164 research outputs found
Application of Discrete Differential Forms to Spherically Symmetric Systems in General Relativity
In this article we describe applications of Discrete Differential Forms in
computational GR. In particular we consider the initial value problem in vacuum
space-times that are spherically symmetric. The motivation to investigate this
method is mainly its manifest coordinate independence. Three numerical schemes
are introduced, the results of which are compared with the corresponding
analytic solutions. The error of two schemes converges quadratically to zero.
For one scheme the errors depend strongly on the initial data.Comment: 22 pages, 6 figures, accepted by Class. Quant. Gra
The premetric approach to electromagnetism in the 'waves are not vectors' debate
A plea for the introduction, in advanced electromagnetics courses, of some basic differential geometric notions: covectors, differential forms, Hodge operators. The main advantages of this evolution should be felt in computational electromagnetism. It may also shed some new light on the concept of material isotropy
Electromagnetic wormholes and virtual magnetic monopoles
We describe new configurations of electromagnetic (EM) material parameters,
the electric permittivity and magnetic permeability , that
allow one to construct from metamaterials objects that function as invisible
tunnels. These allow EM wave propagation between two points, but the tunnels
and the regions they enclose are not detectable to EM observations. Such
devices function as wormholes with respect to Maxwell's equations and
effectively change the topology of space vis-a-vis EM wave propagation. We
suggest several applications, including devices behaving as virtual magnetic
monopoles.Comment: 4 pages, 3 figure
The Simplicial Ricci Tensor
The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of
gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the
moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the
Einstein tensor for such spacetimes. More recently the Ric was used by Hamilton
to define a non-linear, diffusive Ricci flow (RF) that was fundamental to
Perelman's proof of the Poincare conjecture. Analytic applications of RF can be
found in many fields including general relativity and mathematics. Numerically
it has been applied broadly to communication networks, medical physics,
computer design and more. In this paper, we use Regge calculus (RC) to provide
the first geometric discretization of the Ric. This result is fundamental for
higher-dimensional generalizations of discrete RF. We construct this tensor on
both the simplicial lattice and its dual and prove their equivalence. We show
that the Ric is an edge-based weighted average of deficit divided by an
edge-based weighted average of dual area -- an expression similar to the
vertex-based weighted average of the scalar curvature reported recently. We use
this Ric in a third and independent geometric derivation of the RC Einstein
tensor in arbitrary dimension.Comment: 19 pages, 2 figure
Coupling Non-Gravitational Fields with Simplicial Spacetimes
The inclusion of source terms in discrete gravity is a long-standing problem.
Providing a consistent coupling of source to the lattice in Regge Calculus (RC)
yields a robust unstructured spacetime mesh applicable to both numerical
relativity and quantum gravity. RC provides a particularly insightful approach
to this problem with its purely geometric representation of spacetime. The
simplicial building blocks of RC enable us to represent all matter and fields
in a coordinate-free manner. We provide an interpretation of RC as a discrete
exterior calculus framework into which non-gravitational fields naturally
couple with the simplicial lattice. Using this approach we obtain a consistent
mapping of the continuum action for non-gravitational fields to the Regge
lattice. In this paper we apply this framework to scalar, vector and tensor
fields. In particular we reconstruct the lattice action for (1) the scalar
field, (2) Maxwell field tensor and (3) Dirac particles. The straightforward
application of our discretization techniques to these three fields demonstrates
a universal implementation of coupling source to the lattice in Regge calculus.Comment: 10 pages, no figures, Latex, fixed typos and minor corrections
Spaces of finite element differential forms
We discuss the construction of finite element spaces of differential forms
which satisfy the crucial assumptions of the finite element exterior calculus,
namely that they can be assembled into subcomplexes of the de Rham complex
which admit commuting projections. We present two families of spaces in the
case of simplicial meshes, and two other families in the case of cubical
meshes. We make use of the exterior calculus and the Koszul complex to define
and understand the spaces. These tools allow us to treat a wide variety of
situations, which are often treated separately, in a unified fashion.Comment: To appear in: Analysis and Numerics of Partial Differential
Equations, U. Gianazza, F. Brezzi, P. Colli Franzone, and G. Gilardi, eds.,
Springer 2013. v2: a few minor typos corrected. v3: a few more typo
correction
3D Numerical Simulation of Eddy Current Testing of a Block with a Crack
There are many approaches to 3D eddy current analysis. Typical methods for the eddy current analysis are the A-ø method and the T-ω method. Both methods require variables in space as well as in a conductor. We have already proposed the T method [1, 2, 3], where a magnetic scalar potential ω is not included and we do not need variables in space. But the method has a disadvantage that a large core memory is needed due to a dense matrix
Critical state theory for nonparallel flux line lattices in type-II superconductors
Coarse-grained flux density profiles in type-II superconductors with
non-parallel vortex configurations are obtained by a proposed phenomenological
least action principle. We introduce a functional , which is minimized
under a constraint of the kind belongs to for the current density
vector, where is a bounded set. This generalizes the concept of
critical current density introduced by C. P. Bean for parallel vortex
configurations. In particular, we choose the isotropic case ( is a
circle), for which the field penetration profiles are derived when a
changing external excitation is applied. Faraday's law, and the principle of
minimum entropy production rate for stationary thermodynamic processes dictate
the evolution of the system. Calculations based on the model can reproduce the
physical phenomena of flux transport and consumption, and the striking effect
of magnetization collapse in crossed field measurements.Comment: The compiled TeX document length is 10 pages. Two figures (one page
each) are also included The paper is accepted for publication in Phys. Rev.
Let
A simplicial gauge theory
We provide an action for gauge theories discretized on simplicial meshes,
inspired by finite element methods. The action is discretely gauge invariant
and we give a proof of consistency. A discrete Noether's theorem that can be
applied to our setting, is also proved.Comment: 24 pages. v2: New version includes a longer introduction and a
discrete Noether's theorem. v3: Section 4 on Noether's theorem has been
expanded with Proposition 8, section 2 has been expanded with a paragraph on
standard LGT. v4: Thorough revision with new introduction and more background
materia
Nonlinear force-free magnetic field extrapolations: comparison of the Grad-Rubin and Wheatland-Sturrock-Roumeliotis algorithm
We compare the performance of two alternative algorithms which aim to
construct a force-free magnetic field given suitable boundary conditions. For
this comparison, we have implemented both algorithms on the same finite element
grid which uses Whitney forms to describe the fields within the grid cells. The
additional use of conjugate gradient and multigrid iterations result in quite
effective codes. The Grad-Rubin and Wheatland-Sturrock-Roumeliotis algorithms
both perform well for the reconstruction of a known analytic force-free field.
For more arbitrary boundary conditions the Wheatland-Sturrock-Roumeliotis
approach has some difficulties because it requires overdetermined boundary
information which may include inconsistencies. The Grad-Rubin code on the other
hand loses convergence for strong current densities. For the example we have
investigated, however, the maximum possible current density seems to be not far
from the limit beyond which a force free field cannot exist anymore for a given
normal magnetic field intensity on the boundary.Comment: 21 pages, 13 figure
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