Some recent applications of substructuring and

domain decomposition techniques to radiation

and scattering of time-harmonic electromagnetic

waves

N. Balin∗, A. Bendali†, M. Fares, F. Millot, N. Zerbib‡

CERFACS, 42 Avenue Gaspard Coriolis,

31057 Toulouse CEDEX 1, France

November 2, 2005

Abstract

After a rapid survey on the domain decomposition methods, and, more

particularly on their application to problems relative to radiation and

scattering of time-harmonic waves, we describe some contributions of the

authors in this context. Successively are presented a subtructuring pro-

cedure to deal with the scattering of an electromagnetic wave by a large

structure in which is residing an electrically deep cavity, a treatment of

a boundary integral equation by an overlapping domain decomposition

technique and the use of an overlapping domain decomposition method

to extend the so-called adaptive absorbing boundary condition.

1 Domain Decomposition Methods and their use

in radiation and scattering of electromagnetic

waves

Domain Decomposition Methods (DDMs) constitute flexible and powerful tech-niques to numerically solve Partial Differential Equations (PDEs). As clearlydescribed in [42], they can be used as a “process of distributing data from acomputational model among the processors in a distributed memory computer”,or for “the separation of the physical domain into regions” dealt with using dif-ferent and adapted solving methods, and finally as “a process of subdividing thesolution of a large linear system [resulting from the discretization of the PDE

∗N. Balin is with MBDA-France Chatillon (France) and CERFACS†A. Bendali is with UMR˜5640, Laboratoire MIP, INSA-CNRS-UPS and CERFACS‡N. Zerbib is with CNES France & CERFACS

1

on the entire domain] into smaller problems whose solutions can be used as apreconditionner or a solver [to that effect]”.

One generally can distinguish three kinds of DDMs:

• Substructuring methods. After dividing the solution domain in severalpieces without overlap and, in a general meaning, removing the degreesof freedom (dof) internal to each piece through a Schur complement tech-nique, one hence is led to solve a so-called coarse problem of a smaller size(e.g., [39] [22] etc.).

• Nonoverlapping methods. Roughly speaking, substructuring and non-overlapping DDMs are closely related. When the coarse problem is stilltoo large to be handled by a direct method (e.g., [22]), one has to resort toan iterative method. The derivation of efficient iterative procedures, whichis precisely the task of nonoverlapping DDMs, can generally be obtainedonly through a thorough inspection of the physical phenomenon and theproperties of its mathematical model.

• Overlapping methods. In spirit, the overlapping DDMs, proceed in thesame way than the nonoverlapping ones using instead a decomposition ofthe computational domain in overlapping pieces. Following the way thesolution procedure is performed, the iterative process is called a Schwarzmultiplicative or additive alternating method. Because of their greaterflexibility for writing the matching conditions linking the functions ob-tained by solving local problems, the overlapping methods are generallyeasier to carry out. This point will be illustrated below when such tech-niques will be adapted in order to solve some boundary integral equations.

More or less, the guiding principle of a domain decomposition method re-lies on a “static condensation” in an extended sense. This explains why thesemethods were developed mostly in the frame of strongly coercive PDEs likethose involved in structural mechanic problems: to limit the references to somerepresentative or recent developments, we can quote the overlapping alternat-ing Schwarz method [31] [33] [42], the broad class of nonoverlapping methodsbased on a Neumann-Neumann scheme like Finite Element Tearing and Inter-connecting Dual Primal (FETI-DP) [25] and Balancing Domain Decompositionon Constraints (BDDC) [22].

Taking their root in Thevenin’s theorem (e.g., [41]), techniques similar tosubstructuring methods were also used for wave propagation problems in acous-tics and electromagnetism. In the engineering terminology generally used inthis framework, the Neumann-Dirichlet operator or its inverse, which are in-volved in substructuring methods in a basic way, are often called impedance oradmittance operators (e.g., [8] or [35] limiting the references to some techniquesthat were applied to the scattering problems considered below). However, theweak coerciveness properties of the PDEs, governing the phenomena relativeto time-harmonic waves, prevent such operators to be well-defined at some ex-ceptional frequencies at which are occurring the interior resonances of the piece

2

of the computational domain being considered. This perhaps explains whyneither DDMs were broadly used nor a general theory was available for theirconvergence in this context until the break-through done by Despres [21] [20].Roughly speaking, one of his main ideas was to express the matching conditionsequivalently in terms of an absorbing boundary condition instead of the broadDirichlet and Neumann conditions of usual DDMs. In fact, such a way to writeequivalently the matching conditions to construct a nonoverlapping DDM wasearly introduced by P.-L. Lions for the Laplace equation as a limiting case ofan overlapping DDM [32]. In his thesis, Despres has also shown that the samekind of boundary conditions, when used in the context of an overlapping DDM,greatly improves the convergence of the iterative process comparatively to theusual Dirichlet condition. Several works were then devoted to the improve-ment of Despres’ approach (e.g., [18] [26] [36] etc.) and its extension to variousproblems and discretization schemes (e.g., [43] [27] [24] [12] etc.).

However in spite of all this progress, nonoverlapping DDMs have faced a dif-ficulty which remained blocking until recently. The matching conditions cannotbe set in a weak variational form unless resorting to special schemes like themixed formulations of second order elliptic boundary-value problems as it wascarried out in the early work done by Despres [21]. The counterpart for usualfinite element schemes of this drawback is that the usual equations for the coarseproblem cannot be set at the so-called cross-points, that is, points supportinga dof shared by more than two subdomains. The remedy, as this was done in[25] with a FETI-DP method and in [22] for BDDC in the context of PDE withstrong coerciveness properties and in [11] for the Helmholtz equation, is to keepthe strong continuity requirements at cross-points for both the trial functionsand the test ones and to use a suitable Schur complement technique to definethe coarse problem. Even if the stability and the convergence properties of thisapproach were established for the scalar case only [11], mainly because of thehigh level of technicalities related to the coerciveness properties of Maxwell’sequations, its extension to time-harmonic electromagnetic waves in the frame-work of a discretization by Nedelec’s edge elements is straightforward, at leastfor algorithmic purposes.

This paper reports some contributions of the authors in the application andthe development of DDMs for solving problems related to radiation and scatter-ing of time-harmonic electromagnetic waves. More specifically, the first sectionreports a special substructuring technique to efficiently solve a scattering prob-lem involving an perfectly conducting obstacle in which is residing an electri-cally deep cavity by means of the Fast Multipole Method (FMM). The secondsection is devoted to the description of an adaptation of the additive Schwarzalternating method to the so-called Electric Field Integral Equation (EFIE).This adaptation is used either as a fast solver or as a way to couple an exactsolving in some part of the boundary of the obstacle and an asymptotic high-frequency method elsewhere. Finally, the third section presents an extension ofthe localized finite element method of Hazard and Lenoir [28] whose iterativesolution can be interpreted as a special overlapping DDM.

3

Figure 1: Schematic view of a deep cavity residing in a large structure

2 Application of a substructuring method to the

electromagnetic scattering by an electrically

deep cavity

After giving the statement of the problem and making some remarks on thedifficulties arising when undertaking its solution by means of the usual methods,we describe how we have dealt with this task and give the results of somenumerical experiments to illustrate the efficiency of the approach.

2.1 Statement of the problem

Consider an arbitrarily-shaped electrically deep cavity residing in a large struc-ture as schematically depicted in Fig. 1.

Let be given an incident plane wave (Einc,Hinc) such that∣

∣Einc∣

∣ = 1. Re-

call that∣

∣Einc∣

∣ indicates the Euclidian norm of a vector with three Cartesiancomplex components. Usual notation will be used without further comments.

The problem is to compute the bistatic Radar Cross Section (RCS) of thisobstacle. It can be expressed in the direction (cos ϕ sin θ, sin ϕ sin θ, cos θ) fromthe following relationships

s(ϕ, θ) := 10 log10(4π |A(θ, ϕ)|2)

where A(θ, ϕ) is the radiation function of the electromagnetic field, which canbe obtained by solving the following boundary-value problem

∇× E − ikZH = 0∇× H + ikZ−1E = 0

in Ω

n × (E × n) = 0 on ΓE(x) = Einc(x) + (eikr/r)A(ϕ, θ) + O(1/r2)

(1)

in which k is the wavenumber and Z ≈ 120π is the intrinsic impedance of theelectromagnetic waves in the vacuum in SI units.

4

Figure 2: Decomposition of the cavity in nonoverlapping sectional domains

It is well-known that this kind of cavity, residing in perfectly conductingstructure, like for instance an inlet in an aircraft, can significantly contributesto the overall RCS. Standard full-waves methods, like Boundary Element Meth-ods (BEM), or Finite Element Methods (FEM) cannot be used, as they were,because of several difficulties:

• Direct solvers quickly become unusable because of the prohibitory memorystorage and the too costly CPU time, thus required.

• Internal resonances inside the cavity severely slow down the convergenceof an iterative solver coupled with a fast computation of the matrix-vectorproduct by means of a FMM despite the use of the highly performing SPAIpreconditionner (see [3] for details on the definition and the constructionof this kind of preconditionner).

• The geometry of the cavity is well adapted to the use of a FEM and theuse of a frontal forward substitution to remove all the internal dofs [29].However, because of the weak coerciveness of the PDEs related to timeharmonic waves, already mentioned above, the stability of the algorithm isnot guaranteed nor even that it does not breakdown. Moreover, the depthand the narrowness of the cavity prevent the use of low order FEMs oth-erwise the large amount of dispersion errors then makes any computationof the RCS meaningless.

2.2 The substructuring process

A special nonoverlapping decomposition Ωpp=N+1p=1 of the computational do-

main Ω is first carried out. As indicated in Fig. 2, Ωp (p = 1, . . . , N) is asectional domain of the cavity, ΩN+1 is the domain exterior to both the cavityand the structure, and Σp (p = 1, . . . , N) is a sectional surface separating Ωp

from Ωp+1.

5

Denoting by np (p = 1, . . . , N + 1) the unit normal to the boundary ∂Ωp,outwardly directed to Ωp, we state the equivalent formulation of problem (1) asa transmission problem, thus performing a task which can be considered as afirst step common to every substructuring or nonoverlapping DDM,

∇× Ep − ikZHp = 0∇× Hp + ikZ−1Ep = 0

in Ωp

np × (Ep × np) = 0 on ΓEN+1(x) = Einc(x) + (eikr/r)A(ϕ, θ) + O(1/r2),

(2)

with the matching conditions expressed in a way more adapted to the substruc-turing method, we have in view,

Jp + Jp+1 = 0 and Mp + Mp+1 = 0 on Σp (p = 1, , . . . , N) (3)

where Jp = np × Hp|∂Ωpand Mp = Ep × np|∂Ωp

are the equivalent currentsrelative to Ωp according to the engineering terminology. It is worth noting thatthe boundary condition implies Mp = 0 on Γ.

Usual substructuring techniques, including a Finite Element modelling forthe interior of the cavity, determine (an approximation of) the impedance oper-ator np×Mp = ZpJp which plays the role of the Neumann to Dirichlet operatorfor the equations relative to Ωp in problem (2) to carry out the process of re-moving the internal dofs (e.g., [35] [8] etc.). However, observe that the verywriting of this relation amounts to assume that the following boundary-valueproblem is well-posed

∇× Ep − ikZHp = 0∇× Hp + ikZ−1Ep = 0

in Ωp, np × (Ep × np) = 0 on Γ

np × Hp = Jp on Σp−1 ∪ Σp,(4)

with an obvious adaptation for p = 1. It is sufficient to consider a suitablesectional domain of a waveguide in which is standing a correctly chosen system ofstationary waves to check that this property may be false in some configurations.

Two of the authors [5] [6] have designed a procedure that makes it possibleto remove the internal dofs in a stable way and brings back the solution ofthe scattering problem to a well-posed EFIE-like integral equation set on theboundary ∂ΩN+1 of the domain exterior to the cavity. However, the latterequation is suitable for a direct solving only but not for an iterative solutionwhich is the only way to proceed when using the FMM. We now give someindications on how this approach was adapted in [7] to postpone a possiblebreakdown after the elimination of all the interior dofs with the possibility ofchecking whether it is occurring or not before starting the solving process on∂ΩN+1. In this way, one can use the so-called Combined Field Integral Equation(CFIE) whose suitability to iterative methods is well-established. The maintrick is to use the Rumsey reaction concept to equivalently write the matchingconditions at the sectional surfaces Σp (p = 1, . . . , N − 1) at the interior of thecavity and the link between JN and H on ΣN in the form

∑Np=1

∫

∂Ωp

(

Ep · J′

p − Hp · M′

p

)

dS =

∫

ΣN

nN × JN · M′

N dS (5)

6

Figure 3: Structure of the linear system relative to the Rumsey reactions.

for all system of test currents(

J′

p,M′

p

)p=N

p=1satisfying the same relations than

the unknown currents, that is, (3) and M′

p = 0 on Γ, and furthermore such thatJ′

N |ΣN= 0. Likewise, the equations at the interior of the cavity hold by means

of the Stratton-Chu (e.g., [19]) integral representations of Ep|∂Ωpand Hp|∂Ωp

Ep = 12n×Mp + ikZ TpJp +KpMp, Hp = − 1

2n×Jp + ikZ−1 TpMp −KpMp,

Tp and Kp being the surface integral operators

TpJp(x) =1

k2∇S

∫

∂Ωp

G(x, y)∇S · Jp(y) dS(y) + ΠS

∫

∂Ωp

G(x, y)Jp(y) dS(y),

KpMp(x) = ΠS

∫

∂Ωp

∇yG(x, y) × Jp(y) dS(y)

where ∇S , ∇S · and ΠS are respectively the surface gradient and divergence andthe projection onto the tangent plane. Meshing the surfaces in triangles andapproximating the currents by the usual divergence-conforming edge elements,we are led to the system, schematically depicted in Fig. 3, where each blockcorresponds to the matrix associated to the bilinear form which expresses theRumsey reaction in a sectional domain

∫

∂Ωp

(

Ep · J′

p − Hp · M′

p

)

dS =[

X ′

p

]T[Rp] [Xp]

and [Xp] and[

X ′

p

]

are column-wise vectors of complex numbers collecting thedofs of the unknown and the test equivalent currents of Ωp respectively. Wehave also denoted by [NN ] the matrix associated to the bilinear form

[X ′

N ]T

[NN ] [XN ] =

∫

ΣN

nN × JN · M′

N dS.

Introducing successively the sectional domains Ω1, Ω2, . . . , Ωp, it is possibleto remove at each step all the dofs that are not on Σp, with the guaranty that

7

the elimination process cannot breakdown and that it can be done in a stableway. (See [5] [6] for a proof in the 2D case. The same proof applies for the 3Dcase, at least in a formalist way.) As a result, system (5) can be put in theform

[A] [JN ] + [B] [MN ] = 0 (6)

where [JN ] and [MN ] are respectively the dofs of JN |ΣNand MN |ΣN

and [B] isa square matrix.

It is only after this step that the algorithm can breakdown: the impedancematrix of the cavity can be determined in a stable way if and only if matrix[B] in (6) can be inverted in the same way. Since this matrix is moderatelysized, this property can be controlled by making use of a direct solver for densematrices, such as for instance the public domain library Lapack, which, whilecomputing the matrix [Zc] = − [B]

−1[A], also yields an approximate value for

the condition number of [B].Once [Zc] obtained, MN+1 can be expressed in terms of JN+1. One can

then reduce the scattering problem to a CFIE posed on ∂ΩN+1. More pre-cisely, one obtains a perturbation of the usual CFIE for a perfectly conductingboundary condition where the electric and magnetic fields are perturbed by thecontributions of a distribution of magnetic currents on the opening ΣN of thecavity. This makes it possible to solve this equation in a very efficient way byusing a SPAI preconditionner, constructed by assuming that ∂ΩN+1 encloses aperfectly conducting scatterer.

2.3 Some numerical experiments

The first case is destined to compare the performances of this method with abroad approach based on the use of a CFIE taking no care of the particulargeometry of the cavity. To bring out the effect of the cavity, we have considereda box with thick walls and opened at one side as depicted by the first figure inFig. 4 with a variable depth for the cavity. The plot on the second figure inFig. 4 clearly shows that the number of iterations grows almost linearly with thedepth of the cavity in the case of the direct approach while it remains constantfor the method described here.

The second example is related to a cavity COBRA within a fuselage, depictedin Fig. 5, which was designed by MBDA-France as a test-case for the workshopEM-JINA 2004.

Because of the thin wall of the cavity near its opening, only the EFIE canbe used in the context of a direct approach. In Tab. 1, we compares the CPUtime needed for a solution by the direct approach with a usual Krylov iterativesolver, next with a Krylov solver specially customized to handle such kinds ofproblems which can be considered as extreme cases for an iterative method, andfinally with the present procedure.

This method is more than 4 times faster than the most powerful purelyalgebraic process.

8

(a) The box cavity with thick walls

0 5 10 15Depth of the cavity in wavelength

0

25

50

75

100

125

150

Num

ber

of it

erat

ions

Substructuring MethodDirect Solving

(b) Number of iterations versus the depth of thecavity

Figure 4: Behavior of the GMRES iterative solver with and without using thesubstructuring process

Solver Number ofdofs

CPU Time(Cavity)

CPU Time(Exterior)

Total CPUTime

GMRES 216,605 - - 309 hFGMRES 216,605 - - 89 h

This Method 223,907 1 h 17 h 18 h

Table 1: Behavior of various solving procedures for the RCS computation of aCOBRA cavity within a fuselage

9

Figure 5: COBRA cavity within a fuselage

3 Schwarz Additive Methods and Electromag-

netic Boundary Integral Equations

In this section, we show how the Schwarz additive DDMs can be adapted todevelop

• efficient iterative algorithms making it possible to solve largely sized linearsystems obtained from the discretization of the boundary integral equa-tions related to the radiation or the scattering of a time-harmonic electro-magnetic waves,

• efficient methods for coupling alternative methods used in separate zones.

Since this method uses an overlapping decomposition of the boundary in-stead of the domain of the PDE as the usual DDMs, we will designate it as aboundary decomposition method.

3.1 The overlapping boundary decomposition method

For clarity, assume that the boundary integral equation is the EFIE, either in2D or 3D, which, once suitably discretized (cf. e.g., [38]), amounts to solvingthe following dense linear system

[Z] [X] = [U ] . (7)

The method, we develop, can be viewed as an extension of a method introducedby Balabane and Tirel [4] which can be described as follows. Assume that theboundary Γ has two distinct components Γ1 and Γ2. In this case, distinguishingthe dofs on Γ1 and on Γ2 yields a natural partitioning of this system in the form

[

Z11 Z12

Z21 Z22

] [

X1

X2

]

=

[

U1

U2

]

.

10

These authors proved then that the bock Jacobi iterative method (e.g., [40])converges if Γ1 and Γ2 are sufficiently far away each of the other. As this will beshown below, when Γ is connected and admits Γ1 and Γ2 as a nonoverlappingdecomposition, this partioning yields a poorly convergent algorithm even if thefixed point problem associated to the Jacobi iteration is solved by a Krylovmethod instead of a simple successive approximations procedure.

To improve the convergence properties of this method, we extend it as fol-lows. For clarity, we limit ourselves to a decomposition of Γ in two overlappingcomponents Γ1 and Γ2 only. The extension to more general overlapping decom-positions is straightforward.

We first make a geometrical distribution of the dofs as follows. For the 2Dcase, the dofs can be mapped one-to-one to the nodes in a natural way. For the3D one, the same property is also holding if, for each dof, we associate the mid-point of the edge to which it is attached. Let [I] be the identity matrix of thesame size than [Z]. We define a boolean matrix [Ii] (i = 1, 2) by dropping from[I] every row whose number corresponds to a node that is not in the interior ofΓi. This matrix makes it possible to extract the dofs at the interior of Γi from[X] through the matrix-vector product [Xi] = [Ii] [X].

To carry on the construction of the DDM, we now introduce a partition ofunity α1 and α2 associated to this decomposition of Γ. Recall that this meansthat α1 and α2 are positive in the interior, equal to 0 in the exterior of respec-tively Γ1 and Γ2 and that they satisfy α1 + α2 = 1 on Γ. By observing that avector [X] of dofs can be considered as a function defined on the nodes throughthe relation X(x) = [X]`(x) where `(x) is the number the node x, we can further-

more define a diagonal matrix [αi] by the relation (αiXi)(m) = ([αi] [Xi])`(m).In this way, the partition of unity yields an additive decomposition of any vector[X] of dofs in the following manner

[X] = [I1]T

[α1] [I1] [X] + [I2]T

[α2] [I2] [X] . (8)

Clearly, using several times (8), we can now equivalently write system (7) suc-cessively in the following forms

[α1] [I1] [Z] [X] = [U1] , [U1] := [α1] [I1] [U ][α2] [I2] [Z] [X] = [U2] , [U2] := [α2] [I2] [U ]

[I1] [X] + [Z11]−1

[Z12] [I2] [X] = [Z11]−1

[U1]

[Z22]−1

[Z21] [I1] [X] + [I2] [X] = [Z22]−1

[U2]

(

[I] + [I1]T

[α1] [Z11]−1

[Z12] [I2] + [I2]T

[α2] [Z22]−1

[Z21] [I1])

[X]

= [I1]T

[α1] [Z11]−1

[U1] + [I2]T

[α2] [Z22]−1

[U2]

with [Zij ] = [αi] [Ii] [Z] [Ij ]T

[αj ].This last system is solved by a Krylov method (e.g., [40]). Observe that the

main cost of a matrix-vector product is the computation of [Z11]−1

[Z12] [I2] [X]

11

and [Z22]−1

[Z21] [I1] [X] which can be interpreted as the determination of thecurrents relative to a scattering problem set on Γ1 and Γ2 respectively, for theEFIE considered here. Finally, note that this formulation can be extended inan obvious way for equations involving magnetic currents also which are equalto zero on Γ2.

3.2 Validation of the method

To validate the method, we consider the above example of a box cavity withthick walls. Completing (5) by the Rumsey reaction in the exterior domainΩN+1, we obtain a well-posed system of integral equations which, after theelimination of the interior dofs, is reduced to a linear system whose unknownsare the dofs of the electric currents on the opening and the exterior of the cavityand the magnetic ones on the opening of the cavity only. In Fig. 6 are depictedthe meshes used to remove the dofs at the interior to the cavity and the zonewhere the surface Γ1 at the opening of the cavity and the surface Γ2 related tothe exterior side of the cavity walls overlap each other.

Figure 6: The box cavity and the various meshes involved in the solution pro-cedure

In Fig. 7 are reported the history of the residual reduction during the itera-tions for several cases of overlapping. Case (a) corresponds to an overlapping sonarrow that the method reduces to a Krylov iterative solution of the full systemusing the corresponding block Jacobi algorithm as a preconditionner. As an-nounced above, the algorithm is then poorly convergent. Surprisingly enough,case (b) shows that putting the overlapping area far away from the zone wherecomplicate phenomena are occurring slows down the convergence. Cases (c) and

12

(d) show that convergence becomes rather independent of the overlapping zoneonce this one is correctly designed and positioned.

Figure 7: History decreasing of the residual for various overlapping area

The results of several tests reported in [5] show that the error on the RCSis less than 0.5 % as compared to a direct solving.

3.3 Overlapping boundary decomposition and hybridiza-

tion of solving procedures

Of course the above Boundary Decomposition Method (BDM) can be used asa solving process since instead of solving a large linear system, one has to in-vert only very moderately sized ones during the iterations. However, our mainmotivation, at the time of the consideration of such a technique, was a directhandling of the cavity while treating the exterior of the structure by means ofan asymptotic method. Assume that the equations related to Γ1 are dealt withthrough an exact solving and those related to Γ2 by means of an asymptotictechnique. Roughly speaking, this mainly amounts to replace the evaluationof [Z12] ([I2] [X]) and [Z22]

−1[Z21] ([I1] [X]) by an approximation by means of a

high frequency method, respectively [Z12]HF

[X2]HF

and(

[Z22]−1

[Z21])HF

. We

refer for details to Balin’s PhD thesis [5].In Fig. 8, we compare the RCS of the COBRA cavity within a fuselage

achieved using the hybrid approach with the one obtained by means of theEFIE solved by the FMM. The frequency was fixed at 10 GHz. In terms of therequired mesh, this amounts to consider that the length of the fuselage is 40wavelengths. The results clearly demonstrates the reliability of the approach.Moreover the hybrid method used only 20 % of the CPU time that was necessaryto perform the FMM solution. It is worth noting that the FMM is used herewithin its general domain of validity contrary to the hybrid method which cantackle much more higher frequency cases.

13

Observation direction (°)

RC

S(d

Bsm

)

0 10 20 30 40 50 60 70 80 90-25

-20

-15

-10

-5

0

5Fast Multipol MethodHybrid Method

Figure 8: Plots of the RCS computed by a Boundary Integral Equation solvedby the FMM and the present coupling method.

14

4 An extension of the adaptive absorbing bound-

ary condition method

The approach, presented in this section, was first devised and validated in the 2Dcase [9]. It started from the observation that the so-called Adaptive AbsorbingBoundary Condition (AABC) would result in an unnecessary large meshed zoneif the inhomogeneities are located in a small localized area.

The AABC was first introduced in [2] for scalar problems and extendedlater to time-harmonic electromagnetic waves by [34]. A good introduction tothis method can also be found in [16] and [29]. At first sight, it constitutes areally powerful method because of the following advantages:

• At convergence, the radiation condition is exactly satisfied as with a cou-pling of a FEM with a BEM.

• Contrary to the most usual ways for coupling FEM with BEM, neither theresult of the solving process can be corrupted by some spurious solutionsnor one has to use the singular integral operator n × TM (T is definedanalogously to Tp above) which gives rise to concentrated charges alongthe mesh edges for the test currents. We refer to [16] for a presentationof the various ways to perform an hybridisation of a FEM and a BEM. Amethod avoiding concentrated charges along the edges is devised in [44].

• Contrary also to a FEM-BEM coupling, only a sparse linear system hasto be solved at each iteration.

• Only the right-hand side of the linear system is changed from one iterationto another one.

• All the integrals involved in the formulation are not singular and thereforecan be determined by quadrature formulas only.

However, despite these very attractive properties, the above drawback wouldbecome prohibitive because of the dispersion errors inherently attached to aFEM approximation of a wave phenomenon and also for the “manpower” thatcan be spent to obtain a huge volume mesh instead of a surface one.

In fact, the AABC is nothing else but a solving process that consists of suc-cessive approximations of the solution to a fixed-point problem resulting from asuitable formulation coupling a FEM with a BEM. This formulation was indeedintroduced by Jami and Lenoir [30] for a scalar problem and extended later toMaxwell’s system by [28]. A good discussion of these developments and theirlink to the method of Liu and Jin are given in [37]. The AABC was next foundagain from the pioneering work of Jami and Lenoir by Ben Belgacem and hisco-workers [10] who interpreted it as an overlapping DDM and proved the con-vergence of the related successive approximation process for the coercive case ofthe Laplace equation. It is this interpretation which is used in [9] to reduce the

15

Figure 9: Schematic view of a large perfectly conducting structure on which isposed a piece of inhomogeneous dielectric

FEM discretization to a localized zone while keeping the main attractive prop-erties of the AABC. In what follows, we report the extension of this approachto the 3D case.

4.1 The basic DDM principle underlying the construction

of the iterative solving process

Let be given a large perfectly conducting structure on which is posed an inho-mogeneous dielectric, located in a localized zone, as schematically depicted inFig. 9.

We consider a radiation problem where the electromagnetic field results fromlocalized sources (Js,Ms) but the treatment of a scattering problem can be doneexactly along the same lines. Denoting by Ωext, the exterior of the perfectlyconducting metal, we are led to solve the following boundary-value problem

∇× E − ikZµH = −Ms

∇× H + ikZ−1εE = Jsin Ωext

n × (E × n) = 0 on ∂Ωext

E(x) = (eikr/r)A(ϕ, θ) + O(1/r2)

with ε and µ equal to 1 in the exterior of ΩD, the domain filled by possiblyinhomogeneous dielectric and ε, µ satisfying

<ε(x) ≥ 1, =ε(x) ≥ 0, <µ(x) ≥ 1, =µ(x) ≥ 0, for all x ∈ ΩD.

The sources (Js,Ms), producing the radiation, are assumed to have a boundedsupport which is furthermore assumed to be contained in the interior of ΩD forsimplicity.

The first domain in the overlapping DDM is all space but the zone thatis relevant for the solving process reduces to Ω∞ the part of the space free of

16

material. Let us introduce the equivalent currents in Ω∞ defined by

J = n × H|∂Ω∞ , M = E × n|∂Ω∞ .

The following problem

∇× E0 − ikZH0 = −M

∇× H0 + ikZ−1E0 = Jin R

3

E0(x) = Einc(x) + 1rA(ϕ, θ) + O(1/r2)

can be solved explicitly by the Stratton-Chu formula

E0(x) = Einc(x) + ikZ TJ(x) + KM(x),H0(x) = Hinc(x) + ikZ−1 TM(x) − KJ(x),

x /∈ ∂Ω∞, (9)

and givesE0|Ω∞ = E|Ω∞ , H0|Ω∞ = H|Ω∞ ,

and E0 and H0 equal to zero in the domain enclosed by ∂Ω∞.The second domain is defined by introducing a fictitious surface S cutting

∂Ω∞ as indicated in Fig. 10 on which is set the AABC. Denote by Ω the domainlimited partly by ∂Ωext and by S. The second problem involved in the DDM isset in such a way so that the equations relative to ΩD are also holding

∇× EΩ − ikZµHΩ = −Ms

∇× HΩ + ikZ−1εEΩ = Jsin Ω

n × (EΩ × n) = 0 on ∂Ωext

n × (EΩ × n) + Zηn × HΩ = n × (E0 × n) + Zηn × H0 on S,

where η is a given function, defined and positive on S, unless on a neighborhoodof ∂Ωext where it is equal to zero. It can be proved that (EΩ,HΩ) is equal tothe restriction of (E,H) to Ω [44] [14].

This formulation is typically characteristic of an overlapping DDM. Further-more here, in the zone ΩH where Ω∞ and Ω are overlapping, the radiationproblem is modelled equally well by a BEM and by a FEM.

4.2 Implementation of the method

There are two difficulties in the above extension comparatively to the originalmethod as it was described in [28] and in [34]:

• The FEM determines J and M only on the part Σ of ∂Ω∞ inside Ω.

• The term n×(E0 × n) is expressed by a hypersingular integral near ∂Ωext.

These difficulties can be overcome in two steps:

• Set an EFIE in the part Γ of ∂Ω∞ outside the zone Ω where the FEMmodelling is used

n × (E × n) = 0 on Γ. (10)

17

Figure 10: The fictitious surface S enclosing the domain modelled by FEM

• Use integration by parts to remove all the integrals that do not convergein the usual meaning.

As a result, we are led to the following variational system (cf. [44] [14] forthe details): find EΩ defined in ΩD and satisfying n × (EΩ × n) = 0 on ∂Ωext,HΩ defined in ΩH , J vector tangent to ∂Ω∞ linked to HΩ by J = n × HΩ onΣ, such that

a(EΩ,HΩ,J,E′

Ω,H′

Ω,J′) = LE′

Ω,H′

Ω,J′ (11)

for all test functions E′

Ω satisfying the same condition that EΩ, H′

Ω and testcurrent J′ this once linked to H′

Ω by J′ = n × H′

Ω on S, with

a(EΩ,HΩ,J,E′

Ω,H′

Ω,J′) =∫

ΩD

(

1µ∇× EΩ · ∇ × E′

Ω − k2εEΩ · E′

Ω

)

dV + ikZ

∫

ΣD

J · E′

ΩdS

+

∫

ΩH

(

∇× HΩ · ∇ × H′

Ω − k2HΩ · H′

Ω

)

dV + ik

∫

S

n × HΩ · n × H′

ΩdS

− ikZ−1

∫

ΣD

EΩ · n × H′

ΩdS

+

∫

Γ+S

∫

∂Ω∞

G(x, y)(

1k2∇S · J(y)∇S · J′(x) − J(y) · J′(x)

)

dSydSx

+ ikZ−1

∫

Γ+S

∫

ΣD

∇yG(x, y) × M(y) · J′(x)dSydSx + ik

∫

S

n × H0 · J′dS

where H0 is expressed on S in terms of J and M = EΩ × n by (9) with nosingular integral, and

LE′

Ω,H′

Ω,J′ =

∫

ΩH

(ikZJs · E′

Ω − Ms · ∇ × E′

Ω) dS

18

where the integral has to be correctly interpreted in the distributional sensewhen the sources are singular.

Observe that the variational formulation involves no hypersingular integraland thus can be handled by standard codes of BEM for electromagnetic waves.

Let us assume now that Ω and Γ are meshed respectively in tetrahedronsand triangles in a way compatible with the matching of currents at the junctionof Γ and the boundary of Ω. Approximating the fields by the lowest orderedge elements of Nedelec (e.g., [37]) and the currents by the usual lowest orderdivergence-conforming surface edge elements (e.g., [38]) yields a simple way toensure the matching conditions on the currents and the fields. Now denote byXFE and by XBI column-wise vectors collecting the dofs respectively attachedto Ω or its boundary and to the interior of Γ. Making use of the variationalformulation (11), we are hence led to solve the following linear system

[

SEF,EF + DEF,EF DEF,BI

DBI,EF DBI,BI

] [

XEF

XBI

]

=

[

UEF

0

]

(12)

where the symbol S indicates a sparse matrix obtained by the assembly proce-dure usual in a finite element modelling and the symbol D corresponds to densematrices as those obtained in the discretization of boundary integral equations.Using as left preconditionner the block diagonal matrix

[

(

SEF,EF)

−10

0(

DBI,BI)

−1

]

reduces each step of a Krylov iterative process for solving (12) to matrix-vectorproducts by dense matrices and the decoupled solution of a sparse and a denselinear system.

4.3 Validation of the method

The method was used to determine the far-field pattern and the impedance of apatch antenna posed on a complicate perfectly conducting structure as indicatedin Fig. 11. This set-up is aimed to give some informations on the functioning ofthe patch antenna when it is embarked on a satellite.

We do not report here the relative performances of three methods: a directcoupling of a FEM for the substrate of the patch antenna with a BEM (FE-BI),the usual AABC requiring to mesh a complete zone around the structure andthe present method where the additional finite element mesh is used around thepatch antenna only (FE-BI-AABC). This comparison will be done elsewhere[44]. Let us simply mention that the error (comparatively to the FE-BI takenas the reference solution) on the calculation of the impedance of the antenna is0.9 % for the FE-BI-AABC and 2.28 % for the FE-AABC. The far-field patternerror is less than 1 % for the two methods even if the results for the FE-BI-AABC are slightly more accurate. In Fig. 12, we report the results concerningthe cross-circular polarization of the far field obtained by the three methods.

19

Figure 11: Circular patch antenna placed at the center of a complicate perfeclyconducting structure

5 Concluding remarks

Even rapidly covered, the above survey on some applications of DDMs to moreefficiently solve problems related to radiation or scattering of time-harmonicelectromagnetic waves or to couple different solving procedures clearly demon-strates the power and the possibilities of such kinds of techniques. We hopefullyexpect that this review has also shown the great progress that was already car-ried out to improve these methods and to increase their scope of application.

However, in our opinion, there remain some directions needing further in-vestigations.

As iterative procedures, the DDMs basically require a stopping criterion.Usually, this criterion is a reduction of the initial residual either for the coarseproblem or for the direct equations. In many cases, this criterion tests the ac-curacy reached in the computation of the solution to the discrete problem andnot of the actual exact one, that is, the theoretical one with no error comingfrom the discretization. As a result, many costly iterates are computed with-out any true impact on the global accuracy of the complete numerical process.Some possible directions of investigation could be the use of a posteriori errorsestimates like in adaptive FEM (e.g., [23]) or to take advantage of a balance ofenergy as this was carried out in [1].

As above indicated, one of the main technique in the use of DDMs fortime-harmonic electromagnetic problems is to express the matching conditionsthrough, roughly speaking, a combination of the electric field and the electriccurrents. However these quantities are in duality in a natural way instead of

20

-200 -100 0 100 200THETA (Degre)

5

10

15

20

25

FAR

FIE

LD

PA

TT

ER

N (

dB)

FE-IE-AABCFE-AABCFE-IE

Figure 12: Cross-polarization of the far field pattern of the circular patch an-tenna posed on a complicated structure

being in the same vectorial space. Apart from the FEM where this kind ofconditions can be dealt with directly by means of the variational formulation,even if some extra regularity has to be assumed a priori on the trial and testfunctions, it is generally hard to mimic this relationship at the discrete levelin a BEM solution (e.g., [13]) or to perform the rotation by π/2 around thenormal to the surface which makes it possible to pass from fields to currents atthe discrete level (see [17] and [15]). In our opinion, this issue still remains ablocking difficulty which requires to be handled from a completely new point ofview.

Acknowledgements. The authors would like to acknowledge the supportof CINES with processing time on massively parallel platforms which has madepossible the previous studies. They would also like to express their warm thanksto P. Joly and V. Gobin for their kind invitation to submit this contribution.

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