83 research outputs found
Cell averaging two-scale convergence: Applications to periodic homogenization
The aim of the paper is to introduce an alternative notion of two-scale
convergence which gives a more natural modeling approach to the homogenization
of partial differential equations with periodically oscillating coefficients:
while removing the bother of the admissibility of test functions, it
nevertheless simplifies the proof of all the standard compactness results which
made classical two-scale convergence very worthy of interest: bounded sequences
in and are proven
to be relatively compact with respect to this new type of convergence. The
strengths of the notion are highlighted on the classical homogenization problem
of linear second-order elliptic equations for which first order boundary
corrector-type results are also established. Eventually, possible weaknesses of
the method are pointed out on a nonlinear problem: the weak two-scale
compactness result for -valued stationary harmonic maps.Comment: 20 pages, 2 Figure
Energetics and switching of quasi-uniform states in small ferromagnetic particles
We present a numerical algorithm to solve the micromagnetic equations based on tangential-plane minimization for the magnetization update and a homothethic-layer decomposition of outer space for the computation of the demagnetization field. As a first application, detailed results on the flower-vortex transition in the cube of Micromagnetic Standard Problem number 3 are obtained, which confirm, with a different method, those already present in the literature, and validate our method and code. We then turn to switching of small cubic or almost-cubic particles, in the single-domain limit. Our data show systematic deviations from the Stoner-Wohlfarth model due to the non-ellipsoidal shape of the particle, and in particular a non-monotone dependence on the particle size
A simple preconditioned domain decomposition method for electromagnetic scattering problems
We present a domain decomposition method (DDM) devoted to the iterative
solution of time-harmonic electromagnetic scattering problems, involving large
and resonant cavities. This DDM uses the electric field integral equation
(EFIE) for the solution of Maxwell problems in both interior and exterior
subdomains, and we propose a simple preconditioner for the global method, based
on the single layer operator restricted to the fictitious interface between the
two subdomains.Comment: 23 page
Homogenization of Composite Ferromagnetic Materials
Nowadays, nonhomogeneous and periodic ferromagnetic materials are the subject
of a growing interest. Actually such periodic configurations often combine the
attributes of the constituent materials, while sometimes, their properties can
be strikingly different from the properties of the different constituents.
These periodic configurations can be therefore used to achieve physical and
chemical properties difficult to achieve with homogeneous materials. To predict
the magnetic behavior of such composite materials is of prime importance for
applications. The main objective of this paper is to perform, by means of
Gamma-convergence and two-scale convergence, a rigorous derivation of the
homogenized Gibbs-Landau free energy functional associated to a composite
periodic ferromagnetic material, i.e. a ferromagnetic material in which the
heterogeneities are periodically distributed inside the ferromagnetic media. We
thus describe the Gamma-limit of the Gibbs-Landau free energy functional, as
the period over which the heterogeneities are distributed inside the
ferromagnetic body shrinks to zero.Comment: 21 pages, 1 figure. Keywords: Micromagnetics, Periodic
Homogenization, Gamma Convergence, Two-scale Convergence, Demagnetizing Fiel
Polynomial approximations in a generalized Nyman-Beurling criterion
The Nyman-Beurling criterion, equivalent to the Riemann hypothesis, is an
approximation problem in the space of square integrable functions on
, involving dilations of the fractional part function by factors
, . Randomizing the generates new
structures and criteria. One of them is a sufficient condition that splits into
(i) showing that the indicator function can be approximated by convolution with
the fractional part, (ii) a control on the coefficients of the approximation.
This self-contained paper aims at identifying functions for which (i) holds
unconditionally, by means of polynomial approximations. This yields in passing
a short probabilistic proof of a known consequence of Wiener's Tauberian
theorem. In order to tackle (ii) in the future, we give some expressions of the
scalar products. New and remarkable structures arise for the Gram matrix, in
particular moment matrices for a suitable weight that may be the squared
-function for instance.Comment: 12 page
A semi-discrete scheme for the stochastic Landau-Lifshitz equation
We propose a new convergent time semi-discrete scheme for the stochastic Landau-Lifshitz-Gilbert equation. The scheme is only linearly implicit and does not require the resolution of a nonlinear problem at each time step. Using a martingale approach, we prove the convergence in law of the scheme up to a subsequence
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