Helmholtz Equation Least Squares Research Articles

Experimental validation of vibroacoustic reconstruction of a rectangular plate with different boundary conditions.

This paper provides an experimental validation of vibroacoustic responses of a rectangular plate reconstructed using Helmholtz equation least squares (HELS) method. Experimental studies were conducted on baffled rectangular plates of different aspect ratios with free as well as clamped boundary conditions under random excitation. The radiated acoustic pressures were measured using a planar array of microphones at a very close distance to the plate surface and taken as input for the HELS codes. The normal surface velocity distributions were then reconstructed, and the results were compared against the benchmark data obtained using a laser vibrometer. The structural modes of the plate with different boundary conditions were compared with those obtained by experimental modal analysis. Good agreements were obtained for both clamped and free boundary conditions.

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.

The Nyquist spatial sampling requirement in nearfield acoustical holography.

Numerical simulations of Fourier acoustics and Helmholtz equation least squares (HELS) based NAH to visualize incoherent point sources separated by 6.25 mm are presented. The acoustic pressure measurements are taken over a 50 × 50 mm2 plane at different stand‐off distances under various signal to noise ratios (SNRs). The considered source frequencies are well above 9 kHz. For Fourier acoustics based NAH, an 11 × 11 microphone array with 5 mm microphone spacing is used to collect the input data. Results show that Fourier acoustics based NAH encounters great challenges in locating these sources as the frequency and stand‐off distance increase and SNR decreases unless regularization is used. On the other hand, a 5 × 5 microphone array with 12.5‐mm microphone spacing is used for HELS under the same conditions without regularization. Clearly, the Nyquist spatial sampling requirement is violated in this case. Nevertheless, if the goal is to identify the locations of discrete airborne sources rather than reconstructing precise sources amplitudes, HELS can still yield satisfactory results. Results show that HELS is capable of locating the discrete airborne sources with a high spatial resolution (<± 2.5 mm) when stand‐off distance is 10 mm and SNR is 10 dB.

High-frequency reconstruction of radiation patterns from a rectangular baffled plate using the Helmholtz equation least squares method

The Audio Engineering Society (AES) Standards require that full-sphere polar measurements of a sound field be taken at a very fine angular increment (1 deg) to ensure the accuracy in describing high-frequency radiation patterns of a source. This super fine angular resolution coupled with a far-field measurement requirement leads to an excessive number of measurements. For example, at 1 m and 1 degree angular resolution we would need more than 64&amp;lt;th&amp;gt;000 measurement points. To alleviate this difficulty, the HELS (Helmholtz Equation Least Squares)-based near-field acoustical holography is used to reconstruct the radiation patterns. It is shown that HELS can provide accurate reconstruction of radiation patterns for dimensionless frequencies up to ka=25 based on the sound field sampled on a spherical surface at a rate lower than that required by AES Standards. HELS allows for accurate reconstruction even for sample spacing that violates the Nyquist spatial sampling rate in certain directions. In this paper, the convergence of HELS solutions is examined by comparing reconstructed radiation patterns with the analytic solution for a rectangular baffled plate. In particular, the impacts of the number of expansion terms in HELS and measurement points, and that of errors imbedded in the input data on the accuracy of reconstruction are analyzed.

Spatial resolution of Helmholtz equation least-squares-based nearfield acoustical holography

Traditional applications of near-field acoustical holography require that the acoustic pressures be taken at the Nyquist spatial sampling rate. For example, if the intended spatial resolution is 10 mm, then the spacing between measurement points should be 5 mm or less to ensure a target resolution. As a result, the total number of measurement points can be very high when a high spatial resolution in reconstruction is desired. The present paper examines the capability of using the Helmholtz equation least squares (HELS) method to reconstruct acoustic sources based on input data sampled at a rate lower than the Nyquist spatial sampling rate. Experimental results on two speakers separated by 9.53 mm are shown. A 4×4 microphone array is used with a spacing of 12.7 mm in the horizontal axis and 25.4 mm in the vertical axis, which is much larger than the spacing between the sound sources. Results show that if the coordinates of microphones are measured with a high accuracy (less than 2 mm) and the signal-to-noise ratio is 20 dB or higher, then these two sources can be reconstructed regardless whether the sounds are coherent or not. The reason for this phenomenon is explored in this paper.

DISTRIBUTED SOURCE BOUNDARY POINT METHOD-BASED NEARFIELD ACOUSTIC HOLOGRAPHY FOR RECONSTRUCTING THE ACOUSTIC RADIATION FROM ARBITRARILY SHAPED OBJECTS

Nearfield acoustic holography (NAH) is an indirect technique for identifying noise sources and visualizing acoustic field. Recently, several different methods, such as the spatial Fourier transform method, the boundary element method (BEM) and the Helmholtz equation-least squares (HELS) method, have been used to realize the NAH successfully. In the paper, a novel numerical method, the distributed source boundary point method (DSBPM), is proposed to realize the NAH. In the method, the transfer matrices from the reconstructed surface to the hologram surface are constructed indirectly by a set of particular solution sources located inside the vibrating structure, and their inverses are carried out by singular value decomposition (SVD) technique. Additionally, considering the high sensitivity of the reconstructed solution to measurement errors, the Tikhonov regularization method is implemented to stabilize the reconstruction procedure and the regularization parameter is determined by L-curve criterion. Compared with the BEM-based NAH, the variable interpolation, the numerical quadrature, and the treatments of singular integral and nonuniqueness of solution are all avoided in the proposed method. Two numerical examples and an experiment are investigated to validate the feasibility and correctness of the proposed method.

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.]

Acoustical feature extraction of rotating machinery with combined wave superposition and blind source separation

As the result of vibration emission in air, machine sound signal carries affluent information about the working condition of machine and it can be used to make mechanical fault diagnosis. The fundamental problems with fault diagnosis are the estimation of the number of sound sources and the localization of sound sources. The wave superposition can be employed to identify and locate sound sources, which is based on the idea that an acoustic radiator can be approximated and represented by the sum of the fields due to a finite number of interior point sources. But, in practice, a large number of measurements must be used in order to achieve a desired resolution, which makes the reconstruction process very time-consuming and expensive. In this paper, a combined wave superposition method has been developed reconstruct to acoustic radiation from machine acoustical signals. This method combines the advantages of both the wave superposition and Helmholtz equationleast squares methods, and it allows for reconstruction of the acoustic field from an arbitrary object with relatively few measurements, thus significantly enhancing the reconstruction efficiency. After sound source localization, the blind source separation (BSS) is proposed to extract acoustical feature from the mixed measuring sound signals. In a semi-anechoic chamber, a cross-planar microphone array, which consists of 29 microphones, was successfully applied to obtain the two-dimensional mapping of the sound sources. The location, the sound pressure, and the properties in frequency domain of the sound sources can be found through this method precisely. The experimental results demonstrate that the methods presented can potentially become an acoustical diagnosis tool.

Comparison of numerical methods for noise source identification

The identification of sound sources is difficult in the presence of many potential sources that interact. Nearfield Acoustical Holography (NAH), which refers to a process by which all aspects of the sound field in a three-dimensional volume can be reconstructed based on sound pressure measurements on a two-dimensional surface conceptually, allows the identification of complex noise sources that are otherwise difficult to characterize. Conventional NAH has been applied most commonly in simple geometries such as planar, cylindrical and spherical surfaces. Since most noise sources are not simple geometrically, it is not possible to project to the actual surface of the source in most instances, thus limiting the ability of conventional NAH to resolve the source structure accurately. Numerical acoustics procedures based on the Helmholtz integral equation have been developed that allow sound fields to be projected into arbitrarily-shaped interior or exterior domains from measured sound pressure field. Two commonly implemented procedures are the inverse Boundary Element Method (BEM) and the Helmholtz Equation Least-Squares (HELS) method. An Enhanced Surface Potential (ESP) based formulation was recently developed and promised to overcome some of the drawbacks of the BEM-based formulation. In this paper, these three methods are compared through a numerical example.

Comparison of Helmholtz equation least squares and other methods in diagnosing stationary and nonstationary vehicle noises

This paper gives a brief overview of current near-field acoustic holography (NAH) techniques. In particular, it compares the analyses of vehicle transaxle noise under run-up and coast-down conditions using spatial transformation of sound field (STSF) and Helmholtz equation least-squares (HELS) based NAH. It also demonstrates an application of HELS to visualizing brake squeals and compares its advantages and disadvantages to those of a laser vibrometer.

Reconstruction of transient acoustic radiation from a sphere

Transient near-field acoustical holography (NAH) formulation is derived from the Helmholtz equation least squares (HELS) method to reconstruct acoustic radiation from a spherical surface subject to transient excitations in a free field. To facilitate derivations of temporal solutions, we make use of the Laplace transform and expansion in terms of the spherical Hankel functions and spherical harmonics, with their coefficients settled by solving a system of equations obtained by matching an assumed-form solution to the measured acoustic pressure. To derive a general form of solution for a temporal kernel, we replace the spherical Hankel functions and their derivatives by polynomials, recast infinite integrals in the inverse Laplace transform as contour integrals in a complex s-plane, and evaluate it via the residue theorem. The transient acoustic quantities anywhere including the source surface are then obtained by convoluting the temporal kernels with respect to the measured acoustic pressure. Numerical examples of reconstructing transient acoustic fields from explosively expanding, impulsively accelerating, and partially accelerating spheres, and that from a sphere subject to an arbitrarily time-dependent excitation are depicted. To illustrate the effectiveness of HELS-based transient NAH formulations, all input data are collected along an arbitrarily selected line segment and used to reconstruct transient acoustic quantities everywhere.

Reconstruction of vibroacoustic fields in half-space by using hybrid near-field acoustical holography

In this paper we examine the accuracy and efficiency of reconstructing the vibroacoustic quantities generated by a vibrating structure in half-space by using hybrid near-field acoustic holography (NAH) and modified Helmholtz equation least squares (HELS) formulations. In hybrid NAH, we combine modified HELS with an inverse boundary element method (IBEM) to reconstruct a vibroacoustic field. The main advantage of this approach is that the majority of the input data can be regenerated but not measured, thus the efficiency is greatly enhanced. In modified HELS, we expand the field acoustic pressure in terms of outgoing and incoming spherical waves and specify the corresponding expansion coefficients by solving a system of equations obtained by matching the assumed-form solution to the measured acoustic pressure. Here the system of equations is ill conditioned and Tikhonov regularization is implemented through singular value decomposition (SVD) and the generalized cross-validation (GCV) method. Numerical examples of a dilating and oscillating spheres and finite cylinder are demonstrated. Test results show that hybrid NAH can yield a more accurate reconstruction than does a modified HELS, but a modified HELS is more efficient than is hybrid NAH [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.

Visualization of vehicle interior sound field using Nearfield Acoustical Holography based on the Helmholtz-Equation Least-Squares (HELS) method

This paper examines the application of Nearfield Acoustical Holography based on the Helmholtz Equation Least-Squares (HELS) method for visualization of the sound-pressure field in the passenger compartment of a vehicle with hard interior surfaces. In the HELS method, sound pressure is determined by an expansion of a set of orthogonal basis functions that are solutions to the Helmholtz equation. Coefficients associated with these basis functions were determined by a collocation method with errors minimized by the least-squares method. The number of expansion terms in the method was optimized to ensure a convergence of the solution anywhere in the interior region of the vehicle, including the boundary surfaces. The reconstructed sound pressures were compared with benchmark sound pressures calculated by the boundary element method (BEM). Results demonstrated that the HELS method permitted reconstruction of the sound-pressure field with much greater computational efficiency than the BEM procedure. Accuracy of reconstruction was relatively insensitive to the choice of the locations for input data. For hard interior surfaces, a small box-shaped surface on which input data were collected was shown to be sufficient to provide good reconstruction results. Such a configuration could be implemented relatively easily in practice. Reconstruction of sound-pressure fields at resonance frequencies of the vehicle interior cavity was also validated.

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.

Reconstructing transient acoustic field resulting from an explosion

Generalized Helmholtz equation least squares (HELS) formulations for reconstructing transient acoustic radiation from an arbitrary object subject to arbitrarily time-dependent excitations are developed. Laplace transform is employed to build a transfer matrix for a multi-input multi-output system and reconstructed signals are obtained using convolution with respect to the input data in the time domain directly. Stability of solutions is realized by controlling the pole-residue locations in the Laplace domain. To facilitate calculations of the poles and residues, ztransform is utilized. To validate this general transient HELS formulation, the reconstructed pressure histories are compared with those measured by pressure sensors at the same locations. Space-time tomography resulting from different charges is produced, and the acoustic pressure distribution over the entire time record is reconstructed. It is interesting to notice that this transient HELS formulation enables one to trace time histories of blast pressures back to the explosion location. In other words, it is possible to reconstruct an explosion back in space and time based on the time-domain signals measured at a later time. [Work supported by NSF.]

Visualizing automobile squeals and their vibration modes

Automobile brake squeals have always been a major problem for the auto industry because of the high pitch and intensity level. Currently, the vehicle brake squeal diagnostics is carried out by using a scanning laser synchronized with squeals. If squeals occur at multiple frequencies, then filtering must be used so that the scanned out-of-plane vibration corresponds to a single frequency. This process is very time consuming, especially when squeals contain multiple frequencies. Moreover, the results thus obtained are valid at measurement locations only. In this paper, we use Helmholtz equation least-squares (HELS)-based near-field acoustical holography to reconstruct squeal sounds via a custom-designed 48-microphone array conformal to the shape of a brake assembly. This microphone array allows for taking a snapshot of 3D acoustic image on the surface of the brake assembly and produces 3D mappings of all acoustic quantities. In particular, we can identify the out-of-plane vibration modes that are responsible for squeals and reveal the locations where squeals originate from. There is no need for synchronization and filtering. The whole measurement can be done in seconds, and the reconstructed results are valid not only on source surface, but also in the field. [Work supported by NSF.]

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.

Hybrid near-field acoustic holography.

Hybrid near-field acoustical holography (NAH) is developed for reconstructing acoustic radiation from an arbitrary object in a cost-effective manner. This hybrid NAH is derived from a modified Helmholtz equation least squares (HELS) formula that expands the acoustic pressure in terms of outgoing and incoming waves. The expansion coefficients are determined by solving an overdetermined linear system of equations obtained by matching the assumed-form solution to measured acoustic pressures through the least squares. Measurements are taken over a conformal surface around a source at close range so that the evanescent waves can be captured. Next, the modified HELS is utilized to regenerate as much acoustic pressures on the conformal surface as necessary and take them as input to the Helmholtz integral formulation implemented numerically by boundary element method (BEM). The acoustic pressures and normal velocities on the source surface are reconstructed by using a modified Tikhnov regularization (TR) with its regularization parameter determined by generalized cross validation (GCV) method. Results demonstrate that this hybrid NAH combines the advantages of HELS and inverse BEM. This is because a majority of the input data are regenerated but not measured, thus the efficiency of reconstruction is greatly enhanced. Meanwhile, the accuracy of reconstruction is ensured by the Helmholtz integral theory and modified TR together with GCV method, provided that HELS converges fast enough on the measurement surface. Numerical examples of reconstructing acoustic quantities on the surface of a simplified engine block are demonstrated. [Work supported by NSF.]

Reconstruction of the surface normal acoustic intensity from noisy pressure data by regularized Helmholtz equation least squares method

Reconstructing the normal acoustic intensity on a source surface from the field pressure measurements is critical in identifying the noise sources and their transmission paths. In this paper, we consider the reconstruction of time-averaged normal acoustic intensities from noisy pressure data using the Helmholtz equation least squares (HELS) method with regularization by truncated singular value decomposition or the method of Tikhonov. The regularization parameter is calculated by generalized cross validation, L-curve, and quasioptimality criterion and the best solution is chosen. Numerical results show that the normal velocity reconstruction is more sensitive to noise in the input data than the pressure reconstruction, and depends critically on the choice of the regularization parameter. It is found that the HELS-based Dirichlet-to-Neumann (DtN) map has a similar accuracy as compared to the Green’s formula-based DtN map with Tikhonov regularization for 1% noise or larger in the input data, provided that HELS converges sufficiently fast. Moreover, determining the surface normal velocity from the reconstructed surface pressure is more stable than reconstructing it directly from the field pressure data. It is concluded that a regularized HELS, when it converges fast enough, can be an easy-to-use and cost-effective tool for intensity reconstruction [Work supported by NSF.]