Some second-kind integral equations in electromagnetism
Marion DARBAS
∗Abstract
We address the derivation of new second-kind combined field integral equations for the Krylov iterative solution of high-frequency electromagnetic scattering problems by a perfect conductor. The proposed formulations extend the well-known Brakhage-Werner and Combined Field Integral Equations and improve the convergence properties of their numerical solution through a Krylov iterative method. We prove that these integral equa-tions are well-posed for any frequency. Preliminary experiments with spherical harmonics in the case of a spherical scatterer illustrate the good behavior of a Krylov iterative solver used for computing the solution of these new integral equations relatively to an increase of the frequency or/and to the presence of a large number of vectorial spherical harmonics. Keywords. Maxwell equations, second-kind integral equation, transparent operator, high-frequency, iterative Krylov subspace solver
1
Introduction
The numerical solution of high-frequency electromagnetic wave scattering problems is a chal-lenging and active research area (see for example [13]) with many defence and civil applications. Among the different issues, an important investigation topic concerns the development of ef-ficient solvers based on a boundary integral equation. In modern industrial codes, integral equations are solved with a Krylov iterative solver [29] coupled to a fast evaluation of the ele-mentary integral operators applied to a given density (as e.g. the Fast Multipole Method [28]). Generally, the computational cost of the numerical solution of a given integral equation is in
O(niter nlogn) operations if the total number of degrees of freedom is n. The parameter niter
is the number of iterations required to reduce the residual by a prescribed factor. Industrial solvers are based on different kinds of integral equations like the Electric Field Integral Equa-tion (EFIE), the Magnetic Field Integral EquaEqua-tion (MFIE), the Brakhage-Werner or Combined Field Integral Equations (CFIE). Recent numerical experiments and theoretical studies [13] have shown that an increase of a) the temporal frequency or/and of b) the density of discretization
points per wavelength can really increase niter. Therefore, the obtention of integral equations
without these flaws is highly desirable. One possible solution consists in building a suitable preconditioner to improve the convergence rate of the iterative solver. Various strategies can be considered as for instance these based on the algebraic point of view (incomplete factor-ization algorithm, sparse approximate inverses [10, 11],...) or more recently these using the Calderon integral relations to construct an analytical preconditioner for the EFIE as proposed by Christiansen & Nédélec [15, 32].
However, the two above points a) and b) are not the only difficulties to overcome. Whatever a preconditioner is chosen, it cannot improve a formulation which has not a unique solution. Indeed, it is well-known for example that the EFIE has spurious solutions when the wavenumber
∗Université Paris Dauphine, Laboratoire Ceremade, Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16, France. E-mail: darbas@ceremade.dauphine.fr
k corresponds to a resonant frequency of the interior cavity [16]. This limits its application to solve inverse scattering problems. To avoid the problem of interior frequencies, a possibility consists in using some modified or combined integral equations. Essentially, the idea is to use an integral representation of the exterior field involving both the Maxwell single- and double-layer potentials. The Brakhage-Werner integral formulation (1965) [7] and the Combined Field Integral Equation (CFIE) of Harrington and Mautz (1978) [20] rely on this approach and are commonly used in practice. They are uniquely solvable for any frequency but unfortunately are not free of the flaws a) and b).
This paper provides the construction of some well-posed integral equations which do not suffer from both the frequency increase and the mesh refinement. These integral equations are
second-kind Fredholm equations ensuring hence that niter is independent of a mesh refinement
for a fixed wavenumber (when a simple preconditioning by the mass matrix is used) [9, 25]. Their representation incorporates a suitable operator which also leads to a condition number independent of the frequency parameter. For the sake of completeness, let us mention other CFIEs based on different techniques which have been recently proposed [2, 8, 23].
The outline of the paper is the following. In section 2, we recall the electromagnetic scat-tering problem and some related functional analysis results. In section 3, we describe the construction of the new combined field integral equations. The principle follows some ideas de-veloped in acoustics [4, 17]. One keypoint in the construction process of these integral equations is based on the use of an approximate regularizing operator built in the section 4. We prove the existence and uniqueness of the solution to these integral equations in section 5 using the Helmholtz decomposition. Finally, section 6 is devoted to a formalist analytical and numerical study of the spectral properties related to these integral equations for a spherical scatterer. The results indicate that the new integral operators remain particularly adapted to a Krylov (here GMRES) iterative solution either for a high wavenumber or when the solution is composed of a large number of spherical harmonics.
2
Problem setting
2.1
The scattering problem
Let Ω− be a bounded domain of R3with a regular compact boundary Γ = ∂Ω− representing a
perfectly conducting object. We consider the scattering of an incident time-harmonic
electro-magnetic field (Einc, Hinc) by Ω−, with time-dependence exp(−ikct), setting k as the the real
positive wavenumber and c the light speed. If we set n as the unit normal to Γ outwardly
di-rected to Ω−and the exterior domain of propagation as Ω+= R3
\Ω−, the total electromagnetic
field (E, H) is solution to the system [16]
curl E− ikZ0H= 0, curl H+ ikZ0−1E= 0 in Ω+,
ET = n× (E × n) = 0 on Γ. (1)
To get the uniqueness of the solution to the above boundary-value problem [26], we consider the well-known Silver-Müller radiation condition
lim r→+∞r (E − Einc) − Z0(H− Hinc)× r/ |r| = 0. (2)
The vectors E and H represent the electric and magnetic fields respectively. The operator
curl is the vectorial curl operator and Z0 the impedance of the vacuum. Moreover, for two
complex-valued vector fields a and b of R3, we denote by a × b their vectorial cross product
2.2
Functional spaces and traces theorems
Let Ω ⊂ R3be any of the sets Ω±, R3. We define the spaces
Hcurl(Ω) = {v ∈ (L2(Ω)) 3 , curl v∈ (L2(Ω))3}, Hdiv(Ω) = {v ∈ (L2(Ω)) 3 , div v∈ L2(Ω) },
and the map γt: v7→ v|Γ×n. Let L2(Γ), Hs(Γ), s≥ 1, be the usual functional Sobolev spaces.
Then, we can introduce the following vectorial Sobolev spaces
TL2(Γ) ={v ∈ (L2(Γ))3, v
· n = 0},
THs(Γ) ={v ∈ TL2
(Γ), vi∈ Hs(Γ) i = 1, 2, 3}, ∀s ≥ 1,
where videnotes the i-th component of the vector-field v. Let us recall that identifying TL2(Γ)
with its dual allows the indentification of TH−s(Γ),
∀s ∈ N, with the dual (THs
(Γ))0of THs
(Γ) [33]. We consider the following spaces
H−1/2curl (Γ) = {v ∈ TH−1/2(Γ), curl
Γv∈ H−1/2(Γ)},
H−1/2div (Γ) = {v ∈ TH−1/2(Γ), div
Γv∈ H−1/2(Γ)}.
In the above expressions, curlΓ and divΓ denote the surface curl and the divergence operators
respectively. Following [26], the tangential trace γt : Hcurl(Ω)→ H−1/2div (Γ) is continuous and
surjective. Finally, we define the antisymmetric pairing as
{u, v}Γ =
Z
Γ
(u× n) · vdΓ, u, v ∈ TL2(Γ). (3)
2.3
Integral representations
One possible way to solve (1)-(2) is to use an integral representation of E and H by vector fields tangent to Γ. To begin, let us recall the Stratton-Chu formulae [16].
Theorem 1
The solution (E, H) to system (1)-(2) has the following integral representation (Stratton-Chu
formulae)
E = Einc+ ikZ
0T J + KM, in Ω+,
H = Hinc− KJ + ikZ0−1T M, in Ω+.
(4) In the above equations, J and M designate the electric and magnetic equivalent surface currents given by
J(x) = n(x)× H(x) = n(x) × H+(x), x∈ Γ,
M(x) = E(x)× n(x) = E+(x)
× n(x), x ∈ Γ.
The subscripts± correspond to traces onto Γ from Ω±. The potentials
T and K are given by
T J(x) = Z Γ G(x, y)J(y)dΓ(y) + 1 k2∇Γ Z Γ G(x, y)∇Γ· J(y)dΓ(y), x /∈ Γ, KM(x) = Z Γ∇ yG(x, y)× M(y)dΓ(y), x /∈ Γ, (5)
The integral operators T and K are called the Maxwell single- and double-layer potentials.
The operator ∇Γ is the tangential gradient. The surface currents J and M are also called the
Cauchy data and, here, they represent the new unknowns of the diffraction problem.
Since we wish to consider integral equations on Γ, we need to apply the tangential trace
operator γt to the potentials T and K.
Lemma 1
The exterior (+) and interior (-) tangential traces of the potentialsT and K on Γ are given by
(γt±◦ T J)(x) = T J(x) × n(x) and (γ ± t ◦ KM)(x) = ± 1 2M(x) + KM(x)× n(x), x ∈ Γ, with T J(x) = Z Γ G(x, y)J(y)dΓ(y) + 1 k2∇Γ Z Γ G(x, y)∇Γ· J(y)dΓ(y), x ∈ Γ, KM(x) = Z Γ∇ yG(x, y)× M(y)dΓ(y), x ∈ Γ. (6)
Consequently, the exterior and interior traces of the representation formulae (4) yield γ± t E = γt±Einc± 1 2M+ KM× n + ikZ0T J× n, in H −1/2 div (Γ), γ± t H = γt±Hinc∓ 1 2J− KJ × n + ikZ −1 0 T M× n, in H −1/2 div (Γ). (7)
If we introduce the following trace operator γn= γt◦ curl, then we get [16] the jump relations
[γn]Γ◦ T = [γt]Γ◦ K = I and [γn]Γ◦ K = [γt]Γ◦ T = 0, (8)
setting [γ]Γ = γ+− γ− for a given trace γ.
2.4
Properties of the boundary integral operators
Following [26], these mapping properties hold
T : H−1/2div (Γ) → H−1/2curl (Γ),
K : H−1/2div (Γ) → H−1/2curl (Γ).
Moreover, we can state this coerciveness result [36]. Proposition 1
Let T1, T2: H−1/2div (Γ)→ H−1/2curl (Γ) be the two coercive operators with respective orders 1 and
−1 given by
T1=
1
k2∇ΓV0divΓ,
and
T2= curlΓ∆−1Γ curlΓV0curlΓ∆−1Γ curlΓ,
where ∆Γ and curlΓ are respectively the scalar Laplace-Beltrami operator and the tangential
rotational operator. Let V0 be the sound-single-layer
V0j(x) =
Z
Γ
1
Then, the boundary integral operator T : H−1/2div (Γ)→ H−1/2curl (Γ) can be written as
T = T1+ T2+ V0Π + C, (9)
with Π the projection on the space of the harmonics tangential fieldsN = {v ∈ TL2
(Γ), LΓv=
0} where LΓ is the vectorial Laplace-Beltrami operator. The operator V0Π is of finite rank and
C is a compact operator from H−1/2div (Γ) into H−1/2curl (Γ).
The Electric Field Integral Equation (EFIE) is based on the integral operator T . The non-uniqueness of the EFIE at exceptional frequencies corresponds to a kernel of T not reduced to 0 for the corresponding values of k. Furthermore, T is a first-order integral operator. Therefore,
its spectrum is not clustered around a point z0 6= 0. As a result, its condition number is
dependent on both the wavenumber and the density of discretization points per wavelength. The compact operator K is involved in the Magnetic Field Integral Equation. This equation is a of second-kind and naturally yields a good behavior relatively to a mesh refinement for a fixed frequency. However, it is also well-known that the spurious currents associated with this formulation deteriorate the accuracy of the solution. In view only of the uniqueness result, both the EFIE and the MFIE are not satisfactory.
3
Generalized combined field integral equations
We give an answer to the issue of building some suitable well-posed second-kind integral equa-tions. Unlike the Brakhage-Werner (BW) integral equation [7] and the Combined Field Integral Equation (CFIE) [20], the associated integral operators overcome the above mentioned flaws: the high-frequency parameter and the mesh refinement. This section introduces the general lines of the construction of the new combined field integral equations which will be carried out in the next two sections.
3.1
Generalized Brakhage-Werner integral equation
Let us recall the construction of Brakhage and Werner to get a well-posed integral equation at any frequency. To overcome interior resonances, they proposed an integral representation of the exterior field in terms of the two potentials T and K
E(x)− Einc(x) =
Ka(x) − ikZ0ηT (n × a)(x), x ∈ Ω+,
H(x)− Hinc(x) = curlE(x)/(ikZ
0), x∈ Ω+. (10)
The number η is a real coupling parameter and a is a fictitious density. Taking the exterior
tangential trace γ+
t of the electric field (10) and using the boundary condition E+× n =
−Einc
× n on Γ, we get the following boundary integral equation (cf. relations (7))
Ba = 12a+ Ka× n − ikZ0ηT (n× a) × n = −Einc× n, on Γ. (11)
This integral equation is well-posed for any wavenumber k > 0 and any parameter η > 0 [16]. In [22], Kress numerically established that the value η = 1 yields the lowest condition number for (11). However, equation (11) is not a second-kind integral equation since it can be viewed as a perturbation of the EFIE (operator T ) by a lower-order part. As a consequence, we are solving a first-kind integral equation.
To improve this formulation, we propose the following strategy. Assume that we are able
to define the exact operator Λ : H−1/2curl (Γ)→ H
−1/2
div (Γ) linking the Cauchy data of Maxwell’s
equations
J=−Λ(n × M). (12)
Replacing the surface current J by its value (12) in the integral representation of the electric field (4) yields the following integral representation
1
2M+ KM× n − ikZ0T Λ(n× M) × n = M, in H
−1/2
div (Γ). (13)
The operator defining the integral equation (13) is nothing else than the identity. This is an "ideal" situation for an iterative process. However, this approach is quite theoretical because being able to know the operator Λ means that we have already solved the scattering problem.
So, we rather consider an approximation eΛ (see the next section for an explicit construction)
of the exact operator Λ and seek the exterior field under this form
E(x)− Einc(x) = Ka(x) − ikZ
0T eΛ(n× a)(x), x ∈ Ω+,
H(x)− Hinc(x) = curlE(x)/(ikZ
0), x∈ Ω+. (14)
Then, we get the new integral equation
BeΛa=
1
2a+ Ka× n − ikZ0T eΛ(n× a) × n = −E
inc
× n, in H−1/2div (Γ). (15)
Let us remark that the choice of the Silver-Müller condition, that is eΛ = I, gives the usual
BW integral formulation on Γ with the optimal coupling parameter of Kress. Therefore, this method can be considered as a generalization of the classical Brakhage-Werner approach. We
will see that a suitable approximation eΛ results in a well-posed and well-conditioned second-kind
integral equation.
3.2
Generalized Combined Field Integral Equation
A similar approach can be derived to build generalized CFIEs. The usual CFIE introduced by Harrington and Mautz [20] corresponds to a convex combination between the EFIE and the MFIE as follows
CFIE = αEFIE + (1− α)MFIE, (16)
where α is a free parameter. The EFIE and MFIE are expressed respectively as
−ikZ0T J = EincT and
J
2+ n× KJ = n × H
inc, on Γ.
As for the BW formulation, the CFIE remains a first-kind integral equation because its top
part is the EFIE. However, if we apply eΛ to the EFIE before adding the MFIE, we obtain the
generalized CFIE
CeΛJ=
1
2J+ n× KJ − ikZ0ΛT J = ne × H
inc+ eΛEinc
T , on Γ. (17)
The unknown is the current and not a fictitious density as in the BW formulations. This property perhaps explains the success of the utilization of the usual CFIE by the engineering community.
4
Construction of the operator e
Λ
This section proposes the construction of the operator eΛ. To this end, we consider an
approxi-mation of the exact transparent operator using microlocal analysis. This kind of construction is often met in the theory of artificial boundary conditions for hyperbolic pseudodifferential systems and in particular for the solution to the Maxwell equations [3, 6, 18, 19, 21, 27, 31]. Basically, the construction follows the two successive steps.
4.1
The tangent plane approximation
We do not detail here the complete construction of the approximation eΛ of the exact transparent
operator Λ. The approach follows ideas developed in [3]. Essentially, it can be summarized in the following points. The method consists in firstly rewritting the Maxwell equations in a local
coordinates system (r, s) associated with Γ. More precisely, locally near Γ, Ω− is mapped to a
half-space. The variables r and s = (s2, s3) represent respectively the normal and tangential
variables. Next, we use the theory of pseudodifferential operators and particularly Taylor’s diagonalization theorem [34, 35] to extract the highest order classical symbol λ of the operator Λ λ = Z0−1(1−|ξ| 2 k2 ) −1/2 1− ξ2 3 k2 ξ2ξ3 k2 ξ2ξ3 k2 1− ξ2 2 k2 ,
where ξ = (ξ2, ξ3) is the dual Fourier variable of s = (s2, s3) and |ξ|2 = ξ22+ ξ32. Then, we
consider the pseudodifferential operator eΛ of principal symbol λ as a first approximation of the
exact transparent operator Λ e Λ = σp(λ) = Z0−1(I + LΓ k2) −1/2 (I−k12curlΓcurlΓ). (18)
Finally, we get the On-Surface Radiation Condition (OSRC)
n× H + eΛET= 0, on Γ. (19)
4.2
Regularization of the square-root symbol
This step consists in a formal regularization of the approximate operator eΛ which is not defined
for the tangential rays to the surface. Indeed, the construction of the condition (19) is valid for both the hyperbolic zone H = {(k, ξ); k |ξ|} of propagative rays and the elliptic zone E = {(k, ξ); k |ξ|} of evanescent rays [3]. However, it is not valid in the transition region of glancing rays: G = {(k, ξ); k ≈ |ξ|} and is therefore non-uniform since the square-root operator is singular for this set of frequencies. To partly overcome this difficulty, we introduce a damping
kε= k + iε by perturbating the wavenumber k by a small imaginary part ε > 0. As a result,
we get the following new regularized OSRC
n× H + eΛεET = 0, on Γ, or equivalently J + eΛε(n× M) = 0, on Γ, (20) where e Λε= Z0−1(I + LΓ k2 ε ) −1/2 (I−k12 ε curlΓcurlΓ). (21)
To obtain a symmetrical operator in the equation (20)-(21), the complex wavenumber is also
introduced for (I − 1/k2
εcurlΓcurlΓ). The next step consists in providing a suitable choice of
the damping function.
To this aim, let us consider the Helmholtz decomposition of a vector field v ∈ H−1/2curl (Γ)
v= θ + curlΓψ +∇Γϕ, (22)
with θ ∈ N (cf. proposition 1) and (ψ, ϕ) ∈ H3/2(Γ)
× H1/2(Γ) [36]. The application of the
damped operator to the above decomposition yields
Z0Λeεv = θ + (I + LΓ k2 ε ) 1/2 curlΓψ + (I + LΓ k2 ε ) −1/2 ∇Γϕ = θ + 1 ikcurlΓ(ik(1 + ∆Γ k2 ε ) 1/2 ψ) + ik∇Γ( 1 ik(1 + ∆Γ k2 ε ) −1/2 ϕ), (23) because
divΓcurlΓψ = 0 and curlΓ∇Γϕ = 0.
In the case of three-dimensional acoustic scattering [5, 17], an efficient approximation of the
Dirichlet-to-Neumann (DN) operator is given by ik(1 + ∆Γ/kε2)
1/2
while for the
Neumann-to-Dirichlet (ND) operator a suitable approximation is (1 + ∆Γ/k2ε)
−1/2
/(ik). Then, according to
the splitting (23), applying the operator eΛε consists in using an approximation of the acoustic
DN operator and an approximation of the acoustic ND operator according to the scalar un-knowns ψ and ϕ respectively. In the OSRC context for three-dimensional acoustics [5, 17], a suitable value of ε has been computed formally to obtain a good representation of the tangential
modes. This coefficient is given by εopt= 0.4k1/3H−2/3 in (21), where H represents the mean
curvature of Γ. We will see during the numerical experiments (section 5) that this choice is also suitable for the Maxwell system.
5
Well-posedness of the generalized integral equations
We consider now the problem of existence and uniqueness of the solution to the generalized BW integral equations and CFIE considering the coupling operator (20)-(21). To this end, we firstly prove the two following lemmas.
The generalized BW integral equation is expressed by BeΛεa= 1 2a+ Ka× n − ikZ0T eΛε(n× a) × n = −E inc × n, in H−1/2div (Γ). (24) Lemma 2
The generalized Brakhage-Werner integral operatorBΛeε : H−1/2div (Γ)→ H−1/2div (Γ) defining (24)
is a second-kind Fredholm operator.
Proof. Our goal is to prove that the operator BeΛ
ε can be written under the form αI + C,
with α 6= 0 and C a compact perturbation. Let us consider the Helmholtz decomposition of the
density a ∈ H−1/2div (Γ)
a= θ + curlΓψ +∇Γϕ, (25)
with θ ∈ N and (ψ, ϕ) ∈ H3/2(Γ)
× H1/2(Γ). The fields θ, ψ and ϕ become the new unknowns.
Therefore, we can decompose our study following these three fields and analyze the form of the
Let us first consider the integral operator Z0T eΛε. According to Proposition 1, we have
Z0T eΛεθ = V0θ + Cθ,
because divΓv= curlΓv= 0 for any v∈ N . For the density curlΓψ, we can write that
Z0T eΛεcurlΓψ = curlΓV0(I + ∆Γ k2 ε ) 1/2 ψ + C2curlΓψ,
because divΓcurlΓψ = 0. The operator C2= (V0Π + C)(I + LΓ/k2ε)
1/2
is compact. Moreover,
the relation curlΓ∇Γϕ = 0 yields
Z0T eΛε∇Γϕ = 1 k2∇ΓV0∆Γ(I + ∆Γ k2 ε ) −1/2 ϕ + C1∇Γϕ, where C1 = (V0Π + C)(I + LΓ/k2ε) −1/2
is a compact operator. Since the principal symbol of
the pseudodifferential operator V0 of order −1 is given by σp(V0) = 1/(2|ξ|) [12] where ξ is the
dual variable to the variable x on Γ, we get
σp(V0(I +∆Γ k2 ε ) 1/2 ) = i 2kε and σp(V0∆Γ(I +∆Γ k2 ε ) −1/2 ) = ikε 2.
Consequently, we obtain the following relations
−ikZ0T eΛεcurlΓψ =
k
2kε
curlΓψ− ikC2curlΓψ, (26)
and
−ikZ0T eΛε∇Γϕ = kε
2k∇Γϕ− ikC1∇Γϕ. (27)
The generalized BW integral operator can be written as BΛeεa= ( 1 2θ + ( 1 2+ kε 2k)curlΓψ + ( 1 2+ k 2kε )∇Γϕ) +Ca, (28)
and is the sum of an isomorphism between H−1/2
div (Γ) and itself with three eigenvalues and
a compact perturbation C. We conclude that the generalized BW integral operator defines a second-kind Fredholm integral equation. More precisely, it can be decomposed in a system of three second-kind integral equations. This ends the proof of the Lemma 2.
Lemma 3
The generalized Brakhage-Werner integral operatorBΛeε : H−1/2div (Γ)→ H−1/2div (Γ) given in (24)
is an injective operator.
Proof. Let us assume that a ∈ H−1/2div (Γ) is such that BΛeεa = 0. Then, the associated
electromagnetic field (E, H) is solution to the scattering problem with a zero incident field in
Ω+. Hence, E = H = 0 in Ω+. Using the tangential traces (7) and the jump relations (8)
respectively, we can write
An application of the Gauss theorem directly gives −ik{a, Z0eΛε(n× a)}Γ= Z Ω− (|curlE|2− k2|E|2)dΩ− ∈ R. This means that we have
<({a, Z0Λeε(n× a)}Γ) = 0.
The next step consists in showing that this last relation implies a = 0.
To this end, let (Yi)i∈Nbe a basis of eigenvectors associated to the scalar Laplace-Beltrami
operator ∆Γ
−∆ΓYi= λiYi.
It is well-known that the eigenvalues λi are real positive numbers and that λ0 = 0. According
to [33], the following result holds. Theorem 2
Let{u1, ..., uN} be an orthonormal basis of the space N of the harmonics tangential fields and
(Yi)i∈Nbe the orthornormal Hilbertian basis of L2(Γ) described above. Then, the family
{(u1, ..., uN), (∇Γ Yi √λ i ,n× ∇√ ΓYi λi )i≥1}
is an orthornormal Hilbertian basis of TL2
(Γ) and a basis of eigenvectors for the vectorial
Laplace-Beltrami operator LΓ such that
−LΓ(∇Γ Yi √ λi ) = λi(∇Γ Yi √ λi ), −LΓ( n× ∇ΓYi √ λi ) = λi( n× ∇ΓYi √ λi ), for i≥ 1.
According to the above result, a can be written as
a= N X j=1 βjuj+ ∞ X i=1 (αigi+ γiri).
The coefficients βj, αi, γi are real and we have ∀i ≥ 1
gi= ∇√ΓYi λi , ri= n× ∇ΓYi √ λi . Then, we get Z0Λeε(n× a) = N X j=1 βjn× uj+ ∞ X i=1 (αi(1− λi k2 ε ) 1/2 ri− γi(1− λi k2 ε ) −1/2 gi).
Indeed, we have for all i ≥ 1 Z0Λeεri= (1− λi k2 ε ) 1/2 ri, Z0Λeεgi= (1−λi k2 ε ) −1/2 gi.
On N , Z0Λfε is nothing else then the identity operator
Z0Λeε(n× v) = n × v.
As a consequence, we deduce the relation
{a, Z0Λeε(n× a)}Γ= N X j=1 β2j|n × uj|2+ ∞ X i=1 (α2i(1− λi k2 ε ) 1/2 + γi2(1− λi k2 ε ) −1/2 ).
However, since λi> 0 for i∈ N∗, we can directly prove that: <((1 − λi/k2ε)
±1/2
) > 0, for k > 0
and ε > 0. Moreover, the relation <(< a, Z0eΛε(n× a) >Γ) = 0 leads to βj = 0,∀j ∈ {1, ..., N},
and αi= γi= 0,∀i ∈ N∗. This finally shows that: a = 0.
Proposition 2
The generalized Brakhage-Werner integral equation (24) is uniquely solvable for any frequency k > 0 and damping parameter ε > 0.
Proof. Using Lemmas 2 and 3, we deduce the existence of a solution to (24) from its
unique-ness by means of a Fredholm alternative argument. Let us now consider the generalized CFIE given by
CΛeεJ= 1 2J+ n× KJ − ikZ0ΛeεT J = n× H inc+ eΛ εEincT , on Γ. (29) Proposition 3
The generalized CFIE integral equation (29) admits one and one solution for any frequency k > 0 and damping parameter ε > 0.
Proof. Let (a, b) ∈ TH−1/2(Γ)× TH−1/2(Γ). We can write
{CΛeεa, b}Γ=
1
2{a, b}Γ+{a, K(b × n)}Γ− ikZ0{a, T eΛεb}Γ. (30)
From the fact that the commutator AB − BA of two pseudodifferential operators A and B is
compact, we replace K(b × n) by Kb × n and T eΛεb by T eΛε(n× b) × n up to a compact
operator. Consequently, we obtain
{CΛeεa, b}Γ =
1
2{a, b}Γ+{a, Kb × n}Γ− ikZ0{a, T eΛε(n× b) × n}Γ
+ {a, Cb}Γ
= {a, (BΛeε+C)b}Γ,
with C a compact operator. The generalized CFIE operator CΛeε represents the transposed
operator of the BW generalized operator BΛeε up to a compact perturbation. Then, the
well-possedness of the integral equation (29) for any frequency k > 0 and damping parameter ε > 0 is a consequence of proposition 2.
Remark 1
In the case of non-smooth surfaces, the compactness of the integral operator K is no longer valid. In [8], Buffa and Hiptmair have recently proposed a regularization technique to overcome this default and to construct a coercive CFIE. However, this CFIE represents a compact perturbation of the EFIE. Consequently, its condition number deteriorates at high frequencies on refined meshes. Then, the construction of both well-posed and well-conditioned integral equations remains an open issue for non-smooth surfaces.
6
The case of the sphere
In the case of a spherical scatterer, the integral operators can be expressed explicitly in a diagonal form. This property allows a study of the behavior of the spectrum relative to both the usual and generalized integral equations. In particular, an asymptotic analysis can be developed according to the different zones: hyperbolic, elliptic and glancing. Finally, some computations can be done, first, to indicate the eigenvalue cluster or not around some accumulation points, and next, to test the convergence of the iterative solving of the integral equations by the GMRES method.
6.1
Determination of the eigenvalues and some of their asymptotic
properties
Let us consider the sphere SR of radius R. We denote by Ymn the spherical harmonics of order
m, with m≥ 1 and |n| ≤ m. The family (∇SRY
n
m, n× ∇SRY
n
m) forms a basis of eigenvectors
for the integral operators T and K [22] with an exact expression of the related eigenvalues. Proposition 4
Let Yn
m be a spherical harmonic of order m. The eigenvalues of the operators T and K are
given by T∇SRY n m = i kξ (1)0 m (kR)ψ 0 m(kR)∇SRY n m= t+m∇SRY n m T (n× ∇SRY n m) = i kξ (1) m (kR)ψm(kR)(n× ∇SRY n m) = t−m(n× ∇SRY n m) (31) K∇SRY n m = i 2(ξ (1)0 m (kR)ψm(kR) + ξm(1)(kR)ψ 0 m(kR))(n× ∇SRY n m) = km(n× ∇SRY n m) K(n× ∇SRY n m) = km∇SRY n m (32)
for m≥ 1 and |n| ≤ m. The functions ξm(1)(x) = xh(1)m(x) and ψm(x) = xjm(x) stand for the
Ricatti-Hankel and the Ricatti-Bessel functions of order m respectively. The functions h(1)m and
jmare the spherical Hankel and Bessel functions of order m respectively.
We also make use of the following important preliminary result [26, 33]. Theorem 3
The family (∇SRY
n
m, n× ∇SRY
n
m) is a basis of eigenvectors for the vectorial Laplace-Beltrami
operator LSR. Moreover, we have
−LSR∇SRY n m= m(m + 1) R2 ∇SRY n m= λm∇SRY n m −LSR(n× ∇SRY n m) = m(m + 1) R2 (n× ∇SRY n m) = λm(n× ∇SRY n m) (33) and Z SR |∇SRY n m|2dσ = Z SR |n × ∇SRY n m|2dσ = m(m + 1) R2 .
As a consequence, the eigenvalues of the operator eΛεcan be obtained e Λε∇SRY n m= Z0−1(1− λm k2 ε ) −1/2 ∇SRY n m e Λε(n× ∇SRY n m) = Z0−1(1− λm k2 ε ) 1/2 (n× ∇SRY n m) (34) because curlSR∇SRY n
m = 0 and curlSRcurlSR(n× ∇SRY
n
m) = ∆SR∇SRY
n
m× n respectively.
Thanks to Proposition (4) and relations (34), the eigenvalues of the usual and generalized BW integral operators are given by
B∇SRY n m = ( 1 2+ km− ikZ0t − m)∇SRY n m= b+m∇SRY n m B(n × ∇SRY n m) = ( 1 2− km− ikZ0t + m)(n× ∇SRY n m) = b−m∇SRY n m) and BΛeε∇SRY n m = ( 1 2+ km− ikt − m(1− λm k2 ε ) 1/2 )∇SRY n m= b+ε,m∇SRY n m BΛeε(n× ∇SRY n m) = ( 1 2− km− ikt + m(1− λm k2 ε ) −1/2 )(n× ∇SRY n m) = b − ε,m(n× ∇SRY n m).
For the usual and generalized CFIE operators, we have C∇SRY n m = ((1− α)( 1 2− km)− ikZ0αt + m)∇SRY n m= c+m∇SRY n m C(n × ∇SRY n m) = ((1− α)( 1 2+ km)− ikZ0αt − m)(n× ∇SRY n m) = c − m(n× ∇SRY n m) and CeΛε∇SRY n m = b−ε,m∇SRY n m= c+ε,m∇SRY n m CeΛε(n× ∇SRY n m) = b+ε,m(n× ∇SRY n m) = c − ε,m(n× ∇SRY n m).
We propose to analyse the eigenvalue behavior of the usual and generalized integral oper-ators. We consider three study zones: the elliptic zone (m kR) of evanescent modes, the hyperbolic zone (m kR) of propagative modes and the transition zone (m ≈ kR) of surface rays. First, we are interested in the asymptotic behavior of the eigenvalues in the elliptic zone. To this end, we use asymptotic expansions of the spherical Bessel and Hankel functions for large m [1].
Proposition 5
The eigenvalue behavior of the usual BW and CFIE integral operators is given in the elliptic
zone associated with evanescent modes, that is m kR, by
b+ m = 1 2+ ( 3 8− ikZ0 R 2)(m + 1 2) −1 +O( 1 m3) b− m = 1 2+ i Z0 2kR(m + 1 2) +O( 1 m), and c+ m = 1− α 2 + i αZ0 2kR(m + 1 2) +O( 1 m) c− m = 1− α 2 + ( 3(1− α) 8 − ikZ0α R 2)(m + 1 2) −1 +O( 1 m3).
Proposition 6
The eigenvalue behavior of the generalized BW and CFIE integral operators is given in the
elliptic zone associated with evanescent modes, that is m kR, by
b+ ε,m = c−ε,m= 1 2+ k 2kε+O( 1 m)' 1 + O( 1 m) b− ε,m = c+ε,m= 1 2+ kε 2k +O( 1 m)' 1 + O( 1 m).
These results show the clustering of the spectrum of the generalized integral operators at the accumulation point 1. On the contrary, the eigenvalues of the usual operators behave like m. This is characteristic of first-kind Fredholm equations. This property induces the flaw relative to a mesh refinement.
Now, let us observe the behavior of the eigenvalues in the hyperbolic zone. Asymptotic ex-pansions of the spherical Bessel and Hankel functions for large arguments [1] yield the following results expressed in the next proposition.
Proposition 7
The eigenvalue behavior of the usual BW and CFIE integral operators is given in the hyperbolic
zone associated with propagative modes, that is m kR, by
b+ m = 1 2− 1 2e 2i(kR+θm)+ Z 0cos(kR + θm)ei(kR+θm)+O( 1 kR) b− m = 1 2+ 1 2e 2i(kR+θm) − iZ0sin(kR + θm)ei(kR+θm)+O( 1 kR), and c+ m = 1− α 2 + 1− α 2 e 2i(kR+θm)
− iαZ0sin(kR + θm)ei(kR+θm)+O(
1 kR) c− m = 1− α 2 − 1− α 2 e 2i(kR+θm)+ αZ 0cos(kR + θm)ei(kR+θm)+O( 1 kR), with θm=−(m + π)/2. Proposition 8
The eigenvalue behavior of the generalized BW and CFIE integral operators is given in the
hyperbolic zone associated with propagative modes, that is m kR, by
b+ ε,m = c−ε,m= 1 +O( 1 kR) b− ε,m = c+ε,m= 1 +O( 1 kR).
In this zone again, we can observe that the spectrum of the generalized integral operators is well clustered around point 1. Concerning the transition zone, we do not possess sufficiently precise asymptotic expansions of the spherical Bessel and Hankel functions to obtain good estimates of the eigenvalue behavior. But, as we can see on Fig. 1 and 2, the eigenvalues associated with surface modes form a loop around point 1 for the generalized operators. Remark 2
In three-dimensional acoustics [4, 17], the construction of a generalized BW integral equation of second-kind is proposed following a similar method. We plot on Fig.3 the eigenvalue distri-bution of both the usual and generalized BW integral operators for a spherical geometry in the
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 Real part Imaginary part Usual BW formulation Generalized BW formulation 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 Real part Imaginary part Usual BW formulation Generalized BW formulation
Figure 1: Eigenvalues of the usual (Z0= 1) and generalized BW operators associated with the
eigenvectors ∇S1Y
n
m(left) and n × ∇S1Y
n
m(right) for k = 25 and mmax= 4(kR + 5log(kR + π)).
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Real part Imaginary part Usual CFIE (α=0.2) Generalized CFIE 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Real part Imaginary part Usual CFIE (α=0.2) Generalized CFIE
Figure 2: Eigenvalues of the usual (Z0= 1) and generalized CFIE operators associated with the
eigenvectors ∇S1Y
n
m(left) and n × ∇S1Y
n
m(right) for k = 25 and mmax= 4(kR + 5log(kR + π)).
case of Dirichlet (left) and Neumann (right) conditions. According to the Helmholtz
decompo-sition, the behavior of the eigenvalues associated with the eigenvectors∇S1Y
n
m (cf. Fig. 1 left)
corresponds to the Dirichlet acoustic case while the behavior of the eigenvalues associated with
the eigenvectors n× ∇S1Y
n
m coincides with the Neumann acoustic problem (cf. Fig. 1 right).
The same conclusions are observed for the CFIE.
To complete the comparison between usual and generalized integral formulations, we plot on Fig. 4 the behavior of the condition number of the different operators relatively to the frequency and to the number of vectorial spherical harmonics which describes the effect of a mesh refinement. The condition number of the generalized operators not suffers from an increase of these two parameters. Unlike the generalized formulations, the usual ones are not some second-kind integral equations. Therefore, their condition number depends linearly on the number of spherical harmonics. This is linked to the dispersion of the eigenvalues in the elliptic zone. It also shows a dependence on the frequency increase.
0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 Real part Imaginary part Usual BW formulation Generalized BW fomulation 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 −1 0 1 2 3 4 5 6 Real part Imaginary part Usual BW formulation Generalized BW fomulation
Figure 3: Eigenvalues of the usual and generalized BW operators for the 3D acoustic Dirichlet (left) and Neumann (right) problems for k = 25 and a density of discretization points per wavelength equal to 24.
6.2
Convergence of the iterative solver GMRES
We finally address the behavior of the solver GMRES for the iterative solution of the usual (16) and generalized CFIE (29). These integral equations can be diagonalized in the basis
of the vectorial spherical harmonics (∇SRY
1
m, n× ∇SRY
1
m). Working with a finite dimensional
approximation of these equations induces that only a finite number of modes mmaxis taken into
account. Increasing this parameter mmax can be considered as a model of a mesh refinement.
The approximate surface current J is given by
J ≈ mXmax m=1 (ejm+∇SRY 1 m+ ejm−(n× ∇SRY 1 m)), where (ej+
m, ejm−) is solution to the diagonal system of size 2mmaxassociated with the usual CFIE
((1− α)(12− km)− ikZ0αt+m)ej+m = (1− α)(hinc) + m+ α(e inc)+ m 1≤ m ≤ mmax. ((1− α)(12+ km)− ikZ0αt−m)ej − m = (1− α)(hinc) − m+ α(e inc)− m (35)
For the generalized CFIE, the unknown J is approximated by
J ≈ mXmax m=1 (ejm,ε+ ∇SRY 1 m+ ejm,ε− (n× ∇SRY 1 m)), with (1 2− km− ik(1 − λm k2 ε ) −1/2 t+m)ejm,ε+ = (hinc) + m+ Z −1 0 (1− λm k2 ε ) −1/2 (einc)+m 1≤ m ≤ mmax. (1 2+ km− ik(1 − λm k2 ε ) 1/2 t−m)ej − m,ε = (hinc) − m+ Z0−1(1− λm k2 ε ) 1/2 (einc)−m (36)
5 10 15 20 25 30 1 2 3 4 5 6 7 8 Wave number k Condition number Usual BW formulation Usual CFIE (α=0.2) Generalized BW fomulation Generalized CFIE 20 40 60 80 100 120 140 160 180 200 0 5 10 15 20 25 30 35 40 45
Number of spherical harmonics
Condition number
Usual BW formulation Usual CFIE (α=0.2) Generalized BW fomulation Generalized CFIE
Figure 4: Evolution of the condition number of the usual (Z0 = 1) and generalized BW and
CFIE operators relatively to the wavenumber k for mmax = 2(kR + 5log(kR + π)) (left) and
relatively to the number of spherical harmonics mmax for k = 5 (right).
Concerning the right-hand side of the previous systems, one has
n× Hinc ≈ mXmax m=1 ((hinc)+m∇SRY 1 m+ (hinc) − m(n× ∇SRY 1 m)) ≈ mXmax m=1 1 k(−i) mp 2π(2m + 1)(ψm(kR)∇SRY 1 m− iψ 0 m(kR)(n× ∇SRY 1 m)) EincT ≈ mXmax m=1 ((einc)+ m∇SRY 1 m+ (einc) − m(n× ∇SRY 1 m)) ≈ mXmax m=1 1 k(−i) mp 2π(2m + 1)(iψ0m(kR)∇SRY 1 m− ψm(kR)(n× ∇SRY 1 m)).
The two diagonal systems (35) and (36) are solved with the GMRES iterative solver without
restart and a tolerance tol = 10−6 on the reduction of the initial residual. We plot on Fig.5
the history of the residual reduction relatively to the number of iterations needed to reach the
convergence. The total number of vectorial spherical harmonics is equal to mmax = 4(kR +
5log(kR + π)) and Z0 = 1. The generalized CFIE provides the best convergence. They lead
to a significant reduction of the number of iterations compared to the usual formulations. The well clustering of their spectrum at point 1 (cf. Fig.2) explains this superlinear convergence [25].
Remark 3
The parameter ε > 0 allows to cluster the eigenvalues corresponding to the surface modes. Taking ε > 0 is crucial. The convergence is slightly slowed for ε = 0. Two to four supplementary
iterations are required compared to the optimal choice εopt= 0.4k1/3R−2/3. Let us note again
that, here, the main advantage of taking ε > 0 is to construct well-posed generalized integral equations.
Remark 4
In this paper, we do not deal with the numerical implementation of the generalized BW and CFIE equations. However, we can give some elements towards this issue. The Helmholtz
0 5 10 15 20 25 30 −6 −5 −4 −3 −2 −1 0 Number of iterations Residual reduction Usual CFIE (α=0.2) Generalized CFIE
Figure 5: History of the residual norm according to the number of iterations for the unit sphere,
k = 30, and mmax= 4(kR + 5log(kR + π)) (right).
decomposition gives an orthogonal decomposition of the space. It would be certainly interesting to use mixed methods which can take advantage of this decomposition.
Concerning the treatment of the regularizing operator eΛε and the approximation of the
square-root operator, the idea is to consider efficient complex Padé approximants with rotation of the usual branchcut [24]. In acoustics [5], it was shown that this localization of the square-root operator can be obtained by the fast iterative solution of a few sparse linear systems by GMRES preconditioned by an incomplete LU decomposition. Consequently, the consideration
of eΛεwill induce a low numerical extra cost only.
7
Conclusion
Extension and justification of the Brakhage-Werner and CFIE boundary integral equations for the scattering of time-harmonic electromagnetic waves have been derived. The proposed new integral equations are well-posed and well-conditioned. The incorporation of a suitable regularizing operator plays an essential role in improving the efficiency of these formulations relatively to an iterative solution. Numerical experiments have shown that the condition number and the eigenvalue clustering are independent of the frequency and of the mesh refinement for a spherical geometry. The presented numerical and analytical results constitute an encouraging preliminary step towards the construction of very efficient solvers for electromagnetic scattering problems.
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