16,815 research outputs found
High frequency homogenisation for elastic lattices
A complete methodology, based on a two-scale asymptotic approach, that
enables the homogenisation of elastic lattices at non-zero frequencies is
developed. Elastic lattices are distinguished from scalar lattices in that two
or more types of coupled waves exist, even at low frequencies. Such a theory
enables the determination of effective material properties at both low and high
frequencies. The theoretical framework is developed for the propagation of
waves through lattices of arbitrary geometry and dimension. The asymptotic
approach provides a method through which the dispersive properties of lattices
at frequencies near standing waves can be described; the theory accurately
describes both the dispersion curves and the response of the lattice near the
edges of the Brillouin zone. The leading order solution is expressed as a
product between the standing wave solution and long-scale envelope functions
that are eigensolutions of the homogenised partial differential equation. The
general theory is supplemented by a pair of illustrative examples for two
archetypal classes of two-dimensional elastic lattices. The efficiency of the
asymptotic approach in accurately describing several interesting phenomena is
demonstrated, including dynamic anisotropy and Dirac cones.Comment: 24 pages, 7 figure
On Fields with Finite Information Density
The existence of a natural ultraviolet cutoff at the Planck scale is widely
expected. In a previous Letter, it has been proposed to model this cutoff as an
information density bound by utilizing suitably generalized methods from the
mathematical theory of communication. Here, we prove the mathematical
conjectures that were made in this Letter.Comment: 31 pages, to appear in Phys.Rev.
Computational Approaches to Lattice Packing and Covering Problems
We describe algorithms which address two classical problems in lattice
geometry: the lattice covering and the simultaneous lattice packing-covering
problem. Theoretically our algorithms solve the two problems in any fixed
dimension d in the sense that they approximate optimal covering lattices and
optimal packing-covering lattices within any desired accuracy. Both algorithms
involve semidefinite programming and are based on Voronoi's reduction theory
for positive definite quadratic forms, which describes all possible Delone
triangulations of Z^d.
In practice, our implementations reproduce known results in dimensions d <= 5
and in particular solve the two problems in these dimensions. For d = 6 our
computations produce new best known covering as well as packing-covering
lattices, which are closely related to the lattice (E6)*. For d = 7, 8 our
approach leads to new best known covering lattices. Although we use numerical
methods, we made some effort to transform numerical evidences into rigorous
proofs. We provide rigorous error bounds and prove that some of the new
lattices are locally optimal.Comment: (v3) 40 pages, 5 figures, 6 tables, some corrections, accepted in
Discrete and Computational Geometry, see also
http://fma2.math.uni-magdeburg.de/~latgeo
Effects of geometric anisotropy on local field distribution: Ewald-Kornfeld formulation
We have applied the Ewald-Kornfeld formulation to a tetragonal lattice of
point dipoles, in an attempt to examine the effects of geometric anisotropy on
the local field distribution. The various problems encountered in the
computation of the conditionally convergent summation of the near field are
addressed and the methods of overcoming them are discussed. The results show
that the geometric anisotropy has a significant impact on the local field
distribution. The change in the local field can lead to a generalized
Clausius-Mossotti equation for the anisotropic case.Comment: Accepted for publications, Journal of Physics: Condensed Matte
Discretization Errors and Rotational Symmetry: The Laplacian Operator on Non-Hypercubical Lattices
Discretizations of the Laplacian operator on non-hypercubical lattices are
discussed in a systematic approach. It is shown that order errors always
exist for discretizations involving only nearest neighbors. Among all lattices
with the same density of lattice sites, the hypercubical lattices always have
errors smaller than other lattices with the same number of spacetime
dimensions. On the other hand, the four dimensional checkerboard lattice (also
known as the Celmaster lattice) is the only lattice which is isotropic at order
. Explicit forms of the discretized Laplacian operators on root lattices
are presented, and different ways of eliminating order errors are
discussed.Comment: 30 pages in REVTe
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