Time-domain Magnetic Field Integral Equation Research Articles

An Explicit Time Marching Scheme for Efficient Solution of the Magnetic Field Integral Equation at Low Frequencies

An explicit marching-on-in-time (MOT) scheme to efficiently solve the time-domain magnetic field integral equation (TD-MFIE) with a large time step size (under a low-frequency excitation) is developed. The proposed scheme spatially expands the current using high-order nodal functions defined on curvilinear triangles discretizing the scatterer surface. Applying Nyström discretization, which uses this expansion, to the TD-MFIE, which is written as an ordinary differential equation (ODE) by separating self-term contribution, yields a system of ODEs in unknown time-dependent expansion coefficients. A predictor-corrector method is used to integrate this system for the samples of these coefficients. Since the Gram matrix arising from the Nyström discretization is a block-diagonal, the resulting MOT scheme replaces the matrix “inversion” required at each time step by a product of the inverse block-diagonal Gram matrix and the right-hand side vector. It is shown that, on the convergence of the corrector updates, this explicit MOT scheme produces the same solution as its implicit counterpart and is faster for large time step sizes.

On the Stability of Time-Domain Magnetic Field Integral Equation Using Laguerre Functions

The marching-on-in-degree (MOD) method has been proposed for solving time-domain integral equations (TDIEs), and it uses the associated Laguerre functions as temporal basis and testing functions. Although the MOD scheme is assumed to be stable and accurate, there exists little investigation into the accuracy of the integral operations involving the associated Laguerre functions of high degree. It is shown in this paper that the numerical quadrature for integrating the highly oscillatory Laguerre functions may not be accurate and this may result in instability for the MOD solvers. In this paper, the highly oscillatory nature of the high-degree associated Laguerre functions is investigated first. Based on that, the numerical procedures are described which result in the inaccuracy of the numerical quadrature associated with the highly oscillatory nature for the higher order Laguerre functions. This can produce instability for the MOD solvers when a solution requires a high degree of the Laguerre functions. Hence, a novel Filon-type quadrature method and a stabilized MOD technique for time-domain magnetic field integral equations (TD-MFIE-SMOD) are presented in this paper. Some numerical results are presented to illustrate the validity of the proposed Filon-type quadrature rule resulting in stable solution of the TD-MFIE for transient scattering problems.

A Stabilized Marching-on-in-Degree Scheme for the Transient Solution of the Electric Field Integral Equation

Time-domain electric integral equation methods (TD-EFIE) suffer from the numerical stability problem. Numerous schemes have been proposed over the years for the purpose of taking care of this instability. Among them, the marching-on-in-degree (MOD) method is one of the newly proposed algorithms for calculating the various transient responses. The MOD solver employs a set of associated Laguerre functions and incorporates exact temporal Galerkin testing. Although the MOD scheme is assumed to be more stable when compared with marching-on-in-time (MOT), there are few numerical or theoretical investigations on the stability of the various numerical recursive operations for the MOD methodology. It has been shown that the time-domain magnetic field integral equation (TD-MFIE) using the MOD method can be stabilized by using a Filon-type quadrature formula to accurately integrate the highly oscillatory Laguerre functions which has caused instability for the MOD solvers using a higher degree of approximation. On the other hand, the analytical expression of temporal derivatives used in the MOD scheme is strongly affected by an unbounded initial value of the transient current. In this paper, a novel Filon-type radial integration method and a sequential constraint term for TD-EFIE-MOD are proposed, which make the stabilized MOD solver accurate and stable for the arbitrary degree of the Laguerre polynomials. Some numerical results are presented to illustrate the validity of the stabilized MOD solution of TD-EFIE for transient scattering problems.

Refining Transient Electromagnetic Scattering Analysis: A new approach based on the magnetic field integral equation

The magnetic field integral equation (MFIE) in a time-domain form is used to describe transient electromagnetic (EM) scattering by conductors. The MFIE is a second kind of integral equation that can help improve the conditioning of the matrix equation but is less widely addressed in the time domain. The time-domain MFIE (TDMFIE) is generally solved by the spatial-domain method of moments (MoM) at each time step, after being discretized in the time domain with a march-on-in-time (MOT) scheme. The MoM is inconvenient in implementation due to its special basis and testing functions, and, in addition, the MOT will produce the well-known instability problem. This article proposes a novel solving approach by incorporating a spatial-domain Nystr?m scheme with a robust singularity treatment that has a temporal-domain Galerkin method (TDGM) with Laguerre basis and testing functions. The Nystr?m scheme has several advantages, as demonstrated in the frequency domain, but it has not been used for the TDMFIE. The TDGM can fully overcome the drawback of the MOT and simplify the implementation by deriving closed-form formulations. Numerical examples for transient EM scattering by conductors including a concave object are presented here to illustrate the approach and its robust results.

Impulse responses and the late time stability properties of time‐domain integral equations

Impulse responses to the time-domain integral equations (TDIEs) using marching-on-in-time (MOT) scheme are discussed in this study. It can be seen clearly that the interior resonance currents are included in the impulse responses of the time-domain electric field integral equation (TDEFIE) and the time-domain magnetic field integral equation (TDMFIE), but not in those of the time-domain combined field integral equation (TDCFIE), while spurious DC solenoidal currents are only included in the TDEFIE. Analysis shows that the MOT-TDMFIE is basically exponentially convergent, and the MOT-TDCFIE usually converges to a small constant value, while the MOT-TDEFIE is basically unstable. The stability properties are justified by the eigenvalues of the corresponding systems. This study also shows that it is possible to solve time-domain integral equations (TDIEs) using their impulse responses truncated according to the maximum physical size of the scatterer.

An accurate time-domain algorithm using high-order spatial basis functions for bodies of revolution

In this article, a time-domain algorithm using high-order spatial basis functions is proposed to obtain the induced currents and scattering field for bodies of revolution accurately and effectively. The third-order Hermite polynomials are used as the spatial basis functions, and the Laguerre polynomials are employed as the temporal basis functions. The marching-on-in-degree method is applied to solve the time-domain magnetic field integral equation and ensures the late-time stability. The formulas are derived and numerical results are evaluated. We also compared the traditional method which uses the low-order spatial basis functions with the method proposed which shows to be more accurate and needs fewer computational loads.

Efficient Analysis of Shielding Effectiveness of Metallic Rectangular Enclosures Using Unconditionally Stable Time-Domain Integral Equations

This paper proposes an efficient modeling technique for transient electromagnetic analysis of a rectangular metallic enclosure with multiple apertures, using the surface equivalent principle (SEP) and integral equations. The SEP is used to replace the apertures by perfect electric conductor surfaces while introducing appropriate magnetic current densities on both sides of the apertures. As a result, the problem is reduced to two independent regions. The first region is outside the enclosure for which the free-space Green's function with the aid of image theory can be used to obtain the total fields in it. The second region is the inside of the enclosure and a closed-form expression is then derived for its time-domain dyadic Green's function. Finally, the governing time-domain magnetic field integral equation (TD-MFIE) for the structure is derived by enforcing the continuity of the tangential magnetic field at the apertures. To eliminate the late time instabilities in solving the resultant TD-MFIE, an unconditionally stable marching-on-in degree scheme is used. A new accelerated algorithm is proposed for the internal problem that relies on the separation of time and space in the cavity Green's function. The validity of the proposed method is confirmed by comparing the simulation results of several case studies with those obtained using the frequency-domain methods and CST microwave studio.

Marching On-In-Time Solution of the Time Domain Magnetic Field Integral Equation Using a Predictor-Corrector Scheme

An explicit marching on-in-time (MOT) scheme for solving the time-domain magnetic field integral equation (TD-MFIE) is presented. The proposed MOT-TD-MFIE solver uses Rao-Wilton-Glisson basis functions for spatial discretization and a PE(CE)m-type linear multistep method for time marching. Unlike previous explicit MOT-TD-MFIE solvers, the time step size can be chosen as large as that of the implicit MOT-TD-MFIE solvers without adversely affecting accuracy or stability. An algebraic stability analysis demonstrates the stability of the proposed explicit solver; its accuracy and efficiency are established via numerical examples.

Investigation of the stability of time-domain magnetic field integral equations based on marching on-in time algorithm

A sufficient stability condition for time-domain magnetic field integral equations (TDMFIE) based on marching on-in time (MOT) algorithm is obtained through theoretical derivation using the norm. If the condition is satisfied, it can be ensured that the computational results will always be stable in late-time no matter what kind of incident wave is. Moreover, a factor expressing the recursive relationship of the currents is obtained, which can represent the constringency of the currents. Finally, the sufficient stability condition is validated by numerical results. Meanwhile, the numerical results also verify that the factor can be used to represent the stability of the currents.

A Higher-Order Nyström Scheme for a Marching-On-in-Degree Solution of the Magnetic Field Integral Equation

A higher-order Nystrom scheme is developed for the marching-on-in-degree (MOD) solution of the time-domain magnetic field integral equation (TDMFIE) for the analysis of transient electromagentic scattering from a three-dimensional closed conducting object of arbitrary shape. In this method, the surface of the object is discretized into curvilinear triangular patches and the Lagrange interpolation polynomials are utilized to expand the spatial variation of the unknown electric current density in the TDMFIE. The transient variation of the electric current density is expanded in terms of the weighted Laguerre polynomials. With the use of the point-matching spatial and Galerkin temporal testing procedures, the proposed algorithm overcomes the late-time instability problem that often occurs in the marching-on-in-time (MOT) approach. Numerical results are presented to show that the proposed algorithm exhibits a good accuracy, a highly efficient computation of the impedance matrices, and a higher-order convergence with regard to the spatial discretization.

Nullspaces of MFIE and CalderÓn Preconditioned EFIE Operators Applied to Toroidal Surfaces

Magnetic field integral equation (MFIE) and Calderon preconditioned electric field integral equation (EFIE) operators applied to toroidal surfaces have nontrivial nullspaces in the static limit. The nature of these nullspaces is elucidated and a technique for generating a basis for them presented. In addition, the effects of these nullspaces on the numerical solution of both frequency and time-domain MFIE and CalderOacuten preconditioned EFIEs are investigated. The theoretical analysis is accompanied by corroborating numerical examples that show how these operators' nullspaces affect real-world problems.

Marching-on-in-Degree Based Time-Domain Magnetic Field Integral Equation Method for Bodies of Revolution

A marching-on-in-degree (MOD) based time-domain magnetic field integral equation method for bodies of revolution (BOR) is proposed and applied to obtain the induced currents on perfectly electric conducting BOR. Before this work, the time-domain integral equation method for BOR based on a marching-on-in-time procedure cannot really reduce the computational cost, since the number of unknowns cannot really be reduced. But it is the unknown reduction that serves as the key point of cost saving in BOR-problems. The method implemented in this letter can really utilize the symmetric property of BOR by applying two sets of entire domain basis functions. One is a set of scaled Laguerre polynomials inherited from common MOD method and used as temporal basis functions. The other is a Fourier series which comes from frequency domain method for solving BOR-problems. The validity, efficiency, and stability of the method are verified by several numerical examples.

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.

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.

Solution of a time‐domain magnetic‐field integral equation for arbitrarily closed conducting bodies using an unconditionally stable methodology

AbstractIn this work, we present a new and efficient numerical method to obtain an unconditionally stable solution for the time‐domain magnetic‐field integral equation (TD‐MFIE) for arbitrarily closed conducting bodies. This novel method does not utilize the customary marching‐on‐in‐time (MOT) solution method often used to solve a hyperbolic partial‐differential equation. Instead, we solve the wave equation by expressing the transient behaviors in terms of weighted Laguerre polynomials. By using these orthonormal basis functions for the temporal variation, the time derivatives in the TD‐MFIE formulation can be handled analytically. Since these weighted Laguerre polynomials converge to zero as time progresses, the electric surface currents also converge to zero when expanded in a series of weighted Laguerre polynomials. In order to solve the wave equation, we introduce two separate testing procedures: spatial and temporal testing. By introducing the temporal testing procedure first, the marching‐on in time procedure is replaced by a recursive relation between the different orders of the weighted Laguerre polynomials. The other novelty of this approach is that through the use of the entire domain Laguerre polynomials for the expansion of the temporal variation of the currents, the spatial and the temporal variables can be separated. To verify our method, we do a comparison with the results of an inverse Fourier transform of a frequency domain MFIE. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 35: 493–499, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10647

Transient Scattering From Three-Dimensional Conducting Bodies By Using Magnetic Field Integral Equation

In this paper, we present a time-domain magnetic field integral equation (TD-MFIE) to obtain the transient scattering response from three-dimensional closed conducting bodies. The formulation is described for explicit and implicit solutions. This approach results in accurate and comparably stable transient responses from conducting scatterers. Detailed mathematical steps are included, and several numerical results are presented and compared with results from a time-domain electric field integral equation (TD-EFIE) and the inverse Fourier transform solution of the frequency domain results.

Finite-difference approach to the solution of time-domain integral equations for layered structures

A numerical algorithm for the analysis of transient electromagnetic fields in planar structures is proposed based on the time-domain magnetic-field integral equation (MFIE), electric-field integral equation (EFIE), and the marching-on-in-time approach. The field vectors are represented in terms of vector potential functions which are calculated either by integration or by the three-dimensional (3-D) wave equation according to the geometry of the structure. Thus, the algorithm combines the advantages of integral equation techniques and finite-difference schemes. While this approach is applicable to any geometries, it is especially suitable for multilayered planar structures and is competitive to the finite-difference time-domain (FDTD) method in the case of open and radiating problems. Theoretical results are verified by the analysis of a pulse propagation in a homogeneous open-end microstrip line.