A deep autoencoder based approach for the inverse design of an acoustic-absorber

Abstract

This paper proposes an algorithm to perform the inverse design of a low-frequency acoustic absorber using a deep convolutional autoencoder network. A hybrid sound-absorber configuration based on Helmholtz resonators with inserted curvy neck and microperforated panel is suggested and its geometrical properties are inversely forecasted from the targeted signal. A mathematical model is put forwarded to evaluate the absorption characteristics of the introduced geometry by employing the effective medium theory and the electro-acoustic analogy. The large dataset required to train, validate and test the deep neural network is extracted through this analytical procedure. Initially, the proposed inverse technique is successfully applied on a standard Helmholtz resonator based absorber setup with great accuracy. This prediction approach is further extended to suit the inverse design of a hybrid sound absorber with complex geometrical attributes. The encoder maps the input acoustic absorption spectrum to geometrical features of the absorber, and the subsequent decoder recreates the absorption characteristics using convolutional layers. Once the training and testing of the neural network are over, the deep autoencoder inversely predicts the geometrical parameters. In comparison with earlier inverse models which employed deep neural networks, the accuracy of the current scheme is very high and no pre-design information on absorber geometry is required as well. Since the relevant learnable parameters involved are very low, the computational load is also very less for this autoencoder based method. Later, using the new inverse scheme, four representative absorber designs with specific acoustic functionality are deduced. Most importantly, these four compact absorber models produce quasi-perfect absorption in the frequency bands 200–315 Hz, 255–400 Hz, 300–530 Hz, and 350–650 Hz. Notably, the developed absorber versions have great potential in noise reduction applications owing to their deep sub-wavelength thickness ($\lambda$/23 at 200 Hz) and wide absorption spectra.

Appendices

Theoretical model of HRICN

The equivalent medium theory [82] is used to analyze the absorption characteristics of the HRICN. With this theory, the acoustic characteristics of neck and cavity of HRICN are separately modelled. Thus, when considering the visco-thermal losses, the complex density ($\rho _{N,C}^{c}$), complex wave number ($k_{N,C}^{c}$) and complex sound speed ($c_{N,C}^{c}$) of fluid propagating through the neck and cavity are formulated as [82, 83],

$$\begin{aligned}{} & {} k_{N,C}^{c}=k \sqrt{\frac{\gamma -(\gamma -1) \psi _{N,C}^{h}}{\psi _{N,C}^{v}}}, \end{aligned}$$

(19)

$$\begin{aligned}{} & {} \rho _{N,C}^{c}=\frac{\rho }{\psi _{N,C}^{v}}, \end{aligned}$$

(20)

and

$$\begin{aligned} c_{N,C}^{c}=\frac{\omega }{k_{N,C}^{c}}, \end{aligned}$$

(21)

where k is the wave number and $\gamma$ is the specific heat ratio of air. Further, $\psi _{N,C}^{v}$ and $\psi _{N,C}^{h}$ are the functions of viscous and thermal fields inside the neck and cavity, which are formulated as [82, 83],

$$\begin{aligned}{} & {} \psi _{N,C}^{v}=-\frac{J_{2}\left( k_{v} d_{N,C} / 2\right) }{J_{0}\left( k_{v} d_{N,C} / 2\right) }, \end{aligned}$$

(22)

$$\begin{aligned}{} & {} \psi _{N,C}^{h}=-\frac{J_{2}\left( k_{h} d_{N,C} / 2\right) }{J_{0}\left( k_{h} d_{N,C} / 2\right) }, \end{aligned}$$

(23)

where $k_{v}=\sqrt{-j\frac{\omega \rho }{\eta }}$ is the viscous wavenumber and $k_{h}=\sqrt{-j\frac{ \omega \rho c_{p}}{K}}$ is the thermal wavenumber. $J_{2}$ and $J_{0}$ are the Bessel functions of first kind and order two and zero respectively.

Using Eqs. (19)–(23), the acoustic impedance of neck ($Z_{N}$) is formulated as,

$$\begin{aligned} Z_{N}= j~\frac{2 \rho c \sin \left( k_{N}^{c} l_{N} / 2\right) }{\sqrt{\left( \gamma -(\gamma -1) \psi _{N}^{h}\right) \psi _{N}^{v}}}, \end{aligned}$$

(24)

Due to the insertion of the neck into the cavity, the cavity has an irregular shape. Hence, its acoustic impedance is represented using its effective volume. The effective volume of the cavity $V=\pi r_{C}^{2} l_{C}-\pi r_{N}^{2} l_{N}$, where $r_{N}= d_{N}/2$ and $r_{C}=d_{C}/2$ are the neck and the cavity radii, respectively. Thereby, the acoustic impedance of the cavity is given by,

$$\begin{aligned} Z_{V}=-j~\frac{ S_{N} \rho _{C}^{c} (c_{C}^{c})^{2}}{\omega V}. \end{aligned}$$

(25)

From $Z_{N}$ and $Z_{V}$ the total acoustic impedance of HRICN is evaluated as,

$$\begin{aligned} Z_{H R}=\frac{A}{S_{N}}\left( Z_{N}+Z_{V}+2 \sqrt{2 \omega \rho \eta }+j \omega \rho \delta \right) , \end{aligned}$$

(26)

where $A=\pi r_{C}^{2}$ is the cross-sectional area of the HRICN and $S_{N}=\pi r_{N}^{2}$ is the cross-sectional area of the inserted curvy neck. In addition, $\delta =(1+(1-1.25 \varepsilon ))(4 / 3 \pi ) d_{N}$ is the end correction length of inserted neck, in which $\varepsilon =\frac{d_{N}}{d_{C}}$ is the ratio of diameter of the neck to diameter of the cavity. Finally, the frictional resistance $(2 \sqrt{2 \omega \rho \eta })$ offered by the inner boundaries of the inserted neck is also added to the acoustic resistance. From $Z_{\textrm{HR}}$, the absorption coefficient of HRICN is obtained as,

$$\begin{aligned} \alpha =1-|{\frac{Z_{HR}-z_{0}}{Z_{HR}+z_{0}}}|^{2}. \end{aligned}$$

(27)

Inverse design using deep neural network (IDDN)

For the comparative analysis of IDDN and ICAN schemes, the IDDN scheme is applied to the HRICN absorber to forecast the four geometrical parameters such as $l_{N}$, $r _{N}$, $l_{C}$, and $r_{C}$. The absorption coefficients corresponding to 0–700 Hz are given as the input and the IDDN predicts the respective geometric parameters. The details of the IDDN architecture is given in Table 9. For training and testing, mean absolute error loss function is used. The datasets and hyperparameters for both the IDDN and ICAN schemes are same. More details on the implementation are available in the previous work [29].

Table 9 The details of the IDDN model used for the comparative study. Linear activation is used for output layer, whereas rectified linear activation function (ReLU) is used for all other layers