Magnetic Field Integral Equation Research Articles

Integral Equation Residuals for Error Estimation and Internal Resonance Detection

The performance of the four basic residual error estimators for perfectly conducting targets is evaluated in connection with the Rao-Wilton-Glisson method of moments numerical solutions of the electric field, magnetic field, and combined field integral equations. Results for 17 perfectly conducting test targets are used to evaluate the performance. Tangential-field residuals appear to be more reliable than normal-field residuals for assessing accuracy. Residual estimates are shown to be poor when used to assess an equation based on the same boundary condition as the residual. Residuals are sensitive to internal resonance modes associated with the complementary boundary condition and can be used to detect spurious solutions due to their presence.

Analyzing electromagnetic scattering from complex multi-layer patch objects using a multi-trace domain decomposition method

The local-coupling multi-trace and domain decomposition method originally developed to analyze electromagnetic scattering from closed composite objects is extended to solve the scattering from complex multilayer patch objects. Different from the traditional global radiation coupling surface integral equation domain decomposition method and contact-region modeling method (CRM), the local-coupling property yields many excellent properties, highly sparse matrix, better computational performance, etc. In this novel proposed method, the multilayer patch objects are decomposed into many independent sub-domains, and each sub-domain is assumed as a homogeneous dielectric subdomain to formulate the electric-field integral equation (EFIE) and the magnetic-field integral equation (MFIE) for dielectrics as the governing equations. The corresponding boundary condition of the perfect electric conducting (PEC) surface sheet is then achieved by adopting Robin transmission conditions (RTCs) to PEC surfaces, termed PEC-RTCs. The final resulting equation yield a better performance than the traditional method. Numerical examples validate the advantages of the proposed method.

A fast time-domain boundary element method for three-dimensional electromagnetic scattering problems

This paper proposes a fast time-domain boundary element method (TDBEM) to solve three-dimensional transient electromagnetic scattering problems regarding perfectly electric conductors in the classical marching-on-in-time manner. The algorithm of the fast TDBEM is a time-domain variant of the interpolation-based fast multipole method (IFMM), which is similar to the time-domain IFMM for acoustic scattering problems investigated in the author's previous studies. The principle of the present IFMM is to interpolate the kernel functions of the electric and magnetic field integral equations (EFIE and MFIE, respectively) so that every kernel function is expressed in a form of separation of variables in terms of both the spatial and temporal variables. Such an expression enables to construct a fast method to evaluate the scalar and vector potentials in the EFIE and MFIE with using so-called multipole-moments and local-coefficients associated with a space-time hierarchy. As opposed to $O(N_s^2 N_t)$ of the conventional TDBEM, the computational complexity of the fast TDBEM is estimated as $O(N_s^{1+\delta}N_t)$, where $N_s$ and $N_t$ stand for the spatial and temporal degrees of freedom, respectively, and $\delta$ is typically $1/2$ or $1/3$. The numerical examples presented the advantages of the proposed fast TDBEM over the conventional TDBEM when solving large-scale problems.

Near-Field IPO for Analysis of EM Scattering from Multiple Hybrid Dielectric and Conductor Target and High Resolution Range Profiles

Aiming at improving the accuracy and efficiency of scattering information from multiple targets in near-field regions, this paper proposes a near-field iterative physical optics (IPO) method based on a modified near-field Green’s function for the composite electromagnetic scattering analysis of multiple hybrid dielectric and conductor targets. According to the electric field and magnetic field integral equation, the electric and magnetic current were updated utilizing the Jacobi iteration method. Then, by introducing an expansion center lying in the neighborhood of the source point, Green’s function was modified for near-field scattering between multiple hybrid dielectric and conductor targets. To accelerate the implementation of the procedure, the multilevel fast multipole method, the fast far-field approximation, and parallel multicore programming were introduced. Numerical results indicate that there is good agreement between the results calculated by the near-field IPO method and MLFMM solver in commercial software FEKO while significantly reducing the computational burden. To fully exploit the scattering information, the high resolution range profiles (HRRP) of different targets under different conditions were analyzed, which can be further applied for automatic target detection and recognition.

Code-verification techniques for the method-of-moments implementation of the magnetic-field integral equation

For computational physics simulations, code verification plays a major role in establishing the credibility of the results by assessing the correctness of the implementation of the underlying numerical methods. In computational electromagnetics, surface integral equations, such as the method-of-moments implementation of the magnetic-field integral equation, are frequently used to solve Maxwell's equations on the surfaces of electromagnetic scatterers. These electromagnetic surface integral equations yield many code-verification challenges due to the various sources of numerical error and their possible interactions. In this paper, we provide approaches to separately measure the numerical errors arising from these different error sources. We demonstrate the effectiveness of these approaches for cases with and without coding errors.

Accuracy Analysis of Div-Conforming Hierarchical Higher-Order Discretization Schemes for the Magnetic Field Integral Equation

Accuracy Analysis of Div-Conforming Hierarchical Higher-Order Discretization Schemes for the Magnetic Field Integral Equation

Simulation of realistic speckle fields by using surface integral equation and multi-level fast multipole method

This report presents a speckle simulator which calculates rigorously speckle fields with full randomization from rough surfaces. To achieve this, surface integral equation (SIE) method accelerated via multilevel fast multipole method (MLFMM) is implemented. Among several linear formulations to combine the electric and magnetic field integral equations, we demonstrate that one is most suitable for dielectric rough surfaces and one for metallic rough surfaces. As examples, silver and silicon rough surfaces are simulated at a wavelength of 500 nm. To investigate speckles from extended areas, silicon rough surfaces with a size of 30×30 µm2 and two million unknowns are calculated. Based on the resultant speckle fields, speckle contrast, angular speckle-correlation and transverse autocorrelation function of intensity in the van Cittert-Zernike zone are further calculated and known relations versus surface roughness are manifested, indicating that sufficient statistic surface information is encoded in the speckle fields. The SIE-MLFMM method with the suitable formulation proves to be an efficient rigorous simulation tool for relatively large rough surface problems and would enable more profound numerical studies for speckle involved optical metrologies.

Improving the Solution Accuracy of the Volume-Surface Integral Equation for the Open PEC Structures Embedded Within Dielectrics

In the solution of volume-surface integral equation (VSIE) for open perfect electric conductor (PEC) structures embedded within dielectric substrates, how to explicitly enforce the continuity condition (CC) of electric flux on the PEC–dielectric interfaces to improve the numerical accuracy is investigated. In the traditional VSIE formed by combining the surface electric field integral equation (EFIE) and volume integral equation (VIE), the fictitious surface charge, which does not exist physically, is accumulated on the PEC–dielectric interfaces. This will lead to the inaccuracy of numerical results, which can be eliminated by the CC. To correctly enforce the CC, the equivalent surface current distribution on both sides of open PEC should be distinguished. For this purpose, the combined field integral equation (CFIE) consisting of electric field and magnetic field integral equations is used to model the open PEC part. In the proposed approach, the number of surface unknowns needs to be doubled, which can be balanced by reducing the volumetric unknowns through the enforcement of CC. Furthermore, by fully considering the numerical symmetry between both sides of open PEC surface, the filling process of the matrix as well as the excitation vector in implementing the method of moments (MoMs) can be simplified. Numerical experiment demonstrates that the enforcement of CC can improve the VSIE accuracy obviously. For example, when calculating the frequency response of frequency selective surfaces (FSSs), the results obtained from the traditional VSIE often shift to higher frequencies, while the proposed approach gives reasonably good results.

Surface Integral Equation With Multibranch RWG Basis Functions for Electromagnetic Scattering From Dielectric Objects

A surface integral equation method with multibranch Rao–Wilton–Glisson (MB-RWG) basis functions for scattering from dielectric objects is proposed based on the electric and magnetic current combined field integral equations (JMCFIE). The MB-RWGs are defined at the boundaries between dense and coarse meshes and can keep the current continuity. Thus, the JMCFIE with MB-RWGs allows different surfaces of the dielectric models to be independently discretized by triangles with different sizes. Compared with the conventional JMCFIE, the JMCFIE with MB-RWGs increases the flexibility of mesh generation, which is attractive for multiscale dielectric models. Furthermore, the iterative convergence of JMCFIE with MB-RWGs is similar to that of conventional JMCFIE. Numerical examples are presented to demonstrate the advantages of the proposed method.

The Applications of the Symmetric Layered Medium Green’s Function in Magnetic Field Integral Equation

A symmetry relation of the integral operators in a layered medium proved by the reciprocal theory can be applied to improve the accuracy of the magnetic-field integral equation (MFIE). Based on the symmetric layered medium Green’s function, the normal vector can be transferred into the inner integral to keep the field integral as the inner integral of the elements in the impedance matrix. When discretized by the divergence-conforming function, this field-extracted scheme shows better performance than the classic expression in accuracy. Meanwhile, this form of expression can be applied in the combined-field integral equation (CFIE). Several numerical results are provided to validate the formulation of the symmetry relations from the layered medium Green’s function to the impedance matrices and to reveal its advantages in enhancing the accuracy of the field-extracted form of MFIE.

Accelerated Adaptive Error Control and Refinement for SIE Scattering Problems

We present an application of adjoint-based adaptive error control and refinement for scattering problems solved using the method of moments (MoM) in the electric field integral equation (EFIE) and the coupled EFIE and magnetic field integral equation (MFIE) formulations. Specifically, we examine the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) formulation of the EFIE-MFIE. We first construct the adjoint problems of the EFIE and the EFIE-MFIE for estimating the error of radar cross section (RCS) quantities of interest (QoIs) directly. We then introduce an effective adaptive refinement algorithm based on an error prediction heuristic and a posteriori error estimation, which enables rapid and consistent convergence regardless of a chosen tolerance and the coarseness of the starting discretization. The approach, moreover, inherently promotes equilibration of the QoI error contributions and produces, therefore, consistently balanced meshes. Numerical examples with canonical scattering targets and adaptive <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> -refinement confirm the strength of the proposed refinement method, demonstrating the ability to generate high-quality discretizations, both in terms of accuracy and efficiency, without expert user intervention.

Low-Frequency Transmitted Fields of a Source Inside a Magnetic Shell with Large Conductivity

The method to evaluate the transmitted fields of a source inside a simply connected magnetic shell with large but finite conductivity at low frequencies is proposed in this paper. When modeling the magnetic shell with large conductivity, it is regarded as a penetrable object. Electric field integral equation (EFIE) is selected for the exterior region problem and magnetic field integral equation (MFIE) is chosen for the interior region problem. Each operator is decomposed with loop-star functions to overcome the problem of low-frequency breakdown. Numerical results verify the accuracy of the proposed method.

Low-Frequency Analysis of Lossy Interconnect Structures Based on Two-Region Augmented Volume-Surface Integral Equations

Real conductors of interconnect structures are lossy and their skin depth becomes large at low frequencies. The traditional one-region formation with the approximation of perfectly electric conductors (PECs) or surface impedances may not be valid anymore, and two-region integral equation formations are needed in the integral equation approach. Also, the electric field integral equation (EFIE) tends to break down at low frequencies, and augmented electric field integral equation (AEFIE) has been proposed to remedy the problem. In this work, we treat lossy conductors as penetrable objects and propose two-region augmented hybrid field integral equations (AHFIEs) for low-frequency analysis. The HFIEs consist of the EFIE of describing the exterior of a conductor and the magnetic field integral equation (MFIE) of describing its interior. Since the magnetic current density appears in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal{ L}}$ </tex-math></inline-formula> operator in the HFIEs, we select the magnetic charge density as a new unknown function to be solved and introduce the continuity equation of magnetic current density as an extra equation. By incorporating the volume integral equations (VIEs) of describing the substrate with arbitrary penetrable media in the interconnect structures, the two-region augmented volume-surface integral equations (AVSIEs) are formulated for entire structures. The AVSIEs are solved by the method of moments (MoM) where the Rao–Wilton–Glisson (RWG) and Schaubert–Wilton–Glisson (SWG) basis functions are used to represent the surface current densities of AHFIEs and volume current densities of VIEs, respectively, while a pulse basis function is employed to represent the charge densities of AHFIEs. Numerical examples are presented to illustrate the approach and good results have been obtained.

Solving the Surface Current Distribution for Open PEC-Dielectric Objects Using the Volume Surface Integral Equation

For the open perfect electric conductor (PEC)-dielectric objects, a novel type of volume surface integral equation (VSIE) is proposed to solve the current distribution on both sides of the open PEC surfaces, which cannot be obtained by using the traditional VSIE. In the proposed method, both the electric and magnetic field integral equations are established on both sides of the PEC surfaces, whereas the basic functions are also defined at the borders of PEC surfaces to consider the current flowing over the edges of PEC surfaces. Compared with the traditional VSIE, due to the asynchronous solution processes of the surface current, the added CPU time for the proposed method is quite limited, whereas the memory usage is the same. Numerical results verify the validity and efficiency of the proposed method.

High-efficient analysis of metal target electromagnetics above the half-space based on mixed field integral equation

A new acceleration method is proposed for efficiently solving the problem of electromagnetic scattering from metal targets in half-space. The analysis of electromagnetic problems in any structure can be settled by the electric field integral equation. But the generated matrix condition number is large and the iterative solution has poor convergence. The number of the matrix condition generated by the magnetic field integral equation is small and iterative convergence is good. But only the closed structure problems can be worked out. The combined field integral equation is adopted because of the universality of the electric field integral equation and the convergence of the magnetic field integral equation. The gradient term of Green's function is involved in the integral equation of the mixed field. In order to further enhance the calculation efficiency, an efficient four-dimensional spatial interpolation method is introduced for half-space Green's function. Tabulation and lagrange interpolations are performed in the Sommerfeld integrals for the half-space Green's function. The improved efficiency can be 7.5 times higher than that of the traditional combined field integral equation(CFIE). Numerical results show that the computational time can be reduced significantly by the proposed method with encouraging accuracy.

A Discontinuous Galerkin Combined Field Integral Equation Formulation for Electromagnetic Modeling of Piecewise Homogeneous Objects of Arbitrary Shape

We present a novel discontinuous Galerkin (DG) surface integral equation approach, based on the electric and magnetic current combined field integral equations (JMCFIE), for the electromagnetic analysis of arbitrarily shaped piecewise homogeneous objects. In the proposed scheme, nonoverlapping boundary surfaces and interfaces between materials can be handled independently, without any continuity requirement through multimaterial junctions and tear lines between surfaces in contact. The use of nonconformal meshes provides improved flexibility for computer-aided-design (CAD) prototyping and tessellation. The proposed formulation can readily address nonconformal multimaterial junctions, where three or more material regions meet. The continuity of the electric surface current across the junction contours is enforced by the combination of the boundary conditions implicit in the JMCFIE formulation and the weakly imposed interior penalty (IP) between the contacting surfaces within each region. This completely avoids the cumbersome junction problem, which no longer requires any special treatment. Numerical experiments are included to validate the accuracy and demonstrate the great versatility of the proposed JMCFIE-DG formulation for the management and solution of complex composite objects with junctions.

Broadband Analysis of Multiscale Electromagnetic Problems: Novel Incomplete-Leaf MLFMA for Potential Integral Equations

Recently introduced incomplete tree structures for the magnetic-field integral equation are modified and used in conjunction with the mixed-form multilevel fast multipole algorithm (MLFMA) to employ a novel broadband incomplete-leaf MLFMA (IL-MLFMA) to the solution of potential integral equations (PIEs) for scattering/radiation from multiscale open and closed surfaces. This population-based algorithm deploys a nonuniform clustering that enables to use deep levels safely and, when necessary, without compromising the accuracy resulting in an improved efficiency and a significant reduction for the memory requirements (order of magnitudes), while the error is controllable. The superiority of the algorithm is demonstrated in several canonical and real-life multiscale geometries.

Fast Direct Error-Controllable Solution of Integral Equations via H,-matrix accelerated Locally Corrected Nyström Method

Computational framework for error-controlled fast direct solution of large-scale scattering problems is demonstrated. The approach is based on higher-order Locally Corrected Nystrom (LCN) discretization of the Electric and Magnetic Field Integral Equations (EFIE and MFIE) accelerated with hierarchical matrices. The latter enable $O(N\log N)$ solutions for matrices featuring quasi-constant ranks in their compressed blocks and make solutions largely insensitive to poor conditioning of the matrix equations typically resulting from LCN discretization of EFIE and MFIE.

Augmented Hybrid Integral Equations for Low-Frequency Analysis of Lossy Conducting Objects

Electromagnetic (EM) integral equations include either $\mathcal{L}$ operator or $\mathcal{K}$ operator or both. The $\mathcal{L}$ operator has a low-frequency-breakdown (LFB) problem at low frequencies, and as a remedy, the augmented electric field integral equation (AEFIE) for conducting objects was proposed. Later on, the AEFIE was extended to AEFIEs for dielectric objects which include both $\mathcal{L}$ operator and $\mathcal{K}$ operator and the dual basis function (DBF) is used to represent the magnetic current density. In this work, the hybrid field integral equations (HFIEs) are employed for lossy conducting objects, and the HFIEs consist of the EFIE in the exterior and magnetic field integral equation (MFIE) in the interior of the lossy conducting object, respectively. Both electric and magnetic current densities can be expanded by the Rao-Wilton-Glisson (RWG) basis function which is also a testing function and the resultant impedance matrix is well-conditioned due to the feature of HFIEs. At low frequencies, the HFIEs are extended to the augmented HFIEs (AHFIEs) by introducing the continuity equation of magnetic current and magnetic charge density as an extra unknown function. A numerical example is shown to demonstrate the approach and good performance has been verified.

Accurate and Efficient Solutions of Densely Discretized Closed Conductors Using a Combined Potential-Field Formulation

We present an accurate, efficient, and stable formulation for rigorous analyses of electromagnetic problems involving closed conductors. The formulation, namely the combined potential-field formulation (CPFF), is constructed from the conventional potential integral equations and the magnetic-field integral equation, together with an additional integral equation using the boundary condition for the normal component of the magnetic vector potential. Being both low-frequency-stable and resonance-free, CPFF is a broadband formulation, which enables accurate and efficient solutions of objects with diverse dimensions and discretization sizes.