Kirchhoff Integral Equation Research Articles

Prediction of Time Domain Vibro-Acoustic Response of Conical Shells using Jacobi–Ritz Boundary Element Method

Considering the lack of studies on the transient vibro-acoustic properties of conical shell structures, a Jacobi–Ritz boundary element method for forced vibro-acoustic behaviors of structure is proposed based on the Newmark-β integral method and the Kirchhoff time domain boundary integral equation. Based on the idea of the differential element method and the first-order shear deformation theory (FSDT), the vibro-acoustic model of conical shells is established. The axial and circumferential displacement tolerance functions are expressed using Jacobi polynomials and the Fourier series. The time domain response of the forced vibration of conical shells is calculated based on the Rayleigh–Ritz method and Newmark-β integral method. On this basis, the time domain response of radiated noise is solved based on the Kirchhoff integral equation, and the acoustic radiation characteristics of conical shells from forced vibration are analyzed. Compared with the coupled FEM/BEM method, the numerical results demonstrate the high accuracy and great reliability of this method. Furthermore, the semi-vertex angle, load characteristics, and boundary conditions related to the vibro-acoustic response of conical shells are examined.

A Composite Scattering Model of the Vegetated Ground With a Target

In this study, a composite scattering model of the vegetated ground with a target is developed. In the composite scattering model, a Facet-based Modified Two-scale Model (FMTSM), combining the advantages of Kirchhoff Approximate (KA) and Integral Equation Model (IEM), is established to calculate the electromagnetic (EM) scattering of the rough ground, which can enhance precision and extend application range comparing with the traditional TSM. Based on the FMTSM and Vector Radiative Transfer Theory (VRT), an improved two-layered vegetation model is developed to deal with the EM scattering of vegetated ground. Besides, two typical scattering problems, including the rough ground and vegetated ground, are performed to prove the validity of FMTSM and FMTSM-VRT methods. Simulation results agree well with the measured value. Finally, the composite scattering model of the vegetated ground with a target is proposed by combing the FMTSM-VRT method with Shooting and Bouncing Ray (SBR) algorithm. Meanwhile, the composite scattering characteristics of the vegetated ground with a missile are explored.

Aeroacoustic source prediction using material surfaces bounding the flow

This article presents an extension of Liepmann’s characterization of an aeroacoustic source in terms of the motion of a bounding surface containing the source region. Rather than using an arbitrary surface, we express the problem in terms of bounding material surfaces, identified by Lagrangian coherent structures (LCSs), which demarcate flow into regions with distinct dynamics. The sound generation of the flow is written in terms of the motion of these material surfaces using the Kirchhoff integral equation, so that the flow noise problem now appears like that of a deforming body. This approach provides a natural connection between the flow topology, as revealed through LCS analysis, and sound generation mechanisms. As examples, we examine two-dimensional cases of co-rotating vortices and leap-frogging vortex pairs and compare estimated sound sources to vortex sound theory.

Robust reconstruction of scattering surfaces using a linear microphone array

The analysis of sound scattered by a rough surface and measured by multiple microphones positioned in the far field yields an estimate of the unknown scattering surface profile. Expanding from previous work, the approach used in this paper is based on an expansion and linearization of the Kirchhoff integral equation, and applies to a low density of receivers. Here, the original algorithm is modified in order to reduce the measurement bias, and extended to broadband signals to over-constrain the problem and improve its robustness. The improved method is rigorously assessed alongside the original algorithm and its small perturbation version, for a two-dimensional geometry and for scattering surfaces with a spatial power-function spectrum. The impact of the measurement setup and surface characteristics on the reconstruction uncertainty are evaluated by means of numerical simulations. Additional experimental data obtained for three known surface profiles reveal the impact of noise and measurement uncertainties. The optimal measurement configuration requires a trade-off between resolution (higher at high frequencies), and robustness (higher at low frequencies). This limit is overcome at least partially by the proposed multiple-frequency extension. The resulting measured uncertainties were close to the theoretical expectation of approximately 2% of the acoustic wavelength.

Boundary-Element Methods for Field Reconstruction in Accelerator Magnets

Magnetic fields in the aperture of particle accelerator magnets can be represented by boundary potentials, exploiting Kirchhoff's integral equation. Depending on the formulation, magnetic measurement data can be represented by the discrete approximations of Dirichlet or Neumann data at the domain boundary. The missing Cauchy data, which are related to the tangential-field components, can then be computed by the boundary-element method (BEM) in a numerical post-processing step. Evaluating the integral equation for field reconstruction inside the domain of interest will reduce measurement uncertainties and approximation errors due to the smoothing property of Green's kernel. Applications to the reconstruction of 2-D fields (integrated quantities from stretched-wire measurements) and 3-D fields (local quantities from measurements with moving induction-coil magnetometers) are presented.

An explicit marching-on-in-time scheme for solving the time domain Kirchhoff integral equation.

A fully explicit marching-on-in-time (MOT) scheme for solving the time domain Kirchhoff (surface) integral equation to analyze transient acoustic scattering from rigid objects is presented. A higher-order Nyström method and a PE(CE)m-type ordinary differential equation integrator are used for spatial discretization and time marching, respectively. The resulting MOT scheme uses the same time step size as its implicit counterpart (which also uses Nyström method in space) without sacrificing from the accuracy and stability of the solution. Numerical results demonstrate the accuracy, efficiency, and applicability of the proposed explicit MOT solver.

Fast time domain modeling of surface scattering from reflectors and diffusers.

Time-domain prediction models have been developed for auditorium reflectors and room acoustic diffusers. The models are time-domain equivalents of the single-frequency formulations that exploit the Kirchhoff boundary conditions. Consequently, they are approximate, wave-based solutions to the Kirchhoff integral equation using surface meshes. The new time-domain formulations are validated by comparison to their frequency-domain equivalents for three different surfaces: a plane surface, a curved reflector, and a Schroeder diffuser. In terms of computation time and accuracy, the new models lie between the finite difference time domain and geometric room models.

Dynamics and noise radiation of multiple bubbles in compressible fluid using boundary integral equation

Purpose – The purpose of this paper is to research the interaction between multiple bubbles and their noise radiation considering the influence of compressibility. The influences of bubble spacing, initial inner pressure, buoyance and phase difference are presented with different bubbles arrangements. Design/methodology/approach – Based on wave equation, the new boundary integral equation considering the compressibility is given by the matching between prophase and anaphase approximation of bubble motion and solved with boundary element method. The time-domain characteristics of noise induced by multiple bubbles are obtained by the moving boundary Kirchhoff integral equation. With the surface discretization and coordinate transformation, the bubbles surface is treated as a moving deformable boundary and noise source, and the integral is implemented on the surface directly. Findings – Numerical results show the manner of jet generation will be affected by the phase difference between bubbles. With the increasing of phase difference, the directive property of noise becomes obvious. With the enlargement of initial inner pressure, the sound pressure will arise at the early stage of expanding, and the increasing of buoyance parameter will reduce the sound pressure after the generation of jet. Since the consideration of compressibility, the oscillation amplitude of bubbles will be weakened. Originality/value – The paper could provide the reference for the research about the dynamics and noise characteristics of multiple bubbles in compressible fluid. And the new method based on boundary integral equation to simulate the multiple bubbles motion and noise radiation is presented.

Prediction of offshore impact pile driving noise using numerical and analytical approach

Noise generated by offshore impact pile driving can radiate into and propagate through the air, water, and sediment. Predicting noise levels around the support structures at sea is required to estimate the effects of the noise on marine life. We have been investigated acoustic impact using our previously verified coupled Finite Element (Commercial FE code Abaqus) and Monterey Miami Parabolic Equation (2D MMPE) models [J. Acoust. Soc. Am. 131(4), 3392 (2012) and J. Acoust. Soc. Am. 133, 3419 (2013)]. A simple analytical model for the impact pile driving noise is developed to reduce computation cost and modeling effort when compared to the numerical model. In the present study, we are developing a Matlab based analytical model to calculate the Receive Levels (RL) in water. This analytical model solves simplified Donnell’s equation, i.e., governing equation for the thin cylindrical shell using the modal superposition approach assuming azimuthal symmetry. It solves the Helmholtz or Kirchhoff integral equation to evaluate the acoustic pressure field. We will compare the results of the coupled FE-MMPE numerical model and the Matlab based analytical model and discuss their advantages and disadvantages. [Work sponsored by the Link Foundation Ocean Engineering and Instrumentation Ph.D. Fellowship program.]

Comprehensive nanostructure and defect analysis using a simple 3D light-scatter sensor

Light scattering measurement and analysis is a powerful tool for the characterization of optical and nonoptical surfaces. A new 3D scatter measurement system based on a detector matrix is presented. A compact light-scatter sensor is used to characterize the scattering and nanostructures of surfaces and to identify the origins of anisotropic scattering features. The results from the scatter sensor are directly compared with white light interferometry to analyze surface defects as well as surface roughness and the corresponding scattering distributions. The scattering of surface defects is modeled based on the Kirchhoff integral equation and the approach of Beckmann for rough surfaces.

Stabilization of time-domain acoustic boundary element method for sound radiation problems

Time-domain acoustic solution from the Kirchhoff integral equation for the exterior problem is not unique due to the presence of fictitious internal modes and also suffers from the instability that stems from the time marching scheme. In this work, methods to stabilize the time-domain acoustic boundary element calculation were suggested. Low-order fictitious internal modes within the effective frequency range of boundary element calculation were suppressed by the newly formulated time-domain CHIEF (Combined Helmholtz Integral Equation Formulation) method. Additional interior points were included, similar to frequency-domain problems, to satisfy the zero pressure constraint considering the shortest retarded time between boundary nodes and interior points. However, the calculation was yet unstable due to remaining unstable high-order modes beyond the effective frequency limit. To further stabilize the computation, unstable high-order internal modes were nullified using the wave vector filtering method. In comparison with the time-domain Burton-Miller formulation, the proposed method has no hyper-singular integral and the monotonically increasing instability was not observed. As a test example, sound radiation from a pulsating sphere was used and a good stabilization performance was shown. Average relative-difference norm of the stabilized time response from the analytic solution was 2.7%. (This work was partially supported by BK21 project)

시간 영역 음향 경계요소법에서의 비유일성 문제 해결을 위한 방법에 관하여

Kirchhoff 적분식을 이용하여 외부 음향 문제의 시간 영역 응답을 계산하는 경우, 주파수영역 해석과 마찬가지로 가상적인 내부 음향 모드에 기인한 비유일성 문제가 발생한다. 이를 해결하는 방법들 중의 하나로서 CHIEF(Combined Helmholtz Integral Equation Formulation) 방법이 쓰이는데, 이는 몇몇 내부 수음점의 응답을 0으로 추가하여 구속하는 조건을 부가하는 기법이다. 이 기법은 주파수 영역 경계요소법에서는 간편한 수식 때문에 많이 사용되고 있지만, 시간 영역에서는 사용된 예가 없다. 본 연구에서는 대상체 내부의 가상 수음점과 경계 표면의 절점들간의 최소 거리에 대한 지연시간을 고려하여, 계산하고자 하는 미지수인 현재 시간의 경계 표면 음장을 구속함으로써, 시간 영역 해석에 적합하도록 CHIEF 방법을 수식화하였다. 예제로서, 반지름 방향으로 진동하는 구의 음향 방사 문제를 다루었다. CHIEF 방법을 적용함에 따라 저차의 내부 음향 모드에 기인한 비유일성 문제를 해결할 수 있었고, 비요동 모드에 의한 수치적 불안정성을 피할 수 있었다. 그러나, 유효주파수 밖에 남은 내부 음향의 고차모드들에 의한 수치적 불안정성은 증가하였다. The time-domain solution from the Kirchhoff integral equation for an exterior problem is not unique at certain eigen-frequencies associated with the fictitious internal modes as happening in frequency-domain analysis. One of the solution methods is the CHIEF (Combined Helmholtz Integral Equation Formulation) approach, which is based on employing additional zero-pressure constraints at some interior points inside the body. Although this method has been widely used in frequency-domain boundary element method due to its simplicity, it was not used in time-domain analysis. In this work, the CHIEF approach is formulated appropriately for time-domain acoustic boundary element method by constraining the unknown surface pressure distribution at the current time, which was obtained by setting the pressure at the interior point to be zero considering the shortest retarded time between boundary nodes and interior point. Sound radiation of a pulsating sphere was used as a test example. By applying the CHIEF method, the low-order fictitious modes could be damped down satisfactorily, thus solving the non-uniqueness problem. However, it was observed that the instability due to high-order fictitious modes, which were beyond the effective frequency, was increased.

On Acoustic Radiation by Rotating Body Sources in Frequency Domain

For the complexity of the sound generated mechanism of rotating body source, this paper describes a frequency-domain numerical method for predicting sound radiation of rotating body sources based on the Kirchhoff integral equation with analytical Green's function of rotating monopole and dipole source in free space. The sound radiation model is established in free space and the relationship between characteristics of sound field and acoustic nature frequency of source, angular frequency and its harmonics can be revealed by the mathematical solution. Numerical simulation shows that sound field has a strong directivity, fundamental frequency transmitting in the rotary shaft direction and harmonics spreading along radial direction and frequency shift phenomena appearing clearly in higher rotating speed of source. The method has a theoretical significance for exploring the low-noise rotating machinery. Research about sound radiation of moving source originates from Lighthill. The first integral approach for acoustic propagation is the acoustic analogy. In the acoustic analogy, the governing Navier-Stokes equations are rearranged to be in wave-type form. There is some question as to which terms should be identified as part of the sound source and retained in the right-hand side of the equation and which terms should be in the left-hand side as part of the operator. The far-field sound pressure is then given in terms of a volume integral over the domain containing the sound source. However the major difficulty with the acoustic analogy is that the sound source is not compact in supersonic flows. Several modifications to Lighthill's original theory have been proposed to account for the moving sound source or other effects. The Ffowcs Williams and Hawkings (FW-H) equation was introduced to extend acoustic analogy in the case of solid surfaces. However, when acoustic sources are present in the flowfield a volume integration is needed. This volume integration of the quadrupole source term is difficult to compute and is usually neglected in most acoustic analogy codes. Errors could be encountered in calculating the sound field. Another alternative is the Kirchhoff method which assumes that the sound transmission is governed by the simple wave equation. Kirchhoff's method consists of the calculation of the nonlinear near- and mid-field, usually numerically, with the far-field solutions found from a linear Kirchhoff formulation evaluated on a control surface surrounding the nonlinear-field. The control surface is assumed to enclose all the nonlinear flow effects and noise sources. The sound pressure can be obtained in terms of a surface integral of the surface pressure and its normal and time derivatives. This approach has the potential to overcome some of the difficulties associated with the traditional acoustic analogy approach. The method is simple and accurate and accounts for the nonlinear quadrupole noise in the far-field. Full diffraction and focusing effects are included while eliminating the propagation of the reactive near-field. Kirchhoff's formula, published in 1882, is used in the theory of diffraction of light and in other electromaqnetic problems. It also has many applications to problems of wave in acoustics. One of the novel uses of this formula was proposed by Hawkinqs for predicting

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X-ray specular scattering from statistically rough surfaces: a novel theoretical approach based on the Green function formalism

The Green function formalism is applied to the problem of grazing-incidence small-angle X-ray scattering from statistically rough surfaces. Kirchhoff's integral equation is used to describe the X-ray wavefield propagation through a single rough surface separating vacuum and medium. Taking into account multiple diffuse X-ray scattering effects, the reflection R(coh)(θ) and transmission T(coh)(θ) coefficients of the specular wave are obtained using the Gaussian statistical model of rough surfaces in terms of the two-point height-height correlation function. In the limiting cases when the correlation length xi is equal to zero or infinity, analytical formulae for the reflection R(coh)(θ) and transmission T(coh)(θ) coefficients of the specular wave are obtained. It is important that in the case xi --> infinity they coincide with the corresponding reflection R(DW)(θ) and transmission T(DW)(θ) coefficients related to the conventional Debye-Waller approximation for describing the grazing X-ray scattering from a rough surface. In the case of finite values of correlation length \xi the reflection |R(coh)(θ)|(2) and transmission |T(coh)(θ)|(2) scans are numerically calculated.

Microwave Doppler spectra of sea return at small incidence angles: specular point scattering contribution

It is well known that the sea return echo contains contributions from at least two scattering mechanisms. In addition to the resonant Bragg scattering, the specular point scattering plays an important role as the incidence angle becomes smaller (≤ 20 °). Here, in combination with the Kirchhoff integral equation of scattering field and the stationary phase approximation, analytical expressions for Doppler shift and spectral bandwidth of specular point scattering, which are insensitive to the polarization state, are derived theoretically. For comparison, the simulated results related to the two-scale method (TSM) and the method of moment (MOM) are also presented. It is found that the Doppler shift and the spectral bandwidth given by TSM are insufficient at small incidence angles. However, a comparison between the analytical results and the numerical simulations by MOM in the backscatter configuration shows that our proposed formulas are valid for the specular point scattering case. In this work, the dependences of the predicted results on incidence angle, radar frequency, and wind speed are also discussed. The obtained conclusions seem promising for a better understanding of the Doppler spectra of the specular point scattering fields from time-varying sea surfaces.

Temporal prediction of the acoustic radiation of vibrating structures: subjective evaluation of a simplified approach

The goal of the approach presented in this paper is to predict the noise radiated by vibrating structures with acquired vibratory measurements, in the temporal field, on their surface. In the frequential field this prediction is based on the resolution of the Kirchhoff integral equation, which requires a lot of computing time and is thus limited to low and intermediate frequencies; the boundary element method (BEM) is used for the resolution. Simplifications were already brought to low (< Flim1) and to high (> Flim2) frequencies, leading to a simple calculation of a fast integral. The limiting frequencies (Flim1 and Flim2) depend on the structure dimensions and its position towards the observation point. In this paper we will present an approach developed in the temporal field in which a complete calculation (BEM) is avoided by choosing a minimal distance in order to have Flim1 = Flim2 and use only the simplified calculations. The vibratory data is measured in the temporal field and the studied structure is a parallelepipedic box excited by two mechanical excitation pots which excite one or two faces. The acoustic pressures calculated in the temporal field will be confronted with measurements, by using similarity subjective tests and integrating a variation of the principal parameters intervening in this approach.

Implicit methods of solution to integral formulations in boundary element method based nearfield acoustic holography

Two integral equation methods are considered which can be inverted to provide the surface velocity and/or pressure given a measurement of the pressure on an imaginary surface in the nearfield of a vibrating or scattering body. This problem is central to nearfield acoustical holography (NAH). The integral equations are discretized using boundary element methods (BEMs). The integral equation methods considered are (1) an indirect formulation method based on the single layer integral equation and (2) a direct formulation method based on a system of equations derived from the Helmholtz–Kirchhoff integral equation. The formation of integral equations from the mentioned methods will not involve the explicit inversion of matrices, but instead will require this inversion to be done implicitly. Since these methods back-track the sound field from the measurement surface to the surface of the source/vibrator they are ill-posed in nature and Tikhonov regularization are used to stabilize the reconstruction. Problems are noted with the use of higher order shape functions in the BEM discretization which are detrimental to the inversion, and it is shown that the breakdown of the method at the interior resonance frequencies can be mitigated successfully. These two approaches are compared with the conventional direct method using measured experimental data from a point-driven rectangular plate.

Study of the Electromagnetic Scattering from the Very Rough Fractal Sea Surface**The project supported by National Natural Science Foundation of China under Grant No. 60101001 and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China

In this paper, based on the fundamental formulae of the first-order and second-order Kirchhoff approximation and with consideration of the shadowing effect, the backscattering enhancement of the one-dimensional very rough fractal sea surface with Pierson-Moskowitz spectrum is studied under the second-order Kirchhoff approximation at microwave frequency. The numerical results are compared with those of the first-order Kirchhoff approximation and integral equation method. The dependencies of the bistatic scattering cross section and the backscattering enhancement on the incident angle, fractal dimension, and windspeed over the sea surface are analyzed in detail.

Validity study of the distorted wave Born approximation (DWBA) model: Application to the acoustic scattering by marine organisms

Applications of the distorted wave Born approximation (DWBA) to the acoustic scattering by weakly scattering marine organisms have shown much progress [Chu et al., J. Acoust. Soc. Am. 93, 2985–2988 (1993); Stanton and Chu, ibid. 103, 236–253 (1998)]. Although the applicability of the model has been examined for simple objects such as spheres, cylinders, and prolate spheroids, the validity of applying the DWBA model to the acoustic scattering by marine organisms with more complicated shapes has not been studied. To evaluate the performance of the DWBA model, a systematic study of the DWBA will be presented. An in-depth study of the 1D problem reveals the fundamental difference between the exact and DWBA solutions and helps us better understand the scattering problem. By studying the internal field of 1D, 2D, and 3D problems for a more general bistatic geometry and evaluating the volume integral derived from the exact Helmholtz–Kirchhoff integral equation, we are able to estimate the validity of the DWBA model in describing the acoustic scattering by arbitrarily shaped marine organisms. [Work supported by NSF and ONR.]