Henrik Wallén

TRANSLATION PROCEDURES FOR BROADBAND MLFMA

The multilevel fast multipole algorithm (MLFMA) is used in computing acoustic and electromagnetic fields with integral equation methods. The traditional MLFMA, however, suffers from a lowfrequency breakdown that effectively limits the minimum division cube side length to approximately one wavelength. To overcome this low-frequency breakdown and get a broadband MLFMA, we propose an efficient and relatively straightforward implementation of the field translations based on the spectral representation of the Green’s function. As an alternative we also consider the so called uniform MLFMA, which has a lower computational cost but limited accuracy. We consider the essential implementation details and finally provide numerical examples to demonstrate the error controllability of the translations.

Zero Backscattering From Self-Dual Objects of Finite Size

Duality transformation is applied to the theory of zero backscattering from finite objects. It is shown that if the object, defined by medium and/or boundary condition, is self dual, i.e., invariant in the duality transformation, it is invisible to radar if a certain condition for the polarizability dyadic is valid. This is a general statement and includes previous theorems as special cases. As novel self-dual objects those with boundary conditions requiring vanishing of the normal components of the D and B vectors (DB boundary) or their normal derivatives (DB boundary) are introduced. As examples of zero-backscattering objects, scattering properties of a sphere and a cube with DB or DB boundary, as well as an anisotropic asymmetric spheroid, are discussed.

Polarizability of a dielectric hemisphere

This article presents a method for solving the polarizability of a dielectric hemispherical object as a function of its relative electric permittivity. The polarizability of a hemisphere depends on the direction of the exciting electric field. Therefore, the polarizability can be written as a dyadic consisting of two components, the axial and the transversal polarizabilities, which can be solved separately. The solution is based on an analytical approach where the electrostatic potential function is written as a series expansion. However, no closed-form solution for the coefficients of the series is found, so they must be solved from a matrix equation. This method provides very high accuracy. However, it requires construction of large matrices which consumes both time and memory. Therefore, approximative expressions for the polarizabilities with absolute error less than 10−5 are also presented.

General Electromagnetic Boundary Conditions Involving Normal Field Components

A set of electromagnetic boundary conditions requiring vanishing of the normal components of the D and B fields (DB boundary) or their normal derivatives (DB boundary) have been recently under interest. In the present study, more general conditions called those of the generalized DB boundary (GDB boundary) are introduced. The boundary depends on a parameter (GDB parameter) whose two special values yield the DB and DB boundaries. It is shown that, for the plane-wave reflection of a planar GDB boundary, the boundary can be replaced by an anisotropic impedance boundary with impedance dyadic depending on the direction of propagation of the plane wave. Numerical analysis of a sphere with GDB boundary shows that it yields a wide backscattering zero for an incident plane wave, while its forward scattering magnitude depends on the GDB parameter. Finally, yet another set of generalized boundary conditions is introduced that corresponds to that of an isotropic impedance boundary while being represented in terms of normal field quantities.

Scattering by DB Spheres

This letter analyzes scattering properties of spheres with DB boundary conditions. The DB condition is defined by the requirement that the components of electric and magnetic flux densities normal to the boundary are zero. Rayleigh scattering, Mie scattering, and method-of-moments (MoM)-based computations are applied to the problem. The most interesting results are vanishing backscattering and rotational symmetry of the scattering diagram.

Electrostatic resonances of a negative-permittivity hemisphere

This article studies the electric response of an electrically small hemispherical object with negative permittivity by computing its polarizability which is determined by two orthogonal components, the axial one and the transverse one. A certain range of negative permittivity values is found where the mathematical determination of the polarizability becomes impossible due to an unlimited number of singularities. These singularities are due to surface plasmons, also referred to as electrostatic resonances, caused by the sharp edge of the hemisphere. It is also found that the planar surface of the hemisphere may support resonant surface modes. Furthermore, there exists a dipolar resonance determined by the overall geometry. In addition, it is shown that the resonances can be smoothened by introducing losses and, even more importantly, rounding the edge.

Broadband Müller-MLFMA for Electromagnetic Scattering by Dielectric Objects

The multilevel fast multipole algorithm (MLFMA) is very efficient for solving large-scale electromagnetic scattering problems. However, at low frequencies, or when the discretization is small compared with the wavelength, both the MLFMA and the underlying integral equation formulation typically suffer from a subwavelength breakdown. For the electromagnetic scattering from a homogeneous dielectric object, we obtain a stable and well-conditioned surface integral formulation using a variant of the classical Muumlller formulation and linear basis functions. To overcome the subwavelength breakdown of the MLFMA, we use both propagating and evanescent plane waves to represent the fields. The implementation is based on a combination of the spectral representation of the Greens function and Rokhlins translation formula. We also present a new interpolation scheme for the evanescent part, which significantly improves the error-controllability of the MLFMA-implementation. Several numerical results verify both the error-controllability and scalability of the proposed algorithm

BROADBAND MULTILEVEL FAST MULTIPOLE ALGORITHM FOR ACOUSTIC SCATTERING PROBLEMS

A broadband multilevel fast multipole algorithm (MLFMA) for the acoustic scattering from a sound-hard obstacle is presented. The formulation is based on the Burton–Miller boundary integral equation and Galerkins method, avoiding any hypersingular integral operators. The resulting matrix equation has good iterative properties for all frequencies and avoids the interior resonance problem. The main novel feature is the use of a broadband MLFMA to accelerate the iterative generalized minimal residual (GMRES) solver. The algorithm is based on a combination of Rokhlins translation formula for large division cubes and the spectral representation of the Greens function for cubes smaller than one half wavelength, thereby avoiding the sub-wavelength breakdown of the high-frequency MLFMA.

Polarizability of conducting sphere-doublets using series of images

The classical electrostatic problem of two nonintersecting conducting spheres in a uniform incident electric field is considered. Starting from the basic Kelvin’s image principle, the two spheres are replaced with equivalent series of image sources, from which the polarizability is calculated. Explicit expressions for the axial and transversal components of the polarizability dyadic are found by solving the recurrence equations. Efficient numerical evaluation of the different series is also discussed.

How well can a PEC-backed gyrotropic layer approximate the ideal PEMC boundary?

The perfect electromagnetic conductor (PEMC) is a generalization of both the perfect electric conductor (PEC) and the perfect magnetic conductor (PMC). As shown previously, the PEMC boundary can be realized using a gyrotropic layer backed by a PEC plane. If the axial permittivity and permeability are infinite and the other (lossless) parameters are chosen appropriately, the PEMC boundary condition can be satisfied exactly for a fixed frequency. In this paper, we continue the analysis of the PEMC boundary realization, taking into account the effect of finite axial parameters and also losses in the structure. Including these real-world deviations from the ideal case, we still get a useful PEMC approximation.