Local Impedance Boundary Conditions Research Articles

Current Based Automated Design of Realizable Metasurface Antennas With Arbitrary Pattern Constraints

We present a 3-D method to numerically design a realizable metasurface, which transforms a given incident field into a radiated field that satisfies mask-type (inequality) constraints. The method is based on an integral equation formulation, with local impedance boundary condition (IBC) approximation. The procedure yields the spatial distribution of the impedance, yet the process involves the synthesis of the equivalent current only. This current is constrained to correspond to a realizable surface impedance, i.e., passive, lossless, and with reactance values bounded by practical realizability limits. The current-based design avoids any solution to the forward problem, and the impedance is obtained from the synthesized current only at the end of the process. The procedure is gradient-based, with the gradient expressed in closed form. This allows handling large metasurfaces, with full spatial variability of the impedance in two dimensions. The method requires no a priori information, and all relevant operations in the iterative process can be evaluated with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol { O}(N~\log~{N})$ </tex-math></inline-formula> complexity. Application examples concentrate on the case of on-surface excitation and far-field (FF) pattern specifications; they show designs of circular and rectangular metasurface antennas of 20 wavelengths in size, with pencil- and shaped-beam patterns, and for both circular and linear polarizations.

An acoustic boundary integral formulation for open shells allowing different impedance conditions, top and bottom surfaces

In the automotive, aerospace industries as well as the underwater acoustics defense applications the addition of coatings to modify the response of the vibrating structure is commonplace. The formulation developed is an extension of the existing thin body formulation currently found in commercial noise prediction codes such as SYSNOISE in order to allow problems requiring different boundary condition on both sides of the thin shell to be solved. For instance, a generalised thin body acoustic formulation would allow for a local impedance boundary condition to be specified on the top surface and a hard Neumann boundary condition to be specified on the bottom surface. Such a formulation would allow for different local impedance coatings to be specified on the top and bottom surfaces. Due to the lack of analytical solutions for such problems (even for the simplest geometries) it is difficult to validate such a formulation, however, three test cases are presented, two flat plate examples and a open cylinder with a point monopole source excitation at its centre.

An analytic multiple frequency adjoint-based inversion algorithm for parabolic-type approximations in ocean acoustics

Inverse problems in ocean acoustics are generally solved by means of matched-field inversion in combination with meta-heuristic global search algorithms. Recently adjoint-based optimal nonlocal boundary control has been proposed as a possible complement or alternative to the traditional geoacoustic inversion methods. Especially for the assessment of a shallow water environment where field data are recorded on non-fully populated hydrophone arrays there is an inherent need for observation over a broad range of frequencies, as has been widely experimentally demonstrated. This article presents an analytic multiple frequency adjoint formulation for estimating an acoustically equivalent model of the seabed based on local impedance boundary conditions. It is intended as a benchmark solution for an extension of the approach to more sophisticated boundary conditions by means of automatic differentiation. The approach, coming from control theory, aims at inverting the impedance boundary condition along the water–sediment interface via measurements of the acoustic propagation in the water column. The impedance boundary condition thus plays the role of the control parameter of the complex pressure field in the waveguide. The developed multiple frequency adjoint method of optimal control provides an explicit representation of the gradient of the matched-field oriented cost function, and the article further proposes an analytical second-order adjoint formulation which permits the computation of the product of the Hessian matrix by a vector. By means of example inversion results the feasibility of the multi-frequency approach is discussed and the effect of a limited set of receiving locations is illustrated.

Calculation of effective impedance of polycrystals in weak magnetic fields

We present results for the effective surface impedance tensor (EIT) of polycrystals of metals in a weak uniform magnetic field H. The frequency region corresponds to the region in which the local impedance boundary conditions are applicable. We suppose that the resistivity tensor ρ i k ( H ) of the single crystal grains out of which the polycrystal is composed, is known up to the terms of O ( H 2 ) . For polycrystals of metals of arbitrary symmetry, the elements of the EIT can be calculated to the same order in H, even if the tensor ρ i k ( H ) is strongly anisotropic. As examples, we write down the EIT of polycrystals of (i) cubic metals, (ii) metals with ellipsoidal Fermi surfaces, and (iii) metals of tetragonal symmetry whose tensor ρ i k ( 0 ) is strongly anisotropic. Although polycrystals are metals that are isotropic on average, in the presence of a uniform magnetic field the structure of the EIT is not the same as the structure of the impedance tensor of an isotropic metal with a spherical Fermi surface. The results cannot be improved either by taking into account higher powers of H, or with respect to the anisotropy of the single crystal grains.

The finite-volume Dirac–Hartree–Fock method for confined relativistic many-electron systems

An energy eigenproblem for a relativistic N-electron system confined to the interior of a finite volume is considered. The confinement is modelled by imposing a local impedance boundary condition at a hypersurface enclosing the hypervolume in the configuration space. It is shown that energy eigenvalues are non-increasing functions of the hypersurface impedance. Variational principles for energy eigenvalues, admitting the use of trial functions which do not obey the boundary condition imposed on exact eigenfunctions, are constructed in a systematic manner. The Dirac–Hartree–Fock method is applied to derive integro-differential equations and local boundary conditions satisfied by one-electron spin orbitals from which the best determinantal approximations to exact eigenfunctions are built. It is proved that the Dirac–Hartree–Fock estimates of exact energy eigenvalues are also non-increasing functions of the hypersurface impedance.

On calculation of ohmic losses at metallic surface with sharp edges

We discuss the applicability of the perturbation theory in electrodynamic problems where the local impedance boundary conditions are used to calculate the ohmic losses at the metallic surface. As an example, we examine a periodic grating formed from semi-infinite rectangular plates exposed to the s-polarized electromagnetic wave. Two different ways of calculation are presented: (i) the calculation of the reflection coefficient obtained with the aid of the perturbation theory (the impedance is the small parameter) and (ii) the direct calculation of the energy flux through the metallic surface. In the case (ii) to get the answer only the tangential magnetic field at the surface of a perfect conductor of the same geometry has to be known. The results (i) and (ii) differ noticeably. The same difficulty is inherent in all the problems where the metallic surface has rectangular grooves. We show that in this problem the standard first-order perturbation theory is not applicable. The point is that beginning from a number n the first corrections to the modal functions φ n used to calculate the fields, are of the same order as the zero-order modal functions (the impedance is equal to zero). Basing on the energy conservation law we show that the accurate value for the ohmic losses is obtained with the aid of the approach (ii).

Impedance boundary conditions for a metal film with a rough surface

We have obtained local impedance boundary conditions for a metal film characterized by an isotropic, frequency-dependent, complex dielectric function epsilon(omega) that occupies the region zeta(x(1)) < x(3) < D. The surface-profile function zeta(x(1)) is assumed to be a single-valued function of x(1) that is differentiable as many times as is necessary. The electromagnetic field in the system is assumed to be p polarized with the plane of incidence, the x(1)x(3) plane. The results are used to study the scattering of p-polarized light from and its transmission through the metal film when the surface-profile function zeta(x(1)) in these calculations is assumed to be a stationary, zero-mean, Gaussian random process. These calculations are approximately four times faster than rigorous computer simulations, and their results are qualitatively and quantitatively in good agreement with those of the latter simulations.

Derivation of an impedance boundary condition for one-dimensional, curved, reentrant surfaces

A local impedance boundary condition for one-dimensional curved surfaces, applicable to a scattering formalism that permits the study of reentrant surfaces, is derived. The results are applied to the calculation of near- and far-field scattering patterns produced by a surface with a cylindrical cavity. The accuracy of the impedance boundary condition is evaluated with this example, which also illustrates the utility of the approximation.

Theoretical studies of the enhanced backscattering of light from one-dimensional randomly rough metal surfaces by the use of a nonlocal impedance boundary condition

Nonlocal impedance boundary conditions for p- and s-polarized electromagnetic fields at the curved interface between vacuum and a metal, defined by the equation x 3 = ζ( x 1), when the plane of incidence is the x 1 x 3-plane, are derived. The conditions for the validity of a local impedance boundary condition are established. The results are used in computer simulation studies of the scattering of infrared and visible light from a large rms height, large rms slope, one-dimensional, randomly rough, metal surface. It is shown, by comparison with the results of formally exact simulations, that the use of a local impedance boundary condition, even for light in the visible region of the optical spectrum, yields accurate results with a significant saving of computational time.

The impedance boundary condition for a periodically corrugated metal surface

We present a method based on Green's second integral theorem for obtaining a local impedance boundary condition for a p- or s-polarized electromagnetic field at the periodically corrugated interface between vacuum and a metal. The surface is defined by the equation χ 3 = ζ ( χ 1), where the surface profile function ζ (χ 1) is a single-valued, periodic function of χ 1, and the plane of incidence is the χ 1χ 3-plane. The results are used in the determination of the dispersion curves for surface plasmon polaritons propagating normal to the grooves and ridges of the surface.

Scattering of a plane wave from a cylinder partially coated by a layer modeled as a dissipative fluid

A previous investigation of scattering from a partially coated infinite circular cylinder employed a local impedance boundary condition to represent the interface between the coating and the exterior domain. Such a model led to a singularity in the surface response at the discontinuity. In addition, such a model is inadequate to describe dissipative effects of the layer. In the present paper, the partial coating is treated as a fluid layer bounded at its inner radius and azimuthal ends by a rigid cylinder, while continuity of pressure and particle velocity with the exterior fluid is imposed at the outer radius. Dissipative effects are included by allowing the wave number of the layer to be complex. The scattered pressure field in the exterior domain and the pressure within the layer are represented by azimuthal spectral series whose basis functions are solutions of the Helmholtz equation for the respective regions. The series coefficients for those series are obtained by using a collocation technique to enforce the continuity conditions at the outer radius in a point-wise manner. Results for a range of system parameters are used to investigate the effects of the surface discontinuity associated with the partial coating. [Work supported by ONR.]