Helmholtz Equation Least-squares Method Research Articles

Half-Space Sound Field Reconstruction Based on the Combination of the Helmholtz Equation Least-Squares Method and Equivalent Source Method.

In practical conditions, near-field acoustic holography (NAH) requires the measurement environment to be a free sound field. If vibrating objects are located above the reflective ground, the sound field becomes non-free in the presence of a reflecting surface, and conventional NAH may not identify the sound source. In this work, two types of half-space NAH techniques based on the Helmholtz equation least-squares (HELS) method are developed to reconstruct the sound field above a reflecting plane. The techniques are devised by introducing the concept of equivalent source in HELS-method-based NAH. Two equivalent sources are tested. In one technique, spherical waves are used as the equivalent source, and the sound reflected from the reflecting surface is regarded as a linear superposition of orthogonal spherical wave functions of different orders located below the reflecting surface. In the other technique, some monopoles are considered equivalent sources, and the reflected sound is considered a series of sounds generated by simple sources distributed under the reflecting surface. The sound field is reconstructed by matching the pressure measured on the holographic surface with the orthogonal spherical wave source in the vibrating object and replacing the reflected sound with an equivalent source. Therefore, neither technique is related to the surface impedance of the reflected plane. Compared with the HELS method, both methods show higher reconstruction accuracy for a half-space sound field and are expected to broaden the application range of HELS-method-based NAH techniques.

Using learned priors to regularize the Helmholtz equation least-squares method

The Helmholtz equation least-squares (HELS) method is a valuable tool for estimating equivalent sound sources of a radiating object. It solves an inverse problem by mapping measured pressures to a set of basis functions satisfying the Helmholtz equation in spherical coordinates. However, this problem is often ill-posed, necessitating additional regularization methods, in which often variations of Ridge or Lasso are used. These conventional methods do not explicitly consider the distribution underlying the source radiations (besides sparsity) and are often used in the context of obtaining only a point estimate, even in the presence of ambiguity in the data. In this work, we propose the use of empirical priors through a normalizing flow model to enhance the inversion results obtained with the HELS method. We first validate our approach using numerical data and subsequently demonstrate its superior performance in interpolating a measured violin directivity compared to Lasso and Ridge methods, even when optimal regularization parameters are selected.

Analyzing motorcycle low-frequency noise

This paper presents analyses of the 3D acoustic fields generated by motorcycles at very low frequencies ([Formula: see text] Hz) at the idle speed and during sudden acceleration. Diagnosis and analyses of sound sources at low frequencies have always been a significant challenge because the directivity of low-frequency sound is very poor. To date, there are no research papers and/or reports that have demonstrated low-frequency sources localization and radiation patterns of any kind at high spatial resolution in 3D space. This study shows that by using sound viewer technologies, which include the passive sonic detection and ranging (SODAR), the Helmholtz equation least squares (HELS) method, advanced signal processing, denoising, etc., the locations of sound sources and visualization of the sound fields can be determined with high spatial resolution, even at frequencies below 150 Hz. In particular, the HELS method allows for reconstructing all the acoustic quantities, including the acoustic pressure, time-averaged acoustic intensity, time-averaged acoustic power on the source surfaces and in 3D space. The hardware needed consists of a 3D array with six free-field precision microphones with pre-amplifiers, a miniature wide-angle camera, an eight-channel digital signal processor and a laptop computer.

Obtaining far-field spherical directivities from arbitrarily shaped arrays using the Helmholtz equation least-squares method

Commonly, directivities of sound sources are measured using surrounding spherical arrays. However, in some cases, such as when measurement hardware is limited, or when an array shape would ideally conform to the radiating object's geometry, obtaining spherical far-field directivities using an arbitrarily shaped array could be useful. This work illustrates how the Helmholtz equation least-squares method applies to accomplish this task. Both numerical models and the measured directivities of various guitar amplifiers validate the technique.

RECONSTRUCTING ACOUSTIC RADIATION PATTERNS OF AN ELONGATED, BAFFLED PLATE AT HIGH FREQUENCIES WITH THE NYQUIST SPATIAL SAMPLING RATE SIGNIFICANTLY RELAXED

Traditionally, high-accuracy full-sphere polar measurements require dense sampling of the sound field at very-fine angular increments, particularly at high frequencies. The proposed HELS (Helmholtz equation least squares) method allows this restriction to be relaxed significantly. Using this method, far fewer sampling points are needed for full and accurate reconstruction of the radiated sound field. Depending on the required accuracy, sound fields can be reconstructed using only 10% to 20% of the number of sampling points required by conventional techniques. The HELS method allows accurate reconstruction even when the Nyquist spatial sampling rate is violated in certain directions. This paper examines the convergence of HELS solutions via theory and simulation for reconstruction of the acoustic radiation patterns generated by a rectangular plate mounted on an infinite rigid flat baffle. In particular, the impact of the number of expansion terms and that of measurement points as well as errors imbedded in the input data on the resultant accuracy of reconstruction is analyzed.

A new data extrapolation method based on the modified Helmholtz equation least-squares method

Fast Fourier transform-based near-field acoustic holography requires that the measurement aperture should completely enclose the source, which is impractical for large-scale sound sources. Helmholtz equation least-squares method can reconstruct the acoustic field with fewer measurements than fast Fourier transform-based near-field acoustic holography. However, it is not suitable for reconstructing acoustic radiation from multiple sources or a complicated source which consists of several separated parts. To circumvent this difficulty and enhance the reconstruction accuracy, a new data extrapolation method based on the modified Helmholtz equation least-squares method is proposed. The base of this data extrapolation method is a modified Helmholtz equation least-squares method which expresses the acoustic field of a complicated source as the superposition of the acoustic radiation from all of its separated parts. Using the acoustic pressures reconstructed with the modified Helmholtz equation least-squares method, the measurement surface is extended through an iterative procedure. Meanwhile, Tikhonov regularization method together with generalized cross-validation parameter choice method is utilized to treat the ill-posed problem in reconstructions. In the end, taking the extrapolated pressures as input data, the acoustic field can be reconstructed more precisely. Numerical simulation and experimental results demonstrate that this method can significantly enhance the reconstruction accuracy and efficiency.

Reconstruction of vibroacoustic responses of a highly nonspherical structure using Helmholtz equation least-squares method

The vibroacoustic responses of a highly nonspherical vibrating object are reconstructed using Helmholtz equation least-squares (HELS) method. The objectives of this study are to examine the accuracy of reconstruction and the impacts of various parameters involved in reconstruction using HELS. The test object is a simply supported and baffled thin plate. The reason for selecting this object is that it represents a class of structures that cannot be exactly described by the spherical Hankel functions and spherical harmonics, which are taken as the basis functions in the HELS formulation, yet the analytic solutions to vibroacoustic responses of a baffled plate are readily available so the accuracy of reconstruction can be checked accurately. The input field acoustic pressures for reconstruction are generated by the Rayleigh integral. The reconstructed normal surface velocities are validated against the benchmark values, and the out-of-plane vibration patterns at several natural frequencies are compared with the natural modes of a simply supported plate. The impacts of various parameters such as number of measurement points, measurement distance, location of the origin of the coordinate system, microphone spacing, and ratio of measurement aperture size to the area of source surface of reconstruction on the resultant accuracy of reconstruction are examined.

Reconstruction of vibro-acoustic response of a plate using Helmholtz equation least-squares method

A numerical investigation of reconstructing the vibro-acoustic responses of an arbitrary structure subject to vibration excitations based on the Helmholtz equation least-squares (HELS) method is presented. It is emphasized that in many engineering applications, the exact solution to a general vibrating structure does not exist, and the HELS method is one way of getting approximate solutions in a cost-effective manner. In this study, the test object is a simply supported, unbaffled thin plate. The reason for selecting this simply supported plate is that the analytic solutions to the plate vibrations are readily available. The field acoustic pressures generated by the Rayleigh integral are taken as input to HELS algorithms to reconstruct the normal velocity and normal acoustic intensity on the plate surface using Tikhonov regularization associated with generalized cross-validation methods. The reconstructed normal surface velocities are compared with the benchmark values, and the out-of-plane vibration patterns at the first five natural frequencies are compared with natural modes of the simply supported plate. The effects of measurement aperture, stand-off distance, and location of the origin of the coordinate system on the resultant accuracy of reconstruction are examined. [Work supported by NSF.]

On the choice of expansion functions in the Helmholtz equation least-squares method

This paper examines the performance of Helmholtz equation least-squares (HELS) method in reconstructing acoustic radiation from an arbitrary source by using three different expansions, namely, localized spherical waves (LSW), distributed spherical waves (DSW), and distributed point sources (DPS), under the same set of measurements. The reconstructed acoustic pressures are validated against the benchmark data measured at the same locations as reconstruction points for frequencies up to 3275 Hz. Reconstruction is obtained by using Tikhonov regularization or its modification with the regularization parameter selected by error-free parameter-choice methods. The impact of the number of measurement points on the resultant reconstruction accuracy under different expansion functions is investigated. Results demonstrate that DSW leads to a better-conditioned transfer matrix, yields more accurate reconstruction than both LSW and DPS, and is not affected as much by the change in measurement points. Also, it is possible to obtain optimal locations of the auxiliary sources for DSW, LSW, and DPS by taking an independent layer of measurements. Use of these auxiliary sources and an optimal combination of regularization and error-free parameter choice methods can yield a satisfactory reconstruction of acoustic quantities on the source surfaces as well as in the field in the most cost-effective manner.

Reconstruction of vibro-acoustic fields using hybrid nearfield acoustic holography

Hybrid nearfield acoustical holography is utilized to reconstruct the vibro-acoustic fields generated by an arbitrary structure vibrating in an unbounded fluid medium. This hybrid NAH is based on the modified Helmholtz equation least-squares method that expresses the field acoustic pressure in terms of expansions of outgoing and incoming waves. The expansion coefficients are determined by solving an over-determined linear system of equations obtained by matching the assumed-form solution to a finite number of acoustic pressures measured in the field via least squares. The measurements are taken over a conformal surface around the source at close range so that the evanescent waves can be captured. Once the expansion coefficients are specified, enough field acoustic pressures are regenerated on a conformal surface around the source and are taken as input to the Helmholtz integral formulation implemented numerically by the boundary element method. The acoustic pressure and normal velocity on the source surface are reconstructed using singular value decomposition, modified Tikhonov regularization and generalized cross-validation method. This hybrid NAH enables one to obtain a satisfactory reconstruction cost effectively because the majority of the input data are calculated but not measured. Numerical examples of reconstructing the vibro-acoustic quantities on the surfaces of an engine block and a highly elongated cylinder are demonstrated.

The Helmholtz equation least-squares method and Rayleigh hypothesis in near-field acoustical holography

In this paper we present a numerical investigation of reconstructing time-harmonic acoustic pressure field in two dimensional space by using a series expansion—the so-called Helmholtz equation least-squares (HELS) method. Series expansion methods (or the Rayleigh methods) have been widely used in predicting the scattered acoustic pressure. With regularization, they can also be applied to reconstruction of acoustic pressure on the source surface from the measurements taken in the field, and HELS is the first such attempt for these problems. In this paper, we establish HELS in the framework of the Rayleigh methods and reveal its interrelationship with the Rayleigh hypothesis. In particular, to regularize a reconstruction problem, we use the method of quasisolutions, i.e., a Tikhonov regularization with an a posteriori choice of the regularization parameter. It is shown that without regularization HELS can still yield a satisfactory reconstruction of acoustic radiation from an arbitrary object when enough measurements are taken at sufficiently close range to the source. With regularization the number of measurements can be reduced and reconstruction accuracy be enhanced. It is concluded that HELS can be used to reconstruct acoustic radiation from a convex arbitrarily shaped vibrating object regardless of the validity of the Rayleigh hypothesis, although in practice the results will depend on the rate of convergence of the approximating sequence.

On the behavior of solutions obtained by the HELS method inside the minimum sphere

Previous experimental and numerical studies have shown that the Helmholtz equation least-squares (HELS) method [Wu, J. Acoust. Soc. Am. 107, 2511–2522 (2000)] can be an effective methodology for nearfield acoustical holography. However, results also demonstrated that the accuracy of reconstruction on a highly nonspherical surface might be unsatisfactory. The validity of the HELS method on these highly nonspherical surfaces has become a controversial topic, just like the Rayleigh hypothesis for acoustic scattering on a corrugated surface. In this paper, the validity of solutions obtained by the HELS method will be examined for acoustic radiation from an infinite column with a rectangular cross section. The acoustic field inside the minimal circle that encircles the column will be reconstructed. The convergence of solutions both on the surface and inside the minimum circle will be checked. Moreover, a modified HELS formulation will be used, which describes the acoustic pressure as a superposition of both out-going and in-coming spherical waves. The nature of the solutions thus obtained will be examined, and the dependence of convergence of the HELS solution on the validity of the Rayleigh hypothesis and on the measurement locations will be investigated. [Work supported by NSF.]

Visualization of acoustic radiation from a vibrating bowling ball.

This paper presents visualization of acoustic radiation from a vibrating bowling ball using the Helmholtz equation least squares (HELS) method. In conducting the experiments, the ball is excited by a vibration shaker using stationary random signals. The radiated acoustic pressures are measured using two microphones and taken as input to the HELS formulations. The reconstructed acoustic pressures on the bowling ball surface are compared with those measured at the same locations. Also shown are comparisons of the reconstructed and measured acoustic pressure spectra at various locations on the bowling ball surface. Results demonstrate that the accuracy of reconstruction based on measurements over a conformal surface is much higher than that over a finite planar surface. This is because the latter often extends beyond the near-field region, making the accuracy of measurements inconsistent. Nevertheless, satisfactory reconstruction of acoustic pressure fields over the entire bowling ball surface can still be obtained based on the measurements taken over a finite planar surface on one side of the source. In a similar manner, the normal component of the surface velocity is reconstructed. Once these acoustic quantities are determined, the time-averaged acoustic intensity is calculated. Also presented are the formulations for estimating a priori the numbers of expansion functions and measurements required by the HELS method and the guidelines for determining the reconstruction error and optimum measurement locations, given the overall dimensions of the source and the highest frequency of interest in reconstruction.

An outline of a proposed acoustic holography system based on the Helmholtz equation–least-squares (HELS) method

The Helmholtz equation–least-squares (HELS) method [Wang and Wu, J. Acoust. Soc. Am. 102, 2020–2032 (1997); Wu, J. Acoust. Soc. Am. 107, 2511–2522 (2000)] has been proven a reliable acoustic holography method. The main advantages are its simplicity in formulations, flexibility in measurement locations, and uniformity in algorithm for a variety of source geometry and conditions. This paper presents a complete acoustic holography system based on the HELS method that makes full use of these advantages. This system has been under development as part of an ongoing research at the Acoustics, Vibration, and Noise Control Laboratory at Wayne State University. The physical setup will be described which includes the front-end multichannel data acquisition system and the associated hardware. Experimental results on reconstruction of the radiated acoustic fields from a complex source using this system will be demonstrated. Comparisons of this system with other acoustic holography systems currently available will be made. [Work supported by NSF.]

On reconstruction of acoustic pressure fields using the Helmholtz equation least squares method

This paper presents analyses and implementation of the reconstruction of acoustic pressure fields radiated from a general, three-dimensional complex vibrating structure using the Helmholtz equation least-squares (HELS) method. The structure under consideration emulates a full-size four-cylinder engine. To simulate sound radiation from a vibrating structure, harmonic excitations are assumed to act on arbitrarily selected surfaces. The resulting vibration responses are solved by the commercial FEM (finite element method) software I-DEAS. Once the normal component of the surface velocity distribution is determined, the surface acoustic pressures are calculated using standard boundary element method (BEM) codes. The radiated acoustic pressures over several planar surfaces at certain distances from the source are calculated by the Helmholtz integral formulation. These field pressures are taken as the input to the HELS formulation to reconstruct acoustic pressures on the entire source surface, as well as in the field. The reconstructed acoustic pressures thus obtained are then compared with benchmark values. Numerical results demonstrate that good agreements can be obtained with relatively few expansion functions. The HELS method is shown to be very effective in the low-to-mid frequency regime, and can potentially become a powerful noise diagnostic tool.

Analysis of reconstruction of radiated acoustic pressure fields by using the HELS method

This paper presents a detailed analysis of reconstruction of the acoustic pressure fields radiated from a partially vibrating sphere by using the Helmholtz equation-least squares (HELS) method [Z. Wang and S. F. Wu, J. Acoust. Soc. Am. 102, 2020–2032 (1997); 104, 2054–2060 (1998)]. The objectives of this study are to gain insight into an acoustic inverse problem and to develop guidelines that may be helpful in the future reconstruction work. Because of the complexities involved, many questions such as where and how many measurements should be taken, how many expansion terms are necessary for a given number of measurements, etc., remain unanswered. These questions will be tackled here. The reasons for selecting a partially vibrating sphere are twofold: (1) the resulting acoustic pressure field contains a very pronounced near-field effect; and (2) analytic results are known so the reconstructed acoustic pressures can be checked rigorously. The impact of a loss of the near-field effect on reconstruction will be investigated. The effects of measurement distance, area, and shape on the accuracy of reconstruction will be examined. The efficiency of reconstruction using the HELS method as the increase in frequency will also be studied. [Work supported by the NSF, Grant No. CMS-9802847.]

An overview of reconstructing acoustic pressure fields using the HELS method

The acoustic pressures radiated from complex vibrating structures are reconstructed by using the Helmholtz equation least-squares (HELS) method [Wang and Wu, J. Acoust. Soc. Am. 102, 2020–2032 (1997); Wu and Yu, ibid. 104, 2054–2060 (1998)]. Specific examples include an engine block and the interior space of a passenger vehicle. These structures are of arbitrary shapes and geometry, containing sharp edges and corners, and abrupt changes in surface contour. To test the robustness of the HELS method, measurements of field acoustic pressures are taken over a planar surface at a certain distance away from the structure. The reconstructed acoustic pressures, however, extend over the entire (nonplanar) surface of the structure. Note that the input data in these cases are not error free due either to measurement uncertainties or to the loss of the near-field effect. On the other hand, reconstructed acoustic pressures consist of predominantly the near-field effect. Hence the problem becomes mathematically ill-posed. To overcome this ill-posedness difficulty, an optimization scheme is developed which enables one to obtain satisfactory reconstruction results with a relatively few number of measurements in the field. The HELS method is shown to be effective in the low- to mid-frequency range, and can become a robust noise diagnostic tool for analyzing structure-borne sounds. [Work supported by NSF.]

Reconstructing interior acoustic pressure fields via Helmholtz equation least-squares method

This paper extends the Helmholtz equation least-squares (HELS) method previously developed by Wang and Wu [J. Acoust. Soc. Am. 102, 2020–2032 (1997)] to reconstruction of acoustic pressure fields inside the cavity of a vibrating object. The acoustic pressures are reconstructed through an expansion of the acoustic modes generated by the Gram–Schmidt orthonormalization with respect to the particular solutions to the Helmholtz equation. Such an expansion is uniformly convergent because the selected acoustic modes consist of a uniformly convergent series of Legendre functions. The coefficients associated with these acoustic modes are determined by requiring the assumed-form solution to satisfy the pressure boundary condition at the measurement points. The errors incurred in this process are minimized by the least-squares method. Numerical examples of partially vibrating spheres and cylinders with various half-length to radius aspect ratios subject to different frequency excitations are demonstrated. The reconstructed acoustic pressures are compared with the analytic solutions and numerical ones obtained by using the standard boundary element method (BEM) codes.

On reconstruction of acoustic pressure fields using the HELS method

The Helmholtz equation-least squares (HELS) method was recently developed by Wang and Wu [J. Acoust. Soc. Am. 102, 2020–2032 (1997)] to reconstruct the acoustic pressures generated by an arbitrary vibrating structure. In this method the acoustic pressures are expressed in terms of an expansion of ‘‘acoustic modes.’’ The coefficients associated with these acoustic modes are determined by requiring the assumed-form solution to satisfy the pressure boundary condition at measurement points. The errors incurred in this process are minimized by the least-squares method. The advantages of this method are that (1) acoustic pressures can be reconstructed over the entire surface; (2) solutions thus obtained are unique; and (3) efficiency of numerical computation is high, because the number of measurements is determined by that of expansion terms, which is small when a right coordinate system is selected for a particular source geometry under consideration. Such an approach is applicable to both exterior and interior regions. For exterior problems, the reconstructed acoustic pressures can be accurate when the points fall outside the minimum circumscribing spheroidal surface which encloses the vibrating surface, but approximate when they are inside. For interior problems, the accuracy of reconstruction is high because the entire region is inside the minimum circumscribing spheroidal surface. In both cases, the accuracy of reconstruction is limited by excitation frequencies, which is inherent in all expansion theories. [Work supported by NSF.]

Helmholtz equation–least-squares method for reconstructing the acoustic pressure field

A method using spherical wave expansion theory to reconstruct acoustic pressure field from a vibrating object is developed. The radiated acoustic pressures are obtained by means of an expansion of independent functions generated by the Gram–Schmidt orthonormalization with respect to the particular solutions to the Helmholtz equation on the vibrating surface under consideration. The coefficients associated with these independent functions are determined by requiring the assumed form of solution to satisfy the pressure boundary condition at the measurement points. The errors incurred in this process are minimized by the least-squares method. Once these coefficients are specified, the acoustic pressure at any point, including the source surface, is completely determined. In this paper, this method is used to reconstruct the surface acoustic pressures based on the measured acoustic pressure signals in the field. It is shown that this method can be applied to both separable and nonseparable geometries, and the solutions thus obtained are unique. However, the convergence may be worse for an elongated object subject to high-frequency excitations.