The state-variable approach to network analysis

The universality of the state-variable approach to network analysis is demonstrated in general discussions and specific examples. The method of formulation of the state equations for an arbitrary lumped, linear, finite, reciprocal, passive, time-invariant network is presented fully, while the relaxation of these restrictions is indicated in detail; i.e., the state-variable characterization of active, nonreciprocal, time-variable, and nonlinear networks is discussed. Finally, there is a brief guide of the current research where the state-variable analysis is brought to bear upon certain qualitative aspects of classical and nonclassical network behavior.     

透层天线电流积分方程的数值解

在已有的三层有耗媒质中的透层天线（ＢＰＡ）电流积分方程的基础上,给出了电流积分方程中的索末菲积分的低频近似解析公式并给出了其物理解释。利用三项式全域电流基函数,求得了透层天线的输入阻抗。采用静态场近似方法,很方便的求出了透层天线在地面的电场分布。     

Asymptotic expressions for the Fredholm determinant for state-variable covariance functions (Corresp.)

A new asymptotic expression for the Fredholm determinant is derived for stationary separable covariance functions. Solutions are given for the cases of known state-variable model and known separable covariance. The expression is obtained from the solution of a Riccati equation.     

An iterative solution of fredholm integral equations of the first kind

A formal solution is obtained for the Fredholm integral equation of the first kind. The unknown function is represented by a completely general Fourier series, and the Fourier coefficients are obtained by an iterative process. The formulation also yields an easily obtained approximate solution, as well as the estimate of its accuracy.     

Fast solution of electromagnetic integral equations using adaptivewavelet packet transform

The adaptive wavelet packet transform is applied to sparsify the moment matrices for the fast solution of electromagnetic integral equations. In the algorithm, a cost function is employed to adaptively select the optimal wavelet packet expansion/testing functions to achieve the maximum sparsity possible in the resulting transformed system. The search for the best wavelet packet basis and the moment matrix transformation are implemented by repeated two-channel filtering of the original moment matrix with a pair of quadrature filters. It is found that the sparsified matrix has above-threshold elements that grow only as O(N^1.4) for typical scatterers. Consequently the operations to solve the transformed moment equation using the conjugate gradient method scales as O(N^1.4). The additional computational cost for carrying out the adaptive wavelet packet transform is evaluated and discussed     

Numerical solution of integral equations for dielectric objects ofprismatic shapes

A numerical method to investigate scattering from dielectric geometries of prismatic shapes has been developed. The surface integral equations are formulated by Schelkunoff's equivalence principle in terms of equivalent surface electric and magnetic currents. To solve these integral equations for the unknown currents, the object's cross-section is mapped onto a circle. In the transformed space, Fourier type entire-domain basis functions are used in the cross section and triangular subdomain basis functions are selected along the generating curve to represent the currents. A moment method is then used to reduce the integral equations to a matrix equation to compute the current coefficients. It is found that the transformation of the object's surface to a circular shape improves the convergence of the current mode in the cross-section. However, the current modes are coupled on the surface and the matrix equation includes all the modes     

Fast solution of mixed-potential time-domain integral equations for half-space environments

A fast Fourier transform-accelerated integral-equation based algorithm to efficiently analyze transient scattering from planar perfect electrically conducting objects residing above or inside a potentially lossy dielectric half-space is presented. The algorithm requires O(N/sub t/N/sub s/(logN/sub s/+log/sup 2/N/sub t/)) CPU and O(N/sub t/N/sub s/) memory resources when analyzing electromagnetic wave interactions with uniformly meshed planar structures. Here, N/sub t/ and N/sub s/ are the numbers of simulation time steps and spatial unknowns, respectively. The proposed scheme is therefore far more efficient than classical time-marching solvers, the CPU and memory requirements of which scale as O(N/sub t//sup 2/N/sub s//sup 2/) and O(N/sub t/N/sub s//sup 2/). In the proposed scheme, all pertinent time-domain half-space Green functions are (pre) computed from their frequency-domain counterparts via inverse discrete Fourier transformation. In this process, in-band aliasing is avoided through the application of a smooth and interpolatory window. Numerical results demonstrate the accuracy and efficiency of the proposed algorithm.     

The well—posedness of the formulation of diffraction problems reduced to Fredholm integral equations of the first kind with a smooth kernel

The well-posedness of diffraction problems that are reduced to Fredholm integral equations of the first kind with a smooth kernel is analyzed. The auxiliary source method and the method of extended boundary conditions, both of which involve solution of Fredholm integral equations of the first kind with a smooth kernel, are applied to show for specific examples that algorithms of calculation of all physically significant quantities—the scattering pattern, the field at an arbitrary spatial point except current-carrier points, etc.—are quite stable and allow for computation of the aforementioned quantities with a preassigned accuracy.     

Sampling method using prefiltered band-limited Green's functionsfor the solution of electromagnetic integral equations

In the context of far-field scattering, the spatial spectrum of fields or currents on a scatterer may be approximated as band-limited. On this premise one can greatly simplify the computation of radiation integrals by eliminating in advance those spectral components of the Green's function not expected to be present in the fields. This approach enables one to dispense with the notion of basis functions and, instead, represent fields and currents as well as the impedance matrix representing the Green's function entirely in terms of a sparse set of samples, thereby obviating the need for integration. Moreover, by taking full advantage of the band-limited character of the solution, this sampling theoretical approach yields an efficient representation with a minimal number of degrees of freedom. The method was explored for scattering from two-dimensional perfect electric conductors including ellipsoidal cylinders and flat strips, in both polarizations, using uniform as well as nonuniform sampling, and yielded surprisingly good results     

On the use of overdetermined systems in the adaptive numerical solution of integral equations

The residual error incurred when numerically solving integral equations for a number of electromagnetic radiation and scattering problems is calculated with the aid of an overdetermined system. This error is systematically reduced by adaptively refining the model for the surface current. Error reduction is achieved by selectively shrinking cell dimensions (h-refinement), increasing the order of the basis functions representing the current (p-refinement), or a combination of both (hp-refinement). The correlation between residual error and surface current error is investigated and found to be high. The impact of edge singularities and curvature discontinuities is discussed.     

A boundary integral equation approach to oxidation modeling

Thermal oxidation of silicon involves the diffusion of oxidant molecules from the gas-oxide interface to the oxide-silicon interface, and the transport of newly formed oxide away from the latter. Under suitable formulations these two processes can be shown to be boundary-value problems of harmonic and biharmonic nature. Based on these properties, a boundary integral equation method (BIEM) has been developed for modeling oxidation. This method achieves simplicity and efficiency by solving a two-dimensional problem using line integrals on the boundaries. The use of source distributions as intermediary solutions facilitates direct calculations of a wide variety of boundary parameters.     

A comparison of integral equations with unique solution in theresonance region for scattering by conducting bodies

A comparison of integral equations, for problems involving scattering by arbitrary-shape conducting bodies, having a unique solution in the resonance region is presented. The augmented electric and magnetic field integral equations and the combined field integral equation, in their exact and approximate versions, are considered. The integral equations and the basis and test functions used in the method of moments to solve them are reviewed. Their implementation in a computer code is analyzed, mainly the relation between the matrix properties and the CPU time and memory. Numerical results (condition number and backscattering cross section) are presented for the cube. It is shown that the combined field integral equation, and the approximate (symmetric) combined field integral equation, are the most efficient equations to use in the neighborhood of resonant frequencies, because the overdetermined augmented integral equations require an extra matrix multiplication     

A Lyapunov approach to the stability of fractional differential equations

Lyapunov stability of fractional differential equations is addressed in this paper. The key concept is the frequency distributed fractional integrator model, which is the basis for a global state space model of FDEs. Two approaches are presented: the direct one is intuitive but it leads to a large dimension parametric problem while the indirect one, which is based on the continuous frequency distribution, leads to a parsimonious solution. Two examples, with linear and nonlinear FDEs, exhibit the main features of this new methodology.     

Multilevel fast multipole method solution of volume integral equations using parametric geometry modeling

We present a multilevel fast multipole method (MLFMM) solution for volume integral equations dealing with scattering from arbitrarily shaped inhomogeneous dielectrics. The solution accuracy, convergence, computer time and memory savings of the method are demonstrated. Previous works have employed the MLFMM for impenetrable targets. In this paper, we integrate the MLFMM with the volume integral equation method for scattering by inhomogeneous targets. Of particular importance is the use of curvilinear elements for better volume representation and the use of simple basis functions for ease of parallelization.     

求解时域电场积分方程的稳定时间步进算法

利用面积坐标变换、相对坐标变换、积分区域分解和广义Duffy坐标变换将时域电场积分方程中奇异性积分(共面、共边和共单结点的场源三角形单元上)转化成可精确计算的非奇异性积分.在不同时间基函数(导数连续和导数不连续)、不同时间步长情况下对比分析了该方法和现有的常用方法计算奇异性积分的精度.计算实例表明:时域阻抗矩阵的精确计算有效地改善了时间步进算法的后时稳定性.     

An effective wavelet matrix transform approach for efficientsolutions of electromagnetic integral equations

An effective numerical method based on wavelet matrix transforms for efficient solution of electromagnetic (EM) integral equations is proposed. Using the wavelet matrix transform produces highly sparse moment matrices which can be solved efficiently. A fast construction method for various orthonormal or nonorthonormal wavelet basis matrices is also given. It has been found that using nonsimilarity wavelet matrix transforms such as nonsimilarity nonorthonormal cardinal spline wavelet (NSNCSW) transform, one can obtain a much higher compression rate and much better accuracy of the approximate solutions than using similarity wavelet transforms such as Daubechies' (1992) orthonormal wavelet (DOW) transform. Numerical examples are given to show the validity and effectiveness of the method     

Applications of differential forms to boundary integral equations

In this paper we discuss the application of differential forms to integral equations arising in the study of electromagnetic wave propagation. The usual Stratton-Chu integral equations are derived in terms of differential forms and corresponding Galerkin formulations are constructed. All numerical schemes require the specification of basis functions and the use of differential forms provides a very general method for the construction of arbitrary order basis functions on curvilinear geometries. It is noted that the lowest order approximations on flat geometries reduce to forms essential equivalent to the standard Rao-Wilton-Glisson functions. The effect on accuracy is investigated for electric field integral equation and magnetic field integral equation formulations for a range of bases. Hierarchical classes of functions are also developed, as are transition elements useful in p-adaptive schemes where variable orders of approximation are sought.     

A simple approach to obtaining network state equations

A simple approach to obtaining the state equations for electric networks is introduced. This approach can easily accommodate any type of network element. It requires a relatively small number of simultaneous equations. An illustrative example is included.     

The closed-form solution of vector finite element integral equations for three dimensional electromagnetic analysis

In this paper, a set of closed-form formulas for vector Finite Element Method (FEM) to analyze three dimensional electromagnetic problems is presented on the basis of Gaussian quadrature integration scheme. By analyzing the open region problems, the first-order Absorbing Boundary Condition (ABC) is considered as the truncation boundary condition and the equation is carried out in a closed-form. Based on the formulas, the hybrid Expanded Cholesky Method (ECM) and MultiFrontal algorithm (MF) is applied to solve finite element equations. Using the closed-form solution, the electromagnetic field of three dimensional targets can be studied sententiously and accurately. Simulation results show that the presented formulas are successfully and concise, which can be easily used to analyze three dimensional electromagnetic problems.     

时域电场积分方程的稳定求解

提出了一种基于时域电场积分方程的稳定精确数值方法计算任意形状理想导体的时域散射.矢量位的时间导数采用中心差分近似,而标量位采用时间平均表示,并且标量位和矢量位在单元上随时间变化采用单元中心处的值来计算.将该方法与隐式方法相结合可以得到稳定精确的求解结果,而不必对电流密度进行时间或空间平均的处理过程.从模拟结果证明了该方法的稳定性和精确性.