32,311 research outputs found
On non-local variational problems with lack of compactness related to non-linear optics
We give a simple proof of existence of solutions of the dispersion manage-
ment and diffraction management equations for zero average dispersion,
respectively diffraction. These solutions are found as maximizers of non-linear
and non-local vari- ational problems which are invariant under a large
non-compact group. Our proof of existence of maximizer is rather direct and
avoids the use of Lions' concentration compactness argument or Ekeland's
variational principle.Comment: 30 page
A quasi-optimal coarse problem and an augmented Krylov solver for the Variational Theory of Complex Rays
The Variational Theory of Complex Rays (VTCR) is an indirect Trefftz method
designed to study systems governed by Helmholtz-like equations. It uses wave
functions to represent the solution inside elements, which reduces the
dispersion error compared to classical polynomial approaches but the resulting
system is prone to be ill conditioned. This paper gives a simple and original
presentation of the VTCR using the discontinuous Galerkin framework and it
traces back the ill-conditioning to the accumulation of eigenvalues near zero
for the formulation written in terms of wave amplitude. The core of this paper
presents an efficient solving strategy that overcomes this issue. The key
element is the construction of a search subspace where the condition number is
controlled at the cost of a limited decrease of attainable precision. An
augmented LSQR solver is then proposed to solve efficiently and accurately the
complete system. The approach is successfully applied to different examples.Comment: International Journal for Numerical Methods in Engineering, Wiley,
201
Stochastic band structure for waves propagating in periodic media or along waveguides
We introduce the stochastic band structure, a method giving the dispersion
relation for waves propagating in periodic media or along waveguides, and
subject to material loss or radiation damping. Instead of considering an
explicit or implicit functional relation between frequency and
wavenumber , as is usually done, we consider a mapping of the resolvent set
in the dispersion space . Bands appear as as the trace of
Lorentzian responses containing local information on propagation loss both in
time and space domains. For illustration purposes, the method is applied to a
lossy sonic crystal, a radiating surface phononic crystal, and a radiating
optical waveguide. The stochastic band structure can be obtained for any system
described by a time-harmonic wave equation
- …