Modeling and Computations in Electromagnetics: A Volume Dedicated to Jean-Claude Nédélec

240 Pages·2008·2.615 MB·English

by  R. Hiptmair, P. Meury (auth.), Habib Ammari (eds.)

Lecture Notes
in Computational Science 59
and Engineering
Editors
TimothyJ.Barth
MichaelGriebel
DavidE.Keyes
RistoM.Nieminen
DirkRoose
TamarSchlick
Habib Ammari (Ed.)
Modeling and Computations
in Electromagnetics
A Volume Dedicated to Jean-Claude Nédélec
With74Figuresand5Tables
ABC
Editor
HabibAmmari
ÉcolePolytechnique
CNRSUMR7641
CentredeMathématiquesAppliquées
91128PalaiseauCedex
France
email:[email protected]
LibraryofCongressControlNumber:2007933491
Mathematics Subject Classification (2000): 78M05, 78M15, 83C50, 35Q60, 35R30,
65R20,65R32
ISBN978-3-540-73777-3SpringerBerlinHeidelbergNewYork
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To Jean-Claude N´ed´elec
Preface
Modeling and computations in electromagnetics is quite novel and growing
discipline, expanding as a result of the steadily increasing demand for de-
signingelectricaldevices,modelingelectromagneticmaterials,andsimulating
electromagnetic fields in nanoscale structures.
The aim of this volume is to bring together prominent worldwide experts
to review state-of-the-art developments and future trends of modeling and
computations in electromagnetics. This volume is devoted to merging the ex-
pertise of scientists working on this active discipline, and to raise interest for
challengingissues.Themostsignificantadvancesincomputationaltechniques
developedintheverylastyearsandseveralchallengingtechnologicalapplica-
tions such as those related to communications, to biomedical devices, and to
magnetic storage design are presented in this volume. It covers the following
topics: fast algorithms for time-dependent electromagnetic waves, high-order
methods for high-frequency electromagnetic scattering, non-reflecting bound-
aryconditionsfortime-dependentelectromagneticwaves,multi-scaleanalysis
for Maxwell’s equations, time-reversal for electromagnetic waves, and inverse
electromagnetic scattering.
ThisvolumeisdedicatedtoJean-ClaudeN´ed´elecincelebrationofhis65th
birthday and retirement from Ecole Polytechnique in January 2008. Jean-
Claude has been one of the most distinguished scientist in the area of com-
putational electromagnetics. His outstanding contributions on finite-elements
and fast algorithms for Maxwell’s equations have significantly advanced the
field.
Paris, Habib Ammari
May 2007
Contents
1 Stabilized FEM–BEM Coupling for Maxwell
Transmission Problems
R. Hiptmair and P. Meury ........................................ 1
2 A Posteriori Error Analysis and Adaptive Finite Element
Methods for Electromagnetic and Acoustic Problems
Z. Chen ........................................................ 39
3 Time Domain Adaptive Integral Method
for Surface Integral Equations
H. Ba˘gcı, A.E. Yılmaz, J.-M. Jin, and E. Michielssen ................ 65
4 Local and Nonlocal Nonreflecting Boundary Conditions
for Electromagnetic Scattering
M.J. Grote......................................................105
5 High-Order Methods for High-Frequency
Scattering Applications
O.P. Bruno and F. Reitich........................................129
6 Recent Studies on Inverse Medium Scattering Problems
G. Bao, S. Hou, and P. Li ........................................165
7 Time Reversal of Electromagnetic Waves
J. de Rosny, G. Lerosey, A. Tourin, and M. Fink ....................187
8 Addition Theorem
B. He and W.C. Chew............................................203
Index..........................................................227
1
Stabilized FEM–BEM Coupling for Maxwell
Transmission Problems
R. Hiptmair and P. Meury
1.1 Introduction
We consider the electromagnetic scattering of monochromatic incident waves
from a penetrable, three-dimensional bounded object Ω R3, the scat-
⊂
terer.Inapplicationsoneusuallyencountersscattererswithpiecewisesmooth,
Lipschitz continuous boundaries. Thus it is natural to assume the scatterer
to be a curvilinear Lipschitz-polyhedron in the parlance of [28, Sect.1]. For
the sake of simplicity, we assume that its surface Γ := ∂Ω is connected.
However, with slight changes all theorems can be extended to more general
situations. The material parameters ε and µ may display some spatial vari-
r r
ation inside Ω but assume the constant values ε > 0 and µ > 0 in the air
0 0
region Ω+ :=R3 Ω.
\
Let Es denote the complex amplitude of the scattered electric field
in the air region and E the total electric field inside the scatterer Ω,
which emerge as solutions to the Maxwell transmission problem (cf. [49,
Sect.5.6.3])
curlµ (x)−1curlE κ2ε (x)E = F(x) in Ω,
r r
−
curlcurlEs κ2Es = 0 in Ω+,
−
(1.1)
γ+Es γ−E = g on Γ, γ+Es µ−1γ−E = g on Γ ,
t − t D N − r N N
lim curlE x iκxE = 0.
|x|→∞ × − | |
Here, κ := ω√µ0ε0L (with ω > 0 the fixed angular frequency of the exci-
tation, L the characteristic length of the scatterer) denotes the normalized
wave number and should be considered as a real positive parameter. Further-
more, we write γ E for the tangential components of E on Γ and γ E for
t N
the “magnetic components” curlE n on Γ. The exterior unit normal vec-
tor field n on Γ belongs to L∞(Γ)×and is directed from Ω into Ω+. In the
2 R. Hiptmair and P. Meury
case of excitation by plane electric waves, whose complex amplitude will be
denoted by Einc, the generic jump data g and g evaluate to the following
D N
traces
g := γ Einc, g := γ Einc.
D − t N − N
Finally, we designate by
[γU] :=U U and γU := 1 U +U
Γ |Ω+ − |Ω { }Γ 2 |Ω+ |Ω
the jump, respectively the average, of some generic t(cid:1)race γ of a fu(cid:2)nction U
across the boundary Γ.
Using Rellich’s lemma and unique continuation techniques, the following
result can be established (cf. [36, Theorem 3.1]).
Theorem 1.1. Provided that the relative material parameters µ and ε >0
r r
are piecewise smooth and bounded away from zero everywhere in Ω, the prob-
lem (1.1) has a unique solution.
Boundary element methods (BEM) offer the most flexible way to deal
with the homogeneous problem in the unbounded exterior domain Ω+. They
are based on boundary integral operators on the interface Γ. Due to poten-
tially nonconstant material parameters, the field problem inside Ω may not
be amenable to a treatment by means of boundary element methods. Hence,
finite element schemes (FEM) have to be used here. Thus the topic of this
chapter comes into focus, namely how to derive and discretize stable cou-
pled variational formulations, and how to analyze the resulting FEM–BEM
formulation.
The coupling entails expressing the Dirichlet-to-Neumann (DtN) map of
the exterior problem by means of boundary integral operators linking the
Cauchy data γ E and γ E for the electric field. There exists a huge vari-
t N
ety of integral formulations for the exterior electromagnetic boundary value
problem. A comprehensive survey is given in N´ed´elec’s monograph [49]. In
principle all these methods furnish Dirichlet-to-Neumann maps. However, in
many cases, in particular with so-called indirect formulations, the resulting
operator lacks structural properties of the Dirichlet-to-Neumann map, for in-
stancesymmetry.Thisisobviouslythecaseforsecond-orderellipticproblems.
If the structure of the DtN map is not preserved, then the linear systems of
equations obtained by a Ritz–Galerkin boundary element discretization are
adversely affected.
For second-order elliptic problems Costabel [27] discovered that the so-
calleddirectboundaryintegralequationmethodsprovidearemedy.Themain
ideaistoemploytheCaldero´nprojector,whichactsontheCauchydataofthe
problem. For details and theoretical considerations we refer to [21, Sect.4.5]
and[30].Inshort,theCaldero´nprojectoryieldstwosetsofboundaryintegral
equations. Judiciously combining them yields a version of the Dirichlet-to-
Neumann map, which is perfectly suited for a Ritz–Galerkin discretization.
1 Stabilized FEM–BEM Coupling for Maxwell Transmission Problems 3
Costabel’s idea of coupling finite elements with boundary elements is usually
referred to as “the symmetric coupling approach.” It has been applied to a
wide range of strongly elliptic problems; see, among others [19, 38, 44]. For
references to the engineering literature see [53, 55] and the references therein.
Unsurprisingly, the Caldero´n projector for the Maxwell system has been
thoroughly studied, cf. [20, Sect.1.3.2], [32], [49, Sect.5.5], and [43, Sect.3].
Theideaofsymmetriccouplingforthetransmissionproblemwastheoretically
probed in [1, 2, 4], and in [7] for a related problem involving impedance
boundary conditions. All these results employ compactness arguments and
theFredholmalternative. Tothisend,most authorshavestudied theintegral
operators intrinsically on Γ. They have been successful on smooth interface
boundaries, but all efforts to adjust the approach to nonsmooth boundaries
have been in vain.
The fundamental new insights about the traces of electromagnetic fields,
presented in [9, 12, 13, 15], paved the way to further progress. That progress
couldfinallybeachievedbyrememberingahighlyeffectivepolicyinthemod-
ern treatment of boundary integral equations: The guideline is to stay off
the boundary as far as possible by studying variational problems instead of
the boundary integral operators directly. This policy has demonstrated its
efficacy in the work of Costabel [27]. The recent textbook [46] discusses all
nuances of this approach for strongly elliptic systems. Moving off the bound-
ary helps steer clear of its awkward geometric features. Thus, the foundation
for a theory of electromagnetic boundary integral operators could be laid in
[16, 18].
Inaddition,inordertoharnesscompactnessarguments,wehavetoemploy
decompositions of the surface vector fields on Γ. The classical composition is
theso-calledHodgedecomposition[32],whichremainsaveryeffectivetoolon
piecewise smooth boundaries, cf. [14], [18], and, in particular [41]. Its coun-
terpart on domains istheHelmholtz decomposition. It isimportant to realize
that there is some leeway in choosing the decomposition, because the exact
orthogonality featured by Hodge or Helmholtz decompositions is of minor
importance. Instead, we prefer to use related, but simpler, splittings.
Almost all boundary integral equations for the exterior Dirichlet problem
in electromagnetic and acoustic scattering are haunted by the presence of
“spuriousfrequencies”[15,18,22],forwhichtheequationsfailtohaveunique
solutions. Those agree with interior Dirichlet eigenvalues. The symmetrically
coupledvariationalformulationpresentedin[39]exhibitsthesamedrawback.
In this chapter, we propose a stabilized method for FEM–BEM coupling
based on (mixed) Robin-type boundary conditions to ensure unique solvabil-
ityofthecorrespondinginteriorboundaryvalueproblem.Theuseofcomplex
combinations of boundary integral operators has been an invaluable tool for
deriving resonance-free combined field integral equations (CFIE) for electro-
magnetic scattering from a perfect conductor, cf. [35]. Furthermore, our ap-
proach also features regularizing operators, already used to stabilize Maxwell
scatteringproblemsin[17],toensureaG˚ardinginequalityforthesesquilinear
4 R. Hiptmair and P. Meury
form underlying the variational formulation. In our case, both problems are
tackled by introducing modified trace operators.
Basedonthegeneralizedtraces,stabilizedversionsofCalder´onprojectors
canbedefinedforthecouplingofdomainbasedvariationalformulationswith
boundary integral equations. Thus, we can derive new coupled variational
formulations, which feature existence, uniqueness, and stability of solutions
for all wave numbersκ>0. Asimilar approach totheonepresentedherecan
be found in [54].
To discretize the symmetric and the stabilized coupled variational formu-
lations,werelyondiscretedifferentialforms(edgeelements,faceelements)on
triangulations of both Ω and Γ. The Ritz–Galerkin approach is straightfor-
ward,andyet,inthediscretesettinganotherchallengearises.The Helmholtz
and Hodge-type decompositions do not directly carry over to the discrete
spaces. For pure indirect boundary element formulations (Rumsey’s princi-
ple) remedies have been explored in [41] and [22]. Direct boundary integral
equations were tackled in [18]. All these approaches exploit the fact that ap-
propriate discrete splittings can approximate their continuous counterparts
reasonably well. In this chapter we adapt the ideas in [18] and [39] to the
symmetrically coupled FEM–BEM problem. We will use variants of these re-
sults that do not require sophisticated elliptic regularity theory.
The outline of this chapter is as follows: In the following section we will
review the theory of Sobolev spaces and tangential traces. In Sect.1.3 we will
introduce the potentials, which form the building blocks of the Stratton–Chu
representation formula and the boundary integral operators for the electric
field equation. In Sect.1.4 we construct decompositions of the electric field in
Ω. The theoretical results for the symmetrically coupled variational formula-
tion are reviewed in Sect.1.5. In Sect.1.6 the stabilized coupling strategy is
presented.Sofar,allsectionshavebeenmerelyconcernedwiththeanalysisof
thecontinuousvariationalproblems.Then,inSect.1.7,weintroducethefinite
elementandboundaryelementspaces,whichareusedforaRitz–Galerkindis-
cretizationofthecoupledproblem.InSect.1.8wederivediscretecounterparts
to the decompositions on the continuous level and establish discrete inf–sup
estimates for the underlying sesquilinear forms. Finally, in Sect.1.10 we will
establish a priori convergence estimates for the stabilized coupling approach.
1.2 Traces and Spaces
The main purpose of this section is to define suitable Sobolev spaces, which
can be used to derive weak formulations of (1.1), and review some of their
most important properties. The notation and notions we introduce closely
follow the ones of [39, Sect.2].
The natural Hilbert space for an analysis of the Maxwell transmission
problem (1.1) is the space
Hloc(curl, D):= V L2loc(D); curlV L2loc(D) .
∈ ∈
(cid:3) (cid:4)