Graph Convolutional Network Surrogate Model for Mesh-Based Structure-Borne Noise Simulation

Abstract

This study presents a unique method of building a surrogate model using a graph convolutional network (GCN) for mesh-based structure-borne noise analysis of a fluid–structure coupled system. Structure-borne noise generated from irregular shape panel vibration and sound pressure was measured in a closed-volume cavity coupled with the panel. The proposed network was trained to predict the sound pressure level with three steps. The first step is predicting the natural frequency of panels and cavities using the graph convolutional network, the second step is to predict the averaged vibration and acoustic response of the panel and cavity, respectively, in a given excitation condition using a triangular wave-type inference function based on the natural frequency predicted from the first step, and the third step is to predict the sound pressure in a cavity using a panel and cavity average response as an input to a 2D convolutional neural network (CNN). This method is an efficient way to build a surrogate model for predicting the response of a system which consisted of several sub-systems, like a full vehicle system model. We predicted the response of each sub-system and then combined this to obtain the response of the whole system. Using this method, an average 0.86 r-square value was achieved to predict the panel-induced structure-borne noise in a cavity from 10 to 500 Hz range in 1/12 octave band. This study is the first step towards creating a surrogate model of an engineering system with various sub-systems by changing it into a heterogeneous graph.

1. Introduction

Deep learning has been adopted and showed remarkable results in various areas, like object detection and recognition, natural language processing, healthcare and medical applications, self-driving cars, and so on [1]. The commonality of these fields is to simulate human cognitive ability, which is generally a difficult field to solve in a theoretical way. Nowadays, deep learning has begun to be applied to the field of finding solutions, where it has been used as analytical formulas and achieved considerable results. A typical example is physics-informed machine learning [2], while research on building a surrogate model using deep learning and a graph neural network is also actively underway [3,4,5].

A conventional surrogate model requires that all training data should have the same parameterization. And, if the model used for prediction has a different parameter than the pre-trained surrogate model, there is a disadvantage that the pre-trained surrogate model could not be applied to other models which have different parameters. A surrogate model using deep learning can overcome this obstacle.

Building a surrogate model using deep learning has been mainly focused on the parameterized model, for which the system can be represented by moderate amounts of parameters. For example, machine learning algorithms proposed to estimate the time-varying response of a one-dimensional beam [6] and graph-based surrogate model were proposed for trusses [7] to predict the displacement field from static loads given the structure’s geometry as an input and also explore transfer learning within the context of engineering design. But the physics model built for engineering applications generally uses the finite element method (FEM) which has numerous parameters like nodes which have location information and connections to represent the complicated structure’s shape. To learn the parameter of this complicated model shape, graph neural networks (GNNs) [8,9] are the most promising because they can learn in non-Euclidian space and have node-invariant characteristics. Among GNNs, graph convolutional networks (GCNs) [10] were selected for use in this study as a surrogate model of mesh-based structure-borne noise analysis because they can explicitly take into account the connectivity patterns between nodes when performing the convolution operation.

FEM has been used for various engineering applications and also for structure-borne noise analysis. It has numerous numbers of nodes and meshes to represent the structure’s geometry in a non-Euclidian space. An Euclidian-space-based method like a CNN (convolutional neural network) [11] and PointNet [12] has a limitation in terms of the irregular location of each of the nodes of the FEM, but the GNN holds a strength in this area. Surrogate model building using the GNN is an emerging area, and the application of the engineering problem is expanding nowadays [13].

Another main focus of this study is the application of the GCN-based surrogate model to the system, for which several sub-systems are combined. Like a vehicle model, it consists of a lot of sub-systems, and they are combined to make up a total system. Up to now, building a surrogate model for a finite element model has been focused on the single component to which all elements are connected homogeneously, and the excitation and response location which is within the same system [3,13]. However, in engineering problems, several components are connected with various kinds of connection like a spring, bolt, coupling structure with an interior cavity, and so on. This study proposes a GCN-CNN architecture for building a surrogate model for a panel–cavity coupled system response by combining each sub-system response. The concept of building a surrogate model of each sub-system and then combining to the total system is an effective way to predict the total system response of various sub-system combinations.

In this study, irregular shape panels were generated using morphing and predicting natural frequencies using the GCN. Based on the predicted frequency, a triangular shape inference function which was based on predicted frequencies were generated prior to the panel and cavity average response. The inference function was used as an input of the neural network to predict the panel’s averaged vibration response. In the same way, cavity natural frequency was predicted using the GCN and the average pressure transfer function from the area where coupling with the panel to measurement point was predicted using the inference function based on the cavity natural frequency. The inference function is a domain-knowledge-based proposal, for which the main response peak has a strong relationship with the system’s natural frequency, and it works efficiently to predict the system’s response with a limited number of datasets. Panel average vibration response and cavity average acoustic response are aligned to 2D images as input data, and the structure-borne noise of the whole system was predicted using 2D-CNN.

The proposed model trained for the finite element panel–cavity model to minimize the mean squared error loss for the prediction of sound pressure level from an approximately 10 to 500 Hz range in the 1/12 octave band in a cavity. Various kinds of shapes/sizes of the panel and cavity were used for the training, validation, and test. Using this proposed method, predicted structure-borne noise shows an 0.858 R-Square value.

2. Problem Definition

The purpose of this study is to build a surrogate model for a mesh-based structure-borne noise analysis model using a combination of a graph neural network, neural network, and CNN. Figure 1 shows a mesh-based structure-borne noise analysis model for this study, and it consists of a panel and cavity. The panel boundary is connected to the rigid element which has a clamped boundary condition. The impact force is applied in a vertical direction at a rigid element independent node to the excite panel. Panel vibration excites the cavity boundary where the panel is adjacent to the cavity. The panel and cavity are coupled where the panel is adjacent to the cavity. The cavity has a rigid boundary condition, except for the area which is facing the panel.

3. Background Theory

3.1. Graph Convolutional Networks

Graph convolutional neural networks (GCNs) are a type of neural network architecture designed to operate graph-structured data. The key idea of GCNs is to generalize the convolution operation from regular grids, such as images, to irregular graph structures. So, it can work directly on graphs and takes features of their structural information, and graph convolution is defined as Equation (1) [8].

$$H^{\left(k\right)}=\sigma \left({\widehat{D}}^{-\frac{1}{2}}\widehat{A}{\widehat{D}}^{-\frac{1}{2}}{H}^{\left(k-1\right)}{W}^{\left(k\right)}\right)$$

(1)

where H^(k) and H^(k−1) are the k-th k − 1st layer hidden representation, respectively, $\widehat{A }$ is the graph adjacency matrix which A added with self-connections I, and $\widehat{D }$ is the degree matrix of $\widehat{A }$ with the identity matrix. $\widehat{A }$ is normalized using $\widehat{D }$. σ is a non-linear activation function and W is a trainable weight matrix for the k-th layer with the dimension of a number of features and weights.

The GCN equation consists of three parts. Graph convolution, normalization, and non-linearity. The graph convolutional operation is defined as a weighted sum of the neighboring nodes feature vectors, with the weights being determined by the graph structure. The normalization step ensures the outputs have a unit norm, which is important for the stability of the learning process. Non-linearity introduces non-linearities into the model, allowing it to capture complex relationships in the data.

Equation (2) shows the graph neural network (GNN) [8]. A major difference between the graph neural network (GNN) and graph convolutional neural networks (GCNs) is weight parameter sharing.

$$H^{\left(k\right)}=\sigma (A{H}^{\left(k-1\right)}{W}_{neigh}^{\left(k\right)}+H^{\left(k-1\right)}{W}_{self}^{\left(k\right)})$$

(2)

The GCN has a common weight matrix for both the self and neighbor node, but the GNN has a separate weight for them. The GCN down-weights on the high-degree neighbors by normalizing the adjacency matrix, including self-connections.

3.2. Structure-Borne Noise Analysis

To predict the interior sound pressure caused by structural vibration, the finite element models representing the enclosed cavity and the structural panels can be coupled. The coupling accounts for the structural vibrations forcing the interior acoustic response, and also the loading of the acoustic pressure excites the panels.

The coupled acoustic–structure equations of the motion are expressed as Equation (3) [14,15].

$$\left[\begin{array}{cc}M& 0\\ A^T& Q\end{array}\right]\left\{\begin{array}{c}\ddot{w}\\ \ddot{p}\end{array}\right\}+\left[\begin{array}{cc}C& 0\\ 0& D\end{array}\right]\left\{\begin{array}{c}\dot{w}\\ \dot{p}\end{array}\right\}+\left[\begin{array}{cc}K& -A\\ 0& H\end{array}\right]\left\{\begin{array}{c}w\\ p\end{array}\right\} = \left\{\begin{array}{c}F\\ G\end{array}\right\}$$

(3)

where

$$F_i=F_i^E-{\int }_A{p}^E{N}_{i}dA G_i={\int }_V\dot{qi}{N}_i^{A}dV$$

where $\ddot{w}$, $\dot{w}$, and $w$ are the matrix of nodal acceleration, velocity, and displacement at the boundary, respectively, N is the row matrix of interpolation functions for the structural displacements in the normal direction of the surface, D is the impedance damping matrix, superscript E indicates external load or pressure, and q is the volume velocity of any interior sound source. The upper row of Equation (3) represents the structure and the lower row represents the acoustic finite-element equation of motion, respectively, and they are coupled using structural–acoustic coupling matrix A which was obtained by assembling the integrations over the appropriate elements on A.

In this study, the structure and acoustic response is converted to the surrogate model using the GCN, inference function, CNN, and neural network. Coupling matrix A is represented by the CNN, which is input from the structure and acoustic cavity.

4. Proposed Method

4.1. Overview of Proposed Method

The proposed method in this paper is predicting sound pressure level in a closed cavity excited by the panel vibration using a GCN-CNN-based surrogate model, which is shown in Figure 2.

The system to make a surrogate model is composed of two sub-systems, a panel and a cavity. Each sub-system’s natural frequency was predicted using the GCN outlined in Step 1.

The average response of each system was predicted by following Step 2 using the inference function, which is based on the natural frequency from Step 1. The acoustic response in a cavity was caused by panel vibration predicted using the combined average response of each system using the 2D CNN in Step 3.

The proposed model was developed using PyG (PyTorch Geometric) [16], which built upon PyTorch.

4.2. Predicting Natural Freuency Using GCN—Step 1

Graph convolutional neural networks (GCNs) were adopted to predict the natural frequency of panels and cavities. Figure 3 shows the GCN structure for this study. Inputs are the adjacency matrix, which defines the connectivity of nodes, and three features, which means the panel/cavity FE model’s node location (x,y,z). For panel natural frequency prediction, skip connection was added and Relu/Elu activation functions were used alternately to improve the accuracy. Both GCN models use graph classification with a read-out layer, which uses a global mean.

4.3. Predicting Average Response of Sub-system Using Inference Function—Step 2

The panel excited by a unit force in the vertical direction up to a 500 Hz range and the vibration response were averaged of all panel nodes. For cavity average response, the cavity was excited by the acoustic source at the measurement point and response average of the pressure from all nodes where the panel coupled.

The average response function of the panel and cavity was predicted using the inference function as an input of the neural network. The inference function is a type of wave function which was devised by the physical knowledge of the structure response in the frequency domain. The frequency of major response peaks from the panel/cavity are caused by the panel/cavity modes and the corresponding natural frequency of them. The inference function for this study is shown in Figure 4, and it is expressed with Equation (4).

$$\begin{array}{c}0\le f\le f_1 y=\frac{1}{f_1}f\hfill \\ f_i<f\le {(f}_{i+1}{-f}_i)/2 y=-\frac{2}{{(f}_{i+1}{-f}_i)}f+1+\frac{2f_i}{{(f}_{i+1}{-f}_i)}\hfill \\ (f_{i+1}-f_i)/2<f\le f_{i+1} y=\frac{2}{{(f}_{i+1}{-f}_i)}f+1-\frac{2f_{i+1}}{{(f}_{i+1}{-f}_i)}\hfill \\ f_5<f y=1 i=\mathrm{1,2},\mathrm{3,4}\hfill \end{array}$$

(4)

System response peak location in frequency domain depends on the natural frequency, and the magnitude of response depends on the radiation efficiency of each mode.

Figure 5 shows the neural network for predicting average response. The frequency location is guided by the inference function as input data, and the level of response highly depends on the mode shape and the neural network trained to predict the level of response according to the order of the peaks. The number of input and output of the neural network was 62. This is the number of the 1/12 octave band from approximately 10 to 500 Hz.

4.4. Predicting Average Response of Panel–Cavity System Using CNN—Step 3

Step 3 is the final stage of panel–cavity coupled structure-borne noise prediction. The average response from the panel and cavity is 1D data in a frequency domain. One-dimensional data can be stacked to two-dimensional data, which have a 2 × 62 size and are fed into the 2D CNN network to express the coupling behavior of two sub-system responses, as shown in Equation (3). Details of the 2D CNN network are shown in Figure 6. Two-dimensional convolution filters were adopted to the first and second layer, and a one-dimensional filter was used for the third CNN layer. Convolution filter column sizes of 32, 16, and 8 were used from the first to third layer, respectively.

5. Experimentation and Results

5.1. Dataset–Panel

The panel training set has four panel sizes, 0.4 m × 0.6 m, 0.5 m × 0.6 m, 0.6 m × 0.6 m, and 0.7 m × 0.6 m, and its thickness is 2 mm. The number of panels for training and validation is 320, for which there are 80 samples for each panel size and 20 samples for testing. The testing samples were build based on the four panel sizes which are used for training the data, but they have a different curvature.

The panel has an arbitrary shape with morphing. All panels have a different curvature and depth by moving the panel nodes in the panel normal direction randomly using four morphing handles on a panel center area, as shown in a Figure 7. The maximum height of the morphed panel node is 10 mm.

Figure 8 shows the natural frequency and mode shape of different sizes of flat panels and morphed panels. The flat panel’s first natural frequency of each size of the panels is 115 Hz, 84 Hz, 68 Hz, and 60 Hz, respectively, in a clamped boundary condition, and the natural frequency of the panel decreases as the panel size increases. The morphed panel’s first natural frequency of four panel sizes is 124~194 Hz, 93~148 Hz, 84~130 Hz, and 71~113 Hz, respectively.

The morphed panel natural frequency and mode shape in Figure 8 are samples from the training set. The panel mode shape is a major factor which decides the panel radiation efficiency [17]. Morphed panel natural frequency generally increased and the mode shape changed because of the increased effective thickness and curvature caused by morphing. In the flat panel, depending on the frequency, the generated mode shape follows a certain rule according to the aspect ratio of the panel. But in the morphed panel, some mode shapes cannot be seen in the flat panel. This phenomenon is intensified when the effective thickness due to curvature is large compared to the panel size. As shown in Figure 8, it is more possible for the 0.4 m × 0.6 m-sized panel to have an abnormal mode shape than the 0.7 m × 0.6 m-sized panel.

5.2. Dataset—Cavity

The cavity training set also has four cavity sizes, 1.2 m × 1.0 m × 0.8 m, 1.3 m × 0.8 m × 0.8 m, 1.5 m × 1.0 m × 0.9 m, and 1.6 m × 1.3 m × 1.0 m, and the number of training/validation samples is 320 and 20. Samples for test were built based on the four cavity sizes which were used for training the dataset, but the shape is different with training data.

The cavity dataset generated by cutting the volume arbitrary based on the rectangular-box-type cavity is shown in Figure 9. Figure 10 shows the rectangular-box-type cavity finite element model and one sample of the arbitrary shape in the first column. The first to the fifth mode shape and natural frequency are arranged in the order of columns. Cavity natural frequency is inversely proportional to the boundary length of each direction, as explained in Equation (5), and the mode shapes are expressed by Equation (6) [18]. Where Lx, Ly, and Lz are the cavity boundary lengths in each direction, i, j, and k are the number of nodal points of the cavity in each direction, and c is the speed of sound.

$$f_{ijk}=\frac{c}{2}\sqrt{{\left(\frac{i}{L_x}\right)}^2+{\left(\frac{j}{L_y}\right)}^2{+\left(\frac{k}{L_z}\right)}^2}$$

(5)

$${\mathsf{\Psi }}_{ijk}=cos\frac{i\pi x}{L_x}cos\frac{j\pi y}{L_y}cos\frac{k\pi z}{L_z}$$

(6)

The cavity dataset is augmented by cutting the volume from the box-type cavity, so the cavity dataset has a higher frequency than the box-type cavity. But the cavity mode shape does not change significantly compared with the morphed panels.

The first natural frequency range of each size of the cavity dataset is 143~187 Hz, 132~173 Hz, 115~143 Hz, and 108~127 Hz, respectively.

5.3. Training and Results—Step 1

The panel and cavity natural frequency were predicted using the GCN with the following parameters: learning rate 0.0004, weight decay 0.002, and number of epochs 800 for panel and 0.003, 0.0015, and 700 for cavity, respectively. Figure 11 shows the accuracy of the predicted natural frequency of the morphed panels.

The frequency range of the first to fifth natural frequency of all sizes of morphed panels is from 71 Hz to 430 Hz. The r-square value of the proposed method is about 0.96.

The cavity natural frequency and mode shape depends on the effective length of each facet, as explained in Equation (5), and the adjacency matrix for the GCN to predict the cavity natural frequency was setup based on the node connectivity of the 2D element, which represents the outer shape of the cavity to reduce the matrix size and setup time instead of the node connectivity of the 3D element.

Figure 12 shows the accuracy of the predicted natural frequency of the cavity. The frequency range of first to fifth natural frequency of all sizes of cavities which are to be predicted with a proposed GCN is from 71 Hz to 430 Hz. The -r-square value of the proposed method is about 0.97.

Generally, low-frequency modes are less sensitive to shape change than higher-frequency modes. In this study, the cavity first mode frequency is distributed below the 200 Hz range, so the high-frequency range in Figure 12b relatively shows more discrepancy than the low-frequency range.

5.4. Training and Results—Step 2

The inference function based on the predicted natural frequency in Step 1 was used as input data to predict the average response of the panel and cavity.

Figure 13 and Figure 14 show the predicted average response of the panel and cavity, respectively, with a target value for each of the 20 test data. The average r-square values of the 20 test data of the panel and cavity are 0.912 and 0.914, respectively. The predicted response is well correlated with the target data, but some peaks show a difference. This is mainly caused by the difference in the predicted natural frequency in Step 1. Even though the r-square value of the predicted natural frequency is over 0.96, some predicted frequency differences are up to 20 Hz, where the panel natural frequency is around 200~300 Hz in the test data of Figure 11. In the case of the panel response of test no. 1 and test no. 4 in Figure 13, in which the r-square value is 0.8856 and 0.7506, respectively, the target data first peaks are located over the 200 Hz range and are not well correlated.

The average response of the cavity in Figure 14 shows a better correlation than the panel in Figure 13 because the natural frequency prediction in Step 1, as shown in Figure 12b, is better than Figure 11b. The cavity frequency is much more influenced by cutting the cavity volume, but this does not have much effect on the mode shape, as shown in Figure 10.

5.5. Training and Results—Step 3

Step 3 is predicting the panel-vibration-induced structure-borne noise in a cavity. The average response from the panel and cavity in Step 2 is used as the input data. These two data combine to size of 2 × 62 and are input into the 2D CNN. The two-dimensional CNN is used to express the coupled acoustic–structure in Equation (3) as a neural network.

The number of training/validation data was doubled to 640 samples by shifting the input data to the left or right up to 2 Hz randomly as a noise to improve the accuracy. In case of panel–cavity coupled structure-borne noise, the coupled response is affected by the frequency position of each system response peak. Therefore, in order to increase the robustness of the error of the peak position that would be included in the response of each system, the frequency of each system response peak was randomly adjusted up to 2 Hz while the target response was maintained.

The Adam optimizer was used, and hyper parameters of 0.0001, 0.0003, and 1000 denote the learning rate, weight decay, and epoch, respectively.

Using the proposed method, the average r-square value of the predicted structure-borne noise is 0.86. Some test data show a low r-square value, like tests no. 4 and 17 in Figure 15. They are mainly related to the low r-square value of the sub-system, such as in tests no. 4 and 17 in Figure 13, which were used as an input for Step 3.

6. Discussion

Up to now, the surrogate model using a neural network for a finite element model has been focused on a single component, for which all the elements are connected homogeneously. In this study, the GCN-CNN network was proposed to predict panel-vibration-induced structure-borne noise in a cavity by combining the responses from each sub-system to predict the full-system response.

The GCN was used to predict the natural frequency of the morphed panel and various shape of the cavity, which are the finite element model. The prediction accuracy of the panel and cavity was 0.956 and 0.974 r-square values, respectively.

To predict the vibration and acoustic response up to 500 Hz in the 1/12 octave band, a triangular-shaped inference function, for which the peak frequencies are based on the predicted frequency using the GCN, was adopted. This is an efficient way to predict the frequency response using a small number of datasets, and it achieves the r-square value 0.912 for the panel and 0.914 for the cavity.

The 2D-CNN was used to represent the coupling effect between the panel and cavity to predict structure-borne noise in a cavity. Using this proposed method, an 0.858 average r-square value was achieved.

In this study, the graph classification method for the GCN was adopted to predict the natural frequency, and then the knowledge-based inference function was used to predict the components average response. This is a novel approach considering the limited number of datasets in a real engineering problem and how the size of the data can be reduced for the loss function.

Table 1. Proposed method, predictions, and the achieved r-square values at each step.

	Step 1	Step 2	Step 3
Proposed method	GCN	Inference function Neural network	CNN
Predictions	Natural frequency	Average response	Interior noise
R-Square value	0.956 (Panel)	0.912 (Panel)	0.858
	0.974 (Cavity)	0.914 (Cavity)

summarizes the proposed method, predictions, and the achieved r-square values at each step.

Another proposed option is to predict the frequency response directly because the graph classification method aggregates the node embeddings into a single embedding using the global mean in a readout layer, where the single embedding is the target.

A potential further research direction could be adding a node classification method for the GCN to learn the mode shape depending on the changes in the panel and cavity shape, which could lead to a direct prediction of frequency response using the GCN. The graph classification method which was used in this study aggregates the node embeddings into a single embedding using the global mean in a readout layer, so it is unclear whether the trained individual features of the node reflect the actual panel behavior. To predict the frequency response, each mode shape should be predicted because the panel radiation efficiency depends on the mode shape. Adding a node classification to the selected node could be an efficient way. But there is a need to find a balance between the amount of data to compare for the loss function and the prediction accuracy.

This study is the first step towards creating a surrogate model of an engineering system with various sub-systems by changing it into a heterogeneous graph. A graph network for more diverse sub-system combinations and connection methods will be developed in future.