The electromagnetic scattering problem with generalized impedance boundary conditions

ABSTRACTWe show how a Trefftz Discontinuous Galerkin (TDG) method for the displacement form of the Helmholtz equation can be used to approximate problems having a generalized impedance boundary condition (GIBC) involving surface derivatives of the solution. Such boundary conditions arise naturally when modeling scattering from a scatterer with a thin coating. The thin coating can then be approximated by a GIBC. A second place GIBCs arise is as higher order absorbing boundary conditions. This paper also covers both cases. Because the TDG scheme has discontinuous elements, we propose to couple it to a surface discretization of the GIBC using continuous finite elements. We prove convergence of the resulting scheme and demonstrate it with two numerical examples.

The authors present a rigorous uniform theory of diffraction (UTD) solution of the diffraction by a coated convex cylinder simulated with generalized impedance boundary conditions (GIBC). In addition, a uniform asymptotic solution is obtained which remains valid when the observation point is in the close vicinity of the cylinder. An important aspect of this work is the use of second-order GIBCs which are characterized by the inclusion of second-order field derivatives in their definition. Because of this, they are less local, leading to an improved simulation (with respect to the standard impedance boundary conditions) of the coating. The improved accuracy of the GIBC simulation is demonstrated, and it is shown how the presented UTD solution can be extended to treat multilayered coated cylinders. >

The authors present generalized surface impedance conditions that improve medium characterization. One of the most common conditions that is often used to simulate a dielectric coating is the standard impedance boundary condition, and the improved boundary condition discussed can be viewed as a generalization of this, distinguished by the presence of higher-order derivatives of the field. By virtue of the derivatives, the conditions are less local in character, and the additional degrees of freedom can be used to better simulate the material properties of a surface. Following a review of the standard impedance condition, the new conditions are introduced. Several forms are presented, each with coefficients which are functions of the geometry and material properties of the surface. The applications of these to a variety of scattering problems are then discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The notion of impedance boundary condition is generalized and illustrated by the discussion of the electromagnetic field arising from a magnetic line dipole source located above a plane structure. The generalized impedance boundary condition contains a set of parameters whose totality offers a physical description of the configuration. By suitably selecting these parameters, the configuration may be made to correspond to a structure that supports surface waves. We give an exact solution for a plane structure that supports one, two and three surface waves. The magnitudes of the surface waves are obtained and simple formulas for the radiated far field patterns are given. We also show how the methods employed can be extended to the case of any number of surface waves. This involves an n n th order mixed boundary condition for a second order partial differential equation.

We consider the inverse scattering problem of determining the shape of an impenetrable two-dimensional obstacle with a generalized impedance boundary condition from near-field measurements. We propose the reverse time migration method where the imaging functional is defined as the imaginary part of the cross-correlation. Our analysis shows that, on the one hand, the imaging functional has contrast at the boundary of the scatterer and decays outside the scatterer; On the other hand, the imaging functional is always positive thus may have better stability properties. These imply that the approach has powerful imaging quality which is confirmed in our numerical experiments.

Rigorous uniform geometrical theory of diffraction (UTD) diffraction coefficients are presented for a convex coated cylinder simulated with generalized impedance boundary conditions. In particular, ray solutions are obtained which remain valid in the transition region and reduce uniformly to those in the deep lit and shadow regions. These involve new transition functions in place of the usual Fock‐type integrals, characteristic to the impedance cylinder. A uniform asymptotic solution is also presented for observations in the close vicinity of the cylinder. As usual, the diffraction coefficients for the convex cylinder are obtained via a generalization of the corresponding ones for the circular cylinder.

In this paper, we consider the direct and inverse problem for time-fractional diffusion in a domain with an impenetrable subregion. Here we assume that on the boundary of the subregion the solution satisfies a generalized impedance boundary condition. This boundary condition is given by a second order spacial differential operator imposed on the boundary. A generalized impedance boundary condition can be used to model corrosion and delimitation. The well-posedness for the direct problem is established where the Laplace Transform is used to study the time dependent boundary value problem. The inverse impedance problem of determining the parameters from the Cauchy data is also studied provided the boundary of the subregion is known. The uniqueness of recovering the boundary parameters from the Neumann to Dirichlet mapping is proven.

: The report represents the fifteenth annual summary of The Ohio State University Joint Services Electronics Program (JSEP). The transfer of the compact range identification technology initiated under JSEP support for time domain studies continues to make large advances. We are also assisting Rockwell (Tulsa) to update their RCS facilities. This work is on a subcontract to the ESL from the Air Force. This has lead to involvement in the study of Ultra Wide Band radar systems. The research activities devoted to the Generalized Ray and Gaussian Beams continues. Our JSEP research continues to be expanded by external funding. This program is being expanded by use of such funds which are more focussed on the requirements of the sponsors which includes both the Air Force and the Navy. Our JSEP research continues to focus on electromagnetic related topics. There are four major electromagnetics areas that were pursued in the past year. The Diffraction Studies Work Unit has initiated research on a time domain version of the Uniform Theory of Diffraction. A second topic under the Diffraction Studies Work Unit involves further extensions of the generalized resistive boundary condition and the generalized impedance boundary condition. These have been applied to scattering from a chiral slab. A third topic of interest is the diffraction from a corner. A fourth task involves the reflection/diffraction of a Gaussian beam. This represents an approach to replace the usual ray optics solution for very complex geometries where the versatile ray optics solution becomes cumbersome.

In this paper, we are interested in using impedance boundary conditions ( IBCs) to model the scattering from a vibrating object which can be considered a perturbation problem where the perturbation (and the corresponding IBC) varies with time. One of the most basic objects for which a scattering solution can be obtained is an infinite PEC cylinder. Using the exact solution for a circular cylinder, the IBCs needed to model the scattering from a perturbed PEC cylinder are derived for both the TM and TE polarizations. It is shown that the standard IBC (SIBC) is accurate to first order for the TM case, but a second generalized IBC (GIBC) is needed to provide first order accuracy for the TE case. Results from a moments method (MoM) implementation of the IBCs are presented.

The scattering of electromagnetic waves from obstacles with wave-material interaction in thin layers on the surface is described by generalized impedance boundary conditions, which provide effective approximate models. In particular, this includes a thin coating around a perfect conductor and the skin effect of a highly conducting material. The approach taken in this work is to derive, analyse and discretize a system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. In a familiar second step, the fields are evaluated in the exterior domain by a representation formula, which uses the time-dependent potential operators of Maxwell’s equations. The time-dependent boundary integral equation is discretized with Runge–Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. Using the frequency-explicit bounds from the well-posedness analysis given here together with known approximation properties of the numerical methods, the full discretization is proved to be stable and convergent, with explicitly given rates in the case of sufficient regularity. Taking the same Runge–Kutta based convolution quadrature for discretizing the time-dependent representation formulas, the optimal order of convergence is obtained away from the scattering boundary, whereas an order reduction occurs close to the boundary. The theoretical results are illustrated by numerical experiments.

Numerical solution of a generalized boundary value problem for the modified Helmholtz equation in two dimensions

A semi-analytical finite element numerical mode-matching (FNMM) method is developed to simulate microscopic structures of metasurfaces on the interfaces of the 3-D multilayered structure. Similar to the recently developed spectral element NMM method, the FNMM method reduces the original 3-D problem into a series of 2-D waveguide eigenvalue problems plus a 1-D layered medium problem, which can be analytically described by a recursion procedure. The 2-D eigenvalue problems in the horizontal directions are solved numerically by the mixed finite element method (MFEM) to remove the spurious modes in the conventional FEM. Therefore, it can effectively and accurately simulate electromagnetic wave interactions with the detail metasurfaces, especially for thick layers. Metasurfaces embedded on the interfaces of 3-D structures are treated by the generalized impedance boundary conditions, thus enabling their rapid yet accurate computation. This method is applied to characterize metasurfaces consisting of microscopic patterns with complex geometric construction with details much smaller than the wavelength. Numerical experiments indicate that the FNMM method is highly efficient and accurate for the microscopic metasurface models.

The problem of the diffraction of a plane electromagnetic wave incident at an oblique angle on a wedge of arbitrary angle with general tensor impedance boundary conditions is solved using a semi-analytical approach. Application of Maliuzhinets' method transforms the boundary-value problem into coupled functional difference equations (FDEs) for two unknown Sommerfeld integral spectra in a basic strip. By explicitly separating out the singular parts of the spectra in the strip, followed by a partial inversion of the FDEs, we obtain integral representations of the regular parts of the spectra. The regular parts are then expanded in a Taylor series in terms of a new variable that conformally maps the strip on to a disc. This expansion reduces the integral representations to algebraic equations for the series coefficients and these are solved numerically. We examine the convergence of the procedure, compare the numerical solution with an available reference solution and present solutions of new problems.

Second-order boundary conditions are used to simulate a three-layer coating on perfectly conducting cylindrical structures. Both E and H polarizations are considered, and integral equations are presented based on the derived conditions. A moment method implementation of the resulting integral equations is given, and results are presented for various coatings in order to illustrate the accuracy of the higher-order conditions. The improvement that can be achieved with the proposed second-order generalized impedance boundary conditions over the standard boundary conditions is shown.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

This paper develops pseudospectral meshless radial point Hermit interpolation (PSMRPHI) and pseudospectral meshless radial point interpolation (PSMRPI) in order to apply to the elliptic partial differential equations (PDEs) held on irregular domains subject to impedance (convective) boundary conditions. Elliptic PDEs in simplest form, i.e., Laplace equation or Poisson equation, play key role in almost all kinds of PDEs. On the other hand, impedance boundary conditions, from their application in electromagnetic problems, or convective boundary conditions, from their application in heat transfer problems, are nearly more complicated forms of the boundary conditions in boundary value problems (BVPs). Based on this problem, we aim also to compare PSMRPHI and PSMRPI which belong to more influence type of meshless methods. PSMRPI method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of PSMRPI and PSMRPHI methods. While the latter one has been rarely used in applications, we observe that is more accurate and reliable than PSMRPI method.