2,693 research outputs found
Particle Motion in Monopoles and Geodesics on Cones
The equations of motion of a charged particle in the field of Yang's
monopole in 5-dimensional Euclidean space are derived by
applying the Kaluza-Klein formalism to the principal bundle
obtained by radially
extending the Hopf fibration , and solved by elementary methods.
The main result is that for every particle trajectory
, there is a 4-dimensional cone with
vertex at the origin on which is a geodesic. We give an explicit
expression of the cone for any initial conditions
Low-order continuous finite element spaces on hybrid non-conforming hexahedral-tetrahedral meshes
This article deals with solving partial differential equations with the
finite element method on hybrid non-conforming hexahedral-tetrahedral meshes.
By non-conforming, we mean that a quadrangular face of a hexahedron can be
connected to two triangular faces of tetrahedra. We introduce a set of
low-order continuous (C0) finite element spaces defined on these meshes. They
are built from standard tri-linear and quadratic Lagrange finite elements with
an extra set of constraints at non-conforming hexahedra-tetrahedra junctions to
recover continuity. We consider both the continuity of the geometry and the
continuity of the function basis as follows: the continuity of the geometry is
achieved by using quadratic mappings for tetrahedra connected to tri-affine
hexahedra and the continuity of interpolating functions is enforced in a
similar manner by using quadratic Lagrange basis on tetrahedra with constraints
at non-conforming junctions to match tri-linear hexahedra. The so-defined
function spaces are validated numerically on simple Poisson and linear
elasticity problems for which an analytical solution is known. We observe that
using a hybrid mesh with the proposed function spaces results in an accuracy
significantly better than when using linear tetrahedra and slightly worse than
when solely using tri-linear hexahedra. As a consequence, the proposed function
spaces may be a promising alternative for complex geometries that are out of
reach of existing full hexahedral meshing methods
Imaging polarizable dipoles
We present a method for imaging the polarization vector of an electric dipole
distribution in a homogeneous medium from measurements of the electric field
made at a passive array. We study an electromagnetic version of Kirchhoff
imaging and prove, in the Fraunhofer asymptotic regime, that range and
cross-range resolution estimates are identical to those in acoustics. Our
asymptotic analysis provides error estimates for the cross-range dipole
orientation reconstruction and shows that the range component of the dipole
orientation is lost in this regime. A naive generalization of the Kirchhoff
imaging function is afflicted by oscillatory artifacts in range, that we
characterize and correct. We also consider the active imaging problem which
consists in imaging both the position and polarizability tensors of small
scatterers in the medium using an array of collocated sources and receivers. As
in the passive array case, we provide resolution estimates that are consistent
with the acoustic case and give error estimates for the cross-range entries of
the polarizability tensor. Our theoretical results are illustrated by numerical
experiments.Comment: 35 pages, 18 figure
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