Search

This work concerns the design of perfectly conducting objects that are invisible to an incident transverse magnetic plane wave. The object in question is a finite planar waveguide with a finite periodic array of barriers. By optimizing this array, the amplitude of the scattered field is reduced to less than $10^{-9}$ times the amplitude of the incident plane wave everywhere outside the waveguide. To accurately evaluate such minute amplitudes, we employ a recently developed boundary integral equation technique, adapted for objects whose boundaries have endpoints, corners, and branch points., Comment: 19 pages, 8 figures

This paper concerns the inverse problem of determining a planar conductivity inclusion. Our aim is to analytically recover from the generalized polarization tensors (GPTs), which can be obtained from exterior measurements, a homogeneous inclusion with arbitrary constant conductivity. The primary outcome of recovering a homogeneous inclusion is an inversion formula in terms of the GPTs for conformal mapping coefficients associated with the inclusion. To prove the formula, we establish matrix factorizations for the GPTs., Comment: 29 pages, 6 figures

An integral equation reformulation of the Maxwell transmission problem is presented. The reformulation uses techniques such as tuning of free parameters and augmentation of close-to-rank-deficient operators. It is designed for the eddy current regime and works both for surfaces of genus $0$ and $1$. Well-conditioned systems and field representations are obtained despite the Maxwell transmission problem being ill-conditioned for genus $1$ surfaces due to the presence of Neumann eigenfields. Furthermore, it is shown that these eigenfields, for ordinary conductors in the eddy current regime, are different from the classical Neumann eigenfields for superconductors. Numerical examples, based on the reformulation, give an unprecedented $13$-digit accuracy both for transmitted and scattered fields., Comment: Accepted for publication in Computers and Mathematics with Applications

A numerical scheme is presented for the solution of Fredholm second-kind boundary integral equations with right-hand sides that are singular at a finite set of boundary points. The boundaries themselves may be non-smooth. The scheme, which builds on recursively compressed inverse preconditioning (RCIP), is universal as it is independent of the nature of the singularities. Strong right-hand-side singularities, such as $1/|r|^\alpha$ with $\alpha$ close to $1$, can be treated in full machine precision. Adaptive refinement is used only in the recursive construction of the preconditioner, leading to an optimal number of discretization points and superior stability in the solve phase. The performance of the scheme is illustrated via several numerical examples, including an application to an integral equation derived from the linearized BGKW kinetic equation for the steady Couette flow., Comment: 29 pages, 19 figures

Two recently derived integral equations for the Maxwell transmission problem are compared through numerical tests on simply connected axially symmetric domains for non-magnetic materials. The winning integral equation turns out to be entirely free from false eigenwavenumbers for any passive materials, also for purely negative permittivity ratios and in the static limit, as well as free from false essential spectrum on non-smooth surfaces. It also appears to be numerically competitive to all other available integral equation reformulations of the Maxwell transmission problem, despite using eight scalar surface densities., Comment: 36 pages, 10 figures

A new integral equation formulation is presented for the Maxwell transmission problem in Lipschitz domains. It builds on the Cauchy integral for the Dirac equation, is free from false eigenwavenumbers for a wider range of permittivities than other known formulations, can be used for magnetic materials, is applicable in both two and three dimensions, and does not suffer from any low-frequency breakdown. Numerical results for the two-dimensional version of the formulation, including examples featuring surface plasmon waves, demonstrate competitiveness relative to state-of-the-art integral formulations that are constrained to two dimensions. However, our Dirac integral equation performs equally well in three dimensions, as demonstrated in a companion paper., Comment: 36 pages, 9 figures

A boundary integral equation formulation is presented for the electromagnetic transmission problem where an incident electromagnetic wave is scattered from a bounded dielectric object. The formulation provides unique solutions for all combinations of wavenumbers in the closed upper half-plane for which Maxwell's equations have a unique solution. This includes the challenging combination of a real positive wavenumber in the outer region and an imaginary wavenumber inside the object. The formulation, or variants thereof, is particularly suitable for numerical field evaluations as confirmed by examples involving both smooth and non-smooth objects., Comment: 32 pages, 7 figures

An integral equation-based numerical method for scattering from multi-dielectric cylinders is presented. Electromagnetic fields are represented via layer potentials in terms of surface densities with physical interpretations. The existence of null-field representations then adds superior flexibility to the modeling. Local representations are used for fast field evaluation at points away from their sources. Partially global representations, constructed as to reduce the strength of kernel singularities, are used for near-evaluations. A mix of local- and partially global representations is also used to derive the system of integral equations from which the physical densities are solved. Unique solvability is proven for the special case of scattering from a homogeneous cylinder under rather general conditions. High achievable accuracy is demonstrated for several examples found in the literature., Comment: 26 pages, 9 figures

A particular mix of integral equations and discretization techniques is suggested for the solution of a planar Helmholtz transmission problem with relevance to the study of surface plasmon waves. The transmission problem describes the scattering of a time harmonic transverse magnetic wave from an infinite dielectric cylinder with complex permittivity and sharp edges. Numerical examples illustrate that the resulting scheme is capable of obtaining total magnetic and electric fields to very high accuracy in the entire computational domain., Comment: 28 pages, 8 figures

We consider the perturbation of an electric potential due to an insulating inclusion with corners. This perturbation is known to admit a multipole expansion whose coefficients are linear combinations of generalized polarization tensors. We define new geometric factors of a simple planar domain in terms of a conformal mapping associated with the domain. The geometric factors share properties of the generalized polarization tensors and are the Fourier series coefficients of a kind of generalized external angle of the inclusion boundary. Since the generalized external angle contains the Dirac delta singularity at corner points, we can determine the criterion for the existence of corner points on the inclusion boundary in terms of the geometric factors. We illustrate and validate our results with numerical examples computed to a high degree of precision using integral equation techniques, Nystr\"om discretization, and recursively compressed inverse preconditioning., Comment: 25 pages, 6 figures

We study the adjoint of the double layer potential associated with the Laplacian (the adjoint of the Neumann-Poincar\'e operator), as a map on the boundary surface $\Gamma$ of a domain in $\mathbb{R}^3$ with conical points. The spectrum of this operator directly reflects the well-posedness of related transmission problems across $\Gamma$. In particular, if the domain is understood as an inclusion with complex permittivity $\epsilon$, embedded in a background medium with unit permittivity, then the polarizability tensor of the domain is well-defined when $(\epsilon+1)/(\epsilon-1)$ belongs to the resolvent set in energy norm. We study surfaces $\Gamma$ that have a finite number of conical points featuring rotational symmetry. On the energy space, we show that the essential spectrum consists of an interval. On $L^2(\Gamma)$, i.e. for square-integrable boundary data, we show that the essential spectrum consists of a countable union of curves, outside of which the Fredholm index can be computed as a winding number with respect to the essential spectrum. We provide explicit formulas, depending on the opening angles of the conical points. We reinforce our study with very precise numerical experiments, computing the energy space spectrum and the spectral measures of the polarizability tensor in two different examples. Our results indicate that the densities of the spectral measures may approach zero extremely rapidly in the continuous part of the energy space spectrum., Comment: 45 pages, 8 figures. To appear in Journal de Math\'ematiques Pures et Appliqu\'ees

A high-order convergent and robust numerical solver is constructed and used to find complex eigenwavenumbers and electromagnetic eigenfields of dielectric objects with axial symmetry. The solver is based on Fourier--Nystr\"om discretization of combined integral equations for the transmission problem and can be applied to demanding resonance problems at microwave, terahertz, and optical wavelengths. High achievable accuracy, even at very high wavenumbers, makes the solver ideal for benchmarking and for assessing the performance of general purpose commercial software., Comment: 28 pages, 8 figures, errata corrected

We study spectral properties of the Neumann-Poincar\'e operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum is different from that at eigenvalues, and then derive a method to distinguish continuous spectrum from eigenvalues. We perform computational experiments using the method to see whether continuous spectrum and pure point spectrum appear on domains with corners. For the computations we use a modification of the Nystr\"om method which makes it possible to construct high-order convergent discretizations of the Neumann-Poincar\'e operator on domains with corners. The results of experiments show that all three possible spectra, absolutely continuous spectrum, singularly continuous spectrum, and pure point spectrum, may appear depending on domains. We also prove rigorously two properties of spectrum which are suggested by numerical experiments: symmetry of spectrum (including continuous spectrum), and existence of eigenvalues on rectangles of high aspect ratio.

The magnetic field integral equation for axially symmetric cavities with perfectly conducting piecewise smooth surfaces is discretized according to a high-order convergent Fourier--Nystr\"om scheme. The resulting solver is used to accurately determine eigenwavenumbers and normalized electric eigenfields in the entire computational domain., Comment: 34 pages, 6 figures

The magnetic field integral equation for axially symmetric cavities with perfectly conducting surfaces is discretized according to a high-order convergent Fourier--Nystr\"om scheme. The resulting solver is used to determine eigenwavenumbers and normalized magnetic eigenfields to very high accuracy in the entire computational domain., Comment: 23 pages, 4 figures

The incorporation of analytical kernel information is exploited in the construction of Nystr\"om discretization schemes for integral equations modeling planar Helmholtz boundary value problems. Splittings of kernels and matrices, coarse and fine grids, high-order polynomial interpolation, product integration performed on the fly, and iterative solution are some of the numerical techniques used to seek rapid and stable convergence of computed fields in the entire computational domain., Comment: 21 pages, 3 figures

A high-order accurate, explicit kernel-split, panel-based, Fourier-Nystr\"om discretization scheme is developed for integral equations associated with the Helmholtz equation in axially symmetric domains. Extensive incorporation of analytic information about singular integral kernels and on-the-fly computation of nearly singular quadrature rules allow for very high achievable accuracy, also in the evaluation of fields close to the boundary of the computational domain., Comment: 30 pages, 5 figures, errata corrected

This note is about promoting singularity subtraction as a helpful tool in the discretization of singular integral operators on curved surfaces. Singular and nearly singular kernels are expanded in series whose terms are integrated on parametrically rectangular regions using high-order product integration, thereby reducing the need for spatial adaptivity and precomputed weights. A simple scheme is presented and an application to the interior Dirichlet Laplace problem on some tori gives around ten digit accurate results using only two expansion terms and a modest programming- and computational effort., Comment: 7 pages, 2 figures

In this paper we consider the classic problems of scattering of waves from perfectly conducting cylinders with piecewise smooth boundaries. The scattering problems are formulated as integral equations and solved using a Nystr\"om scheme where the corners of the cylinders are efficiently handled by a method referred to as Recursively Compressed Inverse Preconditioning (RCIP). This method has been very successful in treating static problems in non-smooth domains and the present paper shows that it works equally well for the Helmholtz equation. In the numerical examples we specialize to scattering of E- and H-waves from a cylinder with one corner. Even at a size kd=1000, where k is the wavenumber and d the diameter, the scheme produces at least 13 digits of accuracy in the electric and magnetic fields everywhere outside the cylinder., Comment: 19 pages, 3 figures

Recursively compressed inverse preconditioning (RCIP) is a kernel-independent and purely numerical method for solving Fredholm second kind boundary integral equations in situations where the boundary shape induces a non-smooth behavior in the solution. The method originated in 2008 within a scheme for Laplace's equation in two-dimensional domains with corners. In a series of subsequent papers the method was then refined and extended as to apply to integral equation formulations of a broad range of boundary value problems in physics and engineering. The purpose of the present tutorial is threefold: First, to review the RCIP method in a simple setting. Second, to show how easily the method can be implemented in Matlab. Third, to present new applications., Comment: Revised and enlarged edition, 58 pages, 35 figures, demo codes available at http://www.maths.lth.se/na/staff/helsing/Tutor/

An efficient integral equation based solver is constructed for the electrostatic problem on domains with cuboidal inclusions. It can be used to compute the polarizability of a dielectric cube in a dielectric background medium at virtually every permittivity ratio for which it exists. For example, polarizabilities accurate to between five and ten digits are obtained (as complex limits) for negative permittivity ratios in minutes on a standard workstation. In passing, the capacitance of the unit cube is determined with unprecedented accuracy. With full rigor, we develop a natural mathematical framework suited for the study of the polarizability of Lipschitz domains. Several aspects of polarizabilities and their representing measures are clarified, including limiting behavior both when approaching the support of the measure and when deforming smooth domains into a non-smooth domain. The success of the mathematical theory is achieved through symmetrization arguments for layer potentials., Comment: 33 pages, 7 figures

An integral equation based scheme is presented for the fast and accurate computation of effective conductivities of two-component checkerboard-like composites with complicated unit cells at very high contrast ratios. The scheme extends recent work on multi-component checkerboards at medium contrast ratios. General improvement include the simplification of a long-range preconditioner, the use of a banded solver, and a more efficient placement of quadrature points. This, together with a reduction in the number of unknowns, allows for a substantial increase in achievable accuracy as well as in tractable system size. Results, accurate to at least nine digits, are obtained for random checkerboards with over a million squares in the unit cell at contrast ratio 10^6. Furthermore, the scheme is flexible enough to handle complex valued conductivities and, using a homotopy method, purely negative contrast ratios. Examples of the accurate computation of resonant spectra are given., Comment: 28 pages, 11 figures, submitted to J. Comput. Phys

We investigate the optical properties of periodic composites containing metamaterial inclusions in a normal material matrix. We consider the case where these inclusions have sharp corners, and following Hetherington and Thorpe, use analytic results to argue that it is then possible to deduce the shape of the corner (its included angle) by measurements of the absorptance of such composites when the scale size of the inclusions and period cell is much finer than the wavelength. These analytic arguments are supported by highly accurate numerical results for the effective permittivity function of such composites as a function of the permittivity ratio of inclusions to matrix. The results show that this function has a continuous spectral component with limits independent of the area fraction of inclusions, and with the same limits for both square and staggered square arrays., Comment: 17 pages, 6 figures