Augmented Electric Field Integral Equation Research Articles

A-EFIE With Model-Order Reduction Technique for Fast Analysis of Circuit Simulation

A fast frequency-sweep method for microwave circuit simulation based on a full-wave solver is presented in this letter. The complexity of the mathematical model of the augmented electric field integral equation (A-EFIE) system is reduced by employing the model-order reduction (MOR) technique, where the reduced-order subspace is constructed by singular value decomposition (SVD). Compared to the A-EFIE with the asymptotic waveform evaluation (AWE) technique, the A-EIFE with the SVD-based MOR technique can speed up the broadband circuit analysis more efficiently.

Improved A-EFIE System for Electromagnetic Simulation in Low Frequency Regime

In this letter, the quasi-Helmholtz projectors are applied to address numerical cancellation of augmented electric field integral equation (A-EFIE). The projectors rescale the term <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\overline{\bf V} \cdot i{k}_0{\bf j}$</tex-math></inline-formula> and right-hand side to guarantee that the solution is stable at very low frequency. The improved system is full rank without removing the singularity caused by charge neutrality. Compared to the EFIE with quasi-Helmholtz projectors, this letter presents a better convergence performance and takes less computational cost. The numerical results show that the solution of both scattering problems and circuit problems can ensure the accuracy in the low-frequency regime.

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.

Characteristic Mode-Based Efficient Broadband Electromagnetic Scattering Analysis Method for Array Structures

A fast and accurate numerical method based on the theory of characteristic mode (TCM) is proposed for analyzing electromagnetic scattering from repetitive multiscale array structures. In order to overcome the low frequency breakdown problem when interacting at short distances between observation and source basis functions, the CMs are extracted from the impedance matrix generated by the augmented electric field integral equation (AEFIE), which is discretized by the method of moments (MoM). Then the CMs of a single unit are used as the global basis functions to reduce the unknowns of the array structures. In addition, the broadband multilevel fast multipole algorithm (MLFMA) based on approximate diagonalization of the Green’s function is implemented to accelerate the matrix-vector multiplications between the impedance matrix and CM vectors. Several numerical examples are given to demonstrate the high efficiency and accuracy of the proposed scheme.

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.

Study of generalized formulation for augmented electric field integral equation with perturbation method

AbstractIn this work, a generalized form for the augmented electric field integral equation with perturbation method is studied. By implementing Taylor expansion on all the frequency‐dependent matrix terms and the unknowns, the current coefficients on each frequency order can be obtained in the perturbation method, where the accuracy can be guaranteed near the zero frequency or within the static frequency region. When the frequency goes higher, more expanding terms are required to obtain the accurate results. In order to obtain a wideband solution, a general formulation for the perturbation method is derived and analyzed for the first time. As a result, the improved method can be further extended to dynamic region. Numerical results show that the achieved results for both capacitive circuit problems and plane wave scattering problems are more accurate in comparison with those obtained from the traditional solutions, where the number of the required expanding terms can be determined according to the corresponding error analysis.

Entire Domain Basis Function Expansion of the Differential Surface Admittance for Efficient Broadband Characterization of Lossy Interconnects

This article presents a full-wave method to characterize lossy conductors in an interconnect setting. To this end, a novel and accurate differential surface admittance operator for cuboids based on entire domain basis functions is formulated. By combining this new operator with the augmented electric field integral equation, a comprehensive broadband characterization is obtained. Compared with the state of the art in differential surface admittance operator modeling, we prove the accuracy and improved speed of the novel formulation. Additional examples support these conclusions by comparing the results with commerical software tools and with measurements.

A Memory Saving Augmented EFIE With Modified Basis Functions for Low-Frequency Problems

A new augmented electric field integral equation (A-EFIE) formulation is discussed in the low frequency. In the new formulation, both the electric current and charge are considered as unknowns. Different from the conventional A-EFIE, a constraint on the normal component of the magnetic field is added to modifying the formulation of A-EFIE for improving the accuracy and a kind of constant nodal functions is introduced for expanding the electric charge. With the proposed method, besides avoiding the low-frequency breakdown, the storage of impedance matrix is reduced over 35% compared with the conventional A-EFIE, which is owing to the application of constant nodal functions associated with vertices. Numerical examples illuminate the validity and performance of the new A-EFIE with constant nodal functions.

Augmented combined field integral equation for low frequency scattering analysis

In this study, a new augmented combined field integral equation formulation has been proposed to address the low-frequency breakdown problem. The proposed formulation is developed by adding the magnetic field integral operator to the conventional augmented electric field integral equation (AEFIE). In the meanwhile, a saddle point constraint preconditioning is also introduced to improve the convergence property of the system matrix. In some cases, the conventional AEFIE method cannot achieve the accurate solutions when the frequency is extremely low and a perturbation method is needed to tackle this accuracy problem. However, the new method is able to get the accurate solutions without any other strategy in the same cases mentioned above. Some numerical examples have been presented to demonstrate the validity of the proposed method.

An Integral Equation Modeling of Lossy Conductors With the Enhanced Augmented Electric Field Integral Equation

The formulation of the enhanced augmented electric field integral equation for dielectrics is generalized to conductor problems in this paper. The conductive region is simulated as a lossy dispersive medium using a full wave solver. In order to calculate the method of moments matrix elements in the conductive region accurately, we investigate the evaluations of the integrals of Green’s function in lossy media. After comparing with some other integration methods, we propose a new method to evaluate such integrals. This method turns out to improve the accuracy and efficiency. Moreover, the proposed formulation can be regarded as a generalized impedance boundary condition (IBC). This generalized IBC will become global if the skin depth is comparable to the size of the structures/details. The mixed-form fast multipole algorithm is employed for the simulations. Numerical examples of complex circuit structures are given to demonstrate the accuracy and capabilities of the proposed method.

A Discontinuous Galerkin Augmented Electric Field Integral Equation for Multiscale Electromagnetic Scattering Problems

A discontinuous Galerkin (DG) augmented electric field integral equation method based on the domain decomposition is proposed in this paper for full-wave solution of multiscale targets. The conventional surface integral equation-based DG method allowing both conformal and nonconformal discretizations for multiscale structures suffers from low-frequency breakdown. By augmenting the DG-EFIE with current continuity equation, the proposed scheme can alleviate the low-frequency breakdown. In the augmented system, the electric field integral equation and the current continuity equation are discretized by using hybrid basis functions including Rao–Wilton–Glisson (RWG) and half RWG basis functions. Since the half RWG basis is not divergence conforming, line charge degrees of freedom on the adjoining edges are introduced in this paper. It is observed that the resulting linear system is well conditioned at low frequencies, which leads to a rapid convergence over wide frequency band. Numerical examples demonstrate the accuracy and efficiency of the augmented system.

Acceleration of Augmented EFIE Using Multilevel Complex Source Beam Method

The computation of the augmented electric field integral equation (A-EFIE) is accelerated by using the multilevel complex source beam (MLCSB) method. As an effective solution of the low-frequency problem, A-EFIE includes both current and charge as unknowns to avoid the imbalance between the vector potentials and the scalar potentials in the conventional EFIE. However, dense impedance submatrices are involved in the A-EFIE system, and the computational cost becomes extremely high for problems with a large number of unknowns. As an exact solution to Maxwell’s equations, the complex source beam (CSB) method can be well tailored for A-EFIE to accelerate the matrix-vector products in an iterative solver. Different from the commonly used multilevel fast multipole algorithm (MLFMA), the CSB method is free from the problem of low-frequency breakdown. In our implementation, the expansion operators of CSB are first derived for the vector potentials and the scalar potentials. Consequently, the aggregation and disaggregation operators are introduced to form a multilevel algorithm to reduce the computational complexity. The accuracy and efficiency of the proposed method are discussed in detail through a variety of numerical examples. It is observed that the numerical error of the MLCSB-AEFIE keeps constant for a broad frequency range, indicating the good stability and scalability of the proposed method.

Augmented Electric-Field Integral Equation for Inhomogeneous Media

The augmented electric-field integral equation is extended for the analysis of inhomogeneous media at low frequencies. The internal surface integral equation for inhomogeneous region is derived from the vector potential ( ${{\mathbf A}}$ ) and the scalar potential ( ${{\mathbf \Phi }}$ ) equations, which are free of low-frequency breakdown. In the new equations, Green's functions of ${{\mathbf A}}$ and ${{\mathbf \Phi }}$ for inhomogeneous media are incorporated. Due to the absence of the analytic solutions, Green's functions for ${{\mathbf A}}$ and ${{\mathbf \Phi }}$ are solved numerically with the finite-element method and represented in matrix forms. Numerical examples are presented to demonstrate the validity of the proposed scheme in the study of inhomogeneous objects at low frequencies.

An Enhanced Augmented Electric-Field Integral Equation Formulation for Dielectric Objects

A full-wave surface integral equation (SIE) method based on the augmented electric-field integral equation (A-EFIE) for dielectric objects with low-frequency stability is presented in this paper. Motivated by the A-EFIE formulation for perfect electric conductor (PEC), the internal and external problems are both augmented with the current continuity equation and renormalized to eliminate the low-frequency breakdown. Although the magnetic-field integral equation operator $\mathcal{K}$ is free of low-frequency breakdown, its matrix form is ill-conditioned and unsolvable if the traditional Rao–Wilton–Glisson (RWG) basis function is used as the testing and basis functions. As a remedy, the Buffa–Christiansen (BC) basis function is introduced to alleviate this testing issue. After this treatment, the matrix form of operator $\mathcal{K}$ is well conditioned. To solve problems with a large number of unknowns, a preconditioning scheme is introduced to accelerate the convergence and the mixed-form fast multipole algorithm (FMA) is adopted to accelerate the matrix vector product.

Augmented EFIE With Normally Constrained Magnetic Field and Static Charge Extraction

A new surface integral equation formulation for scattering from perfectly conducting objects is presented. The formulation is developed by adding a constraint on the normal component of the magnetic field to the augmented electric field integral equation (AEFIE) and extracting the static charge solution. The resulting AEFIEnH-S formulation is discretized using the method of moments with Rao–Wilton–Glisson (RWG) source functions and Buffa–Christiansen (BC) test functions. An iterative diagonal matrix scaling algorithm is used to improve the conditioning of the discrete system. Numerical examples demonstrate that the AEFIEnH-S is stable and accurate as the frequency is reduced for closed, open, and multiscale multiply connected geometries. The formulation relies only on diagonal preconditioning, it accurately models the near electric, near magnetic, and far fields, it does not require frequency scaling of the unknowns, and it does not incorporate any type of Helmholtz decomposition.

Fast solution of low‐frequency complex problems over a frequency band using enhanced A‐EFIE and FMM

ABSTRACTWe present a fast method for solving low‐frequency complex problems over a frequency band. In this method, we apply fast multipole method to accelerate the solution process of augmented electric field integral equation with perturbation method (enhanced A‐EFIE). As the current is expanded as a power series in frequency in enhanced A‐EFIE, results over a frequency band can be obtained by one low‐cost simulation. Real circuit problem with one million unknowns is used to demonstrate the effectiveness of the method. © 2014 Wiley Periodicals, Inc. Microwave Opt Technol Lett 56:2153–2158, 2014

Loop‐Tree Free Augmented Equivalence Principle Algorithm for Low‐Frequency Problems

ABSTRACTIn this article, we describe a newly developed integral equation domain decomposition method for solving complex low frequency electromagnetic problems. This method is based on low frequency augmented equivalence principle algorithm (EPA) with the augmented electric field integral equation (A‐EFIE). The A‐EFIE provides a stable solution in the low‐frequency regime as it includes both charge and current as unknowns to avoid the imbalance between the vector potential and the scalar potential. Based on similar idea, the augmented EPA can suppress the projection error resulting from low‐frequency problems of the normal EPA. Combining these two methods, the proposed algorithm can be used in domain decomposition problems and is numerically stable at low frequencies. Several numerical examples are provided to verify its validity. © 2013 Wiley Periodicals, Inc. Microwave Opt Technol Lett 55:2475–2479, 2013

Improved Augmented Electric Field Integral Equation Method for Modeling Micro-Scale Structures

An improved augmented electric field integral equation method for modeling micro-scale structures is proposed in this article. The scaling property of the matrix systems of the original electric field integral equation method and the corresponding augmented electric field integral equation method are analyzed in detail. It is found that though the augmented electric field integral equation method works well for ordinary structures at very low frequencies, the matrix is ill conditioned for micro-scale structures with very dense mesh. Different balancing techniques are studied to alleviate this problem. A normalization factor is proposed to make the system stable for electrically small structures, especially for micro-scale structures working at several GHz. Numerical results are presented to demonstrate the accuracy and stability of this method.

Augmented EFIE with Adaptive Cross Approximation Algorithm for Analysis of Electromagnetic Problems

The augmented electric field integral equation (A-EFIE) with charge neutrality enforcement provides a stable formulation to conquer low-frequency breakdown characteristic of conventional EFIE. It is augmented with additional charge unknowns through current continuity equation. The A-EFIE combined with the multilevel adaptive cross-approximation (MLACA) algorithm is developed to further reduce the memory requirement and computation time for analyzing electromagnetic problems. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed method.

Alternative AEFIE‐EFIE method for broadband CEM simulations

AbstractIt is well known that electric field integral equation (EFIE) has the low frequency breakdown problem due to the decoupling of electric and magnetic field. Augmented electric field integral equation (AEFIE) method overcomes the low frequency breakdown. However, its performance degrades with the increasing frequency. To obtain a unified method that works for both low and high frequencies, a novel method using the EFIE complement process inside AEFIE is proposed. Numerical results demonstrated that this alternative AEFIE‐EFIE method is able to extend integral equations to wide band applications. © 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:1737–1740, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26919