Rooftop Functions Research Articles

The Radar Cross Section Analysis of Radome Based on Conjugate Gradient Fast Fourier Transform

This paper presents an algorithm for analysis of the dielectric radomes. In this method, the radome is discretized by a regular grid with rooftop basic functions. The Volume Integral Equation (VIE) for 3D dielectric object is transformed to linear system by Galerkin’s testing formulation. Furthermore, the linear system is presented by Toeplitz matrix which can be solved by the Conjugate Gradient algorithm combined with Fast Fourier Transform (CG-FFT) iteratively. Also, the algorithm requires less computational complexity and memory. This paper simulates the mono-static Radar Cross Section of dielectric radome by the CG-FFT, which was validated against commercial software FEKO.

Fast Solution of Electromagnetic Scattering for 3-D Inhomogeneous Anisotropic Objects Embedded in Layered Uniaxial Media by the BCGS-FFT Method

In this paper, the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is used to solve the electromagnetic scattering problem for 3-D inhomogeneous uniaxial anisotropic objects embedded in a layered uniaxial medium. The optical axis of the background medium is perpendicular to the layer boundaries. However, the optical axis of the scatterer can be rotated through any angles. The electric field integral equation (EFIE) is formulated in the layered uniaxial anisotropic background medium. The volumetric roof-top function is used for the basis function as well as the testing function. The dyadic Green’s functions (DGFs) for both the electric field and the magnetic vector potential are used to solve the EFIE, and the results as well as the performance are compared. Several numerical simulations are carried out to validate the accuracy and efficiency of the proposed method. The major new contribution of this paper is to extend the BCGS-FFT method to accommodate the uniaxial anisotropy of the layered background medium. Therefore, the DGFs for the layered uniaxial medium are evaluated before solving the EFIE. In addition, the DGF for the electric field is also used to solve the 3-D scattering problem through BCGS-FFT iterations, which is different from the previous work.

Fast Simulation of Scattering Problem for Magnetodielectric Materials With General Anisotropy in Layered Media

In this paper, the mixed-order stabilized biconjugate gradient fast Fourier transform (mixed-order BCGS-FFT) method is presented to solve the scattering problem of magnetodielectric materials with general anisotropy in layered media. While the volumetric roof-top functions are used as the testing functions for the coupled field volume integral equation and the basis functions for flux densities, the second-order curl conforming basis functions are applied to expand the vector potentials with the aim of both preserving the continuity of their tangential components and avoiding the zero terms that might otherwise be caused by the divergence operator. The layered medium Green’s function (LMGF) is efficiently evaluated through the recursive matrix method along with an interpolation technique. Several numerical experiments are presented to demonstrate the high accuracy and efficiency of the method. Different from the previously published works that aim to solve the similar problem, the new contribution of this work is to extend the mixed-order BCGS-FFT method to accommodate the layered background medium. Therefore, the 3-D FFT acceleration for integral kernels associated with the LMGF as well as the interpolation technique has been implemented and combined with the mixed-order BCGS-FFT method.

Current Distribution of Dipole Antenna forDifferent Lengths Using Different Types ofBasis Functions Applying To Method ofMoment

In this paper, the generation of current distribution of a planar dipole antenna for different length of the antenna using different types of basic functions applying Method Of Moment (MOM) is discussed. Here, we give a brief description on MOM for generating the current distribution of the dipole antenna over the entire length of the antenna so that we became enable to calculate the self impedance of the dipole antenna. The calculation starts from the basic Pocklingtons' Integral Equation (PIE) of Greens' function. The basic functions are chosen as the rooftop function and the pulse function and the further progress has been done.

Numerical solution of the problem of diffraction of electromagnetic waves by nonplanar screens of complex geometric shapes by means of a subhierarchic method

The problem of diffraction of an electromagnetic wave by a nonplanar screen of a complex geometric shape located in free space is considered. The problem is reduced to an integral equation. A numerical method is proposed and realized with the use of rooftop functions for the solution of the integral equation. A subhierarchic method is applied for the solution of the integral equation. Numerical results are presented.

The RCS of a Resistive Rectangular Patch Antenna in a Substrate-Superstrate Geometry

The scattering radar cross section (RCS) and the resonant frequency problem of a superstrate loaded resistive rectangular microstrip patch which is printed on isotropic or uniaxial anisotropic substrate are investigated, where an accurate design based on the moment method technique is developed. The choice of two types of basis functions (entire domain and roof top functions) is illustrated to develop the unknown currents on the patch. The accuracy of the computed technique is presented and compared with other computed results.

On the Coupling Integrals Arising in the Method of Moments Formulation of Laterally Bounded Structures

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A generic integral equation and method of moments formulation is presented for laterally bounded stratified media including planar metallization. The main asset of the developed approach is its flexibility, as it encompasses generic lateral boundary conditions and explicitly applies to any linear subsectional basis functions with constant surface divergence. This includes the rooftop functions on rectangular and triangular supports currently proposed in standard method of moment meshers. This approach provides closed expressions for the coupling integrals appearing in the method of moments matrix elements. These formulas are based of Green's functions modal expansions and in the possibility, conclusively demonstrated in this paper to transform the surface integrals into contour integrals allowing an efficient and systematic implementation of the procedure. Full derivations are presented for several lateral boundary conditions, including rectangular and circular metallic cavities and periodic structures. Numerical examples including the analysis of real-life planar boxed circuits are presented. In all cases the obtained results compare favourably with other existing techniques. </para>

Radiation and resonant frequency of a resistive patch and uniaxial anisotropic substrate with entire domain and roof top functions

This paper develops and describes briefly the spectral domain analysis of microstrip patch antennas in which two sets of basis functions are used to model the patch current: the entire domain sinusoid basis functions and roof-top sub-domain basis functions. The moment method technique based on Galerkin's procedure is developed to examine the resonant frequency of a perfectly or an imperfectly conducting microstrip rectangular patch, which is printed on isotropic or uniaxial anisotropic substrate. It is seen that the results obtained by the roof top basis functions provides a significant improvement in the computations time with less iterations in the evaluation of the resonant frequency of a microstrip patch compared with the entire domain basis functions. Numerical results for the effects of surface resistance and uniaxial substrate on the radiation and the directivity of the rectangular microstrip structure are also presented. The accuracy of the computed technique is presented and compared with other computed results.

Bistatic Scattering and Emissivities of Lossy Dielectric Surfaces With Exponential Correlation Functions

Bistatic scattering and emissivities of surfaces with exponential correlation functions are studied numerically for 2-D geometries in a numerical Maxwell model with 2-D simulations. Surfaces with exponential correlation functions are important for the active and passive microwave remote sensing of land surfaces. Because of the fine-scale features with large slopes of such surfaces, numerical accuracy, which is particularly important for the calculation of emissivity in passive remote sensing, is ensured by a variety of procedures in this paper. The rooftop function and Galerkin's method with numerical integration of near-field impedance matrix elements are used. Cubic spline interpolation is employed to connect knots on random rough surfaces. Numerical accuracy convergence tests are performed for numerical solutions of Maxwell equations by varying the number of points from 13 to 103 points per wavelength in the dielectric medium corresponding to 50-400 points per free wavelength. Surface lengths of up to 100 and 200 free wavelengths and root mean square heights of up to 0.4 and 0.8 free wavelengths, respectively, are used at 5 and 10 GHz to capture all the essential features. Because of the large number of surface unknowns (up to 80 000), the multilevel UV method is further used to accelerate the matrix equation solver. Numerical results are illustrated for both bistatic scattering and emissivities as functions of frequencies and incidence and scattering angles for cases of interests in microwave remote sensing. Comparisons are made with the second-order small perturbation method and Kirchhoff's approximation to reestablish the regimes of validity of these methods

Application of two-dimensional nonuniform fast Fourier transform (2-D NUFFT) technique to analysis of shielded microstrip circuits

A two-dimensional nonuniform fast Fourier transform (2-D NUFFT) technique is developed for analysis of microstrip circuits in a rectangular enclosure. The 2-D Fourier transform of a nongrid point is approximated by Fourier bases in a square neighborhood with (q+1) by (q+1) grid points. The square neighborhood can be reduced to an octagonal region with q/sup 2//2+3q+1 grid points without sacrificing accuracy if q is sufficiently large. This technique allows an arbitrary discretization scheme on conductors and shows a great flexibility for the analysis. Asymmetric rooftop functions are inevitably used to expand surface current densities on conductors. Based on the spectral-domain approach, all elements of the final method-of-moments matrix are double summations of products of a weighted Green's function and trigonometric functions. By using the proposed technique, the double summations at all sampled points can be obtained via the 2-D NUFFT. The scattering parameters of a compact miniaturized hairpin resonator, an interdigital capacitor, and a wide-band filter are calculated. The calculated results show good agreement with measurements.

Analysis of microwave tomography systems using the wire‐volume integral equations method

AbstractA numerical scheme to analyse the forward problem in microwave tomography systems is introduced in this paper. Wire‐volume electric‐field integral equations are given to describe the electromagnetic environment in microwave tomography systems. The 1D triangular function and the 3D volumetric rooftop function are chosen as the basis functions to approximate the unknown electric current in the antennas and electric flux density in the object. The bi‐conjugate gradient method is employed to solve the system of linear equations. A comparison of the measured results shows that this scheme is efficient for simulating microwave tomography systems. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 43: 250–253, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20434

Full-Wave Spectral Analysis and Design of Annular Patch Antenna With Electromagnetically Coupled Microstrip Feed Line

This paper presents a full-wave analysis, and the ensuing design considerations, of the annular patch antenna fed by an electromagnetically coupled microstrip line. The numerical analysis employs the spectral domain version of the integral equation, with the method of moments formulation; the basis functions on the ring patch are the eigenmodes of the corresponding magnetic wall cavity, and subsectional (rooftop) functions are employed on the access line. Both TM and TE modes are used, along with more-efficient "collective" solenoidal modes; mode sorting criteria are given to ensure uniform convergence. Efficient schemes are presented for the calculation of the spectral-domain reaction integrals, along with criteria to avoid calculation of negligible mode-to-mode couplings, based on the spectral analysis of the matrix condition number. Very good agreement is shown with measured data, and design considerations are given for matching the radiator.

Loop-Tree Implementation of the Adaptive Integral Method (AIM) for Numerically-Stable, Broadband, Fast Electromagnetic Modeling

The adaptive integral method (AIM) is implemented in conjunction with the loop-tree (LT) decomposition of the electric current density in the method of moments approximation of the electric field integral equation. The representation of the unknown currents in terms of its solenoidal and irrotational components allows for accurate, broadband electromagnetic (EM) simulation without low-frequency numerical instability problems, while scaling of computational complexity and memory storage with the size of the problem are of the same order as in the conventional AIM algorithm. The proposed algorithm is built as an extension to the conventional AIM formulation that utilizes roof-top expansion functions, thus providing direct and easy way for the development of the new stable formulation when the roof-top based AIM is available. A new preconditioning strategy utilizing near interactions in the system which are typically available in the implementation of fast solvers is proposed and tested. The discussed preconditioner can be used with both roof-top and LT formulations of AIM and other fast algorithms. The resulting AIM implementation is validated through its application to the broadband, EM analysis of large microstrip antennas and planar interconnect structures.

Generalized Poisson–Neumann Polygonal Basis Functions for the Electromagnetic Simulation of Complex Planar Structures

Rooftop functions are commonly used for the discretization of planar currents in electromagnetic (EM) simulators. We describe the generalization of the rectangular and triangular rooftop functions to arbitrary polygonally shaped subdomains. It is shown that these generalized basis functions are solutions to a pertinent Neumann-Poisson problem, and we derive the integral equations satisfied by these basis functions. The new generalized polygonal functions allow for a more efficient meshing of complex geometrical structures in terms of polygonally shaped cells. They naturally model the current flow in the polygonal cells, satisfy the current continuity relation, and significantly enhance the EM simulation performance for complex geometrical structures. The increased simulation performance is demonstrated for a complex radio-frequency board interconnection layout and a spiral inductor on a silicon substrate.

Radiation and scattering by complex conformal antennas on a circular cylinder

A technique for characterizing and designing complex conformal antennas flush-mounted on a singly-curved surface is presented. This approach is based on the hybrid finite element–boundary integral (FE–BI) method. A related method was proposed in the past utilizing cylindrical-shell finite element and roof-top rectangular basis functions for the boundary integral. Although that method proved very powerful for analyzing cylindrical–rectangular patch arrays flush-mounted to a circular cylinder, the requirement for uniform meshing in the aperture ultimately limited its usefulness. In this present formulation, tetrahedral elements are used to expand the volumetric electric fields while similar basis functions are used for the boundary integral. The curvature of the aperture is explicitly included via the use of the circular cylinder dyadic Green's function. After presentation of the formulation and validation using several well-understood examples, an example is presented that illustrates the capabilities of this method for modeling complex conformal antennas heretofore not examined by rigorous methods due to inherent limitations of the various published methods.

A Weak Form of the Conjugate Gradient FFT Method for Two-Dimensional Elastodynamics

The problem of two-dimensional scattering of elastic waves by an elastic inclusion can be formulated in terms of a domain integral equation, in which the grad-div operator acts on a vector potential. The vector potential is the spatial convolution of a Green's function with the product of the density and the displacement over the domain of interest. A weak form of the integral equation for the unknown displacement is obtained by testing it with rooftop functions. This method shows excellent numerical performance.

Rational rooftop functions for convex quadrilaterals

Rational rooftop functions for convex quadrilaterals

An Improved Model for Full Wave Analysis of Multilayered Frequency Selective Surface With Gridded Square Element - Abstract

Analysis of frequency selective surface with gridded square element using accurate integral equation technique is considered in this paper. An improved subsectional current approximation model is proposed. Two more terms of basis function, the downward half triangle (DHT) term and the upward half triangle (UHT) term, besides the commonly adopted rooftop function, are included to expand the induced current. The additional terms are used to account for the effect of the induced currents at the corners of the outer square of the unit cell. Green's functions are derived by using spectral domain immittance approach and the incident fields are derived by using the z-directed potential. The computed results are in good agreement with the measured results.

A New Formula for the Evaluation of the Impedance Matrix in the Method of Moments

In this article, a new formula for computing the impedance matrix in the moment method is presented. By use of the properties of the basis functions, Green function, and the Fourier transformation, the integrals of the impedance matrix element can be simplified into one double integral (for two dimension) and one triple integral (for three dimension) when using the Galerkin's method. The singular point in the integrals can easily be removed through a suitable variable transform. The impedance matrix for the two dimensional case are computed by use of two kind of basis (One is the product of two triangular functions. Another is the roof-top function). Three examples of scattering from conducting plates are presented.

Fast algorithm for electromagnetic scattering by buried 3-D dielectric objects of large size

A fast algorithm for electromagnetic scattering by buried three dimensional (3-D) dielectric objects of large size is presented by using the conjugate gradient (CG) method and fast Fourier transform (FFT). In this algorithm, the Galerkin method is utilized to discretize the electric field integral equations, where rooftop functions are chosen as both basis and testing functions. Different from the 3-D objects in homogeneous space, the resulting matrix equation for the buried objects contains both cyclic convolution and correlation terms, either of which can be solved rapidly by the CG-FFT method. The near-scattered field on the observation plane in the upper space has been expressed by two-dimensional (2-D) discrete Fourier transforms (DFTs), which also can be rapidly computed. Because of the use of FFTs to handle the Toeplitz matrix, the Sommerfeld integrals' evaluation which is time consuming yet essential for the buried object problem, has been reduced to a minimum. The memory required in this algorithm is of order N (the number of unknowns), and the computational complexity is of order N/sub iter/N log N, in which N/sub iter/ is the iteration number, and N/sub iter//spl Lt/N is usually true for a large problem.