Higher Order Impedance Boundary Conditions Research Articles

New approximation method for high order impedance boundary condition with surface integral equations for the scattering problem

AbstractIn this paper, we propose a new method to approximate operators resulted from solving the scattering Problem in electromagnetism by dielectrically coated conducting bodies, using integral equations and high order impedance boundary condition. We introduce the variational Problem and we prove its well‐posedness. After discretization, we find that operators arising from the high‐order impedance boundary conditions are not well‐defined. We present the new theoretical approach and highlight its potential through numerical experiments.

Boundary element method with high order impedance boundary condition for Maxwell’s equations

A boundary-element method is introduced for solving electromagnetic scattering problems for coated objects. We develop a new approximate impedance boundary condition with Hodge operator and propose a variationnal formulation for the physical problem. The existence and uniqueness of the solution are studied and we propose a discretization of the unknowns which are magnetic and electric currents with Rao-Wilton-Glisson basis functions to lead to the correct field continuity properties at the interface of different media. And the unknowns are the fluxes of the magnetic and electric currents across the edges of the mesh. Numerical results from the formulation is presented and measurements are compared in order to obtain test cases for the development and validation of numerical method. In the first time, we compare standard impedance boundary condition and high order impedance boundary condition on bodies of revolution with simple topology and we show the increased accuracy of the high order impedance boundary condition solution relative to the standard impedance boundary condition solution. Finally, a benchmark test is treated to show the interest of our method.

Efficient EDM-PO Method for the Scattering From Electrically Large Objects With the High-Order Impedance Boundary Condition

This work proposes an efficient hybrid method consisting of the efficient iterative method of moments–physical optics (EI-MoM-PO) and the equivalent dipole moment (EDM) with the high-order impedance boundary condition (HOIBC) to explore the electromagnetic scattering from the object with an isotropic or anisotropic surface coating. By introducing the HOIBC into the EDM method, the relationship between the equivalent electric and magnetic dipole moments is established. Therefore, the EDM method is extended to general cases. For the composite scattering from electrically large objects including MoM and PO regions, the EDM method is implemented to fill the MoM impedance matrix and to accelerate the evaluation of the coupling between the MoM and PO regions, which considerably improves the computing efficiency. The numerical results indicate that the proposed hybrid method allows for a substantial decrease in computation time. Also, they also agree well with the traditional finite element method, MoM, or MoM-PO method, which proves the validity of the proposed method.

A Wave Diffraction Problem with Higher Order Impedance Boundary Conditions

In this paper, we consider an impedance boundary transmission problem for the Helmholtz equation originated by a problem of wave diffraction by an infinite strip with higher order imperfect boundary conditions. Operator theoretical methods and relations between operators are built to deal with the problem and, as consequence, a transparent interpretation of the problem in an operator theory framework are associated to the problem. In particular, different types of operator relations are exhibited for different types of operators acting between Lebesgue and Sobolev spaces on a finite interval and the positive half-line. All this has consequences in the understanding of the structure of this type of problems. At the end, we describe when the operators associated with the problem enjoy the Fredholm property in terms of the initial space order parameters.

Sufficient uniqueness conditions for the solution of the time harmonic Maxwell’s equations associated with surface impedance boundary conditions

We consider the time-harmonic Maxwell’s equations for the scattering or radiating problem from a 3-D object that is either a metallic surface coated with material layers (MCS) or a dichroic structure (DS) made up of multiple frequency selective surfaces (FSS) embedded in materials. Low or high order impedance boundary conditions (IBC) are employed to reduce the numerical complexity of the solution of this problem via an integral equation or a finite element formulation. An IBC links the tangential components of the electric field to those of the magnetic field on the outer surface of the MCS, or on the FSSs, and avoids the solution of Maxwell’s equations inside the inhomogeneous domain for a MCS or, for a DS, the meshing of the numerous unit cells in a FSS. Sufficient uniqueness conditions (SUC) are established for the solutions of Maxwell’s equations associated with these IBCs, the performances of which, when constrained by the corresponding SUCs, are numerically evaluated for an infinite or finite planar structure.

Comments on electromagnetic scattering from inhomogeneous impedance periodic rough surfaces by transform technique

AbstractIn an recent article by Tezel (Microwave and Optical Technology Letters 51 (2009), 116–119) an electromagnetic scattering problem related to a periodic rough surface having an inhomogeneous standard impedance boundary condition is considered. The author proposes a solution method, which is based on the equivalent representation of the rough surface in terms of a planar one with higher order impedance boundary condition. We have already noticed that the mathematical formulation of the problem is completely incorrect and such a rough surface cannot be equivalently represented in terms of higher order impedance boundary condition. Therefore, the numerical results has no reliability, and they are fictitious. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 52: 245, 2010; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24862

Comments on solution of electromagnetic scattering problems by transformation into high order impedance boundary condition

AbstractIn an recent article by Tezel (Microwave and Optical Technology Letters 50 (2008), 831–836), it is claimed that a new method is derived for the scattering of electromagnetic waves from cylindrical objects having standard impedance boundary condition on its boundary. This article has two fundamental flaws: First, the approach that the author tries to apply is not new; Second, as we prove below the formulation, consequently, the numerical results are completely incorrect. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 52: 244–245, 2010; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24861

FDTD Implementation and Application of High Order Impedance Boundary Condition Using Rational Functions

An algorithm is proposed for the implementation of the high-order surface impedance boundary condition using the finite-difference time-domain method. The surface impedance function of a lossy medium is approximated by a series of rational functions in the Laplace domain, whereas the dyadic differential operator is approximated by a second-order power series. By assuming that the fields are piecewise linear, the time-domain convolution integrals are computed using a recursive formula. The impedance function of a coating layer is approximated by a third-order power series. The algorithm can be applied to scattering problems of a three-dimensional coating for both vertically and horizontally polarized waves. The advantage of the proposed method is that the result can be applied to media of arbitrary conductivities, with a wide range of incident angles from zero to graze. Some numerical examples are given to substantiate the theory.

Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches

This paper introduces simple analytical formulas for the grid impedance of electrically dense arrays of square patches and for the surface impedance of high-impedance surfaces based on the dense arrays of metal strips or square patches over ground planes. Emphasis is on the oblique-incidence excitation. The approach is based on the known analytical models for strip grids combined with the approximate Babinet principle for planar grids located at a dielectric interface. Analytical expressions for the surface impedance and reflection coefficient resulting from our analysis are thoroughly verified by full-wave simulations and compared with available data in open literature for particular cases. The results can be used in the design of various antennas and microwave or millimeter wave devices which use artificial impedance surfaces and artificial magnetic conductors (reflect-array antennas, tunable phase shifters, etc.), as well as for the derivation of accurate higher-order impedance boundary conditions for artificial (high-) impedance surfaces. As an example, the propagation properties of surface waves along the high-impedance surfaces are studied.

Solution of electromagnetic scattering problems by transformation into high order impedance boundary condition

AbstractIn this study, electromagnetic scattering from inhomogeneous impedance cylinder of arbitrary shape have been solved by means of transformation of problem into equivalent problem, that is scattering from circle represented by high order inhomogeneous impedance boundary condition (IBC). High order impedance functions are determined by using shape of object and its surface impedance. Then, equivalent problem is solved by means of series expansion method. This transformation makes the problem simple formulation, which can be solved computational effectively. Results and computational times obtained by transform method and those obtained by Method of Moment (MoM) technique are compared. Good agreements are observed in results. It is also observed that transform method needs less computational time than MoM method. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 832–836, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23200

Higher Order Inhomogeneous Impedance Boundary Conditions for Perfectly Conducting Objects

A new, simple, and fast method for the solution of electromagnetic scattering problems for perfectly conducting objects of arbitrary shape is presented. The method is based on an equivalent representation of the conducting object in terms of a circular one having a higher order impedance boundary condition on its surface. As a first result, a new universal relation between the higher order surface impedances and the shape of the object is obtained. Then, by taking advantage of this relationship, the scattering problem related to a conducting object is recast as the solution of a matrix equation whose coefficients are determined from the higher order impedances. Numerical simulations show that the method yields to accurate results, and that, it is computationally effective

Surface and internal waves in a stratified layer of liquid and an analysis of the impedance boundary conditions

The surface and internal waves in a multilayered ideal liquid for specified displacements of the bottom are considered. The upper surface of the liquid is either free or displacements are specified on it. In the long-wave approximation, asymptotically accurate models of the waves propagating along the surface layers of an incompressible liquid are constructed both for intense stratification (the ratio of the densities of neighbouring layers are associated with a small parameter) and for weak stratification. The important case of this problem of the dynamic contact of rigid bodies through a layer of incompressible or compressible liquid is investigated. High-order impedance boundary conditions are constructed and the results of testing them using the exact solutions are presented.

Impedance boundary conditions for finite planar or curved frequency selective surfaces embedded in dielectric Layers

We consider the time-harmonic electromagnetic scattering problem from a finite planar or curved structure made up of infinitesimally thin frequency-selective surfaces (FSSs) embedded in dielectric layers, with possibly nearby located objects. In order to avoid the meshing of the unit cells that constitute the FSSs, this problem is solved by employing an integral equation (IE) or finite-element (FE) formulation in conjunction with approximate impedance boundary conditions (IBCs) prescribed on the sheets that model the FSSs. The impedances in the IBCs are derived from the exact reflection and transmission coefficients calculated for the fundamental Floquet mode on the infinite planar structure illuminated by a planewave at a given incidence. When the structure is curved and/or the direction of the incident wave is unknown, higher order IBCs are proposed that are valid in a large angular range and can be implemented in a standard IE or FE formulation. Their numerical efficiencies are evaluated for finite planar or curved two-dimensional structures, or radomes, where the FSSs are strip gratings. As an example, for a curved radome surrounding a conducting plate, it is shown that, when the Floquet modes of the gratings are evanescent, these IBCs allow an accurate calculation of the radar cross section of the whole structure with far smaller computing resources than would have been required by a full-wave formulation.

Impedance boundary conditions for finite planar and curved frequency selective surfaces

We consider the time-harmonic electromagnetic scattering problem from a finite planar or curved, infinitesimally thin, frequency selective surface (FSS), the periodic unit cells of which are constituted, exclusively, by electric conductors and free-space. In order to avoid the meshing of these cells, the problem is solved by employing an integral equation formulation in conjunction with approximate impedance boundary conditions (IBC) prescribed on the sheet that models the FSS. The impedance in the IBC is derived from the exact reflection coefficient calculated, for the fundamental Floquet mode, on the infinite planar FSS illuminated by a plane-wave at a given incidence. When the FSS is curved, and/or the direction of the incident wave is unknown, higher order IBCs are proposed that are valid in a large angular range and can be implemented in a standard method of moments formulation. Also, a simple technique is presented that allows to reproduce the radiating Floquet modes in the scattered field even though those are not accounted for in these IBCs. Their numerical efficiencies are evaluated for a curved strip grating translationally invariant along one direction. Finally, we present an alternative approach where the impedance is approximated by its truncated Fourier series, that considerably enhances the accuracy of the results at the cost, however, of a denser mesh of the sheet.

Narrow-Waisted Gaussian Beams for Aperture-Generated Scattering From Planar Conducting Surfaces With Complex Coatings Described by Higher Order Impedance Boundary Conditions

In this paper, we use higher order impedance boundary conditions (HOIBCs) for studying high frequency asymptotic two-dimensional (2-D) scattering of truncated aperture-generated electromagnetic fields from planar conducting surfaces coated by multiple layers of homogeneous bi-anisotropic media. The reflected field syntheses are carried out via asymptotic reduction of rigorous plane-wave spectral integrals, which are subsequently discretized and transformed to the spatial domain through use of a Gabor-based narrow-waisted (NW) Gaussian beam (GB) algorithm. In this discretized algorithm, the GB propagators are approximated by previously explored standard and modified (uniform) complex-source-point paraxial asymptotic techniques. Example applications are restricted to zeroth and second order IBCs for single- and multilayer complex coatings, with emphasis on the adaptation of the NW-GBs to the HOIBC launch conditions in the presence of localizing (e.g., focused and/or abruptly truncated) illumination. The results confirm that the previously established utility of the NW-GB algorithm with respect to accuracy and computational feasibility continues to hold for this fairly general combination of environmental complexity and strongly inhomogeneous (localizing) illumination.

Finite-difference time-domain model of interfaces with metals and semiconductors based on a higher order surface impedance boundary condition

A new numerical finite-difference time-domain (FDTD) model of interfaces with metals and semiconductors is developed. The model uses a higher order impedance boundary condition to simulate the fields in conductive media. The approach has been found to be considerably more accurate than the known models based on the simple Leontovich impedance boundary condition. This is because the new model takes the incidence angle of the incident waves into account, and it is valid for all values of conductivity. The derivation of the method is presented in the two-dimensional (2D) case, and the performance is studied over a wide range of conductivity. The validation of the method is made by comparing with the exact results in a 2D example problem. The proposed method is also compared with some other methods available in the literature.

Higher order impedance boundary conditions for sparse wire grids

Higher order impedance boundary conditions designed for modeling wire grids of thin conducting wires are established. The derivation is based on the exact analytical summation of the individual wire fields. This allows one to write an approximate boundary condition on the grid surface, which connects the averaged electric field and the averaged current (or the electric field and the averaged magnetic fields on the two sides of the grid surface). The condition depends on the tangential derivatives of the averaged current (up to the sixth order). This approach provides an extension of the averaged boundary conditions method (well established for dense grids) to sparse grids. Numerical examples demonstrate very good accuracy of the solutions for the field reflected from grids with the wire separation as large as half of the wavelength.

Impedance Boundary Conditions for a Thin Periodically Inhomogeneous Dielectric Layer Coated On a Curved Pec

In this paper an asymptotic solution to the wave equation for a thin curved layer with periodically inhomogeneous permittivity is derived. The wave equation and field components are expanded in the thickness h of the layer. A propagator, i.e., an operator that maps the fields from one side of the surface to the other, is derived. This propagator is used to derive a higher order impedance boundary condition. The solution is expressed as an asymptotic expansion in the thickness h for a layer coated on a perfect electric conductor (PEC). The problem treated is two-dimensional and the TE case is considered. A couple of numerical examples are given with comparisons to solutions obtained by integral equation based techniques.

Higher order impedance boundary conditions for metal-backed inhomogeneous dielectric layers

A systematic spectral-domain approach is described for obtaining higher order electromagnetic impedance boundary conditions for isotropic, longitudinally inhomogeneous, dielectric coatings on a polarization-preserving impedance plane. Representative computational examples including single- and two-layer inhomogeneous coatings are discussed to illustrate the accuracy of the proposed approach. ©1999 John Wiley & Sons, Inc. Microwave Opt Technol Lett 22: 249–254, 1999.

Sdra Approach for Higher-Order Impedance Boundary Conditions for Complex Multi-Layer Coatings on Curved Conducting Bodies

Sdra Approach for Higher-Order Impedance Boundary Conditions for Complex Multi-Layer Coatings on Curved Conducting Bodies