Research on the acoustic performance of an anechoic coating composed of cavities in a viscoelastic material has recently become an area of great interest. Traditional forward research methods are unable to manipulate sound waves accurately and effectively, are difficult to analyse, have time-consuming solution processes, and have large optimization search spaces. To address these issues, this paper proposes a deep learning-based inverse research method to efficiently invert the material parameters of Alberich-type sound absorption coatings and rapidly predict their acoustic performance. First, an autoencoder (AE) model is pretrained to reconstruct the viscoelastic material parameters of an Alberich-type sound absorption coating, the material parameters are extracted into the latent feature space by the encoder, and the decoder model is saved. The internal relationship between the reflection coefficient and latent feature space is trained to establish a multilayer perceptron (MLP). Then, the reflection coefficients in the test set are input to the trained MLP and decoder models to automatically invert the material parameters. The accuracy of the inversion result is 95.08%. Finally, a predictive model is trained to rapidly predict the acoustic performance of the inverted material parameters. The speed of a single test target is 80 times faster than that of the finite element method (FEM). Furthermore, sound absorber material parameters with the best sound absorption performance and a three-band sound absorber are inverted, and their actual sound absorption performance is predicted by the proposed method. The proposed deep learning-based inversion research method provides a solution for low-frequency, wide-band, strong attenuation, and precisely controlled sound waves. It achieves an efficient inversion of material parameters and the rapid forecasting of acoustic performance. The training model can be used for a sound absorbing coating composed of irregular cavities in a viscoelastic material and predict its acoustic performance by only modifying the dataset.
A sound-absorbing coating composed of cavities in a viscoelastic material has application potential in vibration and noise reduction, sound insulation, filtering, acoustic stealth, etc. The applications of these materials in information, communication, and military are of great significance. A cavity in a viscoelastic material, such as rubber, is formed to have the basic structure of the anechoic coating. The basic shape of the cavity is a cylinder with a uniform cross section or a cone with a cross section that varies with the thickness. The main function of the cavity is to make the anechoic coating resonate in the low-frequency range to expand the effective sound absorption frequency band. The technique of laying a structure with periodically arranged cavities on a target surface to improve the acoustic performance, such as a reflective baffle or sound-absorbing cover on stealth submarines, has been widely used [
A theoretical analysis method [
However, traditional methods are unable to manipulate sound waves accurately and effectively; and have analysis difficulties, time-consuming solution processes, and large optimization search spaces. Therefore, it is of great significance to find a research method that can quickly, efficiently and automatically generate the structure and material parameters of a sound-absorbing cover layer according to the expected acoustic performance target. In this case, a suitable design for an anechoic coating can be obtained for different needs. Currently, a data-driven artificial neural network (ANN) combined with a sound-absorbing coating composed of cavities in a viscoelastic material has advantages that traditional methods lack. For example, no feature engineering is required, which effectively overcomes the shortcomings of the difficulty of meshing in the finite element method (FEM); data-driven methods apply end-to-end learning without intermediate processes; and large amounts of information can be generated in batches, which saves time and economic costs [
To date, deep learning has made the reasoning ability of computers a reality through dataset optimization and has shown results in acoustic performance prediction. Jeon et al. [
In the case of the Alberich-type sound absorption coating, as the basis for the study of other forms of anechoic coatings, this paper proposes a reverse research method based on the combination of an autoencoder (AE) and a multilayer perceptron (MLP), called the AEMLP method, to efficiently invert the material parameters of the Alberich-type sound absorption coating. On this basis, a predictive model called the simulator was trained to rapidly predict the acoustic performance. For other anechoic coatings with different cavity forms, the parameters can also be inverted efficiently, the acoustic performance can be predicted rapidly according to the method proposed in this paper, and only the structure and material parameters need to be modified accordingly.
The method of directly generating material parameters according to the anticipated acoustic performance targets automates the design process with higher efficiency and less time consumption. It is an effective research tool, especially for lay users who do not have professional acoustic theory knowledge. The organization structure of this paper is as follows:
This section mainly introduces the efficient AEMLP research method. We use the deep learning method to link the material parameter characteristics of the coating with the reflection coefficient characteristics. First, we introduce the structure and material parameter characteristics of the cell unit. Then, the overall design idea of AEMLP is introduced from the training and generation aspects of the reflection coefficient and material parameters.
The outer diameter of the cell unit and the material parameters of the viscoelastic Alberich-type sound absorption coating are shown in
Material scale | Geometric size | |||||
---|---|---|---|---|---|---|
Attribute | Unit | Min | Max | Num | b* | 100 mm |
Density | kg/m3 | 1023 | 1216 | 6 | a* | 20 mm |
Modulus | MPa | 19.50 | 328.57 | 16 per density | H* | 50 mm |
Loss factor | / | 0.0782 | 0.8492 | 7 per modulus | d1* | 5 mm |
Poisson's ratio | / | 0.4692 | 0.4970 | 7 per loss factor | d2* | 5 mm |
Total num | 4704 | h* | 10 mm |
Obviously, there is a close relationship between the characteristic column vector of each material parameter and the reflection coefficient, and each column vector corresponds to a set of reflection coefficients. The rules between the reflection coefficient and the feature column vector of the material parameters can be determined with a deep learning model, and the material parameter features can be automatically inverted from the reflection coefficient. To describe the design principle more clearly, based on the FEM, we performed a parametric scan of the 4704 feature column vectors to calculate the reflection coefficient of the viscoelastic Alberich-type sound absorption coating by using COMSOL Multiphysics 5.6. According to the literatures [
To obtain an AEMLP deep learning model, we used the normalized material parameters of the Alberich-type sound absorption coating in matrix
This section mainly describes the technical details of the training process for the AEMLP model. The training process can be divided into three parts: material parameter preprocessing, feature extraction and data reconstruction, and feature matching. In terms of material parameter preprocessing, we used the maximum-minimum method to normalize the parameters. During feature extraction, an AE-based method was used to map the material parameters into a latent feature space with 30 dimensions and reconstruct them. To match the reflection coefficient with the mapped latent feature space, we use a fully connected neural network with an MLP to obtain connections.
Since the inputs include the density, elastic modulus, loss factor and Poisson's ratio of the coating and have different measurement units and value ranges, normalization preprocessing of the inputs is critical to the success of the model. We used the maximum-minimum normalization preprocessing method in
The next step after the normalization of the material parameters is to use the AE deep learning method to extract features. We established an AE model, which is an ANN composed of two parts: the encoder function z = f(x) and the decoder function x = g(z). The output dimension of the encoder is 30 to facilitate feature matching with the output values of the MLP, and the decoder is used to reconstruct the input data. The process of material parameter feature extraction and data reconstruction based on AE is shown in
Our main purpose is to map the material parameters into the latent feature space and retain the data of the space by the encoder while recording the decoder part of the AE model. The main component of the AE model is called a fully connected ANN, and information is forwarded through the network layer. The propagation process is as
In back propagation, a gradient descent algorithm is used to update the weight matrix and bias to minimize the loss function. According to the backpropagation principle, the loss function gradient of the output layer is
For a given training set, the label value of On is a constant. In actual training, we initialize the bias
In terms of deep learning methods, for the MLP network, the inputs are the feature vectors of the reflection coefficient. Since the latent feature space and the reflection coefficient are matched through the last layer of the MLP, the accuracy of the matching has been effectively improved. When the features match, the loss function is defined as
From the point of view of digital signals, for our task, the target output range is from 0 to 1. Therefore, we choose the sigmoid function as the activation function of the last layer, which is defined as
The sigmoid function can handle any output from 0 to 1; specifically, for a larger negative input, the output is 0, and for a larger positive input, the output is 1. The derivative of the sigmoid is easy to calculate. Using the sigmoid activation function, the final output can be strictly limited between 0 and 1 according to the goal we set. The process of feature matching is shown in
In addition, to solve the overfitting problem, we introduce a dropout layer into the model. This is a technique to avoid overfitting in training, which randomly stops the coefficients in the hidden layer, thereby avoiding the dependence of coefficient updates on the connection effect of fixed hidden nodes.
AEMLP is a supervised deep learning model that requires a dataset containing the material parameters of the Alberich-type sound absorption coating and the corresponding reflection coefficients, where the material parameters are used as the label values of the dataset. Here, we use the cell unit structure shown in
Name | Parameters | Unit | Inverted value |
Density | kg/m3 | 1123.8 | |
Optimal absorber | Modulus | MPa | 69.819 |
Loss factor | / | 0.4241 | |
Poisson | / | 0.4931 |
The AEMLP model is established in a Windows 10 operating system. The computer configuration is an Intel(R) Core (TM) i5–8265U CPU @ 1.6 GHz/8 GB/512 G SSD. The deep learning algorithm is implemented on the Anaconda platform with Python version 3.7, and the model is built using TensorFlow 2.0 and the Keras framework.
The detailed training process of the AEMLP method is shown in
First, we define the ability in which a slight change in material parameters does not change the acoustic performance of the coating as “material robustness”. Therefore, the greater the influence of material parameters on the acoustic performance is, the weaker the material robustness; in contrast, the smaller the influence of the material parameters on the acoustic performance is, the stronger the material robustness. Obviously, when the material parameters are inverted, the smaller the error is, the weaker the material robustness, and the larger the error is, the stronger the material robustness. There are 48 sets of material parameters sampled in the test set, and the result of the inversion using AEMLP is shown in
In
The FEM is often very time-consuming when it is used to determine the acoustic performance of a coating, and the experimental measurement of the reflection coefficient is very expensive. For these reasons, based on the AEMLP model, we trained a simulator model to directly obtain the reflection coefficients corresponding to the inverted material parameters. In this way, the shortcomings of time-consuming FEM and expensive experimental measurements were solved. The design process is shown in
In general, under different density conditions, the reflection coefficient predicted by the simulator model does not change with the peak and trough frequencies of the label value, and the change trend is basically the same, as shown by the black line and blue stippled line in
When the elastic modulus is constant, the smaller the loss factor is, the larger the error between the label value and the inversion result, as shown by the black and blue stippled lines in
When the elastic modulus is small, the predicted reflection coefficient and the label value have a certain error, as shown in
By analysing the results of
To further illustrate the advantages of the simulator more intuitively, we compared it with the FEM in terms of computational time consumption and the number of degrees of freedom to be solved, as shown in
As an example, to obtain the desired low-frequency, broadband, strong attenuation properties and effectively manipulate sound waves, we designed a sound absorber with optimal sound absorption performance and a three-band absorber of an Alberich-type sound absorption coating.
First, to obtain the low-frequency, wide-band, and strong attenuation characteristics required by the Alberich-type sound absorption coating, we refer to the literature [
Then, we applied the simulator model to predict the actual reflection coefficient and calculated the sound absorption coefficient, as shown in
The above inversion results of an optimal absorber on the material parameters of the Alberich-type sound absorption coating with the best sound absorption performance illustrate the effectiveness of the method in this paper. On this basis, to better explain how to effectively manipulate sound waves, we set the sound absorption coefficient of the coating with the expected three-band sound absorption performance, as shown in
Name | Parameters | Unit | Inverted value |
---|---|---|---|
Density | kg/m3 | 1083.4 | |
Three-band absorber | Modulus | MPa | 30.325 |
Loss factor | / | 0.1107 | |
Poisson | / | 0.4764 |
The comparison of the acoustic performance predicted by the material parameters of the three-band sound absorber using the simulator and the FEM are shown in
Due to the small elastic modulus and loss factor of the three-band sound absorber, a certain deviation was in the amplitudes of the peaks and troughs, as shown in
Although there is a certain deviation between the black solid line and blue short dashed-dotted line in
Obviously, through the above two examples, it is shown that the method proposed in this paper achieves the low-frequency, broadband, and strong attenuation design goals and effectively manipulates sound waves. Therefore, it is effective to use the method proposed in this paper to invert the material parameters of an Alberich-type sound absorption coating for the expected acoustic performance targets.
This paper first proposes the AEMLP research method to invert material parameters based on the acoustic performance of an Alberich-type sound absorption coating. Once the acoustic performance design goals of the coating are input into the trained deep learning model, the corresponding material parameters are automatically generated, and the accuracy of the inversion result of the material parameters is as high as 95.03%. Subsequently, we introduce a simulator method based on AEMLP, which addresses the time-consuming shortcomings of the FEM. As examples, we used the AEMLP method to invert the material parameters of an Alberich-type sound absorption coating for an optimal sound absorption coefficient and three-band sound absorption performance. The consistency between the acoustic performance of the optimal sound absorber and the design goal proves the effectiveness of the method; however, the deviation in the three-band sound absorber from the design target indicates that the AEMLP model has the disadvantage of insufficient generalization because the AEMLP model only considers the inversion of material parameters with a cylindrical cavity structure and a fixed scale. To further enhance the model's reverse design capability and accurately manipulate sound waves in a larger range, the next step will be to enrich the cavity structure and expand size parameters on the basis of existing work.
Compared with traditional methods, in the study of the acoustic performance of Alberich-type sound absorption coatings, the AEMLP method is superior to the general methods in terms of the number of calculation iterations, design time consumption, and result accuracy. Moreover, the efficiency is significantly improved, the design process is accelerated, the calculation and human resources are significantly reduced, and the trained model can be applied to new functions with different acoustic performance goals without additional training. Theoretically, the inversion of the material parameters of different structures can be realized by only completing the transformation of the dataset. In addition, the AEMLP method requires less professional knowledge in the field of acoustic wave theory and acoustic materials, and the material parameters can be automatically generated on demand. It provides an effective design tool for engineering and technical personnel, especially lay users. Users only need to focus on the expected acoustic performance design goals, rather than researching optimal design theory for optimization search calculations.
We would like to thank Cunhong Yin and other members of the materials team for helpful discussions. We are grateful for the support of the Doctoral Workstation of Guizhou Open University. We thank the developers of TensorFlow 2.0, which we used for all our experiments.