Equivalent Source Method Research Articles

Fourier acoustics: Uncovering the origins of sound

No doubt, Jean-Baptiste Joseph Fourier has had a great impact in science. At the root of Fourier Acoustics, the Fourier series, when employed as a starting point and manipulated, allows us to uncover the origins of sound in a simple and elegant way, not only in acoustics but also in electromagnetics. A well-known stepping stone on the path is nearfield acoustical holography (NAH) and its theory can be recast almost completely, and simply, in terms of two-dimensional (2D) Fourier series and applied to any geometry. For separable coordinate systems, each of the components of the series can be propagated to a surface conformal to the measurement surface using an analytically defined propagator. But even when the surfaces are not part of a separable system, the spirit of Fourier’s expansions is mirrored with the singular value decomposition, applied to a numerically derived propagator constructed using the equivalent source method. A comprehensive demonstration of Fourier acoustics applied to aeroacoustics was in the cabin of a Boeing aircraft, in flight, using a two-dimensional spherical array. Reconstructions of vector intensity in a volume from correlations of the measured pressure to panel accelerometers showed the direction and location of the dominant fuselage sources of noise—uncovering the origin of sound. [Work supported by the Office of Naval Research.]

Predicting range-dependent underwater sound propagation from structural sources in shallow water using coupled finite element/equivalent source computations

Predictions of underwater sound propagation (USP) from structural sources in complex shallow-water environments are crucial for underwater navigation, communication, and localization. Modeling range-dependent USP in shallow water remains challenging because structural acoustic radiation is coupled with complex waveguide physics. This paper presents a coupled finite element (FE)/equivalent source (ES) computation scheme for predicting the range-dependent USP from a structural source. The scheme involves coupled vibroacoustic FE/ES analyses and waveguide-field ES computations. The former computes the structural vibration response and reproduces the structural-acoustic radiation at arbitrary spatial positions. The coupled vibroacoustic FE/ES analysis provides the input for the waveguide-field ES computations, which couple the structural-acoustic radiation with the shallow-water environment. A multilayer acoustic–elastic ES method (ESM) is developed to accommodate sound speed inhomogeneities and a range-dependent elastic seabed. Numerical simulations demonstrate the interactions of structural-acoustic radiation with two-dimensional topographies and internal solitary waves. The proposed scheme is extended to three dimensions by combining the coupled vibroacoustic FE/ES analysis with a pre-corrected fast Fourier transform-accelerated ESM. The results validate the proposed scheme and demonstrate its benchmark-quality solutions and high numerical efficiency, suggesting great application potential for optimizing the sonar performance at the preliminary design stage.

Multizone sound reproduction with adaptive control of scattering effects.

The aim of multizone sound reproduction is to reproduce different sound programs to different spatial zones in a common space using a set of loudspeakers. Several multizone sound reproduction methods have been developed over the past few decades, and most of them assume that there are no listeners in the listening area. However, when listeners enter the listening area, the scattering effects will reduce the performance of the multizone sound reproduction system. In this paper, an adaptive control method of scattering effects is proposed for multizone sound reproduction. The proposed method models the scattering effects based on the equivalent source method and uses adaptive filters to estimate the scattering effect by a small number of additional microphones. To validate the proposed method, several simulations are presented. The results of the simulations indicate that the proposed adaptive method exhibits good performance when used in multizone sound reproduction with the scattering effects.

Machine learning aided near-field acoustic holography based on equivalent source method.

In recent times, equivalent source method-based near-field acoustic holography methods have been extensively applied in sound source localization and characterization. The most commonly used equivalent sources are spherical harmonics. In a non-reverberant environment with no reflections, these equivalent sources could be the best choice since spherical harmonics are derived for the Sommerfeld boundary condition. However, these methods are not the best fit for reverberating environments. In such cases, a new relationship can be calculated between the field weights and the measured pressure with enough training examples. The proposed machine learning models include linear regression (LR) with adaptive moment estimation (Adam), LR with limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS), and multi-layer perceptron with one and two hidden layers. These methods are tested for multiple monopoles and vibrating plate simulations in a room with different wall absorption coefficients. The data-driven methods are also studied on loudspeakers numerically and experimentally in a free field environment. The results from these methods are compared with the results of one norm convex optimization (L1CVX). LR with L-BFGS performed the best among all the methods studied and performed better than L1CVX for less absorption coefficient for geometrically separable sources. LR with L-BFGS also has much faster inference times.

Spatial Interpolation of Early Room Impulse Responses Using Equivalent Source method based on Grouped Image Sources

The Room Impulse Response (RIR), which indicates the propagation characteristics of sound, is used in various acoustical applications. In particular, multi-point measurement of RIRs is important for applications that require spatial information, and measurement of these is difficult. Previously, spatial interpolation methods of RIRs based on physical models have been proposed. Especially, the sparse Equivalent Source Method (ESM) with Image Source Method (ISM) acheive the spatial interpolation of the early reflected sounds with higher accuracy at high frequency. However, this method has a problem that when the order of reflected sounds to be estimated increases, it becomes difficult to find the optimal solutions because the number of bases increases. In this study, to reduce the number of bases in each optimization problem, we proposed a sparse ESM with the image sources grouped by the arrival times of reflected sounds. The optimization problem for each group of image sources is individually solved. In the simulation experiment, it is revealed that the proposed method can estimate higher-order reflections with higher accuracy compared to the conventional method. In addition, we found that the estimated region with high accuracy became broader.

Equivalent source method based Near-field acoustic holography using machine learning

The equivalent source method has been one of the most commonly used methods for sound source localization. It involves equivalent sources spread over the source plane (or region). The pressure fields from these equivalent sources are usually spherical harmonics. But, the spherical harmonic fields are derived for the Sommerfeld boundary condition with no reflection or reverberation. Data-driven methods help perform sound source localization in a reverberant environment when no prior information about the surroundings is available. The methods studied are linear regression (LR) with Adam, linear regression with L-BFGS, multi-layer perceptron (MLP) with one and two hidden layers. The simulations are conducted for two monopoles in rooms with different reverberation times and compared with one norm convex optimization (L1CVX). It is observed that overall, LR with L-BFGS gave the best results. Also, for low reverberation time, LR with L-BFGS was able to localize the sources better than L1CVX.

Sound field reproduction based on Pressure Matching with transfer functions modeled by equivalent sources and image sources

Pressure matching (PM) method is an effective method to control a sound field using many loudspeakers. It is difficult to implement PM method because it requires multi-point room impulse response (RIR) measurements. Multi-point measurements are challenging because they require a large number of microphones and a narrow sampling interval when dealing with high frequencies. Therefore, to synthesize a sound field in a local region, we proposed a PM method using estimated RIRs from only a small number of measurements based on the sparse Equivalent Source Method (ESM). In the previous method, we modeled the transfer functions considering only the direct sound from the loudspeaker. However, when the sound field is synthesized in the actual environment, the reflected sounds must be considered. In this study, we propose a PM method using the estimated transfer functions by sparse ESM based on the image source method considering the direct sound and early reflected sounds. In the experiment, we evaluated the accuracies of synthesized sound field by the proposed method compared to the methods only considering the direct sound.

Correction to: An equivalent source method for removal of attitude-induced responses in drone-towed magnetic scalar gradiometry data

Correction to: An equivalent source method for removal of attitude-induced responses in drone-towed magnetic scalar gradiometry data

АНАЛІЗ ЕЛЕКТРОМАГНІТНИХ ПЕРЕХІДНИХ ПРОЦЕСІВ В РОЗПОДІЛЬЧІЙ МЕРЕЖІ 10 КВ ПРОМИСЛОВОГО ПІДПРИЄМСТВА

The article is devoted to the issue of short-circuit analysis in the high-voltage network of the power supply system of an industrial enterprise. A short-circuit analysis is based on the methodology recommended by the IEC 60909 International Standard. The method of equivalent voltage source at a shortcircuit point was used to calculate a short-circuit current. The necessity to analyze short-circuit currents at all levels of the power supply system was used as justification of the feasibility of automation of the calculation process. Module "Shortcircuit analysis" of the ETAP software was used as an analysis tool. The nominal parameters of the elements of the power supply system of the industrial enterprise and its topological structure are the initial data for the calculation. A single-line scheme of a 10 kV distribution network of power supply system is designed in ETAP program for implementation of the calculation process. The calculation of the value of the initial and maximum current for symmetric (three-phase) and asymmetrical ("phase-phase" and "phase-land") short circuits at the specified points of the electrical network, as well as the detection of inconsistency of characteristics installed equipment to short-circuit currents and displaying inadequate places are the result of the module "Short-circuit analysis" of the ETAP program. The results of the analysis of short circuit in the investigated distribution network of the 10 kV industrial enterprise did not find inconsistencies of the equipment characteristics of the requirements for the stability of short-circuit currents.

An Improved Time-Domain Equivalent Source Method for Accurate Indoor Simulation of Pass-By Noise

Abstract Time-domain equivalent source method (TDESM) has been employed for indoor simulation of pass-by noise (PBN). Despite removing the requirement of dedicated large semi-anechoic chambers for the microphone array technique, it is still difficult to obtain high simulation accuracy due to the existence of signal splicing error and ill consideration of Doppler effect induced by the vehicle motion. In this article, the TDESM is developed to improve the simulation accuracy of PBN. Unlike the previous TDESM, the improved TDESM (I-TDESM) models the PBN by superposing the contributions of an array of moving equivalent sources, which can take the influence of vehicle motion into account. The PBN at the receivers can be directly reconstructed via the sound field calculation of moving sources. Numerical studies and real vehicle experiments are implemented to demonstrate the feasibility of the I-TDESM. Simulation results show that the I-TDESM can naturally incorporate the Doppler effect and effectively eliminate the signal splicing error, thus improving the simulation accuracy. The experiment results of real vehicle further validate the I-TDESM.

An equivalent source method for removal of attitude-induced responses in drone-towed magnetic scalar gradiometry data

SUMMARY Drone-towed scalar field gradiometry surveys conducted in windy conditions or under self-excited oscillations generate attitude-induced responses that can hinder the geological interpretation. Here, we present a gradiometric equivalent source method (GESM) to remove these attitude-induced responses by interpolating and continuing the measured gradiometry data to new idealized pseudo-sensor positions free of any attitude deviations. In addition, we present transverse horizontal difference (THD) data from a precisely positioned drone-towed horizontal gradiometry survey collected in Nautanen, northern Sweden. Analysing the Nautanen survey’s positional data revealed that the gradiometer system exhibited directional-dependent yaw deviations with periods of unpredictable attitude deviations. Based on synthetic THD data created using the Nautanen survey’s positional data, these deviations manifest as line-to-line striping and short-wavelength oscillations in the THD maps. Applying GESM to the synthetic THD data removes these attitude-induced THD responses with satisfactory accuracy compared to the true THD values. Furthermore, on the actual THD data collected in Nautanen, applying GESM improved the continuity of anomalies, significantly improving the interpretation of the data. The results suggest that including GESM in the data processing of drone-towed gradiometry surveys, given precise positional information via an onboard GNSS-IMU system, ensures high-quality geological interpretation even in windy conditions or in high-gradient areas.

Galerkin equivalent sources method for sound field reconstruction around diffracting bodies

The rising interest for three-dimensional acoustic imaging requires the improvement of the numerical models describing the propagation between a radiating body and a microphone array. The commonly used free field transfer functions boil down to assume a full acoustic transparency of the radiating object, which, in some cases, may lead to misleading outcomes for their characterization. Among other approaches, equivalent sources methods (ESM) emerged as a convenient and powerful approach to simulate scattered sound fields. In this paper, an acoustic imaging algorithm, named Galerkin ESM, where equivalent sources are tailored to concomitantly match with microphone pressures and a Neumann boundary condition, is proposed. By means of a projected matrix inversion and backpropagation of the equivalent sources, Galerkin ESM aims at the direct synthesis of the pressure field around a diffracting body by making the most of an array measurement. This method is compared with two other existing imaging algorithms fueled by free field and computed transfer functions. The impact of the chosen transfer model is discussed, and Galerkin ESM performances are evaluated based on numerical and experimental test cases.

Extension of sound field reconstruction based on element radiation superposition method in a sparsity framework

Nearfield acoustic holography (NAH) is a powerful tool for realizing source identification and sound field reconstruction. The wave superposition (WS)-based NAH is appropriate for the spatially extended sources and does not require the complex numerical integrals. Equivalent source method (ESM), as a classical WS approach, is widely used due to its simplicity and efficiency. In the ESM, a virtual source surface is introduced, on which the virtual point sources are taken as the assumed sources, and an optimal retreat distance needs to be considered. A newly proposed WS-based approach, the element radiation superposition method (ERSM), uses piston surface source as the assumed source with no need to choose a virtual source surface. To satisfy the application conditions of piston pressure formula, the sizes of pistons are assumed to be as small as possible, which results in a large number of pistons and sampling points. In this paper, transfer matrix modes (TMMs), which are composed of the singular vectors of the vibro-acoustic transfer matrix, are used as the sparse basis of piston normal velocities. Then, the compressive ERSM based on TMMs is proposed. Compared with the conventional ERSM, the proposed method maintains a good pressure reconstruction when the number of sampling points and pistons are both reduced. Besides, the proposed method is compared with the compressive ESM in a mathematical sense. Both simulations and experiments for a rectangular plate demonstrate the advantage of the proposed method over the existing methods.

Compensating Method of Equivalent Current Sources of LSI-Core Macromodel Considering Voltage Fluctuations in On-Chip Power Distribution Network

Electro-magnetic compatibility (EMC) macromodels of a large-scale integrated (LSI) circuit core have been proposed to predict conducted RF emission and power integrity issues not only on a printed circuit board but also inside an LSI. They are equivalent circuit models that consist of lumped element impedance blocks of an LSI-core and equivalent current sources expressing internal switching activities. In conventional macromodeling, equivalent current sources have been extracted under ideal power supply conditions, i.e., with constant supply voltage. However, there exists transient supply voltage fluctuations in an actual on-chip power distribution network (PDN), and the operating current and its equivalent current sources are expected to vary with such fluctuations. Such issues with transient supply voltage fluctuations have not been extensively investigated. We propose a method of compensating the equivalent current sources of a macromodel by introducing amplitude and time correction factors, taking into account the voltage fluctuations in an on-chip PDN.

Inferring the characteristics of marine sediments from the acoustic response of a known, partially buried object.

We investigate the feasibility of using a known elastic target located near the seabed for the purpose of inferring characteristics of marine sediment. In the problem considered the object position and its burial depth are not known with precision. First, the admittance matrix of the elastic object is determined (numerically or experimentally) over a wide frequency range in the structural acoustic regime. Then, the equivalent source method (ESM) coupled with a spectral representation of the Green's functions in stratified domains is used to predict the object acoustic signature in various environments and experimental configurations. The resulting solver takes into account all multiple scattering between target (buried or not), sea floor, and sea surface and is not limited to short distances. After presenting the solution to the forward problem several synthetic inversions for sediment characteristics are shown. They are based upon a resonance-based misfit function we describe. The Bayesian procedure also infers object burial and source-object range, broadening its range of application.

An efficient frequency-domain prediction method for aeroacoustic radiation and scattering using equivalent sources

An efficient radiation and scattering prediction method is proposed in the frequency domain based on the convective Ffowcs Williams–Hawkings (FW–H) equation and the convective equivalent-source method (ESM). In this work, the convective FW–H equation and its spatial derivatives are adopted to develop ESM-I and ESM-S for predicting incident acoustic pressure and acoustic pressure gradient, respectively. The equivalent sources strength of ESM-I and ESM-S are determined by matching the pressure and pressure gradient boundary conditions on the integral surface of FW–H equation. The pressure gradient from ESM-S constructs a hard-wall boundary on a solid surface for the prediction of scattered pressure. Since then the total acoustic pressure can be efficiently evaluated from the incident and scattered components. An energy-based singular value decomposition method is adopted to filter the small singular values that probably lead to solution instability. Convective Green functions are suggested, which allow the final prediction methodology to be suitable for situations in different dimensions with the inclusion of moving-medium effects. Available analytical solutions for point sources as well as a monopole scattered by a cylinder, a sphere and a plane are used to examine the parameter sensitivity and accuracy of the proposed methodology.

A Dual-Layer Equivalent-Source Method for Deriving Gravity Field Vector and Gravity Tensor Components from Observed Gravity Data

A Dual-Layer Equivalent-Source Method for Deriving Gravity Field Vector and Gravity Tensor Components from Observed Gravity Data

Coupling 3D Liquid Simulation with 2D Wave Propagation for Large Scale Water Surface Animation Using the Equivalent Sources Method

AbstractThis paper proposes a method for simulating liquids in large bodies of water by coupling together a water surface wave simulator with a 3D Navier‐Stokes simulator. The surface wave simulation uses the equivalent sources method (ESM) to efficiently animate large bodies of water with precisely controllable wave propagation behavior. The 3D liquid simulator animates complex non‐linear fluid behaviors like splashes and breaking waves using off‐the‐shelf simulators using FLIP or the level set method with semi‐Lagrangian advection.We combine the two approaches by using the 3D solver to animate localized non‐linear behaviors, and the 2D wave solver to animate larger regions with linear surface physics. We use the surface motion from the 3D solver as boundary conditions for 2D surface wave simulator, and we use the velocity and surface heights from the 2D surface wave simulator as boundary conditions for the 3D fluid simulation. We also introduce a novel technique for removing visual artifacts caused by numerical errors in 3D fluid solvers: we use experimental data to estimate the artificial dispersion caused by the 3D solver and we then carefully tune the wave speeds of the 2D solver to match it, effectively eliminating any differences in wave behavior across the boundary. To the best of our knowledge, this is the first time such a empirically driven error compensation approach has been used to remove coupling errors from a physics simulator.Our coupled simulation approach leverages the strengths of each simulation technique, animating large environments with seamless transitions between 2D and 3D physics.

The adaptability of IEC 60909: 0-2016 to power grid with voltage levels above 400kV

The equivalent voltage source method is recommended in IEC 60909 for short-circuit current calculation, and the accuracy is verified by 400 kV and below voltage level power grids. There are 500 kV, 750 kV or 1000 kV power grids in some countries where IEC standards are applied, but there is no relevant research to verify the accuracy of the calculation method. In this paper, verification methods for IEC 60909 calculation accuracy for the three voltage level power grids are adopted. Calculation examples based on the operation and equipment parameters of the 500 kV, 750 kV and 1000 kV power grids are established. The results show that IEC 60909 short-circuit current calculation method is conservative, and the calculation accuracy meets the requirements. The research expands the application scope of this standard to all AC power grids, which is conducive to the engineering application of IEC 60909.

Acoustic radiation from a sphere pulsating near an impedance plane using a boundary integral equation method

In this study, a boundary integral equation method is proposed for investigating acoustic pressure radiation from a sphere pulsating near a free surface or an impedance plane. The half-space and free-space problems are investigated for the acoustic radiation of pulsating sphere. The effects of free surface and impedance boundaries are introduced into the mathematical model by employing three different half-space Green’s functions, respectively. These Green’s functions are derived, respectively, using the single image-source method, multiple equivalent-source method, and complex equivalent-source method. Green’s functions are implemented into the boundary element (BE) formulation. The surface of the pulsating sphere is discretized with linear and quadratic BEs, and the Combined Helmholtz Integral Equation Formulation (CHIEF) is employed to overcome the non-uniqueness problem. Four different case studies are considered for the sphere pulsating near a free surface or an impedance plane. The first case study involves the sphere pulsating near a free surface (perfectly reflective) and the single image-source method is used in the boundary element method (BEM) formulation. In the second case study, the sphere is assumed as pulsating near a perfectly reflecting and perfectly absorbing impedance planes, respectively. The multiple equivalent-source method is employed for the perfectly reflecting plane, but the multiple equivalent-source method and complex equivalent-source methods for the perfectly absorbing plane. The third case study involves a general impedance plane, and all the methods are employed, respectively, in the BE formulation. The final case study assumes a general impedance plane forming a perpendicular incidence and the complex equivalent-source method is used in this particular case. It is observed that there is a very good comparison between the results obtained from all these methods.