A new bistatic model for electromagnetic scattering from perfectly conducting random surfaces: numerical evaluation and comparison with SPM

This paper is a companion to our previous contribution deriving a new approximate bistatic model for electromagnetic scattering from perfectly conducting rough surfaces. We evaluate this model numerically and compare it with an 'exact' numerical solution of the scattering problem. This comparison shows good agreement between our approximation and numerical solution for a wide range of incident and scattering angles. However, for horizontal-incident horizontal-scattered polarization (HH-pol), the model exhibits strong deviation from the 'exact' solution for near-grazing scattering angles. The model shows a similar divergence at HH-pol when compared with the small-perturbation method (SPM). The cause of this divergence is explained. During the SPM comparison, we noticed that the integral equation method model also does not reproduce the SPM limit except for forward and backscatter geometries. We propose in this paper a simple modification of our model to ensure agreement with the bistatic SPM approximation when applicable, and show that the modified model also yields close agreement with numerical computations even when the surface roughness does not satisfy the SPM condition.     

A practical second-order electromagnetic model in the quasi-specular regime based on the curvature of a 'good-conducting' scattering surface

This letter presents an approximate second-order electromagnetic model where polarization coefficients are surface dependent up to the curvature order in the quasi-specular regime. The scattering surface is considered 'good-conducting' as opposed to the case for our previous derivation where perfect conductivity was assumed. The model reproduces dynamically, depending on the properties of the scattering surface, the tangent-plane (Kirchhoff) or the first-order small-perturbation (Bragg) limits. The convergence is assumed to be ensured by the surface curvature alone. This second-order model is shown to be consistent with the small-slope approximation of Voronovich (SSA-1+SSA-2) for perfectly conducting surfaces. Our model differs from SSA-1 + SSA-2 in its dielectric expression, to correct for a full convergence toward the tangent-plane limit under the 'good-conducting' approximation. This new second-order formulation is simple because it involves a single integral over the scattering surface and therefore it is suitable for a vast array of analytical and numerical applications in quasi-specular applications.     

Application of the Beilis-Tappert parabolic equation method to sound propagation over irregular terrain

The Beilis-Tappert (1979) parabolic equation method is attractive for irregular terrain because it treats surface variations in terms of a simple multiplicative factor ("phase screen"). However, implementing the exact sloping-surface impedance condition is problematic if one wants the computational efficiency of a Fourier parabolic equation algorithm. This article investigates an approximate flat-ground impedance condition that allows the Beilis-Tappert phase screen method to be used with a Fourier algorithm without any added complications. The exact sloping-surface impedance condition is derived and applied to propagation predictions over hills with maximum slopes from 5° to 22°. The predictions with the exact impedance condition are compared to predictions using the approximate flat-ground impedance condition. It is found that for slopes less than 15°-20°, the flat-ground impedance condition is sufficiently accurate. For slopes greater than approximately 20°, the limiting factor on numerical accuracy is not the flat-ground impedance approximation, but rather the narrow-angle approximation required by the Beilis-Tappert method. Thus, within the 20° limitation and using the flat-ground impedance condition with a Fourier parabolic equation, sound propagation over irregular terrain can be computed simply, efficiently, and accurately.     

A practical second-order electromagnetic model in the quasi-specular regime based on the curvature of a ‘good-conducting’ scattering surface

Abstract

This letter presents an approximate second-order electromagnetic model where polarization coefficients are surface dependent up to the curvature order in the quasi-specular regime. The scattering surface is considered ‘good-conducting’ as opposed to the case for our previous derivation where perfect conductivity was assumed. The model reproduces dynamically, depending on the properties of the scattering surface, the tangent-plane (Kirchhoff) or the first-order small-perturbation (Bragg) limits. The convergence is assumed to be ensured by the surface curvature alone. This second-order model is shown to be consistent with the small-slope approximation of Voronovich (SSA-1+SSA-2) for perfectly conducting surfaces. Our model differs from SSA-1 + SSA-2 in its dielectric expression, to correct for a full convergence toward the tangent-plane limit under the ‘good-conducting’ approximation. This new second-order formulation is simple because it involves a single integral over the scattering surface and therefore it is suitable for a vast array of analytical and numerical applications in quasi-specular applications.     

Statistical characteristics of photon distributions in a semi-infinite turbid medium: Analytical expressions and their application to optical tomography

The paper focuses on the determination of statistical characteristics of photon distributions in a semi-infinite turbid medium, specifically the photon average trajectory and the root-mean-square deviation of photons from the average trajectory, with an approach based on the diffusion approximation to the radiative transfer equation. We show that the Dirichlet and Robin boundary conditions used for this purpose give close results. We derive exact analytical expressions for the case of the Dirichlet boundary condition. To demonstrate the practical value of our results we consider approximate solution of the inverse problem of time-domain diffuse optical tomography with the flat layer transmission geometry. The problem is solved with the method of photon average trajectories which are constructed with analytical expressions derived for a semi-infinite medium.     

MHD Boundary Layer Flow of a Non-Newtonian Fluid on a Moving Surface with a Power-Law Velocity

A theoretical analysis for MHD boundary layer flow on a moving surface with the power-law velocity is presented. An accurate expression of the skin friction coefficient is derived. The analytical approximate solution is obtained by means of Adomian decomposition methods. The reliability and efficiency of the approximate solutions are verified by numerical ones in the literature.     

Time-domain impedance formulation for transmission line matrix modelling of outdoor sound propagation

Outdoor sound propagation modelling has attracted considerable attention in the past and has lead to many analytical and numerical models. More recently, the increase of computational power has led to consider time-domain methods that enable to consider transient phenomena. Among these models, the transmission line matrix (TLM) method has been proposed, but the sound absorption at boundaries appears to be a somewhat underdeveloped aspect of this approach. In this paper, a specific implementation of impedance boundary condition is proposed. The method is based on the approximation of the impedance as a sum of linear systems, which allows the formulation of an equivalent impedance model in the time-domain. The proposed implementation is applied for two common impedance models of porous material. Numerical simulations have been carried out in the case of sound propagation over a flat ground with and without an impedance discontinuity, and for several values of specific airflow resistivity. TLM numerical predictions expressed in terms of excess attenuation relative to free field show a good agreement with analytical solutions.     

High order numerical approximation of minimal surfaces

We present an algorithm for finding high order numerical approximations of minimal surfaces with a fixed boundary. The algorithm employs parametrization by high order polynomials and a linearization of the weak formulation of the Laplace–Beltrami operator to arrive at an iterative procedure to evolve from a given initial surface to the final minimal surface. For the steady state solution we measure the approximation error in a few cases where the exact solution is known. In the framework of parametric interpolation, the choice of interpolation points (mesh nodes) is directly affecting the approximation error, and we discuss how to best update the mesh on the evolutionary surface such that the parametrization remains smooth. In our test cases we may achieve exponential convergence in the approximation of the minimal surface as the polynomial degree increases, but the rate of convergence greatly differs with different choices of mesh update algorithms. The present work is also of relevance to high order numerical approximation of fluid flow problems involving free surfaces.     

Boundary conditions for differential approximations

Low order spherical harmonic (P-N) approximations are applied to a radiative transfer Marshak wave problem. A modified Milne boundary condition is developed for the P-2 approximation, similar to one suggested earlier for the P-1 approximation. Comparison with exact Monte Carlo results suggests that this modified P-2 method may be an accurate and generally applicable differential approximation to the equation of transfer. The Monte Carlo results presented should be useful for testing other approximate formulations of radiative transfer and validating time dependent numerical solution methods for the equation of transfer.     

Exact boundary condition for time-dependent wave equation based on boundary integral

An exact non-reflecting boundary conditions based on a boundary integral equation or a modified Kirchhoff-type formula is derived for exterior three-dimensional wave equations. The Kirchhoff-type non-reflecting boundary condition is originally proposed by L. Ting and M.J. Miksis [J. Acoust. Soc. Am. 80 (1986) 1825] and numerically tested by D. Givoli and D. Cohen [J. Comput. Phys. 117 (1995) 102] for a spherically symmetric problem. The computational advantage of Ting–Miksis boundary condition is that its temporal non-locality is limited to a fixed amount of past information. However, a long-time instability is exhibited in testing numerical solutions by using a standard non-dissipative finite-difference scheme. The main purpose of this work is to present a new exact boundary condition and to eliminate the long-time instability. The proposed exact boundary condition can be considered as a limit case of Ting–Miksis boundary condition when the two artificial boundaries used in their method approach each other. Our boundary condition is actually a boundary integral equation on a single artificial boundary for wave equations, which is to be solved in conjunction with the interior wave equation. The new boundary condition needs only one artificial boundary, which can be of any shape, i.e., sphere, cubic surface, etc. It keeps all merits of the original Kirchhoff boundary condition such as restricting the temporal non-locality, free of numerical evaluation of any special functions and so on. Numerical approximation to the artificial boundary condition on cubic surface is derived and three-dimensional numerical tests are carried out on the cubic computational domain.     

On the numerical characteristics of an inverse solution for three-term radiative transfer

Certain numerical characteristics of an inverse formulation for three-term scattering radiative transfer are investigated. Specifically, approximate solutions to the direct problem are constructed by the F_N and Monte Carlo methods, allowing approximation of the various surface angular moments and related quantities needed for the inverse calculation. Several numerical schemes are employed in order of demonstrate the computational characteristics for some specific phase functions. The numerical results indicate that the single-scatter albedo can be calculated fairly consistently and accurately, but higher order coefficients of the scattering law are more difficult to obtain by this method.     

随机表面散射光场的格林函数法与基尔霍夫近似的比较

从简谐光波满足的亥姆霍兹方程出发，将由格林定理得到的介质分界面上的积分方程转化为以表面上的光波及其导数为未知量的线性方程组，并对其进行数值求解，实现了光场的数值计算. 同时，由透射光场的格林函数积分得出了基尔霍夫近似下光场的表达式. 通过类比推导夫琅和费面上散斑场自相关函数的方法，提出了产生随机表面及其导数的傅里叶变换方法. 在此基础上，对采用基尔霍夫近似进行自仿射分形随机表面的散射光场数值计算的精确程度进行了研究. 发现在随机表面粗糙度比较小时，基尔霍夫近似的精度比较高；在粗糙度相同的情况下，表面的分形 关键词： 格林函数积分 基尔霍夫近似 自仿射分形随机表面     

Low-frequency radio wave propagation in the earth-ionosphere waveguide disturbed by a large-scale three-dimensional irregularity

In this paper, we develop further the analytical and numerical method of solving three-dimensional problems in the theory of radio wave propagation, including three-dimensional local inhomogeneities (ionospheric disturbances or Earth’s surface irregularities). To model the Earth-ionosphere waveguide, we use the surface impedance concept, by which the irregularity extending beyond one waveguide wall has an arbitrary smooth shape, and its surface can be described by the impedance. In the scalar approximation, this problem is reduced to a two-dimensional integral equation for the irregularity surface, which, by asymptotic (kr ≫ 1) integration over the coordinate transverse to the propagation path (with allowance for terms of the order of (kr)⁻¹), is reduced to a one-dimensional integral equation, in which the integration contour is the linear contour of the irregularity. The equation is solved numerically, combining the inversion of a Volterra integral operator and successive approximations. By reducing the computer times, this method enables one to study both small-scale and large-scale irregularities. The results of numerical simulation of radio wave propagation in the presence of a powerful three-dimensional ionospheric disturbance are presented as an example. State University, St. Petersburg, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 5, pp. 588–604, May, 1998.     

K2_SPH方法及其对二维非线性水波的模拟

K2_SPH方法通过泰勒级数展开和联立求解积分方程组的办法得到具有二阶精度的核近似方法.随着核近似精度的提高,K2_SPH需要对-些关键数值技术进行改进才能成功模拟非线性水波问题,例如自由表面边界和固壁边界.通过与传统SPH方法计算结果比较,K2_SPH方法在非线性自由表面计算精度和整个粒子系统中有关变量分布都有显著提高.     

An approximate Green's function for a locally excited fluid-loaded thin elastic plate

In the classic treatment of the line-driven, fluid-loaded, thin elastic plate, a branch cut integral typically needs to be evaluated. This branch cut arises due to a square root operator in the spectral form of the acoustic impedance. In a previous paper [J. Acoust. Soc. Am. 110, 3018 (2001)], DiPerna and Feit developed a methodology, complex layer analysis (CLA), to approximate this impedance. The resulting approximation was in the form of a rational function, although this was not explicitly stated. In this paper, a rational function approximation (RFA) to the acoustic impedance is derived. The advantage of the RFA as compared to the CLA approach is that a smaller number of terms are required. The accuracy of the RFA is examined both in the Fourier transform domain and the spatial domain. The RFA is then used to obtain a differential relationship between the pressure and velocity on the surface of the plate. Finally, using the RFA in conjunction with the equation of motion of the plate, an approximate expression for the Green's function for a line-driven plate is obtained in terms of a sum of propagating and evanescent waves. Comparisons of these results with the numerical inversion of the exact integral show reasonable agreement.     

Surface impedance design with ground corrugation for mitigation of large-calibre gun blast noise

The surface impedance design approach is proposed for mitigating large-calibre gun blast noise. Surrounding the blast noise, we employ a group of concentric trenches with critical depths to dampen the propagation of the acoustic wave. These trenches behave like quarter-wavelength resonators and produce acoustic soft surfaces at their openings. The sound pressure is then mitigated over these soft surfaces by destructive interference and the wave attenuates rapidly along the ground surface. To evaluate the overall acoustic performance of such a design, we develop an efficient numerical solver by treating the geometry as a body of revolution (BOR). The symmetry of the structure in the revolution direction allows the 3D boundary integral equation (BIE) for acoustic wave scattering to be reduced to a 2D integral equation by the use of Fourier series expansions. Numerical experiments show that this model can effectively suppress the acoustic wave propagation horizontally and the reduction can reach about 15 dB for large-calibre gun noise with very low-frequency components.     

Boundary element-free method for elastodynamics

1 Introduction In recent years, more and more attention has been paid to researches on the meshless (or meshfree) method, which makes it a hot direction of computational mechanics[1,2]. The meshless method is the approximation based on nodes, then the large deformation and crack growth problems can be simulated with the method without the re-meshing technique. And the meshless method has some advantages over the traditional computa- tional methods, such as finite element method (FEM) and boun…     

Semiclassical regime of Regge calculus and spin foams

Recent attempts to recover the graviton propagator from spin foam models involve the use of a boundary quantum state peaked on a classical geometry. The question arises whether beyond the case of a single simplex this suffices for peaking the interior geometry in a semiclassical configuration. In this paper we explore this issue in the context of quantum Regge calculus with a general triangulation. Via a stationary phase approximation, we show that the boundary state succeeds in peaking the interior in the appropriate configuration, and that boundary correlations can be computed order by order in an asymptotic expansion. Further, we show that if we replace at each simplex the exponential of the Regge action by its cosine—as expected from the semiclassical limit of spin foam models—then the contribution from the sign-reversed terms is suppressed in the semiclassical regime and the results match those of conventional Regge calculus.     

Green's function for arbitrarily shaped dielectric bodies of revolution

In this paper, a solution is developed to calculate the electric field at one point in space due to an electric dipole exciting an arbitrarily shaped dielectric body of revolution (BOR). Specifically, the electric field is determined from the solution of coupled surface integral equations (SIE) for the induced surface electric and magnetic currents on the dielectric body excited by an elementary electric current dipole source. Both the interior and exterior fields to the dielectric BOR may be accurately evaluated via this approach. For a highly lossy dielectric body, the numerical Green's function is also obtainable from an approximate integral equation (AIE) based on a surface boundary condition. If this equation is solved by the method of moments, significant numerical efficiency over SIE is realized. Numerical results obtained by both SIE and AIE approaches agree with the exact solution for the special case of a dielectric sphere. With this numerical Green's function, the complicated radiation and scattering problems in the presence of an arbitrarily shaped dielectric BOR are readily solvable by the method of moments.     

Escape rates in periodically driven Markov processes

We present an approximate analytical expression for escape rates of time-dependent driven stochastic processes with an absorbing boundary such as the driven leaky integrate-and-fire model for neural spiking. The novel approximation is based on a discrete state Markovian modeling of the full long-time dynamics with time-dependent rates. It is valid in a wide parameter regime beyond the restraining limits of weak driving (linear response) and/or weak noise. The scheme is carefully tested and yields excellent agreement with three different numerical methods based on the Langevin equation, the Fokker–Planck equation and an integral equation.