms 1-3; knowledge inclusion is neglected, i.e.,  (see Section 4.3 for details). Based on the CMQ dataset,  and =1.3 (Equations 8 and 9). Details of Graph-GEN architecture can be found in Appendix B.

The model performance values on the test set are shown in Table 3. The prediction accuracies of MLP and ResNet are similar, with concrete MAPE ranging from 5.2% to 6.8% and steel MAPE ranging from 10.8% to 12.2%. In contrast, GNN models exhibit significantly improved performance, with concrete MAPE ranging from 2.3% to 3.2% and steel MAPE ranging from 8.6% to 9.5%. This improvement can be attributed to GNN*s ability to extract features in topology (component layout) more effectively than CNN and ANN. Additionally, Graph-GEN outperforms Graph-SF in the estimation accuracy of both concrete and steel, indicating the proposed architecture*s superior capacity in handling heterogeneous data. Overall, the six-layer Graph-GEN architecture achieves the lowest MAPE (2.3% for concrete and 8.6% for steel) and SDAPE (1.8% for concrete and 7.3% for steel) and will be used in subsequent experiments.

Table 3. Performance of different data-driven modeling options on CMQ test set

Data-driven model

Concrete

Steel

MAPE (%)

SDAPE (%)

MAPE (%)

SDAPE (%)

MLP (Garc赤a de Soto et al., 2015)

3

5.2

5.7

11.1

8.1

5

6.8

7.7

12.2

9.7

7

6.0

6.2

11.0

8.5

ResNet (Lu et al., 2022)

18

5.6

5.9

10.8

9.0

50

5.5

5.3

10.8

9.0

101

6.0

6.8

11.4

9.4

Graph-SF (Zhao et al., 2023a)

4

2.7

2.2

9.3

7.2

6

2.6

2.0

9.5

7.3

8

3.2

2.5

9.5

7.5

Graph-GEN

4

2.5

2.0

8.6

7.3

6

2.3

1.8

8.6

7.3

8

3.1

2.4

9.2

7.4

7.2    Discussion on feature representation options

The representation details of the input and output features are further discussed based on the six-layer Graph-GEN. The input features are either normalized or scaled, as given by Equations (15) and (16):

 

(15)

 

(16)

where  is the normalized input feature;  is the original input feature (for coordinates, assuming that the center coordinates are at the origin (Zhao et al., 2023b));  and  are the mean and standard deviation of , respectively;  is the scaled input feature; and  is the scale factor such that . For the coordinates, length, width, and height, ; for , ; and for  and , .

The output features adopt one of the following CMQ vectors: (1) two-dimensional vector, i.e., noutput=2 in Appendix B, representing the combined quantity of concrete and steel and (2) six-dimensional vector, i.e., noutput=6 in Appendix B, representing the separated quantities of concrete and steel for shear walls, beams, and slabs. These two CMQ vectors are normalized.

The experimental results using different feature representations are given in Table 4. It can be observed that the output feature type (combined or separated) has a minor impact on the estimation accuracy for a given input feature type (normalized or scaled), with a maximum variation in MAPE of 0.4% (i.e., 3.2%-2.8%). However, for a given output feature type, the normalized input feature yields better estimation accuracy than the scaled input feature, with up to 1% (i.e., 9.6%-8.6%) improvement in MAPE. The model with normalized input features and combined output features achieves the lowest MAPE and SDAPE for both concrete and steel. Thus, these options are chosen for subsequent experiments.

Table 4. Performance of different feature representation options on CMQ test set

Input

feature

Output

feature

Concrete

Steel

MAPE

(%)

SDAPE

(%)

MAPE

(%)

SDAPE

(%)

Normalized

Combined

2.3

1.8

8.6

7.3

Scaled

Combined

3.2

2.8

9.6

7.4

Normalized

Separated

2.4

2.0

8.9

7.3

Scaled

Separated

3.4

3.2

9.2

7.3

7.3    Discussion on knowledge inclusion strategies

The hyperparameters for knowledge inclusion are further tuned based on the best, purely data-driven model. The model performance values on the test set are listed in Table 5. They are elaborated as follows.

(1) When , the model is purely data-driven through the loss function .

(2) When , prior knowledge only controls the model through the loss function . The predictions of the model cannot be guaranteed to fully satisfy the prior knowledge. As listed in Table 5, the MAPE is increased by a minimum of 0.1% (i.e., 2.4%-2.3%) and a maximum of 0.7% (i.e., 9.3%-8.6%) because of the interference of the two loss functions (i.e.,  and ).

(3) When , the model directly replaces the erroneous prediction results at the output layer to ensure that the prior knowledge is 100% satisfied; meanwhile, the corresponding gradient in  disappears. In this case, the model parameters related to the incorrect prediction are updated through  to a certain extent (depending on the  value). It can be observed that when , the MAPE and SDAPE values of concrete are 30.4% and 38.9% smaller relative to those of the purely data-driven model, respectively. Regretfully, the MAPE of steel increases by 2.3% relatively, and its SDAPE remains unchanged.

The results obtained when  imply that the maximum concrete consumption can effectively govern the model prediction, whereas, in general, minimum steel consumption is unable to provide useful information. This is due to the fact that in the CMQ dataset used in this study, the minimum steel consumption is only about 2/3 of the actual steel consumption (label) calculated by commercial design software. Since the minimum bound is relatively low, this prior knowledge is only beneficial at the beginning of the training when the error is huge. Nevertheless, the minimum steel consumption can still be used to prevent the model from producing major errors under certain circumstances (see Section 8.2).

Table 5. Performance of different knowledge inclusion strategies on CMQ test set

Concrete

Steel

MAPE (%)

SDAPE (%)

MAPE (%)

SDAPE (%)

0

0

2.3

1.8

8.6

7.3

0

0.01

2.5

2.4

9.2

7.2

0

0.1

2.6

2.5

9.3

7.2

0

1

2.4

1.8

9.2

7.3

1

0.01

2.0

1.0

8.8

7.3

1

0.1

1.6

1.1

8.8

7.3

1

1

1.8

1.4

9.0

7.2

Note that Graph-GEN has low computational requirements and good scalability. All Graph-GEN configurations described in this section use less than 2GB GPU memory during training. The saved Graph-GEN models are smaller than 1GB and can run on any consumer grade CPU.

8        Case study

To verify the effectiveness of the proposed method, a trained Graph-GEN is applied in the structural design of three typical shear wall structures of residential buildings and results are compared with existing CMQ estimation methods. Residential buildings with seismic intensity, site condition, floor area, and number of floors considered in this case study are commonly built in China.

Figure 7. Structural design schemes of shear wall structures of residential buildings: (a) scheme of Building A; (b) scheme of Building A; (c) Building B; (d) Building C.

8.1    Comparison of alternative design schemes

During the structural design of shear wall structures of residential buildings, various design schemes may be proposed and compared to obtain an optimal one. In this case study, the material cost and carbon emission of two design schemes of Building A are estimated and compared to guide the direction of design optimization.

Essential information of the project is given below. Building A has 11 floors, and each has a floor area of 220 m2. The seismic intensity is 7 degrees ( ), the site is class II, and the seismic design group is 2 ( ). The structural layouts of the two design schemes with different shear-wall lengths are shown in Figures 7(a) and 7(b). The shear-wall thicknesses are both 180 mm. The underground structure is not considered in this study.

A refined finite element model of the building is established using the parametric modeling procedure (see Section 5.3) in the structural design software YJK-GAMA (see Section 7). Structural analysis and reinforcement design are performed, and the obtained CMQs are considered as the ground truth. An MLP, a Graph-SF, a Graph-GEN, and a simplified method are used to estimate the CMQs. Each data-driven model adopts the best configuration listed in Tables 3-5, according to MAPEs. Specifically, the simplified method considers the calculated volume of all structural components as concrete consumption, and assumes the reinforcement ratio as 0.1-ton steel per 1 m3 concrete, which is an empirical ratio obtained from the training dataset explained in Section 5.1. Based on the CMQs, material cost, and carbon emission are calculated based on China*s statistical data (Zhang & Zhang, 2021). Assuming 1 USD = 7 CNY. For C35 concrete, the unit cost is 544 CNY/m3 (~78 USD/m3) and the unit emission is 387 kgCO2e/m3. For HRB400 steel, the averaged cost is 6266 CNY/ton (~895 USD/ton) and the averaged emission is 2262 kgCO2e/ton. Comparisons of the estimation accuracy of CMQ, and material cost and carbon emission by different methods are summarized in Tables 6 and 7, respectively.

It can be concluded from Table 6 that the CMQ error of the simplified method ranges from 6.5% to 24.1%, that of MLP ranges from -2.3% to 13.8%, that of Graph-SF ranges from -2.2% to 1.7%, and that of Graph-GEN ranges from -0.7% to 1.9%. It can also be determined from Table 7 that the cost and emission error of the simplified method ranges from 11.9% to 15.3%, that of MLP ranges from 1.0% to 8.9%, that of Graph-SF ranges from -1.9% to 0.3%, and that of Graph-GEN ranges from 0.3% to 1.0%. Therefore, the proposed Graph-GEN has a significant accuracy advantage over Graph-SF, MLP, and the simplified method.

In terms of time efficiency, the times used for a single CMQ estimation are: ~3 minutes (YJK); 0.36 seconds (Graph-GEN); 0.35 seconds (Graph-SF); 0.19 seconds (MLP); ~0.001 seconds (simplified method). It can be observed that the proposed Graph-GEN is remarkably advantageous (an increase of 500 times), over YJK, which calculates the CMQs of each structural component after detailed structural analysis and reinforcement design. When using genetic algorithms for structural optimization, hundreds of times of CMQ estimation are typically needed (Zhou et al., 2022; Orhan & Taşkın, 2021). For example, a population size of 20 and an iteration number of 50 will result in 1000 possible schemes and 1000 times of CMQ estimation. If YJK is used, CMQ estimation alone will take 50 hours, which is unacceptable for urgent design tasks. However, if Graph-GEN is used, CMQ estimation will only take 6 minutes and will not cause a bottleneck for structural optimization. Although Graph-GEN is not the quickest method, its efficiency and accuracy can well satisfy the needs of engineering practices.

Therefore, the proposed method demonstrates a highly desirable balance between accuracy and efficiency.

8.2  Advantages of knowledge inclusion

In this case study, the CMQ estimations by different methods are compared with the maximum concrete consumption and the minimum steel consumption according to the prior knowledge specified in Section 4.3.

Essential information of the two projects is given below. Building B has 8 floors, and each has a floor area of 210 m2. Building C has 25 floors, and each has a floor area of 800 m2. They are both considered for a seismic intensity of 8 degrees ( ), a site class of II, and a seismic design group of 2 ( ). The structural layouts of the two buildings are shown in Figures 7(c) and 7(d). The shear-wall thicknesses of Buildings B and C are 180 mm and 240 mm, respectively. The underground structure is not considered herein.

The CMQs of Buildings B and C are estimated using the five methods in Section 8.1. The maximum concrete consumption ( ) of Building B is calculated as 68.0 m3 per floor, and the minimum steel consumption ( ) of Building C is calculated as 25.7 tons per floor. A comparison of the prior knowledge and the CMQ estimated by different methods is summarized in Table 8.

For Building B, the concrete usages estimated by the simplified method, MLP, and Graph-SF are larger than the maximum concrete consumption, whereas only the concrete usage estimated by Graph-GEN satisfies the prior knowledge. For Building C, the steel usages estimated by MLP and Graph-SF are smaller than the minimum steel consumption. In contrast, steel usage estimated by Graph-GEN satisfies the prior knowledge. This case study shows that the proposed prior knowledge inclusion strategy can effectively avoid the fundamental errors that might be encountered by existing data-driven methods.


Table 6. CMQ estimation accuracy of different methods (per floor)

Method

Scheme i of Building A

Scheme j of Building A

Concrete

(m3)

Percentage

error (%)

Steel

(ton)

Percentage

error (%)

Concrete

(m3)

Percentage

error (%)

Steel

(ton)

Percentage

error (%)

YJK

74.0

0.0

6.5

0.0

75.1

0.0

6.4

0.0

Simplified

78.9

6.6

7.9

22.3

79.9

6.5

8.0

24.1

MLP

72.3

-2.3

6.9

7.4

78.1

4.0

7.3

13.8

Graph-SF

72.8

-1.6

6.3

-2.2

74.2

-1.1

6.5

1.7

Graph-GEN

74.9

1.3

6.4

-0.7

75.1

0.1

6.6

1.9

Table 7. Cost and emission estimation accuracy of different methods (per floor)

Method

Scheme i of Building A

Scheme j of Building A

Cost

(USD)

Percentage

error (%)

Emission

(kgCO2e)

Percentage

error (%)

Cost

(USD)

Percentage

error (%)

Emission

(kgCO2e)

Percentage

error (%)

YJK

11546.4

0.0

43232.5

0.0

11617.4

0.0

43611.4

0.0

Simplified

13214.7

14.4

48377.7

11.9

13390.7

15.3

49022.0

12.4

MLP

11841.3

2.6

43655.9

1.0

12649.4

8.9

46793.1

7.3

Graph-SF

11324.6

-1.9

42442.4

-1.8

11648.4

0.3

43530.6

-0.2

Graph-GEN

11582.8

0.3

43504.2

0.6

11731.8

1.0

43908.1

0.7

Table 8. Influence of knowledge inclusion on CMQ estimations (per floor)

Method

Building B

Building C

Concrete

(m3)

Percentage

error (%)

Knowledge

satisfied?

Steel

(ton)

Percentage

error (%)

Knowledge

satisfied?

YJK

67.3

0.0

Yes

28.2

0.0

Yes

Simplified

71.6

6.4

No

33.4

18.4

Yes

MLP

72.7

8.0

No

22.0

-22.1

No

Graph-SF

72.3

7.4

No

23.6

-16.3

No

Graph-GEN

68.0

1.1

Yes

25.7

-8.9

Yes

Note: The maximum concrete consumption of Building B is 68.0 m3 per floor and the minimum steel consumption of Building C is 25.7 ton per floor.


9        Conclusion

This study proposes an accurate and efficient CMQ estimation method for RC buildings. The major contributions are as follows. First, a data augmentation method is proposed for the CMQ dataset, incorporating GANs and parametric modeling. Second, a novel GNN architecture, Graph-GEN, is constructed to enable feature extraction and fusion of heterogeneous data, including component layout (topology), component section sizes (geometry), and design conditions (text). Third, prior knowledge is introduced into Graph-GEN to further improve the model accuracy and thus prevent potential major and fundamental errors. The following specific conclusions are also drawn.

(1) Through the unique cooperation of GANs and parametric modeling, synthetic structural designs can be effectively generated and CMQs can be automatically calculated, thereby enlarging the data size by 10 times, which greatly facilitate the training of the CMQ estimation model.

(2) The heterogeneous feature fusion mechanism of the proposed Graph-GEN architecture can facilitate accurate mapping between the input information and CMQs. Graph-GEN is significantly superior to the existing ANN, CNN, and GNN methods in terms of MAPE and SDAPE. These strongly convey the novelty of this study.

(3) Unlike the existing studies, by modifying the loss function and output layer in the present study, the proposed knowledge inclusion strategy can further reduce the MAPE and SDAPE of concrete consumption by 30.4% and 38.9%, respectively. Notably, it can ensure that the estimated steel consumption exceeds the minimum steel consumption required by the design codes.

(4) The case study shows that Graph-GEN outperforms three existing methods in terms of estimation accuracy, and is 500 times more efficient than the commercial design software. It also demonstrates the advantages of knowledge inclusion in avoiding fundamental errors. To this end, Graph-GEN exhibits a highly desirable balance between computational accuracy and efficiency.

The limitations of this study are recognized as: (1) the minimum reinforcement requirement embedded in the Graph-GEN is a simplification to the complex reinforcement design; (2) the influence of the underground structures on CMQs is ignored. In the future study, Graph-GEN shall incorporate more domain knowledge to improve its practicality. Further, it should be applied in the evaluation stage of evolutionary algorithms and deep learning models to explore its applicability.

AppendIx a

The seismic design condition is represented by the design seismic acceleration, , which corresponds to a 10% probability of exceedance over 50 years.

The site design condition is represented by the site characteristic period, , as listed in Table A1.

The height design condition is represented by the number of superstructure floors, , excluding the underground floors, which are not considered in this study.

Table A1. Site characteristic period (s) (MOHURD, 2016)

Seismic design group

Site class

0

1

1

0.20

0.25

0.35

0.45

0.65

2

0.25

0.30

0.40

0.55

0.75

3

0.30

0.35

0.45

0.65

0.90

AppendIx B

Details of Graph-GEN architecture are listed in Table B1. According to the feature representation outlined in Section 4.1, nnode = 2, nedge = 3, and nglobal = 3. According to the output feature type in Section 7.2, noutput =2 or 6. The dropout ratio (p) is 0.2, and the nonlinear activation function (考(﹞)) adopts ReLU(﹞).

Each GNNLayer(nnode,in, nedge,in, nnode,out) consists of two weighting matrices: Linear(2℅nnode,in + nedge,in, nnode,out) and Linear(nnode,in + nnode,out, nnode,out), see Algorithm 1.

Table B1. Architecture of GNN models

Parameters

Architecture

K=4

M=4

N=1

GNNLayer(nnode, nedge, 8); Dropout(p);

GNNLayer(8, nedge, 8); Dropout(p);

BatchNorm(8); Dropout(p);

GNNLayer(8, nedge, 16); Dropout(p);

GNNLayer(16, nedge, 16);

考(Linear(32 + nedge, 64));

考(Linear(64, 64));

BatchNorm(64); Dropout(p);

考(Linear(64, 32)); 考(Linear(32, 8));

考(Linear(8, 8)); 考(Linear(nglobal, 8));

Linear(16, noutput)

K=6

M=6

N=2

GNNLayer(nnode, nedge, 8); Dropout(p);

GNNLayer(8, nedge, 8); Dropout(p);

BatchNorm(8); Dropout(p);

GNNLayer(8, nedge, 16); Dropout(p);

GNNLayer(16, nedge, 16); Dropout(p);

BatchNorm(16); Dropout(p);

GNNLayer(16, nedge, 32); Dropout(p);

GNNLayer(32, nedge, 32);

考(Linear(64 + nedge, 128));

考(Linear(128, 128));

BatchNorm(128); Dropout(p);

考(Linear(128, 64)); 考(Linear(64, 64));

BatchNorm(64); Dropout(p);

考(Linear(64, 32)); 考(Linear(32, 8));

考(Linear(8, 8)); 考(Linear(nglobal, 8));

考(Linear(16, 8)); Linear(8, noutput)

K=8

M=8

N=3

GNNLayer(nnode, nedge, 8); Dropout(p);

GNNLayer(8, nedge, 8); Dropout(p);

BatchNorm(8); Dropout(p);

GNNLayer(8, nedge, 16); Dropout(p);

GNNLayer(16, nedge, 16); Dropout(p);

BatchNorm(16); Dropout(p);

GNNLayer(16, nedge, 32); Dropout(p);

GNNLayer(32, nedge, 32); Dropout(p);

BatchNorm(32); Dropout(p);

GNNLayer(32, nedge, 64); Dropout(p);

GNNLayer(64, nedge, 64);

考(Linear(128 + nedge, 256));

考(Linear(256, 256));

BatchNorm(256); Dropout(p);

考(Linear(256, 128)); 考(Linear(128, 128));

BatchNorm(128); Dropout(p);

考(Linear(128, 64)); 考(Linear(64, 64));

BatchNorm(64); Dropout(p);

考(Linear(64, 32)); 考(Linear(32, 8));

考(Linear(8, 8)); 考(Linear(nglobal, 8));

考(Linear(16, 8)); 考(Linear(8, 4));

Linear(4, noutput)

ApPENDIX C

For shear walls, the minimum reinforcement within the end restraint zone and the minimum reinforcement ratio of the wall distribution rebars are listed in Tables C1 and C2, respectively.

Table C1. Minimum reinforcement within end restraint zone (MOHURD, 2010)

Seismic grade

Minimum longitudinal bars

Minimum tie bar diameter (mm)

Maximum vertical spacing of tie bars (mm)

1

Max(0.008 , 6 14)

8

150

2

Max(0.006 , 6 12)

8

200

3

Max(0.005 , 4 12)

6

200

4

Max(0.004 , 4 12)

6

250

Note:  is the section area of the end restraint zone;  denotes rebar diameter.

Table C2. Minimum reinforcement ratio of wall distribution rebars (MOHURD, 2016)

Seismic grade

Minimum reinforcement ratio of horizontal and vertical wall distribution rebars

1

0.25%

2

0.25%

3

0.25%

4

0.20%

For beams (including frame beams and coupling beams), the minimum requirements of the longitudinal tensile rebars, erecting rebars, and stirrups are listed in Tables C3每C6.

Table C3. Minimum reinforcement ratio of longitudinal tensile rebars of frame beams (MOHURD, 2010)

Seismic grade

Beam end (%)

Beam midspan (%)

1

Max(0.40, 80 )

Max(0.30, 65 )

2

Max(0.30, 65 )

Max(0.25, 55 )

3 or 4

Max(0.25, 55 )

Max(0.20, 45 )

Note:  is the tensile strength of concrete, and  is the yield strength of steel rebar.

Table C4. Minimum reinforcement ratio of longitudinal tensile rebars of coupling beams (MOHURD, 2010)

Span每depth ratio

Minimum reinforcement ratio (%)

Max(0.20, 45 )

Max(0.25, 55 )

Note:  is the tensile strength of concrete, and  is the yield strength of steel rebar.

Table C5. Minimum erecting rebars (MOHURD, 2015)

Seismic grade

Minimum number of rebars at top and bottom

Minimum rebar diameter (mm)

1 or 2

2

14

3 or 4

2

12

Table C6. Minimum stirrups in beams (MOHURD, 2010)

Seismic grade

Length of dense zone (mm)

Maximum spacing of stirrups (mm)

Minimum stirrup diameter (mm)

1

Max(2 , 500)

Max( , 6d, 100)

10

2

Max(1.5 , 500)

Max( , 8d, 100)

8

3

Max(1.5 , 500)

Max( , 8d, 150)

8

4

Max(1.5 , 500)

Max( , 8d, 150)

6

Note:  is the beam section height, and d is the longitudinal rebar diameter.

For slabs, the minimum reinforcement ratio (%) of the longitudinal tensile rebars is Max(0.20, 45 ). (MOHURD, 2015).

ACKNOWLEDGMENT

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