Green's Functions

Exact results for the stress intensity factor are presented for an external circular crack with oppositely directed concentrated loads applied to the crack surfaces. This result is specialized to the case of a semi-infinite crack in an infinite body with concentrated loads on the crack. A procedure is then suggested by which one can obtain from the corresponding plane result the approximate three-dimensional Green's function (concentrated load result) for any straight crack in an infinite elastic body. This procedure is used to determine the Green's functions for a finite-length crack in an infinite body, and is then used in conjunction with a suggested “slicing” procedure to obtain approximate three-dimensional Green's function for plates of finite thickness and infinite extent, containing finite length cracks. Previously existing solutions for crack problems are compared with results obtained by application to plate tension and bending problems of the three-dimensional Green's functions. The results indicate that the procedure yields satisfactory results when stress gradients through the plate thickness are not excessive. However, an accurate assessment of the validity of the slicing procedure awaits further progress in three-dimensional crack analysis.

Green's functions play an important role in electroelastic analyses of piezoelectric media. However, most works available on this topic are on the case of identical temperature. Based on the compact general solution of transversely isotropic piezothermoelectric media, which is expressed in harmonic functions, and employing the trial-and-error method, the three-dimensional Green's function for a steady point heat source in a two-phase piezothermoelectric media is presented by eight newly induced harmonic functions. Numerical results are given graphically by contours.

Knowing Green's function and the charge density found on different conductors, the diverse capacities can eventually be calculated by solving an integral equation. This has been dealt with only for simple dielectric-conductor configurations. In Coen's article, the integral representation of log (Z) is employed in calculating Green's function for microstrips (with or without an upper ground plane). Electrostatically speaking, the boundary conditions along conductors or dielectric interfaces are represented by means of infinite charge series.

The linear Boltzmann equation with constant coefficients in the three-dimensional infinite space is revisited. It is known that the Green's function can be calculated via the Fourier transform in the case of isotropic scattering. In this paper, we show that the three-dimensional Green's function can be computed with the Fourier transform even in the case of arbitrary anisotropic scattering.

On a link between the network model and the group-theoretical perturbation treatment in the theory of the de Haas-van Alphen effect

The commenters state that the one- and two-dimensional frequency domain Green's functions for isotropic chiral media derived by Engheta and Bassiri (see ibid., vol.37, p.512-5, Apr. 1989) were reported first by the commenters. The authors, in a paper submitted in December 1986 which was published in February 1988, point out that the one- and two-dimensional dyadic Green's functions for isotropic chiral materials, derived by them, extend and continue their original work on three-dimensional chiral dyadic Green's function, which first appeared in March 1986, and was used and acknowledged by the commenters in some of their work.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Unique, explicit, and exact expressions for the first- and second-order derivatives of the three-dimensional Green's function for general anisotropic materials are presented in this paper. The derived expressions are based on a mixed complex-variable method and are obtained from the solution proposed by Ting and Lee (Ting and Lee, 1997,“The Three-Dimensional Elastostatic Green's Function for General Anisotropic Linear Elastic Solids,” Q. J. Mech. Appl. Math. 50, pp. 407–426) which has three valuable features. First, it is explicit in terms of Stroh's eigenvalues pα (α=1,2,3) on the oblique plane with normal coincident with the position vector; second, it remains well-defined when some Stroh's eigenvalues are equal (mathematical degeneracy) or nearly equal (quasi-mathematical degeneracy); and third, they are exact. Therefore, both new proposed solutions inherit these appealing features, being explicit in terms of Stroh's eigenvalues, simpler, unique, exact and valid independently of the kind of degeneracy involved, as opposed to previous approaches. A study of all possible degenerate cases validate the proposed scheme.

Linearized shallow-water wave theory is used to investigate the interaction of surface waves with multiple rectangular submarine pits in water of otherwise uniform depth. The solution is obtained by a boundary element technique using a two-dimensional Green's function. It is shown that appropriate selection of pit dimensions and placement may lead to a significant reduction in wave heights behind these structures. Numerical results have been presented that illustrate the influence of the various pit characteristics on the diffracted wave field. Two pits can provide a shadow region in which wave heights are reduced to 10–20% of the incident wave height. The shadow region, with wave heights reduced to 30% of the incident wave height, is approximately the width of the pits and five wavelengths long. Guidance is given on how to select pit geometries and placement for optimal breakwater performance. An example is also presented that shows how to select a pit to reduce wave heights in a navigation channel.

A survey is presented of recent related work (comprising numerical modeling and field experiments) on the determination of seafloor geoacoustic properties in shallow water from monochromatic pressure field measurements obtained using synthetic aperture and towed arrays. Two related techniques are discussed, their common foundation lying in the Hankel transform inversion of the pressure versus range data to obtain a depth-dependent Green's function versus horizontal wave number. Characteristic features of the Green's function, such as the peaks associated with normal mode propagation in the shallow water waveguide, are used to extract the geoacoustic properties (compressional and shear) as a function of depth in the bottom. INTRODUCTION Two related methods for determining geoacoustic properties in shallow water waveguides have recently been under development (Frisk and Lynch, 1984; Frisk et al., 1985; Kuperman et al., 1985; Werby and Tango, 1985). In both cases, the required input data are the magnitude and phase versus range of the pressure field due to a monochromatic (CW) point source. These data are numerically Hankel transformed to obtain the depth-dependent Green's function versus horizontal wave number. In the context of normal mode theory, the Green's function contains information about the nature of the discrete and continuous modal spectra as well as the plane-wave reflection coefficients of the waveguide boundaries. Specifically, the Green's function contains prominent peaks at horizontal wave numbers corresponding to the eigen-values for any trapped and virtual modes excited in the waveguide. The positions and magnitudes of these modal peaks are sensitive to the acoustic properties of the bottom, and this sensitivity is the basis of the two inverse methods for determining geoacoustic models in shallow water. The WHOI method utilizes an exact Hankel transform inversion of synthetic aperture array data obtained by towing a CW source away from two moored receivers. A geoacoustic model is obtained by computing the theoretical Green's function for various values of the bottom parameters and determining the parameter set which provides the best agreement with the experimental Green's function, particularly in the positions and relative magnitudes of the modal peaks. The NORDA technique involves an approximate Hankel transform inversion in which the beam formed output of a towed array due to the self-noise of the tow ship is computed. Layer sound velocity estimates are obtained from the identification of modal critical angles in the variable-offset beam former response. THEORY Consider a water column of thickness h, characterized by a constant density and a sound speed c(z), and bounded at the surface and bottom by horizontally stratified media as shown in Fig. 1. Then the spatial part of the acoustic pressure p, due to a point source at r =.0 and z = Zo with harmonic time satisfies the inhomogeneous Helmholtz equation(Mathematical equation available in full paper) The Green's function satisfies along with impedance boundary conditions at the surface and bottom. These boundary conditions can be ex pressed in terms of the plane-wave reflection coefficients RS(kr) and RB(kr) of the surface and bottom respectively (Frisk et al., 1980).

It is well known that the conventional diagonalization used in the multilevel fast multipole algorithm (MLFMA) suffers from low-frequency breakdowns, which limit the application of this powerful method to multiscale problems involving large objects with small (but important) details in terms of wavelength. Using multipoles explicitly or resorting to alternative expansions incorporating evanescent waves for subwavelength interactions are common techniques for broadband implementations of MLFMA, while all these techniques need extensive efforts and re-programming. Recently, we presented an approximate diagonalization of the three-dimensional Green's function using scaled Hankel functions and plane waves (O. Ergul and B. Karaosmanoglu, IEEE Antennas Wireless Propag. Lett., 13, 2014, pp. 1054–1056). While its accuracy is limited (e.g., 3–4% in the worst case for the challenging one-box-buffer scheme), the approximate diagonalization provides stable computations of interactions at arbitrarily short distances. As a major advantage, the approximate diagonalization needs only minor modifications in the existing codes, including the parallel programs, to stabilize the conventional implementations for low frequencies.

본 논문에서는 전자파를 이용하여 인체 내 암 조직을 찾아내기 위해 역산란 방법인 BIM(Born Iterative Method)을 이용한 알고리즘으로 구현하였다. Born 반복법을 이용하여 2차원 유방암 진단에 적용하였으며, 전자파산란 해석에는 적분방정식과 2차원 그린함수를 이용한 MoM(Method of Moment)을 이용하였다. 또한, 유방암 진단에 있어 적용된 역산란 알고리즘의 계산 결과를 확인하고, 암진단 의료기기로의 적용가능성과 알고리즘 사용에 제한을 분석하였다. In this paper, we used MOM-based BIM(Born Iterative Method) algorithm to implement the inverse scattering for the detection of cancer. We adopted two-dimensional breast structure, integral equations and two-dimensional Green's function is solved with MoM(Method of Moment) to analyzing electromagnetic scattering phenomena. In addition, verifying the calculation of developed inverse scattering algorithm and analyzing medical applicability and limitations of the algorithm.

Folding is a process in which bending is localized at sharp edges separated by almost undeformed elements. This process is rarely encountered in Nature, although some exceptions can be found in unusual layered rock formations (called ‘chevrons’) and seashell patterns (for instance Lopha cristagalli). In mechanics, the bending of a three-dimensional elastic solid is common (for example, in bulk wave propagation), but folding is usually not achieved. In this article, the route leading to folding is shown for an elastic solid obeying the couple-stress theory with an extreme anisotropy. This result is obtained with a perturbation technique, which involves the derivation of new two-dimensional Green's functions for applied concentrated force and moment. While the former perturbation reveals folding, the latter shows that a material in an extreme anisotropic state is also prone to a faulting instability, in which a displacement step of finite size emerges. Another failure mechanism, namely the formation of dilation/compaction bands, is also highlighted. Finally, a geophysical application to the mechanics of chevron formation shows how the proposed approach may explain the formation of natural structures.

The goal of this paper is to provide an intuitive and useful tool for analyzing the impurity-bound-state problem. We develop a semiclassical approach and apply it to an impurity in two-dimensional systems with parabolic or Dirac-like bands. Our method consists of reducing a higher-dimensional problem into a sum of one-dimensional ones using the two-dimensional Green's functions as a guide. We then analyze the one-dimensional effective systems in the spirit of the wave-function-matching method as in the standard one-dimensional quantum model. We demonstrate our method on two-dimensional models with parabolic and Dirac-like dispersion, with the later specifically relevant to topological insulators.

Power and ground planes are the major noise sources in the electronic package and multi-layered printed circuit boards. We propose a novel imaging method to simulate the impedance of power and ground planes. In the proposed method, the noise propagation between the power and ground planes is equivalent to a two-dimensional transverse magnetic problem. The electromagnetic wave multi reflections from the periphery of the power and ground planes are represented by the images of the source current, then a two-dimensional Green's function is used to efficiently calculate the contributions from the source and all its images. The major advantage of the proposed imaging method over the available mode expansion method is that it converges quickly at high frequencies, and it can be easily extended to simulate power and ground planes with arbitrary shapes. By the comparison with the full-wave method and measurement results, the accuracy and efficiency of the proposed imaging method are validated.

In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that Maxwell's equations may be viewed as an evolution equation in one of the spatial directions. With such applications in mind, we propose a new Eulerian geometrical-optics method, dubbed the fast Huygens sweeping method, for computing Green's functions of Maxwell's equations in inhomogeneous media in the high-frequency regime and in the presence of caustics. The first novelty of the fast Huygens sweeping method is that a new dyadic-tensor-type geometrical-optics ansatz is proposed for Green's functions which is able to utilize some unique features of Maxwell's equations. The second novelty is that the Huygens--Kirchhoff secondary source principle is used to integrate many locally valid asymptotic solutions to yield a globally valid asymptotic solution so that caustics associated with the usual geometrical-optics ansatz can be treated automatically. The third novelty is that a butterfly algorithm is adapted to carry out the matrix-vector products induced by the Huygens--Kirchhoff integration in $O(N\log N)$ operations, where $N$ is the total number of mesh points, and the proportionality constant depends on the desired accuracy and is independent of the frequency parameter. To reduce the storage of the resulting traveltime and amplitude tables, we compress each table into a linear combination of tensor-product based multivariate Chebyshev polynomials so that the information of each table is encoded into a small number of Chebyshev coefficients. The new method enjoys the following desired features: (1) it precomputes a set of local traveltime and amplitude tables; (2) it automatically takes care of caustics; (3) it constructs Green's functions of Maxwell's equations for arbitrary frequencies and for many point sources; (4) for a specified number of points per wavelength it constructs each Green's function in nearly optimal complexity $O(N\log N)$ in terms of the total number of mesh points $N$, where the prefactor of the complexity depends only on the specified accuracy and is independent of the frequency parameter. Three-dimensional numerical experiments are presented to demonstrate the performance and accuracy of the new method.