Second‐kind integral solvers for TE and TM problems of diffraction by open arcs

Abstract

[1] We present a novel approach for the numerical solution of problems of diffraction by open arcs in two dimensional space. Our methodology relies on composition of weighted versions of the classical integral operators associated with the Dirichlet and Neumann problems (TE and TM polarizations, respectively) together with a generalization to the open-arc case of the well known closed-surface Calderón formulae. When used in conjunction with spectrally accurate discretization rules and Krylov-subspace linear algebra solvers such as GMRES, the new second-kind TE and TM formulations for open arcs produce results of high accuracy in small numbers of iterations-for low and high frequencies alike.Citation: Bruno, O. P., and S. K. Lintner (2012), Second-kind integral solvers for TE and TM problems of diffraction by open arcs, Radio Sci., 47, RS6006,

Scattering by large periodic surfaces: Novel numerical method and applications

“…As this expression only contains tangential derivatives (s denotes the arclength), it can be implemented via numerical differentiation with FFTs [11].…”

High-Performance Computational Electromagnetics in Frequency-Domain and Time-Domain

“…Complete proofs of the well conditioned character of the aforementioned integral equation formulation for open surfaces was provided in [13,14]. A numerical analysis of the related closed surface solvers [22], in turn, was for the first time put forth in [21].…”

“…This problem has many applications in optics, from the very small to the very large (from nano-scale optical devices up and including the discovery of planets) as well as applications in stealth, antenna design, electronic and photonic devices, etc. For the first time a regularized integral equation yielding a Fredholm equation of the second kind for opensurface problems was put forth in the contributions [12][13][14], including rigorous analysis of the equations and efficient implementations in two-and three-dimensional space. In practice, the second-kind character of the equation enables solution by iterative solvers in very small numbers of iterations and it thus enables effective solution of the problem.…”

“…A variety of problems and configurations were thus considered, including problems of scattering by open surfaces [12][13][14]; improved integral methods for closed surfaces [15]; problems concerning propagation and scattering by penetrable scatterers [16]; studies of absorption properties of conducting materials containing asperities [17,18]; as well as new methods for evaluation of Laplace eigenfunctions on general domains and under challenging boundary conditions [19,20]. A rigorous convergence proof for the original methods [22] was provided in [21].…”

“…Meaningful progress concerning well conditioned integral algorithms took place as a result of work sponsored by this contract, including rigorous mathematical theory and powerful numerical algorithms with applicability in a number of fields of science and engineering [12][13][14][15][16][17][18][19][20][21]. A variety of problems and configurations were thus considered, including problems of scattering by open surfaces [12][13][14]; improved integral methods for closed surfaces [15]; problems concerning propagation and scattering by penetrable scatterers [16]; studies of absorption properties of conducting materials containing asperities [17,18]; as well as new methods for evaluation of Laplace eigenfunctions on general domains and under challenging boundary conditions [19,20].…”