1,046 research outputs found
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
High-order integral equation methods for problems of scattering by bumps and cavities on half-planes
This paper presents high-order integral equation methods for evaluation of
electromagnetic wave scattering by dielectric bumps and dielectric cavities on
perfectly conducting or dielectric half-planes. In detail, the algorithms
introduced in this paper apply to eight classical scattering problems, namely:
scattering by a dielectric bump on a perfectly conducting or a dielectric
half-plane, and scattering by a filled, overfilled or void dielectric cavity on
a perfectly conducting or a dielectric half-plane. In all cases field
representations based on single-layer potentials for appropriately chosen Green
functions are used. The numerical far fields and near fields exhibit excellent
convergence as discretizations are refined--even at and around points where
singular fields and infinite currents exist.Comment: 25 pages, 7 figure
Fast integral equation methods for the Laplace-Beltrami equation on the sphere
Integral equation methods for solving the Laplace-Beltrami equation on the
unit sphere in the presence of multiple "islands" are presented. The surface of
the sphere is first mapped to a multiply-connected region in the complex plane
via a stereographic projection. After discretizing the integral equation, the
resulting dense linear system is solved iteratively using the fast multipole
method for the 2D Coulomb potential in order to calculate the matrix-vector
products. This numerical scheme requires only O(N) operations, where is the
number of nodes in the discretization of the boundary. The performance of the
method is demonstrated on several examples
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