A Two- or Three-Dimensional Green's Function Which can be Applied to Hyperfrequency Microelectronic Transmission Lines (Letters)

As in the case of time-harmonic waves (see Chapter 1), we begin with the simplest model, replacing a ship by a point source in the uniform forward motion in calm water. The corresponding velocity potential is sometimes referred to as the Kelvin source, but to keep the terminology unified we call it the Green's function in what follows. Similar to the theory of time-harmonic waves developed in Part 1, the theory of ship waves presented here relies essentially on Green's functions. They are of importance not only for proving solvability theorems (see Chapters 7 and 8) but also for constructing examples of trapped waves (nontrivial solutions to homogeneous boundary value problems) in Section 8.4. The three-dimensional Green's function of a point source in deep water is considered in detail in Sections 6.1 and 6.2. General facts about the three-dimensional Green's function are considered in Section 6.1 and the far-field expansions for Green's function and the corresponding elevation of the free surface are obtained in Section 6.2. Two-dimensional Green's functions are treated in Section 6.3, which we begin with the simpler case of deep water (Subsection 6.3.1). For water of finite depth, which will be referred to as shallow water, we consider Green's function in Subsection 6.3.2.

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.

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.

Artificial boundary conditions, which can be identified as Robin boundary conditions positioned at a complex space coordinate, are introduced in order to obtain pertinent approximations for the Green's functions in grounded layered media. These artificial boundary conditions include perfectly matched layers backed by perfectly electric or magnetic conductors. As a first result, we obtain analytical expressions for the differences of Green's functions subject to different boundary conditions. Since weighted sums of Green's functions are again Green's functions, the need arises to solve an optimization problem, in the sense of obtainingthe optimal weig hted mixture of Green's functions, as compared to the exact Green's function. Comprehensive eigenexpansions for the Green's functions are given in the general case, and a few examples illustrate the goodness of fit between the approximate Green's functions and the exact Green's function.

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.

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.

SECTION 12 - Green's Function

We present a computer assisted method for generating existence proofs and a posteriori error bounds for solutions to two point boundary value problems (BVPs). All truncation errors are accounted for and, if combined with interval arithmetic to bound the rounding errors, the computer generated results are mathematically rigorous. The method is formulated for $n$-dimensional systems and does not require any special form for the vector field of the differential equation. It utilizes a numerically generated approximation to the BVP fundamental solution and Green's function and thus can be applied to stable BVPs whose initial value problem is unstable. The utility of the method is demonstrated on a pair of singularly perturbed model BVPs and by using it to rigorously show the existence of a periodic orbit in the Lorenz system.

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

In the use of the time-domain integral equation (TDIE) method for the analysis of layered media, it is important to have the time-domain layered medium Green's function computed for many source-to-field distances /spl rho/ and time instants t. In this paper, a numerical method is used that computes the mixed potential Green's functions G/sub v/(/spl rho/,t) and G/sub A/(/spl rho/,t) for a multilayered medium for many /spl rho/'s and t's simultaneously. The method is applicable to multilayered media and for lossless or lossy dispersive media. Salient features of the method are: 1) the use of complex /spl omega/ so that the surface wave poles are lifted off the real k/sub /spl rho// axis such that pole extractions are not required; 2) the use of half-space extraction so that the integrand for the Sommerfeld integral decays exponentially along the k/sub /spl rho// axis to obtain fast convergence of the integral; and 3) the use of the fast Hankel transform so that the Green's function is calculated for many values of /spl rho/ simultaneously. For a four-layer medium, we illustrate the numerical results by a three-dimensional plot of /spl rho/G/sub v/(/spl rho/,t) versus /spl rho/ and t and demonstrate the space-time evolution of these Green's functions. For a maximum frequency range of 8 GHz, the method requires only a few CPU minutes to compute a table of 100 (points in /spl rho/) /spl times/ 168 (points in t) uniformly spaced values of G/sub v/(/spl rho/,t) on an 867-MHz Pentium PC.

Three-dimensional Green's functions of thermoporoelastic axisymmetric cones

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).

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>

Three-dimensional static Green's function for a half space over mixed PEC and dielectric wedges