Equivalent Source Method Research Articles

A study on aerodynamic pass-by noise based on equivalent source method

Pass-by noise generated by high-speed trains is an important influencing factor on the surrounding environment and is getting more and more attention with the increase of speed. In the past, the aerodynamic pass-by noise prediction was usually conducted with the wind tunnel mode, ignoring the Doppler effect. In this paper, an equivalent source method considering the Doppler effect is proposed to study the aerodynamic pass-by noise. A cylinder case is considered first to verify the effectiveness of the method. Then a DSA 380 pantograph of a Chinese high-speed train is considered. The pantograph is divided into three parts, and each part is equivalent to a monopole or dipole sound source. Results show that the upper part of the pantograph is closer to a dipole source, while the middle and lower parts are closer to a monopole source. When the measurement point is directly above the pantogragh, the contribution of the upper part is the largest, while for both sides measurement points of the pantograph, the contribution of the upper part is the smallest.

The relationship between singular value decomposition orthogonal space vector and sound field reconstruction in equivalent source method

In acoustic holography, the equivalent source method represents a typical ill-posed inverse problem, characterized by the significant impact of measurement errors on the sought solution. To minimize the interference of noise with the sought solution, we analyze the relationship between orthogonal space vectors obtained through singular value decomposition and acoustic field reconstruction. The research findings indicate that the primary information of the acoustic field exhibits strong correlation with a small number of space vectors, while noise information demonstrates random correlation with all space vectors. Therefore, we retain the space vectors with higher correlation to the acoustic field while filtering out redundant ones, thereby stabilizing and accurately reconstructing the acoustic field. Numerical analysis of the acoustic field reconstruction demonstrates that compared to regularization methods such as Tikhonov and truncated singular value decomposition (TSVD), this approach achieves higher computational accuracy. Furthermore, the text combines practical examples to further explain the reasons behind the superior accuracy of this method.

An equivalent source method for acoustic problems with thermoviscous effects.

This paper presents an equivalent source method (ESM) for analyzing sound propagation in small-scale acoustic structures with thermoviscous effects. The formulations that describe the thermal, viscous, and acoustic modes for thermoviscous acoustic problems are introduced. The concept of ESM is then applied to solve these formulations, resulting in an efficient numerical computation and implementation procedure. Based on two different strategies, the obtained ESM formulations are coupled at the boundary using the isothermal, non-slip, and null-divergence conditions. The coupling based on the first strategy is efficient for solving thermoviscous acoustic problems with few matrices required. However, this procedure faces the evaluation of the tangential derivatives of the boundary velocity. Coupling the ESM formulations directly for each component of the total particle velocity at the boundary has no such problem, which leads to the second strategy. However, it entails a larger memory usage compared to the former. Additionally, the coupled finite element method (FEM)-ESM formulations based on the above strategies are developed for acoustic-structural interaction. The validity of the presented ESM formulations is demonstrated through benchmark examples, and that of the coupled FEM-ESM formulation is illustrated by the numerical analysis of a simplified microphone.

Research on multi-heat source arrangement optimization based on equivalent heat source method and reconstructed variational autoencoder

The variational autoencoder (VAE) architecture has significant advantages in predictive image generation. This study proposes a novel RFCNN-βVAE model, which combines residual-connected fully connected neural networks with VAE to handle multi-heat source arrangements. By integrating analytical solutions, polynomial fitting, and temperature field superposition, we accurately simulated the temperature rise distribution of a single heat source. We further explored the use of multiple equivalent heat sources to replace adiabatic boundary conditions. This enables the analytical method to effectively solve the two-dimensional conjugate convective heat transfer problem, providing a reliable alternative to traditional numerical solutions. Our results show that the model achieved high predictive accuracy. By adjusting the β parameter, we balanced reconstruction accuracy and latent space generalization. During the stable phase of the multi-heat source optimization iteration, 73.4% of the results outperformed the dense dataset benchmark, indicating that the model successfully optimized heat source coordinates and minimized peak temperature rise. This validates the feasibility and effectiveness of deep learning in thermal management system design. This research lays the groundwork for optimizing complex thermal environments and contributes valuable perspectives on the effective design of systems with multiple thermal sources or for applications like multi-beam lighting equalization.

A reconstruction method for harmonic measurement in distribution networks based on equivalent harmonic source approach

Abstract In recent years, with a large number of nonlinear devices connected to the distribution network, harmonic problems have been inevitably generated during operation, which makes harmonic problems more complex in the distribution network. However, it is important to master the real-time harmonic level of the distribution network for the analysis and management of the harmonic problems in the grid. However, because of economic reasons and decentralized nodes, it is impractical to install a harmonic measurement device at each node in engineering. Moreover, it is important to know how to measure harmonics at important nodes in a distribution network using a limited number of measurement devices. Therefore, a harmonic measurement reconstruction method for distribution networks is proposed based on the equivalent harmonic source method. This method aims to obtain more harmonic source state information with a minimum economic cost. First, a mathematical model of the harmonic currents is established. Subsequently, an equivalent harmonic source method is proposed, and a harmonic reconstruction model based on the equivalent harmonic source method is presented. Finally, an example of an IEEE 14-node distribution network is used to verify the effectiveness of the equivalent harmonic source method.

Sound source localization in non-line-of-sight environment based on equivalent source method combined with time reversal method

Sound source localization is a critical task in various applications. However, it becomes challenging when the sound sources are not directly visible to the sensors due to obstacles or other environmental factors. This paper proposes a novel approach for locating sound sources in non-line-of-sight environments by combining the equivalent source method (ESM) with the time reversal method (TRM). The ESM is used to compute the frequency responses between the focusing points and the measurement points numerically. These frequency responses are then substituted to the TRM to localize the sound sources. To evaluate the performance of the proposed method, it is tested through numerical simulations and an experiment. The results prove that the ESM can compute the frequency responses accurately, and the combination of TRM and ESM effectively localizes sound sources in non-line-of-sight environments.

Reconstruction of nonstationary sound field based on group sparsity time domain equivalent source method

To improve the accuracy and stability of reconstruction of nonstationary sound field, a novel time domain equivalent source method based on non-convex overlapping group sparsity (NOGS-TESM) is proposed. In the proposed NOGS-TESM, the time-varying pressure in the near field is first modeled by a set of equivalent sources positioned inside the vibrating structure. Then, the NOGS regularization is applied for solving the inverse problem, and a novel algorithm derived on the basis of Majorize-Minimization theory is utilized to obtain the optimal solution of equivalent sources strengths. Finally, the optimal equivalent sources strengths at all times are substituted into a time-domain convolution to calculate the time-varying pressure of reconstruction surface. The proposed NOGS-TESM enriches the prior information by considering the group sparsity of spatial pressure distribution, and enhances the accuracy of estimated equivalent sources strength via its alternative calculation of l0−norm. A vibrating plate to produce a nonstationary sound field, which is designed as a numerical simulation, and the time-varying pressures of measurement surface are input into the proposed NOGS-TESM. The quantitative and qualitative comparisons confirm that the proposed NOGS-TESM can greatly reconstruct the non-stationary sound field in time–space domains, and its reconstruction accuracy is higher than that of TESM with Tikhonov regularization (TR-TESM). Meanwhile, several parameters, for example, the noise and position of equivalent source surface, are discussed in detail, and the superiorities of the proposed NOGS-TESM in terms of stability and robustness are proved by comparing to TR-TESM. The predominance of the proposed NOGS-TESM is further validated by conducting one experiment on the vibrating steel plate.

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.

Sparse reconstruction of sound field using pattern-coupled Bayesian compressive sensing.

Conventional near-field acoustic holography based on compressive sensing either does not fully exploit the underlying block-sparse structures of the signal or suffers from a mismatch between the actual and predefined block structure due to the lack of prior information about block partitions, resulting in poor accuracy in sound field reconstruction. In this paper, a pattern-coupled Bayesian compressive sensing method is proposed for sparse reconstruction of sound fields. The proposed method establishes a hierarchical Gaussian-Gamma probability model with a pattern-coupled prior based on the equivalent source method, transforming the sound field reconstruction problem into recovering the sparse coefficient vector of the equivalent source strengths within the compressive sensing framework. A set of hyperparameters is introduced to control the sparsity of each element in the sparse coefficient vector of the equivalent source strengths, where the sparsity of each element is determined by both its own hyperparameters and those of its immediate neighbors. This approach enables the promotion of block sparse solutions and achieves better performance in solving for the sparse coefficient vector of the equivalent source strengths without prior information of block partitions. The effectiveness and superiority of the proposed method in reconstructing sound fields are verified by simulations and experiments.

A regional approach for high-resolution gravity anomaly recovery from full airborne gravity gradient tensor

SUMMARY A regional approach is developed for high-resolution gravity anomaly recovery from the full airborne gravity gradient tensor (GGT) based on the radial basis function (RBF) technique. The analytical expressions that link the full GGT to the gravity anomaly based on Poisson wavelets are developed, where the closed formulae of the associated derivatives of Poisson wavelets are deduced. Based on this approach, the gravity anomalies at a mean resolution of ∼0.15 km over the Kauring Test Range in Australia are recovered by using the local airborne GGT. The results show that the solution computed from the vertical component provides the best quality when a single component is used, whereas the model computed from the curvature component performs the worst. Moreover, the incorporation of two components magnifies the gravity anomalies and further improves the fit with the terrestrial and airborne gravity data, compared with the solutions computed from individual components. However, the solutions calculated by additionally merging one or more components provide comparable qualities with the models calculated by fusing two components only. Finally, the solution is computed by merging the full airborne GGT, and the standard deviation of the misfits against the terrestrial gravity data is 0.788 mGal. Further comparisons with the Fourier transformation and equivalent source method demonstrate that the proposed approach has slightly better performance. The proposed method is numerically efficient and offers a better data adaptation, which is useful for high-resolution gravity data recovery in managing huge number of gravity gradient data.

Surface velocity reconstruction of a vibrating structure based on dictionary learning and sparse sampling

The sparse regularization has been successfully applied to near-field acoustic holography to provide the reconstruction accuracy of sound field with limited number of measurements. However, most of the applications are concentrated on the reconstruction of the sound pressure and the corresponding sparse bases are designed for the sound pressure. In this study, dictionary learning is introduced and K-SVD is utilized to generate a sparse basis for the velocity. Then the reconstruction of surface velocity of a vibrating structure can be realized in a sparse framework to improve the reconstruction accuracy with limited number of measurements. In the process of data sample selection, the equivalent source method is used to generate the velocity samples according to the feature of the sound field and samples can be obtained by numerical simulations. The results of numerical simulations and experiment demonstrate the validity of the learned dictionary and the advantage of the proposed method.

Application of compressive sensing to equivalent source method on near-field acoustic holography

Near-field acoustic holography (NAH) is an acoustic visualization technique that uses a microphone array to measure the radiated sound waves from a source in the near-field. In the present study, the equivalent source method (ESM) and the 2D spatial Fourier transform based on the generalized discrete Fourier series (GDFS) are compared, and the reconstruction performance of NAH using compressed sensing (CS) combined with both theories for NAH measurements is discussed. Additionally, the influence of different array designs (regular and pseudo-random) on the reconstruction performance of NAH is investigated. Simulations and experimentation are used to demonstrate its feasibility. In the simulation, three source models (dipole, longitudinal quadrupole and simply supported rectangular plate) are tested over a range of frequencies, SNR and reconstruction distances. The consequence indicates that use of GDFS results in a large reconstruction error in all test conditions under the rectangular array, especially in the high frequency range. The combination of a pseudo-random array and ESM-L1 provides the best reconstruction performance in most test conditions. In the experiment, two sets of pseudo-random microphone arrays with different diameters were used. They were tested separately for the dual loudspeakers and vibrating plate. ESM-L1 is suitable for reconstructing sound fields with sparse source distributions, whereas ESM-L2 is suitable for reconstructing sound fields with continuous source distributions, such as the radiated acoustic field of a vibrating plate. However, the use of pseudo-random arrays in combination with ESM is indeed a better solution for NAH.

High-resolution acoustic imaging method based on equivalent source method and reweighted l1 minimization

Sparse regularization has been successfully applied to equivalent source method (ESM) in order to improve the acoustic imaging resolution. However, the application is not always feasible, especially at low frequencies. To overcome the problem, this paper proposes a high-resolution acoustic imaging method. In this method, reweighted l1 minimization is introduced to ESM to deal with the ill-posed inverse problems. Then the obtained equivalent source strengths are used to locate the sound sources. Compared to the sparse regularization-based ESM, the proposed method can provide a low side lobe and higher spatial resolution of acoustic imaging. Meanwhile, by arranging equivalent sources in three-dimensional space, the proposed method can also realize the acoustic imaging in three-dimensional sound field with high resolution. The results of the simulation and experiment demonstrate the validations.

High-Resolution localization of broadband sound sources in a duct using out-duct array measurements

This paper investigates using an Iterated Bayesian Focusing Algorithm and out-duct array measurements to identify multiple broadband sources placed in a cylindrical duct. The problems of large main lobe and low resolution due to localization of sources using conventional methods are investigated in this paper. To solve these problems, an Iterated Bayesian Focusing (IBF) method is utilized to localize the broadband sources. To evaluate and confirm the efficacy of the method, simulation tests and experimental validation are performed. In our experiments, we used a 56-channel dobby spiral array for out-duct measurements and beamforming method, equivalent source method (ESM), deconvolution approach for the mapping of acoustic sources (DAMAS), and Iterated Bayesian Focusing (IBF) method to localize broadband sources inside the duct. Simulation and experimental results demonstrate that the Iterated Bayesian Focusing (IBF) method significantly reduces the main lobe width and minimizes the number of minor lobes. This leads to improved accuracy in localizing double sources.

Multizone sound field reproduction using pressure matching with sparse equivalent source

Multizone sound field reproduction aims to create different acoustic environments in multiple spatial zones, allowing listeners to enjoy their individual sounds without being disturbed by sound from other zones. The conventional pressure matching method is used in practice to reproduce the desired sound field by minimizing the sound field error with a limited number of control points, whose reproduction performance decreases above the cutoff frequency. In this paper, a multizone sound field reproduction method based on a sparse equivalent source method is proposed to reproduce the sound field in the whole target region. The acoustic transfer function of each loudspeaker and the desired sound field are represented by the sparse equivalent sources. The whole target region is controlled by interpolating fine virtual control points within the region of the uncontrolled points, and the acoustic transfer functions between the virtual control points and the loudspeaker array are interpolated by the sparse equivalent sources. An optimization reproduction model with the objective of reproducing a desired sound field in the bright zone and constraining the acoustic energy in the dark zone is formulated to find the sparse loudspeaker weights. The simulation results and experimental results show that the proposed method achieves better performance than the conventional pressure matching method in free field and reverberant environments.

A Continental Model of Curie Point Depth for China and Surroundings Based on Equivalent Source Method

AbstractThe Curie Point Depth (CPD) marks a significant temperature boundary (∼580°C) within the Earth's lithosphere. However, there has been ongoing debate regarding its spatial distribution. In this research, we utilized the Equivalent Source Method (ESM) based on Gauss‐Legendre integration and data obtained from the EMM2017 model, along with a five‐layer susceptibility model, to generate a 0.5° × 0.5° grid of continental CPD distribution for China and surroundings. The average CPD in the study area is 30.4 km, which is slightly shallower than the average depth of global continental Moho (∼33 km). Notably, stable and cold cratonic basins, such as the Tarim Basin and the Sichuan Basin, exhibit deep CPD of ∼45 km. In contrast, the North China Craton, which has experienced significant tectono‐thermal activity since the Late Mesozoic, shows moderate CPD of ∼30 km and a gradual uplift from west to east. The Tuva‐Mongol orocline within the Central Asian Orogenic Belt, the Deccan Volcanic Province in the Indian subcontinent and the Eastern Yangtze Craton have shallow CPD of ∼20 km. We estimate the surface heat flow by CPD, and the result is consistent with measurements within a RMSE of 18.1 mW/m2. When comparing the CPD with Moho, we find that the CPD may lie below Moho in stable and cold cratonic areas. In comparison to two recent global CPD models, our regional model demonstrates better alignment with tectonic features.

Sound field separation based on dictionary learning and sparse sampling

Sound field separation techniques are an advancing tool for extracting a target sound field from a mixed sound field. However, the methods bear a high measurement cost due to the restriction of the sampling theorem. In this study, a sound field separation method based on sparse sampling is established. The method initially utilizes dictionary learning to generate a sparse basis of the sound field. Then, a mixed sound field can be precisely recovered from sparse sampling of sound pressure and the target sound field can be extracted based on the recovered sound field by means of the theory of equivalent source method. The method is validated by numerical simulations. Compared to sound field separation based on the equivalent source method, the proposed method has advantage in terms of both the accuracy and the stability for sparse sampling.

A new seperation method for rotationg and static sound source localization

Traditional sound source localization methods encounter significant challenges in simultaneously locating rotating and static sources. These challenges arise from the differing motion patterns of these two types of sound sources, and they are typically not situated on the same plane. To address this issue, a method based on Modal Composition Beamforming (MCB) and the equivalent source method is proposed for separating rotating and static sound sources that can fully utilize priori knowledge of the spatiotemporal properties of the sources. The proposed approach involves establishing a Rotating-Static Sources Power Propagation (R-S2P) model, utilizing the relationship between the beamforming output of equivalent sources and the actual beamforming output. Solving this model allows for matching the contributions of rotating and static equivalent sources. Simulations for three cases with different source strengths are presented, and the R-S2P model is solved using the Least Absolute Shrinkage and Selection Operator (LASSO). This method enables accurate separation and localization of rotating and static sources on different planes with varying relative intensities, even if the background noise is strong

Noise Separation Technique for Enhancing Substation Noise Assessment Using the Phase Conjugation Method

The intrinsic noise of different transformers in the same substation belongs to the same type of noise, which is strongly coherent and difficult to separate, greatly increasing the cost of substation noise assessment and treatment. To solve the problem, the present paper proposes a noise separation technique using the phase conjugation method to separate the intrinsic noise signals of different transformers: firstly, the reconstruction of sound source information is realized by the phase conjugation method based on the measurement and emission of a line array; secondly, the intrinsic noise signals of the sound source are obtained by the equivalent point source method. The error of the separation technique is analyzed by point source simulation, and the optimal arrangement form of the microphone line array is studied. A validation experiment in a semi-anechoic chamber is also carried out, and the results prove that the error of separation technique is less than 2dBA, which is the error tolerance of engineering applications. Finally, a noise separation test of three transformers is performed in a substation using the proposed technique. The results show that the proposed technique is able to realize the intrinsic noise separation of each transformer in the substation, which is of positive significance for substation noise assessment and management.

A Forward−Backward Splitting Equivalent Source Method Based on S−Difference

The regularization method has a direct impact on the accuracy of the reconstructed sound field in the process of inverse calculation of near−field acoustic holography using the equivalent source method. To expand the frequency range of sound field reconstruction and improve computational accuracy, a forward-backward splitting equivalent source method based on s−difference was proposed, which uses the ratio of the output results of the broadband acoustic holography algorithm as the regularization parameter. Numerical simulations of single source and coherent source sound fields were conducted under different frequency conditions to analyze the performance of the forward-backward splitting regularization algorithm based on s−difference, and experimental verification was performed. The simulation results show that the proposed method can accurately reconstruct the sound field in a wider frequency range, and has high accuracy in reconstructing the sound field of low−frequency coherent sources. The experimental results demonstrate the accuracy and effectiveness of this method in reconstructing mid−to−low−frequency sound fields.