Method of partial overlapping regions for solving electromagnetic problems

: This report considers techniques used to solve the Navier field equations of viscoelasticity in the Kelvin-Voigt or Maxwell models, for cylindrical of spherical geometries. Introducing scalar and vector potentials into the viscoelasticity equations formulation, ultimately yields telegraph-type partial differential equations governing those potentials. For harmonic time- dependence, these reduce to scalar and vector Helmholtz's equations with complex propagation constants. These constants are shown to be related to the viscoelastic material-constants in a more or less complicated fashion depending on the viscoelastic model used. The stresses, strains and displacements are then found from these potentials for a dozen cases of interest in those two coordinate systems. The formulation resembles that of electrodynamics in a Coulomb gauge. The above information is vital to set-up and solve various kinds of boundary-value-problems of dynamic viscoelasticity which appear when studying cases of acoustic scattering from sound-absorbing structures.

Trace formulas are derived for the coefficients of a matrix differential equation representing wave propagation in a medium which supports N types of waves. In general these waves are coupled to each other but are assumed to become uncoupled as lzl -> ?o. The two coefficients of the ordinary differential equation, N x N matrices, are assumed known for z 0. A point source is located in the region z 0. The unique features of this work are a derivation of the trace formula in which the Jost functions are not needed, an alternate measurement method based on an impedance concept and a trace equation based on the impedance concept. Introduction. Motivated by the work of Trubowitz (1), trace formula methods were introduced by Deift and Trubowitz (2) to study one-dimensional inverse quantum mechanical scattering problems on the line. The trace formula expresses the potential as a functional of the reflection coefficient. Greene (3) used the same techniques to study the one-dimensional acoustical wave equation with the angular frequency playing the role of the quantum mechanical energy. Stickler and Deift (4) examined the acoustic problem of a point source in an inhomogeneous half space bounded by a pressure release surface. The sound speed was recovered from a measurement of the normal component of the acoustic velocity at the pressure release surface. The role of the quantum mechanical energy was played by a vertical wave number, which does not correspond to either a spatial or temporal derivative. Using again the analogy between vertical wave number (the wave number in the direction of the coefficients' variation) and the quantum mechanical energy, Stickler (5) showed how the trace formula methods could be used to recover the sound speed when it approaches different values at plus and minus infinity. In a quantum mechanical context this means that the potential approaches zero, say at minus infinity, but some positive constant, not necessarily known, at plus infinity. (In the Deift-Trubowitz analysis the potential approached zero in both limits.) Stickler (6) also discussed this same problem in an electromagnetic context. Tomei (7) has used the trace methods to recover the coefficients of a linear, third order equation associated with the Boussinesq equation. In the present paper the trace method is extended to a one-dimensional vector Helmholtz equation which supports N types of waves and for which the waves are coupled to each other except at infinity. The trace method yields a functional relation- ship between the coefficients of the Helmholtz equation, a reflection coefficient described below and assumed known and a specific solution of the vector Helmholtz equation with known and prescribed initial conditions. These relationships are called the trace formulas and are derived here. The coefficients of the vector Helmholtz differential equation can be replaced by these trace formulas and this results in a system of coupled, nonlinear ordinary differential equations with known initial condi- tions. Once this solution is determined, the coefficients can be recovered. The existence of a solution of this coupled system is not given.

SPHEREPACK 3.0 is a collection of FORTRAN programs that facilitates computer modeling of geophysical processes. The package contains programs for computing certain common differential operators including divergence, vorticity, gradients, and the Laplacian of both scalar and vector functions. Programs are also available for inverting these operators. For example, given divergence and vorticity, the package can be used to compute the velocity components. The Laplacian can also be inverted and therefore the package can be used to solve both the scalar and vector Helmholtz equations. Use of the package is illustrated by three sample programs that solve the scalar Helmholtz equation, the time-dependent linear advection equation, and the time-dependent nonlinear shallow-water equations. Accurate solutions are obtained via the spectral method that uses both scalar and vector spherical harmonic transforms that are available to the user. The package also contains utility programs for computing the associated Legendre functions, Gauss points and weights, and multiple fast Fourier transforms. Programs are provided for both equally spaced and Gauss distributed latitudinal points as well as programs that transfer data between these grids.

Scalar and vector Helmholtz equations and their applications in electromagnetics are discussed. Relevance of the Helmholtz equation to wave motion is explained. Derivation involving Maxwell's equations and auxiliary scalar and vector potentials is presented. Basic analytical solution techniques for homogeneous and nonhomogeneous cases are introduced and illustrated in simple problems such as guided waves and radiation from a finite‐length dipole. Numerical methods and the operator approach to investigation of the Helmholtz equation are briefly outlined.

С помощью непрерывно-неоднородных упругих покрытий можно эффективно изменять характеристики рассеяния тел в определенных направлениях, если подобрать соответствующие законы неоднородности для механических параметров покрытия. В статье рассматривается задача дифракции сферической звуковой волны на однородном изотропном упругом цилиндре с радиально-неоднородным упругим покрытием. Полагается, что бесконечный круговой цилиндр с покрытием помещен в идеальную безграничную жидкость, законы неоднородности материала покрытия описываются дифференцируемыми функциями, на тело падает гармоническая сферическая звуковая волна, излучаемая точечным источником. В случае установившихся колебаний распространение малых возмущений в идеальной жидкости описывается скалярным уравнением Гельмгольца, а в упругом однородном изотропном цилиндре — скалярным и векторным уравнениями Гельмгольца. Колебания неоднородного изотропного упругого цилиндрического слоя описываются общими уравнениями движения сплошной среды. Аналитическое решение рассматриваемой задачи получено на основе известного решения аналогичной задачи дифракции плоской волны. Потенциал скорости сферической волны представляется в интегральной форме в виде разложения по цилиндрическим волновым функциям. При этом подынтегральное выражение оказывается аналогичным по форме выражению потенциала скорости плоской волны. Поэтому потенциал скорости рассеянной волны в случае падения сферической волны на цилиндр с покрытием записывается в виде интеграла, подынтегральное выражение которого аналогично по форме выражению потенциала скорости рассеянной волны при падении плоской волны на тело. Для вычисления подынтегральной функции необходимо определить поле смещения в неоднородном покрытии, решая построенную краевую задачу для системы обыкновенных дифференциальных уравнений второго порядка. Рассматриваются вычислительные аспекты оценки интеграла.

A new vector volume integral inequality has been derived using calculus of variations techniques [1]. This inequality was used to determine conditions for waveguide cutoff. The use of the inequality requires only the knowledge of the smallest eigenvalue of the vector Helmholtz's equation for the geometry of the waveguide. The inequality can also be of value in applications of Liapunov Stability Theory to systems of partial differential equations. This result is illustrated by an example.

Radiation boundary conditions developed for use with the vector Helmholtz equation in three dimensions are reviewed, including an alternate second-order radiation condition that preserves the symmetry of the matrix operator arising within a numerical implementation. The accuracy of these conditions is systematically tested as a function of the boundary location using outward-propagating spherical harmonics. >

The electro-optical properties of nanowire array solar cells are analyzed with a microscopic simulation model. For the electromagnetic part, the vectorial Helmholtz equation is solved for the three-dimensional structure, and a modal analysis reveals the absorption mechanisms. The electronic part is analyzed with a drift-diffusion approach. It is shown that with a proper design, a III-V nanowire array solar cell can approach the efficiency of a thin-film solar cell, while only a fraction of the active material volume is used.

Although some calculations will be specialized for toroidal systems, some conclusions will be valid independent of the coordinate system. The manuscript contains 1. a proof that on the basis of a solution of Maxwell's equations a solution of curl B = const B may be found, 2. a method to transform the vector Helmholtz equation into a scalar one, 3. three solutions of the scalar Helmholtz equation, of Maxwell's equations and of curl B = const B in quasi‐toroidal coordinates, 4. a method to construct solutions of curl B × B = 4πΔp on the basis of solutions of curl B = const B (including two special solutions and a derivation of the toroidal Shafranov, equation).

Numerical simulation of Bessel beams by FDTD employing the superposition principle

This program produces assembled finite-element matrices for solving scalar two-dimensional Laplace's, Poisson's, or Helmholtz's equations. It is written in FORTRAN IV language.

In this note it is demonstrated how the sampling theorem may be used to reduce the problems of transmission line taper synthesis and antenna aperture field synthesis to that of optimizing a polynomial. In both cases the problem may be stated mathematically in the following form: find the function g(x) which is zero outside the domain - /spl pi//spl les/x/spl les//spl pi/ that will yield a desired f(u) where f(u) is given by the fourier transform of g(x).

We report a novel broadband slot-spiral antenna that can be integrated with high-electron-mobility transistor (HEMT) terahertz (THz) detectors. The effect of various antenna parameters on the transmission efficiency of the slot-spiral structure at 150–450 GHz is investigated systematically. The performances of the slot-spiral antenna and the spiral antenna both integrated with HEMTs are compared. The results show that the slot-spiral structure has a better transmission and miniaturization capability than the spiral structure. A formula for the responsivity is derived based on the transmission line principle and antenna theory, and results show that the detector responsivity is correlated with the antenna absorptivity. Additionally, guidelines for HEMT THz detector design are proposed. The results of this study indicate the excellent application prospects of the slot-spiral antenna in THz detection and imaging.

The problem of diffraction of a plane wave by a cone with impedance condition on its surface is studied, for a scalar wave field satisfying the Helmholtz equation. The integral Kontorovich-Lebedev, Sommerfeld-Maliuzhinets and Fourier transformations are exploited to investigate the problem and to separate the radial variable. The problem in question is reduced to that for a spectral function satisfying a Laplace-Beltrami type equation on the unit sphere with a hole cut out by the conical surface. An impedance type boundary condition with a nonlocal impedance operator is determined on the boundary of the hole. Then the problem for the spectral function can be transformed to a second kind integral equation with a nonoscillatory kernel. In the particular case of a circular impedance cone the problem is simplified. As an application, a closed form expression for the scattering diagram (or pattern) is deduced in the narrow cone approximation. The leading and first correction terms are represented.

Sensitivity analysis for active control of the Helmholtz equation