Kirchhoff Helmholtz Research Articles

Accurate and computationally efficient basis function generation using physics informed neural networks

Basis functions that can accurately represent simulated or measured acoustic pressure fields with a small number of degrees of freedom is of great use across various applications, including finite element methods, model order reduction, and compressive sensing. In a previous work [B. M. Goldsberry, J. Acoust. Soc. Am. 153, A193 (2023)], basis functions were derived for an element in a given mesh using a combination of interpolation functions defined on the boundaries of the element and the Helmholtz-Kirchhoff (HK) integral. This forms a new interpolatory basis set that efficiently and accurately represents the interior of the element. However, the previous analysis was limited to a two-dimensional rectangular element. In this work, physics informed neural networks (PINN) are investigated as a means to generate HK basis functions for general element shapes. PINNs have been previously shown to accurately learn solutions to parameterized partial differential equations. The element geometry parameterization and the boundary interpolation functions are given as inputs to the PINN, and the output of the PINN consists of the physically accurate basis functions within the element. Details on the implementation and training requirements on the PINN to achieve a desired accuracy will be discussed. [Work supported by ONR.]

An efficient offline scheme to compute an FIR controller for active reduction of acoustic reflections in an anechoic chamber

This paper presents an offline iterative scheme to efficiently obtain a fixed controller for larger-scale active noise control systems. The approach reduces the computational complexity of the scheme by applying preconditioning and prewhitening via the frequency domain, followed by conversion of the filters to the time domain using the inverse Fourier transform. Although the filters are not necessarily causal, causality is ensured by utilizing delay and truncation techniques. The use of these filters in the iterative scheme leads to improved convergence, lowering computational effort while also being memory efficient. Regularization is applied to maintain convergence while allowing for larger step sizes between iterations. The controller minimizes the reflected sound field, which is measured by the performance sensors. The signals of the performance sensors are computed with the Kirchhoff–Helmholtz integral. The results show that this allows for global control of the reflected sound field within the microphone area. A simulation is shown of a noise control setup with 12 primary sources, 12 secondary sources, 12 reference sensors, 12 error sensors and 37 performance sensors. Although the delay of the filters in the direct path is reflected in the filter coefficients, this delay can be truncated to obtain filter coefficients without any delay. Despite the truncation resulting in an average loss in performance of 1.6 dB, the control system without any delays is able to reduce the average reflected sound at the performance sensors by 13.8 dB, when the reference signal is known.

An Acoustic Simulation Method of the Japanese Vowels /i/ and /u/ by Using the Boundary Element Method

This study aimed to establish and verify the validity of an acoustic simulation method during sustained phonation of the Japanese vowels /i/ and /u/. The study participants were six healthy adults. First, vocal tract models were constructed based on computed tomography (CT) data, such as the range from the frontal sinus to the glottis, during sustained phonation of /i/ and /u/. To imitate the trachea, after being virtually extended by 12 cm, cylindrical shapes were then added to the vocal tract models between the tracheal bifurcation and the lower part of the glottis. Next, the boundary element method and the Kirchhoff–Helmholtz integral equation were used for discretization and to represent the wave equation for sound propagation, respectively. As a result, the relative discrimination thresholds of the vowel formant frequencies for /i/ and /u/ against actual voice were 1.1–10.2% and 0.4–9.3% for the first formant and 3.9–7.5% and 5.0–12.5% for the second formant, respectively. In the vocal tract model with nasal coupling, a pole–zero pair was observed at around 500 Hz, and for both /i/ and /u/, a pole–zero pair was observed at around 1000 Hz regardless of the presence or absence of nasal coupling. Therefore, the boundary element method, which produces solutions by analysis of boundary problems rather than three-dimensional aspects, was thought to be effective for simulating the Japanese vowels /i/ and /u/ with high validity for the vocal tract models encompassing a wide range, from the frontal sinuses to the trachea, constructed from CT data obtained during sustained phonation.

Far field acoustic radiation and vibration analysis of combined shells submerged at finite depth from free surface

A completely submerged underwater vehicle is modeled as a conical-cylindrical-spherical combined shell, the vibro-acoustic response near free surface of which is rarely analyzed in existing studies. In this paper, a Ritz-Legendre spectral method is proposed to analyze the vibro-acoustic response of combined shells under different submerged depths from the free surface. A theoretical structural model is established based on the Love shell theory, virtual spring technology and coordination relationship between displacements and rotation angle of adjacent subshells. The half-space external acoustic field model with the free surface is formulated by the Legendre spectral method, image method and Kirchhoff–Helmholtz boundary integral equation. The unified displacement admissible function of each subshell is expanded as Legendre orthogonal polynomial along generatrix direction and Fourier series along circumferential direction. The free surface treated as an ideal infinite acoustic soft boundary is substituted into acoustic wave equation and the half-space Green's function is obtained. The key technology to construct final coupled governing equation is how to expand the half-space Green's function with two-dimensional Fourier transform. Computed results are compared with those of the references and finite element method. The convergence, applicability, and correctness of the present method are verified. Influence of the free surface on the vibration and far-field acoustic radiation of combined shells under different submerged depths is analyzed, providing engineering significance for predicting external acoustic radiation of complicated combined shells with free surface.

A Physics-Informed Neural Network Approach for Nearfield Acoustic Holography.

In this manuscript, we describe a novel methodology for nearfield acoustic holography (NAH). The proposed technique is based on convolutional neural networks, with autoencoder architecture, to reconstruct the pressure and velocity fields on the surface of the vibrating structure using the sampled pressure soundfield on the holographic plane as input. The loss function used for training the network is based on a combination of two components. The first component is the error in the reconstructed velocity. The second component is the error between the sound pressure on the holographic plane and its estimate obtained from forward propagating the pressure and velocity fields on the structure through the Kirchhoff–Helmholtz integral; thus, bringing some knowledge about the physics of the process under study into the estimation algorithm. Due to the explicit presence of the Kirchhoff–Helmholtz integral in the loss function, we name the proposed technique the Kirchhoff–Helmholtz-based convolutional neural network, KHCNN. KHCNN has been tested on two large datasets of rectangular plates and violin shells. Results show that it attains very good accuracy, with a gain in the NMSE of the estimated velocity field that can top dB, with respect to state-of-the-art techniques. The same trend is observed if the normalized cross correlation is used as a metric.

Boundary integrals for oscillating bodies in stratified fluids

The theoretical foundations of the boundary integral method are considered for inviscid monochromatic internal waves, and an analytical approach is presented for the solution of the boundary integral equation for oscillating bodies of simple shape: an elliptic cylinder in two dimensions, and a spheroid in three dimensions. The method combines the coordinate stretching introduced by Bryan and Hurley in the frequency range of evanescent waves, with analytic continuation to the range of propagating waves by Lighthill's radiation condition. Not only are the waves obtained for arbitrary oscillations of the body, with application to radial pulsations and rigid vibrations, but also the distribution of singularities equivalent to the body, allowing later inclusion of viscosity in the theory. Both a direct representation of the body as a Kirchhoff–Helmholtz integral involving single and double layers together, and an indirect representation involving a single layer alone, are considered. The indirect representation is seen to require a certain degree of symmetry of the body with respect to the horizontal and the vertical. As the surface of the body is approached the single- and double-layer potentials exhibit the same discontinuities as in classical potential theory.

Near-field target strength of finite cylindrical shell in water

In this study, near-field acoustic scattering from a finite elastic cylindrical shell is investigated based on a previous cylinder method. An analytic expression for scattered pressure is derived in the form of a Kirchhoff–Helmholtz integral with the assumption that the field induced on the cylinder surface is determined by the canonical solution of an infinite elastic cylindrical shell. A useful approximate solution of the integral is obtained in the near-field zone of the receiver and then applied to formulate the near-field target strength. The results of the approximate solution are compared with the full numerical solution and experimental data measured using a linear frequency-modulated continuous-wave pulse at a frequency range of 5 to 25 kHz in a water tank in a laboratory. The results indicate consistency, except for some discrepancies due to experimental restrictions. The approximate solution is validated via finite element modeling using COMSOL Multiphysics. Moreover, the range dependency of the near-field formula and the effect of the end-caps in the finite cylindrical shell are investigated through simulations.

Thermo-Acoustic Effects on the Natural Frequencies of Vibration of an Elastic Rectangular Panel

In this paper, the thermo-acoustic behavior of a rectangular panel fully immersed in a compressible fluid at rest is investigated. A boundary element method (BEM) has been employed taking into account the Kirchhoff–Helmholtz (K-H) integral equation for the acoustic pressure and with the fluid-plate interface boundary condition the acoustic pressure jump over the panel is calculated. The thermal effects are considered regarding in the form of a uniform increment of temperature of the panel and are analyzed in order to prevent the buckling phenomena. The deformation modes of the panel correspond to the vacuum case. Applying a collocation method for the panel equation, the natural frequencies are obtained. The effects of several geometric parameters regarding different thermal loads on these frequencies are evaluated. Furthermore, the influence of the wave number for different temperatures of the panel on the acoustic damping ratio is evaluated, as well as the acoustic radiation efficiency for the different modes. The verification of the method is proven with other works.

Digital lensless holographic microscopy: numerical simulation and reconstruction with ImageJ.

The description and validation of an ImageJ open-source plugin to numerically simulate and reconstruct digital lensless holographic microscopy (DLHM) holograms are presented. Two modules compose the presented plugin: the simulation module implements a discrete version of the Rayleigh-Somerfield diffraction formula, which allows the user to directly build a simulated hologram from a known phase and/or amplitude object by just introducing the geometry parameters of the simulated setup; the plugin's reconstruction module implements a discrete version of the Kirchhoff-Helmholtz diffraction integral, thus allowing the user to reconstruct DLHM holograms by introducing the parameters of the acquisition setup and the desired reconstruction distance. The plugin offers the two said modules within the robust environment provided by a complete set of built-in tools for image processing available in ImageJ. While the simulation module has been validated through the evaluation of the forecasted lateral resolution of a DLHM setup in terms of the numerical aperture, the reconstruction module is tested by means of reconstructing experimental DLHM holograms of biological samples.

Exact extrapolation and immersive modelling with finite-difference injection

SUMMARY In numerical modelling of wave propagation, the finite-difference (FD) injection method enables the re-introduction of simulated wavefields in model subdomains with machine precision, enabling the efficient calculation of waveforms after localized model alterations. By rewriting the FD-injection method in terms of sets of equivalent sources, we show how the same principles can be applied to achieve on-the-fly wavefield extrapolation using Kirchhoff–Helmholtz (KH)-like integrals. The resulting extrapolation methods are numerically exact when used in conjunction with FD-computed Green’s functions. Since FD injection only relies on the linearity of the wave equation and compactness of FD stencils in space, the methods can be applied to both staggered and non-staggered discretizations with arbitrary-order spatial operators. Examples for both types of discretizations show how these extrapolators can be used to truncate models with exact absorbing or immersive boundary conditions. Such immersive modelling involves the evaluation of KH-type extrapolation and representation integrals in the same simulation, which include the long-range interactions missing from conventional FD injection.

Wavefield reconstruction for velocity–stress elastodynamic full-waveform inversion

SUMMARY Gradient computations in full-waveform inversion (FWI) require calculating zero-lag cross-correlations of two wavefields propagating in opposite temporal directions. Lossless media permit accurate and efficient reconstruction of the incident field from recordings along a closed boundary, such that both wavefields propagate backwards in time. Reconstruction avoids storing wavefield states of the incident field to secondary storage, which is not feasible for many realistic inversion problems. We give particular attention to velocity–stress modelling schemes and propose a novel modification of a conventional reconstruction method derived from the elastodynamic Kirchhoff–Helmholtz integral. In contrast to the original formulation (in a previous related work), the proposed approach is well-suited for velocity–stress schemes. Numerical examples demonstrate accurate wavefield reconstruction in heterogeneous, elastic media. A practical example using 3-D elastic FWI demonstrates agreement with the reference solution.

Numerical computation on the scattering sound field distribution of rigid sphere in shallow water waveguide

Numerical computational model about the scattering fields of rigid sphere in shallow water waveguide is accomplished using boundary element method and normal mode model. A kind of modified CHIEF boundary element method is used to solve the non-uniqueness problem at characteristic frequency for exterior sound field computation. The green function for shallow water waveguide acoustic wave propagation is modified based on the normal mode model. The acoustic pressure in the far field yields Kirchhoff Helmholtz equation and the scattering sound field distribution of rigid sphere in a Pekeris waveguide is achieved. Also the scattering sound field distribution of rigid sphere in the free field is calculated. The results indicate that the waveguide influences the sound propagation, and it need to pay attention during the test.

Broadband monostatic acoustic scattering simulations of underwater unexploded ordnance using time-domain spectral-elements

Acoustic detection of unexploded ordnance (UXO) that contaminate the world’s waterways is vital. Critical to the implementation of detection methods is the numerical simulation of the scattering response of these targets using finite element methods. We introduce a parametric approach based on the time-domain spectral finite element method (SEM). This technique allows for the computation of broadband acoustic response of complex heterogeneous 3-D fluid-solid objects, in particular a 5 inch rocket UXO used for this study. The scattered field is obtained at arbitrary distances using the Kirchhoff–Helmholtz integral. The main interest of the SEM is that it is particularly adapted to high-performance computing (CPU or GPU). Contrary to most of the other finite element methods it scales perfectly; using twice as many processors leads to a halving of computing time. In addition, working in the time domain is a direct simulation of common sonars and experiments. The method is first benchmarked against the commercial finite-element code COMSOL on the monostatic response of a rigid target over a full 180 deg. Finally, results obtained for the 5 inch rocket are compared to actual measurements obtained in the NRL underwater acoustics tank facility. [Work partially supported by the Office of Naval Research.]

Boundary Matching Filters for Spherical Microphone and Loudspeaker Arrays

Conversion of microphone array signals into loudspeaker array signals is an essential process in high-definition spatial audio. This paper presents the theory of boundary matching filters (BMFs) for spherical array signal conversion. BMFs adapt the physical boundary conditions used during recording to the ones required for reproduction by relying on a theoretical framework provided by the Kirchhoff–Helmholtz integral equation (KHIE). Computationally, array signal conversion is performed in a transform domain where sound fields are represented in terms of spherical harmonic functions. Related research on transform-domain signal conversion filters is interpreted in the context of the KHIE. The case of a rigid recording boundary and an open reproduction boundary is addressed. The proposed rigid-to-open BMFs provide a suitable basis for designing gain-limited filters to deal with the problem of excessive gains at certain frequency bands, observed when using high-resolution arrays. Spatial, spectral, and temporal effects in sound field reconstruction when finite numbers of transducers are used in anechoic conditions are investigated analytically and exemplified numerically. Results show that the proposed gain-limited rigid-to-open BMFs outperform the existing gain-limited filters based on Tikhonov regularization because they reduce the spatial discretization effects and yield impulse responses that are more localized around their main peaks.

Time-domain Helmholtz-Kirchhoff integral for surface scattering in a refractive medium.

The time-domain Helmholtz-Kirchhoff (H-K) integral for surface scattering is derived for a refractive medium, which can handle shadowing effects. The starting point is the H-K integral in the frequency domain. In the high-frequency limit, the Green's function can be calculated by ray theory, while the normal derivative of the incident pressure from a point source is formulated using the ray geometry and ray-based Green's function. For a corrugated pressure-release surface, a stationary phase approximation can be applied to the H-K integral, reducing the surface integral to a line integral. Finally, a computationally-efficient, time-domain H-K integral is derived using an inverse Fourier transform. A broadband signal scattered from a sinusoidal surface in an upwardly refracting medium is evaluated with and without geometric shadow corrections, and compared to the result from a conventional ray model.

Modeling surface scatter in the channel impulse response with a Helmholtz-Kirchhoff integral

Several experimental measurements of channel impulse responses (CIRs) using underwater communication bandwidths have shown distinct striation features arriving up to 10 milliseconds after the main surface bounce, e.g. Badiely et al. [J. Acoust. Soc. Am. 132, EL290 (2012)]. Choo et al. [J. Acoust. Soc. Am. 136, 1046 (2014)] used an intuitive geometric ray model to argue that the source of these arrivals was the reflection from a wave trough near the receiver. This presentation extends Choo's approach and focuses on a quadrature solution of the time domain Helmholtz-Kirchhoff (HK) integral in a half space. This approach provides the same intuitive geometric ray perspective on the surface scattering, but additionally models arrival amplitudes near caustics and higher order catastrophe phenomena induced by surface waves. It is shown that a simple shadowing correction of the HK result produces all of the major features of the full Helmholtz scattering integral solution. The HK integral is computationally tractable, accurate, and can show the path from the surface that contributes to each CIR arrival.

Prediction of acoustic radiation from functionally graded shells of revolution in light and heavy fluids

This paper presents a semi-analytical method for the vibro-acoustic analysis of a functionally graded shell of revolution immersed in an infinite light or heavy fluid. The structural model of the shell is formulated on the basis of a modified variational method combined with a multi-segment technique, whereas a spectral Kirchhoff–Helmholtz integral formulation is employed to model the exterior fluid field. The material properties of the shell are estimated by using the Voigt׳s rule of mixture and the Mori–Tanaka׳s homogenization scheme. Displacement and sound pressure variables of each segment are expanded in the form of a mixed series using Fourier series and Chebyshev orthogonal polynomials. A set of collocation nodes distributed over the roots of Chebyshev polynomials are employed to establish the algebraic system of the acoustic integral equations, and the non-uniqueness solution is eliminated using a combined Helmholtz integral equation formulation. Loosely and strongly coupled schemes are implemented for the structure-acoustic interaction problem of a functionally graded shell immersed in a light and heavy fluid, respectively. The present method provides a flexible way to account for the individual contributions of circumferential wave modes to the vibration and acoustic responses of functionally graded shells of revolution in an analytical manner. Numerical tests are presented for sound radiation problems of spherical, cylindrical, conical and coupled shells. The individual contributions of the circumferential modes to the radiated sound pressure and sound power of functionally graded shells are observed. Effects of the material profile on the sound radiation of the shells are also investigated.

Comment on "Reconstructing surface wave profiles from reflected acoustic pulses" [J. Acoust. Soc. Am. 133(5), 2597-2611 (2013)

A computationally efficient, time-domain Helmholtz-Kirchhoff (H-K) integral was derived and applied to reconstructing surface wave profiles from reflected acoustic pulses [Walstead and Deane, J. Acoust. Soc. Am. 133, 2597-2611 (2013)]. However, the final form of the integral equation incorporating a stationary phase approximation contained a complex phase term exp(iπ/4), which cannot be treated as a simple time delay. In this work, a real time-domain H-K integral is presented that includes an additional Hilbert transform of the time-derivative of the transmitted pulse. Numerical simulation with a sinusoidal surface shows good agreement between the real time-domain formulation and exact H-K integral, while achieving a significant improvement in computational speed (e.g., 2 orders of magnitude).

Active structural acoustic control of a smart cylindrical shell using a virtual microphone

This paper investigates the active structural acoustic control of sound radiated from a smart cylindrical shell. The cylinder is equipped with piezoelectric sensors and actuators to estimate and control the sound pressure that radiates from the smart shell. This estimated pressure is referred to as a virtual microphone, and it can be used in control systems instead of actual microphones to attenuate noise due to structural vibrations. To this end, the dynamic model for the smart cylinder is derived using the extended Hamilton’s principle, the Sanders shell theory and the assumed mode method. The simplified Kirchhoff–Helmholtz integral estimates the far-field sound pressure radiating from the baffled cylindrical shell. A modified higher harmonic controller that can cope with a harmonic disturbance is designed and experimentally evaluated. The experimental tests were carried out on a baffled cylindrical aluminum shell in an anechoic chamber. The frequency response for the theoretical virtual microphone and the experimental actual microphone are in good agreement with each other, and the results show the effectiveness of the designed virtual microphone and controller in attenuating the radiated sound.

The Burton and Miller Method: Unlocking Another Mystery of Its Coupling Parameter

The phenomenon of irregular frequencies or spurious modes when solving the Kirchhoff–Helmholtz integral equation has been extensively studied over the last six or seven decades. A class of common methods to overcome this phenomenon uses the linear combination of the Kirchhoff–Helmholtz integral equation and its normal derivative. When solving the Neumann problem, this method is usually referred to as the Burton and Miller method. This method uses a coupling parameter which, theoretically, should be complex with nonvanishing imaginary part. In practice, it is usually chosen proportional or even equal to [Formula: see text]. A literature review of papers about the Burton and Miller method and its implementations revealed that, in some cases, it is better to use [Formula: see text] as coupling parameter. The better choice depends on the specific formulation, in particular, on the harmonic time dependence and on the fundamental solution or Green’s function, respectively. Surprisingly, an unexpectedly large number of studies is based on the wrong choice of the sign in the coupling parameter. Herein, it is described which sign of the coupling parameter should be used for different configurations. Furthermore, it will be shown that the wrong sign does not just make the solution process inefficient but can lead to completely wrong results in some cases.