Matti Taskinen

Calculation of CFIE impedance matrix elements with RWG and n/spl times/RWG functions

The method of moments (MoM) solution of combined field integral equation (CFIE) for electromagnetic scattering problems requires calculation of singular double surface integrals. When Galerkins method with triangular vector basis functions, Rao-Wilton-Glisson functions, and the CFIE are applied to solve electromagnetic scattering by a dielectric object, both RWG and n/spl times/RWG functions (n is normal unit vector) should be considered as testing functions. Robust and accurate methods based on the singularity extraction technique are presented to evaluate the impedance matrix elements of the CFIE with these basis and test functions. In computing the impedance matrix elements, including the gradient of the Greens function, we can avoid the logarithmic singularity on the outer testing integral by modifying the integrand. In the developed method, all singularities are extracted and calculated in closed form and numerical integration is applied only for regular functions. In addition, we present compact iterative formulas for computing the extracted terms in closed form. By these formulas, we can extract any number of terms from the singular kernels of CFIE formulations with RWG and n/spl times/RWG functions.

Application of combined field Integral equation for electromagnetic scattering by dielectric and composite objects

Combined field integral equation (CFIE) is applied for computing electromagnetic scattering by arbitrarily shaped three dimensional dielectric and composite objects. The objectives of this paper are as follows. First, to present a CFIE formulation which can be used in the analysis of piecewise dielectric and composite metallic and dielectric objects with junctions. Second, to show that properly choosing the coupling coefficients of the equations the conditioning of the discretized matrix equation can be essentially improved and rapidly converging iterative solutions can be obtained even without preconditioning.

Surface Integral Equation Method for General Composite Metallic and Dielectric Structures with Junctions

The surface integral equation method is applied for the electromagnetic analysis of general metallic and dielectric structures of arbitrary shape. The method is based on the EFIE-CFIE-PMCHWT integral equation formulation with Galerkins type discretization. The numerical implementation is divided into three independent steps: First,the electric and magnetic field integral equations are presented and discretized individually in each non-metallic subdomain with the RWG basis and testing functions. Next the linearly dependent and zero unknowns are removed from the discretized system by enforcing the electromagnetic boundary conditions on interfaces and at junctions. Finally,the extra equations are removed by applying the wanted integral equation formulation,and the reduced system is solved. The division into these three steps has two advantages. Firstly,it greatly simplifies the treatment of composite objects with multiple metallic and dielectric regions and junctions since the boundary conditions are separated from the discretization and integral equation formulation. In particular,no special junction basis functions or special testing procedures at junctions are needed. Secondly,the separation of the integral equation formulation from the two previous steps makes it easy to modify the procedure for other formulations. The method is validated by numerical examples.

Current and charge Integral equation formulation

A new stable frequency domain surface integral equation formulation is proposed for the three dimensional electromagnetic scattering of composite metallic and dielectric objects. The developed formulation does not suffer from the low frequency breakdown and leads to a well balanced and stable system on a wide frequency band. Surface charge densities are used as unknowns in addition to the traditional surface current densities. The balance of the system is achieved by using normalized field quantities and by enforcing the continuity of the fields across the boundaries with carefully chosen scaling factors. The linear dependence between the currents and charges is taken into account with an integral operator, and the linear dependence in charges is removed with the deflation method. A combined field integral equation form of the formulation is proposed to remove the internal resonance problem associated to the closed metallic objects. Due to the good balance in the new formulation, fast converging iterative solutions on a very wide frequency band can be obtained. The new formulation and its convergence is verified with numerical examples.

Well-conditioned Muller formulation for electromagnetic scattering by dielectric objects

Numerical solution of electromagnetic scattering by homogeneous dielectric objects with the method of moments (MoM) and Rao-Wilton-Glisson (RWG) basis functions is discussed. It is shown that the low-frequency breakdown associated to the MoM solution of scattering by dielectric objects can be avoided by the classical Müller formulation without the loop-tree or loop-star basis functions. Two variations of the Müller formulation, T-Müller and N-Müller, are considered. It is demonstrated that only the N-Müller formulation with the Galerkin method and RWG functions gives stable solution. Discretization of the N-Muller formulation leads to a well-conditioned matrix equation and rapidly converging iterative solutions on a wide frequency range from very low frequencies to microwave frequencies. At zero frequency, the N-Müller formulation decouples into the electrostatic and magnetostatic integral equations.

SINGULARITY SUBTRACTION INTEGRAL FORMULAE FOR SURFACE INTEGRAL EQUATIONS WITH RWG, ROOFTOP AND HYBRID BASIS FUNCTIONS

Numerical solution of electromagnetic scattering problems by the surface integral methods leads to numerical integration of singular integrals in the Method of Moments. The heavy numerical cost of a straightforward numerical treatment of these integrals can be avoided by a more efficient and accurate approach based on the singularity subtraction method. In the literature the information of the closed form integral formulae required by the singularity subtraction method is quite fragmented. In this paper we give a uniform presentation of the singularity subtraction method for planar surface elements with RWG, ˆ n×RWG, rooftop, and ˆ n×rooftop basis functions, the latter three cases being novel applications. We also discuss the hybrid use of these functions. The singularity subtraction formulas are derived recursively and can be used to subtract more than one term in the Taylor series of the Greens function.

Improving Conditioning of Electromagnetic Surface Integral Equations Using Normalized Field Quantities

When the surface integral equation method is applied to study electromagnetic scattering by dielectric or composite metallic and dielectric objects, the unknowns, i.e., the electric and magnetic surface current densities, and the elements of the system matrix, are often of the very different scales. As a consequence, the system matrix may have a high (singular value) condition number. An efficient method is presented to balance the unknowns and the integral equations, and the elements of the system matrix, too. The method is based on the use of normalized field quantities and unknowns, and carefully chosen scaling factors. In the case of dielectric and composite objects the condition numbers of the SIE matrices can be reduced with several orders of magnitudes by the developed method. In the case of high contrast objects, or if the frequency is very low, the developed method leads also to a clear improvement on the convergence of iterative solutions

Current and Charge Integral Equation Formulations and Picard's Extended Maxwell System

In this paper, the relation between the current and charge formulations and the Picards extended Maxwell system is studied. It is shown, that the new formulation can be derived from the Picards system and the linear dependence between the currents and charges is taken into account with integral operators

MULTILAYERED MEDIA GREEN'S FUNCTIONS FOR MPIE WITH GENERAL ELECTRIC AND MAGNETIC SOURCES BY THE HERTZ POTENTIAL APPROACH

A complete set of three dimensional multilayered media Greens functions is presented for general electric and magnetic sources. The Greens functions are derived in the mixed potential form, which is identical with the Michalski-Zheng C-formulation. The approach applied in this paper is based on the classical Hertz potential representation. A special emphasis is on the formulation of the dyadic Greens functions GHJ and GEM. In these functions the derivatives due to the curl operator are taken in the spectral domain. This avoids the need of the numerical differentiation. Furthermore, it is found that the Hertzian potentials satisfy several useful duality and reciprocity relations. By these relations the computational efficiency of the Hertz potential approach can be significantly improved and the number of required Sommerfeld integrals can be essentially reduced. We show that all spectral domain Greens functions can be obtained from only two spectral domain Hertzian potentials, which correspond to the TE component of a vertical magnetic dipole and the TM component of a vertical electric dipole. The derived formulas are verified by numerical examples.

Scattering by DB Spheres

This letter analyzes scattering properties of spheres with DB boundary conditions. The DB condition is defined by the requirement that the components of electric and magnetic flux densities normal to the boundary are zero. Rayleigh scattering, Mie scattering, and method-of-moments (MoM)-based computations are applied to the problem. The most interesting results are vanishing backscattering and rotational symmetry of the scattering diagram.