Uncertainty Quantification of Vibroacoustics with Deep Neural Networks and Catmull–Clark Subdivision Surfaces

Zhou, Zhongbin; Gao, Yunfei; Cheng, Yu; Ma, Yujing; Wen, Xin; Sun, Pengfei; Yu, Peng; Hu, Zhongming

doi:https://doi.org/10.1155/2024/7926619

Shock and Vibration

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 7926619 | https://doi.org/10.1155/2024/7926619

Uncertainty Quantification of Vibroacoustics with Deep Neural Networks and Catmull–Clark Subdivision Surfaces

Zhongbin Zhou,^1,2Yunfei Gao ,²Yu Cheng,^2,3Yujing Ma,^2,3Xin Wen,⁴Pengfei Sun,¹Peng Yu,⁵and Zhongming Hu²

Academic Editor: Arcanjo Lenzi

Received29 Sept 2023

Revised15 Dec 2023

Accepted25 Jan 2024

Published10 Feb 2024

Abstract

This study proposes an uncertainty quantification method based on deep neural networks and Catmull–Clark subdivision surfaces for vibroacoustic problems. The deep neural networks are utilized as a surrogate model to efficiently generate samples for stochastic analysis. The training data are obtained from numerical simulation by coupling the isogeometric finite element method and the isogeometric boundary element method. In the simulation, the geometric models are constructed with Catmull–Clark subdivision surfaces, and meantime, the physical fields are discretized with the same spline functions as used in geometric modelling. Multiple deep neural networks are trained to predict the sound pressure response for various parameters with different numbers and dimensions in vibroacoustic problems. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

1. Introduction

Thin-shell structures manifest an evident vibroacoustic interaction phenomenon, which has a significant impact on structural safety and noise control. Considering the inherent randomness of material properties, geometry, and loading [1–7], it is necessary to quantify these uncertainties to evaluate structural reliability in engineering [8]. The commonly used uncertainty quantification methods include Monte Carlo simulation (MCs) [9–15], stochastic spectral methods [16, 17], and perturbation methods [18]. Among them, MCs is the most versatile and easiest to implement, but it requires a large number of samples to ensure sufficient accuracy [19]. Hence, MCs is feasible in practice only when simulation can be conducted very efficiently.

A surrogate model can accurately fit numerical calculation results and consume less computing power, thereby speeding up the sample generation process and alleviating the issue of inadequate samples during Monte Carlo simulation. Well-known surrogate models include radial basis functions (RBFs) [20–23], polynomial chaos expansions [24–26], Gaussian processes [27–30], and relevance vector machines [31, 32]. Each of these models has its own advantages and is widely used. Nonetheless, when the number of random variables increases, the effectiveness of these approaches significantly diminishes.

The state-of-the-art surrogate model, deep neural network (DNN), can effectively address the issue of excessive input dimensions. Huang [33] et al. conducted a comparative analysis for different regression surrogate models in energy transmission quality forecasting, which reveals that DNN serves as the most suitable regression model. Jung [34] et al. constructed a DNN surrogate model to predict propellant combustion characteristics, in which the high-dimensional inputs are captured throughout stochastic analysis. Kontou [35] et al. have effectively minimized the optimization turnaround time by employing DNN for turbulence closure in Reynolds-averaged Navier–Stokes equations. The results obtained by the DNN model agree well with the solution by classical numerical methods. Shahriari [36] et al. used DNN as a surrogate model of Maxwell’s equations to infer geological conditions from resistivity. Sun [37] et al. used 15 input variables to estimate the variable infiltration capacity of soil moisture. Comparative analysis of models validates the superior performance of DNN. Moreover, many efforts are made to improving DNN by incorporating physical information [38–43].

In order to construct a reliable DNN model, high-quality training data should be generated from high-performance simulation. The finite element method (FEM) is a powerful method in vibration analysis. However, the boundary element method (BEM) is superior to FEM in calculating sound waves in exterior domains [44–49], as it discretizes only the structural surfaces without truncating the unbounded domain. Therefore, combining these two methods can significantly enhance calculation efficiency and accuracy [50]. Moreover, traditional FEM/BEM employs polygonal elements, which deteriorate geometric accuracy and lead to heavy meshing burden. The isogeometric analysis (IGA) [51–54] can address this challenge by integrating numerical analysis and computer-aided design (CAD). Coupling the FEM/BEM in the context of isogeometric analysis further improves the computing performance.

In this paper, a novel DNN-accelerated MCs framework is proposed for uncertainty quantification. Following the work of [55], the Catmull–Clark subdivison is applied as a surface refinement method. We add density and Young’s modulus as independent variables in DNN inputs and expand the input dimension to 5 in calculation of a sphere model. We add density, Young’s modulus, and Poisson ratio in the analysis of the submarine model. As shown in Figure 1, the method is mainly comprised of two parts. The first part is isogeometric FEM/BEM, which simulates vibroacoustic systems from the geometric model of Catmull–Clark subdivision surfaces. The simulation results provide training data for DNN. It is noted that compared to loop subdivision surfaces as used in [55], the Catmull–Clark subdivision surface allows for quadrilateral elements. The second part of the uncertainty analysis framework is that DNN is employed as a surrogate model to rapidly generate samples for MCs. Multiple DNN models are created due to different types of input variables, and the statistical indicators of MCs include expectation and standard deviation.

The remainder of this paper is as follows: Section 2 describes MCs and presents the physics problem. Section 3 describes a brief DNN theory. Section 4 presents the IGA FEM/BEM with the Catmull–Clark subdivision surface. Section 5 applies sphere and submarine models to verify the proposed method. Finally, in Section 6, the conclusion is provided.

2. MCs and Structureacoustic Coupling Problems

The Monte Carlo simulation (MCs) can learn about a system through a large number of random samples affected by stochastic parameters. By calculating statistics, for example, expectation and standard deviation, MCs can be applied to stochastic analysis, which is shown as follows:where is the number of samples, is the independent variable, and denotes the -th sample.

In general, MCs mainly consists of the following four processes:(1)Determine the physics problem and its influencing factors(2)Collect or generate sample data through various means(3)Calculate the physical responses of each sample point(4)Calculate statistical indicators for uncertainty quantification

The MCs is suitable for multidimensional input problems and less limited by geometric conditions. Its convergence speed is independent of the dimension, and errors are easy to determine. Thus, it is suitable for stochastic analysis of vibroacoustic coupling problems. The mathematical model of the problem is shown in Figure 2. In the figure, is the thin-shell structure, is the fluid domain, and is the coupling surface. The Kirchhoff–Love shell theory and the Helmholtz equation are used as a governor for thin-shell mechanical behavior and radiation or scattered acoustic field, respectively.

We consider the input plane wave or a time-harmonic force with an angular frequency and use subscript as random parameters. It is essential to solve the following coupled equations [56, 57]:where equations (3) and (4) are the governing equations, (5) and (6) represent force balance and displacement continuity, , , and are the Laplace operator, stress tensor, and displacements, respectively, , , and are the density, the sound pressure, and the the fluid normal velocity, and are the normal vectors, and and are the imaginary unit and the wave number, respectively.

Before presenting the specific numerical calculation methods, we will show how to calculate the statistical characteristics of responses.

Let denote the random variables, which iswhere and are input bounds and is the input dimension. We select values from to form samples , that iswhere . Then, utilizing IGA FEM/BEM, the response of each point can be evaluated as follows:

Finally, the dataset can be generated by assembling and as follows:

Although MCs has numerous advantages, its accuracy heavily relies on the number of samples, which results in significant wastage of computing resources. With complexity and influencing factors increasing, numerical calculations become highly time-consuming and put a heavy burden on computer performance. Furthermore, to maintain the accuracy of stochastic analysis, increasing the number of sampling points becomes necessary, which makes it challenging to obtain the response merely through numerical simulation. Therefore, a cost-effective and time-efficient method is required to overcome this drawback.

3. DNN Surrogate Model

As mentioned previously, the process of finding solutions for all samples in Monte Carlo simulation is typically time-consuming and requires significant computing power. Thus, to solve this problem, in this section, we propose a DNN surrogate model to accelerate stochastic analysis. It can enlarge the initial dataset of generated by numerical calculations to dataset of as follows:

Now, let us provide a brief introduction for DNN (please see [58–61] for detailed information). By utilizing a specific mapping function , DNN can use input of multitype and any amount to predict output of multitype and any amount. To acquire a best , the original DNN model needs to be trained, and this process is usually guided by the loss function. Assuming that the initial dataset with samples is the training dataset, in which is the input data, is the real value (target) needed to be compared with predictions (outputs). After training, the best model can then use larger amounts of inputs to generate corresponding outputs, which is the process of expansion of the initial dataset for MCs.

Moreover, a general DNN model is shown in Figure 3. According to it, let , , and denote the vector of input layer, hidden layer, and output layer, respectively, the forward calculation process of DNN can be expressed by following equations:where and are the weights and bias of linear transformation between layers and , , and are the number of neurons, number of hidden layers, and activation function, respectively.

The response predicting by DNN of the vibroacoustic problem is a regression process. The mean square error (MSE) is a typical loss function for this process. Assuming that the prediction is ( for -th unit), MSE is as follows:

By selecting an appropriate learning rate, the optimal weights and biases can be acquired through a chosen optimizer. It is difficult to fully evaluate the performance of the DNN model from all perspectives with a single evaluation metric. Therefore, additional metrics such as RMSE (root mean squared error), MAE (mean absolute error), and MAPE (mean absolute percentage error) are introduced to assess the performance of the trained model, which arewhere and are the prediction and the real value, respectively, and and are the number of testing points and testing data, respectively.

By employing both MAE and RMSE, one can assess the extent of dispersion of sample error. Likewise, when using MAPE and MAE in conjunction with the average value of prediction, the fitting accuracy across samples of varying magnitudes can be evaluated. More clearly, a DNN-based stochastic analysis process is illustrated in Figure 4. The steps in the figure can be concluded into following three parts:(i)Dataset preparation: Using statistical data to standardize the dataset , where and represent the input and output of , respectively, the corresponding and denote the standardized value, and denote the mean value and standard deviation of inputs, and is the coefficient of variation. Meanwhile, and are the mean value and the standard deviation of outputs. Finally, the exact dataset can be rewritten as(ii)Training and testing: The dataset in equation (18) is divided into two groups for training and testing. The mapping function is acquired through the training dataset, and then, the testing dataset is used to assess the trained model. Finally, postprocessing of is conducted, which is as follows: where denotes the j-th postprocessed value, denotes the initial output of the DNN model, and and are priori data in equation (17).(iii)Application: After the function has been constructed, it can be used to predict the responses using existing or randomly generated data as inputs . Then, postprocessing should be conducted to get the meaningful value. Finally, the dataset with samples obtained by numerical calculations is enlarged to one with samples generated by DNN through the above operations.

4. IGA FEM/BEM with Catmull–Clark Subdivision Surfaces

In order to obtain the initial samples of good quality for DNN training, it is necessary to choose an appropriate numerical calculation method. In this work, IGA FEM/BEM with the Catmull–Clark subdivision surface method is introduced to acquire responses of random generated samples. In the following subsections, the Catmull–Clark subdivision surfaces will be introduced first; then, IGA FEM and IGA BEM methods are demonstrated, respectively. Finally, the coupled method is presented.

4.1. Catmull–Clark Subdivision Surface Scheme

The subdivision surface method has drawn significant attention due to its effectiveness in shaping smooth limit surfaces. Moreover, the compatibility of the subdivision surface method with the shell finite element method or the boundary element method is natural since they all focus on the structural surface. The Catmull–Clark subdivision surface method is one of the self-adaptive mesh technologies. Compared to other similar methods, it can maintain structural model consistency with no geometric model error during automatic mesh refinement (increasing the mesh number). Thus, the Catmull–Clark subdivision algorithm with the bicubic B-spline is applied to refine the control meshes.

The subdivision process divides each element into four subquadrilateral elements, as illustrated in Figure 5. The refinement of the quadrilateral mesh can be seen in Figure 6. To explain its rules, let us assume the initial mesh subdivision level , and after one subdivision, it becomes the level . The vertices denoted by “V” on the level are weighted to determine the -level vertices. At each element, four new vertices are inserted at the middle of edges, denoted by “E.” In addition, a new vertex is inserted at the center of the -level element, denoted by “F.” The edge passing through “V” on the level is the adjacent edge. The valence of a vertex represents the number of adjacent edges. A vertex is classified as a regular vertex if ; otherwise, it is an irregular vertex, as shown in Figure 7. The interpolation weights are illustrated in Figure 6.

To build smooth surfaces, bicubic box splines are applied. The patch of a regular element contains 16 vertices (Figure 7). The shape of the mapped surface is expressed as follows [62]:where is the point coordinates, is its parameter coordinate, is the basis function, and is the control vertex coordinate. Detailed formulas for can be found in [62].

Figure 8 shows how to deal with irregular elements, and the fitting point will finally be located in regular subelements after a limited number of subdivisions. Then, equation (20) can be used to acquire the shape of the mapped surface.

4.2. IGA FEM with Catmull–Clark Subdivision Surfaces

According to the Kirchhoff–Love shell theory, the thin-shell variational formulation can be written as follows [57, 63]:where the three terms , , and are the virtual work contributions, internal, and external forces, respectively. If the midsurface displacements are time harmonic, they can be rewritten aswhere is material constants, , , and are Young’s modulus, Poisson ratio, and the shell thickness, respectively, and are the membrane and bending strains, respectively, and is the boundary.

After the governor equation is determined, the Catmull–Clark subdivision surface is introduced to FEM [64]. To discrete the displacement, the basis functions are used as follows:where is the nodal parameter. Therefore, after element matrices are assembled, the equations system is as follows:where is the coefficient matrix of FEM, and are the stiffness matrix and mass matrix, is the displacement, and the load is the sum of (the mechanical force) and (the sound pressure load) as follows:

4.3. IGA BEM with Catmull–Clark Subdivision Surfaces

Based on the Burton–Miller formulation, Helmholtz equation (4) can be transformed to the following boundary integral equation [65–69]:where (for ) is the combining parameter, is the jump term, is the acoustic pressure of the incident wave, and are the normal derivative of and , respectively, and , , , and are the boundary integral operators, which are as follows:where and are unit outward normal, respectively. (Green’s function) and the derivative are defined as follows:where .

Then, the boundary is discreted by quadrilateral elements and constructed by IGA principles. In this work, the applications of basis functions of the Catmull–Clark subdivision surface are as follows:where and are sound pressure and normal flux and subscripts and are for the element and node. Afterwards, the following equation is assigned to a collection of collocation points, which arewhere the quantity represents the element number. The placement is acquired through fitting operations. Moreover, by gathering the equations for all collocation points and presenting them in matrix form, the linear system of equations can be obtained, which is as follows:where and are the coefficient matrices, and are the sound pressure and its flux, respectively, and is the incident wave.

It should be emphasized that solving equation (30) directly using the Gaussian quadrature method can be challenging as a result of singular integrals present in the equation. Thus, we use the regularization technique to deal with the problem. For more detailed information, please refer to [70].

4.4. IGA FEM/BEM with Catmull–Clark Subdivision Surfaces

Since , , , and are unknown, equations (24) and (31) cannot be solved individually. They can be bond together through the boundary conditions in equations (5) and (6). In addition, the nodal force in equation (25) is related to as follows:where , is the global vector, and is the normals matrix which is defined as [57, 63]:where are the orthogonal normal Cartesian base vectors. Therefore, the new expression of equation (25) can be acquired through equation (32) as follows:where .

Substituting equation (34) into equation (24), the structural system equation can be expressed as follows: (sound flux) in equation (31) is given as , where is the fluid density and . Then, equation (31) can be rewritten as

Substituting equation (35) into equation (36), the displacement can be eliminated; then, the boundary element formulation is as follows [57, 63]:with

In this work, direct computation of the inverse matrix in equation (38) is avoided due to its high time complexity. Instead, the solution of can be obtained by directly solving the algebraic equation .

An important drawback of the boundary element method is the formation of and , which leads to algorithmic operations when the number of degrees of freedom is large. In this work, the FMM is utilized to address this issue for its low memory consumption and high accuracy. In addition, to further improve the speed of the algorithmic operations, a multipole expansion formulation based on partial wave expansion with rotation-coaxial translation is implemented. The original boundary integral can be divided into two parts: the near and the far field. The former is computed using the boundary element method (BEM), while the latter is computed using the accelerated BEM based on the fast multipole method (FMBEM). For more detailed information about the FMM, please refer to [70–73].

5. Calculation Examples for Verification

5.1. Verification of IGA FEM/BEM

In this section, a series of spherical models are used to validate the numerical simulation algorithm. The model is present in Figure 9. The external effect is incident plane waves with amplitude 1. The IGA FEM/BEM method is employed to handle the problem, and the obtained solution is compared with the analytical one. We use and Gaussian quadrature points for regular and singular integrals, respectively.

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Figure 10 shows the results of sound pressure calculated with different variables and frequency at the computing point located on the fluid domain. Table 1 displays the variable settings for each case, where each case has a different variable value relative to Case 1. In Figure 10, the symbol “Simulation” represents the IGA FEM/BEM solution and “Analysis” represents the analytical solution. All calculations are performed in the lower frequency band, such as Hz. Since the error becomes larger at higher frequencies (for details, see [55]), the high frequency cases are not considered in this work.

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Observing the figure, it can be found that the two solutions exhibit a high level of agreement, which provides evidence for the effectiveness of the proposed method. The deviations at some certain frequencies like Hz are caused by nonunique solutions, and the details can also be seen in [55]. In addition, with the change of different variables, the change trend of sound pressure as well as the peak position and occurrence times will change. In comparison, the radius and Young’s modulus have a more substantial impact on sound pressure values, while other parameters have an impact that is less apparent. Sound pressure cloud diagrams are given in Figure 11. The figure once again confirms that different random parameters have a significant impact on the sound pressure value and its distribution.

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In order to get further information about the specific evolution trends of sound pressure with different variables, analytical calculations and numerical calculations are conducted employing the radius and other variables as independent variables. Figures 12 and 13 illustrate the sound pressure changes with different variables at the fixed calculating point (5, 0, 0). It shows that there is a decreasing trend of sound pressure with radius increasing, while with other variables increasing, the sound pressure is growing as a whole. Due to the strong correlation between sound pressure and shown variables, it is worth using these variables to perform stochastic analysis of sound pressure. At the same time, it can also be found that the results of numerical simulation and theoretical analysis are still in agreement with different independent variables.

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5.2. DNN-MCs with Sphere Models

5.2.1. Dataset Generation

The simulation results in Section 5.1 are chosen to form the datasets for DNN. Therefore, the output of both model training and model testing is merely sound pressure. The differences between the datasets are the type and number of input variables. All input variables are generated with a normal distribution by the built-in random function. In order to avoid that the sound pressure distribution is not obvious due to excessive parameter changes, the maximum of the coefficient is 0.2, and the input variables change within a reasonable range. Table 2 lists all the parameters details. The principle is used to determine parameter variation ranges.

The datasets are specifically divided into two types: one with a single input is called “1-D” dataset, and the other with 5 inputs is called “5-D” dataset. These inputs are variables mentioned in Section 5.1 and also present in Table 2. The number of initial sampling points is 100 and 500 for 1-D and 5-D analysis, respectively. The calculation frequency is 50 Hz and 300 Hz because the numerical calculation method has high accuracy and the sound pressure changes smoothly at these two frequencies. The calculation point of the sound pressure is still at (5, 0, 0), which is outside the sphere and not at infinity.

According to equation (17), all inputs and outputs are standardized using built-in functions for DNN training and testing. The processed data conform to the standard normal distribution.

5.2.2. Model Training and Testing

Each dataset is automatically divided into training and testing datasets by a random process. Some training parameters are shown in Table 3. The ratio of the training set to test set is 9 : 1. The number of hidden layers for 1-D input and 5-D inputs is 2 and 3, respectively. Since it is a simple regression problem, the optimizer is chosen as SGD. Different learning rates are picked for datasets of different dimensions. All processes are performed by an Intel(R) Xeon(R) Bronze 3204 CPU with PyTorch.

The loss function is MSE. Table 4 gives the loss change. For the 1-D input network, its loss converges faster than that of the 5-D input network (mainly because a bigger learning rate). Usually after 10 iterations, MSE can reach a quite low level for 1-D inputs, while the number of iterations is 100 for 5-D inputs. Finally, all losses have reached a low level, and all model training processes take less than 10 minutes.

For 1-D input and 5-D inputs, the prediction accuracies are verified by 10 and 50 data, respectively. Figures 14 and 15 give the comparison between the simulation and the DNN prediction at different frequencies. Except that in the comparison chart with shell thickness as a variable (Figures 14(b) and 15(b)) where there is a certain gap between the predicted results and the numerical calculation results in a small range, all of the rest of the subfigures show that the two solutions are in good agreement.

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More specific statistical results calculated through equations (14)–(16) for DNN prediction evaluation are shown in Table 5 (all values are calculated after prediction destandardization), from which it could be found that the dispersion of prediction errors of each model is small and the model fits well to samples of different orders of magnitude. To sum up, the prediction effects of the trained models for different situations are good, and DNN can be used as an effective surrogate model to replace IGA FEM/BEM to generate samples.

5.2.3. MCs with Samples Generated by DNN

After obtaining a DNN model with a good fitting effect, it can be further utilized for uncertainty quantification analysis. Here, results of MCs with sound pressure generated by IGA FEM/BEM are used as a comparison. Table 6 shows the details for this. The new input data are also generated through the principle. For one-dimensional input, in equations (1) and (2) are 100 and 1,000 for IGA FEM/BEM and DNN, respectively. For five-dimensional inputs, is 500 and 2000.

Table 6 shows that the results of DNN are consistent with the IGA FEM/BEM results. As for calculating time contrast, the time of 1-dimensional input (per group) and 5-dimensional inputs of the IGA FEM/BEM method is about 3 hours and 12 hours, respectively, while for DNN, the calculation time is less than 1 minute regardless of dimensions (meanwhile, the amount of data is several times that of the numerical method). This shows the time advantage of DNN computing.

In conclusion, the DNN model is well-suited for uncertainty quantification, especially for multidimensional and large-scale inputs. The proposed method offers the advantage of eliminating the scale problem associated with the IGA FEM/BEM method and significantly reducing the calculation time.

5.3. DNN-MCs with the Complicated Model

The complicated thin-shell submarine model is used to prove the stochastic analysis ability of MCs with DNN in this section. The geometry model and its subdivision surface are shown in Figure 16. The loading conditions are similar to those of the sphere model. The response is calculated at fixed points (i.e., ). DNN training data are still generated by IGA FEM/BEM, and numerical calculation result cloud diagrams are shown in Figure 17. It can be found that the sound pressure distribution varies across different frequencies. Figure 18 shows the evolution curve of sound pressure with frequency, from which we can see that 100 Hz is suitable for sampling. Because the sound pressure exhibits an obvious change and clear trend near this point, consequently, conducting calculations at this point can obtain data of higher quality.

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For DNN training, the input variables are shell thickness, material density, Young’s modulus, and Poisson ratio. The output value is sound pressure. All input variables are generated with a normal distribution by the built-in random function. The mean of four types of inputs is 0.05 m, 7860 , 210 GPa, and 0.3, respectively. The coefficients of variation for each variable include 0.05, 0.1, 0.015, and 0.2. 50 samples are generated through IGA FEM/BEM for each coefficient of variation.

Due to the complexity of the model, adjusting the hyperparameters of the DNN training process becomes necessary for achieving the optimal fitting network. Figure 19 shows how the training loss varies with different hyperparameters. Among different activation functions, tanh demonstrates faster convergence speed and avoids loss oscillation in later stages. As the learning rate decreases, the convergence speed of the model drops significantly, but the accuracy does not show a significant improvement. In comparison to the other two optimizers, despite not being the fastest in terms of convergence speed, SGD does not exhibit oscillation in later iterations, making it more effective overall. As the complexity of the model increases, the convergence speed initially rises and then declines, and the model with four hidden layers has the most favorable effect. Based on this, the activation function, learning rate, optimizer, and number of hidden layers of the trained DNN model are tanh, 0.01, SGD, and 4, respectively.

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Table 7 shows the sound pressure comparison. For several points, the relative error is close to 10 percent, but for most points, the relative error is stable within 3 percent and mostly below 1 percent. This shows that the calculation results of the trained DNN surrogate model are in good agreement with the numerical simulation results.

Finally, the DNN model is used to generate 2000 datasets for MCs. Comparing the calculation time of IGA FEM/BEM with DNN, the former takes around 647.66 h, while the latter is only 18.7 s. This shows that DNN is far ahead of IGA FEM/BEM in efficiency. In the end, the expectation and standard deviation are calculated, and their values at the point (100, 0, 0) are 6.62 Pa and 3.21 Pa, respectively.

6. Conclusion

In this work, our purpose is to enhance the sampling speed of MCs and minimize the resource consumption of traditional numerical calculation methods. To achieve this, we utilize a DNN as a surrogate model to generate a substantial volume of multidimensional data. We employ the IGA FEM/BEM method to compute the sound pressure of the vibroacoustic coupling problem influenced by multidimensional uncertain parameters. Subsequently, we process these parameters and sound pressure values to construct the training dataset for DNN. The sound pressure predicted by DNN shows excellent agreement with the results of numerical calculations. Compared to IGA FEM/BEM, DNN also exhibits a significantly higher sample generation rate. This suggests that DNN holds great potential as a surrogate model. Furthermore, the MCs result comparison between DNN samples and numerical calculation samples is conducted, and the two match well, which affirms the effectiveness of the DNN surrogate model in performing stochastic analysis.

Data Availability

The data that support the findings of this study are available on request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors appreciate the financial support of the National Natural Science Foundation of China (NSFC) under Grant no. 12102095.

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Copyright © 2024 Zhongbin Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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