scholarly journals Efficient Evaluation of Slowly Converging Integrals Arising from MAP Application to a Spectral-Domain Integral Equation

Electronics ◽  
2019 ◽  
Vol 8 (12) ◽  
pp. 1500
Author(s):  
Mario Lucido ◽  
Marco Donald Migliore ◽  
Alexander I. Nosich ◽  
Gaetano Panariello ◽  
Daniele Pinchera ◽  
...  
Keyword(s):  
Integral Equation ◽  
Electromagnetic Scattering ◽  
Surface Current ◽  
Coefficient Matrix ◽  
Spectral Domain ◽  
Surface Current Density ◽  
Integral Theorem ◽  
Analytical Technique ◽  
Domain Integral ◽  
Cauchy Integral Theorem

In this paper, we devised an analytical technique to efficiently evaluate the improper integrals of oscillating and slowly decaying functions arising from the application of the method of analytical preconditioning (MAP) to a spectral-domain integral equation. The reasoning behind the method’s application may consistently remain the same, but such a procedure can significantly differ from problem to problem. An exhaustive and understandable description of such a technique is provided in this paper, where we applied MAP for the first time to analysis of electromagnetic scattering from a zero-thickness perfectly electrically conducting (PEC) disk in a planarly layered medium. Our problem was formulated in the vector Hankel transform domain and discretized via the Galerkin method, with expansion functions reconstructing the physical behavior of the surface current density. This ensured fast convergence in terms of the truncation order, but involved numerical evaluation of slowly converging integrals to fill in the coefficient matrix. To overcome this problem, appropriate contributions were pulled out of the kernels of the integrals, which led to integrands transforming into exponentially decaying functions. Subsequently, integrals of the extracted contributions were expressed as linear combinations of fast-converging integrals via the Cauchy integral theorem. As shown in the numerical results section, the proposed technique drastically outperformed the classical analytical asymptotic-acceleration technique.

scholarly journals Analysis of the Propagation in High-Speed Interconnects for MIMICs by Means of the Method of Analytical Preconditioning: A New Highly Efficient Evaluation of the Coefficient Matrix

2021 ◽  
Vol 11 (3) ◽  
pp. 933
Author(s):  
Mario Lucido
Keyword(s):  
Integral Equation ◽  
Integrated Circuits ◽  
High Speed ◽  
Coefficient Matrix ◽  
Single Step ◽  
Integral Theorem ◽  
Acceleration Technique ◽  
Cauchy Integral Theorem ◽  
Integral Equation Formulation ◽  
High Speed Interconnects

The method of analytical preconditioning combines the discretization and the analytical regularization of a singular integral equation in a single step. In a recent paper by the author, such a method has been applied to a spectral domain integral equation formulation devised to analyze the propagation in polygonal cross-section microstrip lines, which are widely used as high-speed interconnects in monolithic microwave and millimeter waves integrated circuits. By choosing analytically Fourier transformable expansion functions reconstructing the behavior of the fields on the wedges, fast convergence is achieved, and the convolution integrals are expressed in closed form. However, the coefficient matrix elements are one-dimensional improper integrals of oscillating and, in the worst cases, slowly decaying functions. In this paper, a novel technique for the efficient evaluation of such kind of integrals is proposed. By means of a procedure based on Cauchy integral theorem, the general coefficient matrix element is written as a linear combination of fast converging integrals. As shown in the numerical results section, the proposed technique always outperforms the analytical asymptotic acceleration technique, especially when highly accurate solutions are required.

scholarly journals Analysis of the Scattering from a Two Stacked Thin Resistive Disks Resonator by Means of the Helmholtz–Galerkin Regularizing Technique

2021 ◽  
Vol 11 (17) ◽  
pp. 8173
Author(s):  
Mario Lucido
Keyword(s):  
Cross Section ◽  
Near Field ◽  
Surface Current ◽  
Coefficient Matrix ◽  
Hankel Transform ◽  
General Purpose ◽  
Analytical Technique ◽  
Planar Surfaces ◽  
Natural Modes ◽  
Resonance Frequencies

In this paper, the scattering of a plane wave from a lossy Fabry–Perót resonator, realized with two equiaxial thin resistive disks with the same radius, is analyzed by means of the generalization of the Helmholtz–Galerkin regularizing technique recently developed by the author. The disks are modelled as 2-D planar surfaces described in terms of generalized boundary conditions. Taking advantage of the revolution symmetry, the problem is equivalently formulated as a set of independent systems of 1-D equations in the vector Hankel transform domain for the cylindrical harmonics of the effective surface current densities. The Helmholtz decomposition of the unknowns, combined with a suitable choice of the expansion functions in a Galerkin scheme, lead to a fast-converging Fredholm second-kind matrix operator equation. Moreover, an analytical technique specifically devised to efficiently evaluate the integrals of the coefficient matrix is adopted. As shown in the numerical results section, near-field and far-field parameters are accurately and efficiently reconstructed even at the resonance frequencies of the natural modes, which are searched for the peaks of the total scattering cross-section and the absorption cross-section. Moreover, the proposed method drastically outperforms the general-purpose commercial software CST Microwave Studio in terms of both CPU time and memory occupation.

scholarly journals Решение электродинамической задачи для микрополосковой излучающей структуры с киральной подложкой

2018 ◽  
Vol 44 (11) ◽  
pp. 80
Author(s):  
М.А. Бузова ◽  
Д.С. Клюев ◽  
М.А. Минкин ◽  
А.М. Нещерет ◽  
Ю.В. Соколова
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Surface Current ◽  
Longitudinal Component ◽  
Cauchy Type ◽  
Surface Current Density ◽  
Electrodynamic Problem ◽  
Type Singularity ◽  
Different Types

AbstractWe present a solution of the electrodynamic problem for a microstrip radiating structure with a substrate of a chiral metamaterial using the singular integral representation of the field, which in turn is reduced to a singular integral equation with the Cauchy-type singularity relative to the longitudinal component of the surface current density. Graphs of the current distribution for different types of substrates and the chirality parameters of a substrate are given.

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