896 research outputs found
The effective conductivity of arrays of squares: large random unit cells and extreme contrast ratios
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. Phy
A higher-order singularity subtraction technique for the discretization of singular integral operators on curved surfaces
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 figure
On a Helmholtz transmission problem in planar domains with corners
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 figure
An explicit kernel-split panel-based Nystr\"om scheme for integral equations on axially symmetric surfaces
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 correcte
Determination of normalized electric eigenfields in microwave cavities with sharp edges
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 figure
Determination of normalized magnetic eigenfields in microwave cavities
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 figure
An accurate boundary value problem solver applied to scattering from cylinders with corners
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 figure
On the polarizability and capacitance of the cube
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 figure
Corner effects on the perturbation of an electric potential
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 figure
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