Shape derivatives of boundary integral operators in electromagnetic scattering. Part II : Application to scattering by a homogeneous dielectric obstacle

Abstract

We develop the shape derivative analysis of solutions to the problem of
scattering of time-harmonic electromagnetic waves by a bounded penetrable
obstacle. Since boundary integral equations are a classical tool to solve
electromagnetic scattering problems, we study the shape differentiability
properties of the standard electromagnetic boundary integral operators. The
latter are typically bounded on the space of tangential vector fields of mixed
regularity $TH\sp{-1/2}(\Div_{\Gamma},\Gamma)$. Using Helmholtz decomposition,
we can base their analysis on the study of pseudo-differential integral
operators in standard Sobolev spaces, but we then have to study the G\^ateaux
differentiability of surface differential operators. We prove that the
electromagnetic boundary integral operators are infinitely differentiable
without loss of regularity. We also give a characterization of the first shape
derivative of the solution of the dielectric scattering problem as a solution
of a new electromagnetic scattering problem.

Full text

Page 1

arXiv:1105.2479v1 [math.NA] 12 May 2011
Shape derivatives of boundary integral operators in
electromagnetic scattering. Part II : Application to
scattering by a homogeneous dielectric obstacle
Martin Costabel∗
Frédérique Le Louër†
Abstract
We develop the shape derivative analysis of solutions to the problem of scattering
of time-harmonic electromagnetic waves by a bounded penetrable obstacle. Since
boundary integral equations are a classical tool to solve electromagnetic scattering
problems, we study the shape differentiability properties of the standard electromag-
netic boundary integral operators. The latter are typically bounded on the space
of tangential vector fields of mixed regularity TH−1
composition, we can base their analysis on the study of pseudo-differential integral
operators in standard Sobolev spaces, but we then have to study the Gâteaux dif-
ferentiability of surface differential operators. We prove that the electromagnetic
boundary integral operators are infinitely differentiable without loss of regularity.
We also give a characterization of the first shape derivative of the solution of the
dielectric scattering problem as a solution of a new electromagnetic scattering prob-
lem.
2(divΓ,Γ). Using Helmholtz de-
Keywords :
operators, shape derivatives, Helmholtz decomposition.
Maxwell’s equations, boundary integral operators, surface differential
1Introduction
Consider the scattering of time-harmonic electromagnetic waves by a bounded obstacle
Ω in R3with a smooth and simply connected boundary Γ filled with an homogeneous
dielectric material. This problem is described by the system of Maxwell’s equations with
piecewise constant electric permittivity and magnetic permeability, valid in the sense of
distributions, which implies two transmission conditions on the boundary of the obstacle
guaranteeing the continuity of the tangential components of the electric and magnetic
fields across the interface. The transmission problem is completed by the Silver–Müller
radiation condition at infinity (see [26] and [27]). Boundary integral equations are an
efficient method to solve such problems for low and high frequencies. The dielectric
∗IRMAR,
martin.costabel@univ-rennes1.fr
†Institut für Numerische und Andgewandte Mathematik, Universität Göttingen, 37083 Göttingen,
Germany, f.lelouer@math.uni-goettingen.de
Institut Mathématique, Université de Rennes1,35042 Rennes,France,
1

Page 2

scattering problem is usually reduced to a system of two boundary integral equations for
two unknown tangential vector fields on the interface (see [5] and [27]). We refer to [8] for
methods developed by the authors to solve this problem using a single boundary integral
equation.
Optimal shape design with a goal function involving the modulus of the far field
pattern of the dielectric scattering problem has important applications, such as antenna
design for telecommunication systems and radars. The analysis of shape optimization
methods is based on the analysis of the dependency of the solution on the shape of the
dielectric scatterer, and a local analysis involves the study of derivatives with respect
to the shape. An explicit form of the shape derivatives is desirable in view of their
implementation in shape optimization algorithms such as gradient methods or Newton’s
method.
In this paper, we present a complete analysis of the shape differentiability of the
solution of the dielectric scattering problem and of its far field pattern, using integral rep-
resentations. Even if numerous works exist on the calculus of shape derivatives of various
shape functionals [11, 12, 13, 15, 33], in the framework of boundary integral equations
the scientific literature is not extensive. However, one can cite the papers [28], [30] and
[29], where R. Potthast has considered the question, starting with his PhD thesis [31],
for the Helmholtz equation with Dirichlet or Neumann boundary conditions and the per-
fect conductor problem, in spaces of continuous and Hölder continuous functions. Using
the integral representation of the solution, one is lead to study the Gâteaux differentia-
bility of boundary integral operators and potential operators with weakly singular and
hypersingular kernels.
The natural space of distributions (energy space) which occurs in the electromagnetic
potential theory is TH−1
are in the Sobolev space H−1
main difficulties: On one hand, the solution of the scattering problem is given in terms of
products of boundary integral operators and their inverses. In order to be able to construct
shape derivatives of such products, it is not sufficient to find shape derivatives of the
boundary integral operators, but it is imperative to prove that the derivatives are bounded
operators between the same spaces as the boundary integral operators themselves. On
the other hand, the very definition of shape differentiability of operators defined on the
shape-dependent space TH−1
in using the Helmholtz decomposition of this Hilbert space which gives a representation
of a tangential vector field in terms of (tangential derivatives of) two scalar potentials. In
this way, we split the analysis into two steps: First the Gâteaux differentiability analysis
of scalar boundary integral operators and potential operators with strongly and weakly
singular kernels, and second the study of shape derivatives of surface differential operators.
This work contains results from the thesis [24] where this analysis has been used to
develop a shape optimization algorithm of dielectric lenses in order to obtain a prescribed
radiation pattern.
This is the second of two papers on shape derivatives of boundary integral operators,
the first one [9] being aimed at a general theory of shape derivatives of singular integral
2(divΓ,Γ), the set of tangential vector fields whose components
2(Γ) and whose surface divergence is in H−1
2(Γ). We face two
2(divΓ,Γ) poses non-trivial problems. Our strategy consists
2

Page 3

operators appearing in boundary integral equation methods.
The paper is organized as follows:
In Section 2 we recall some standard results about trace mappings and regularity
properties of the boundary integral operators in electromagnetism. In Section 3 we define
the scattering problem for time-harmonic electromagnetic waves at a dielectric interface.
We then give an integral representation of the solution — and of the quantity of interest,
namely the far field of the dielectric scattering problem — following the single source
integral equation method developed in [8].
The remaining parts of the paper are dedicated to the shape differentiability analysis
of the solution of the dielectric scattering problem. We use the results of our first paper [9]
on the Gâteaux differentiability of boundary integral operators with pseudo-homogeneous
kernels. We refer to this paper for a discussion of the notion of Gâteaux derivatives in
Fréchet spaces and of some of their basic properties. In Section 4 we discuss the difficulties
posed by the shape dependency of the function space TH−1
operators are defined, and we present a strategy for dealing with this difficulty, namely
using the well-known tool [10] of Helmholtz decomposition. In our approach, we map
the variable spaces TH−1
preserves the Hodge structure. This technique involves the analysis of surface differential
operators that have to be considered in suitable Sobolev spaces. Therefore in Section (5)
we recall and extend the results on the differentiability properties of surface differential
operators established in [9, Section 5]. Using the rules on derivatives of composite and
inverse functions, we obtain in Section 6 the shape differentiability properties of the
solution of the scattering problem. More precisely, we prove that the boundary integral
operators are infinitely Gâteaux differentiable without loss of regularity, whereas previous
results allowed such a loss [29], and we prove that the shape derivatives of the potentials
are smooth away from the boundary but they lose regularity in the neighborhood of the
boundary. This implies that the far field is infinitely Gâteaux differentiable, whereas the
shape derivatives of the solution of the scattering problem lose regularity.
These new results generalize existing results: In the acoustic case, using a variational
formulation, a characterization of the first Gâteaux derivative was given by A. Kirsch
[21] for the Dirichlet problem and then by Hettlich [16, 17] for the impedance problem
and the transmission problem. An alternative technique was introduced by Kress and
Päivärinta in [23] to investigate Fréchet differentiability in acoustic scattering by the
use of a factorization of the difference of the far-field pattern of the scattered wave for
two different obstacles. In the electromagnetic case, Potthast used the integral equation
method to obtain a characterization of the first shape derivative of the solution of the
perfect conductor scattering problem. In [22], Kress improved this result by using a
far-field identity and in [14] Kress and Haddar extended this technique to acoustic and
electromagnetic impedance boundary value problems.
At the end of Section 6 we obtain a characterization of the first shape derivative of
the solution of the dielectric scattering problem as the solution of a new electromagnetic
transmission problem. We show by deriving the integral representation of the solution
that the first derivative satisfies the homogeneous Maxwell equations, and by directly
2(divΓ,Γ) on which the integral
2(divΓr,Γr) to a fixed reference space with a transformation that
3

Page 4

deriving the boundary values of the solution itself we see that the first derivative satisfies
two new transmission conditions on the boundary.
In the end we will have obtained two different algorithms for computing the shape
derivative of the solution of the dielectric scattering problem and of the far field pattern:
A first one by differentiating the integral representations and a second one by solving the
new transmission problem associated with the first derivative.
The characterization of the derivatives as solutions to boundary value problems has
been obtained in the acoustic case by Kress [6], Kirsch [21], Hettlich and Rundell [18]
and Hohage [19] and has been used for the construction of Newton-type or second degree
iterative methods in acoustic inverse obstacle scattering. Whereas the use of these char-
acterizations requires high order regularity assumption for the boundary, we expect that
the differentiation of the boundary integral operators does not require much regularity.
Although in this paper we treat the case of a smooth boundary, in the last section we give
some ideas on possible extensions of the results of this paper to non-smooth domains.
2 Boundary integral operators and their main properties
Let Ω be a bounded domain in R3and let Ωcdenote the exterior domain R3\Ω. Through-
out this paper, we will for simplicity assume that the boundary Γ of Ω is a smooth and
simply connected closed surface, so that Ω is diffeomorphic to a ball.
We use standard notation for surface differential operators and boundary traces. More
details can be found in [27]. For a vector function v ∈ Ck(R3,C3) with k ∈ N∗, we denote
by [∇v] the matrix the i-th column of which is the gradient of the i-th component of v,
and we set [Dv] = [∇v]T. Let n denote the outer unit normal vector on the boundary Γ.
The tangential gradient of a complex-valued scalar function u ∈ Ck(Γ,C) is defined by
∇Γu = ∇˜ u|Γ−?∇˜ u|Γ· n?n,
and the tangential vector curl is defined by
(2.1)
curlΓu = ∇˜ u|Γ∧ n,
(2.2)
where ˜ u is a smooth extension of u to the whole space R3. For a complex-valued vector
function u ∈ Ck(Γ,C3), we denote [∇Γu] the matrix the i-th column of which is the
tangential gradient of the i-th component of u and we set [DΓu] = [∇Γu]T.
The surface divergence of u ∈ Ck(Γ,C3) is defined by
divΓu = div ˜ u|Γ−?[∇˜ u|Γ]n · n?,
and the surface scalar curl is defined by
(2.3)
curlΓu = n · (curl ˜ u) .
These definitions do not depend on the choice of the extension ˜ u.
4

Page 5

Definition 2.1 For a vector function v ∈ C∞(Ω,C3) a scalar function v ∈ C∞(Ω,C)
and κ ∈ C \ {0}, we define the traces :
γv = v|Γ,
γnv =
∂
∂nv = n · (∇v)|Γ,
γDv := n ∧ (v)|Γ(Dirichlet),
γNκv :=1
κn ∧ (curlv)|Γ(Neumann).
We define in the same way the exterior traces γc, γc
n, γc
Dand γc
Nκ.
For a domain G ⊂ R3we denote by Hs(G) the usual L2-based Sobolev space of
order s ∈ R, and by Hs
subdomain B of G belong to Hs(B). Spaces of vector functions will be denoted by
boldface letters, thus
Hs(G) = (Hs(G))3.
loc(G) the space of functions whose restrictions to any bounded
If D is a differential operator, we write:
Hs(D,Ω)
Hs
loc(D,Ωc)
=
{u ∈ Hs(Ω) : Du ∈ Hs(Ω)}
{u ∈ Hs
=
loc(Ωc) : Du ∈ Hs
loc(Ωc)}.
The space Hs(D,Ω) is endowed with the natural graph norm. When s = 0, this defines in
particular the Hilbert spaces H(curl,Ω) and H(curlcurl,Ω). We introduce the Hilbert
spaces Hs(Γ) = γ
?
mappings
γ : Hs+1
Hs+1
2(Ω)
?
, and THs(Γ) = γD
?
Hs+1
2(Ω)
?
. For s > 0, the trace
2(Ω) → Hs(Γ),
2(Ω) → Hs(Γ),
2(Ω) → THs(Γ)
γn: Hs+3
γD: Hs+1
are then continuous. The dual of Hs(Γ) and THs(Γ) with respect to the L2(or L2) scalar
product is denoted by H−s(Γ) and TH−s(Γ), respectively.
The surface differential operators defined above can be extended to Sobolev spaces:
For s ∈ R the tangential gradient and the tangential vector curl are obviously linear and
continuous operators from Hs+1(Γ) to THs(Γ). The surface divergence and the surface
scalar curl can then be defined on tangential vector fields by duality, extending duality
relations valid for smooth functions
?
Γ
(divΓj) · ϕds = −
?
Γ
j · ∇Γϕds
for all j ∈ THs+1(Γ), ϕ ∈ H−s(Γ),
(2.4)
?
Γ
(curlΓj) · ϕds =
?
Γ
j · curlΓϕds
for all j ∈ THs+1(Γ), ϕ ∈ H−s(Γ).
(2.5)
5