Time‐domain analysis of unbounded media using mixed‐variable formulations

The constant term of certain rational functions has attracted much attention recently. For example the Dyson conjecture; that the constant term ofis the multinomial coefficienthas spawned many generalizations (see [2], [7]). In this paper we consider some other families of rational functions which have interesting constant terms. For example, Corollary 4 states that the constant term of(1.1)is . Here, and throughout this paper, A and B denote fixed positive integers.In order to prove this result, we consider the rational function in two variables

An eigensolution expansion of dyadic Green's function for cylindrical sector structures is presented. The cylindrical structure is limited in azimuthal direction by electric or magnetic walls. Due to orthogonality of the eigensolution functions between the walls, the derivation of the dyadic Green's function can be done in spectral domain. The Green's dyad is used to set up an electric field integral equation for the microstrip structure. The solution is carried out with the method of moments. Numerical results for input impedance and radiation pattern are shown.

This paper contains further steps towards a Szegö theory for orthogonal rational matrix‐valued functions on the unit circle 𝕋. It continues the investigations started in [18]–[20]. Hereby we are guided by former work of Bultheel, González–Vera, Hendriksen, and Njåstad on scalar orthogonal rational functions on the one side and by research of Delsarte, Genin, and Kamp on matrix polynomials on the other side. In this paper we derive recursion formulas for Christoffel–Darboux pairs of rational matrix functions which lead us to jqq ‐recursively connected pairs of rational matrix functions. Moreover, we prove a Favard‐type theorem for rational matrix functions. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The scattering and radiation from large finite arrays of microstrip patches is analyzed by an improved spectral-domain moment method. Based on the electric field integral equation (EFIE) and using rooftop expansion and testing functions in a uniform discretization scheme, the elements of the resulting equation system are represented by inverse Fourier integrals. In the new approach, the application of the two-dimensional sampling function leads to equivalent finite integral expressions which are efficiently evaluated using Filon type quadrature rules in combination with the FFT algorithm. This reduces the number of sample points in the spectral domain drastically. The final discrete equation system is solved iteratively using the biconjugate gradient-FFT (BCG-FFT) algorithm. This results in a fast and accurate computation code appropriate for application on a PC. Several numerical examples for arrays up to N=32/spl times/16 patches are presented demonstrating the versatility of this technique.

A clever way to simulate Numerically Matched Loads is presented and applied in the mathematical framework of 3-D spectral domain analysis. The numerically matched loads already have been developed in the particular case of the spectral domain approach using a set of subsectional rooftop expansion and testing functions in the moment method. In order to properly model more complicated structures, we propose in this letter an extension of the NML simulation where triangular subdomain functions are used as both expansion and testing functions in the moment method. © 1997 John Wiley & Sons, Inc. Microwave Opt Technol Lett 15: 373–377, 1997.

Abstract. Representation of data on the sphere is conventionally done using spherical harmonics. Making use of the Fourier series of the Legendre function in the SH representation results in a 2D Fourier expression. So far the 2D Fourier series representation on the sphere has been confined to a scalar field like geopotential or relief data. We show that if one views the 2D Fourier formulation as a representation in a rotated frame, instead of the original Earth-fixed frame, one can easily generalize the representation to any gradient of the scalar field. Indeed, the gradient and the scalar field itself are simply linked in the spectral domain using spectral transfers. We provide the spectral transfers of the first-, second- and third-order gradients of a scalar field in a local frame. Using three numerical examples based on gravity and geometrical quantities, we show the applicability of the presented formulation.

In this paper we generalize for matrix valued functions a number of well known interpolation problems for scalar rational functions and obtain explicit formulas for the solutions. The realization approach toward the study of rational matrix functions from systems theory serves here as the main tool. The main results recently appeared in the literature; here we give a more systematic and transparent exposition based exclusively on analysis in finite dimensional spaces.

This paper presents an efficient uncertain analysis method for estimating the bounds on the reliability of structural systems in the presence of mixed uncertain variables. The proposed method involves high dimensional model representation (HDMR) for the limit state function approximation, transformation technique to obtain the contribution of the fuzzy variables to the convolution integral and fast Fourier transform for solving the convolution integral. In the proposed method, efforts are required in evaluating conditional responses at a selected input determined by sample points, as compared to full scale simulation methods. Therefore, the proposed technique estimates the failure probability accurately with significantly less computational effort compared to the direct Monte Carlo simulation. The methodology developed is applicable for structural reliability estimation involving any number of fuzzy variables and random variables with any kind of distribution. The accuracy and efficiency of the proposed method is demonstrated through numerical examples.

The integration of observations issued from a satellite-based system (GNSS) with an inertial navigation system (INS) is usually performed through a Bayesian filter such as the extended Kalman filter (EKF). The task of designing the navigation EKF is strongly related to the inertial sensor error modeling problem. Accelerometers and gyroscopes may be corrupted by random errors of complex spectral structure. Consequently, identifying correct error-state parameters in the INS/GNSS EKF becomes difficult when several stochastic processes are superposed. In such situations, classical approaches like the Allan variance (AV) or power spectral density (PSD) analysis fail due to the difficulty of separating the error processes in the spectral domain. For this purpose, we propose applying a recently developed estimator based on the generalized method of wavelet moments (GMWM), which was proven to be consistent and asymptotically normally distributed. The GMWM estimator matches theoretical and sample-based wavelet variances (WVs), and can be computed using the method of indirect inference. This article mainly focuses on the implementation aspects related to the GMWM, and its integration within a general navigation filter alibration procedure. Regarding this, we apply the GMWM on error signals issued from MEMS-based inertial sensors by building and estimating composite stochastic processes for which classical methods cannot be used. In a first stage, we validate the resulting models using AV and PSD analyses and then, in a second stage, we study the impact of the resulting stochastic models design in terms of positioning accuracy using an emulated scenario with statically observed error signatures. We demonstrate that the GMWM-based calibration framework enables to estimate complex stochastic models in terms of the resulting navigation accuracy that are relevant for the observed structure of errors.

The pattern synthesis based on the spectral Green's functions for multi-layered perforated dielectric loaded patch antenna is presented. It is conducted by optimising the dielectric superstrate parameters, i.e. relative permittivity (εr) and thickness. To solve the problem of the optimised εr unavailable from market, perforated dielectric superstrate (PDS) is utilised. The equivalence between PDS and homogeneous dielectric superstrate (HDS) in spectral domain is investigated and the effective relative permittivity (εreff) of PDS is studied. Moreover, by introducing εreff, the general optimisation procedure for HDS loaded antenna is also applicable for PDS loaded antenna, which reduces the computational cost significantly. To validate their method, a two-layer PDS loaded patch antenna with E/H beamwidth equalisation was designed, fabricated and measured. Simulations and measurements show that the synthesised antenna has satisfactory performances over the ITU-RS.1068-1994 band from 13.75 to 14 GHz. The half 10 dB E/H beamwidths of 19°/20°, −27 dB cross polarisation and −15.5 dB side lobe level have been achieved.

The Spectral Differential Equation Approximation Method (SDEAM) provides a closed-form approximation of the Sommerfeld integral and serves as an attractive alternative to existing Green's function approximation techniques. Recently, a high-order Discontinuous-Galerkin-Method (DGM) implementation of SDEAM for the Michalski-Zheng's mixed potential Green's function in shielded layered medium has been developed. For unbounded layered media, the radiation boundary condition (RBC) introduces an irrational dependence on the spectral lateral distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k_{p}$</tex> , whereas SDEAM typically exploits a polynomial dependence on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k_{p}$</tex> . This issue can be addressed by fitting this irrational dependence with rational functions. In this work, the high-order DGM-based SDEAM is augmented with the rational function fitting method in an effort to achieve an error-controllable framework for evaluating the mixed potential Green's function for RBCs. Consideration is limited to those Green's function components needed to solve 2.5D problems.

In this chapter we study an interpolation problem of the following type. A finite set of interpolating points is given, and the first several terms of the Taylor expansion of a rational function about each interpolation point are prescribed. As we have seen in Section 2 of Chapter I, there exists a polynomial solution to this interpolation problem. Here, however, we shall require that the degree of the rational function be sufficiently low for it to be admitted as a solution to the interpolation problem. We can state our problem more formally as follows: Let there be given l interpolation points in the complex plane $${z_1},{z_2}, \cdot \cdot \cdot ,{z_l}$$ and equally many nonnegative integers $${v_1},{v_2}, \cdot \cdot \cdot ,{v_l}$$ and N numbers f ij where 1≦i≦l and 0≦ j ≦ v i with \(N = \sum\limits_{i = 1}^l {({v_i} + 1)}\). It is required to find a rational function f (z) of degree no greater than N/2 for which the N equations $${f^{(j)}}({z_i}) = {f_{ij}}$$ are satisfied.

A general framework for deriving multi-parameter classes of methods of any presribed order is presented. In particular, two classes of cubically and fourth order convergent iterative methods for simple zeros of polynomial equations are derived. As special cases, the cubically convergent class includes the one-parameter Hansen-Patricks family and other well-known Laguerre’s, Halley’s and Ostrowski’s methods. Additionally, methods of any order which are based on Taylor expansion of certain rational functions are developed. Modified methods for extracting multiple zeros are briefly discussed.

The graded-index (GRIN) media have been widely used in modern optical systems, such as imaging lenses, optical beam delivery components, and beam shaping elements. Accurate and efficient modeling of electromagnetic fields propagating in GRIN media is essential for developing innovative and high-quality optical products. It is the goal of this thesis. We first give an overview of several existing modeling techniques, including the rigorous ones to model symmetric GRIN structures, e.g., Mie theory (spherically symmetric) and several beam propagation methods (BPMs) with approxiamations to model general GRIN structures, which are numerically more efficient than rigorous methods. However, many of BPMs are limited by considering only a small variance of the refractive index. To overcome the limitations, we develop a unified field solver, known as the Runge-Kutta (RK) k-domain field propagation method. It can be used to calculate general input fields propagating through an arbitrary GRIN medium, accurately, without physical approximations, such as the scalar field approximation or the paraxial approximation. We convert Maxwells equations in spatial (x-) domain into ordinary differential equations (ODEs) in the spatial-frequency (k-) domain, which can be solved iteratively using the RK method. In our numerical calculation, taking advantage of the convolution theorem, the total numerical effort is linear in the number of sampling points. As many GRIN components are modeled or designed using ray tracing techniques, the field passing through the GRIN components must show negligible diffraction, which satisfies the geometric field assumption. By substituting geometric field ansatz into the ODEs of RK k-domain field propagation method and Maxwells equations, we obtain two ODEs in x-domain. After solving them by using again RK method, the RK x-domain field propagation method is established.

The frequency-dependent spectral element matrix (or exact dynamic stiffness matrix) has been known to provide very accurate solutions in structural dynamics by using only a minimum number of degrees of freedom which may resolve the computation and cost problems. Thus, this paper presents a spectral element model for thin plates by using the concept of plane-waves superposition.