Wave Propagation in Complex Media

293 Pages·1998·9.417 MB·English

by  W. C. Chew, J. M. Song, C. C. Lu, R. Wagner, J. H. Lin (auth.), George Papanicolaou (eds.)

The IMA Volumes
in Mathematics
and its Applications
Volume 96
Series Editors
Avner Friedman Robert Gulliver
Springer Science+Business Media, LLC
Institute for Mathematics and
its Applications
IMA
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Continued at the back
George Papanicolaou
Editor
Wave Propagation in
Complex Media
With 68 Illustrations
Springer
George Papanicolaou
Department of Mathematics
Stanford University
Stanford CA 94305, USA
Series Editors:
Avner Friedman
Robert Gulliver
Institute for Mathematics and its
Applications
University of Minnesota
Minneapolis, MN 55455 USA
Mathematics Subject Classifications (1991): 34L25, 35P25, 35Q60, 47A40,
60H25, 73D25, 73D70, 78-08, 78A35, 78A40, 78A45, 81 U40, 82D30
Library of Congress Cataloging-in-Publication Data
Wave propagation in complex media I [edited by] George Papanicolaou.
p. cm. - (The IMA volumes in mathematics and its
applications ; 96)
ISBN 978-1-4612-7241-0 ISBN 978-1-4612-1678-0 (eBook)
DOI 10.1007/978-1-4612-1678-0
1. Wave-motion, Theory of. 1. Papanicolaou, George. II. Series:
IMA volumes in mathematics and its applications ; v. 96.
QA927.W3784 1997
531'.1133-dc21 97-26382
Printed on acid-free paper.
O 1998 Springer Science+Business Media New York
Originally published by Springer-Verlag New York,lnc. in 1998
Soltcover reprint of the hardcover I st edition 1998
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ISBN 978-1-4612-7241-0
FOREWORD
This IMA Volume in Mathematics and its Applications
WAVE PROPAGATION IN COMPLEX MEDIA
is based on the proceedings of two workshops:
• Wavelets, multigrid and other fast algorithms (multipole, FFT)
and their use in wave propagation and
• Waves in random and other complex media.
Both workshops were integral parts of the 1994-1995 IMA program on
"Waves and Scattering." We would like to thank Gregory Beylkin, Robert
Burridge, Ingrid Daubechies, Leonid Pastur, and George Papanicolaou for
their excellent work as organizers of these meetings.
We also take this opportunity to thank the National Science Foun
dation (NSF), the Army Research Office (ARO, and the Office of Naval
Research (ONR), whose financial support made these workshops possible.
A vner Friedman
Robert Gulliver
v
PREFACE
During the last few years the numerical techniques for the solution of
elliptic problems, in potential theory for example, have been drastically
improved. Several so-called fast methods have been developed which re
duce the required computing time many orders of magnitude over that
of classical algorithms. The new methods include multigrid, fast Fourier
transforms, multi pole methods and wavelet techniques. Wavelets have re
cently been developed into a very useful tool in signal processing, the solu
tion of integral equation, etc. Wavelet techniques should be quite useful in
many wave propagation problems, especially in inhomogeneous and nonlin
ear media where special features of the solution such as singularities might
be tracked efficiently.
Waves propagation in random and other complex media exhibit effects
of inhomogeneities that are challenging both theoretically and computa
tionally. Areas of interest include long wave propagation in periodic and
random media, effective media theory and homogenization, nonlinear wave
propagation, localization in strongly inhomogeneous media, geometrical
optics (short waves) in randomly inhomogeneous media, multiple scatter
ing by discrete scatterers, transport theory for waves in random media,
dispersion and randomness in long wave transmission, and reflection and
transmission of waves by nonlinear random media. Applications include op
tical fibers, radio wave propagation in the atmosphere, sound propagation
in the ocean, and seismic waves propagation in the earth.
During the Fall of 1994 the Institute for Mathematics and its Appli
cations held two workshops; one devoted to wavelets, multigrid and other
fast algorithms (multipole, FFT) and their use in wave propagation, and
another devoted to waves in random waves and other complex media.
Both workshops focused on applications to problems in wave propa
gation. The first workshop dealt primarily with fast numerical methods,
whereas the second workshop concentrated on the effects of inhomogeneities
on wave propagation.
Most of the articles in this volume deal with the effects of inhomo
geneities of wave propagation both theoretically and computationally. They
include topics such as waves in random media, coherent effects in scattering
for random systems with discrete spectrum, interaction of microwaves with
sea ice, scattering in magnetic field, surface waves, seismograms envelopes,
backscattering, polarization mode dispersions, and spatio-temporal distri
bution of seismic power. Several articles describes numerical methods, such
as fast algorithm for solving electromagnetic scattering problems, and the
panel clustering methods in 3-d BEM.
George Papanicolaou
vii
CONTENTS
Foreword ......................... .................... ................ v
Preface ....................... .. .. ......... ......................... vii
Fast algorithms for solving electromagnetic scattering problems ....... 1
W.C. Chew, J.M. Song, C.C. Lu, R. Wagner, J.H. Lin,
H. Gan, and M. Nasir
2d photonic crystals with cubic structure: asymptotic analysis. . . . . . 2.3.
A. Figotin and P. Kuchment
On waves in random media in the diffusion-approximation
regime ............................ .................. ...... ........... 31
Jean-Pierre Fouque and Josselin Garnier
Coherent effects in scattering from bounded random
systems with discrete spectrum ....................................... 49
V. Freilikher, M. K aveh, M. Pustilnik, I. Yurkevich,
J. Sanches-Gil, A. Maradudin, and Jun Q. Lu
The interaction of microwaves with sea ice ............................ 75
Kenneth M. Golden
Electron in two-dimensional system with point scatterers
and magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .9 5. . . . . . . . . . . . . . .
Sergey Gredeskul, Masha Zusman, Yshai Avishai,
and Mark Ya. Azbel
On the propagation properties of surface waves .................... " 143
V. JakSic S. Molchanov, and L. Pastur
Green's function, lattice sums and Rayleigh's identity
for a dynamic scattering problem. . . . . . . . . . . . . . . . . . . . . . . .1 . 5. 5 . . . . . . . ..
R.C. McPhedran and N.A. Nicorovici, L.C. Botten,
and Bao Ke-Da
Study of seismogram envelopes based on the energy
transport theory .............. ........................ ............. 187
Haruo Sato
ix
x CONTENTS
The panel clustering method in 3-d bern. . . . . . . . . . . . . . . . . . ... .1 9. .9 . . . . .
Stefan A. Sauter
Propagation of electromagnetic waves in two-dimensional
disordered systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 .2 .5 . . . . . . . . . ..
M.M. Sigaias, C. T. Chan, and C.M. Soukouiis
Reciprocity and coherent backscattering of light ........ ............. 247
Bart A. van Tiggelen and Roger Maynard
Spatio-temporal distribution of seismic power for
a random absorptive slab in a half space . . . . . . . . . . . . . . . . . . .. . 2.7 3. . . . . .
Ru-Shan Wu
FAST ALGORITHMS FOR SOLVING ELECTROMAGNETIC
SCATTERING PROBLEMS·
W.C. CHEWt , J.M. SONGt , C.C. LUt , R. WAGNERt
J.H. LINt , H. GANt , AND M. NASIRt
Abstract. A review of various methods to solve electromagnetic wave-scattering
problems efficiently is presented. Electromagnetic scattering problems are divided into
surface scattering problems and volume scattering problems. Different methods have to
be used depending on the nature of the scatterer. We review several fast methods to
solve scattering problems of volumetric type and surface type.
1. Introduction. Computation of electromagnetic fields (computa
tional electromagnetics) is a fascinating discipline that has drawn the at
tention of mathematicians, engineers, physicists, and computer scientists
alike. It is a discipline that finds a symbiotic marriage between mathe
matics, physics, computer science, and applications. Computational elec
tromagnetics methods to solve wave-scattering problems of large complex
bodies, and to calculate the propagation of waves through turbulent media,
have been fervently studied by many researchers in the past [1-5]. This is
due to the importance of this research in many practical applications, such
as the prediction of the radar scattering cross section of complex objects
like aircraft, interaction of antenna elements with aircraft and ships, en
vironmental effect of vegetation, clouds, and aerosols on electromagnetic
wave propagation, interaction of electromagnetic waves with biological me
dia, and propagation of signals in high-speed circuits and millimeter wave
circuits.
Due to the large electrical dimensions of typical aircraft, past efforts
to ascertain their scattering cross section and the interaction of antennas
with them have exploited approximate high frequency techniques like the
shooting and bouncing ray (SBR) method [6]. However, the recent phe
nomenal growth in computer technology, coupled with development of fast
algorithms possessing reduced computational complexity and memory re
quirements, have made a rigorous numerical solution to the problem of scat
tering from large scatterers feasible. These numerical techniques involve
either using integral equations by converting them into matrix equations
using the method of moments (MOM) [1], or solving partial-differential
equations using finite-difference or finite-element methods whereby sparse
matrices are obtained .
• This work was supported by Office of Naval Research under grant N00014-89-J1286,
the Army Research Office under contract DAAL03-91-G-0339, and the National Science
Foundation under grant NSF ECS 93-02145. The computer time was provided by the
National Center for Supercomputing Applications (NCSA) at the University of illinois,
Urbana-Champaign.
t Electromagnetics Laboratory, Department of Electrical and Computer Engineering,
University of illinois, Urbana, IL 61801.
1
G. Papanicolaou (ed.), Wave Propagation in Complex Media
© Springer-Verlag New York, Inc. 1998
2 W.C. CHEW ET AL.
2. Differential equation solvers. A popular way to solve electro
magnetic scattering problems is to solve the associated partial differen
tial equation directly. The pertinent matrix equation is usually sparse
with O(N) elements. This implies that a matrix-vector multiply can be
performed in O(N) operations. Partial differential equations (PDE's) for
electromagnetics can be roughly categorized into elliptic type for low fre
quencies, and hyperbolic type for high frequencies, and parabolic type for
intermediate frequencies. Elliptic PDE's have the advantage of positive
definiteness, and hence, when iterative methods are used to solve the as
sociated matrix equation, a definite statement can be made about their
convergence rate. For instance, when CG is used to solve Poisson type
equations, it converges in O(No.S) steps in two dimensions, and in O(NO.33)
in three dimensions. When multigrid is used to solve the same equation,
the number of iterations is independent of the size of the problem [7]. As a
consequence, the total computational labor associated with the conjugate
gradient method to solve such problems scales as Nl.S in 2D and N1.33 in
3D, while it scales as N for multigrid methods. For hyperbolic problems,
these computational complexities can only be regarded as lower bounds,
as with the change of geometry, resonance can occur, and the number of
iteration needed for convergence can diverge.
When applied to a scattering problem, a PDE solver needs absorbing
boundary conditions (ABC's) [8] to truncate the simulation region. Many
ABC's have been proposed so that the sparsity of the matrix can be main
tained. However, these ABC's are approximate and have to be imposed at
a substantial distance from the scatterer to reduce the errors incurred by
them. Recently, an absorbing material boundary condition (AMBC) called
perfectly matched layer (PML) has been suggested by Berenger [9]. This
AMBC is particularly well suited for parallel implementation of differen
tial equation solvers because it permits parallel computers to operate in a
single-instruction-multiple-data (SIMD) mode [10].
Alternatively, surface integral equations (which can be considered to
be numerically exact ABCs) can be used to truncate the mesh of the dif
ferential equation solvers [11]. By so doing, the boundary of the simulation
region can be brought much closer to the surface of the scatterer, thereby
reducing the size of the simulation region and the number of associated
unknowns. However, such a method of "absorbing" the outgoing wave re
sults in a partially dense matrix in the final system matrix for the problem.
But, the recent advances in fast methods for solving dense matrices result
ing from integral equations of scattering can be used to solve the system
matrix efficiently.
By a proper ordering of the elements in FEM [11,12]' the dense matrix
will reside only at the bottom right-hand corner of the system matrix as
shown in Figure 1. In this manner, the inverse of the system matrix can be
found by the matrix-partitioning method. When nested-dissection ordering
[7] is applied to the sparse part, and LU decomposition is applied to the