# Ultrathin Acoustic Absorbing Metasurface Based on Deep Learning Approach

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## Thermal and Viscous Losses

When an acoustic wave propagates through a fluid medium with a normal incidence on a solid wall, the impedance mismatch is very large so that the wall can be treated as acoustically perfect rigid. It means that the sound cannot penetrate the wall and get totally reflected. As illustrated in Fig. 1.12, consider a lossless plane wave propagating along the x-direction. A wall is placed at z=0. According to the ‘no-slip’ boundary condition, the component of the velocity, vx parallel to the wall must equal zero at the boundary.

### Numerical Modeling- The Finite Element Method

Throughout this thesis, theoretical and experimental results are compared with the numerical simulations performed using the finite element method. We have used commercially available FEM-based software COMSOL Multiphysics for this purpose. The FEM is a powerful numerical analysis technique for obtaining approximate solutions of partial differential equations (PDEs) arising in physics and engineering . For the vast majority of geometric problems, partial differential equations (PDEs) cannot be solved with analytical methods. Instead, approximate equations are constructed based on various discretization by using numerical model equations to approximate the PDE which is then solved using numerical methods . The FEM is used to compute such approximations. For example, consider that a function 𝑢 is defined to be a dependent variable in a PDE (i.e., pressure, temperature, volume, etc.). It is possible to approximate the function 𝑢≈𝑢ℎ using linear combinations of basis functions, 𝑢ℎ=Σ𝑢𝑖ψ𝑖𝑖(2.1).
where 𝑢𝑖 is the coefficient of the functions that approximate u with 𝑢ℎ and ψ𝑖 is the basis function. An example to illustrate this for a 1D problem is shown in Fig.2.1. The function 𝑢 (blue line) is approximated by 𝑢ℎ (red line) consists of 8 basis functions. Here, the value of the linear basis functions is 1 at their respective nodes and 0 at other nodes. This discretization depends on the solution required, not necessarily to be linear. Distribution of the basis function is used to resolve parts of the function where a higher resolution is needed. In COMSOL Multiphysics, this discretization is defined by mesh elements. A geometry is divided into a large number of mesh elements and a system of equations is solved in each element which combines to produce the final result.

#### Complex Frequency Plane Analysis

We use the concept of the zeros and the poles of the reflection coefficient in the complex frequency plane for the study of the temperature effect on the multicoiled metasurface absorber (chapter 3). This concept has been widely used as an efficient tool to study and design the broadband and metasurface absorbers in the low-frequency regime [67,68]. Using this method, the reflection coefficients of the whole system can be evaluated in the complex frequency plane in which the real part of the frequency is represented in the abscissas, and the corresponding imaginary part is represented on ordinates. This concept can be explained by considering the simple example of a slot with a quarter wavelength resonance  that is related to the concept of coiling up space for designing subwavelength acoustic metamaterials. As the interest lies in low frequency absorption, attention is paid to the frequency range smaller than the cutoff frequency of the waveguide and therefore the problem can be considered as 1D. The structure of interest is shown in Fig.2.3(a). It is equivalent to an incident wave on a slot having length 𝑙𝑏 and section 𝐵2 at the end of a waveguide of section 𝐵1 or to a wave normally incident on a wall with periodic slots. Considering that a plane wave is incident from the left direction such that a standing wave is formed in 𝑦<0 which can be written as, 𝑝=𝑒𝑖𝑘𝑦+𝑅𝑒−𝑖𝑘𝑦(2.3) Where 𝑘 is the wave number, 𝑐 is the speed of wave, and 𝑟 is the reflection coefficient. For the rigid wall at the end of the slot, 𝑝′(𝑙𝑏)=0 and 𝑝′(0+)𝑝(0+)=𝑘tan(𝑘𝑙𝑏). Here, the prime indicates the differentiation with respect to 𝑦.

Acoustic Absorption Measurement

For the acoustic absorption measurement of the designed metasurface, we use two microphone method which is discussed in the subsequent section. The experimental apparatus consists of an impedance tube (inner size is of 10×10 𝑐𝑚2), two Bruel & Kjær 1/4-in.-diameter microphones (M1 and M2), and Bruel & Kjær measuring module “Acoustic Material Testing” are used to measure the absorption of the metasurface . Two fixed microphones are flush mounted on the wall of a normal incidence acoustic impedance tube. The thickness of the waveguide wall is 6mm.
We fabricate homemade waveguide to fix the sample at the end of the tube as shown in Fig.2.5; the absorption spectra can be measured for the corresponding metasurface. On the other end, a sound source is connected to an amplifier which is connected to a waveform generator that generates the broadband signal. It is important to note that this method requires very precise transfer function measurements, which require accurate amplitude and phase calibrations for the microphones. Before each absorption measurement, the calibration procedure was carried out which is explained in section 2.4.2. Since the rear of the wall is a hard wall condition, it can be assumed that there is no transmission. A digital signal (white noise) powered by the amplifier is sent to the loudspeaker. The absorption coefficient was obtained by analyzing the signal by two microphones. Each experiment was repeated multiple times to validate its reproducibility.

The Two-Microphone Method

To measure the acoustic absorption, the two-microphone method includes the decomposition of the source generating broadband stationary random signal into its incident (𝑃𝑖) and reflected (𝑃𝑟) components . These components are determined from the relationship between the acoustic pressure measured by microphones (M1 and M2) at two locations on the wall of the tube as shown in Fig.2.6.

Discriminative and Generative Neural Networks

To understand the discriminative and generative neural networks, consider the acoustic metasurface which can be described by two types of labels as shown in Fig.2.7(a). The first type includes physical variables such as geometrical parameters. Here, they are labeled by 𝑥. The second type comprises the physical response corresponding to the given spectral range which is labeled by the variables. Normally, physical responses can be expressed as a single-valued function of the physical variables means that given an input value maps a single value of 𝑦 in the output. For example, an acoustic absorbing metasurface having a fixed geometrical configuration produces a single absorption spectrum. However, the opposite is not true, a given input physical response 𝑦 can map to multiple 𝑥𝑠. For example, the same absorption spectrum can be generated using the different combination of physical variables too. Depending on the type of device labels, different neural networks can be implemented. The most commonly used neural network classes in electromagnetism and acoustics are based on discriminative and generative networks [117,118,120,121]. Discriminator neural network can interpolate complex and nonlinear input-output relationship by modeling 𝑦=𝑓(𝑥) and solve the forward problem as shown in Fig.2.7(b). This input-output mapping is single valued function which supports one-to-one or many-to-one mappings. Discriminator neural network can be generically utilized for fast and accurate modeling of acoustic metasurface devices with minimum human intervention. Because it uses the same deep-network concepts as discriminative neural networks, generative neural networks appear deceptively similar to discriminative neural networks architecture . However, there is a main difference between both the networks. One of the inputs to the generative network is a latent variable, 𝑧, which is a random variable to the neural network. Here, ‘latent’ mention that variable, 𝑧 does not have an explicit physical meaning. Either uniform or Gaussian distribution is used to sample the latent variables. A single instance of sampling z maps to a single network output and continuum of z samplings maps to a distribution of outputs of network. In the vast majority of cases, generative networks are used to generate a distribution of device layouts, as shown in Fig.2.7(c). The inputs to the network include a 𝑧 and 𝜃 which represents the set of the label. This type of networks is known as ‘conditional’ as the output distributions can be considered as probability distributions conditioned on the set of the labels, 𝜃. The network learns from a training set that consists of a collection of discretely labeled devices, also called as samples from the distribution 𝑃(𝑥│𝜃). Once properly trained with enough data, the network outputs a device distribution 𝑃̂(𝑥│𝜃) that matches 𝑃(𝑥│𝜃). For the unconditional network, there is only one input, latent variable, 𝑧. Similar to the conditional networks, such kind of network can also be trained and it produces devices. However, there is no control over modes of the data to be generated. The stochastic nature of generative neural networks distinguishes them from discriminative neural networks. Based on the given input 𝜃, generative network can execute one-to-many mappings.

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Principle of the Neural Network

Deep learning is a powerful class of ML algorithms that models extremely nonlinear data relationships by serially stacking nonlinear processing layers. Such networks consist of a series of many hidden layers of the artificial neurons through which the network can learn complex pattern. A neuron can be considered as a mathematical function that takes one or more inputs which are multiplied by a value called ‘weights’ and added together, resulting in a single output value. For example, consider a network that receives an input vector 𝑥={𝑥1,𝑥2,𝑥3} and in output, it gives a scalar value 𝑦. As shown in Fig.2.8(a), the neuron executes two operations: a weighted sum and nonlinear mapping . The weighted sum, s can be expressed as, 𝑠=Σ𝑤𝑖𝑥𝑖+𝑏=𝑤𝑇𝑥+𝑏 where, 𝑤 is the trainable weight vector having the similar dimension as 𝑥. Here, 𝑏 represents trainable bias value. A set of weights is used to initialize a neural network when it is trained on the training set. During the training phase, theses weights are optimized. Next, nonlinear mapping is performed on a weighted sum, 𝑠 using a nonlinear activation function 𝑓, such that the output of the neuron is 𝑦=𝑓(𝑠). The neurons in a network can be connected in a various way to design the network. The most commonly used network design to solve physics related problems are fully connected (FC) and convolution layers [112,114,115,117,118,121,124]. As shown in Fig.2.8(a), FC layers consist of a vector of neurons. Each neuron in a layer receives its inputs from the preceding layer’s output values. For different layers, the number of neurons may be varied. When a large number of layers are stacked in a sequence, data features with higher degrees of abstraction are recorded from lower-level features. Each layer has a nonlinear activation function 𝑓 applied to it, ensuring that stacking various FC layers adds computational complexity that can’t be represented using only one layer.

1. State of the Art
1.1 Acoustic Metamaterials and Metasurfaces
1.3 Background Theory
1.3.1 Equation of State
1.3.2 Conservation of Mass
1.3.3 Conservation of Momentum
1.3.4 The Navier-Stokes Equation
1.3.5 Thermal and Viscous Losses
1.4 Conclusions
2. Methods
2.1 Numerical Modeling- The Finite Element Method
2.2 Complex Frequency Plane Analysis
2.3 Sample Fabrication
2.4 Acoustic Absorption Measurement
2.4.1 The Two-Microphone Method
2.4.2 Calibration Process
2.5 Machine Learning Algorithms
2.5.1 Discriminative and Generative Neural Networks
2.5.1.1 Principle of the Neural Network
2.5.1.2 Convolutional Neural Network
2.5.3. Note on Data Augmentation
2.6 Conclusions
3. Multi-coiled Metasurface for Extreme Low-frequency Absorption
3.1 Introduction
3.2 Coiled Metasurface Absorber
3.2.1 Two-coiled Metasurface Absorber
3.2.2 Three-coiled Structure
3.3 Multi-coiled Structure
3.3.1 Implementation of the MCM
3.3.2 Experimental Measurements
3.3.3 Bandwidth Improvement
3.3.4 Temperature Effects on the MCM
3.4 Conclusions
4. Ultrathin Acoustic Absorbing Metasurface Based on Deep Learning Approach
4.2 Structure of the Metasurface Absorber
4.3 Forward Design
4.3.1 1D Convolutional Neural Network
4.3.1.1 Network Architecture
4.3.1.2 Training and Result Analysis
4.3.2 2D Convolutional Neural Network
4.3.2.1 Network Architecture
4.3.2.2 Training and Result Analysis
4.4. Comparison with Classical Machine Learning Techniques
4.5 Acoustic Absorption Measurement
4.6 Bandwidth Improvement
4.7 Conclusions
5. Forward and Inverse Design of Metasurface Absorber for Oblique Wave Incidence
5.1 Introduction
5.2 Structure Design of Acoustic Metasurface Absorber
5.3 Data Augmentation and Preprocessing
5.4 Forward design: Convolutional Neural Network
5.4.1 Convolutional Neural Network (Processing 1D and 2D properties separately)
5.4.2 Modified Convolutional Neural Network
5.4.2.1 Network Architecture
5.4.2.2 Training Process and Result Analysis
5.4.2.3 Ablation Analysis
5.5 Inverse Design: Conditional Generative Adversarial Network
5.5.1 Network Architecture
5.5.2 Training Process and Result Analysis
5.6 Acoustic Absorption Measurements
5.7 Conclusions
General Conclusions
References

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