Time-domain Equivalent Source Method Research Articles

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.

Krylov subspace iterative methods for time domain equivalent sources method based nearfield acoustical holography

This paper studies the acoustic field reconstruction on an arbitrarily shaped vibrating surface using the time-domain equivalent source method (ESM) based near-field acoustic holography (NAH) from measurements of the pressure field on a nearby conformal surface. For the classical frequency-dependent ESM, we represent the acoustic field by a set of frequency-dependent monopole sources strategically distributed over a close and conformal surface to the vibrating structure. We demonstrated that in this setup, the ESM representation approximates the classical integral representations, which justifies the ESM as an extension to the classical integral theory. For time-domain ESM in a similar manner, we demonstrate that a set of time-dependent monopole sources approximates the single layer retarded potential and then characterize the acoustic field. The discretization of time-domain ESM produces a matrix system with a particular sparse structure that can be efficiently solved using Krylov subspace iterative methods. The iterative methods that will be of our interest are Conjugate Gradients Normal equation Residual, Conjugate Gradients Normal equation Error and Least Squares QR. To reduce the amplification of measurement in the numerical reconstructions, we combine the proposed iterative methods with heuristic stopping rules like L-curve, Quasi-Optimality, Hanke–Raus, Brezinski–Rodriguez–Seatzu and a posteriori stopping rule. To determine the best combination of reconstruction technique, iteration and stopping rule, we develop a statistical study that uses multiple samples of numerically generated data by Gaussian pulses over a sphere. We produce these samples using a random source position and different noise levels. Finally, we validate our proposed methodology using scattering data from an elastic spherical shell.

Separation of nonstationary sound fields based on the time-domain equivalent source method with single layer pressure-velocity measurements.

This paper proposes a sound field separation technique based on the time-domain equivalent source method with single layer pressure-velocity measurements to extract the nonstationary sound field radiated by the target source in a reverberant environment. This technique constructs a formulation that relates the pressures and particle velocities on a measurement surface to the strengths of time-domain equivalent sources arranged for modelling the outgoing and incoming waves. By solving the strengths of time-domain equivalent sources, the sounds coming from different sides of the measurement surface can be separated independently. In the proposed technique, the use of a time-domain equivalent source model allows the measurement surface to be arbitrarily shaped, thus providing the ability to analyze the arbitrarily shaped sources in a reverberant environment. Numerical simulations investigated the performance of the proposed technique when using different types of arrays, including planar, semi-cylindrical, and semi-spherical arrays, and an experiment with three loudspeakers located at two sides of the measurement surface was carried out to test the validity of the proposed technique. Both numerical and experimental results demonstrate that the proposed technique can remove the influence of disturbing sources in both time and space domains and separate out the target sound fields effectively.

Research on Calculation Method for Acoustic Scattering of Helicopter Noise

A new calculation method of helicopter rotor/fuselage acoustic scattering is developed. Firstly, a CFD analysis model is developed to simulate flow field of the rotor, which is based on the motional embedded grid system and RANS equations, and provides aerodynamic data for rotor noise calculation. Then, FW-H equations are employed to calculate the aeroacoustic characteristics of isolated rotor, and G 1A formulas are applied to calculate the rotor acoustic gradient to provide boundary condition for acoustic scattering. Based on these, the time-domain equivalent source method is applied to calculate acoustic scatter field, and the total acoustic field that considered the fuselage scatter is superposed by isolated rotor acoustics and the scatter one. Finally, the numerical simulations of helicopter main-rotor/fuselage and tail-rotor/fuselage scatter effect are conducted by using the developed models. The results indicate that the helicopter fuselage has important scatter effect on the high frequency acoustics of main rotor and tail rotor, and the acoustic scatter effect become more obvious with the smaller space between the main rotor (tail rotor) and fuselage.

An approach for indoor prediction of the pass-by noise of a vehicle based on the time-domain equivalent source method

An approach based on the time-domain equivalent source method (TESM) is developed for the indoor prediction of the pass-by noise of a vehicle in a small-scale semi-anechoic chamber. In the approach, a vehicle drives on a chassis dynamometer, and a number of microphones are distributed around the vehicle to measure the pressure radiated by the vehicle in the near field. Based on the idea of TESM, the sound sources generated by the vehicle are modeled by a series of time-domain equivalent sources, and the sound radiated from these sources to the microphones is depicted by the half-space Green’s function that takes the reflections from the rigid ground into account. To evaluate the pass-by noise, the strengths of equivalent sources are first solved by using the measured pressures as the input of TESM, and then the pressures on the target points specified by the ISO 362-3 Standard are reconstructed by the superposition of sound fields generated by the equivalent sources. Finally, the sound pressure level (SPL) of the pass-by noise can be calculated from the reconstructed pressures. In particular, the proper arrangement scheme for the equivalent sources is investigated in numerical simulations to ensure the prediction accuracy. Numerical and experimental results demonstrate that the proposed approach can effectively predict the SPL of the pass-by noise of a vehicle for both constant-speed and accelerated test conditions in a small-scale semi-anechoic chamber.

On the stability of transient nearfield acoustic holography based on the time domain equivalent source method.

Transient nearfield acoustic holography based on the time domain equivalent source method suffers from the instability that is caused by the use of the time marching scheme. In this paper, the time marching scheme is reformulated to a large iterative scheme. By conducting the eigenanalysis of this large iterative scheme, a necessary condition for stability, i.e., the maximum magnitude of eigenvalues should not be larger than one, can be obtained. Moreover, the causes of instability are analyzed according to the eigenvalues distribution. By virtue of the eigenanalysis, the mechanisms and drawbacks of three previous stabilization methods based on the Tikhonov regularization, the truncated singular value decomposition (TSVD), and the multistep approach are analyzed. To overcome their drawbacks, the classical golden section method is applied to search the regularization parameters and filter parameters based on the necessary condition for stability. Furthermore, the time averaging technique is introduced into the stabilization methods based on the Tikhonov regularization and the TSVD to eliminate the high-frequency oscillation and release the difficulty of searching the filter parameter, respectively. Numerical simulation results indicate that all the improved methods can realize the stabilization of solutions.

A time-domain equivalent source method for acoustic scattering based on acoustic velocity

In this paper, a time-domain equivalent source method (ESM) for acoustic scattering prediction is suggested, based on acoustic velocity formulations. Incident and scattered acoustic predictions are realized by acoustic velocity formulation V1A of Ghorbaniasl. Formulation V1A is based on the FW-H equation and allows direct evaluation of the boundary condition, without requiring acoustic pressure gradient computation. The scattering approach is implemented and validated by the analytical case of scattering of a pulsating monopole source by a rigid sphere. The developed methodology is efficient for scattering problems where broadband noise sources are considered.

Reconstruction of instantaneous surface normal velocity of a vibrating structure using interpolated time-domain equivalent source method

Abstract Interpolated time-domain equivalent source method is extended to reconstruct the instantaneous surface normal velocity of a vibrating structure by using the time-evolving particle velocity as the input, which provides a non-contact way to overall understand the instantaneous vibration behavior of the structure. In this method, the time-evolving particle velocity in the near field is first modeled by a set of equivalent sources positioned inside the vibrating structure, and then the integrals of equivalent source strengths are solved by an iterative solving process and are further used to calculate the instantaneous surface normal velocity. An experiment of a semi-cylindrical steel plate impacted by a steel ball is investigated to examine the ability of the extended method, where the time-evolving normal particle velocity and pressure on the hologram surface measured by a Microflown pressure-velocity probe are used as the inputs of the extended method and the method based on pressure measurements, respectively, and the instantaneous surface normal velocity of the plate measured by a laser Doppler vibrometry is used as the reference for comparison. The experimental results demonstrate that the extended method is a powerful tool to visualize the instantaneous surface normal velocity of a vibrating structure in both time and space domains and can obtain more accurate results than that of the method based on pressure measurements.

Suppressing the onset of instabilities in the time-domain equivalent source method using a multistep approach.

Numerical instability is an important issue that should be addressed in the time-domain equivalent source method (TESM). This study proposes a multistep method to stabilize TESM when using the measured acoustic pressure data to optimize equivalent source strengths. Unlike the conventional single-step method that solves each time step in the time-marching process of TESM, the proposed method performs a one-time solution for several time steps. The multistep solution can potentially reduce the accumulation rate of error, and improves filtering effects by changing the structure of the matrix that needs to be inverted when the truncated singular value decomposition or Tikhonov regularization is used in the time-marching process. Numerical simulations with three examples demonstrate the effectiveness of the multistep method in improving the stability of solutions compared with the single-step method. Effects of the number of merged time steps on the solutions are also discussed to guide the selection process. Finally, the sensitivity of the multistep method to numerical parameters is investigated to demonstrate its consistency under different configurations of numerical parameters.

Sound Field Prediction and Separation in a Moving Medium Using the Time-Domain Equivalent Source Method

Sound Field Prediction and Separation in a Moving Medium Using the Time-Domain Equivalent Source Method

Review: The Use of Equivalent Source Method in Computational Acoustics

This paper reviews the equivalent source method (ESM), an attractive alternative to the standard boundary element method (BEM). The ESM has been developed under different names: method of fundamental solutions, wave superposition method, equivalent source method, etc. However, regardless of the method name, the basic concept is very similar; that is to use auxiliary points called equivalent sources to reconstruct the acoustic pressure for radiation or scattering problems. The strength of the equivalent sources are then determined via various approaches such that the boundary conditions on the boundary surface are satisfied. This paper reviews several frequency-domain and time-domain ESMs. There are several distinct advantages in these types of methods: (1) the method is a meshless approach so that it is easy and simple to implement; (2) it does not have a numerical singularity problem that occurs in the BEM; (3) the number of equivalent sources can be fewer than the number of surface collocation points so that the matrix size is reduced and a fast computation is achieved for large problems. The main issue of the ESM is that there is no rule to find out the optimal number and position of equivalent sources. In addition, the ESM suffers from the numerical instability that is associated with the ill-conditioned matrix. Some guidelines have been suggested in terms of finding the number and position of the sources, and several numerical techniques have been developed to resolve the numerical instability. This paper reviews the common theories, numerical issues and challenges of the ESM, and it summarizes recent developments and applications of the ESM to aircraft noise.

A time-domain inverse technique for the localization and quantification of rotating sound sources

A time-domain inverse technique based on the time-domain equivalent source method is proposed for the localization and quantification of rotating sound sources. In this technique, the actual rotating sound sources are modeled by a series of rotating equivalent sources distributed on the source surface. The strengths of these equivalent sources are solved based on the exact transfer relationship between the measured pressure at the receiver time and the desired equivalent source strengths at the source time. Compared to the known time-domain rotating beamforming that just owns the function of source localization, the proposed inverse technique not only can locate rotating sources accurately but also can predict sound fields quantitatively. Moreover, due to the use of retarded time approach, the proposed inverse technique avoids the interpolation of measured pressure that is needed in the time-domain rotating beamforming, thus providing the ability of real-time calculation of source strengths. Numerical simulations and experiments examine the validity of the proposed technique and demonstrate its advantages of locating sources more accurately and enabling to predict sound fields quantitatively by comparing with the time-domain rotating beamforming.

Separation of non-stationary multi-source sound field based on the interpolated time-domain equivalent source method

In the sound field with multiple non-stationary sources, the measured pressure is the sum of the pressures generated by all sources, and thus cannot be used directly for studying the vibration and sound radiation characteristics of every source alone. This paper proposes a separation model based on the interpolated time-domain equivalent source method (ITDESM) to separate the pressure field belonging to every source from the non-stationary multi-source sound field. In the proposed method, ITDESM is first extended to establish the relationship between the mixed time-dependent pressure and all the equivalent sources distributed on every source with known location and geometry information, and all the equivalent source strengths at each time step are solved by an iterative solving process; then, the corresponding equivalent source strengths of one interested source are used to calculate the pressure field generated by that source alone. Numerical simulation of two baffled circular pistons demonstrates that the proposed method can be effective in separating the non-stationary pressure generated by every source alone in both time and space domains. An experiment with two speakers in a semi-anechoic chamber further evidences the effectiveness of the proposed method.

Sound source identification and sound radiation modeling in a moving medium using the time-domain equivalent source method.

Planar near-field acoustic holography has been successfully extended to reconstruct the sound field in a moving medium, however, the reconstructed field still contains the convection effect that might lead to the wrong identification of sound sources. In order to accurately identify sound sources in a moving medium, a time-domain equivalent source method is developed. In the method, the real source is replaced by a series of time-domain equivalent sources whose strengths are solved iteratively by utilizing the measured pressure and the known convective time-domain Green's function, and time averaging is used to reduce the instability in the iterative solving process. Since these solved equivalent source strengths are independent of the convection effect, they can be used not only to identify sound sources but also to model sound radiations in both moving and static media. Numerical simulations are performed to investigate the influence of noise on the solved equivalent source strengths and the effect of time averaging on reducing the instability, and to demonstrate the advantages of the proposed method on the source identification and sound radiation modeling.

An Efficient Time-Domain Equivalent Source Method for Acoustic Scattering

The prediction of acoustic scattering from hard surfaces such as the fuselage or deck near an exhausting jet aircraft or rotorcraft is important to attempts at predicting the spatial effects of various engine and nozzle configurations. A time-domain equivalent source method similar to Lee et al.'s method is presented for predicting acoustic scattering using pressure gradient predictions obtained from the G1A formulation. In contrast to Lee's method, the approach proposed here involves using coincident source and control points, leading to improved performance for some geometries. The greatest advantage appears to be for planar surfaces, where interaction between panels is avoided. Validation cases considered include a series of planar segments for analytical validation as well as cases to validate the use of the G1A inputs in connection with the proposed method, and a series of scattering cases using input from an acoustic data surface.

Modeling transient sound propagation over an absorbing plane by a half-space interpolated time-domain equivalent source method.

A half-space interpolated time-domain equivalent source method (ITDESM) is proposed to model the transient sound propagation over an absorbing plane. In this approach, a closed-form transient half-space Green's function (i.e., the impulse response function in three-dimensional space) for a pure absorbing infinite plane is introduced to develop the half-space ITDESM formulation. Instead of the free transient Green's function employed in the conventional ITDESM, such Green's function contains the reflection effect of the absorbing plane. As a numerical example, reconstructing the transient pressure fields from two monopole sources is depicted, where both monopoles are located in front of an infinite plane with absorbing impedance. Simulation results indicate that the half-space ITDESM can reconstruct the half-space transient sound fields in both the space and time domains very well. The proposed method is also investigated by taking into account the measurement noise in the reconstruction process. An experiment of an impacted steel plate above a glass wool board is presented to illustrate the validity of the proposed method under actual conditions.

Theoretical and experimental study of transient sound field reconstruction in semi-free field

The interpolated time-domain equivalent source method (ITDESM) is based on the assumption of a free field, which might not be suitable for the direct application to a semi-free field. Here, an extended ITDESM procedure is proposed to reconstruct the semi-free transient sound field directly in the time domain. In this method, the time-domain equivalent sources are placed not only near the actual sound source, but also near its image source. It is assumed that the number of the equivalent sources distributed around the image source (virtual equivalent sources) is identical to that distributed around the actual source (actual equivalent sources), and their locations are mirror-symmetric with the symmetry plane being the reflecting surface. The acoustic contribution from each virtual equivalent source is further formulated with the strengths of the corresponding actual equivalent source to save the computational cost. Reconstruction of the semi-free transient sound fields from three monopoles is contrastively studied by numerical simulations to demonstrate the superiority of the proposed method. This method is also examined by adding noise and measurement point offsets into the reconstruction process, respectively. An experiment with a loudspeaker is conducted in a semi-anechoic chamber, and the results show the validity and applicability of the proposed method.

An interpolated time-domain equivalent source method for modeling transient acoustic radiation over a mass-like plane based on the transient half-space Green’s function

Interpolated time-domain equivalent source method (ITDESM) is based on the assumption of free space, which makes it not suitable for reconstructing the transient acoustic quantities in the half space. Here, a half-space ITDESM is proposed to model the transient acoustic radiation over a mass-like plane. In this method, the free transient Green’s function existing in the conventional ITDESM is replaced by a closed-form transient half-space Green’s function for a mass-like plane. Such transient Green’s function enables one to take the reflection effect of the mass-like plane into consideration. Modeling acoustic radiation from three transient monopoles above an infinite plane with mass-like behavior is studied by numerical simulations to demonstrate the feasibility of the half-space ITDESM. The proposed method is also examined by comparing the reconstruction accuracy among a free-field model, a rigid plane model and a mass-like plane model. An experiment with an impacted steel plate lying above a table plate is conducted in the semi-anechoic room, and the results further verify the effectiveness of the proposed method.

Reconstruction of semi-free transient sound field under different reflection conditions using an extended interpolated time-domain equivalent source method

Transient acoustic field can be rebuilt directly in the time-domain via the interpolated time-domain equivalent source method (ITDESM). However, this method requires that the reconstruction should be addressed in the free-field only, which can hardly be met in the engineering noise problems. To circumvent this difficulty, an extended ITDESM procedure is developed by extending the ITDESM from the free-field to the semi-free-field. In this approach, the time-domain equivalent sources are placed not only near the actual sound sources but also around their image sources with respect to the planar reflecting surface. The solving procedure of the equivalent source strengths is improved to decrease the computing load. The reflection conditions treated here can be arbitrary, i.e. both perfectly rigid and impedance-effected. Reconstruction results of the transient sound field radiated from three monopoles under different reflection conditions demonstrate the validity and applicability of the proposed method.

A hybrid approach to reconstruct transient sound field based on the free-field time reversal method and interpolated time-domain equivalent source method

Abstract A hybrid approach is presented in the current work, which reconstructs the transient sound field radiated from the two-dimensional sources with unknown locations and sizes, by combining the free-field time reversal method and the interpolated time-domain equivalent source method (TDESM). In the first step of the proposed method, the time reversal focusing algorithm is performed to estimate the source locations on the source plane. And then, the interpolated TDESM is applied to reconstruct the transient sound field on the reconstruction plane by assuming that the equivalent sources are located near the estimated source locations found in the previous step. The proposed technique, in principle, requires fewer microphones in the measurement since the equivalent sources are only placed in the vicinity of the ‘real’ sound sources. Reconstruction of the transient sound field radiated from the dual-planar-piston model is studied by numerical simulation for feasibility demonstration. A measurement of the sound fields radiated from two baffled loudspeakers is performed in the anechoic chamber, which shows that a better reconstruction result can be achieved by using the proposed hybrid scheme than the original interpolated TDESM with relatively the same number of sampling channels.