2,142 research outputs found
Dynamics of Hilbert nonexpansive maps
In his work on the foundations of geometry, Hilbert observed that a formula
which appeared in works by Beltrami, Cayley, and Klein, gives rise to a
complete metric on any bounded convex domain. Some decades later, Garrett
Birkhoff and Hans Samelson noted that this metric has interesting applications,
when considering certain maps of convex cones that contract the metric. Such
situations have since arisen in many contexts, pure and applied, and could be
called nonlinear Perron-Frobenius theory. This note centers around one
dynamical aspect of this theory.Comment: 10 pages. To appear in the Handbook of Hilbert Geometr
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
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
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
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