Kirchhoff-Helmholtz Integral Research Articles

A small-scale active anechoic chamber

This paper presents a solution method to suppress reflections from the walls in an anechoic room using active control, in which the control coefficients are obtained with an efficient algorithm. One of the main challenges in suppressing the reflections, is the estimation of the reflected sound field, which is a non-measurable quantity. The reflected sound field is computed using the Kirchhoff-Helmholtz integral with a set of microphones on a circle. The effectiveness of the method is shown on an experimental setup. The experimental setup is a small-scale version of an acoustic anechoic chamber. An anechoic chamber, with larger dimensions and higher frequencies, will result in a system with a relatively large number of sources and sensors. It is computationally expensive to find a set of control filters for larger-scale systems. The conjugate gradient algorithm using block-circulant preconditioning is an efficient method to solve for the optimal set of control filters. The preconditioned conjugate gradient algorithm is used together with the Kirchhoff-Helmholtz integral. Real-time measurements show that the reflections are effectively suppressed, expressed in terms of the reverberation time. This results in a small-scale active anechoic chamber.

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.]

Two dimensional elastic dynamically focused beam migration in transversely isotropic (TI) media

The anisotropic ray tracing equations of elastic waves are derived for calculating the central ray's travel time, trajectory, and parameters P and Q. Based on the anisotropic ray tracing algorithm, we proposed a transversely isotropic elastic dynamically focused beam migration for two dimensional media. Based on the two dimensional Kirchhoff-Helmholtz integral of elastic waves, the elastic imaging weight coefficients are used to reduce the crosstalk noise. Then, based on the previous research, we introduced a sign function to eliminate the effect of polarity reversal in qPqSV wave migration. Through model tests, it can be found that the proposed method can process the multicomponent seismic data and image the complex structures for anisotropic media. The optimized dynamic ray tracing algorithm is superior to the conventional one in computational efficiency. Furthermore, the proposed method eliminates the limitations of the Gaussian beam imaging. This method not only does not affect the accuracy of shallow imaging, but also enhances the energy focusing and the amplitude of deep layers. The model tests have shown the accuracy and effectiveness of our proposed method.

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 improved lumped parameter model of the single free-flooded ring transducer using Helmholtz–Kirchhoff integral solution

The Helmholtz-Kirchhoff integral (HKI) may be considered as the boundary element method (BEM) used to calculate the acoustic pressure at any position in the 3D-radiation problem from a vibrating surface. This method provides valuable insights into the radiation characteristics of vibrating structures, which may offer significant information and tools to underwater transducer design engineers during the early stages of development. It may save considerable computational costs in the design processes for the underwater transducer such as Hull Mount Sonar. This study illustrates usefulness of the HKI-based approach by developing a simple model of the Free Flooded Ring (FFR) transducer using a piezoelectric ring model and the HKI solution on the surface acoustic pressure of the ring through their physical conditions on their connected interfaces. The FFR transducer is widely utilized as a low-frequency acoustic source in underwater environments due to its broad operating frequency bandwidth and relatively small size. The developed approach aims to construct a precise model that considers the sound-structure interactions between the piezoelectric ring and the acoustic medium around it. To validate the accuracy of the proposed method, the acoustic pressure and electrical admittance are compared with those obtained through finite element method (FEM) simulations.

3D finite element modeling techniques and application to underwater target scattering

Underwater acoustic target scattering measurements rely on high fidelity modeling for experimental comparison and understanding. Three-dimensional (3D) finite element models are well suited for this purpose, as they can account for arbitrary or unknown target properties and configurations/orientations within complex and asymmetrical seafloor environments. High acoustic frequencies and large physical distances associated with in situ scattering measurements pose challenges to 3D modeling efforts in terms of model sizes and runtimes. Certain model considerations must be made to keep the 3D model computationally efficient, yet accurate in predictive capability. Numerically determined Green’s functions are demonstrated to permit 3D model reduction, while still preserving far-field scattering prediction capability through the Helmholtz-Kirchhoff integral. By determining Green’s functions within the model, they need not be known or estimated for complex ocean environments a priori. Nontraditional scattering formulations and a survey of boundary truncation methods also are explored and implemented for maximal accuracy within small 3D computational domains. Model results for canonical elastic targets within varying seafloor environments are shown and compared to theory and experimentation. [This work has been supported by the Strategic Environmental Research and Development Program, and by the Office of Naval Research, Ocean Acoustics.]

3D finite element modeling techniques and application to underwater target scattering

Underwater acoustic target scattering measurements rely on high-fidelity modeling for experimental comparison and understanding. Three-dimensional (3D) finite element models are well suited for this purpose, as they can account for arbitrary or unknown target properties and configurations/orientations within complex and asymmetrical seafloor environments. High acoustic frequencies and large physical distances associated with in situ scattering measurements pose challenges to 3D modeling efforts in terms of model sizes and runtimes. Certain model considerations must be made to keep the 3D model computationally efficient, yet accurate in predictive capability. Numerically determined Green’s functions are demonstrated to permit 3D model reduction, while still preserving far-field scattering prediction capability through the Helmholtz–Kirchhoff integral. By determining Green’s functions within the model, they need not be known or estimated for complex ocean environments a priori. Nontraditional scattering formulations and a survey of boundary truncation methods also are explored and implemented for maximal accuracy within small 3D computational domains. Model results for canonical elastic targets within varying seafloor environments are shown and compared to theory and experimentation. [Work supported by the Strategic Environmental Research and Development Program and by the Office of Naval Research, Ocean Acoustics.]

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.

Elastic immersive wavefield modelling

Many modelling studies of wave scattering require repeated numerical simulations through models with properties that differ only in a small sub-domain. Hence, it is of interest to recompute the wavefields that account for wave propagation through the whole domain, using simulations that are performed only in the sub-domain. Immersive boundary conditions (IBCs) can be used to establish such a local wavefield modelling scheme which enables accurate wavefield recomputation, including all interactions between the locally-perturbed medium and the full domain. We develop IBC theory for elastic wave propagation, in which the boundary conditions are updated dynamically at each time step of a simulation in the local domain. These updates are calculated by wavefield extrapolation based on the Kirchhoff-Helmholtz integrals using Green's functions in the background medium. Wavefield recording and injection in IBCs can be implemented either using finite-difference (FD) injection methods, or using the method of multiple point sources (MPS). The latter method is significantly less computationally demanding in terms of both memory and number of calculations. We therefore both extend acoustic FD injection methods to elastic media, and propose a new second-order accurate MPS method to implement elastic IBCs, which is numerically exact. In higher-order FD modelling, the MPS method is not numerically exact but still produces highly accurate IBC wavefields when compared to global-domain simulations.

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.

Experimental Investigation of Diffraction caused by Transparent Barriers

In addition to wave-particle duality, the contributions of Kirchhoff-Helmholtz are fundamental to the scalar theory of diffraction. The mathematical results of their formulae help predict the maximum intensity of light at the center of the far-field diffraction pattern that coincides with the optical axis. This study demonstrates, via a series of the single-slit experiments, that the Helmholtz–Kirchhoff integral is invalid for transparent barriers. In fact, the experimental results show that the main factors determining the appearance of the diffraction pattern are the refractive index contrast between the barrier and the medium, including the physical invariance of the medium in response to factors such as temperature and pressure, and the dimensions of the barriers.

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.

Reciprocity and Representation Theorems for Flux- and Field-Normalised Decomposed Wave Fields

We consider wave propagation problems in which there is a preferred direction of propagation. To account for propagation in preferred directions, the wave equation is decomposed into a set of coupled equations for waves that propagate in opposite directions along the preferred axis. This decomposition is not unique. We discuss flux-normalised and field-normalised decomposition in a systematic way, analyse the symmetry properties of the decomposition operators, and use these symmetry properties to derive reciprocity theorems for the decomposed wave fields, for both types of normalisation. Based on the field-normalised reciprocity theorems, we derive representation theorems for decomposed wave fields. In particular, we derive double- and single-sided Kirchhoff-Helmholtz integrals for forward and backward propagation of decomposed wave fields. The single-sided Kirchhoff-Helmholtz integrals for backward propagation of field-normalised decomposed wave fields find applications in reflection imaging, accounting for multiple scattering.

Compensating for source directivity in immersive wave experimentation.

A physical boundary mounted with active sources can cancel acoustic waves arriving at the boundary, and emit synthesized waves into the neighboring medium to fully control the acoustic wavefield in an experimental setup such as a water tank or air-filled cavity. Using the same principles, a physical experiment can be artificially immersed within an extended virtual (numerical) domain so that waves propagate seamlessly between the experimental setup and virtual domain. Such an immersive wave control experiment requires physical monopolar sources on the active boundary. However, real physical sources (e.g., piezoelectric transducers) project waves at middle-to-high sonic frequencies (e.g., 1-20 kHz) that do not fully conform to the theoretically required monopolar radiation pattern; if left uncorrected, this causes controlled wavefields to deviate from those desired in immersive experiments. A method is proposed to compensate for the non-monopole-like radiation patterns of the sources, and can be interpreted physically in terms of Huygens principle. The method is implemented as a pre-computation procedure that modifies the extrapolation Green's functions in the Kirchhoff-Helmholtz integral before the actual experiments take place. Two-dimensional finite-difference simulations show that the processing method can effectively suppress the undesired effect caused by non-monopolar active sources in immersive wave control experiments.

Helmholtz-Kirchhoff integral formula for predicting the acoustic pressure from a radiator including irregular surface

The Helmholtz-Kirchhoff integral (HKI) for the radiation problem from a finite source with a vibrating surface is a method of calculating the acoustic pressure at any position in the three-dimensional space using the velocity conditions of the radiating surface. Previous studies predicted the acoustic pressure from the radiator based on the HKI. However, the acoustic pressure should be evaluated first on the radiation surface in order to calculate the acoustic pressure in the space. Nonetheless, it is difficult to evaluate the acoustic pressure on the radiation surface accurately if there are non-smooth points. Because the exact HKI is not easily found to be applicable for the evaluation, in this study, we investigated the HKI from the beginning. Through the process of deriving the formula, we proposed the form of the HKI that can be used for calculating the acoustic pressure at both smooth and non-smooth points of the radiation surface. The proposed formula is also proved mathematically. As a result, the acoustic pressures calculated by the proposed formula are compared to those using a FEM. The results yield that the overall relative errors are less than 2% even on the radiator surface including vertices. [Work supported by Samsung Display Corporation.]

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.]

Immersive wave control experimentation using compensated directive sources

A physical boundary with embedded active sources can cancel acoustic waves incident on the boundary and also synthesize waves to fully control the acoustic field in an experimental setup (e.g., a water tank) such that the physical experiment is artificially immersed into a virtual domain and waves propagate seamlessly between the physical experiment and virtual domain. The acoustic representation theorem at the heart of the immersive experiment requires physical monopole sources to be deployed on the active boundary. However, real physical sources (e.g., piezoelectric transducers) project waves at sonic frequencies (e.g., 2–20 kHz) that do not fully conform to the theoretically required radiation pattern. If left uncorrected, using these physical sources causes the wavefield to deviate from those desired in immersive experiments. A method is proposed to compensate for the non-monopolar radiation pattern of the sources, and the compensation is incorporated into the Kirchhoff-Helmholtz extrapolation, which is used to determine the controlling field at the active boundary in real-time. Numerical simulations show that the method can effectively compensate for the undesired effects caused by such sources. The method is implemented as a pre-processing step that modifies the extrapolation Green's functions in the Kirchhoff-Helmholtz integral before the actual experiments take place and can be physically interpreted in terms of Huygens’ principle.

Immersive wave propagation experiments in a two-dimensional acoustic waveguide

The physical implementation of immersive boundary conditions (IBCs) allows acoustic or elastic waves to propagate seamlessly between a physical domain, such as a wave propagation laboratory, and a numerical simulation virtually enclosing the physical domain. IBCs correctly account for all wavefield interactions between both domains, including higher-order long-range scattering. In this contribution, IBCs are physically implemented in a two-dimensional (2-D) acoustic waveguide. The boundary surrounding the waveguide is densely populated with hundreds of loudspeakers that apply the necessary boundary conditions. The required signals to be injected at the boundary are predicted in real-time by (1) measuring the pressure field and its gradient on two acoustically transparent auxiliary surfaces of microphones inside the waveguide and (2) extrapolating the wavefields to the boundary by evaluating a Kirchhoff-Helmholtz integral using a low-latency, FPGA-enabled data acquisition, computation and control system. Here, we demonstrate the first real-time, 2-D physical immersive wave propagation experiments. We present the setup, as well as a suite of experiments that demonstrate the ability of IBCs to actively suppress broadband incident fields at the boundary of the waveguide and to correctly reproduce all orders of wavefield scattering between the physical experiment and the numerical simulation.The physical implementation of immersive boundary conditions (IBCs) allows acoustic or elastic waves to propagate seamlessly between a physical domain, such as a wave propagation laboratory, and a numerical simulation virtually enclosing the physical domain. IBCs correctly account for all wavefield interactions between both domains, including higher-order long-range scattering. In this contribution, IBCs are physically implemented in a two-dimensional (2-D) acoustic waveguide. The boundary surrounding the waveguide is densely populated with hundreds of loudspeakers that apply the necessary boundary conditions. The required signals to be injected at the boundary are predicted in real-time by (1) measuring the pressure field and its gradient on two acoustically transparent auxiliary surfaces of microphones inside the waveguide and (2) extrapolating the wavefields to the boundary by evaluating a Kirchhoff-Helmholtz integral using a low-latency, FPGA-enabled data acquisition, computation and control system. H...

A robust optimum controller for suppressing radiated sound from an intelligent cylinder based on sliding mode method considering piezoelectric uncertainties

In this study, a robust controller against the uncertainties in piezoelectric patches including sensor and actuator is designed based on sliding mode method to control the radiated sound from cylindrical shells. Accordingly, in order to extract and discretize the dynamic equations of a smart cylinder equipped with piezoelectric patches, the Hamilton’s principle and the Rayleigh-Ritz method are, respectively, used . The radiated sound is estimated by the Kirchhoff-Helmholtz integral and the acoustic structural sensing method. Furthermore, an innovative approach is proposed on sliding mode control to model system uncertainties and design robust control signals against these disturbances. Using effective control signals for each mode is the applied methodology for establishing independent sliding surfaces. In fact, it is attempted to relate between actuator matrix determinant and system control ability in generating the efficient control signals and error reduction due to actuators uncertainties. By the aid of this relation, optimization of the actuators position according to the genetic algorithm is implemented. The obtained results show that by optimizing the actuators position not only the appropriate performance of the system in controlling the radiated sound from the structure is enhanced but also the essential control voltage for each actuator is significantly decreased.