Generalized Impedance Boundary Conditions Research Articles

Time-dependent electromagnetic scattering from thin layers

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.

English

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.

Modes analyses of cylindrical waveguides using the MFCM

An efficient modes analyses technique for isotropic or anisotropic material filled 2D metallic waveguides with an arbitrary contour using the multifilament current method (MFCM) is presented. The ideal PEC boundary of a 2D waveguide is replaced by a shell with a high conductivity and electrical small thickness. The thin lossy shell not only can well approximate the boundary condition of PEC waveguide wall therefore without altering the initial waveguide modes, but also can let the external excitation penetrate through to excite the inside modes, resulting in a high internal field intensity at the frequency of each mode. In this case, the modes are revealed by the peaks of field intensity responses, and the spurious modes which existed in traditional source-free modes determination techniques can be completely avoided. Based on this idea, a generalized impedance boundary condition (GIBC) is formulated to represent the lossy waveguide wall and further utilized in the MFCM for simulating the internal field intensity over frequency. Three different configurations of a 2D waveguide are considered. The computed modes are compared with that obtained from commercial software, and an excellent agreement is achieved, yet an competitive advantage on simulation performances is observed by using the proposed technique.

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

We propose numerical schemes for solving the boundary value problem for the modified Helmholtz equation and generalized impedance boundary condition. The approaches are based on the reduction of the problem to the boundary integral equation with a hyper-singular kernel. In the first scheme the hyper-singular integral operator is treated by splitting off the singularity technique whereas in the second scheme the idea of numerical differentiation is employed. The solvability of the boundary integral equation and convergence of the first method are established. Exponential convergence for analytic data is exhibited by numerical examples.

An Accelerated Surface Integral Equation Method for the Electromagnetic Modeling of Dielectric and Lossy Objects of Arbitrary Conductivity

Surface integral equation (SIE) methods are of great interest for the numerical solution of Maxwell's equations in the presence of homogeneous objects. However, existing SIE algorithms have limitations, either in terms of scalability, frequency range, or material properties. We present a scalable SIE algorithm based on the generalized impedance boundary condition, which can efficiently handle, in a unified manner, both dielectrics and conductors over a wide range of conductivity, size, and frequency. We devise an efficient strategy for the iterative solution of the resulting equations, with efficient preconditioners and an object-specific use of the adaptive integral method (AIM). With a rigorous error analysis, we demonstrate that the AIM can be applied over a wide range of frequencies and conductivities. Several numerical examples, drawn from different applications, demonstrate the accuracy and efficiency of the proposed algorithm.

Time-dependent acoustic scattering from generalized impedance boundary conditions via boundary elements and convolution quadrature

Abstract Generalized impedance boundary conditions are effective approximate boundary conditions that describe scattering of waves in situations where the wave interaction with the material involves multiple scales. In particular, this includes materials with a thin coating (with the thickness of the coating as the small scale) and strongly absorbing materials. For the acoustic scattering from generalized impedance boundary conditions, the approach taken here first determines the Dirichlet and Neumann boundary data from a system of time-dependent boundary integral equations with the usual boundary integral operators, and then the scattered wave is obtained from the Kirchhoff representation. The system of time-dependent boundary integral equations is discretized by boundary elements in space and convolution quadrature in time. The well-posedness of the problem and the stability of the numerical discretization rely on the coercivity of the Calderón operator for the Helmholtz equation with frequencies in a complex half-plane. Convergence of optimal order in the natural norms is proved for the full discretization. Numerical experiments illustrate the behaviour of the proposed numerical method.

An Inverse Parameter Problem with Generalized Impedance Boundary Condition for Two-Dimensional Linear Viscoelasticity

We analyze an inverse boundary value problem in two-dimensional viscoelastic media with a generalized impedance boundary condition on the inclusion via boundary integral equation methods. The model...

Reverse time migration for inverse obstacle scattering with a generalized impedance boundary condition

ABSTRACT 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.

Microscopic Modeling of Metasurfaces by the Mixed Finite Element Numerical Mode-Matching Method

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.

General expression of the diffracted waves by a half-screen with generalized impedance boundary conditions

The diffraction process of plane waves by a half-screen with generalized impedance boundary conditions is studied. Firstly, the scattered geometrical optics waves are obtained by subtracting the incident field from the total geometrical optics wave. The diffracted field expressions are derived by taking into account the construction of the scattered geometrical optics wave for first, second and third order impedance half-planes. Then the diffracted field expression is generalized for the Nth order half-screen. The uniform representation of the edge diffracted wave is obtained by considering the exact solution of Sommerfeld. The behaviors of the total, total GO and diffracted fields are analyzed numerically.

Shape Sensitivity Analysis for Elastic Structures with Generalized Impedance Boundary Conditions of the Wentzell Type—Application to Compliance Minimization

This paper focuses on Generalized Impedance Boundary Conditions (GIBC) with second order derivatives in the context of linear elasticity and general curved interfaces. A condition of the Wentzell type modeling thin layer coatings on some elastic structure is obtained through an asymptotic analysis of order one of the transmission problem at the thin layer interfaces with respect to the thickness parameter. We prove the well-posedness of the approximate problem and the theoretical quadratic accuracy of the boundary conditions. Then we perform a shape sensitivity analysis of the GIBC model in order to study a shape optimization/optimal design problem. We prove the existence and characterize the first shape derivative of this model. A comparison with the asymptotic expansion of the first shape derivative associated to the original thin layer transmission problem shows that we can interchange the asymptotic and shape derivative analysis. Finally we apply these results to the compliance minimization problem. We compute the shape derivative of the compliance in this context and present some numerical simulations.

A Trefftz Discontinuous Galerkin method for time-harmonic waves with a generalized impedance boundary condition

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.

An Integral Equation Modeling of Lossy Conductors With the Enhanced Augmented Electric Field Integral Equation

The formulation of the enhanced augmented electric field integral equation for dielectrics is generalized to conductor problems in this paper. The conductive region is simulated as a lossy dispersive medium using a full wave solver. In order to calculate the method of moments matrix elements in the conductive region accurately, we investigate the evaluations of the integrals of Green’s function in lossy media. After comparing with some other integration methods, we propose a new method to evaluate such integrals. This method turns out to improve the accuracy and efficiency. Moreover, the proposed formulation can be regarded as a generalized impedance boundary condition (IBC). This generalized IBC will become global if the skin depth is comparable to the size of the structures/details. The mixed-form fast multipole algorithm is employed for the simulations. Numerical examples of complex circuit structures are given to demonstrate the accuracy and capabilities of the proposed method.

Generalized impedance boundary conditions and shape derivatives for 3D Helmholtz problems

This paper is concerned with the shape sensitivity analysis of the solution to the Helmholtz transmission problem for three-dimensional sound-soft or sound-hard obstacles coated by a thin layer. This problem can be asymptotically approached by exterior problems with an improved condition on the exterior boundary of the coated obstacle, called generalized impedance boundary condition (GIBC). Using a series expansion of the Laplacian operator in the neighborhood of the exterior boundary, we retrieve the first-order GIBCs characterizing the presence of an interior thin layer with a constant thickness. The first shape derivative of the solution to the original Helmholtz transmission problem solves a new thin layer transmission problem with non-vanishing jumps across the exterior and the interior boundary of the thin layer. We show that we can interchange the first-order differentiation with respect to the shape of the exterior boundary and the asymptotic approximation of the solution. Numerical experiments are presented to highlight the various theoretical results.

The electromagnetic scattering problem with generalized impedance boundary conditions

In this paper we consider the electromagnetic scattering problem by an obstacle characterised by a Generalized Impedance Boundary Condition in the harmonic regime. These boundary conditions are well known to provide accurate models for thin layers or imperfectly conducting bodies. We give two different formulations of the scattering problem and we provide some general assumptions on the boundary condition under which the scattering problem has at most one solution. We also prove that it is well-posed for three different boundary conditions which involve second order surface differential operators under weak sign assumptions on the coefficients defining the surface operators.

Recent progress on the factorization method for inverse acoustic scattering problems

In this paper, we review recent progress on the factorization method for inverse acoustic scattering problems. Included are the classical factorization method for penetrable scatterers and generalized impedance boundary conditions, some modi cations for mixed-typed scatterers, near-field measurements and avoiding interior eigenvalues.

Generalized impedance boundary conditions for strongly absorbing obstacle: The full wave equation

This paper is devoted to the study of the generalized impedance boundary conditions (GIBCs) for a strongly absorbing obstacle in the time regime in two and three dimensions. The GIBCs in the time domain are heuristically derived from the corresponding conditions in the time harmonic regime. The latter is frequency-dependent except the one of order 0; hence the formers are non-local in time in general. The error estimates in the time regime can be derived from the ones in the time harmonic regime when the frequency dependence is well controlled. This idea is originally due to Nguyen and Vogelius [Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807] for the cloaking context. In this paper, we present the analysis to the GIBCs of orders 0 and 1. To implement the ideas in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807], we revise and extend the work of Haddar, Joly, and Nguyen, [Generalized impedance boundary condition for scattering by strongly absorbing obstacles: the scalar case, Math. Models Methods Appl. Sci.15 (2005) 1273–1300], where the GIBCs were investigated for a fixed frequency in three dimensions. Even though we heavily follow the strategy in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807], our analysis on the stability contains new ingredients and ideas. First, instead of considering the difference between solutions of the exact model and the approximate model, we consider the difference between their derivatives in time. This simple idea helps us to avoid the machinery used in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807] concerning the integrability with respect to frequency in the low frequency regime. Second, in the high frequency regime, the Morawetz multiplier technique used in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variable, SIAM J. Math. Anal.44 (2012) 769–807] does not fit directly in our setting. Our proof makes use of a result by Hörmander in [Lp estimates for (pluri-) subharmonic functions, Math. Scand.20 (1967) 65–78]. Another important part of the analysis in this paper is the well-posedness in the time domain for the approximate problems imposed with GIBCs on the boundary of the obstacle, which are non-local in time.

Multilayered Scattering Problem with Generalized Impedance Boundary Condition on the Core

This paper is concerned with the scattering problem of time-harmonic acoustic plane waves by an impenetrable obstacle buried in a piecewise homogeneous medium. The so-called generalized impedance boundary condition is imposed on the boundary of the obstacle. Firstly, the well posedness of the solution to the direct scattering problem is established by using the boundary integral method. Then a uniqueness result for the inverse scattering problem is proved; that is, both of the obstacle’s shape and the impedances (μ,λ) can be uniquely determined from far field measurements. Furthermore, a mathematical basis is given to reconstruct the shape of the obstacle by using a modified linear sampling method.

Variational approach to scattering by unbounded rough surfaces with Neumann and generalized impedance boundary conditions

This paper is concerned with problems of scattering of time-harmonic electromagnetic and acoustic waves from an innite penetrable medium with a certain non-trapping and geometric conditions.

Microwave Imaging of a Slightly Varying 2-D Conducting Object Through Generalized Impedance Boundary Conditions

We consider reconstructing the shape of a perfect electric conducting object and developed a novel imaging method based on generalized impedance boundary conditions (GIBCs). The method relies on selecting a fictitious surface encircling the unknown object in which the surface impedance is reconstructed from the scattered field measurements. Later, the shape reconstruction problem is cast into an equivalent problem where the distance variation between the fictitious impedance surface and the unknown target is determined from the reconstructed surface impedance. Since the performance of the method depends on the selection of the impedance surface, we present a selection criterion according to the validity conditions of GIBCs. The method is capable of reconstructing both convex and concave structures whose shape deviates, at most, a tenth of the wavelength from the impedance surface, as demonstrated with numerical results.