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.

Designing a sparse sensor array for sound field reconstruction using compressive-equivalent source method.

The compressive-equivalent source method (C-ESM) can reconstruct the sound field radiated by sparsely distributed sound sources with a reduced number of sensors. To ensure the performance of the C-ESM, the transfer matrix between the sensors and equivalent point sources should exhibit sufficient incoherence. Given that the configuration of the sensor array affects this incoherence condition, concern regarding the sensor array design would arise. To address such concern, this paper proposes a sensor array design approach. The primary objective of this approach is to minimize the mean coherence of the transfer matrix within the developed iterative framework, providing the incoherence condition required by the C-ESM. Subsequently, the designed sensor array is utilized by the C-ESM for the reconstructions. The effectiveness and practicality of the proposed approach are validated through numerical simulations and experiments.

Repairing distorted hologram data for sound field reconstruction.

This paper proposes a distorted hologram data repair approach for sound field reconstruction. In this approach, an equivalent source model is established by placing a set of equivalent sources near the hologram surface to represent the measured hologram pressures. Each hologram pressure is simultaneously assigned an indicator to describe whether its measurement is corrupted by errors or not. This model is then formulated within a modal framework by utilizing the modes generated through the singular value decomposition of the transfer matrix between the hologram and nearby equivalent source surfaces. Subsequently, the indicators and modal coefficients are assigned the 0-1 and Gaussian prior distributions, respectively, and their posterior distributions are derived using the Bayesian method. The means of the posterior distributions are calculated to discriminate corrupted measurements and repair distorted hologram pressures. Repaired hologram pressures are finally utilized for reconstructions using the equivalent source method. Results from both numerical simulations conducted under various parameter settings and two experiments demonstrate the effectiveness of the proposed approach in automatically discriminating all the corrupted measurements and accurately repairing the distorted hologram pressures. Furthermore, the accuracy of the reconstructions using the repaired hologram pressures is comparable to that achieved with the correctly measured pressures.

Real-time nearfield acoustic holography using particle velocity.

In this paper, a series of impulse response functions between acoustic quantities on the source plane and particle velocity on the hologram plane are derived. In virtue of these functions, real-time nearfield acoustic holography (RT-NAH) is extended from pressure-based to particle velocity. Pressure, normal velocity, acceleration, and displacement radiated from planar sources can be reconstructed by measuring time-dependent particle velocity signals on the hologram plane. A simulation of an excited aluminum plate is performed to evaluate the difference in accuracy between RT-NAHs based on pressure and based on particle velocity. This study also examines the impact of impulse response functions on the reconstruction results, allowing for detailed analysis of the reconstruction accuracy based on these functions. The simulation results demonstrate that using RT-NAH based on particle velocity obtains significantly higher-accuracy reconstruction results when reconstructing normal velocity and displacement and slightly more accurate reconstructed pressure and normal acceleration.

Analytical method for studying acoustic fields radiated by general spherical sound sources in thermoviscous fluids.

The study of acoustic radiation from spherical sound sources plays a crucial role in understanding the thermoviscous effects in practical acoustic problems. However, finding a general solution of acoustic radiation from spherical sound sources in thermoviscous fluids remains a formidable challenge. To advance this issue, an analytical method is developed in this paper to calculate the acoustic field radiated by spherical sound sources with the isothermal boundary condition and arbitrary velocity boundary condition. The developed method is utilized to present the solutions of the acoustic field generated by an oscillating sphere and a general spherical sound source, and the accuracy and validity of these solutions are verified through analytical and numerical methods.

Passive directivity detection of acoustic sources based on acoustic Luneburg lens.

This article reports an acoustic Luneburg lens (ALL) design with graded refractive index for passive directivity detection of acoustic sources. The refractive index profile of the lens is realized based on square pillars with graded variation of their dimensions. Numerical and experimental studies are conducted to investigate the performance of directivity detection. The results demonstrate that the lens designed and developed in this study is capable of precisely detecting the directivity of one acoustic source. Furthermore, the directivities of two acoustic sources can also be detected with a resolution of 15°. In addition, different methods are investigated, including introducing phase difference by tuning input signals or moving ALL, and increasing the aperture size of ALL, to improve the resolution of dual sources directivity detection.

Active control of airfoil turbulent boundary layer noise with trailing-edge blowing.

Large Eddy Simulation (LES) and Ffowcs Williams-Hawkings acoustic analogy are performed to study the effect of trailing-edge blowing on airfoil self-noise. Simulations were conducted using a National Advisory Committee for Aeronautics 0012 airfoil at zero angle of attack and a chord-based Reynolds number of 4 × 10 5. The aerodynamic and aeroacoustic characteristics of the baseline airfoil were thoroughly verified by comparison with previous numerical and experimental data. The noise reduction effects of continuous and local blowing with different blowing ratios and blowing momentum coefficients were compared. A maximum noise reduction of 20 dB was achieved via trailing-edge blowing and the noise reduction mechanisms of the two blowing methods were discussed. The LES results show a pair of recirculation bubbles in the airfoil wake which are suppressed by trailing-edge blowing. As the blowing vortices convect into the wake, they stretch and stabilize the shear flows from airfoil surfaces. Instantaneous vorticity and root mean square velocity fluctuations are also weakened. There is a decrease in the spanwise coherence and an increase in the phase difference, which contribute to noise reduction. It is concluded that the suppression of turbulence fluctuations in the near wake is the main mechanism of noise reduction for airfoil trailing-edge blowing.

Sound field reconstruction using block sparse Bayesian learning equivalent source method.

Nearfield acoustic holography based on the compressed sensing theory can realize the accurate reconstruction of sound fields with fewer measurement points on the premise that an appropriate sparse basis is obtained. However, for different types of sound sources, the appropriate sparse bases are diverse and should be constructed elaborately. In this paper, a block sparse Bayesian learning (SBL) equivalent source method is proposed for realizing the reconstruction of the sound fields radiated by different types of sources, including the spatially sparse sources, the spatially extended sources, and the mixed ones of the above two, without the elaborate construction of the sparse basis. The proposed method constructs a block sparse equivalent source model and promotes a block sparse solution by imposing a structured prior on the equivalent source model and estimating the posterior of the model by using the SBL, which can achieve the accurate reconstruction of the radiated sound fields of different types of sources simply by adjusting the block size. Numerical simulation and experimental results demonstrate the validity and superiority of the proposed method, and the effects of two key parameters, the block size, and sparsity pruning threshold value are investigated through simulations.

A free field recovery technique based on the boundary element method and three-dimensional scanning measurements.

The boundary element method- (BEM-) based free field recovery technique (FFRT) has been proposed to recover the free field radiated by an arbitrarily shaped source from the mixed field that would be measured in a noisy environment. However, that technique requires that the boundary integral equation should be established on an enclosed hologram surface surrounding the source, which means that the hologram surface should be discretized into elements and the measurement points should be located on the nodes of the elements. For large-scale or mid-high frequency problems, it makes the total number of measurement points huge since it should obey the criterion of more than six elements per wavelength, which put forward very high requirements for holographic data measurement. To overcome this problem, a more flexible BEM-based FFRT without the restriction on the locations of measurement points is proposed in this study. In virtue of this, a three-dimensional scanning measurement method can be applied to acquire holographic data with high efficiency. The effectiveness of the proposed method is validated by two numerical simulations and an experiment.

Reconstruction of time-dependent forces acting on a vibrating structure from pressure measurements.

This paper proposes an approach to reconstruct the time-dependent forces acting on a vibrating structure from pressure measurements. In the approach, the pressures measured in the near field of the structure are related to the exciting forces at the reconstruction points by the transfer functions determined in an experimental way, whereupon the time-dependent forces can be reconstructed with these pressures as inputs. In the reconstruction process, an additional regularization with a mixed lp , q-norm term is introduced to resolve the ill-posed inverse problem, which is able to take advantage of the prior knowledge of space and time characteristics of the forces. A numerical simulation of reconstructing the time-dependent forces acting on a plate and two experiments of reconstructing the impact forces acting on a semi-cylindrical shell and an elliptically shaped structure are carried out. The results demonstrate the feasibility of the proposed approach for reconstructing the forces in both temporal and spatial domains from pressure measurements. The proposed approach provides a non-contact and real-time way to identify the locations of forces and reconstruct their time histories, which can be further used to reveal the mechanical cause of the radiated noise.

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.

Sensitivity analysis of acoustic eigenfrequencies by using a boundary element method.

This paper presents a boundary element-based scheme for the sensitivity analysis of acoustic eigenfrequencies of both interior and exterior acoustic systems. The nonlinear eigenvalue problem generated by the acoustic boundary element method is first reformulated into a generalized eigenvalue problem of reduced dimension through a contour integral approach. The sensitivity formulations for acoustic eigenfrequencies are then derived based on an adjoint method that uses both the right and left eigenvectors. The adaptive cross approximation in conjunction with the hierarchical matrices is used to reduce the solution burden of the boundary element systems. The Burton-Miller-type combined formulation is applied to shift the spurious eigenfrequencies and their sensitivities, and the strategies to identify the spurious results are suggested. Three numerical examples are used to verify the accuracy and applicability of the developed scheme.

Reconstruction of nonstationary sound fields based on a time domain angular spectrum method.

A time domain angular spectrum method is proposed to reconstruct nonstationary sound fields. In this method, the sound field is expressed as a superposition of a series of plane wave bases, and the plane wave basis is constructed by an impulse response function that relates the time domain angular spectrum to the field point pressure. The impulse response function consists of two parts, the propagating plane waves and the evanescent plane waves, and their physical interpretation is provided. By discretizing the time convolution between the plane wave strength and the impulse response function, the reconstruction can be carried out at each time step, thus providing the advantage of real-time reconstructing sound fields. Since the real-time reconstruction process is non-recursive, it can provide a stable reconstruction. In the reconstruction process, the Tikhonov regularization is introduced at each time step to obtain an appropriate estimation of the plane wave strength. Numerical simulations with an unsteady excitation plate and an experiment with an impacted plate were carried out to demonstrate the feasibility of the proposed method on reconstructing nonstationary sound fields. The effect of numerical parameters on the reconstruction accuracy was also investigated in the numerical simulations.

Estimating the acoustical properties of locally reactive finite materials using the boundary element method.

The finite size of a sound-absorbing material may lead to inaccurate results when measuring the acoustical properties of the material using the free-field measurement methods. In this study, a method of estimating the acoustical properties of locally reactive finite materials is proposed by combining a sound field model established by the boundary element method with an iteration algorithm. The proposed method takes the finiteness of the material into account, meaning that the size effect is removed and accurate results can be obtained. Numerical simulations and experiments of two kinds of materials, including a rigid floor and a porous material, are carried out to verify the validity of the proposed method. Results demonstrate that the proposed method is effective in estimating the acoustical properties of these two kinds of materials. Besides, a detailed analysis of the influences of the sample size, the source location, and the receiving point position is done in the simulations.

A boundary element eigensolver for acoustic resonances in cavities with impedance boundary conditions.

This letter presents a boundary element scheme for analysis of acoustic resonances in cavities with impedance boundary conditions. The resultant eigenproblem, which is nonlinear and difficult to solve directly, is transformed to a linear one through a contour integral method. A variant-parameter scheme based on the Burton-Miller combined formulation is given to identify spurious eigenfrequencies, which are complex and similar to true eigenfrequencies. A numerical example is used to show the accuracy and effectiveness of the proposed method.

Sound field reconstruction using inverse boundary element method and sparse regularization.

The inverse boundary element method (IBEM) is a powerful tool for realizing sound field reconstruction of sources with arbitrarily-shaped surfaces. In the conventional IBEM, the Tikhonov regularization is generally used and the number of sampling points is required to be larger than that of nodes on the boundary surface to guarantee to obtain a unique solution. Meanwhile, it requires that the minimum discretization interval on the boundary surface should be less than one-sixth wavelength to ensure to obtain enough calculation accuracy. Therefore, the number of sampling points may be dramatically large at high frequencies. In this paper, acoustic radiation modes, which are composed of the eigenvectors of the resistive impedance matrix, are used as the sparse basis of source surface velocities. Based on this sparse basis, sparse regularization is introduced into the IBEM. Compared to the Tikhonov regularization, the sparse regularization can provide a higher accuracy for the reconstruction of source surface velocities and can reduce the number of sampling points by taking advantage of the theory of compressive sensing. Both numerical simulation and experimental results demonstrate the superiority of the proposed method. Meanwhile, the effects of the number of sampling points and the signal-to-noise ratio on the reconstruction accuracy are analyzed numerically.

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.

Stability analysis of inverse time domain boundary element method for near-field acoustic holography.

Stability of the inverse time domain boundary element method (ITBEM) for near-field acoustic holography is investigated. An eigenvalue system is built by reformulating the ITBEM to an iterative format. Through the analysis of the eigenvalue system, a stabilization criterion is derived. Then the stabilization criterion is utilized to reveal the stabilization mechanism of the TSVD method which plays an important role in the ITBEM. Furthermore, a method for properly choosing the ratio of truncated singular values to ensure the stability is provided. Although stability can be managed by using the TSVD method, the accuracy of the results cannot always be guaranteed. To overwhelm this difficulty, an averaging technique is further introduced, and its stabilization mechanism is investigated by incorporating it into the ITBEM formulations. Numerical simulations are carried out to validate the stabilization criterion, and the stabilization mechanisms of TSVD and averaging are shown specifically with extensive eigenvalue analyses.

The sound field of a rotating dipole in a plug flow.

An analytical far field solution for a rotating point dipole source in a plug flow is derived. The shear layer of the jet is modelled as an infinitely thin cylindrical vortex sheet and the far field integral is calculated by the stationary phase method. Four numerical tests are performed to validate the derived solution as well as to assess the effects of sound refraction from the shear layer. First, the calculated results using the derived formulations are compared with the known solution for a rotating dipole in a uniform flow to validate the present model in this fundamental test case. After that, the effects of sound refraction for different rotating dipole sources in the plug flow are assessed. Then the refraction effects on different frequency components of the signal at the observer position, as well as the effects of the motion of the source and of the type of source are considered. Finally, the effect of different sound speeds and densities outside and inside the plug flow is investigated. The solution obtained may be of particular interest for propeller and rotor noise measurements in open jet anechoic wind tunnels.

Sound field reconstruction using compressed modal equivalent point source method.

The accuracy, resolution, and economic cost of near-field acoustic holography (NAH) are highly dependent on the number of spatial sampling points. Generally, higher accuracy and resolution require more spatial sampling points, which may increase the workload of measurement or the hardware cost. Compressive sensing (CS) is able to solve the underdetermined problems by utilizing the sparsity of signals, and thus it can be applied to NAH to reduce the number of spatial sampling points but at the same time provide a high-resolution reconstruction image. Based on the CS theory, this paper proposes a compressed modal equivalent point source method (CMESM). In the method, a sparse basis that is obtained from the eigen-decomposition of the power resistance matrix is introduced to compress the equivalent point source strengths, and the ℓ1-norm minimization is used to promote sparse solutions. Both numerical simulation and experimental results demonstrate the validity of the proposed CMESM and show its advantage over the existing methods when the number of spatial sampling points is reduced.