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Higher order nodal basis functions for representing equivalent surface currents on antennas and scatterers are introduced. The performance of the nodal basis is evaluated by comparing two existing higher order edge bases, using the magnetic field integral equation (MFIE) formulation for scattering by a perfect electric conductor (PEC) sphere and icosahedron as test problems. Both nodal and edge bases are implemented on rational Bezier patches, giving an exact representation of the surfaces, free from geometrical error. The accuracy of the numerical solutions obtained with the three different bases for both the surface current and the radar cross section (RCS) are compared, and it is shown that in general the nodal bases give better accuracy than the edge bases for equal computational cost. Index Terms--Boundary element methods (BEMs), electromagnetic scattering, magnetic field integral equation (MFIE).

A combined field integral equation (CFIE) with single operator, which is named equalized CFIE (ECFIE) is presented for solving the electromagnetic scatterings from conducting targets with cavities on which surface both the electrical and magnetic currents exist simultaneously. In ECFIE, the different integral operators imposed on unknown electric and magnetic currents are simplified to a single one by properly choosing the coefficients in conventional CFIE. As a result, the computation complexity is dramatically reduced comparing with EFIE, MFIE, and conventional CFIE. Combined with multilevel fast multipole algorithm (MLFMA) and connection scheme (CS), ECFIE is used to simulate the scattering from conducting objects with cavities. The numerical examples presented in this paper demonstrate the validity and high efficiency of this approach. Index Terms--Cavity scattering, electromagnetic scattering, integral equation (IE) method, multilevel fast multipole algorithm (MLFMA).

The exact closed-form expression for the electromagnetic field excited by pulse-shaped and triangular line currents is presented. The analytical formulation, based on the incomplete Hankel functions, shows that the field is composed of cylindrical waves excited near the current axis and spherical waves arising from the source critical points. Spherical waves related with pulse-shaped current basis functions are shown to have stronger field singularities than for triangular basis functions because of the impulsive charges present at the current truncations. For triangular line currents, it is further shown that an additional spherical wave arises from the critical point featuring a charge jump discontinuity. Near and high-frequency asymptotic field expressions show explicitly the nature of the field singularities. Using the analytical properties of the incomplete Hankel functions, the Galerkin's impedance matrix coefficients, useful to solve radiation and scattering problems in truncated cylindrical structures using the method of moments, are finally derived in a closed-form. Numerical examples show the accuracy of the proposed field representation. Index Terms--Galerkin's method of moments (MoM), incomplete Hankel functions, near-field, mixed potential integral equations (MPIE), pulse-shaped basis functions, triangular basis functions.

The Green's functions employed in the method of moments (MoM) diverge when observation and source points coincide; this is at the origin of the difficulties in computing the MoM matrix entries, and in handling the near-field interactions in fast Fourier transform (FFT)-based fast methods and other sampling-based methods. In this paper, we show that this singularity can be avoided, and a modified regular Green's function can be used instead. This latter is obtained from the spectral representation of the usual Green's function via windowing of its spectrum; the width of the spectral window depends on the size of the mesh employed for discretizing the problem, so that the proposed regular Green's function is a mesh-adapted regular kernel. We address a general 3D problem; we relate the MoM reaction integrals to the 2D Fourier spectrum of the Green's function, that allows to discuss the necessary spectral bandwidth for the windowed Green's function. We employ a tapered window, and present a closed-form expression for the spatial Green's function. Numerical results are presented for 3D antenna and scattering problems discretized with Rao--Wilton--Glisson (RWG) functions, and for uniform and nonuniform meshing. They show that the proposed method yields accurate solutions also for the antenna input impedance. The meaning of the regularized Green's function is also discussed and put in perspective. Index Terms--Antennas, Green's function, method of moments (MoM).

The time-harmonic electromagnetic scattering or radiation problem is considered. The singular value decomposition (SVD) is applied to the radiation operator that maps the set of electric and magnetic currents defined on the surface of an inhomogeneous object onto the set of the far-fields scattered (or radiated) from this object. The SVD yields orthonormal bases for both sets. Because the radiation operator is compact and regularizing, it is demonstrated that the far-field calculated from the series expansions of the currents on these bases converges exponentially fast to the exact one if a sufficient number of terms is considered in these series. This number is closely related to the degrees of freedom that characterize the far-field. The latter can be computed from a reduced number of unknowns in the discretized integral equation that links electric and magnetic surface currents by writing it in the new currents bases. Also, it allows the reduction of the far-field in a given angular sector. The numerical complexity of this technique is addressed, and 2D numerical examples are presented that illustrate its potentialities. Index Terms--Impedance boundary condition, integral equation, model-order reduction (MOR), radar cross section (RCS), radiation operator, singular value decomposition (SVD).

The idea of so-called macromodels is applied to problems of computational electromagnetics involving integral-equation/method-of-moments formulations. It is shown that with the use of equivalence principles, it is possible to exclude a part of the problem, calculate its wideband response without an a part knowledge of the excitations, and then efficiently combine the result into the original situation. The idea is validated using examples involving slot-excited dielectric resonator antennas. Index Terms--Domain decomposition, integral equations, method of moments (MoM).

For electrically large complex electromagnetic (EM) scattering problems, huge memory is often required for most EM solvers, which is too difficult to be handled by a personal computer (PC) even a workstation. Although the multilevel fast multipole algorithm (MLFMA) effectively deals with electrically large problems to some extent, it is still time and memory consuming for very large objects. In order to further reduce the CPU time and the memory requirement, a hybrid algorithm, based on the overlapped domain decomposition method for integral equations (IE-ODDM), MLFMA and block-diagonal, incomplete lower and upper triangular matrices (DILU) preconditioner, is proposed for the analysis of electrically large problems. The dominant memory requirement for plane wave expansions in the three processes of aggregation, translation and disaggregation in the MLFMA is drastically reduced by the first two techniques. The iterative procedure for each overlapped subdomain solved by the MLFMA is effectively sped up by the DILU preconditioner. After integrating these techniques, the proposed hybrid algorithm is more efficient in computing time and memory requirement compared to the conventional MLFMA and is more suitable for analyzing very large EM scattering problems. Enough accurate solution can be obtained within quite a few outer iterations, where an outer iteration means a complete sweep for all the subdomains. Some numerical examples are presented to demonstrate its validity and efficiency. Index Terms--Block-diagonal, incomplete lower and upper triangular matrices (DILU) preconditioner, electromagnetic (EM) scattering, integral equations (IE), multilevel fast multipole algorithm (MLFMA), overlapped domain decomposition method (ODDM).

An improved multilevel Green's function interpolation method (MLGFIM) with adaptive phase compensation (APC) is proposed. The difficulty in applying interpolation approaches to the fast varying phase term in the integral equation kernel for full-wave electromagnetic (EM) simulations is eradicated by using the phase compensation and adaptive direction separation (ADS). The multilevel tree structure in MLGFIM keeps the number of direction separation invariant at all levels, attributing to the recursive interpolation with multilevel phase compensation. The proposed MLGFIM-APC in conjunction with the Lagrange-Chebyshev interpolation yield an O(N log N) CPU time and O(N) computer memory requirement for surface scattering problems. By introducing a transition level, the MLGFIM-APC can adaptively incorporate interpolation techniques of conventional interpolation (CI), transition interpolation (TI) and phase-compensation interpolation (PI), corresponding to electromagnetic simulation of problems of small, moderate, and large electrical sizes, respectively. Large-scale microstrip antenna arrays are simulated to illustrate the accuracy and efficiency of the proposed method. It is found that the CPU time scales better than O(N log N) for these co-planar problems. Index Terms--Adaptive direction separation, adaptive transition, Lagrange-Chebyshev interpolation, multilevel Green's function interpolation method with adaptive phase compensation (MLGFIM-APC), phase compensation (PC).

We study TM and TE diffraction by a perfectly conducting half-plane in presence of a perfectly conducting flat strip of finite width and infinite length; the edges of the strip are parallel to the edge of the half-plane. The relevant integral equations with unknowns the induced current densities on the strip are solved by a Nystrom and a Galerkin method that fully account for the singular nature of the kernels and the singularities of the solution at the edges. The proposed algorithms are highly accurate, and appear to converge exponentially as shown by detailed numerical examples and case studies. The Nystrom method is the most efficient of the two methods because of its simple formulation, fast computer implementation, and much less effort in computing the matrix elements. When the strip is coplanar to the half-plane, an additional formulation method in terms of equivalent surface magnetic currents is possible in the framework of the field equivalence principles. This alternative method is applied in order to validate the algorithms and test their accuracy. Index Terms--Electromagnetic diffraction, Galerkin method, integral equations, Nystrom method, strip scatterers.

A new set of higher order hierarchical basis functions based on curvilinear triangular patch is proposed for expansion of the current in electrical field integral equations solved by method of moments. The hierarchical two-level spectral preconditioning technique is developed for the generalized minimal residual iterative method, in which the multilevel fast multipole method is used to accelerate matrix-vector product. The sparse approximate inverse (SAI) preconditioner based on the higher order hierarchical basis functions is used to damp the high frequencies of the error and the low frequencies is eliminated by a spectral preconditioner in a two-level manner defined on the lower order basis functions. The spectral preconditioner is combined with SAI preeonditioner to obtain a hierarchical two-level spectral preconditioner. Numerical experiments indicate that the new preconditioner can significantly reduce both the iteration number and computational time. Index Terms--Eleetromaguetic scattering, generalized minimal residual (GMRES) algorithm, multilevel fast mnitipole method (MLFMM), preconditioning techniques, sparse approximate inverse (SAD.

The diffusion behavior of electromagnetic (EM) waves in two dimensional (2-D) multipath media is studied through integral equation based full wave Monte Carlo simulations. The influences of some physical factors are explored, among which the area density of the embedded obstacles manifests itself to be the most important one in determining wave diffusion. A lossy system starts to behave diffusively when the area density approximately exceeds 5%, and the diffusion equations are generally applicable for predicting power decay. At low densities, the power-distance relation of the waves appears to follow power laws. The sizes and shapes of the obstacles have a secondary effect on the diffusion of waves. Whenever a system contains small objects or objects with reflecting sides, the waves therein are more diffusive and the diffusion equation approximates the reality more accurately. Absorption loss decreases wave diffusion in general, but our results show that the diffusion equation for a system with very lossy but small obstacles can work very well for predicting power decay. Index Terms--Diffusion, Monte Carlo simulation, multiple scattering, random media, transport theory.

A new technique for machine precision evaluation of singular and nearly singular potential integrals with 1 / R singularities is presented. The numerical quadrature scheme is based on a new rational expression for the integrands, obtained by a cancellation procedure. In particular, by using library routines for Gauss quadrature of rational functions readily available in the literature, this new expression permits the exact numerical integration of singular static potentials associated with polynomial source distributions. The rules to achieve the desired numerical accuracy for singular and nearly singular static and dynamic potential integrals are presented and discussed, and several numerical examples are provided. Index Terms--Boundary element methods, finite element methods, finite volume methods, integral equations, method of moment (MoM).

The interaction of electromagnetic waves with an arbitrarily shaped three-dimensional (3-D) homogeneous chiral object located above a lossy half space is investigated using the method of moments (MoM) via the coupled mixed potential integral equations (MPIEs). Based on the surface equivalence principle, the equivalent surface electric and magnetic currents are used to replace the homogeneous chiral target in the presence of the half space. Two coupled MPIEs are developed for the unknown equivalent surface electric and magnetic currents by utilizing the continuity condition of the tangential total electric and magnetic field components on the chiral body's surface. The well-known Galerkin procedure with Rao-Wilton-Glisson (RWG) basis functions is applied to solve this problem. The spatial domain half-space Green's functions are obtained from the corresponding spectral domain Green's functions via the discrete complex image method (DCIM) combined with the generalized-pencil of function (GPOF) technique. The reciprocity theorem is employed to calculate the far-zone scattered field. Numerical results are presented for characterizing electromagnetic scattering by a 3-D arbitrarily shaped homogenous chiral object located above a lossy half space so as to demonstrate the accuracy and efficiency of the proposed technique. Index Terms--Discrete complex image method, electromagnetic scattering, homogeneous chiral target, integral equations, method of moments (MoM).

A spatially hierarchical current expansion is used to construct a marching-on-in-time based time domain electric field integral equation solver. The formulation gives rise to well-conditioned linear systems and is immune to time domain low-frequency breakdown phenomena. Numerical results that demonstrate the effectiveness of the proposed hierarchical approach in comparison with existing technologies are presented. Index Terms--Electric field integral equation (EFIE), multiresolution techniques, time domain analysis, wavelets.

An efficient algorithm for electromagnetic wave scattering from rough dielectric surfaces is developed. The algorithm is based on the single magnetic field integral equation (SMFIE) and the surface is discretized using Rao-Wilton-Glisson (RWG) triangular basis functions. The new feature of the algorithm is the application of the adaptive integral method (AIM) with SMFIE for speeding up the calculation. The developed new method utilizes the flexibility of RWG functions to model arbitrary rough surface and the speed of FFT, enabling accurate simulations over large surfaces. Index Terms--Adaptive integral method (AIM), fast Fourier transform (FFT), Rao-Wilton-Glisson (RWG) triangular basis, rough surface scattering, single magnetic field integral equation (SMFIE).

A new analytical approach for obtaining the time samples of the magnetic field intensity due to an impulsively excited Rao-Wilton-Glisson (RWG) basis function is presented. The approach is formulated directly in the time domain. It is shown that the magnetic field is related to the arc segments formed by the intersection of the triangular patch of the RWG basis with the sphere that is centered at the observation point and that has a radius of R = ct, where c is the speed of light. In particular, the magnetic field can be expressed as the variations of two quantities with respect to R. The first quantity is the arc segment length, and the second quantity is the bisecting vector of the arc segment. Analytical representations of these quantities are presented. Contrary to previous studies, these representations do not require the calculation of the intersection points of the sphere with the boundaries of the bases. The validity of the obtained time domain formulae is demonstrated through comparison of the results with those obtained in the frequency domain by using numerical quadrature. Finally, it is demonstrated that the derived formulae yield closed-form expressions when convolved with piecewise polynomial temporal basis functions. Index Terms--Integral equations, Magnetic field integral equation (MFIE), magnetic field intensity, marching-on-in-time (MOT) method, Rao-Wilton-Glisson (RWG) basis, time domain analysis.

Important connection between computational and mathematical electromagnetics is presented. The newly developed well-conditioned electromagnetic frequency domain surface integral equation formulations, the current and charge integral equations, are shown to be related to Picard's extended Maxwell system, an extended partial differential equation system that has the correct static behavior. Electromagnetic surface integral representations are derived in this paper for traditional surface integral equation formulations and for the Picard system using the fundamental solution approach, i.e., from the definition of Dirac's delta function. The surface integral representations are constructed with proper solid angle coefficients starting from the scalar Helmholtz equation. The traditional surface integral equation formulations are shown to be derived from Maxwell's curl equations and are thus lacking the contribution of the divergence equations at zero frequency. It is shown that the new current and charge formulations can be derived from the surface integral representation of the Picard system. Index Terms--Low frequency breakdown, Maxwell's equations, surface integral equations.

A technique for the determination of the equivalent currents distribution from a known radiated field is described. This Inverse Radiation Problem is solved through an Integral Equation algorithm that allows the characterization of antennas of complex geometry both for near field to far field (NF-FF) transformation purposes as well as for diagnostic tasks. The algorithm is based on the representation of the radiating structure by means of a set of equivalent currents over a three-dimensional (3-D) surface that can be fitted to the arbitrary geometry of the antenna. The innovative formulation uses an integral equation involving the electric field due to the currents tangential components to the represented antenna 3-D surface. For that purpose, both the magnetic and electric equivalent currents are considered in the integral equations. Regularization techniques are also introduced to improve the convergence of the proposed iterative solution. The paper concludes with several results related to the practical verification of the Equivalence Principle and the characterization of a horn antenna. Index Terms--Equivalence principle, equivalent sources, integral equations, inverse radiation problem, near-field far-field transformation (NF-FF).

A generalized impedance boundary condition is developed to rigorously model on-chip interconnects in the full-wave surface integral equation by a two-region formulation. It is a combination of the electric-field integral equation for the exterior region and the magnetic-field integral equation for the interior conductive region. The skin effect is, therefore, well captured. A novel integration technique is proposed to evaluate the Green's function integrals in the conductive medium. Towards tackling large-scale problems, the mixed-form fast multipole algorithm and the multifrontal method are incorporated. A new scheme of the loop-tree decomposition is also used to alleviate the low-frequency breakdown for the formulation. Numerical examples show the accuracy and reduced computation cost. Index Terms--Full-wave solver, generalized impedance boundary condition, impedance boundary condition, interconnects, loop tree, mixed-form fast multipole algorithm, skin effect, surface integral equation.

This paper describes an efficient numerical approach for the calculation of the per-unit length earth-return impedance of a conductor inside a tunnel. The influence of the tunnel on the per-unit length impedance is replaced by a set of equivalent current sources via the fictitious medium method. The main advantage of the proposed approach is that it can solve the boundary value problem within the tunnel region. The proposed method can reduce the calculation region and, thus, leads to time savings. A comparison of the numerical results with published data shows very good agreement. The proposed method is a good candidate for impedance calculation of a conductor inside a tunnel. Index Terms--Earth return impedance, fictitious medium, integral equation.