Fast evaluation of time domain fields in sub-wavelength source/observer distributions using accelerated Cartesian expansions (ACE)

Time domain integral equation solvers for transient scattering from electrically large objects have benefitted significantly from acceleration techniques like the plane wave time domain (PWTD) algorithm; these techniques reduce the asymptotic CPU and memory cost. However, PWTD breaks down when used in the analysis of structures that have subwavelength features or features whose length scales are orders of magnitude smaller than the smallest wavelength in the incident pulse. Instances of these occurring in electromagnetics range from antenna topologies, to feed structures, etc. In this regime, it is the geometric constraints that dictate the computational complexity, as opposed to the wavelength of interest. In this work, we present an approach for efficient analysis of such sub-wavelength source/observer distributions in time domain. The methodology that we seek to exploit is the recently developed algorithm based on Cartesian expansions for accelerating the computation of potentials of the form R^ν. In this paper, we present an efficient methodology for computing these polynomials for two different scenarios; where the size of the domain spans the distance travelled by light in (i) one time step and (ii) multiple time steps. These algorithms are cast within the framework of both uniform and non-uniform distributions. Results that demonstrate the efficiency and convergence of the proposed algorithm are presented.     

Fast evaluation of time domain fields in sub-wavelength source/observer distributions using accelerated Cartesian expansions (ACE)

Time domain integral equation solvers for transient scattering from electrically large objects have benefitted significantly from acceleration techniques like the plane wave time domain (PWTD) algorithm; these techniques reduce the asymptotic CPU and memory cost. However, PWTD breaks down when used in the analysis of structures that have subwavelength features or features whose length scales are orders of magnitude smaller than the smallest wavelength in the incident pulse. Instances of these occurring in electromagnetics range from antenna topologies, to feed structures, etc. In this regime, it is the geometric constraints that dictate the computational complexity, as opposed to the wavelength of interest. In this work, we present an approach for efficient analysis of such sub-wavelength source/observer distributions in time domain. The methodology that we seek to exploit is the recently developed algorithm based on Cartesian expansions for accelerating the computation of potentials of the form R^ν. In this paper, we present an efficient methodology for computing these polynomials for two different scenarios; where the size of the domain spans the distance travelled by light in (i) one time step and (ii) multiple time steps. These algorithms are cast within the framework of both uniform and non-uniform distributions. Results that demonstrate the efficiency and convergence of the proposed algorithm are presented.     

Applying time domain physical optics to acoustic wave backscattering problem

The interest in transient analysis of acoustic waves has been growing in recent years, due to the advance of wide-band sonars. In this paper, a transient analysis method for acoustic backscattering signals is proposed based on the time domain physical optics (TDPO). TDPO is formulated via a theoretical inverse Fourier transform of the conventional physical optics formula used in the frequency domain wave scattering analyses. A hidden surface removal algorithm using an adaptive triangular beam method and a virtual surface concept are adopted to explain shadow effects and multiple reflections among elements, respectively. Numerical analyses are carried out for two kinds of underwater targets: a submarine pressure hull and an idealized submarine, in order to validate the proposed method. The result of the submarine pressure hull shows good agreements between the proposed method and conventional physical optics based on inverse fast Fourier transform. Additionally, the result of the idealized submarine shows that the proposed method is efficient for finding highlights including their contribution to the whole backscattering signal.     

Multilevel fast multipole algorithm for acoustic wave scattering by truncated ground with trenches

The multilevel fast multipole algorithm (MLFMA) is extended to solve for acoustic wave scattering by very large objects with three-dimensional arbitrary shapes. Although the fast multipole method as the prototype of MLFMA was introduced to acoustics early, it has not been used to study acoustic problems with millions of unknowns. In this work, the MLFMA is applied to analyze the acoustic behavior for very large truncated ground with many trenches in order to investigate the approach for mitigating gun blast noise at proving grounds. The implementation of the MLFMA is based on the Nystrom method to create matrix equations for the acoustic boundary integral equation. As the Nystrom method has a simpler mechanism in the generation of far-interaction terms, which MLFMA acts on, the resulting scheme is more efficient than those based on the method of moments and the boundary element method (BEM). For near-interaction terms, the singular or near-singular integrals are evaluated using a robust technique, which differs from that in BEM. Due to the enhanced efficiency, the MLFMA can rapidly solve acoustic wave scattering problems with more than two million unknowns on workstations without involving parallel algorithms. Numerical examples are used to demonstrate the performance of the MLFMA with report of consumed CPU time and memory usage.     

Rigorous and approximate methods for modeling wave scattering from a locally perturbed perfectly conducting surface

Rigorous and approximate methods are considered for solving the problem of harmonic plane wave scattering from a plane surface arbitrarily perturbed along one dimension on a finite interval. This problem is treated using the Fredholm integral equations of the second kind and the Kirchhoff and Rayleigh approximations. The estimates of the computational efficiency of the integral equation method and the Rayleigh approximation are compared by calculating fields scattered from random rough surfaces in the resonance region (i.e., when the roughness height is comparable to or smaller than the incident wavelength) for an arbitrary incidence of a plane wave. Scattering patterns calculated using the integral equations and the Kirchhoff approximation are discussed in the case of large-scale random rough surface scattering. Particular attention is paid to scattering at near-grazing incidence.     

Time-Domain Scattering Modelling of Two-Dimensional Cylinder Located on a Rough Surface

Numerical modelling on the transient electromagnetic scattering by a two-dimensional （21）） cylinder located on a time-evolving rough surface is presented by using time-domain integral equations. The proposed special choice of a tapered Gauss pulse incident wave removes the truncation error from the rough surface. Additionally, a two-level averaging technique is utilized to overcome the instability from the time marching procedure of solving integral equations. Excellent correspondences between the surface current distributions, as well as the far-zone fields, computed by the proposed method and that obtained by the traditional method of moments associated with the inverse discrete Fourier transformation scheme demonstrate the accuracy of the modelling.     

Two-frequency mutual coherence function for electromagnetic pulse propagation over rough surfaces

The propagation of a transient electromagnetic pulse over irregular terrain is considered. We model the wave propagation using the parabolic wave equation, which is valid for near-horizontal propagation. We model the effect of scattering from the rough terrain by introducing a surface-flattening coordinate transform. This coordinate transform simplifies the boundary condition of our problem, and introduces an effective refractive index into our wave equation. As a result, the problem of propagation over an irregular surface becomes equivalent to the problem of propagation through random media. The parabolic equation is solved analytically using the path integral method. Both vertically polarized and horizontally polarized signals are treated. Cumulant expansion is introduced to obtain an approximate expression for the two-frequency mutual coherence function. From the mutual coherence function, spatial and temporal dependence of the propagating signal can be determined. It can be shown that scattering from the irregular surface can cause broadening of the transient signal. This can have a significant impact on the performance of radio communication systems.     

Two-frequency mutual coherence function for electromagnetic pulse propagation over rough surfaces

The propagation of a transient electromagnetic pulse over irregular terrain is considered. We model the wave propagation using the parabolic wave equation, which is valid for near-horizontal propagation. We model the effect of scattering from the rough terrain by introducing a surface-flattening coordinate transform. This coordinate transform simplifies the boundary condition of our problem, and introduces an effective refractive index into our wave equation. As a result, the problem of propagation over an irregular surface becomes equivalent to the problem of propagation through random media. The parabolic equation is solved analytically using the path integral method. Both vertically polarized and horizontally polarized signals are treated. Cumulant expansion is introduced to obtain an approximate expression for the two-frequency mutual coherence function. From the mutual coherence function, spatial and temporal dependence of the propagating signal can be determined. It can be shown that scattering from the irregular surface can cause broadening of the transient signal. This can have a significant impact on the performance of radio communication systems.     

Mass operator for wave scattering from a slightly random surface

This paper deals with the mass operator representing multiple-scattering effects in the theory of wave scattering from a slightly random surface. By means of the stochastic-functional approach, a recurrence equation for the mass operator is obtained in the form of an iterative integral. However, its solution oscillates in a non-physical manner against the number of iterations. Next, the recurrence equation may be regarded as a nonlinear integral equation, when the number of iterations goes to infinity. An analytical solution of the nonlinear integral equation is presented for a special case in which the roughness spectrum is the Dirac delta function. Then, the nonlinear integral equation is solved numerically for the Gaussian roughness spectrum by iteration, starting from such an analytical solution. It is shown that only a few iterations are required to obtain the mass operator, even when the correlation distance is small. Effects of the mass operators on the coherent reflection coefficient and the incoherent scattering cross section are calculated and shown in figures.     

Phase integral theory,coupled wave equations,and mode conversion

Phase integral or WKB theory is applied to multicomponent wave equations, i.e., wave equations in which the wave field is a vector, spinor, or tensor of some kind. Specific examples of physical interest often have special features that simplify their analysis, when compared with the general theory. The case of coupled channel equations in atomic or molecular scattering theory in the Born-Oppenheimer approximation is examined in this context. The problem of mode conversion, also called surface jumping or Landau-Zener-Stuckelberg transitions, is examined in the multidimensional case, and cast into normal form. The group theoretical principles of the normal form transformation are laid out, and shown to involve both the Lorentz group and the symplectic group.     

Functional integral formulation of classical wave equations

It is shown in this paper that classical wave equations admit path integral formulations. For this, the evolution of the system is first set-up in terms of a fundamental solution or propagator. We choose this last name because it suggests a connection with functional integrals, which are exploited in this work. A functional integral in terms of non-singular functions is then proposed and shown to converge to the propagator in the appropriate limit for the case of scalar wave equations. One of the advantages of such formulation is that it provides an adequate framework for mesh-free numerical methods. This is demonstrated through a computational implementation that combines a simple second-degree polynomial local approximation of the continuous field and an approximate statement of the exact evolution equations. Numerical simulations of modal analysis and transient dynamics indicate the feasibility of the technique.     

Regularization of boundary integral equations in the problems of wave field diffraction by curved boundaries

A regularization of the exact Fredholm integral equations for the field or its derivative on a scattering surface is proposed. This approach allows one to calculate the scattering or diffraction of pulsed wave fields by curved surfaces of arbitrary geometry. Mathematically, the method is based on the replacement of the exact Fredholm integral equations by their truncated analogs, in which the contributions of the geometrically shadowed regions are cancelled. This approach has a clear physical meaning and provides stable solutions even when the direct numerical solution of mathematically exact initial integral equations leads to unstable results. The method is mathematically substantiated and tested using the problem of plane-wave scattering by a cylinder as an example.     

雪层覆盖地面电磁散射的矩量法

为实现分层介质粗糙面电磁散射的矩量法研究,给出一种分层介质粗糙面电磁积分方程的区域分解方法.将格林定理应用于粗糙面所分的各子空间,结合波动方程和格林函数推导分层粗糙面的电磁积分方程,利用矩量法对其进行离散,数值计算得到雪层覆盖地面散射系数的角分布曲线,其中,粗糙表面由一维带限Weierstrass分形谱和Monte Carlo方法模拟.通过与时域有限差分法数值结果的比对,验证该方法的准确性,并分析散射系数随雪和地面参数、介质参数以及入射波参数的变化,获得了较完整的散射特征.     

A study for sound wave scattering by corrugated ground with complex trench structures

Several trench structures in corrugated ground are investigated for the possibility of mitigating gun blast noise by numerical simulations. The blast noise usually includes large explosive energy with nonlinearity in the near field and exhibits a very low-frequency spectrum. In this study, the linearity approximation for the noise is taken because the nonlinearity of the wave reaching the scatterer is not serious for many proved guns and the low-frequency characteristic is concentrated. The structures are designed based on the surface impedance design approach proposed in our previous work and arbitrary three-dimensional (3D) geometries within a truncated ground are now assumed. The acoustic characteristic of the structures is evaluated by using a fast numerical solver. The solver employs the multilevel fast multipole algorithm (MLFMA) as an accelerator and can solve very large acoustic wave scattering problems with millions of unknowns on workstations within several days. This tool allows us to truncate the ground as large as needed for accurate modeling. Four structures are mainly considered in the design, namely, concentric trenches, sectorial trapezoidal trenches, interlaced arc trenches and parabolic reflectors. Some of them may have a sloped inner wall or tilted surface as a means of adjustment. Numerical simulations show that the concentric trench design has a very good mitigation behavior for linear and continuous noise sources and the structure is further studied for mitigating real-world gun blast noise.     

Time reversal and its application to tomography with diffracting sources

An exact time-domain method is proposed to time reverse a transient scalar wave using only the field measured on an arbitrary closed surface enclosing the initial source. Under certain conditions, a time-reversed field can be approximated by retransmitting the measured signals in a reversed temporal order. Exact reconstruction for three-dimensional broadband diffraction tomography (a linearized inverse scattering problem) is proposed by time-reversing the measured field back to the time when each secondary source is excited. The algorithm is verified by a numerical simulation. Extension to the case using Green's function in a heterogeneous medium is discussed.     

A multilevel Cartesian non-uniform grid time domain algorithm

A multilevel Cartesian non-uniform grid time domain algorithm (CNGTDA) is introduced to rapidly compute transient wave fields radiated by time dependent three-dimensional source constellations. CNGTDA leverages the observation that transient wave fields generated by temporally bandlimited and spatially confined source constellations can be recovered via interpolation from appropriately delay- and amplitude-compensated field samples. This property is used in conjunction with a multilevel scheme, in which the computational domain is hierarchically decomposed into subdomains with sparse non-uniform grids used to obtain the fields. For both surface and volumetric source distributions, the computational cost of CNGTDA to compute the transient field at N_s observation locations from N_s collocated sources for N_t discrete time instances scales as O(N_tN_slogN_s) and O(N_tN_slog²N_s) in the low- and high-frequency regimes, respectively. Coupled with marching-on-in-time (MOT) time domain integral equations, CNGTDA can facilitate efficient analysis of large scale time domain electromagnetic and acoustic problems.     

Surface wave scattering by a nano-object situated on a surface

The scattering of the surface electromagnetic waves by a nano-defect (object) on a surface was calculated. The scattered field has been considered as a field caused by the current generated by the self-consistent local field inside the defect. In turn, the self-consistent local field has been determined as a result of solution of the integral Lippmann-Schwinger equation. The effective susceptibility of the object has been calculated using a self-consistent procedure. The corrections of self-energy part due to direct and indirect electromagnetic interactions, as well as due to interaction with surface wave field are taken into account. The self-energy part is calculated analytically within the framework of the near-field approximation. The scattering indicatrisses in reciprocal space have been computed for different shapes of the scatterer. Strong dependence of the scattered field on geometry of the scatterer has been found and explained.     

Structure of boundary diffraction wave revisited

The physically appealing boundary diffraction wave theory which suggests that diffraction patterns arise due to interference of an undisturbed (geometrical) wave and the boundary diffraction wave generated by edge of the diffracting aperture, simplifies the solution of diffraction problems by reducing the Fresnel–Kirchhoff surface integral into a line integral over the illuminated boundary of the diffracting aperture. The present work reports experimental investigations carried out on the structure of the boundary diffraction wave. It has been shown that the boundary diffraction wave is continuous behind the diffracting aperture and apparently there does not exist any discontinuity at the geometrically light to shadow transition boundary, as is required by the theory. PACS 42.25.-p; 42.25.Fx; 42.25.Hz     

Direct Solution of the Inverse Problem for Rough Surface Scattering

We consider the inverse scattering problem for a scalar wave field incident on a perfectly conducting one-dimensional rough surface. The Dirichlet Green function for the upper half-plane is introduced, in place of the free-space Green function, as the fundamental solution to the Helmholtz equation. Based on this half-plane Green function, two reasonable approximate operations are performed, and an integral equation is formulated to approximate the total field in the two-dimensional space, then to determine the profile of the rough surface as a minimum of the total field. Reconstructions of sinusoidal, non-sinusoidal and random rough surface are performed using numerical techniques. Good agreement of these results demonstrates that the inverse scattering method is reliable.     

Elastic scattering amplitude of a scalar particle in a plane wave field

The elastic scattering amplitude of a scalar particle in an arbitrary plane wave electromagnetic field is obtained in the form of a double integral by the method of dispersion relations. Particular cases of giving the plane wave field are investigated. It is shown that the existence of scalar particle radiation in an arbitrary plane wave electromagnetic field results in elastic scattering, whose amplitude determines the change in particle mass in this field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 32–37, May, 1990.