Abstract
A mapping relationship-based near-field acoustic holography (MRS-based NAH) is a kind of innovative NAH by exploring the mapping relationship between modes on surfaces of the boundary and hologram. Thus, reconstruction is converted to obtain the coefficients of participant modes on holograms. The MRS-based NAH supplies an analytical method to determine the number of adopted fundamental solution (FS) as well as a technique to approximate a specific degree of mode on patches by a set of locally orthogonal patterns explored for three widely used holograms, such as planar, cylindrical, and spherical holograms. The NAH framework provides a new insight to the reconstruction procedure based on the FS in spherical coordinates. Reconstruction accuracy based on two types of errors, the truncation errors due to the limited number of participant modes and the inevitable measurement errors caused by uncertainties in the experiment, are available in the NAH. An approach is developed to estimate the lower and upper bounds of the relative error. It supplies a tool to predict the error for a reconstruction under the condition that the truncation error ratio and the signal-to-noise ratio are given. The condition number of the inverse operator is investigated to measure the sensitivity of the reconstruction to the input errors.
Keywords
- near-field acoustic holography
- mapping relationship
- integral identity
- acoustic measurement
- spherical fundamental solutions
1. Introduction
To locate the position and target the strength of noise for a vibrating structure, near-field acoustic holography (NAH) had been widely adopted as an effective tool. It has a significant influence on the noise diagnostics, which gives a permission to get all desired acoustic quantities, such as pressure, particle velocity, and sound power, from a number of discrete field measurement.
It was originally developed by Willams, Manynard, etc., to reconstruct surface velocity of a rectangular plane with Fourier transform technique [1, 2, 3]. Initially, the Fourier-based NAH decomposes the field pressure into k-space (wave number space) for baffled problems. In other words, the field pressure is expanded into plane waves, and the reconstruction procedure is to obtain coefficients of the plane waves based on measured pressure. Although different from the k-space decomposition, concept of Fourier transformation was inherent to the 3D cylindrical and spherical NAH problems as the in-depth discussions in Ref. [4].
Since it was proposed [1], varieties of approaches had been proposed and their superiorities had been proven in various applications, which resulted in several categories according to their underlying theories. Statistical optimal NAH [5, 6, 7] uses the elemental waves to approach the acoustic field, in which the surface-to-surface projection of the sound field is performed by using a transfer matrix defined in such a way that all propagating waves and a weighted set of evanescent waves are projected with optimal average accuracy [6]. Boundary element method (BEM)-based NAH [8, 9, 10, 11, 12, 13] is appropriate for arbitrarily shaped model in which a general transformer matrix between the surfaces of structure and hologram is derived from the integral equation. Among the BEM-based NAH, two types of integration equation are adopted: the directive formulation (Helmholtz integral equation) and indirect formulation (single- or double-layer integral equation). The quantities reconstructed by the NAH derived from directive formulation have clear physical meaning [8, 9, 10], while the ones obtained by NAH derived from the indirect formulation are not the real physical quantities [11, 12, 13]. The equivalent source method (ESM) [14, 15, 16, 17, 18, 19], also named as wave superposition algorithm (WSA) [16, 20, 21], was proposed by Koopman [22] for solving acoustic radiation problems of closed sources. ESM assumes that the field is generated by a series of simple sources such as monopoles and dipoles, and numerical integration is not needed in determining the source strength for a set of prescribed positions. Despite versatility of the ESM and various successful applications, “retreat distance” between the actual source surface and the virtual source cannot be well defined and deserves more attention in the application [23]. The Helmholtz equation least square method (HELS) [24, 25, 26] adopted the spherical wave expansion theory to reconstruct acoustic pressure field from a vibrating structure. Coefficients of the spherical wave function, the fundamental solution (FS) for the Helmholtz equation, are determined by requiring the assumed form of solution to satisfy the pressure boundary condition at the measurement points. Since the spherical wave functions solve the Helmholtz equation directly, it is immune to the nonuniqueness difficulty inherent in BEM-based NAH [27]. However, HELS works better for spherical or chunky model than elongated model due to the specific basis function [25].
Essentially speaking, NAH is to achieve the desired acoustic quantities by the measured physical quantities such as sound pressure in the field. Most of the methods explicitly require the transfer operator
Unfortunately, all the proposed methods are very sensitive to errors which may cause reconstruction to fail. It is primarily due to abundant adoption of basis functions in the transfer operator which amplifies the errors in the inverse process. That is the reason why there have been numerous studies focusing on the development of regularization methods to stabilize this inverse problem, such as truncated singular value decomposition [28] and the Tikhonov regularization [29]. Thus, construction of transfer operator is not a trivial process but is crucial to the feasibility and accuracy of the NAH. Concerning the theory development and practical measurement, it naturally arises a question whether there exists a guideline to determine the number and location of generalized basis function as well as measurement to obtain their coefficients for a given shape of source surface and prescribed tolerance.
The number of FS as well as number and position of the microphones array in the measurement are not well studied for the category of NAH based on the FS. Thus, one advantage of the mapping relationship-based NAH (MRS-based NAH) is the available guideline to the determination of the number of FS and measurement configuration in the FS-based NAH by exploring the mapping relationship between the modes in FS between surface and hologram, and investigating approximation of the modes with a set of locally orthogonal patterns.
As errors are inevitable in the practical measurement, it is curious to know how the errors go through the inverse operation and what influence imposed on the accuracy of the reconstruction results. To the best knowledge of authors, few works are devoted to the errors analysis of the NAH by comparing with that for the regularization methods. It is because that the NAH was usually viewed as a very ill-posed inverse problem for which regularized solution is the primary task. Thus, it is difficult to predict or estimate the reconstruction accuracy. Instead of a predictable way, numerical simulation and experimental validation are two frequently adopted methods to investigate the performance of NAH for different parameters [19, 30, 31]. For practical problems, it is hard to estimate the accuracy of the reconstructed results. Thus, one merit of our approach is the availability for predicting the reconstructed accuracy for a specific setup of the MRS-based NAH.
2. The mapping relationship-based NAH
2.1 Theorem development
As shown in Figure 1, assume that the fluid is homogenous, inviscid, and compressible and only undergoes small translation movement. The time harmonic sound pressure radiated from a vibrating structure into an infinite domain
where
The fundamental solution of the governing formulation Eq. (1) in the spherical coordinates is
where variables
where
Normal gradient in the direction
which is related to the normal velocity
It should be noted that Eqs. (2) and (5) are related as a solution pair for exterior acoustic problems, which means giving one as the boundary condition, the other will be the solution. They form a set of pressure/velocity modes on the boundary of a vibrating structure, which are generally independent on nonspherical surfaces and orthogonal on spherical surfaces. To facilitate derivations, we refer the velocity modes as the normal gradient
Based on the model decomposition theorem and the mapping relationship, the boundary velocity
where
Eq. (6) is the basement of the MRS-based NAH but must be properly truncated. The subscription convention in Eq. (6) is convenient for the discretized linear operation. Obviously, the truncation number
Determination of the number of most efficient modes is converted to seeking the truncation number of radiation efficiency
where
where
Fortunately, for a specific dimensionless value
2.2 The NAH procedure
Suppose a set of at least independent velocity modes on the boundary, denoted as
Generally, the pressure patterns
where the inner product
where
Assume the radiated pressure
where
Once those participant coefficients are obtained by the measured pressure, substituting Eq. (10) into Eq. (12) yields
where the coefficients are
Due to the unique mapping relationship between surfaces of vibrating structure and hologram, reconstruction for the boundary quantities can be performed by multiplying the corresponding modes with the same set of participant coefficients
Thus, acoustic holography is converted to seek explicit descriptions of the mapping relationship between the modes on the boundary and modes on the field and design a proper experimental setup for obtaining the participant coefficients of the modes on the measurement surface. The modes on the boundary are free of restrictions for their form of expression, which could be in any well-studied analytical functions or in generally numerical representations. However, it should be expected to have a capacity of fast convergent ratio in the decomposing of boundary quantities and generate a radiated pressure on the hologram which is easy to be determined by the experiment. In the current work, the FS in spherical coordinates for the Helmholtz equation Eq. (2) and its normal gradient Eq. (5) are chosen as the pressure and velocity modes. Merits of choosing those forms of modes are twofold. First, the radiated modes on the field are also the in the same form; and second, the most effective modes contributing to the field pressure are easy to be determined. Henceforth, the radiated pressure modes in Eq. (13) are chosen as
2.3 Setup of the microphone array
Since the modes are distributed on an enclosing surface, the holograms should form an enveloping surface enclosing the vibrating structure. Otherwise, partially measured pressure cannot represent the modes completely and consequently cannot be applied to reconstruct the boundary information based on the mapping relationship.
The distribution of a specific mode varies on different holograms. Generally, the measurements are subject to the experimental resource such as microphones and permissible space. How to accurately recognize the field pressure modes is one of the crucial factors to NAH. In practice, microphones are preferred to be placed on planar, cylindrical, or spherical surfaces which are easy to be set up but generally not conformal to the vibrating structure, as shown in Figure 3.
For the enclosing planar holograms, as shown in Figure 3a, each pressure mode is divided and projected onto six patches. On each patch, the measured pressure should be able to accurately represent the projected pressure modes. However, once the pressure is discretely sampled, the spectrum or the number of participant modes on that patch is truncated. Therefore, the primary task in the measurement is to set up the microphone arrays properly with an aim to approximate all the projected pressure modes on each patch actually. On each planar surface, the pressure modes can be expressed by two sets of locally orthogonal polynomials such as polynomial
where
A closed cylindrical measurement surface, as shown in Figure 3b, has three patches, one left circular planar patch
A spherical measurement surface is shown in Figure 3c, which is a conformal patch to the spherical coordinates upon which the FS is obtained. Field modes on the spherical surface are orthogonal. Determination of the field modes on the spherical hologram is actually to identify the spherical harmonic functions based on the measured pressure. Due to conformality of the hologram to the coordinate system of the spherical FS, an analytical way is available to determine the number and position of the measurement. The quadrature technique on a sphere is well studied and widely used in the computational acoustics [34]. Therefore, the participant coefficients in Eq. (12) can be accurately evaluated by
3. Error analysis
3.1 Error bounds on pressure energy
The NAH is an inverse problem and thus poses significant challenges to the stable and accurate solution. However, a practical measurement is prone to errors and always incorporates uncertainties, such as random fluctuations, effect of rapid decay of the evanescent waves. Generally, the great affection to the reconstruction by the inevitable measurement errors is largely due to over-selected number of the basis (either in numerical or analytical form) which results in an ill-posed inverse operator. Fortunately, the number of basis or modes can be well estimated by an analytical way as introduced in Section 2. Thus, a pre-regularization process is embedded in the MRS-based NAH.
On the holograms, the error included pressure is simply modeled as:
where
where
where
By taking advantages of the mapping relationships, Eq. (11) is used to evaluate the reconstructed pressure on the surface of vibrating structure after the coefficients
where
Therefore, it could be observed that the reconstruction process is to translate the local coefficients
According to the Parseval law, the reconstructed pressure energy on the surface of vibrating structure is
where
where
where
in which
3.2 The modified error bounds
Above analysis is based on an assumption that the pressure can be completely decomposed by a set of modes. Otherwise, the Parseval law cannot be applied equivalently in evaluating the pressure energy. However, the complete set of modes is hardly to be satisfied in decomposing the radiated pressure of a realistic radiator, but an incomplete set is applied to approximately decompose the radiated pressure within a given tolerance. Therefore, a compromise on accuracy and robustness is made by truncating the series expansion
Suppose the exact pressure on the surface of the vibrating structure is
According to the derivation in the appendix in the ref. [35], the relative error
It reaches the lower bound at
where the objective function is
In the above analysis, the variables SNR and
which is only related to the
The variables
3.3 Characteristics of the translator
To investigate how much the output value of a function, such as the reconstructed quantities, can change for a small variation, such as the errors introduced in the experiment, in the input arguments, condition number of the function is one of the frequently used measure. Therefore, investigation of the condition number of translators in the NAH can somehow describe the stability of the reconstruction. Generally, numerical approach is applied to compute the condition number. However, if both shapes of the structure and holograms are conformal to sphere, a simple asymptotic expression of the condition number is available. The radii of spherical structure and holograms are denoted as
where
According to the analysis in Ref. [36], the asymptotic expression of
which is actually the absolute value of the imagine part of the spherical Hankel function, since the real part goes rapidly to zero for
due to that
To investigate how much the reconstructed coefficients and in turn the pressure can change for a small variation in the local coefficients
Actually, the condition number of translator
The asymptotic expression of the condition number of the translator
4. Experiment study
4.1 Numerical simulation
The necessary number of participant modes is hard to be obtained exactly for a realistic problem, and truncation error is introduced. In this case, the radiated source pressure on holograms is generated from a vibrating cubic model driven by a harmonic excitation. As shown in Figure 6, the cubic model is of size
where
In this section, investigations are devoted to the reconstructions of the NAH with spherical holograms. To use the MRS-based NAH, the primary task is to determine the number of necessary modes. According to the method introduced in Section 2.1, the equivalent radius of the cubic model is
There are errors due to the truncation of the participant modes. Therefore, reconstructions are firstly performed for the pressure
Relative errors of reconstructed pressure energy on the model’s surface is defined as:
where
based on the simulated results. According to bounds Eq. (26), normalized relative errors of pressure energy are defined as:
for different SNRs and holograms, which should satisfy
The errors are plotted in Figure 8. Figure 8a depict that the most necessary number of modes for the boundary pressure reconstruction is
The normalized errors are presented in of Figure 8b. The normalized errors are all less than 1 and increase along with the SNRs. The small errors for small SNRs and participant number of modes are due to the fact their lower bounds are underestimated, while the upper bounds are overestimated by the approach in Section 3.3. The extreme small normalized errors for case
4.2 A practical experiment
An experiment is set up to explore the performance of the MRS-based NAH in this section. The source is in the same size and possesses the same material property as the one in the Section 4.1. Reconstruction is only preformed on the spherical hologram, since it requires the minimum number of microphones by comparing with the other two types of holograms.
An equipment is designed to facilitate the measurement. As shown in Figure 9a, the measurement on a spherical hologram is realized by rotating a half circular album arm, on which the microphones are mounted, around an axial which is the z-axial. The cubic model is placed at the center of the spherical hologram by hanging in a portal frame with a rigid hollow rod. A single-point drive is applied to the model by a small exciter on the top surface, as shown in Figure 9b. To make sure a uniform velocity distribution is generated on the surface, the analyzing frequency is selected closely to one of modal frequencies, which is 634 Hz. The model has an equivalent radius
To validate the reconstructed results, the same 5 by 9 measurements are performed on a spherical validation surface
where
The experiment is done in a semi-anechoic chamber which can reduce the influences of the environmental noise. Even that, positional errors and some other uncertainties are inevitable to be included in the measured signals which in turn affects the reconstruction results. The error analysis in Section 3 ascribes them to the SNR. How those relate to the SNR is crucial to the error estimation, which needs more investigation.
5. Conclusions
A NAH based on the mapping relationship between modes on surfaces of structure and hologram is introduced. The modes adopted in the NAH are FS of the Helmholtz equation in spherical coordinates which are generally independent and not orthogonal except on the spherical surface. The NAH framework provides a new insight to the reconstruction procedure based on the FS in spherical coordinates. The modes on the surface of structure and hologram form a bijective mapping. Number of modes prescribed in the MRS-based NAH is crucial to total number of measurement as well as the final reconstruction accuracy. An approach is proposed to estimate the necessary degree of effective modes. It is built on the energy criteria by exploring the radiation efficiency of the modes on the equivalent spherical source. An upper bounded error is derived for the radiated sound power of a vibrating structure with degree of modes up to a specific value. A relatively small value of degree is given by this approach.
Once the necessary degree of modes is determined, the number and position of microphones, which are also very crucial to the NAH, are investigated. Techniques to approximate the modes on three types of holograms by a set of locally orthogonal patterns are developed. A numerical algorithm is needed to determine the tight bounds for two locally orthogonal patterns on the planar patch. Due to the completeness of polar angles on cylindrical holograms, the algorithm is reduced by one dimension and the number of degree and positions are analytically determined for the local patterns along the polar direction. The number and position of measurement on the spherical hologram are determined by a purely analytical method because of its conformity to the coordinates of modes.
Errors are inevitable to be encountered in the NAH experiment. It is found that the reconstruction accuracy is subjected to two kinds of errors, one is the SNR and another one is the truncation error due to the limited number of participant modes adopted in the MRS-based NAH. An error model is built, and the relative error of the reconstructed pressure energy on the surface of the vibrating structure is derived. The lower and upper bounds of the relative error can be achieved numerically by a constrained nonlinear optimization algorithm. However, the approach generally yields underestimation of the lower bound and the overestimation of the upper bound, especially for MRS-NAH with large condition numbers. Alternatively, a reasonable lower bound is obtained by considering the case without noise or equivalently with positive infinite SNR. It eliminates the influence of the condition number of the inverse translator and is only related to the truncation errors. Thus, it is feasible to predicate the lower error of a reconstruction with the MRS-based NAH once the truncation error is given, which is validated by numerical examples. Proper estimation of the truncation errors is highly related to the reasonable estimation of the lower bound, which deserves more investigation.
Numerical examples are set up to validate the error analysis of the MRS-based NAH. It clearly demonstrates that the reconstructed results agree well with the simulated results. Physical experiment is designed to further demonstrate the feasibility and performance of the MRS-based NAH. The reconstructed results demonstrate a very satisfactory agreement with the direct measured one with respective to the quantities as well as the distribution on the validated surface. However, to estimate the performance with respective to the actual quantities by the proposed approach, it is desirable to investigate the influences of positional errors and other uncertainties on the SNR.
Acknowledgments
The work is supported by the National Natural Science Foundation of China (Grant No. 11404208).
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