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Tikhonov regularization method is effective in stabilizing reconstruction process of the near-field acoustic holography (NAH) based on the equivalent source method (ESM), and the selection of the optimal regularization parameter is a key problem that determines the regularization effect. In this work, a new method for determining the optimal regularization parameter is proposed. The transfer matrix relating the source strengths of the equivalent sources to the measured pressures on the hologram surface is augmented by adding a fictitious point source with zero strength. The minimization of the norm of this fictitious point source strength is as the criterion for choosing the optimal regularization parameter since the reconstructed value should tend to zero. The original inverse problem in calculating the source strengths is converted into a univariate optimization problem which is solved by a one-dimensional search technique. Two numerical simulations with a point driven simply supported plate and a pulsating sphere are investigated to validate the performance of the proposed method by comparison with the L-curve method. The results demonstrate that the proposed method can determine the regularization parameter correctly and effectively for the reconstruction in NAH.

Near-field acoustic holography (NAH) is an effective technique for noise sources identification and acoustic field visualization. By measurements on a hologram surface near the sound source, the acoustic quantities such as pressure, particle velocity, and intensity anywhere including the source surface can be reconstructed, and the three-dimensional acoustic filed can be predicted. In the past years, many alternative methods of transform algorithm for realizing the NAH have been developed, for example, the spatial Fourier transform [

However, the reconstruction of acoustic quantities of sound sources from near-field measurement data is an inverse problem, which is different from the traditional acoustic radiation calculation. Consequently, the reconstruction stability is a key problem in NAH technology due to the fact that the realization process is ill-posed and the reconstructed results are highly sensitive to signal-to-noise ratio (SNR) [

Currently, the direct regularization methods and the iterative regularization methods are mostly used for dealing with ill-posed problem of NAH [

The methods for selection of optimal regularization parameter can be classified into prior [

This paper focuses on a new method for determining the optimal parameters of Tikhonov regularization in NAH technique based on the ESM. In the proposed method, the transfer matrix from the source to the hologram surface is updated by augmenting a fictitious point source with zero strength near the locations of the equivalent sources. The minimization of the norm of this fictitious point source strength can be as the criterion for selection of the optimal regularization parameter since the reconstructed value should tend to zero. Thus the original inverse problem is converted into a univariate optimization problem that can be solved by one-dimensional search technique. The numerical simulations are investigated to demonstrate the validity of the proposed method, and the results show that the novel proposed method is able to select the regularization parameter correctly and effectively.

The basic idea of the NAH based on ESM is that the actual radiated sound field can be replaced by superposing the acoustic field produced by a number of equivalent sources distributed inside the source surface. The source strengths of the equivalent sources can be evaluated by matching the acoustic pressures measured on the hologram surface, and the acoustic quantities in the acoustic field can be obtained by the source strengths and the transfer matrices constructed by the equivalent sources.

As shown in Figure

Schematic diagram of NAH based on ESM.

Given that

If the number of measurement points is not less than the number of equivalent sources, that is,

By using the source strength column vector

Similarly, the pressure and normal velocity on the surface of the acoustic source can be reconstructed in the matrix form as

In practice, there always exist errors in the measured pressures on the hologram surface. If (

The normal equation associated with (

The Tikhonov regularization solution can be expressed as

The regularization parameter

Many researchers have developed a number of selection criteria and algorithms from different points of view, and each of them has their merit, shortcoming, and applicable condition [

After determining the positions of all the equivalent sources, the corresponding source strengths will be obtained by matching the acoustic pressures measured on the hologram surface. In this case, there is no sound source or source strength is zero except for the positions of the equivalent sources. Here, the proposed method is to add a fictitious point source to the location where there is no equivalent source near the surface

Accordingly, the augmented transfer matrix can be written as

From (

Qualitatively, the change trend of the norm of strength of the fictitious source

(a) When

(b) With the increase of

(c) With the decrease of

(d) Note that the norm of strengths of the equivalent sources

Therefore, it can be seen that the optimal regularization parameter should be at the point corresponding to the first local minimum where

The minimization problem related to (

Furthermore, considering the initial situation without the fictitious point source, the corresponding solution equation can be expressed as

According to (

Thus, by using the reconstructed source strength column vector

In addition, the essence of the proposed method to select the regularization parameter can be illustrated by the physical process of the equivalent source method. If the original

When

By comparing (

According to the properties of 2-norm, there is an inequality as

From (

To examine the performance of the proposed method to determine the optimal regularization parameter in NAH, two numerical simulations were investigated, and the reconstruction results by using the proposed method were compared with those by using the L-curve method.

This simulation case is a point driven simply supported aluminum plate with the thickness of 3 mm mounted in an infinite baffle. The dimension of the plate is 0.3 m × 0.3 m, and a harmonic force with an amplitude of 1 N is exerted vertically to the plate at the center. The radiated sound field is calculated by using Rayleigh’s integral formulation [

Figure ^{−2} and 3.7901 × 10^{−3} at the corner of the L-curve, respectively, and the relative errors between the reconstructed pressure and the theoretical value are 6.68% and 6.37% by using the parameters for regularization, respectively.

L-Curve of the pressure reconstruction for the vibrating plate at (a) 500 Hz and (b) 800 Hz.

Figure ^{−2} and 3.5481 × 10^{−3} at the minimum of the curve, and the relative errors are 6.63% and 6.44%, respectively. For comparison, the locations of the parameter determined by using the L-curve method are also marked in Figure

Object function versus the regularization parameter of the pressure reconstruction for the vibrating plate at (a) 500 Hz and (b) 800 Hz.

Figure

Comparison of reconstructed pressures. (a) Theoretical value at 500 Hz, (b) reconstructed by using the L-curve method at 500 Hz, (c) reconstructed by using the proposed method at 500 Hz, (d) theoretical value at 800 Hz, (e) reconstructed by using the L-curve method at 800 Hz, and (f) reconstructed by using the proposed method at 800 Hz.

L-Curve of the pressure reconstruction for the vibrating plate at 1000 Hz.

As a comparison, Figure ^{−3}, and the reconstructed relative error is 6.68%, which is much smaller than that by using the L-curve method. By the comparison of the two errors, it can be seen that the proposed method is valid for the pressure reconstruction when the L-curve method fails to determine the regularization parameter. The comparison of the reconstructed pressure with the corresponding theoretical value at 1000 Hz is shown in Figure

Object function versus the regularization parameter of the pressure reconstruction for the vibrating plate at 1000 Hz.

Comparison of reconstructed pressures at 1000 Hz. (a) Theoretical value, (b) reconstructed by using the L-curve method, and (c) reconstructed by using the proposed method.

Figure ^{−2} by detecting the minimum of the curve. The relative error of the normal surface velocity in the proposed method is 5.97%, which is much smaller than that in the L-curve method.

L-Curve of the normal surface velocity reconstruction for the vibrating plate at 600 Hz.

Object function versus the regularization parameter of the normal surface velocity reconstruction for the vibrating plate at 600 Hz.

Figure

Comparison of reconstructed normal surface velocities at 600 Hz. (a) Theoretical value, (b) reconstructed by using the L-curve method, (c) reconstructed by using the proposed method, and (d) normal surface velocity along the middle row.

The influence of the position of the additional fictitious source on the reconstruction accuracy is also investigated to validate the performance of the proposed method. Assume that the position of the additional fictitious point source moves along from −0.5 m to 0.5 m in

Reconstruction errors for different positions of the additional fictitious source at 500 Hz.

A pulsating sphere with a radius of 0.1 m is investigated as depicted in Figure

The nodes distribution on the source surface of the pulsating sphere.

In the simulation, the radial velocity ^{3}, and the speed of sound

Figure ^{−4}, 5.003 × 10^{−4}, 8.1372 × 10^{−4}, and 1.1166 × 10^{−3} at the corner of the L-curve, respectively, and the relative errors between the reconstructed pressure and the theoretical value are 40.91%, 22.28%, 47.31%, and 58.59% by using the parameters for regularization, respectively.

L-Curve of the pressure reconstruction for the pulsating sphere. (a) 300 Hz, (b) 500 Hz, (c) 700 Hz, and (d) 900 Hz.

By using the proposed method, the curves for choosing regularization parameter at 300 Hz, 500 Hz, 700 Hz, and 900 Hz are shown in Figure ^{−3}, 6.1662 × 10^{−3}, 1.3832 × 10^{−2}, and 2.6923 × 10^{−2} corresponding to the bottom of the curve, respectively, and the reconstruction relative errors are 9.89%, 3.22%, 11.86%, and 13.38%, respectively, all of which are much smaller than those by using the L-curve method. It can be seen that the proposed method can give a good regularization parameter in NAH technique based on the ESM.

Object function versus the regularization parameter of the pressure reconstruction for the pulsating sphere at (a) 300 Hz, (b) 500 Hz, (c) 700 Hz, and (d) 900 Hz.

Figure

Reconstruction errors of pressure by using the two methods versus frequency.

The influence of the SNR on the reconstructed pressure at different frequencies is also examined. Figure

Reconstruction errors of pressures by using the proposed method versus SNR.

A new optimal parameter selecting method for the Tikhonov regularization was proposed in NAH based on the ESM. In the proposed method, the criterion for determining the optimal regularization parameters was given by adding a fictitious point source with zero strength to make a part of the solution of the augmented problem known, which has a clear physical meaning. A one-dimensional search technique was used to minimize the norm of the fictitious point source strength for determination of the optimal parameter in the Tikhonov regularization method. The validity of the proposed method has been demonstrated by numerical simulations with kinds of sound sources including a point driven simply supported plate and a pulsating sphere. It was shown that the proposed method can get a better regularization parameter for the reconstruction with good accuracy and stability in NAH compared with the L-curve method. Future research will emphasize the existence and uniqueness of the minimum point on the curve of the norm of the fictitious point source strength with the regularization parameter in mathematics. The present work also can provide a new idea for solving other inverse problems, that is, try to add new known information to augment the transfer matrix and then make full use of the known information to constrain the solution to approximate the true value.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the Science and Technology Project of Education Department of Jiangxi Province (Grant no. GJJ151121) and the National Natural Science Foundation of China (Grant no. 51565037).