Exponentially Converging NystrÖm's Methods for Systems of Singular Integral Equations With Applications to Open/Closed Strip- or Slot-Loaded 2-D Structures

Abstract

This paper concerns the fast, highly accurate, and exponentially convergent solution of three sets of integral equations pertaining to two-dimensional generalized microstrip or microslot structures. The analysis relies on Nyström's method with due regard to all singular integrals and slowly converging series that appear in the kernels; the sophisticated treatment and efficient computation of such series and integrals is of paramount importance, entailing the use of advanced techniques. As a result of our specialized treatment, the discretization procedures developed herein: a) fully account for both the singular nature of the kernels and the singularities of the solution at the edges; b) obviate the need for taking inner products with testing functions; c) appear to converge exponentially versus matrix size; and d) enable expressing all matrix elements via single, finite sums. Detailed numerical examples and case studies illustrate the simplicity, flexibility, and remarkable efficiency of the algorithms.

Method of singular integral equations in diffraction by semi-infinite grating: $H$-polarization case

“…The derivatives in the electric field integral equations are evaluated using the finite-difference scheme, and then sinc functions are used to approximate the unknown current distribution on strips. Other papers give the Nystrom-type algorithm [27][28][29][30][31][32][33]. It proposes direct discretization of the singular or hyper-singular integral equations.…”

Validity and Limitations of the Median-Line Integral Equation Technique in the Scattering by Material Strips of Sub-Wavelength Thickness

“…All data presented below have been computed with accuracy ( ). Note that unlike PEC-strip SIE [22], (16), (17) cannot provide exponential convergence.…”

Numerical analysis and synthesis of 2D quasi-optical reflectors and beam waveguides based on an integral-equation approach with Nystrom's discretization

“…As a result, we obtain a set of linear equations and solve it with a simple Gaussian scheme. The method's details and extensive numerical validations can be found in [22]; paper [23] is also a relevant source. This method enables one to study the effects of the wave radiation, guidance, and scattering for reflectors up to 100 and more in size, with high accuracy and small computer resources.…”

“…In [22], we presented basic ideas of such a discrete model based on the method of discrete singularities (MDS) and gave some examples of the accurate 2D analysis of curved reflector antennas. In parallel, a similar approach was developed recently in [23] for flat 2D strips and slots in a layered environment. Note that these works use different edge-corrected quadrature formulas, although both consider the whole reflector as an entire domain and improve the accuracy by increasing the order of interpolation polynomials, i.e., the reflector is not meshed with a finer and finer set of subdomains.…”