Electromagnetic boundary value problem in the presence of a partlylossy dielectric: considerations about the uniqueness of the solution

This paper deals with the uniqueness of the solution of a boundary value problem defined by specifying the tangential components of the electric field over the closed regular boundary (or the tangential components of the magnetic field over the boundary, or the former components over part of the boundary and the latter components over the rest of the boundary) of a limited region containing a linear dielectric material not lossy everywhere. In particular, the uniqueness of the solution is proved in the case where the dielectric is everywhere linear, homogeneous, and lossless, except for a subregion where the dielectric is lossy, linear but not necessarily homogeneous     

Extension of the MoM Laplacian solution to the general Helmholtzequation

A new boundary integral method for solving the general Helmholtz equation has been developed. The new formulation is based on the method of moments Laplacian solution. The main feature of this new formulation is that the boundary conditions are satisfied independent of the region node discretizations. The numerical solution of the present method are compared with finite difference and finite element solutions     

An application of the boundary element method to the magnetic fieldintegral equation

A boundary element method (BEM) for the solution of electromagnetic scattering problems using the magnetic field integral equation (MFIE) is discussed. The discretized form of the MFIE is written in indicial notation with no limitations placed on the order of either the geometric or functional approximation. By considering several different types of boundary elements, it is determined that geometric errors can be significant and degrade the accuracy of the numerical solution. It is shown that a higher-order approximation for the current could significantly improve the accuracy of the numerical solution. The superparametric boundary element in which the geometry was given quadratic approximation and the current was given linear approximation was more efficient than elements using lower-order approximations. The BEM results are compared to the results obtained using the dielectric bodies of revolution (DBR) code     

On a generalization of the extended boundary condition method

The elementary problem of scattering of an E-polarized plane wave by a circular perfectly conducting cylinder is used for verifying the efficiency of the boundary deformation method (BDM) that has recently been proposed as a generalization of the extended boundary condition method (EBCM). It is shown that the BDM can ensure the highly accurate fulfillment of the null-field condition on an interior contour spaced from the boundary in accordance with this method. However, when the accuracy of the fulfillment of the aforementioned condition on the contour near the boundary or on the boundary itself corresponds to the solution found for the current, this accuracy is substantially worse than the accuracy achieved in the case when the standard EBCM with a contour located near the boundary is applied. In addition, it is shown that, when the singularities of the wave field are enclosed by the contour chosen according to the BDM, a correct solution to the problem is not guaranteed.     

A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface

The problem of the diffraction of an arbitrary ray optical electromagnetic field by a smooth perfectly conducting convex surface is investigated. A pure ray optical solution to this problem has been developed by Keller within the framework of his geometrical theory of diffraction (GTD). However, the original GTD solution fails in the transition region adjacent to the shadow boundary where the diffracted field plays a significant role. A uniform GTD solution is developed which remains valid within the shadow boundary transition region, and which reduces to the GTD solution outside this transition region where the latter solution is valid. The construction of this uniform solution is based on an asymptotic solution obtained previously for a simpler canonical problem. The present uniform GTD solution can be conveniently and efficiently applied to many practical problems. Numerical results based on this uniform GTD solution are shown to agree very well with experiments.     

方形自聚焦透镜的折射率分布研究

为了获得方形自聚焦透镜的折射率分布,提出了一种求解其折射率分布的半经验方法。该方法利用圆形边界条件下获得的扩散方程的解去近似方形边界条件下扩散方程的解,该近似解中的4个待定系数用雅明干涉法测得的方形自聚焦透镜4个位置点上的折射率来确定。该方法避免了在方形边界条件下求解扩散方程的复杂过程,得到的半经验公式形式简单、计算方便,利用半经验公式计算得到的折射率与实验结果吻合得较好,二者之间的最大相对误差为0.94%,平均相对误差不超过0.3%。该公式为以后研究方形自聚焦透镜阵列成像问题提供了可供参考的理论依据。     

Propagation modeling over terrain using the parabolic wave equation

We address the numerical solution of the parabolic wave equation over terrain using the Fourier/split-step approach. The method, referred to as a shift map, generalizes that of Beilis and Tappert (1979) who introduced a coordinate transformation technique to flatten the boundary. This technique is extended to a wide-angle form, allowing larger propagation angles with respect to the horizon. A new impedance boundary condition is derived for electromagnetic waves incident on a finitely conducting surface that enables solution of the parabolic wave (PWE) using the previously developed mixed Fourier transform. It is also shown by example that in many cases of interest, the boundary may be approximated by discrete piecewise linear segments without affecting the field solution. A more accurate shift map solution of the PWE for a piecewise linear boundary is, therefore, developed for modeling propagation over digitally sampled terrain data. The shift-map solution is applied to various surface types, including ramps, wedges, curved obstacles, and actual terrain. Where possible, comparisons are made between the numerical solution and an exact analytical form. The examples demonstrate that the shift map performs well for surface slopes as large as 10-15° and discontinuous slope changes on the order of 15-20°. To accommodate a larger range of slopes, it is suggested that the most viable solution for general terrain modeling is a hybrid of the shift map with the well-known terrain masking (knife-edge diffraction) approximation     

Use of the generalized eikonal in the analytic solution of the problem of electromagnetic-wave diffraction by a 2D perfectly conducting half-plate

The method of the generalized eikonal is applied to obtain a general analytic solution to the problem of electromagnetic-wave diffraction by a 2D semi-infinite perfectly conducting scatterer with an open-polygon boundary. A specific solution is found for the problem of diffraction by a 2D perfectly conducting half-plate. Analytic formulas and calculation results are presented.     

Higher order asymptotic boundary condition for the finite elementmodeling of two-dimensional transmission line structures

The general form of the solution to Laplace's equation is used to derive a higher-order asymptotic boundary condition. The boundary condition is then implemented in the finite element scheme to model two-dimensional transmission line structures operating in the quasi-TEM mode. The boundary condition is generalized and made valid for an arbitrary outer boundary. The operator is applied on a rectangular outer boundary because that is the most conformable outer boundary for the structures considered. The numerical results of two- and six-conductor configurations showed that the higher-order asymptotic boundary condition yielded a significant improvement over the simple asymptotic boundary condition     

Two mathematical models of unconfined detonation and their numerical solution

We consider the numerical solution of an unconfined detonation problem modelled by differential/algebraic equations in a boundary value format. The problem has index one throughout the interval of integration, except at a free boundary point, where it has index two. Moreover, a nonlinear constraint is active at the free boundary point. We consider a simple mathematical model, as well as a more realistic and functionally complex model. In both cases, the resulting boundary value problem is solved directly, using a collocation method. This approach compares favorably with the solution techniques proposed previously in the literature and enables us to examine some interesting properties of the solution to the detonation problem.     

Higher order impedance boundary conditions applied to scattering bycoated bodies of revolution

In this paper higher order impedance boundary conditions will be employed in the solution of scattering by coated conducting bodies of revolution. The higher order impedance solution reduces the total number of unknowns relative to the exact solution, and produces a system matrix which is less dense than that of the exact solution. The construction of the solution involves two distinct steps. In the first step the body of revolution is replaced by an equivalent set of electric and magnetic currents on its exterior surface which generate the true fields outside the body. An integral equation relating these currents through the free space Green's function is derived. Step two employs the higher order impedance boundary condition to relate the electric and magnetic currents on the surface of the body. This replaces the rigorous solution of the interior problem. The higher order impedance boundary conditions are derived by obtaining an exact impedance boundary condition in the spectral domain for the coated ground plane, approximating the impedances as ratios of polynomials in the transform variables, and employing the Fourier transform. The resulting spatial domain differential equations are solved in conjunction with the integral equation using the method of moments. Several examples of bistatic and monostatic radar cross section for coated bodies of revolution are used to illustrate the accuracy of the higher order impedance boundary condition solution relative to the standard impedance boundary condition solution and the exact solution. The effects of coating thickness, loss, and curvature on the accuracy of the solution are discussed     

静态场边界条件有间断时边值问题的解法

静态场边值问题的解法可以分为解析法和数值法两大类.在解析法中,分离变量法是广泛应用的一种方法.当边界条件有间断时,在应用分离变量法过程中如何适当处理问题是值得注意的.本文讨论了在直角坐标和球坐标中边界存在跃变时静态场计算的两个问题,说明利用间断边界条件确定系数与单纯的傅里叶级数展开的区别,以免得出矛盾的结论.本文就此提出的一点见解,对正确应用分离变量法具有一定参考价值.     

Generalization of the method of extended boundary conditions

The method of extended boundary conditions is modified on the basis of the well-known null-field relationship and applied for solution of problems of diffraction by bodies with an analytic boundary. It is shown that the null-field condition is correctly formulated when it is imposed on a surface enclosing the set of the singularities of the analytically extended diffraction field. In addition, it is demonstrated that the best (in terms of the highest accuracy of computations) method of constructing the aforementioned surface involves analytic deformation of a scatterer’s boundary. The simulation results illustrate the efficiency of the proposed approach.     

The circular harmonic transform for SPECT reconstruction and boundary conditions on the Fourier transform of the sinogram

The circular harmonic transform (CHT) solution of the exponential Randon transform (ERT) is applied to single-photon emission computed tomography (SPECT) for uniform attenuation within a convex boundary. An important special case also considered is the linear (unattenuated) Radon transform (LRT). The solution is on the form of an orthogonal function expansion matched to projections that are in parallel-ray geometry. This property allows for efficient and accurate processing of the projections with fast Fourier transform (FFT) without interpolation or beam matching. The algorithm is optimized by the use of boundary conditions on the 2-D Fourier transform of the sinogram. These boundary conditions imply that the signal energy of the sinogram is concentrated in well-defined sectors in transform space. The angle defining the sectors depends in a direct way on the radius of the field view. These results are also obtained for fan-beam geometry and the linear Radon transform (the Fourier-Chebyshev transform of the sinogram) to demonstrate that the boundary conditions are a more general property of the Radon transform and a not a property unique to rectangular coordinates.     

Modeling of the electrodynamic contact of two materials under the action of electromagnetic waves

A mathematical model of bilateral boundary conditions coupling monochromatic electromagnetic fields on both sides of the transition layer between two materials is developed. The field-induced double-layer surface charges and currents and complex surface capacitances and inductances of the transition region are taken into account. An analytic solution to the boundary value problem of penetration of an arbitrarily polarized plane electromagnetic wave through a two-layered screen with a transition plane region between the layers is obtained. The transmission and reflection coefficients for a plane wave are calculated.     

Boundary waves along an impedance plane with a linearly varying impedance

A formal and asymptotic solution has been obtained for boundary waves which are supported by an infinite plane impedance boundary with a linearly varying impedance. The asymptotic solution (with respect to large distance from the origin) has three terms which are recognized as an incident, a reflected boundary wave, and a radiated wave. It is found that an increasing rate of change of the boundary impedance corresponds to more tightly bound boundary waves, to a power reflection coefficient approaching unity, and to a radiation efficiency approaching zero.     

Investigation of diffraction of an electromagnetic wave arbitrarily incident on a locally inhomogeneous medium interface

The 3D problem of reflection of a plane electromagnetic wave by an irregular boundary containing a locally inhomogeneous well conducting section is considered. The boundary value problem for the system of Maxwell equations in an infinite section with an irregular boundary is reduced to the solution of systems of singular equations. A numerical algorithm for their solution is developed. Results of calculation of the currents induced on the irregularity and reflected field patterns are presented for the case of the E polarization.     

Variational Solution of Integral Equations

A variational solution of the Fredholm integral equation of the first kind resulting from Laplace's equation with Dirichlet boundary conditions is discussed. Positive-definiteness of the integral operator is used to guarantee convergence. The square parallel plate capacitor is given as an example with several different types of trial functions. Special singular functions to handle known field behavior are shown to result in improved accuracy with reduced computing cost. The air-dielectric interface condition is related to a general Neumann-mixed boundary condition for which a variational method with a positive-definite integral operator is presented. Multiple boundary conditions are handled by mutually constraining separate variational expressions for each boundary condition. A T-shaped conductor on a dielectric slab, representative of quasi-static solutions of microstrip discontinuities, is presented as a three-dimensional example with multiple boundary conditions. Generally, it is shown how the finite-element method for the solution of partial differential equations may be extended to handle integral equation formulations.     

On the use of SVD-improved point matching in the current-modelmethod [EM scattering]

An approach which uses the singular value decomposition (SVD) to improve the accuracy of the numerical solution obtained with fictitious current models is introduced. In this approach, the SVD is essentially facilitating a systematic way to optimally reduce the generalized inverse matrix used in the solution to a submatrix of smaller rank. This reduction strikes a balance between the fulfillment of the boundary conditions at the matching points and that between them. Clearly, the boundary conditions errors at the matching points are no longer strictly zero. However, the previously discernible errors between the matching points are markedly suppressed. The approach is efficacious not only when the impedance matrix is inherently singular or highly ill conditioned, but also when this matrix is entirely well conditioned. It can be generalized and implemented in any method of moments code which uses point matching for testing. The approach has been incorporated into an existing solution based on the current-model method for the problem of scattering from periodic sinusoidal surfaces, and is shown to render the solution more accurate     

Analysis of a shielded microstrip line with finite metallizationthickness by the boundary element method

A boundary element method is presented for the quasi-static analysis of a shielded microstrip line with finite metallization thickness. The analysis is based on the solution of a system of boundary integral equations which are derived from Green's second identity. Numerical results for the charge distribution along the strip and the effects of metallization thickness on line characteristics are presented. The results show good agreement with data available in the literature