Triangular Patch Basis Functions Research Articles

MoM과 PMCHW 적분방정식 융합에 의한 유전체 육면체의 유도전류 계산

In this paper, we analysis the electromagnetic scattering of an arbitrary shape dielectric cube subjected to plane wave incidence in three dimensions. MoM(Method of Moments)in which a surface of a body is divided with small triangular patches and equivalence principle are used to fuse the PMCHW(Poggio, Miller, Chang, Harrington, and Wu) Integral Equations with respect to equivalent currents on a dielectric body. Triangular patch and loop-patch basis functions that is robust in wide frequency ranges are used for MoM formulations. Proposed method is very useful to analysis the induced current of arbitrary dielectric bodies and numerical results for a dielectric cube are presented.

A STUDY ON THE NUMERICAL ACCURACY OF THE MATRIX ELEMENTS IN A TIME DOMAIN MOD METHODOLOGY

In a time domain Marching-on-in-degree (MOD) solver based on a Galerkin implementation of the Method of Moments (MoM), it is observed that the matrix elements for the matrix to be inverted contain integrals that are similar to the ones encountered in a frequency domain MoM solver using the piecewise triangular patch basis functions. It is also observed that the error in the evaluation of the matrix elements involving these integrals are larger in the time domain than those involved in the frequency domain MoM solvers. The objective of this paper is to explain this dichotomy and how to improve upon them when using the triangular patch basis functions for both the time and the frequency domain techniques. When the distance between the two triangular patches involved in the evaluation of the matrix elements, are close to each other or when the degree of the Laguerre polynomial in a MOD method is high, the integral accuracy will be compromised and the number of sampling points to evaluate the integrals need to be increased. Numerical results are presented to illustrate this point.

Transient electromagnetic scattering from dielectric objects using the electric field Integral equation with Laguerre polynomials as temporal basis functions

In this paper, we propose a time-domain electric field integral equation (TD-EFIE) formulation for analyzing the transient electromagnetic response from three-dimensional (3-D) dielectric bodies. The solution method in this paper is based on the Galerkin's method that involves separate spatial and temporal testing procedures. Triangular patch basis functions are used for spatial expansion and testing functions for arbitrarily shaped 3-D dielectric structures. The time-domain unknown coefficients of the equivalent electric and magnetic currents are approximated using a set of orthonormal basis function that is derived from the Laguerre functions. These basis functions are also used as the temporal testing functions. Use of the Laguerre polynomials as expansion functions for the transient portion of response enables one not only to handle the time derivative terms in the integral equation in an analytic fashion but also completely separates the space and the time variables. Thus, the time variable along with the Courant condition can be eliminated in a Galerkin formulation using this procedure. We also propose an alternative formulation using a different expansion of the magnetic current. The total computational cost for this new method is similar to that of an implicit marching-on in time (MOT)-EFIE scheme, even though at each step this procedure requires more computations. Numerical results involving equivalent currents and far fields computed by the two proposed methods are presented and compared.

Integral Equation Modeling of Multilayered Doubly-Periodic Lossy Structures Using Periodic Boundary Condition and a Connection Scheme

This paper presents a boundary integral formulation to analyze multilayered doubly-periodic lossy structures with arbitrary geometry. The formulation is based on the moment method using first-order triangular patch basis functions. Each individual layer is analyzed separately using the simple free-space Green's function. After discretization, periodic boundary conditions are imposed on each region and a connection scheme is used to connect the regions. Metallic patches between layers or on the periodic boundary are also included in the model. Several examples are presented showing both the flexibility and the accuracy of the method.

Time‐domain combined field integral equation using Laguerre polynomials as temporal basis functions

AbstractIn this paper, we propose a novel formulation to solve the time‐domain combined field integral equation (TD‐CFIE) for analysing the transient electromagnetic response from three‐dimensional (3D) closed conducting bodies. Instead of the conventional marching‐on in time (MOT) technique, the solution methods in this paper are based on the Galerkin's method that involves separate spatial and temporal testing procedure. Triangular patch basis functions are used for spatial expansion and testing functions for arbitrarily shaped 3D closed structures. The time‐domain unknown coefficient is approximated as an orthonormal basis function set that is derived from the Laguerre functions. These basis functions are also used as the temporal testing. With the representation of the derivative of the transient coefficient in an analytic form, the time derivative terms in the integral equations can be handled analytically. We also propose an alternative formulation to solve the TD‐CFIE. Two methods are presented that results in very accurate and stable transient responses from conducting objects. Numerical results are presented and compared with the inverse discrete Fourier transform (IDFT) of frequency‐domain combined field integral equation (FD‐CFIE) solutions. Copyright © 2004 John Wiley & Sons, Ltd.

Analysis of transient electromagnetic scattering from dielectric objects using a combined‐field integral equation

AbstractIn this paper, we analyze the transient electromagnetic response from three‐dimensional (3D) dielectric bodies using a time‐domain combined‐field integral equation. The solution method in this paper is based on the method of moments (MoM), which involves separate spatial and temporal testing procedures. Triangular‐patch basis functions are used for spatial expansion and testing functions for arbitrarily shaped 3D dielectric structures. The time‐domain unknown coefficients of the equivalent electric and magnetic currents are approximated by a set of orthonormal basis functions derived from the Laguerre polynomials and exponential functions. These functions are also used as temporal testing. In addition to use of the Laguerre polynomials as expansion functions for the transient portion of the response, two new source vectors related to the equivalent currents enables one to handle the time derivative terms of the vector potential in the integral equation in an analytic fashion. Numerical results computed by the proposed method are presented and compared. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 40: 476–481, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20009

Solving the Time-Domain Magnetic Field Integral Equation for Dielectric Bodies without the Time Variable through the Use of Entire Domain Laguerre Polynomials

In this paper, we propose a time-domain magnetic field integral equation (TD-MFIE) formulation for analyzing the transient electromagnetic response from three-dimensional (3-D) dielectric bodies. The solution method in this paper is based on the Galerkin method that involves separate spatial and temporal testing procedures. Triangular patch basis functions are used for spatial expansion, and testing functions for arbitrarily shaped 3-D dielectric structures. The time-domain unknown coefficients of the equivalent electric and magnetic currents are approximated as an orthonormal basis function set that is derived from the entire domain Laguerre functions. These basis functions are also used as the temporal testing. Through the use of the Laguerre functions, it is possible to perform the time derivatives in the integral equations analytically. In addition, due to the orthonormality and additivity properties of these basis functions, it is possible to eliminate the time variable completely from the computations, and therefore one does not have to worry about the Courant condition. We also propose an alternative formulation using a different expansion of the electric current. Numerical results involving equivalent currents and far fields computed by the two different implementations of the TD-MFIE are presented and compared.

SOLUTION OF TIME DOMAIN PMCHW FORMULATION FOR TRANSIENT ELECTROMAGNETIC SCATTERING FROM ARBITRARILY SHAPED 3-D DIELECTRIC OBJECTS

In this paper, we analyze the transient electromagnetic response from three-dimensional (3-D) dielectric bodies using a time domain PMCHW (Poggio, Miller, Chang, Harrington, Wu) integral equation. The solution method in this paper is based on the Galerkin's method that involves separate spatial and temporal testing procedures. Triangular patch basis functions are used for spatial expansion and testing functions for arbitrarily shaped 3-D dielectric structures. The time domain unknown coefficients of the equivalent electric and magnetic currents are approximated by a set of orthonormal basis functions that are derived from the Laguerre functions. These basis functions are also used as the temporal testing. Use of the Laguerre polynomials as expansion functions characterizing the time variable enables one to handle the time derivative terms in the integral equation and decouples the space-time continuum in an analytic fashion. We also propose an alternative formulation using a differential form of time domain PMCHW equation with a different expansion for the equivalent currents. Numerical results computed by the two proposed methods are presented and compared.

TIME DOMAIN EFIE AND MFIE FORMULATIONS FOR ANALYSIS OF TRANSIENT ELECTROMAGNETIC SCATTERING FROM 3-D DIELECTRIC OBJECTS

In this paper, we investigate various methods for solving a time-domain electric field integral equation (TD-EFIE) and a time- domain magnetic field integral equation (TD-MFIE) for analyzing the transient electromagnetic response from three-dimensional (3-D) dielectric bodies. The solution method in this paper is based on the method of moments (MoM) that involves separate spatial and temporal testing procedures. Triangular patch basis functions are used for spatial expansion and testing functions for arbitrarily shaped 3-D dielectric structures. The time-domain unknown coefficients of the equivalent electric and magnetic currents are approximated using a set of orthogonal basis functions that is derived from the Laguerre functions. These basis functions are also used as the temporal testing. Numerical results involving equivalent currents and far fields computed by the proposed TD-EFIE and TD-MFIE formulations are presented and compared.

Time-Domain EFIE, MFIE, And CFIE Formulations Using Laguerre Polynomials as Temporal Basis Functions for the Analysis of Transient Scattering from Arbitrary Shaped Conducting Structures — Abstract

In this paper, we present time-domain integral equation (TDIE) formulations for analyzing transient electromagnetic responses from three-dimensional (3-D) arbitrary shaped closed conducting bodies using the time-domain electric field integral equation (TDEFIE), the time-domain magnetic field integral equation (TD-MFIE), and the time-domain combined field integral equation (TD-CFIE). Instead of the conventional marching-on in time (MOT) technique, the solution methods in this paper are based on the Galerkin's method that involves separate spatial and temporal testing procedure. Triangular patch basis functions are used for spatial expansion and testing functions for arbitrarily shaped 3-D structures. The timedomain unknown coefficient is approximated by using an orthonormal basis function set that is derived from the Laguerre functions. These basis functions are also used as temporal testing. Using these Laguerre functions it is possible to evaluate the time derivatives in an analytic fashion. We also propose a second alternative formulation to solve the TDIE. The methods to be described result in very accurate and stable transient responses from conducting objects. Detailed mathematical steps are included and representative numerical results are presented and compared.

Adaptive multiscale moment method (AMMM) for analysis of scattering from three-dimensional perfectly conducting structures

The adaptive multiscale moment method (AMMM) is presented for the analysis of scattering from three-dimensional (3D) perfectly conducting bodies. This algorithm employs the conventional moment method (MM) using the subsectional triangular patch basis functions and a special matrix transformation, which is derived from solving the Fredholm equation of the first kind by the multiscale technique. This methodology is more suitable for problems where the matrix solution time is much greater than the matrix fill time. The widely used triangular patch vector basis functions developed by Rao et al., (1982), are used for expansion and testing functions in the conventional MM. The objective here is to compress the unknowns in existing MM codes, which solves for the currents crossing the edges of the triangular patch basis functions. By use of a matrix transformation, the currents, source terms, and impedance matrix can be arranged in the form of different scales. From one scale to another scale, the initial guess for the solution can be predicted according to the properties of the multiscale technique. AMMM can reduce automatically the size of the linear equations so as to improve the efficiency of the conventional MM. The basic difference between this methodology and the other wavelet-based techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly in an iterative fashion even though it is an unknown. Two numerical results are presented, which demonstrate that the AMMM is a useful method for analysis of electromagnetic scattering from arbitrary shaped 3D perfectly conducting bodies.

First Order Triangular Patch Basis Functions for Electromagnetic Scattering Analysis

This paper analyzes a set of first order triangular patch basis functions for the method of moments (MoM) solution of electromagnetic scattering problems, and compares its performance to the traditional Rao-Wilton-Glisson (RWG) basis. These basis functions provide a faster convergence and a smoother surface current representation. It is shown that the computational complexity involved is about the same as the RWG basis, which means that existing computer codes can be adapted to use them with minimal modifications.

Rigorous modeling of ultrawideband VHF scattering from tree trunks over flat and. sloped terrain

Three electromagnetic models are employed for the investigation of ultrawideband VHF scattering from tree trunks situated over flat and sloped terrain. Two of the models are numerical, each employing a frequency-domain integral-equation formulation solved via the method of moments (MoM). A body-of-revolution (BoR) Mote formulation is applied for a tree trunk on a flat terrain, implying that the BoR axis is perpendicular to the layers of an arbitrary layered-earth model. For the case of sloped terrain, the BoR model is inapplicable, and therefore the MoM solution is performed via general triangular-patch basis functions. Both MoM models are very accurate but are computationally expensive. Consequently, the authors also consider a third model, employing approximations based on the closed-form solution for scattering from an infinite dielectric cylinder in free space. The third model is highly efficient computationally and, despite the significant approximations, often yields accurate results relative to data computed via the reference MoM solutions. Data from the three models are considered, and several examples of application to remote sensing are addressed.

Method of moments analysis of electromagnetic scattering from a general three‐dimensional dielectric target embedded in a multilayered medium

The method of moments (MOM) is applied to the problem of electromagnetic scattering from general three‐dimensional dielectric targets in an arbitrary multilayered environment. The dyadic multilayered Green's function is computed via the method of complex images, and the Galerkin MOM solution is effected by employing triangular patch basis functions. Several example frequency and time domain results are presented, with application to radar‐based sensing of plastic land mines.

Analysis of transient scattering from composite arbitrarily shaped complex structures

A time-domain surface integral equation approach based on the electric field formulation is utilized to calculate the transient scattering from both conducting and dielectric bodies consisting of arbitrarily shaped complex structures. The solution method is based on the method of moments (MoM) and involves the modeling of an arbitrarily shaped structure in conjunction with the triangular patch basis functions. An implicit method is described to solve the coupled integral equations derived utilizing the equivalence principle directly in the time domain. The usual late-time instabilities associated with the time-domain integral equations are avoided by using an implicit scheme. Detailed mathematical steps are included along with representative numerical results.

Analysis of scattering by surfaces using a wavelet-transformed triangular-patch model

When analyzing electromagnetic scattering by three-dimensional perfectly conducting bodies of arbitrary shape, the surface modeling is often affected by triangulation, and in turn, triangular-patch basis functions are used for expanding the unknown surface current. In this paper, we apply a wavelet transformation to transform the triangular-patch basis functions to a new set of basis functions, which can be interpreted as wavelet combinations of the original basis functions. The new basis functions can lead to a matrix representation of the operator equation that is more localized and which, by proper thresholding, can be rendered sparse. Alternatively, they can lead, via an impedance matrix compression (IMC) approach, to a matrix equation that is compressed (rank reduced). The solution of either the sparse or the compressed matrix equation can yield fairly accurate results with less computational effort. The dependence of the results on ordering of the original triangular-patch basis functions prior to the transformation is discussed. ©1999 John Wiley & Sons, Inc. Microwave Opt Technol Lett 21: 359–365, 1999.

Thin-stratified medium fast-multipole algorithm for solving microstrip structures

An accurate and efficient technique called the thin-stratified medium fast-multipole algorithm (TSM-FMA) is presented for solving integral equations pertinent to electromagnetic analysis of microstrip structures, which consists of the full-wave analysis method and the application of the multilevel fast multipole algorithm (MLFMA) to thin stratified structures. In this approach, a new form of the electric-field spatial-domain Green's function is developed in a symmetrical form which simplifies the discretization of the integral equation using the method of moments (MoM). The patch may be of arbitrary shape since their equivalent electric currents are modeled with subdomain triangular patch basis functions. TSM-FMA is introduced to speed up the matrix-vector multiplication which constitutes the major computational cost in the application of the conjugate gradient (CG) method. TSM-FMA reduces the central processing unit (CPU) time per iteration to O(N log N) for sparse structures and to O(N) for dense structures, from O(N/sup 3/) for the Gaussian elimination method and O(N/sup 2/) per iteration for the CG method. The memory requirement for TSM-FMA also scales as O(N log N) for sparse structures and as O(N) for dense structures. Therefore, this approach is suitable for solving large-scale problems on a small computer.

Stripline-fed arbitrarily shaped printed-aperture antennas

A full-wave analysis method is presented for modeling the radiation properties of a stripline-fed planar printed-aperture antenna element. In this formulation, both the finite length of stripline and the finite aperture may be of any arbitrary shape since their equivalent electric and magnetic currents are modeled with triangular patch basis functions. Galerkin's method is applied to numerically solve the coupled mixed-potential integral equation (MPIE). Exact spatial-domain Green's functions are used to account for all radiation, surface-wave, and mutual-coupling effects. Interactions between the stripline feed and the radiating aperture are rigorously included. Numerical analysis is presented for the following nonrectangular shapes: an exponentially tapered slot, an annular slot, an annular slot with opposing stubs, and a monofilar Archimedean spiral slot element. Results are shown for input return loss, radiation patterns, and axial ratio for the circularly polarized elements.

Integral equation analysis of dielectric and conducting bodies of revolution in the presence of arbitrary surfaces

A formulation is developed to treat radiation from structures consisting of conducting and/or dielectric bodies of revolution (BOR) in the presence of multiple arbitrary shaped three-dimensional objects. A set of integral equations is developed on the surfaces of the combined structure and the resulting integro-differential equations are solved using the method of moments. On the BOR, harmonic entire domain expansion functions are used for the circumferential dependence, while overlapping sub-domain functions are used to model axial curvature. The arbitrary shaped portions of the structure are modeled using triangular surface patch basis functions. The resulting matrix has a partial block diagonal nature which provides a more economical solution for structures which have some rotational symmetry. The accuracy of the BOR and arbitrary surface formulation is verified using the self-consistency method and measured data.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Electromagnetic scattering from 3-D arbitrary coated cavities via a connection scheme using triangular patches

The electromagnetic scattering from arbitrarily shaped 3-D cavities with coated walls is presented. In this formulation, the cavity is divided into smaller sections. Each section is individually analyzed with a boundary integral equation, followed by the standard moment method procedure to arrive at an admittance matrix for each section. These admittance matrices for different sections are then cascaded using a connection scheme based on microwave network theory. Once the admittance matrix of the whole cavity is obtained, the electric field on the aperture and thus the backscattered field can be obtained by the generalized network formulation. Triangular patch basis functions are used to provide easy mesh generation using standard CAD tools. To model wall treatment, the impedance boundary condition is applied. The numerical results generated by the present approach show good agreement with the modal solutions for regular geometries. An example for a cavity with coated walls is also presented.