Kirchhoff Integral Method Research Articles

Prediction method for tire air-pumping noise using a hybrid technique

Air-pumping noise from a car tire is investigated with a hybrid technique composed of three stages: (1) small-scale air-pumping noise generation process is modeled as a piston-like movement of the base-side of the tire groove and then numerically simulated; (2) the flow properties in the tire groove are used as air-pumping sources and noise propagation is simulated with emphasis placed on scattering process with full tire/road geometry; (3) the far-field acoustic pressure is predicted from a Kirchhoff integral method by using unsteady flow data in space and time which is provided by the computational fluid dynamics (CFD) calculation of full tire-road domain. The comparison of predicted results shows that the nonlinearity of the air-pumping noise generation mechanism affects not only noise characteristics in frequency domain but also in the directivity pattern. It seems that this approach can overcome the weakness of the acoustic monopole theory which stems from the usual assumption of a small amplitude acoustic wave equation while using nonlinear governing equation for the CFD calculation. Furthermore, through the use of a computational domain which covers tire and road surface, the geometric effects on air-pumping noise generation and propagation are taken into account in the source modeling process.

자동차 타이어의 Air-Pumping소음 예측을 위한 수치적 기법

The monopole theory has long been used to model air-pumped effect from the elastic cavities in car tire. This approach models the change of an air as a Piston moving backward and forward on a spring and equates local air movements exactly with the volume changes of the system. Thus, the monopole theory has a restricted domain of applicability due to the usual assumption of a small amplitude acoustic wave equation and acoustic monopole theory This paper describes an approach to predict the air-pumping noise of a car tyre with CFD/Kirchhoff integral method. The tyre groove is simply modeled as piston-cavity-sliding door geometry and with the aid of CFD technique flow properties in the groove of rolling car tyre are acquired.`rhese unsteady flow data are used as a air-pumping source in the next CFD calculation of full tyre-road geometry. Acoustic far field is predicted from Kirchhoff integral method by using unsteady flow data in space and time which is provided by the CFD calculation of full tyre-road domain. This approach can cover the non-linearity of acoustic monopole theory with the aid of Non-linear governing equation in CFD calculation. The method proposed in this paper is applied to the prediction of air-pumping noise of simply modeled car tyre and through the predicted results, the influence of nonlinear effect on air-pumping noise propagation is investigated.

Digital holography particle image velocimetry for the measurement of 3D t-3c flows

In this paper a digital in-line holographic recording and reconstruction system was set up and used in the particle image velocimetry for the 3D t-3c (the three-component (3c), velocity vector field measurements in a three-dimensional (3D), space field with time history ( t)) flow measurements that made up of the new full-flow field experimental technique—digital holographic particle image velocimetry (DHPIV). The traditional holographic film was replaced by a CCD chip that records instantaneously the interference fringes directly without the darkroom processing, and the virtual image slices in different positions were reconstructed by computation using Fresnel–Kirchhoff integral method from the digital holographic image. Also a complex field signal filter (analyzing image calculated by its intensity and phase from real and image parts in fast fourier transform (FFT)) was applied in image reconstruction to achieve the thin focus depth of image field that has a strong effect with the vertical velocity component resolution. Using the frame-straddle CCD device techniques, the 3c velocity vector was computed by 3D cross-correlation through space interrogation block matching through the reconstructed image slices with the digital complex field signal filter. Then the 3D-3c-velocity field (about 20 000 vectors), 3D-streamline and 3D-vorticiry fields, and the time evolution movies (30 field/s) for the 3D t-3c flows were displayed by the experimental measurement using this DHPIV method and techniques.

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.

Seismic imaging and velocity analysis for an Alberta Foothills seismic survey

Seismic imaging of complex structures from the western Canadian Foothills can be achieved by applying the closely coupled processes of velocity analysis and depth migration. For the purposes of defining these structures in the Shaw Basing area of western Alberta, we performed a series of tests on both synthetic and real data to find optimum imaging procedures for handling large topographic relief, near‐surface velocity variations, and the complex structural geology of steeply dipping formations. To better understand the seismic processing problems, we constructed a typical foothills geological model that included thrust faults and duplex structures, computed the model responses, and then compared the performance of different migration algorithms, including the explicit finite difference (f-x) and Kirchhoff integral methods. When the correct velocity was used in the migration tests, the f-x method was the most effective in migration from topography. In cases where the velocity model was not assumed known, we determined a macrovelocity model by performing migration/velocity analysis by using smiles and frowns in common image gathers and by using depth‐focusing analysis. In applying depth imaging to the seismic survey from the Shaw Basing area, we found that imaging problems were caused partly by near‐surface velocity problems, which were not anticipated in the modeling study. Several comparisons of different migration approaches for these data indicated that prestack depth migration from topography provided the best imaging results when near‐surface velocity information was incorporated. Through iterative and interpretive migration/velocity analysis, we built a macrovelocity model for the final prestack depth migration.

Dimensionality reduction in the numerical evaluation of electro-magnetic scattering: from the Kirchhoff's integral method to a zero-dimensional approach

A new, simple estimate of the diffraction coefficients, to be used in a ray tracing calculation, is derived applying the stationary phase technique to the boundary wave integral. This approach is shown to be the last step (0-dimensions) of a dimensionality reduction procedure which begins with the Kirchhoff's integral method (2-dimensions) and proceeds with the boundary wave approach (1-dimension). While the reduction – from three to two – of the dimensionality, requires the physical optics approximation and the reduction from two to one holds only for perfectly conducting solids with flat faces and spherical or plane incident waves, no additional price is paid to pass from one to zero dimensions. The Zero-Dimensional (ZD) approach is validated through numerical comparison with higher dimensionality methods and analytical (exact) results in a number of test cases.

An equivalent field principle and the multiple Kirchhoff integral method for UV blazed gratings

In order to solve the surface-shadowing problems that take place in the scattering of electromagnetic waves from rough or period surfaces, an equivalent field principle for plane waves scattering from plane surfaces is given in this paper. Supported by this principle, the reciprocity theorem, and the boundary conditions that relate the scattering field to the incident field through the Fresnel coefficient, a multiple Kirchhoff integral method for metal blazed gratings is developed. Under the conditions of Littrow mounting or fixed angular deviation mounting, illuminated by plane waves, the −1 order's and/or the total efficiencies of perfectly conducting or UV aluminum blazed gratings are calculated and the stability and the self-consistency of this method are discussed. Under near-Littrow mounting (the fixed angular deviation mounting but where the angular deviation is very small) conditions, the calculations are compared with the experimental results.

Wave-vector-time domain and Kirchhoff integral equation methods to determine the transient acoustic radiation loading on circular cylinders

Two new methods are presented to determine the transient acoustic radiation loading on circular cylinders. The acoustic modal forces are expressed as a sum of the convolution of the modal radiation impulse responses and the modal velocities. In the wave-vector-time domain method, the modal radiation impulse responses are obtained by using the radiation impulse responses of an infinite cylinder and the modes in wave-vector space. The approach can thus be easily used for parametric studies on the effects of boundary conditions and normalized lengths on the modal radiation impulse responses. In the Kirchhoff integral equation method, space and time are discretized. The pressure and acceleration in each element are then expressed in the form of a Taylor-series expansion about points at which the pressure is obtained by marching forward in time. The radiation impulse responses of both infinite and simply supported finite cylinders are obtained by the two methods and compared.

Seismic imaging of deep crust

We introduce here an integral two‐dimensional (2-D) scheme for the processing of deep crustal reflection profiles. This approach, in which migration occurs before stacking, is tailored to the unique character of the data in which nonvertically propagating energy is as important as vertically propagating energy. Since reflector depths range beyond 30 km, the horizontal displacement of reflections which occurs in migration can be as large as reflector depths; under these circumstances, the common‐midpoint (CMP) stack is inadequate. In our scheme, each common‐source trace gather is transformed into a set of traces (beams) corresponding to set of different incidence angles. A correction for wavefront curvature similar to the normal moveout (NMO) correction yields traces (focused beams) which are focused at image points along the direction of arrival. While the method is equivalent to the Kirchhoff integral migration method, and therefore to any complete continuation method, it gives rise to an intermediate data set which is characterized by the direction of arrival of the upward propagating energy. By a geometrical transformation of the beams and summation, we may synthesize images composed of a specified range of Fourier spatial components. Geologic examples suggest that complex structures in the basement may be most easily characterized by their local direction of layering, a quantity we may determine by this approach. Noise‐free synthetic data examples illustrate the limits of horizontal and vertical resolving power at mid‐crustal depths for any imaging method. Velocity determination is difficult at these depths due to the small NMO and may be possible only by evaluating the effects of velocity models on the imaged data. Examples of the imaged section from the COCORP test profile in Hardeman County, Texas, show a combination of horizontally continuous reflectors and an irregular pattern of scatterers with locally horizontal layering.

Scattering function for underwater acoustic scatter channels with several bounces

The scattering function, or range-Doppler plot for an underwater acoustic channel involving a single surface bounce was derived in a recent paper [J. F. McDonald and F. B. Tuteur, J. Acoust. Soc. Am. 57, 1025–1029 (19xx)] and was shown to have a characteristic horse-shoe shape when the surface waves have a well-defined direction. This shape indicates strong correlation between Doppler shifts and range delays. In practice underwater acoustic channels often involve several reflections, some from the surface and some from the bottom. A calculation of the scattering function for such channels has been performed using an extension of the Kirchhoff integral method described in [F. B. Tuteur, J. Acoust. Soc. Am. 60 (October 1976)]. For a channel involving a single bottom bounce and surface bounce the reverberation time and the Doppler spread are found to be weighted averages of the respective quantities for single bounces, with weighting factors that depend on the geometry and on the statistics of both surface and bottom. The shape of the scattering function can therefore be expected to be qualitatively similar to the one for the single surface bounce.