A high-order wideband direct solver for electromagnetic scattering from bodies of revolution

Abstract

The generalized Debye source representation of time-harmonic electromagnetic fields yields well-conditioned second-kind integral equations for a variety of boundary value problems, including the problems of scattering from perfect electric conductors and dielectric bodies. Furthermore, these representations, and resulting integral equations, are fully stable in the static limit as ω → 0 in multiply connected geometries. In this paper, we present the first high-order accurate solver based on this representation for bodies of revolution. The resulting solver uses a Nyström discretization of a onedimensional generating curve and high-order integral equation methods for applying and inverting surface differentials. The accuracy and speed of the solvers are demonstrated in several numerical examples. Keywords:Maxwell's equations, second-kind integral equations, magnetic field integral equation, generalized Debye sources, perfect electric conductor, penetrable media, body of revolution IntroductionIn isotropic homogenous media, the time-harmonic Maxwell's equations (THME) governing the propagation of electric and magnetic fields are given by [41]where E and H are the electric and magnetic fields, respectively, and ω is the angular frequency. The electric charge is denoted by ρ and the electric current by J . The electric permittivity and magnetic permeability are given by and µ, respectively, and assumed to be piecewise constant in all that follows. It will be useful to define the wavenumber k as k = ω √ µ, and we assume that (k) ≥ 0 as well. The implicit time dependence of e −iωt on E, H has been suppressed. Maxwell's equations in the above form provide a surprisingly accurate approximation to physical phenomena, and therefore their accurate numerical solution is of the utmost importance in many fields, such as radio-frequency component modeling [30], meta-material simulation [22], and radar cross-section characterizations [52]. Several computational methods for the solution to (1.1) are used in practice, including finite difference, finite element, spectral methods, and integral equation methods [31,35,37,50]. Finite difference and finite element methods are particularly useful when the material parameters vary in space, especially when that variation is anisotropic.On the other hand, when applicable -namely in complex geometries with piecewise constant material parameters -integral equation methods have significant advantages. They reduce the dimensionality of the problem to the boundary alone, satisfy the necessary radiation conditions in exterior domains, do not suffer from grid-based dispersion errors (particularly problematic at high frequencies), and are compatible with a variety of fast algorithms, leading to asymptotically optimal schemes. Two standard problems in electromagnetics -scattering from perfect electric conductors and dielectric bodies -fit exactly these assumptions [28]. While high-order accurate methods in arbitrary geometries in three dimensions are still under active development, there are seve...

An FFT-accelerated direct solver for electromagnetic scattering from penetrable axisymmetric objects

“…For χ ≈ 1, the forward recurrence is only mildly unstable, and can be used with caveats. See [20] for an estimate on the number of terms that can be evaluated accurately in this regime. In order to evaluate g 1 m for k = 0, we apply the convolution technique proposed in [61] with a slight modification.…”

“…See [6,29] for an in-depth discussion of generalized Gaussian quadrature schemes. The panel-based discretization scheme of this paper, as opposed to that based on hybrid-Gauss trapezoidal rules [1], as presented in [20,54], allow for adaptive discretizations, in particular, axisymmetric surfaces in three dimensions with edges and points.…”

An Extended Charge-Current Formulation of the Electromagnetic Transmission Problem

“…The formulation in [15,Section 4.2], the Müller formulation, and the formulations described in this paper are direct formulations, meaning that the surface densities are related to boundary limits of fields, or derivatives of fields. This is in contrast to indirect formulations [5,6,16,26], where the surface densities have no immediate physical interpretation. Our paper, and many other papers [9,17,20,21,23,26], use integral representations of the electric and magnetic fields, but it is also possible to start with representations of scalar and vector potentials and antipotentials [5,6,18].…”

Taylor states in stellarators: A fast high-order boundary integral solver

“…We now turn to a very brief overview of this representation, and the derivation of an integral representation and integral equation for Taylor states in magnetically confined plasmas. Various theoretical and numerical aspects of the generalized Debye source representation can be found in [13,14,16,17,44].…”

“…The space of harmonic vector fields along Γ has dimension 2G, where G is the genus of the boundary. In [17,44], a basis for these harmonic vector fields is known analytically because the boundary Γ is a surface of revolution. However, in the present work, they must be obtained computationally.…”