BlastGraphNet: An Intelligent Computational Method for the Precise and Rapid Prediction of Blast Loads on Complex 3D Buildings Using Graph Neural Networks

Zhiqiao Wang; Jiangzhou Peng; Jie Hu; Mingchuan Wang; Xiaoli Rong; Leixiang Bian; Mingyang Wang; Yong He; Weitao Wu

doi:10.1016/j.eng.2025.03.007

Engineering ›› 2025, Vol. 49 ›› Issue (6) :217 -237. DOI: 10.1016/j.eng.2025.03.007

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BlastGraphNet: An Intelligent Computational Method for the Precise and Rapid Prediction of Blast Loads on Complex 3D Buildings Using Graph Neural Networks

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Abstract

Accurate and efficient prediction of the distribution of surface loads on buildings subjected to explosive effects is crucial for rapidly calculating structural dynamic responses, establishing effective protective measures, and designing civil defense engineering solutions. Current state-of-the-art methods face several issues: Experimental research is difficult and costly to implement, theoretical research is limited to simple geometries and lacks precision, and direct simulations require substantial computational resources. To address these challenges, this paper presents a data-driven method for predicting blast loads on building surfaces. This approach increases both the accuracy and computational efficiency of load predictions when the geometry of the building changes while the explosive yield remains constant, significantly improving its applicability in complex scenarios. This study introduces an innovative encoder–decoder graph neural network model named BlastGraphNet, which uses a message-passing mechanism to predict the overpressure and impulse load distributions on buildings with conventional and complex geometries during explosive events. The model also facilitates related downstream applications, such as damage mode identification and rapid assessment of virtual city explosions. The calculation results indicate that the prediction error of the model for conventional building tests is less than 2%, and its inference speed is 3–4 orders of magnitude faster than that of state-of-the-art numerical methods. In extreme test cases involving buildings with complex geometries and building clusters, the method achieved high accuracy and excellent generalizability. The strong adaptability and generalizability of BlastGraphNet confirm that this novel method enables precise real-time prediction of blast loads and provides a new paradigm for damage assessment in protective engineering.

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Keywords

Blast load prediction / Graph neural networks / Data-driven learning / Real-time prediction / Protective engineering

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Zhiqiao Wang, Jiangzhou Peng, Jie Hu, Mingchuan Wang, Xiaoli Rong, Leixiang Bian, Mingyang Wang, Yong He, Weitao Wu. BlastGraphNet: An Intelligent Computational Method for the Precise and Rapid Prediction of Blast Loads on Complex 3D Buildings Using Graph Neural Networks. Engineering, 2025, 49(6): 217-237 DOI:10.1016/j.eng.2025.03.007

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1. Introduction

Efficient and accurate determination of the complete blast loads acting on building structures under specific explosive conditions is essential for blast-resistant design [1,2], civil defense engineering [3], and resilient protection [4,5]. Currently, blast loads on target structures are obtained primarily through empirical models or direct numerical simulations. Empirical models are effective for calculating blast loads in various straightforward explosive scenarios, such as free-air, near-ground, and enclosed explosions, where high computational efficiency can be achieved [6]. However, in complex engineering applications, empirical models exhibit poor geometric adaptability and struggle to accommodate buildings with intricate geometries. In cases involving multiple buildings, the diffraction and reflection effects of blast waves can significantly impact the accuracy of empirical prediction methods [7], limiting their applicability in real-world engineering scenarios. Finite element analysis (FEA)-based numerical simulation is one of the most accurate methods for predicting the propagation of blast waves [8]. Through mesh discretization and solving related governing equations corresponding to fluid dynamics and chemical kinetics, numerical simulations provide a spatiotemporal distribution of physical quantities, such as pressure and temperature. These simulations can reveal details, including the spatiotemporal propagation of shock waves and the reflective effects on building surfaces. However, numerical simulations also have limitations. High-resolution flow field data typically require a dense mesh. Even with high-performance computing resources, obtaining results can still be time-consuming, posing challenges for real-time calculations. Furthermore, numerical simulations generally necessitate a deep understanding of engineering principles. Expert knowledge is required to define boundary conditions, material parameters, discretization schemes, and other factors, which creates substantial barriers to practical engineering assessments.

With the rapid development of computer hardware and artificial intelligence techniques, many researchers have proposed various methods to address the problem of simulating blast waves with machine learning. Xi et al. [9] proposed the use of a multilayer perceptron (MLP) to learn from their constructed blast database and AUTODYN (ANSYS Inc., USA) to fit the peak overpressure of near-ground explosions. The approach is aimed at addressing the inefficiencies of empirical models in predicting reflected overpressure. Pannell et al. [10] investigated explosions under different shaped charges and used transfer learning to reduce the amount of data needed for constructing new blast models. Advanced neural network architectures, such as Transformer, have shown excellent performance in predicting blast loads and subsequent structural responses [11]. The integration of physical constraints has also been demonstrated to be effective. Zhou et al. [12] incorporated physical information about the structure into their model, resulting in more accurate damage predictions under seismic shock waves. However, these existing machine learning methods are indeed advanced empirical formulas that treat neural networks as implicit function fitters to establish relationships among the explosive yield, standoff distance, peak overpressure, and peak impulse [13]. Their predictions often fail to accurately capture the local details of the overpressure field and are limited in their global spatiotemporal representation of the blast load field. To capture shock wave details around obstacles, convolutional neural network (CNN) frameworks have been applied to predict blast spatiotemporal distributions [14]. Huang et al. [15] used a unified network (U-Net)-based model to predict high-resolution two-dimensional (2D) blast flow field distributions around obstacles. Kang and Park [16] directly used a three-dimensional (3D) convolution model to predict the peak pressure between buildings in exterior explosions. Although CNNs have increased the ability of the model to capture building edge features, the convolutional structure is only suitable for handling grid-based data with relatively fixed spatial relationships [17,18] and suffers from performance degradation when unstructured complex data are encountered [19].

Therefore, a more suitable framework is required to manage unstructured data. Graph data, owing to their excellent ability to interpret non-Euclidean data, have been extensively utilized in fields such as heat transfer [20], fluid dynamics [21], and electromagnetics [22]. Graph neural networks (GNNs) utilize a graph structure composed of nodes and edges as input, updating the embedded features of these nodes and edges through operations such as message passing and aggregation [23]. For instance, Sanchez-Gonzalez et al. [24] applied graph messages to mechanical structure simulations and various complex physical systems. Pfaff et al. [19] proposed MeshGraphNet, which applies a graph model to unstructured meshes, addresses classic aerodynamics challenges such as flow around cylinders and airfoil aerodynamic analysis, and elucidates adaptive learning capabilities across different resolution data. In the realm of blast mechanics, Hao et al. [25,26] implemented an autoregressive strategy to apply graph neural networks to the spatiotemporal simulation of blast loads and structural responses on a 2D scale and achieved significant acceleration in processing time. Peng et al. [27] addressed the issue of sparse data in 2D explosion flow fields by integrating graph neural network design with physical principles, which resulted in accurate predictions of explosion overpressure and pressure–time change distributions. This approach provides a novel method for developing intelligent models suitable for explosions.

These studies utilize primarily data-driven methods to construct explosion simulation models that accurately capture the evolution of shock waves resulting from explosions. However, existing transient prediction strategies [28] typically employ a recursive iterative approach, using the current state to predict the physical field distribution in the subsequent time step. Although this method captures detailed flow field information effectively, it tends to accumulate errors over long-term recursive predictions. To mitigate error accumulation between time steps, smaller intervals must be used, which significantly increases computational costs and diminishes real-time prediction capabilities. This method is inadequate for addressing complex and variable parameters, such as geometry and explosive yield, in practical engineering applications. Moreover, peak overpressure and peak impulses are the primary criteria for assessing explosion damage. Therefore, achieving accurate real-time predictions of peak overpressure and peak impulses via steady-state strategies can more effectively meet the assessment needs for evaluating the effects of explosion damage. Currently, there is no data-driven model suitable for explosions in complex 3D structures. Thus, it is essential to conduct research on steady-state predictions of explosion impacts on complex buildings. The approach should enable rapid and efficient assessment of damage to critical targets during explosion events without compromising computational costs and real-time capabilities.

This study introduces a new graph neural network model, BlastGraphNet, designed for predicting blast loads on buildings. This model leverages the powerful capabilities of graph neural networks to handle non-Euclidean data structures, enabling fast and accurate prediction of blast load fields on structures. This study simplifies the positive phase of the overpressure curve by focusing on three key parameters: peak overpressure, peak impulse, and wavefront arrival time. This targeted approach captures the blast pulse wave impacting the building surface effectively. Rather than learning the entire spatiotemporal field, the model constructs an end-to-end autoencoder graph predictor and pairs nodes on the basis of their connectivity radius to form graph data oriented toward building explosion events. Each node exchanges dynamic information with its adjacent nodes and continuously updates its state, ultimately achieving accurate predictions of the target physical field.

The primary objective of developing the BlastGraphNet model is to offer a tool for rapidly predicting blast load fields on various building structures. BlastGraphNet shows not only exceptional capabilities in predicting overpressure details, effectively replacing empirical formulas and numerical simulation methods, but also significant potential in practical applications such as structural damage assessment and regional explosion simulation. The key contributions of this article are summarized as follows:

•A graph neural network prediction model for building exterior blasts, named BlastGraphNet, is proposed. This model predicts key parameters of blast physical fields, such as peak overpressure and impulses, for various building structures.

•Compared with traditional numerical simulations, BlastGraphNet has strong predictive performance, achieves a relative error of less than 2%, and improves the inference time by three orders of magnitude. This advancement provides essential technical support for downstream structural damage assessments.

•Combined with virtual city technology, BlastGraphNet can function as a real-time decision support tool for swiftly evaluating the response of virtual building facilities during explosion events, thereby establishing a foundation for the rapid deployment of protective measures.

2. Methodologies

The issues related to explosions and their impact on buildings, as discussed in this article, can be summarized as follows: given a set of explosion conditions—including the geometric dimensions of a building, the location of the explosion center, the explosive yield, the type of explosive, and other relevant parameters—various physical outputs related to explosions, such as overpressure and impulses, can be derived. By employing precise computational methods grounded in physical models, an external explosion dataset for buildings can be created to train our proposed model. By utilizing a graph neural network framework and a gradient-based training method, the BlastGraphNet model is developed to accurately predict explosive loads on the surfaces of buildings with varying structures, effectively replacing traditional numerical simulation methods.

2.1. Physical model

The propagation of shock waves is a compressible flow process that necessitates solving the compressible Navier–Stokes equations. The size of the computational domain is

100 m (l e n g t h) \times 100 m (w i d t h) \times 50 m (h e i g h t)

. The open-source software OpenFOAM (OpenCFD Ltd., UK) is employed for mesh discretization of the explosion area, and its built-in density-based solver is used to calculate the corresponding fields. The buildings of various shapes and sizes are randomly distributed within the computational domain. Since this study focuses primarily on the adaptability of the model to buildings with different geometries, the explosion center position in the established Cartesian coordinate system is fixed at the point

(X, Y, Z) = (50 m, 50 m, 5 m)

. Here, X, Y, and Z represent the locations of the explosion, grid nodes, and building centers, respectively. The trinitrotoluene (TNT) equivalent for the central explosion is 200 kg, and the relevant dimensions of the computational domain are shown in Fig. 1. Additionally, to simulate real-life conditions more accurately, the boundary of the computational domain (with the exception of the ground) is a nonreflecting outflow condition. With this setup, the distributions of pressure and other parameters for different buildings during an explosion event can be effectively simulated.

Exploded chemicals release a significant amount of energy when they are detonated in the air, with most of this energy being converted to shock waves. These blast waves are critical considerations in protective design. Fig. 2 shows the pressure–time history at a specific distance from the detonation center. The pressure variation during an explosion typically consists of positive and negative phases. In the positive phase, the pressure exceeds the atmospheric pressure

p_{0}

. The shock wave exerts a strong thrust on the building surface, and the pressure

p

decreases exponentially during propagation until it returns to atmospheric pressure at the end of this phase. The positive phase lasts for approximately

t_{+}

. In the negative phase, the pressure gradually decreases below atmospheric pressure, creating a suction force on the building surface. The negative phase duration

t_{-}

is longer than the positive phase duration

t_{+}

. In most protective studies, positive phase pressure is the primary factor contributing to building damage. Therefore, our study focuses primarily on collecting data and constructing models for positive phase pressure. To maximize the predictive efficiency of the model, the study simplifies the overpressure time history curve, as shown in Fig. 2. When the overpressure time sequence during the positive phase of the shock wave is analyzed, the key parameters are the peak relative overpressure

p_{r, m a x}

, the peak positive phase impulse

i_{+}

(

i_{+} = \int_{t_{a}}^{t_{a} + t_{+}} p_{r} d t

), and the wavefront arrival time

t_{a}

p_{r}

represents the relative pressure,

t

is the time, and

\int_{t_{a}}^{t_{a} + t_{+}} p_{r} d t

denotes the time integral of the relative pressure from

t_{a}

t_{a} + t_{+}

. The positive phase of the shock wave is further approximated as a triangular wave. For simplicity, the peak relative overpressure

p_{r, m a x}

, peak positive phase impulse

i_{+}

, and wavefront arrival time

t_{a}

will be referred to as peak overpressure, peak impulse, and arrival time, respectively, throughout this paper. By determining the peak overpressure, peak impulse, and wavefront arrival time, the model can accurately reconstruct the overpressure time history for all nodes.

2.2. Data construction

The construction of the training dataset significantly impacts the generalization ability of the explosion prediction model. Therefore, this paper presents a building generator designed to create constant-section columnar buildings of various sizes. The main process involves ① determining the number of endpoints to generate a random polygon and ② establishing the building height to stretch the building in the vertical direction. The polygonal column cross section has endpoints ranging from 3 to 7, while the building height varies from

15

30 m

. Consequently, 4000 samples of diverse building configurations were generated for training.

To better evaluate the geometric adaptability of the explosion framework, this work prepared three types of test datasets, each featuring distinct geometric characteristics of the buildings. The dimensions and spatial distribution of the buildings within the explosion domain are shown in Fig. 3. The details are as follows:

(1) The geometric features align with the training set, which consists of 500 samples. Each sample includes columnar buildings with randomly generated polygonal cross-sections, which are utilized to evaluate the model convergence during training and to reconstruct the global node overpressure time history.

(2) Rotational geometry. The building size is fixed at

30 m (l e n g t h) \times 20 m (w i d t h) \times 30 m (h e i g h t)

, and the building rotates around its geometric center by an angle

θ

, which represents the counterclockwise rotation angle of the structure about its center. The building center is located at (X, Y) =

(50 m, 75 m)

, as shown in Fig. 3(a).

(3) Multibody geometry. As shown in Fig. 3(b), the number of buildings varies from 2 to 4. The buildings are cuboids with arbitrary rectangular cross-sections, with lengths and widths ranging from 20 to 40 m and heights ranging from

15

30 m

. Preventing building overlap during the generation of multiple structures is crucial. A total of 100 multibuilding samples were generated for testing.

(4) Complex geometry. The building cross-sections reveal complex geometries, and the center of the building is situated at (X, Y) =

(50 m, 75 m)

and has a fixed height of

30 m

. Fig. 3(c) shows several relatively irregular building shapes, such as petal-shaped, oval, “V,” and “bullet-shaped.”

Notably, complex buildings were excluded from the model training process for several reasons. First, this decision was made to expedite the training, as complex buildings typically introduce more nodes and intricate connections. The use of simple buildings for training increases the stability and efficiency of the model development process. Furthermore, in real-world engineering, the complexity of buildings often varies significantly from that of the training dataset. The objective is to demonstrate that the model exhibits strong robustness and generalizability when it is confronted with more complex structures. Consequently, simple buildings were employed for training, whereas multibody and other structurally complex samples were reserved for testing.

2.3. Model framework

The proposed BlastGraphNet is fundamentally a graph autoencoder that encodes a known graph into a high-dimensional latent vector and decodes the learned latent features to reconstruct target attributes. The target geometry is represented by a set of

N

nodes

V_{x}

, where N represents the number of nodes and x = 1, 2, …, N. Specifically,

V_{x} = (v_{1}, v_{2}, ..., v_{N})

. The model directly predicts and reconstructs the target physical fields

V_{y} = (y_{1}, y_{2}, ..., y_{N})

, where y = 1, 2, …, N. Physical quantities like overpressure are among these target physical fields. In this context,

x

represents the model’s input and

y

represents its output. The main components of the model include graph construction, an encoding layer, a processing layer, and a decoding layer, as shown in Fig. 4.

2.3.1. Graph construction

The construction of graph data is essential for understanding the interactions and dependencies among nodes. Therefore, creating graph data suitable for blast simulations from the mesh nodes extracted during numerical simulation is crucial for achieving an accurate blast field. The original mesh generated during traditional numerical simulations can be viewed as a graph

G_{m e s h} = G (v_{m e s h}, e_{m e s h})

, where

v_{m e s h}

corresponds to the physical domain node and

e_{m e s h}

corresponds to the physical domain edges. All the physical domain nodes were used as input nodes

V_{x}

. To establish edge connections, a “connectivity radius”

R

is defined for all nodes. If the distance between nodes features

v_{i}

and

v_{j}

$‖ p_{i} - p_{j} ‖ < R$ , where

$p_{i}$ and

$p_{j}$ represent the 3D spatial coordinates of nodes

$i$ and

$j$ , respectively, they are considered paired nodes, and a connecting edge is added. The larger the connectivity radius is, the more nodes are connected to the target node. To achieve absolute spatial invariance [28,29], the edge feature between the

$i$ th and

$j$ th nodes is defined as

$r_{ij} = [(p_{i} - p_{j}), ‖ p_{i} - p_{j} ‖]$ , where

$(p_{i} - p_{j})$ is the relative displacement between nodes and

$‖ p_{i} - p_{j} ‖$ is the norm of the relative displacement. The input node feature

$v_{i}$ encompasses the relative position, distance from the blast center, and class information. The relative position is represented as a six-dimensional feature vector that indicates the distance between the node and the boundaries in the computational domain, and the class information is used to differentiate between the ground and the buildings in the computational domain.

2.3.2. Encoder

The primary function of the encoder is to provide an embedded representation of the initial graph, encoding the node features and connection information in the initial graph, capturing their local features, and ultimately returning the latent graph

$G_{0}$ . The encoder includes a node encoding

$f_{v} (v_{i})$ and an edge encoding

$f_{e} (r_{ij})$ , which embed the node feature

$v_{i}$ and the edge feature

$r_{ij}$ , respectively. The embedded node feature is represented as

$v_{i}^{0} = f_{v} (v_{i})$ , and the embedded edge latent feature is represented as

$e_{ij}^{0} = f_{e} (r_{ij})$ . In this study, the node and edge encoding functions are composed of a two-layer MLP, with 64 neurons in the hidden layer.

2.3.3. Processor

The processor is composed of

$M$ cascaded message-passing blocks with identical structures. M represents the number of message-passing iterations. Each message-passing block is a Graph-Net (GN) with independent parameters. The input of the processor is the latent graph

$G_{0}$ , which is embedded by the encoder. After

$M$ iterations of message passing, the processor returns the final latent graph

$G_{M}$ . For the

$m$ th GN block, with the input latent graph

$G_{m}$ , after information aggregation and updates, it outputs a new latent graph

$G_{m + 1}$ . Specifically,

$G_{m + 1} = G N_{m + 1} (G_{m})$ . During this process, the embedded edge features

$e_{ij}^{m + 1}$ and the embedded node features

$v_{i}^{m + 1}$ are updated as follows:

(1)$e_{i j}^{m+1} \leftarrow g_{\mathrm{e}}\left(e_{i j}^{m}, v_{i}^{m}, v_{j}^{m}\right), v_{i}^{m+1} \leftarrow g_{\mathrm{v}}\left(v_{i}^{m}, \sum e_{i \Delta}^{m+1}\right)$

where

$v_{i}^{m}$ and

$v_{j}^{m}$ represent the features of the nodes

$i$ and

$j$ at the latent graph

$G_{m}$ ,

$e_{i Δ}^{m + 1}$ represents any edge connecting node

$i$ to other nodes,

$Δ$ represents the index of any node connected to node

$i$ . Both

$g_{e}$ and

$g_{v}$ are two-layer MLPs with residual networks. During each message-passing process, information from adjacent nodes is aggregated into the embedded feature of the connecting edge, all edge embedding information connected to the target node is aggregated, and then the embedded feature of the target node is updated. From a “field of view” perspective, the number of message updates

$M$ determines the range of information received by the nodes, which enables the processing layer to observe the overall structure of the graph during training. Similarly, the value of

$M$ determines the model complexity. In this study, the number of hidden layer neurons in

$g_{e}$ and

$g_{v}$ within the GN block is set to 128, whereas the number of message-passing iterations

$M$ is 6.

2.3.4. Decoder

To obtain the desired physical information of the nodes, the latent graph

$G_{M}$ returned by the processing layer is transformed into target features, namely,

$y_{i} = h_{v} (v_{i}^{M})$ , where

$y_{i}$ is the ground truth obtained from numerical simulations at node

$i$ ,

$h_{v}$ is the node decoding mechanism for extracting the target information, and

$v_{i}^{m}$ is the feature of the ith node in the final latent graph

$G_{M}$ . Through the decoding layer, the model outputs the actual physical field information, thereby completing the prediction tasks for peak overpressure, peak impulse, and wavefront arrival time. In this study, the decoding layer comprises a two-layer MLP with 64 neurons in the hidden layer.

2.4. Training

To increase the training efficiency of the model and ensure spatial rotational invariance, the study implemented augmentation operations on the input data. Given the symmetric characteristics of the computational domain, the input samples were rotated at angles of 90°, 180°, and 270°. This graph data augmentation accelerates the model convergence and provides a degree of robustness to spatial rotation. Furthermore, considering the substantial differences in the data scale distributions among the samples, this work applied standardization to the physical field to reduce the impact of the data scale on model training and further expedite model convergence.

The mean squared error (MSE) was utilized as the objective function, defined as

$L = \frac{1}{N} \sum_{i}^{N} {(y_{i} - {\hat{y}}_{i})}^{2}$ , and the model parameters were optimized using the Adam optimizer.

${\hat{y}}_{i}$ represents the predicted physical field results from the BlastGraphNet model, and

$L$ represents the loss function used to update the model parameters. The training process consisted of 300 000 gradient descent updates. During training, the learning rate decayed exponentially, starting at

$10^{- 4}$ , and decreased by a factor of 0.1 every 100 000 updates. The explosion simulation framework was developed using PyTorch and PyTorch Geometric [30]. Additionally, distributed training [31,32] was employed across two NVIDIA RTX 4090 graphics processing units (GPUs) to train the model for overpressure, impulse, and wavefront arrival times, resulting in a total training duration of approximately 60 h.

3. Results

To evaluate the predictions of the physical fields, the relative mean squared error (RMSE) and the coefficient of determination

$R^{2}$ were employed as evaluation metrics to effectively compare the model performance across different test datasets [33].

The RMSE (

$ε$ ) is calculated as follows:

(2)$\varepsilon=\frac{\sum\left(y_{i}-\widehat{y}_{i}\right)^{2}}{\sum\left(\widehat{y}_{i}\right)^{2}}$

where

${\hat{y}}_{i}$ and

$y_{i}$ are nonstandardized data, which reveals that they are reverted to their original scale for error calculation. As training progresses, the RMSE decreases significantly, as shown in Fig. 5.

$ε_{m e a n}$ represents the average RMSE of the dataset, including both the train and test data.

The peak overpressure and peak impulse decay in a power-law fashion with increasing standoff distance. Using the overpressure as an example, the region near the blast center is a high-pressure area, where the overpressure is typically two to three orders of magnitude greater than that in low-pressure areas. This discrepancy ultimately results in significantly higher prediction error scales in high-pressure regions than in low-pressure regions. Therefore, by utilizing the RMSE, the impact of uneven data scales can be effectively mitigated, facilitating a more accurate comparison of model performance under varying conditions.

The

$R^{2}$ coefficient measures the linear correlation between the predicted value and the true value in the test samples. The calculation is as follows:

(3)$R^{2}=1-\frac{\sum_{i=1}^{N}\left(y_{i}-\widehat{y}_{i}\right)^{2}}{\sum_{i=1}^{N}\left(y_{i}-\bar{y}_{i}\right)^{2}}$

where

${\bar{y}}_{i}$ is the average of all true values. The closer the

$R^{2}$ value is to one, the greater the linear correlation between the predicted value and the true values, indicating a more accurate prediction.

3.1. Performance of BlastGraphNet on the test set

3.1.1. Prediction results

The proposed built explosion graph neural network model, BlastGraphNet, employs an end-to-end prediction strategy. This approach allows it to provide the global node physical field distribution during an explosion event directly, using relevant building geometric information and explosion conditions as inputs. During the evaluation process, three samples were extracted from the test set, and their physical field distribution cloud maps were plotted, as shown in Fig. 6. Fig. 7 presents their

$R^{2}$ values. The BlastGraphNet models have a strong ability to capture fine-grained features of the shock wave propagation process and exhibit consistent predictive capability across different attributes, such as ground and buildings.

During the propagation of shock waves, the coupling phenomenon between the incident wave from an explosion and the ground-reflected wave is particularly pronounced, resulting in Mach pressure waves that act on the building surface. Within the height range of the Mach wave front, the synthetic wave can be approximated as a uniformly distributed plane wave. When this plane wave reaches the building surface, a reflection effect occurs, which increases the peak pressure and impulse experienced by the structure. In our dataset, all test samples have a blast center height of 5 m, which is lower than the height of the building, and the explosion occurs relatively close to the structure. Consequently, the coupling effect of the synthetic wave is quite significant. The proposed BlastGraphNet simulates the interaction between the building and the ground through information exchange among different nodes, demonstrating the enhanced load effect on the structure due to shock wave coupling. As shown more intuitively in Fig. 8, significant pressure is generated at the base of the building as a result of a near-ground explosion. Along the building height, the ground reflection effect diminishes rapidly, and the overpressure decreases continuously. Notably, BlastGraphNet achieved an average error of 1.8% on the test set, with a single physical field inference time of less than 0.1 s. This performance is highly significant for downstream applications, such as multiple explosion simulations.

Unlike typical autoregressive-based transient explosion simulations [26], an end-to-end learning strategy is used in this work. By simplifying the explosion model and directly learning key parameters of the global physical field, such as peak overpressure, accurate reconstruction of the global node overpressure curve can be achieved. Two nodes on the blast-facing surface of the building were selected, and the pressure time–history curve at these specific locations was reconstructed by integrating their key information, including peak overpressure, peak impulse, and wavefront arrival time, as shown in Fig. 9. The red and black lines represent the shock triangular waveforms reconstructed from the key parameters, such as peak overpressure, obtained through numerical simulation and BlastGraphNet, respectively. The gray line depicts the pressure sampling time series at the target point during the numerical simulation process. Although our method of selecting key parameters may not be suitable for addressing multiple peaks and overlooks the pressure rise process of the impulse, the primary focus for some subsequent overpressure applications remains the initial peak overpressure and impulse. Therefore, our reconstruction of the building surface pressure time series adequately meets the requirements of state-of-the-art damage assessments.

3.1.2. Sampling method

The advantage of graph neural networks lies in their ability to handle unstructured data effectively. In this study, the building file needs to be converted to a mesh space. The current process of converting geometries to nodes is carried out primarily through meshing, which is essentially an extension and iterative process based on the initial mesh, ensuring structural consistency and regularity. However, for new buildings, node generation needs to be performed again, and the time cost of this process cannot be disregarded. To further test the expressive ability of the model when unstructured data are involved and improve its prediction efficiency, the surface random sampling method was adopted, i.e., randomly selecting points on the building surface to generate nodes and inputting them into the BlastGraphNet model. Fig. 10 shows the overpressure distributions for meshing and different random sampling quantities (10 000 and 50 000 samples, respectively).

Compared with the precision and uniformity of grid-generated nodes, randomly sampled nodes are more disordered. Nonetheless, BlastGraphNet employs a “dynamic” processing strategy that determines the edge points connected to the target node on the basis of the connectivity radius

$R$ . This approach enables the model to effectively capture global geometric and physical information between nodes, even with disordered node data. Furthermore, by sacrificing some precision and consistency, the time required for random surface sampling is significantly reduced. Generating nodes through meshing requires approximately 90 s, whereas random sampling of 50 000 nodes only requires 0.8 s, resulting in a speed increase of two orders of magnitude. This improvement is highly significant for predicting explosions in unknown buildings.

The model prediction error was further tested with different numbers of sampled nodes for peak overpressure and peak impulse. As shown in Fig. 11, as the number of sampled nodes increases, the distance between nodes decreases, and their spatial distribution becomes denser. The model can observe more node dependencies, and the error with the baseline decreases. BlastGraphNet still maintains a certain prediction accuracy overall on the randomly sampled node samples, effectively reducing the time compared with that associated with meshing. This provides greater flexibility and creativity for simulating explosions in unknown buildings, thereby enhancing the inference efficiency of the model during real-life engineering deployment.

3.1.3. Model comparison and parameter optimization

Models capable of predicting blast load distributions on complex buildings include convolutional neural networks and transformers. CNNs excel at processing regular grids and capturing local information, whereas transformers are effective at capturing global dependencies but have high computational complexity and lower efficiency when large-scale data are managed. In contrast, graph neural network models are better suited for unstructured data, as the information propagation between nodes more closely resembles numerical simulation methods.

This work focused on the evaluation of GNN-based models, with a particular emphasis on comparing the traditional graph convolutional network (GCN) to the proposed BlastGraphNet. The tests revealed that the GCN model exhibited an average error of 3.32% on the test set samples, whereas BlastGraphNet achieved a significantly lower prediction error of 1.8%. As shown in Fig. 12, BlastGraphNet outperformed GCN in areas with dense grid nodes, such as building edges, and provided more accurate overpressure predictions in localized regions. These results highlight the substantial advantages of BlastGraphNet in managing complex building geometries and capturing detailed local information.

Additionally, the work conducted a series of parameter settings, focusing primarily on hyperparameter searches for the information update steps

$M$ and the connection radius

$R$ , as shown in Table 1. First, increasing the number of message-passing steps

$M$ improved the performance on the test set. This enhancement is likely attributed to the larger

$M$ values, which capture longer-range and more complex interactions between nodes. However, increasing

$M$ also results in longer computation times, and the performance gains are not substantial. Consequently, the article ultimately identified six as the optimal number of message-passing steps. Similarly, a broader connection radius

$R$ resulted in lower errors. A larger neighborhood facilitates long-distance communication between graph nodes, thereby expanding the scope of the model and reducing the average error on the test set. However, as the connection radius

$R$ increases, the number of edges also increases, which requires more computation and memory. Therefore, the connection radius

$R$ is ultimately set to one in this paper.

3.2. Rotational geometries

To analyze the effects of the blast center position and building position on the explosion effects, a special numerical experiment was designed in this work, as shown in Fig. 13. By keeping the geometric dimensions of the building fixed and rotating them around its central axis, the orientation of the building relative to the explosion source can be altered. This method allows us to systematically analyze the response of the model to explosions from different angles while maintaining the inherent characteristics of the building.

When the building rotates around its central axis, the relative positions of the building nodes and the blast center change significantly. This alteration affects the reflected shock wave impacting the building structure, particularly the destructive effects of the reflected shock wave on the overpressure at the base of the building. Fig. 14 shows the physical fields, including overpressure and impulses, at various rotation angles. The visualization results clearly reveal how the rotation angle influences the response of the building to the reflected shock wave, especially with respect to overpressure effects at the base. Table 2 presents the performance of BlastGraphNet for buildings at different rotation angles, with overall errors remaining below 4%. To further analyze and validate the model performance, the predicted results for the nodes on the building surface were extracted, and linear fitting was performed, as shown in Fig. 15. The model performance decreases in certain low-pressure areas, which is considered inevitable because of the significant scale differences in the explosion physical field data, particularly with large numerical values in high-pressure regions. This results in a phenomenon where more attention is given to high-pressure areas in BlastGraphNet, resulting in relatively higher accuracy in those regions.

3.3. Complex geometries

To further evaluate the generalization performance of BlastGraphNet on various building designs, this work proactively supplied the model with more challenging test samples that are entirely distinct from the training set. The uniqueness of these samples lies in their unconventional cross-sectional shapes. As shown in Fig. 16, the cross-sections of buildings are shaped like bullets, V-shapes, ellipses, petals, or circles. For all test samples, the height and central position of the buildings were held constant to ensure that the model focused on changes in the building shapes during inference.

Table 3 presents the prediction errors of BlastGraphNet for various complex geometries, whereas Fig. 17 shows the distribution of the physical fields of the load. The increased complexity of buildings poses a considerable challenge to model predictions. For the petal-shaped structure, the error in the overpressure prediction task increases significantly primarily due to the enhanced reflection effect of shock waves in the recessed areas of the building surface, resulting in a coupling effect of the shock waves, which decreases the model performance. Nevertheless, since the model has learned sufficient combinatorial dependencies between points and edges, it still shows considerable extrapolation ability for relatively complex building explosions.

The longitudinal axis of the physical field distribution at the closest point of the building closest to the explosion center is plotted in Fig. 18. BlastGraphNet captures the gradient changes in overpressure at the base of the building effectively, further confirming that it establishes an intrinsic connection between the building configuration and the distribution of physical fields, such as overpressure.

3.4. Multibody geometries

In real-life explosion scenarios, multiple buildings (obstacles) are always presented, so it is essential to test the performance of the BlastGraphNet model with building cluster samples. The number of buildings in the test set ranges between two and four, as shown in Fig. 19. Each individual building is a standard cuboid that is randomly distributed within the computational domain.

When the focus of study shifts from a single building to clusters of buildings, the difficulty of making predictions significantly increases. With more buildings, the blocking effects on shock waves, as well as the reflection and diffraction effects of shock waves among adjacent structures, become more pronounced, resulting in more irregular distributions of physical fields, such as overpressure and impulses. Additionally, the increase in the number of buildings affects the scale of the graph data directly. For example, with four buildings, the number of sampled nodes can exceed 100 000, and the graph topology will change significantly. Despite these challenges, BlastGraphNet still shows satisfactory prediction performance, as shown in Fig. 20. For both the reflected overpressure on the buildings and ground and the pressure in adjacent areas, the pressure distribution cloud maps remain highly consistent with those of the numerical simulations. During training, the model effectively captured the interactions between ground nodes and building nodes through multiple rounds of information aggregation. These interactions persist in multibuilding samples and are even stronger than the relationships among building nodes, ultimately ensuring the extrapolation ability of BlastGraphNet in multibuilding scenarios.

When multibuilding inference is performed, the inference efficiency of the data-driven model is even more remarkable. Table 4 lists the average error and computation time for the test sets with different numbers of buildings. With two buildings, the inference speed of the model increased by three orders of magnitude, and with four buildings, the inference speed of the model decreased because of the increased scale of the graph data. However, the inference time is less than one second, which is still within the acceptable range compared with that of numerical simulations.

4. Engineering applications of BlastGraphNet

Accurate prediction of blast overpressure loads is essential for downstream engineering applications, including structural damage assessment and civil defense engineering. To further elucidate the feasibility of implementing the proposed building blast prediction model, BlastGraphNet, in engineering, it was tested in two specific applications: structural damage assessment and virtual city explosion effect evaluation.

4.1. Damage assessment

Buildings experience varying degrees of damage after an explosion. Quickly assessing structural damage is necessary for building design and explosion event evaluation. To establish the damage conditions of building structures under different explosion energies and distances, the academic community generally uses experiments, finite element analysis, and empirical relationships to investigate blast response modes. The pressure–impulse (P–I) curve has become a widely utilized design tool [2,34] for simplifying the blast effects through a series of hyperbolic curves that help ascertain the degree of structural damage. The P–I damage curve proposed by Prugh [35] is used in this study to evaluate damage under different blast overpressure fields, as shown in Table 5. The shock wave relative pressure

$p_{r}$ is expressed in

$k P a$ , and the positive impulse

$i_{+}$ is expressed in

$k P a \cdot s$ .

A test sample from Section 3.1 was extracted to assess the structural damage in buildings during an explosion event. As shown in Fig. 21(a), the study focused on the damage distribution on the ground and the blast-facing surface of the building using the P–I curve. Fig. 21(b) shows the different damage levels within the P–I space. By projecting building elements into the P–I space, the potential damage level of the structure under the explosion load was predicted on the basis of the regions they fall into, enabling rapid assessment of the damage levels for all building elements.

Fig. 22, Fig. 23 show the overpressure, impulse, and shock wave arrival time contours on the ground and building surfaces, respectively, and different damage curves. These figures intuitively reveal the impact of overpressure on the structural integrity of the building and depict the potential damage levels the building might experience under different explosion intensities. The importance of this study lies in the precise real-time calculation characteristics of BlastGraphNet, which provides critical theoretical foundations and practical tools for building design, urban planning, and emergency management. This allows experts in relevant fields to assess and respond to potential explosion risks much more effectively. These damage curves reveal the high accuracy and practicality of the model on a technical level and play a crucial role in safety assessment and prevention strategy formulation.

4.2. Virtual city explosion simulation

The emergence of urban virtualization technology provides important technical support for assessing the interactive building explosion effect. In this section, a new virtual city is constructed based on Beirut Port in Lebanon. Using the CADMapper framework [36,37], city area information, including details such as buildings, roads, and terrain, was captured. Appropriate surface smoothing, building optimization, and redundant clipping were also performed [38]. Additionally, some features of the original Beirut Port were modified by adding new elements such as lakes and rivers while preserving the distinctive characteristics of the port and removing certain buildings. Ultimately, a new virtual city named MetaBeirut, approximately

$1.5 k m$ in diameter and modeled after Beirut, was created.

In MetaBeirut, eight locations store the equivalent of

$200 k g$ of TNT in energetic materials. To conduct a safety assessment of these locations, researchers have integrated the proposed data-driven building explosion prediction model BlastGraphNet to simulate explosion incidents rapidly within the MetaBeirut virtual city.

The configurations of the buildings in the explosion areas were input into BlastGraphNet to predict the effects of explosions. Consequently, the predicted overpressure fields are displayed in the digital city scene, as shown in Fig. 24. Table 6 further compared the computational efficiency of two explosion assessment methods: numerical simulation and data-driven modeling. Our test results showed that BlastGraphNet integrates effectively with the virtual city twin system. Although performance decreases in certain multibuilding samples, such as Explosions 6 and 8, where an increase in the number of buildings reduced the efficiency of capturing the overpressure field, the initial capabilities of the model for predicting explosion effects on real buildings were elucidated, especially for single buildings. Furthermore, compared with numerical simulations, our data-driven evaluation method based on BlastGraphNet is much more efficient, with the evaluation time for a single city explosion event being less than one second. The precise real-time prediction capabilities of BlastGraphNet can be leveraged for analyzing the effects of multiple explosion events in large cities, provided that essential technical support for subsequent vulnerability assessments of urban buildings is provided.

5. Conclusions

This paper presents a method for constructing a digital twin model of explosion fields based on graph neural networks designed to predict the distribution of blast overpressure on building exteriors rapidly. By simplifying the explosion model, identifying key parameters in the positive phase of the overpressure time history curve, and utilizing a message-passing graph neural network in an autoencoder format, the model can accurately and efficiently predict global physical fields, including the overpressure, impulse, and wavefront arrival times. For the test set whose distribution is similar to that of the training set, the prediction error is less than 2%, and the inference speed is 3 to 4 orders of magnitude faster than that of the numerical simulation. In generalization tests, the model showed strong extrapolation ability, performing well on challenging test samples such as rotated buildings, buildings with complex geometries, and building clusters. Compared with autoregressive graph neural network models that simulate full spatiotemporal physical fields, our method simplifies the model by directly predicting key parameters in the overpressure curve, such as peak overpressure and impulses. This simplification effectively reduces the computational cost associated with focusing on time terms. As a result, our method achieves rapid and accurate reconstruction of the pressure time history curve. Additionally, initial deployment capabilities for engineering applications of the model, which is capable of being integrated with downstream applications such as structural damage assessment, were highlighted. By coupling the P–I damage criterion with the intelligent building explosion prediction model, the building damage mode can be directly assessed. Using a real-world background, this work constructed a digital city that virtualizes infrastructure such as buildings and road networks. By combining the digital city platform with the proposed intelligent building explosion prediction model, rapid assessment of the explosion effects on virtual buildings can be achieved, providing a new methodology for future urban safety protection.

This groundbreaking research offers new insights for future studies on blast damage. For example, incorporating explosion events with varying heights and yields into a dataset could facilitate the development of a more generalized and physically adaptable data-driven intelligent model. Furthermore, by integrating optimization algorithms with BlastGraphNet, a rapid inversion framework (effects→causalities) [39] can be established. This framework enables the reverse assessment of explosion yield, size, and location on the basis of blast load distribution, which is crucial for postexplosion predictions and site investigations. Additionally, using a cascaded network architecture in conjunction with large-scale graph data training techniques could enhance the predictive capabilities of the model for blast field issues in extensive building areas, which is vital for creating more comprehensive explosion prediction models.

CRediT authorship contribution statement

Zhiqiao Wang: Writing – original draft, Methodology, Data curation. Jiangzhou Peng: Software. Jie Hu: Software. Mingchuan Wang: Writing – review & editing, Supervision. Xiaoli Rong: Writing – review & editing. Leixiang Bian: Supervision. Mingyang Wang: Supervision. Yong He: Supervision. Weitao Wu: Writing – review & editing, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (U2241285).

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Abstract

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Cite this article

1. Introduction

2. Methodologies

2.1. Physical model

2.2. Data construction

2.3. Model framework

2.3.1. Graph construction

2.3.2. Encoder

2.3.3. Processor

2.3.4. Decoder

2.4. Training

3. Results

3.1. Performance of BlastGraphNet on the test set

3.1.1. Prediction results

3.1.2. Sampling method

3.1.3. Model comparison and parameter optimization

3.2. Rotational geometries

3.3. Complex geometries

3.4. Multibody geometries

4. Engineering applications of BlastGraphNet

4.1. Damage assessment

4.2. Virtual city explosion simulation

5. Conclusions

CRediT authorship contribution statement

Declaration of competing interest

Acknowledgments

References

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