Half-space Structure Research Articles

An Improved Boundary Element Method for Predicting Half-Space Scattered Noise Combined with Permeable Boundaries

The boundary element method is widely used in practical engineering problems, especially in the field of acoustics. For flow-induced noise, the main target of acoustic calculations is to solve the wave equation with the flow field information. However, the sound field distribution of noncompact structures in half-space is especially complex because of the strong scattering effect, while the object surface boundary integration often brings a large workload and generates numerical singularities. In this paper, an improved boundary element method for predicting the aeroacoustic noise of noncompact structures is proposed, which can consider the characteristic distribution of sound field induced by complex structures in half-space. The smooth permeable boundary surrounding the object is used as the integration boundary, while the scattering effect of the ground boundary is investigated by combining the mirror Green’s function method, and the numerical prediction of aeroacoustic noise is carried out for the dipole source and NACA0012 airfoil in half-space. Numerical results show that the far-field noise obtained by using the permeable surface is consistent with that obtained by integrating the direct object boundary under the influence of ground boundary scattering. The mirror image Green’s function method is able to finely capture the ground scattering effect, which has a significant effect on the sound field as the frequency increases.

Fast and Accurate Transient Analysis of Thin-Wire Structure in Half-Space With PEEC Method

This article presents an efficient full-wave partial element equivalent circuit (PEEC) method of thin-wire structure in half-space for lightning transient analysis. In the framework of the PEEC formulation for thin-wire conductors, the partial elements are computed by double-folded line integrals. In layered media, the interaction integrals involve dyadic Green's functions, which contain a closed-form part and a Sommerfeld-integral part. The direct numerical computation of these integrals is very challenging and time-consuming, which hinders the practical application of the full-wave method. However, the efficient and accurate evaluation of partial elements has not been investigated in a comprehensive way so far. Accordingly, this article presents an efficient method for the evaluation of the partial elements between elementary segments in half space. The main novelty is that the interaction integrals are computed through a combination of the Taylor series expansion of the full-wave free space Green's function and the Sommerfeld-integral adaptive interpolation technique, which significantly reduces the computation time. The new method is systematically validated on examples from finite-difference time-domain (FDTD) method and method of moments and experimental results. The proposed method is validated with the NEC code in the frequency domain and the FDTD in the time domain.

Direct Estimation of Explosion Pn Green’s Functions Applied to Teleseismic P-wave Intercorrelations for North Korean Nuclear Tests

Abstract Seismic yields and explosion source time functions for the six declared underground nuclear tests at the North Korean test site are estimated using teleseismic waveform equalization (intercorrelation) incorporating improved source structure inferred from regional distance Pn phases. Explosion source time functions estimated by previous modeling efforts are deconvolved from the broadband Pn observations to extract direct estimates of the Pn Green’s functions. Very consistent signals are found for station MDJ (to the north) for all but the 2006 event, and a 1D layered elastic model is determined that matches the corresponding pPn signals for the larger events. The much simpler Pn Green’s functions found for INCN (to the south) are well modeled using a half-space structure. The 1D structures are incorporated into intercorrelations for teleseismic P observations. Very similar results are found using the layered MDJ model for all azimuths, or when azimuthally varying the source velocity model to have a half-space structure for stations to the south, and/or using a half-space structure for the 2006 event at all azimuths. The relative yield estimates from intercorrelation are thus stable with respect to specific 1D source structures; the full effects of 3D elastic and nonelastic structures are yet to be considered.

Static deformation of a multilayered magneto-electro-elastic half-space structures due to vertical inner loading

The static deformation in a multilayered magneto-electro-elastic half-space under vertical inner loading is calculated using a vector function system approach and a stiffness matrix method. Firstly, the displacement, stress, and inner loading are expanded using the vector function system, and the N-type and L&M-type problems related to the expansion coefficient are constructed. Secondly, the stable stiffness matrix method is used to solve the expansion coefficients of the L&M-type problem. After introducing the boundary condition and the discontinuity of the stress caused by inner loading, the displacement and stress are calculated through adaptive Gaussian quadrature. Finally, the numerical examples considering the circular load and point load are designed and analyzed, respectively.

Green’s function technique to study the influence of heterogeneity on horizontally polarised shear-wave propagation due to a line source in composite layered structure

The present paper investigates Green’s function technique to study the impact of the non-linear heterogeneity on the propagation of the horizontally polarised shear (SH) wave in a composite layered structure due to the line source. The composite structure is comprised of two distinct isotropic homogeneous elastic layers of finite width lying over an isotropic heterogeneous semi-infinite elastic medium. A general quadratic heterogeneity (including the linear heterogeneity term) in the rigidity of the semi-infinite medium has been taken into account. An efficient analytical treatment involving Green’s function technique along with the application of Fourier transformation has been employed to establish the closed form of the dispersion equation for the propagating wave. As a special case of the problem, the closed form of the dispersion equation has been found in well agreement with the pre-established and classical results. The influence of linear and non-linear heterogeneities on the phase velocity of the horizontally polarised shear-wave has been analysed comparatively for numerous cases associated with the considered model and the corresponding single layer half-space model. It is reported that the phase velocity of the horizontally polarised shear-wave is more pronounced in the case of the corresponding single layer half-space structure as compared to the considered structure. Also, the impact of the general quadratic heterogeneity (supported by the linear heterogeneity) enhances the phase velocity most favourably as compared to the case of the simple quadratic heterogeneity, linear heterogeneity and homogeneity in the semi-infinite medium.

Dispersion and attenuation of torsional wave in a viscoelastic layer bonded between a layer and a half-space of dry sandy media

Propagation of a torsional wave in a doubly-layered half-space structure of an initially stressed heterogeneous viscoelastic layer sandwiched between a layer and a half-space of heterogeneous dry sandy media is studied. A closed form complex expression for the velocity profile is obtained under effective boundary conditions. The real part of the complex expression provides a dispersion equation, and the imaginary part yields a damping equation. The derived dispersion and damped equations are in well agreement with the classical Love wave condition. In addition, to study the effect of the dissipation factor, the attenuation coefficient, the sandy parameters, the initial stress, the heterogeneity parameters, and the thickness ratio parameter, some noteworthy contemplations are made by numerical calculations and graphical visuals. The results of this paper may present a deeper insight into the behaviour of propagation phenomena in heterogeneous viscoelastic and heterogeneous dry sandy materials that can provide a theoretical guide for the design and optimization in the field of earthquake engineering. The study also reveals that the presence of a damping part due to viscoelasticity affects the torsional wave propagation significantly.

Static response of a layered magneto-electro-elastic half-space structure under circular surface loading

A cylindrical system of vector functions, the stiffness matrix method and the corresponding recursive algorithm are proposed to investigate the static response of transversely isotropic, layered magneto-electro-elastic (MEE) structures over a homogeneous half-space substrate subjected to circular surface loading. In terms of the system of vector functions, we expand the extended displacements and stresses, and deduce two sets of ordinary differential equations, which are related to the expansion coefficients. The solution to one of the two sets of these ordinary differential equations can be evaluated by using the stiffness matrix method and the corresponding recursive algorithm. These expansion coefficients are then integrated by adaptive Gaussian quadrature to obtain the displacements and stresses in the physical domain. Two types of surface loads, mechanical pressure and electric loading, are considered in the numerical examples. The calculated results show that the proposed technique is stable and effective in analyzing the layered half-space MEE structures under surface loading.

IE-DDM with a novel multiple-grid p-FFT for analyzing multiscale structures in half space

We present an integral equation domain decomposition method accelerated by a novel multiple-grid precorrected fast Fourier transform (MG-p-FFT) for the efficient analysis of multiscale structures in a half space. Based on the philosophy of DDM, the original computational domain is partitioned into several non-overlapping sub-domains. By employing non-conformal discretizations to each domain boundaries, combined field integral equation with half-space dyadic Green’s function is proposed for each individual sub-domain. Subsequently, the MG-p-FFT with auxiliary Cartesian grids with different size, order, location, and spacing, is adopted in each sub-domain independently to account for the self-interactions. Here, the proposed MG-p-FFT scheme outperforms the existing single-grid p-FFT scheme for multiscale problems by reducing the computational time and memory consumption. The proposed method can also be viewed as an effective preconditioning scheme for multiscale problems in a half space. The validity and advantages of the proposed method are illustrated by several representative numerical examples.

An Improved Formula of Fundamental Resonance Frequency of a Layered Half-Space Model Used in H/V Ratio Technique

The resonance frequency of the transmission response in layered half-space model is important in the study of site effect because it is the frequency where the shake-ability of the ground is enhanced significantly. In practice, it is often determined by the H/V ratio technique in which the peak frequency of recorded H/V spectral ratio is interpreted as the resonance frequency. Despite of its importance, there has not been any formula of the resonance frequency of the layered half-space structure. In this paper, a simple approximate formula of the fundamental resonance frequency is presented after an exact formula in explicit form of the response function of vertically SH incident wave is obtained. The formula is in similar form with the one used in H/V ratio technique but it reflects several major effects of the model to the resonance frequency such as the arrangement of layers, the impedance contrast between layers and the half-space. Therefore, it could be considered as an improved formula used in H/V ratio technique. The formula also reflects the consistency between two approaches of the H/V ratio technique based on SH body waves or Rayleigh surface waves on the peak frequency under high impedance contrast condition. This formula is in explicit form and, therefore, may be used in the direct and inverse problem efficiently. A numerical illustration of the improved formula for an actual layered half-space model already investigated by H/V ratio technique is presented to demonstrate its new features and its improvement to the currently used formula.

Propagation characteristic of laser-generated visco-elastic Rayleigh-like waves in stratified half-space

This paper reports on a study of the propagation characteristics of visco-elastic, Rayleigh-like waves in stratified half-space structures. Beginning with the Kelvin model, the characterization equation and the normal displacement of visco-elastic Rayleigh waves in stratified half-space structures are derived and the influence of the visco-elastic modulus on dispersion and attenuation is discussed. Theoretical calculations show that the attenuation-frequency curves perfectly match the phase-frequency curves. The effect of visco-elasticity on the attenuation of the Rayleigh-like wave is larger than its effect on dispersion. For “weak viscosity,” the attenuation is directly proportional to the viscosity modulus and the shear viscosity has a greater impact on the dispersion curves than does the bulk viscosity. The transient response of a visco-elastic Rayleigh wave is also simulated by means of Laplace and Hankel inversion transforms. The results are in good agreement with the theoretic predictions. It is believed the paper’s results and conclusions will provide insights and guidance for estimating visco-elastic parameters and for assessing adhesive quality.

Characteristics of Laser-Generated Visco-Elastic Rayleigh Waves

This paper reports on a study of the propagation characteristics of visco-elastic, Rayleigh waves induced by laser ultrasonics in half space structures. Beginning with the Kelvin model, the characterization equation and the normal displacement of visco-elastic Rayleigh waves in are derived and the influence of the visco-elastic modulus on dispersion and attenuation are discussed. The transient response of a visco-elastic Rayleigh wave is also simulated by means of Laplace and Hankel inversion transforms. The papers results and conclusions will provide insights and guidance for estimating visco-elastic parameters

Dynamic Response of Complex Structure in Half Space

The problems of SH-wave scattering caused by a subsurface circular lining structure and a beeline crack with arbitrary length at an arbitrary position were studied by using the methods of Green's function, complex variables and multi-polar coordinates. A adaptive Green's function, an essential solution to the displacement field for the elastic space possessing circular lining structure while bearing out-plane harmonic lining loads at an arbitrary point, was constructed firstly, and then a crack was created using “crack-division”. Thus the expressions of displacement and stress were established while the crack and the inclusion both existed. Finally, we give some numerical examples to discuss the variety of the horizontal surface displacement in the case of different parameters.

Perturbation kernels for generalized seismological data functionals (GSDF)

In seismic waveform analysis and inversion, data functionals can be used to quantify the misfit between observed and model-predicted (synthetic) seismograms. The generalized seismological data functionals (GSDF) of Gee & Jordan quantify waveform differences using frequency-dependent phase-delay times and amplitude-reduction times measured on time-localized arrivals and have been successfully applied to tomographic inversions at different geographic scales as well as to inversions for earthquake source parameters. The seismogram perturbation kernel is defined as the Fréchet kernel of the data functional with respect to the seismic waveform from which the data functional is derived. The data sensitivity kernel, which is the Fréchet kernel of the data functional with respect to structural model parameters, can be obtained by composing the seismogram perturbation kernel with the Born kernel through the chain rule. In this paper, we extend GSDF analysis to broad-band waveforms by removing constraints on two control parameters defined in Gee & Jordan and derive the seismogram perturbation kernels for the modified GSDF analysis. The modifications given in this paper are consistent with the original GSDF theory in Gee & Jordan around the centre frequency and improve the stability of GSDF analysis towards the edges of the passband. We also present numerical examples of perturbation kernels for the modified GSDF analysis and their data sensitivity kernels using a homogenous half-space structure model and a complex 3-D structure model.

Some useful approximations to generalized ray theory for regional distance seismograms

Seismograms recorded at regional distances (2°–12°) are quite complicated due to the waveguide nature of the crust. Generalized ray theory can be used to model the body waves in this distance range but a very large number of rays is required. Here I present a series of approximations to streamline generalized ray theory for the waveguide problem. If a layer over a half-space is used for the structure, then the de Hoop contour for a given ray is most strongly dependent on the fastest velocity of any leg of the ray. This results in analytic approximations to locate the contour. Each ray has two body wave arrivals (a headwave and a reflected arrival) so the displacement response of the ray need only be evaluated at a few points in time about the two arrival times and interpolated in between. A change in structure (increasing crustal thickness or Pn velocity) most strongly affects the relative timing of the headwave and the reflected arrival, so it is possible to ‘stretch’ or ‘squeeze’ the waveform of a representative model to simulate a whole suite of models. Also discussed is the applicability of a single layer over a half-space structure for modelling the observed regional distance waveforms for shallow earthquakes. At periods greater than a few seconds crustal layering can be replaced by a single layer having the appropriate average velocities. Lateral variations in crustal thickness with scale lengths of less than about 100 km can also be modelled with a simple horizontal layer of appropriate average thickness.

Scattering of SH-Waves and Dynamic Stress Intensity Factor by an Interacting Mode III Crack and a Lining Structure in Half Space

Scattering of SH wave by an elastic half space with a lining structure and a crack in any position and direction is studied with Green’s function, complex function and multi-polar coordinate method. First, a suitable Green’s function is constructed, which is a fundamental solution to the displacement field for the elastic space possessing circular lining structure while bearing out-of-plane harmonic line source load at arbitrary point. Then a crack in any position and direction is constructed by means of crack-division in half space. Finally the displacement field and stress field are established in the case of coexistence of circular lining structure and crack, and the expression of dynamic stress intensity factor (DSIF) at the tip of crack is given. According to numerical examples, the influences of different parameters on DSIF are discussed.

Reconstruction of vibroacoustic fields in half-space by using hybrid near-field acoustical holography

In this paper we examine the accuracy and efficiency of reconstructing the vibroacoustic quantities generated by a vibrating structure in half-space by using hybrid near-field acoustic holography (NAH) and modified Helmholtz equation least squares (HELS) formulations. In hybrid NAH, we combine modified HELS with an inverse boundary element method (IBEM) to reconstruct a vibroacoustic field. The main advantage of this approach is that the majority of the input data can be regenerated but not measured, thus the efficiency is greatly enhanced. In modified HELS, we expand the field acoustic pressure in terms of outgoing and incoming spherical waves and specify the corresponding expansion coefficients by solving a system of equations obtained by matching the assumed-form solution to the measured acoustic pressure. Here the system of equations is ill conditioned and Tikhonov regularization is implemented through singular value decomposition (SVD) and the generalized cross-validation (GCV) method. Numerical examples of a dilating and oscillating spheres and finite cylinder are demonstrated. Test results show that hybrid NAH can yield a more accurate reconstruction than does a modified HELS, but a modified HELS is more efficient than is hybrid NAH [Work supported by NSF].

Reconstruction of acoustic radiation from vibrating objects in a half space using hybrid nearfield acoustical holography

Hybrid nearfield acoustical holography (NAH) [Wu, J. Acoust. Soc. Am. 113, 2252 (2003)] is utilized to reconstruct the vibro-acoustic fields generated by vibrating structures in half space. These hybrid formulations are derived from a modified Helmholtz equation least-squares method, where the field acoustic pressure is expanded in terms of both outgoing and incoming spherical waves, and the Helmholtz integral formulations implemented by the boundary element method (BEM). To overcome the ill-conditioning difficulties in the resultant matrix equations, regularization techniques such as the method of Tikhonov and generalized cross validation are employed. The main advantage of this hybrid approach is that satisfactory reconstruction can be obtained with relatively few acoustic pressure measurements taken over a conformal surface around the source. This is because a large portion of the input data is regenerated by the modified HELS, but not actually measured. Hence, the efficiency of reconstruction is enhanced. If the same number of measurements is used in an inverse BEM code, the reconstructed vibro-acoustic quantities on the source surface may be distorted. Numerical examples of a dilating and an oscillating sphere and a cylinder with spherical endcaps in half space are demonstrated. [Work supported by NSF.]

A generalized source method for wave propagation

A generalized source method is presented for the resolution of the problem of monochromatic wave propagation in non-homogeneous, isotropic structures. When implemented in the form of an iterative technique, it is demonstrated to give the exact analytical solution of the known problem of propagation in a two half-space structure. When implemented in the form of an integral expression, it is shown to give an exact solution under a normalized numerical convergence criterion. It thus represents a new powerful method for electromagnetic wave propagation in arbitrary structures, and also for the assessment of other resolution techniques proposed so far in optical wave propagation, diffraction and scattering.