Equivalent Source Method Research Articles

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.

ANALYTICAL CALCULATIONS OF SOLIDIFICATION AND COOLING PROCESSES IN LARGE- TONNAGE COMMERCIAL INGOTS. REPORT 1. MATHEMATICAL FORMULATION OF THE PROBLEM AND DERIVATION OF CORRELATIONS FOR A THERMAL LAYER SCHEME

Abstract. The technique to compute the overall time of bloom ingot solidifi cation was developed. To solve the stated problem a method of equivalent sources (MES) by Yu.S. Postolnik was applied, which has proven its effectiveness for solving problems in the fi eld of heat engineering. The solution has been achieved in terms of the thermal layer format, for two successive stages of heat exchange, namely: – inertial and regular. The comparison of calculated data concerning a commercial bloom ingot, as derived by the MES method, with the results of experiments and those obtained by the L.S. Leybenzon method, has shown a reasonable correlation as to the time of complete solidifi cation. Thus, the obtained solution is of value in heat-engineering computations of commercial ingot solidifi cation conditions. The proposed technique can be useful in heat-engineering calculations as applied to solidifi cation and cooling schedules for ingots and billets when poured into molds and in the process of continuous casting, to get an estimate of the time of complete freezing and of temperature patterns.

THE CALCULATIONS OF STEEL FLAT BRIQUETTES MELTING IN LIQUID BATH OF A STEELMAKING UNIT. REPORT 2. CALCULATIONS OF HEATING AND MELTING MODES OF AN ALLOYED BRIQUETTE

Abstract. The calculations of heating and cooling modes for a thermomassive slab, made of ShKh15 steel in the form of a briquette, are presented, using predictive expressions as obtained following the method of equivalent sources. Three cases of implementation of a “through” (continuous) “heating–melting” operation practices are covered, with concurrent variation of thermal and physical properties (TPP) of the thermally treated steel, namely, – those of the thermal conductivity coefficient and heat capacity per unit volume. To analyze briquette melt-down time errors, thermophysical properties have been specified at the initial temperature, final temperature (melting), and at the briquette average temperature in the course of heating and melting. The results indicate the need for taking into account changes in thermophysical properties of briquettes due to temperature. In computational relations, the external heat exchange, αk , has been varied. A graph of the dynamics of briquette melting vs melting intensity (αk ) and the briquette actual size has been plotted.

An equivalent source method for modelling the global lithospheric magnetic field

An equivalent source method for modelling the global lithospheric magnetic field

THE CALCULATIONS OF STEEL FLAT BRIQUETTES MELTING IN LIQUID BATH OF A STEELMAKING UNIT. Report 1. Mathematical formulation of the problem and its analytical solution

A mathematical model of heating and melting sequences of thermo-massive slabs (fl at steel briquettes) with variable thermal and physical properties (TPP) at boundary conditions of genus III, was formulated. Setting the genus III boundary conditions allows to analyze the impact of the external heat exchange coefficients of briquette melting processes.The calculation technique for solving the stated problem is the equivalent sources method (MES) as applied to a thermal layer scheme. Predictive expressions for slab through-heating (thermal lag stage), heating (ordered stage) time and at the time of melting, were derived. A comparison of solutions was obtained with the MES method and numerical realization of heating and melting the slab with a satisfactory convergence. It was proofed that the obtained mathematical relations may be used in steelmaking practices to perform calculations for lump materials melting in thermal engineering (e.g., when using briquettes in arc-furnaces).

Application of Bayesian Regularization Criterion in near Field Acoustic Holography Based on Equivalent Source Method

In order to choose an optimal regularization parameter for near field acoustic holography (NAH) based on equivalent source method (ESM), Bayesian regularization criterion method was introduced to solve the inverse problem. Comparison simulations were conducted with generalized cross validation method (GCV), L-curve method and Hald regularization formula method at various sound source frequencies, hologram distances, and Signal-to-Noise Ratios (SNR). The results show that GCV, L-curve, and Hald regularization formula method cannot achieve the optimal regularization parameter at large hologram distances, high noise levels, high frequencies and input SNR errors. However, Bayesian regularization criterion method is not affected by these restrictions and has excellent reconstruction performance in wide frequency band. Further experiment validates the correctness and effectiveness of the application of Bayesian regularization criterion method in ESM-based NAH, thus providing a new guidance for the solving of inverse problem in ESM-based NAH.

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.

Equivalent sources method for supersonic intensity of arbitrarily shaped geometries

Supersonic acoustic intensity is utilized to locate radiating regions on a complex vibrating structure. The supersonic intensity is obtained by a special process that removes the evanescent waves from the near-field acoustical holography measurement. The filtering process is well understood for separable geometries, but unfortunately, there are few results for arbitrarily shaped objects. This work proposes a methodology based on a stable invertible representation of the radiated power operator. The power operator is approximated numerically by the equivalent source formulation and the appropriate complete spectral basis is employed to form the stable invertible operator. The operator is formed with the most efficient radiation modes and these modes are utilized to obtain the supersonic solution for the near-field holographic problem. This concept is tested using numerically generated data in a spherical geometry and the results are validated with the spherical harmonic, supersonic filter. Finally, a vibrating ship-hull structure provides a physical example for application and validation of the proposed methodology in a more complex geometry.

Identification of active sources inside cavities using the equivalent source method-based free-field recovery technique

In previous studies, an equivalent source method (ESM)-based technique for recovering the free sound field in a noisy environment has been successfully applied to exterior problems. In order to evaluate its performance when applied to a more general noisy environment, that technique is used to identify active sources inside cavities where the sound field is composed of the field radiated by active sources and that reflected by walls. A patch approach with two semi-closed surfaces covering the target active sources is presented to perform the measurements, and the field that would be radiated by these target active sources into free space is extracted from the mixed field by using the proposed technique, which will be further used as the input of nearfield acoustic holography for source identification. Simulation and experimental results validate the effectiveness of the proposed technique for source identification in cavities, and show the feasibility of performing the measurements with a double layer planar array.

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.

Direct Evaluation of Acoustic-Intensity Vector Field Around an Impedance Scattering Body

The paper presents a direct method for predicting the acoustic-intensity vector field around an impedance scattering body. The results obtained from the present method, combined with the equivalent-source method, graphically display the acoustic-intensity vector field around the scattering body, and reveal the effect of surface impedance on the absorption of the acoustic energy. Three test cases (i.e., the plane wave scattered by a rigid sphere, and the sound of the stationary monopole and dipole point sources scattered by an impedance sphere) are carried out to validate the method. From the analysis of the computational error, it is suggested that the maximum length of the surface element should be less than one-tenth of the wavelength to ensure that the relative error is less than 3%.

Bayesian inversion and sequential Monte Carlo sampling techniques applied to nearfield acoustic sensor arrays

This paper demonstrates that inverse source reconstruction can be performed using a methodology of particle filters that relies primarily on the Bayesian approach of parameter estimation. The proposed approach is applied in the context of nearfield acoustic holography based on the equivalent source method (ESM). A state-space model is formulated in light of the ESM. The parameters to estimate are amplitudes and locations of the equivalent sources. The parameters constitute the state vector which follows a first-order Markov process with the transition matrix being the identity for every frequency-domain data frame. The implementation of recursive Bayesian filters involves a sequential Monte Carlo sampling procedure that treats the estimates as point masses with a discrete probability mass function (PMF) which evolves with iteration. It is evident from the results that the inclusion of the appropriate prior distribution is crucial in the parameter estimation.

Investigation on the reproduction performance versus acoustic contrast control in sound field synthesis

A sound reconstruction system is proposed for audio reproduction with extended sweet spot and reduced reflections. An equivalent source method (ESM)-based sound field synthesis (SFS) approach, with the aid of dark zone minimization is adopted in the study. Conventional SFS that is based on the free-field assumption suffers from synthesis error due to boundary reflections. To tackle the problem, the proposed system utilizes convex optimization in designing array filters with both reproduction performance and acoustic contrast taken into consideration. Control points are deployed in the dark zone to minimize the reflections from the walls. Two approaches are employed to constrain the pressure and velocity in the dark zone. Pressure matching error (PME) and acoustic contrast (AC) are used as performance measures in simulations and experiments for a rectangular loudspeaker array. Perceptual Evaluation of Audio Quality (PEAQ) is also used to assess the audio reproduction quality. The results show that the pressure-constrained (PC) method yields better acoustic contrast, but poorer reproduction performance than the pressure-velocity constrained (PVC) method. A subjective listening test also indicates that the PVC method is the preferred method in a live room.

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.

Sound Scattering and Its Reduction by a Janus Sphere Type

Sound scattering by a Janus sphere type is considered. The sphere has two surface zones: a soft surface of zero acoustic impedance and a hard surface of infinite acoustic impedance. The zones are arranged such that axisymmetry of the sound field is preserved. The equivalent source method is used to compute the sound field. It is shown that, by varying the sizes of the soft and hard zones on the sphere, a significant reduction can be achieved in the scattered acoustic power and upstream directivity when the sphere is near a free surface and its soft zone faces the incoming wave and vice versa for a hard ground. In both cases the size of the sphere’s hard zone is much larger than that of its soft zone. The boundary location between the two zones coincides with the location of a zero pressure line of the incoming standing sound wave, thus masking the sphere within the sound field reflected by the free surface or the hard ground. The reduction in the scattered acoustic power diminishes when the sphere is placed in free space. Variations of the scattered acoustic power and directivity with the sound frequency are also given and discussed.

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.

Research on the Electromagnetic Field Distribution of the Static Horizontal Electric Dipole in Shallow Sea

As a result of ship’s corrosion and anti-corrosion in shallow sea, the static electric field and static magnetic field can be produced around ships, which are referred to the static corrosion-related electromagnetic field. In order to analyze the field characteristics, a static horizontal electric dipole which is located in shallow sea is usually adopted to simulate the field. So,the electromagnetic field distribution of the electric dipole is very important because it is the base of analysis. In this paper, some research has been performed on the electromagnetic field of an electric dipole in three-layered conductive media. At first, equivalent source method has been used to derive the distribution expressions of the electromagnetic field in seawater. Then numerical simulation method has been used to do a contrastive analysis of the distribution characteristics of the electric field and magnetic field. At last, some experiments have been done in laboratory to prove the correctness of the theoretical derivation and the simulation analysis. The research results establish the foundation for the further application research.

Study on Acoustic Model of Transformer

The model of transformer as a noise source is very critical for substation noise prediction. The transformer is equivalent to several point sources on the basis of regarding the transformer as a combination of several planar sources. This equivalent model is based on equivalent source method and it is convenient and easy. The model of 9 equivalent point sources is simulated to verify that the rebuilt sound field is roughly the same as the actual sound field generated by the plane source. Moreover, the accuracy of the model with different settings was discussed. The acoustic model is accurate and feasible to calculate the noise level radiated by transformer and it is meaningful for substation noise control.

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.