Half-space Green's Function Research Articles

Efficient preconditioner for the finite-element boundary integral analysis of electromagnetic scattering in the half space

In this study, the finite-element boundary integral (FE-BI) method is applied to analyse the electromagnetic scattering from arbitrary objects in the half-space environment. Combined with the real image method to approximate the half-space Green's function, the half-space FE-BI can be extended for electrically large problems by the fast multipole method. An efficient preconditioner is constructed with both the finite-element matrix and the near-field part of the boundary integral equation operator for the FE-BI analysis. The numerical examples demonstrate that this preconditioner is efficient for electromagnetic scattering in the half space.

Electromagnetic Scattering From Objects Above a Rough Surface Using the Method of Moments With Half-Space Green's Function

An efficient approach is proposed to analyze the electromagnetic scattering from objects above a 2-D perfectly electric conducting rough surface. A half-space Green's function with the rough-surface interface is first derived from the Kirchhoff approximation (KA). The method of moments is then applied to analyze the scattering problem of 3-D arbitrarily shaped objects above the rough surface. Since only the objects need to be discretized, the computational time and memory requirement are greatly reduced. The radar cross sections of typical objects above the rough surface have been computed using the proposed method. Numerical results show that the proposed method has good accuracy in the valid range of KA.

Adaptive fast multipole boundary element method for three-dimensional half-space acoustic wave problems

A new adaptive fast multipole boundary element method (BEM) for solving 3-D half-space acoustic wave problems is presented in this paper. The half-space Green's function is employed explicitly in the boundary integral equation (BIE) formulation so that a tree structure of the boundary elements only for the boundaries of the real domain need to be applied, instead of using a tree structure that contains both the real domain and its mirror image. This procedure simplifies the implementation of the adaptive fast multipole BEM and reduces the CPU time and memory storage by about a half for large-scale half-space problems. An improved adaptive fast multipole BEM is presented for the half-space acoustic wave problems, based on the one developed recently for the full-space problems. This new fast multipole BEM is validated using several simple half-space models first, and then applied to model 3-D sound barriers and a large-scale windmill model with five turbines. The largest BEM model with 557470 elements was solved in about an hour on a desktop PC. The accuracy and efficiency of the BEM results clearly show the potential of the adaptive fast multipole BEM for solving large-scale half-space acoustic wave problems that are of practical significance.

High-frequency method for scattering from coated targets with electrically large size in half space

The high-frequency method for solving the scattering from coated targets of electrically large size in half space is presented. In order to consider the scattering fields of the targets in half space, the half-space physical-optic (PO) method is deduced by introducing the half-space Green's function into the conventional PO method. Combined with the graphical-electromagnetic computing method to read the geometric information of all visible facets, the impedance boundary conditions are utilised to account for the targets-covered with radar-absorbing material in half space. The numerical results show that this method is efficient and accurate.

Forward modelling the multiply scattered 2.5-D teleseismicPwaves accelerated by the multilevel fast multipole method

SUMMARY The present study introduces an algorithm to compute the forward modelling of multiple scattering. The propagating medium is defined, in general, by 3-D heterogeneities embedded in a 1-D layered structure, assumed as the reference structure. The 3-D formulation is consequently modified to suit 2.5-D scattering modelling. The basic formulation relies on the Green's function method and employs (1) the reflectivity method (RM) to acquire the 1-D multilayered, half-space Green's function, (2) the multilevel fast multipole method (MLFMM) to gain speed and (3) the generalized minimal residual method (GMRES) to iteratively solve the linear system in a memory efficient manner. The heterogeneities are modelled as point scatterers, creating a secondary wavefield that superimposes on the waves in the reference structure. The algorithm is efficient since the CPU time increases logarithmically with the size of the model. The test cases included various anomalous lithosphere structures, ranging from a simple point scatterer to a suture zone.

High-frequency analysis of scattering from coated targets with electrically large size in half space

The high-frequency method for solving the scattering of electrically large coated targets in half space is presented in this paper. In order to consider the influence of environment, the half-space physico-optical integral equation is deduced by introducing the half-space Green's function into the physico-optical method. Combined with the graphical-electromagnetic computing method, the shadow regions are eliminated quickly and the geometry information is attained by reading the color and depth of each pixel. Meanwhile, we use the impedance boundary conditions to account for the target covered RAM in half space, and the radar cross section of coated targets can be exactly calculated in half space. The numerical results show that this method is efficient and accurate.

Convergence and Numerics of a Multisection Method for Scattering by Three-Dimensional Rough Surfaces

We introduce a novel multisection method for the solution of integral equations on unbounded domains. The method is applied to the rough surface scattering problem in three dimensions, in particular to a Brakhage-Werner-type integral equation for acoustic scattering by an unbounded rough surface with Dirichlet boundary condition, where the fundamental solution is replaced by some appropriate half-space Green's function. The basic idea of the multisection method is to solve an integral equation $A\varphi = f$ by approximately solving the equation $P_{\varrho}A P_{\tau}\varphi = P_{\varrho} f$ for some positive constants $\varrho, \tau$. Here $P_{\varrho}$ is a projection operator that truncates a function to a ball with radius $\varrho>0$. For a very general class of operators $A$, for which the Brakhage-Werner equation from acoustic scattering is a particular example, we will show existence of approximate solutions to the multisection equation and show that approximate solutions to the multisection equation approximate the true solution $\varphi_0$ of the operator equation $A\varphi = f$. Finally, we describe a numerical implementation of the multisection algorithm and provide numerical examples for the case of rough surface scattering in three dimensions.

HIGH-FREQUENCY METHOD ANALYSIS ON SCATTERING FROM HOMOGENOUS DIELECTRIC OBJECTS WITH ELECTRICALLY LARGE SIZE IN HALF SPACE

The high-frequency method for solving the scattering from homogeneous dielectric objects with electrically large size in half space is presented in this paper. In order to consider the scattering fields of the targets in half space, the half-space physical optics method is deduced by introducing the half-space Green's function into the conventional physical optics method (PO). Combined with the graphical-electromagnetic computing method to read the geometry information of all visible facets, the equivalent currents and the reflection coefficients are utilized to account of the homogenous dielectric objects with half-space physical optics method in half space. The numerical results show that this method is efficient and accurate.

Interaction of Electromagnetic Waves With 3-D Arbitrarily Shaped Homogeneous Chiral Targets in the Presence of a Lossy Half Space

The interaction of electromagnetic waves with an arbitrarily shaped three-dimensional (3-D) homogeneous chiral object located above a lossy half space is investigated using the method of moments (MoM) via the coupled mixed potential integral equations (MPIEs). Based on the surface equivalence principle, the equivalent surface electric and magnetic currents are used to replace the homogeneous chiral target in the presence of the half space. Two coupled MPIEs are developed for the unknown equivalent surface electric and magnetic currents by utilizing the continuity condition of the tangential total electric and magnetic field components on the chiral body's surface. The well-known Galerkin procedure with Rao-Wilton-Glisson (RWG) basis functions is applied to solve this problem. The spatial domain half-space Green's functions are obtained from the corresponding spectral domain Green's functions via the discrete complex image method (DCIM) combined with the generalized-pencil of function (GPOF) technique. The reciprocity theorem is employed to calculate the far-zone scattered field. Numerical results are presented for characterizing electromagnetic scattering by a 3-D arbitrarily shaped homogenous chiral object located above a lossy half space so as to demonstrate the accuracy and efficiency of the proposed technique.

Using Josephson junctions to determine the pairing state of superconductors without crystal inversion symmetry

Theoretical studies of a planar tunnel junction between two superconductors with antisymmetric spin-orbit coupling are presented. The half-space Green's function for such a superconductor is determined. This is then used to derive expressions for the dissipative current and the Josephson current of the junction. Numerical results are presented in the case of the Rashba spin-orbit coupling, relevant to the much studied compound $\mathrm{Ce}{\mathrm{Pt}}_{3}\mathrm{Si}$. Current-voltage diagrams, differential conductance and the critical Josephson current are presented for different crystallographic orientations and different weights of singlet and triplet components of the pairing state. The main conclusion is that Josephson junctions with different crystallographic orientations may provide a direct connection between unconventional pairing in superconductors of this kind and the absence of inversion symmetry in the crystal.

Scale Model Experimental Validation and Calibration of the Half-Space Green's Function Born Approximation Model Applied to Cross-Well Radar Sensing

Efficient forward models that describe the physical nature of the geophysical problem are desired for subsurface sensing and reconstruction of a contrasting contaminant pool volume. An analytical model to approximate sensing with radar is developed and implemented in the frequency domain in terms of the half-space lossy dyadic Green's function. The Born approximation is employed as a linear forward model, which will eventually be used for tomographic inversion for object detection. The forward model is compared with measurements generated by a cross-well radar (CWR) experiment in a controlled soil test tank using broadband borehole antennas. Soil parameter (dielectric constant and loss tangent) variance with frequency is represented by a quadratic polynomial. Calibration for soil parameters is performed via CWR data using an iterative nonlinear parameterized inversion technique. With the appropriate calibration, good agreement is obtained with wideband experimental measurements for several different borehole antenna placements, confirming the accuracy of the model.

Electromagnetic Scattering Characteristics of Vegetation in a Half‐Space Model

AbstractElectromagnetic scattering characteristics of ground vegetation in a half‐space model are studied. Instead of free‐space Green's functions used in previous literatures, half‐space Green's functions are applied here to solve the scattering fields of a single scatterer. Monte‐Carlo method is employed to build the vegetation over the ground in order to obtain the electromagnetic scattering characteristics of the ground vegetation. Finally, backscattering and bistatic Radar cross sections of the vegetation canopy are calculated. Comparing with the scattering fields obtained by free‐space Green's functions, the scattering fields obtained by half‐space Green's functions are more suitable to describe the electromagnetic scattering characteristics of the ground vegetation.

High-Frequency Method for Scattering From Electrically Large Conductive Targets in Half-Space

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The high-frequency method for solving the scattering of electrically large conductive targets in half-space is presented in this letter. The half-space physical optics method is deduced by introducing the half-space Green's function into the conventional physical optics method. Combined with the graphical-electromagnetic computing method, the radar cross-section of conductive targets can be calculated in half-space. The numerical results show that this method is efficient and accurate. </para>

Sound radiation from a turbulent boundary layer

Sound radiation due to fluctuating viscous wall shear stresses in a plane turbulent boundary layer is investigated by a two-stage procedure using direct numerical simulation (DNS) databases for incompressible turbulent Poiseuille flow in a plane channel, at Reynolds numbers up to Reτ=1440. The power spectral density of radiated pressure and spectra of sound power per unit wall area are calculated in the low Mach number limit by substituting source terms obtained from DNS into a Ffowcs Williams–Hawkings wave equation and using a half-space Green function. The same DNS data are used to predict the spectrum of turbulent boundary layer noise measured in a diffuser downstream of a fully developed channel flow [Greshilov and Mironov, Sov. Phys. Acoust. 29, 275 (1983)]. The measured spectrum is ∼15dB higher at low frequencies, but converges with the prediction at high frequencies.

A well-posed integral equation formulation for three-dimensional rough surface scattering

We consider the problem of scattering of time-harmonic acoustic waves by an unbounded sound-soft rough surface. Recently, a Brakhage–Werner type integral equation formulation of this problem has been proposed, based on an ansatz as a combined single- and double-layer potential, but replacing the usual fundamental solution of the Helmholtz equation with an appropriate half-space Green's function. Moreover, it has been shown in the three-dimensional case that this integral equation is uniquely solvable in the space when the scattering surface does not differ too much from a plane. In this paper, we show that this integral equation is uniquely solvable with no restriction on the surface elevation or slope. Moreover, we construct explicit bounds on the inverse of the associated boundary integral operator, as a function of the wave number, the parameter coupling the single- and double-layer potentials, and the maximum surface slope. These bounds show that the norm of the inverse operator is bounded uniformly in the wave number, κ , for κ &gt;0, if the coupling parameter η is chosen proportional to the wave number. In the case when is a plane, we show that the choice is nearly optimal in terms of minimizing the condition number.

Atomistic scattering close to an interface

Half-space Green functions and T-matrix theory is used to predict that scattering on discrete dopants close to a perfect interface generates strong deviations from scattering in the bulk. Interference between the scattered wave and its reflection from the interface leads to a dipolar scattering rate. By introducing de-coherence the transition to bulk scattering is described quantitatively for strongly screened Coulomb scattering.

Modelling Rupture Dynamics of a Planar Fault in 3-D Half Space by Boundary Integral Equation Method: An Overview

In this article, we first reviewed the method of boundary integral equation (BIEM) for modelling rupture dynamics of a planar fault embedded in a 3-D elastic half space developed recently (ZHANG and CHEN, 2005a,b). By incorporating the half-space Green's function, we successfully extended the BIEM, which is a powerful tool to study earthquake rupture dynamics on complicated fault systems but limited to full-space model to date, to half-space model. In order to effectively compute the singular integrals in the kernels of the fundamental boundary integral equation, we proposed a regularization procedure consisting of the generalized Apsel-Luco correction and the Karami-Derakhshan algorithm to remove all the singularities, and developed an adaptive integration scheme to efficiently deal with those nonsingular while slowly convergent integrals. The new BIEM provides a powerful tool for investigating the physics of earthquake dynamics. We then applied the new BIEM to investigate the influences of geometrical and physical parameters, such as the dip angle (δ) and depth (h) of the fault, radius of the nucleation region (R asp), slip-weakening distance (D c ), and stress inside (T i ) and outside (T e ) the nucleation region, on the dynamic rupture processes on the fault embedded in a 3-D half space, and found that (1) overall pattern of the rupture depends on whether the fault runs up to the free surface or not, especially for strike-slip, (2) although final slip distribution is influenced by the dip angle of the fault, the dip angle plays a less important role in the major feature of the rupture progress, (3) different value of h, δ, R asp, T e , T i and D c may influence the balance of energy and thus the acceleration time of the rupture, but the final rupture speed is not controlled by these parameters.

Acoustic Scattering by Mildly Rough Unbounded Surfaces in Three Dimensions

For a nonlocally perturbed half-space we consider the scattering of time-harmonic acoustic waves. A second kind boundary integral equation formulation is proposed for the sound-soft case, based on a standard ansatz as a combined single- and double-layer potential but replacing the usual fundamental solution of the Helmholtz equation with an appropriate half-space Green's function. Due to the unboundedness of the surface, the integral operators are noncompact. In contrast to the two-dimensional case, the integral operators are also strongly singular, due to the slow decay at infinity of the fundamental solution of the three-dimensional Helmholtz equation. In the case when the surface is sufficiently smooth (Lyapunov) we show that the integral operators are nevertheless bounded as operators on $L^2(\Gamma)$ and on $L^2(\Gamma)\cap BC(\Gamma)$ and that the operators depend continuously in norm on the wave number and on $\Gamma$. We further show that for \emph{mild} roughness, i.e., a surface $\Gamma$ which d...

Fast solution of mixed-potential time-domain integral equations for half-space environments

A fast Fourier transform-accelerated integral-equation based algorithm to efficiently analyze transient scattering from planar perfect electrically conducting objects residing above or inside a potentially lossy dielectric half-space is presented. The algorithm requires O(N/sub t/N/sub s/(logN/sub s/+log/sup 2/N/sub t/)) CPU and O(N/sub t/N/sub s/) memory resources when analyzing electromagnetic wave interactions with uniformly meshed planar structures. Here, N/sub t/ and N/sub s/ are the numbers of simulation time steps and spatial unknowns, respectively. The proposed scheme is therefore far more efficient than classical time-marching solvers, the CPU and memory requirements of which scale as O(N/sub t//sup 2/N/sub s//sup 2/) and O(N/sub t/N/sub s//sup 2/). In the proposed scheme, all pertinent time-domain half-space Green functions are (pre) computed from their frequency-domain counterparts via inverse discrete Fourier transformation. In this process, in-band aliasing is avoided through the application of a smooth and interpolatory window. Numerical results demonstrate the accuracy and efficiency of the proposed algorithm.

The complex equivalent source method for sound propagation over an impedance plane

The sound field caused by a monopole source above an impedance plane can be calculated by using a superposition of equivalent point sources located along a line in the mirror space below the plane. Originally, such an approach for representing the half-space Green's function was described by Sommerfeld at the beginning of the last century, in order to treat half-space problems of heat conduction. However, the representation converges only for masslike impedances and cannot be used for the more important case of reflecting planes with springlike surface impedances. The singular part of the line integral can be transformed into a Hankel function, which shows that surface waves are contained in the whole solution. Unfortunately, this representation suffers from the lack of validity at certain receiver points and from restrictions on wave number and impedance range to ensure the necessary convergence. The main idea of the present method is to use also a superposition of equivalent point sources, but to allow that these sources can be located at complex source points. The corresponding form of the half-space Green's function is suitable for both masslike and springlike surface impedances, and can be used as a cornerstone for a boundary element method.