17 research outputs found
Effective models for the multidimensional wave equation in heterogeneous media over long time and numerical homogenization
A family of effective equations that capture the long time dispersive effects of wave propagation in heterogeneous media in an arbitrary large periodic spatial domain Omega subset of R-d is proposed and analyzed. For a wave equation with highly oscillatory periodic spatial tensors of characteristic length epsilon, we prove that the solution of any member of our family of effective equations is epsilon-close to the true oscillatory wave over a time interval of length T-epsilon = O(epsilon(-2)) in a norm equivalent to the L-infinity(0, T-epsilon; L-2(Omega)) norm. We show that the previously derived effective equation in [T. Dohnal, A. Lamacz and B. Schweizer, Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Model. Simulat. 12 (2014) 488-513] belongs to our family of effective equations. Moreover, while Bloch wave techniques were previously used, we show that asymptotic expansion techniques give an alternative way to derive such effective equations. An algorithm to compute the tensors involved in the dispersive equation and allowing for efficient numerical homogenization methods over long time is proposed
A Bloch wave numerical scheme for scattering problems in periodic wave-guides
We present a new numerical scheme to solve the Helmholtz equation in a
wave-guide. We consider a medium that is bounded in the -direction,
unbounded in the -direction and -periodic for large ,
allowing different media on the left and on the right. We suggest a new
numerical method that is based on a truncation of the domain and the use of
Bloch wave ansatz functions in radiation boxes. We prove the existence and a
stability estimate for the infinite dimensional version of the proposed
problem. The scheme is tested on several interfaces of homogeneous and periodic
media and it is used to investigate the effect of negative refraction at the
interface of a photonic crystal with a positive effective refractive index.Comment: 25 pages, 10 figure
Dispersive homogenized models and coefficient formulas for waves in general periodic media
We analyze a homogenization limit for the linear wave equation of second
order. The spatial operator is assumed to be of divergence form with an
oscillatory coefficient matrix that is periodic with
characteristic length scale ; no spatial symmetry properties are
imposed. Classical homogenization theory allows to describe solutions
well by a non-dispersive wave equation on fixed time intervals
. Instead, when larger time intervals are considered, dispersive effects
are observed. In this contribution we present a well-posed weakly dispersive
equation with homogeneous coefficients such that its solutions
describe well on time intervals . More
precisely, we provide a norm and uniform error estimates of the form for . They are accompanied by computable formulas for all
coefficients in the effective models. We additionally provide an
-independent equation of third order that describes dispersion
along rays and we present numerical examples.Comment: 28 pages, 7 figure
Multiscale Modeling, Homogenization and Nonlocal Effects: Mathematical and Computational Issues
In this work, we review the connection between the subjects of homogenization
and nonlocal modeling and discuss the relevant computational issues. By further
exploring this connection, we hope to promote the cross fertilization of ideas
from the different research fronts. We illustrate how homogenization may help
characterizing the nature and the form of nonlocal interactions hypothesized in
nonlocal models. We also offer some perspective on how studies of nonlocality
may help the development of more effective numerical methods for
homogenization