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Research ArticleSimple and Efficient Preconditioner for Surface IntegralSolution of Scattering from Multilayer Dielectric Bodies

B. B. Kong and X. Q. Sheng

Center for Electromagnetic Simulation, School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to X. Q. Sheng; [email protected]

Received 20 April 2016; Revised 18 July 2016; Accepted 26 July 2016

Academic Editor: Lorenzo Crocco

Copyright © 2016 B. B. Kong and X. Q. Sheng. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The computation of scattering from multilayer dielectric bodies is studied by using the combined tangential formulation (CTF)of surface integral solution. A simple and efficient preconditioner is designed for the surface integral solution of multilayerdielectric bodies and validated by numerical experiments. Compared with the traditional near field preconditioner, the proposedpreconditioner significantly reduce CPU time and memory requirement. Furthermore, the multilevel fast multipole algorithm(MLFMA) is employed to improve the capability of the solutions. The trick of efficiently implementing MLFMA is presented formultilayer dielectric bodies. Numerical examples are presented to verify the accuracy and efficiency of the approach for computingscattering from multilayer dielectric problems.

1. Introduction

Electromagnetic scattering from multilayer dielectric bodiesis an important problem in the computational electromag-netics (CEM) since it is widely used in real-life applicationssuch as Luneburg lens, radomes, where each dielectric isfully included in the outer one. The method of moment(MoM) based on surface integral equations (SIE) has shownto be an efficient solution of this problem, since the volumediscretization is avoided in this solution. The formulationsof SIE have been well studied for homogeneous bodies inthe literatures [1–5]. However, the numerical performanceof surface integral solutions is not well investigated formultilayer dielectric bodies.

It is known that the preconditioners are usually requiredfor obtaining an efficient surface integral solution of homo-geneous bodies [6–8]. There are many preconditioners.A conventional preconditioner is constructed by directlyemploying the inverse of near field matrix, which is exactlycompatible with the multilevel fast multipole algorithm(MLFMA). But it is very time-consuming and large mem-ory requirement. Another general preconditioner is theblock-diagonal preconditioner [9], which is constructed bychoosing the block-diagonal part of the near-interaction.

However, it does not work well for CTF solution as verifiedin literature [5]. A more efficient preconditioner is theSchur complement preconditioner [7]. Since the discretizedmatrix of multilayer dielectric problems contains both theinteractions of equivalent electric and magnetic currentson the same layer and those on different layers, it is hardto efficiently extend the Schur complement preconditionerfrom homogeneous bodies to multilayer bodies. In this letter,the combined tangential formulation (CTF) is employed toformulate scattering from multilayer dielectric bodies due toits high accuracy and efficient memory. Furthermore, CTFis free of internal resonances [4]. A simple preconditioner isdesigned for the efficient CTF solution of multilayer bodies.Compared with the existing preconditioners [6–8], it iseasier to be implemented to the multilayer dielectric bodies.MLFMA [10, 11] is efficiently incorporated into the solution toimprove its efficiency. Numerical experiments are presentedto investigate the performance of the proposed approach.

2. Theory and Formulation

2.1. Surface Integral-Equation Formulations. Consider elec-tromagnetic scattering from a multilayer dielectric objectplaced in free space and illuminated by an incident plane

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2016, Article ID 4134327, 8 pageshttp://dx.doi.org/10.1155/2016/4134327

2 International Journal of Antennas and Propagation

n̂1

n̂2

R1

RN

R2 ⋱

Figure 1: A multilayer dielectric scatterer with𝑁−1 layer medium.

wave (E𝑖,H𝑖). The exterior free space region is denoted as 𝑅1

and the interior regions are denoted as 𝑅𝑖with 𝑖 = 2, 3, . . . , 𝑁

for 𝑁 − 1 dielectric layers from outside to inside, with eachdielectric being fully included in the outer one, as illustratedin Figure 1.The permittivity and permeability of region𝑅

𝑖are

𝜀𝑖and 𝜇

𝑖, respectively. The interface between two regions 𝑅

𝑖

and 𝑅𝑖+1

is denoted by 𝑆𝑖and �̂�𝑖denotes the unit normal of 𝑆

𝑖

pointing toward the interior of𝑅𝑖.The equivalent electric and

magnetic currents on 𝑆𝑖are denoted as J

𝑖andM

𝑖, respectively.

The electric field integral equation in the region 𝑅𝑖(𝑖 =

1, 2, 3, . . . , 𝑁) can be formulated as [12]

E𝑖= (1 − 𝛿

1𝑖)

⋅ [𝑍𝑖L𝑖(−J𝑖−1

) − K𝑖(−M𝑖−1

) +Ω

4𝜋�̂�𝑖−1

×M𝑖−1

]

+ (1 − 𝛿𝑁𝑖) [𝑍𝑖L𝑖(J𝑖) − K𝑖(M𝑖) +

Ω

4𝜋�̂�𝑖×M𝑖]

+ 𝛿1𝑖E𝑖.

(1)

The magnetic field integral equation in the region 𝑅𝑖can be

formulated as

H𝑖= (1 − 𝛿

1𝑖)

⋅ [1

𝑍𝑖

L𝑖(−M𝑖−1

) + K𝑖(−J𝑖−1

) −Ω

4𝜋�̂�𝑖−1

× J𝑖−1

]

+ (1 − 𝛿𝑁𝑖) [

1

𝑍𝑖

L𝑖(M𝑖) + K𝑖(J𝑖) −

Ω

4𝜋�̂�𝑖× J𝑖]

+ 𝛿1𝑖H𝑖,

(2)

where𝑍𝑖= √𝜇𝑖/𝜀𝑖, (E𝑖,H𝑖)denote the total field in𝑅

𝑖andΩ is

the solid angle which equals 2𝜋 for common smooth surface.𝛿𝑗𝑖

= 1 for 𝑖 = 𝑗 and 𝛿𝑗𝑖

= 0 for other cases. The operators Land K are defined as

L𝑖 (X) = −𝑗𝑘𝑖 ∫[X +

1

𝑘2

𝑖

∇ (∇⋅ X)]𝐺

𝑖𝑑𝑟, (3)

K𝑖 (X) = ∫

PVX × ∇𝐺

𝑖𝑑𝑟. (4)

Here PV indicates the principal value of the integral, 𝐺𝑖is

Green’s function in region 𝑅𝑖, and 𝑘

𝑖= 𝜔√𝜀𝑖𝜇𝑖.

By combining the equations in region 𝑅𝑖with those in

region 𝑅𝑖+1

, we obtain the following CTF formulation [5]:

�̂� ⋅ [L𝑖+ L𝑖+1

] (J𝑖) − �̂�

⋅ [1

𝑍𝑖

(K𝑖+

1

2I𝑛) +

1

𝑍𝑖+1

(K𝑖+1

−1

2I𝑛)] (M

𝑖)

− (1 − 𝛿𝑁𝑖) [�̂� ⋅ L

𝑖+1(J𝑖+1

) − �̂� ⋅1

𝑍𝑖+1

K𝑖+1

(M𝑖+1

)]

− (1 − 𝛿1𝑖) [�̂� ⋅ L

𝑖(J𝑖−1

) − �̂� ⋅1

𝑍𝑖

K𝑖(M𝑖−1

)] = 𝛿1𝑖𝑒𝑖,

(5)

�̂� ⋅ [𝑍𝑖(K𝑖+

1

2I𝑛) + 𝑍

𝑖+1(K𝑖+1

−1

2I𝑛)] (J𝑖) + �̂�

⋅ [L𝑖+ L𝑖+1

] (M𝑖)

− (1 − 𝛿𝑁𝑖) [̂𝑡 ⋅ 𝑍

𝑖+1K𝑖+1

(J𝑖+1

) + �̂� ⋅ L𝑖+1

(M𝑖+1

)]

− (1 − 𝛿1𝑖) [̂𝑡 ⋅ 𝑍

𝑖K𝑖(J𝑖−1

) + �̂� ⋅ L𝑖(M𝑖−1

)] = 𝛿1𝑖ℎ𝑖,

(6)

where 𝑖 = 1, 2, 3, . . . , 𝑁− 1, the operator I𝑛(X𝑖) = �̂�𝑖×X𝑖, and

𝑒1= −(1/𝑍

1)̂𝑡 ⋅ E𝑖, ℎ

1= −𝑍1�̂� ⋅H𝑖.

2.2. Discretization of Surface Integral-Equation Formulations.To discretize (4) and (5), each dielectric surface/interface𝑆𝑖is meshed by planner triangular patches. The surface

currents are expandedwith Rao-Wilton-Glisson (RWG) basisfunctions [13]. Applying Galerkin’s testingmethod, we obtainthe following discretized matrix equation:

[[[[[[[[[[[[

[

A1

U1

0 0 ⋅ ⋅ ⋅ 0 0 0

Q2

A2

U2

0 ⋅ ⋅ ⋅ 0 0 0

0 Q3

A3

U3

⋅ ⋅ ⋅ 0 0 0

.

.

. d d d d d d...

0 0 0 0 ⋅ ⋅ ⋅ Q𝑁−2

A𝑁−2

U𝑁−2

0 0 0 0 ⋅ ⋅ ⋅ 0 Q𝑁−1

A𝑁−1

]]]]]]]]]]]]

]

((((((

(

𝑥1

𝑥2

𝑥3

.

.

.

𝑥𝑁−2

𝑥𝑁−1

))))))

)

=

((((((

(

𝑏1

0

0

.

.

.

0

0

))))))

)

,

(7)

where 𝑥𝑖

= (𝐽𝑖,𝑀𝑖)𝑇 and 𝐽

𝑖and 𝑀

𝑖are the unknowns of

electric and magnetic currents, respectively. To be more

International Journal of Antennas and Propagation 3

specific, the submatrices of A𝑖, U𝑖, Q𝑖, 𝑏1have the following

structures:

A𝑖= [

A1𝑖+ A2𝑖A3𝑖+ A4𝑖

A5𝑖+ A6𝑖A7𝑖+ A8𝑖

] ,

U𝑖= [

U1𝑖U2𝑖

U3𝑖U4𝑖

] ,

Q𝑖= [

Q1𝑖Q2𝑖

Q3𝑖Q4𝑖

] ,

𝑏1= [

𝑏𝑒

1

𝑏𝑚

1

] ,

(8)

where

A1𝑖= ⟨g, L

𝑖(g)⟩ ,

A2𝑖= ⟨g, L

𝑖+1(g)⟩ ,

A3𝑖= ⟨g, − 1

𝑍𝑖

(K𝑖+

1

2I𝑛) (g)⟩ ,

A4𝑖= ⟨g, − 1

𝑍𝑖+1

(K𝑖+1

−1

2I𝑛) (g)⟩ ,

A5𝑖= ⟨g, 𝑍

𝑖(K𝑖+

1

2I𝑛) (g)⟩ ,

A6𝑖= ⟨g, 𝑍

𝑖+1(K𝑖+1

−1

2I𝑛) (g)⟩ ,

A7𝑖= ⟨g, L

𝑖(g)⟩ ,

A8𝑖= ⟨g, L

𝑖+1(g)⟩ ,

U1𝑖= ⟨g, −L

𝑖+1(g)⟩ ,

U2𝑖= ⟨g, 1

𝑍𝑖+1

K𝑖+1

(g)⟩ ,

U3𝑖= ⟨g, −𝑍

𝑖+1K𝑖+1

(g)⟩ ,

U4𝑖= ⟨g, −L

𝑖+1(g)⟩ ,

Q1𝑖= ⟨g, −L

𝑖(g)⟩ ,

Q2𝑖= ⟨g, 1

𝑍𝑖

K𝑖(g)⟩ ,

Q3𝑖= ⟨g, −𝑍

𝑖K𝑖(g)⟩ ,

Q4𝑖= ⟨g, −L

𝑖(g)⟩ ,

𝑏𝑒

1= ⟨g, − 1

𝑍1

E𝑖⟩ ,

𝑏𝑚

1= ⟨g, −𝑍

1H𝑖⟩ ,

(9)

where g denotes the RWG basis and testing functions on 𝑆𝑖

and ⟨⋅, ⋅⟩ stands for the inner product.

3. Implementation of MLFMA forMultilayer Bodies

To efficiently solve (7), MLFMA is employed to speed up thematrix-vector multiplication in iterative solutions. Usually,each sub-matrix-vectormultiplication of (7) is independentlyspeeded up by using MLFMA. Each dielectric layer sharesthe same cubic clusters, but different truncation numbers inMLFMA are employed. However, if the special structure of(9) is employed, a more efficient MLFMA implementationcan be achieved. For example, since the far-field interactionsof A1𝑖and A5

𝑖in (7) act on the same equivalent electric

currents and also Green’s function for A1𝑖and A5

𝑖is the same,

the aggregation and translation components for their far-fieldinteractions are the same; only the receiving components aredifferent. Thus, the calculation of the far-field interactions ofA1𝑖andA5

𝑖in each matrix-vector multiple of the iteration can

be finished with only once computation of the aggregationand translation and twice disaggregations.The same trick canbe applied to A2

𝑖and A6

𝑖, A3𝑖and A7

𝑖, A4𝑖and A8

𝑖, U1𝑖and U3

𝑖,

U2𝑖and U4

𝑖,Q1𝑖andQ3

𝑖, andQ2

𝑖andQ4

𝑖.

4. Construction of Preconditioner

Preconditioners are usually required for iterative solutions of(7) since the increase of the number of dielectric layers leadsto the rapid growth of the condition number of impedancematrix. Employing a preconditioner on the matrix equationof (7) yields

P−1M𝑥 = P−1𝑏. (10)

A conventional preconditioner is constructed by employ-ing the inverse of the near field matrix, which is callednear field preconditioner (NFP). NFP is exactly compatiblewith MLFMA. However, the construction of NFP is time-consuming and large memory is required. More efficient pre-conditioners are proposed by making use of the approximateSchur complement [7]. Since (7) is not a 2 × 2 partitionedsystem [14, 15], these preconditioners are difficult to beextended from homogeneous bodies to multilayer dielectricobject.

In this letter, a simple preconditioner is designed for theiterative solution of (7). In the construction of NFP, the nearfield matrix is obtained by maintaining the entries of self-interaction in one box and near-interactionwith the neighborboxes and omitting other entries of the original matrix. Theboxes are ones in the lowest level of MLFMA. Since thenumber of the entries of near-interaction is quite large, NFPis resource-consuming, especially for multilayer dielectrictargets since multilayer dielectric bodies have interactionsbetween equivalent currents on different interfaces. To over-come such drawbacks, we employ the interaction distances 𝐿of entries between source and field as criterion to determinewhether the entries are retained or omitted, as shown in

P𝑖𝑗=

{

{

{

M𝑖𝑗

r𝑖− r𝑗

≤ 𝐿

0 elsewhere,(11)

4 International Journal of Antennas and PropagationRC

S (d

Bsm

)

𝜃 (degree)1801501209060300

−10

0

10

20

30

MieNo PC

NFPDSP

Figure 2: 𝜃𝜃-polarized bistatic RCS of a multilayer sphere.

where r𝑖and r

𝑗represent the location of the testing and

bases elements. To bemore specific, if the interaction distancebetween source andfield is less than 0.15𝜆

0, thematrix entries

are retained; otherwise the matrix entries are omitted. Here𝜆0is the free space wavelength. For multilayer bodies, the

distance between source and basis functions is importantbecause the interactions between different layers are strongerwhen the layers are more close to each other.The maintainedentries include interactions between different layers if thedistance between layers is small. Thus the sparsified schemecan retain the main interactions between source and basisfunctions. Since only those entries within self-elements andneighbor elements are maintained in this sparsified scheme,the number of the maintained entries is much smaller thanthat in NFP. We employ the inverse of this sparsified matrixas preconditioner and call it distance sparse preconditioner(DSP).

5. Numerical Results

In this section, we present several examples to demonstratethe accuracy and efficiency of the presented approach. Allthe simulations are at frequency of 300MHz.The generalizedminimum residual (GMRES) solver is used and a residualerror of 10−3 is set to terminate iterations. MUMPS is usedto calculate the inverse of the sparsified matrix in the pre-conditioner construction. MUMPS, short for “MultifrontalMassively Parallel Solver,” is a direct sparse solver based onthe parallel multifrontal method [16].

First, we consider a dielectric multilayer body consistingof 5 concentric spheres with radius of 0.6𝜆

0, 0.7𝜆

0, 0.8𝜆

0,

0.9𝜆0, and 𝜆

0for each interface from inside to outside. The

relative permittivity of regions 𝑅2, 𝑅3, 𝑅4, 𝑅5, and 𝑅

6is 3, 2, 3,

2, and 1, respectively.The body is illuminated by a plane wavepropagating in 𝑧 direction with the electric field polarized in𝑥 direction.The body is meshed with 𝜆

0/10 element size and

has 16,128 unknowns. Figure 2 presents 𝜃𝜃-polarized bistaticRCS in 𝑥𝑧-plane for this multilayer sphere. It shows that

Solu

tion

time (

s)

No PCNFPDSP

4 6 8 102

𝜀r for R4

0

2000

4000

6000

8000

Figure 3: Total solution time as a function of 𝜀𝑟under different

preconditioner.

the numerical results agree well with the analytical Mie-series solution. The numerical performance of MLFMA withNFP and DSP and no-preconditioner (No PC) is presentedin Table 1. The CPU time and the memory requirement inTable 1 are those for the analysis and factorization step ofMUMPS for constructing the inverse matrix of the precondi-tioners. It can be seen that iteration number is greatly reducedby using preconditioners. The total time required by DSPis only about 1/4 of that required by NFP. Furthermore, thememory required by DSP is less than 1/5 of that requiredby NFP. In addition, the efficient MLFMA implementationpresented in this letter can reduce the iteration CPU timefrom 2,000 s required by the conventional MLFMA imple-mentation to 1,060 s.

To investigate the numerical performance of the solutionfor bodies with higher dielectric constant, the same multi-layer sphere but with different permittivity 𝜀

𝑟of the layer 𝑅

4

is calculated. The parameters of 𝜀𝑟are listed in Table 2. The

element sizes used in the meshing 𝑆3and 𝑆

4with different

𝜀𝑟are shown in Table 2. The resulted numbers of unknowns

are also presented in Table 2. The total solution time as afunction of 𝜀

𝑟is plotted in Figure 3. It can be seen that DSP

has the least total solution time for all dielectric constants.Furthermore, DSP has more advantage over NFP with higherdielectric constant.

The next three examples are computed to further demon-strate the capability of the presented approach for multilayerdielectric objects with different shapes, compared with thecommercial software of FEKO. The first object is a cubicshell having three dielectric layers of 𝜀

𝑟= 2, 3, 4 from

inside to outside. Each of the layers has a thickness of 0.1𝜆0

and the side length of the cubic shell is 4𝜆0, as shown in

Figure 4(a). The average mesh size is 0.1𝜆0and the number

of unknowns is 122,544.The bistatic RCS patterns for 𝜃𝜃- and𝜑𝜑-polarizations are computed. Figures 4(b) and 4(c) presentthe comparison of bistatic RCS obtained by DSP with theresults fromFEKO,where themesh size is also set at 0.1𝜆

0and


x

y

z

(a)RC

S (d

Bsm

)

FEKODSP

0 60 90 120 150 18030

𝜃 (degree)

−20

0

20

40

(b)

RCS

(dBs

m)

0 60 90 120 150 18030

𝜃 (degree)

−20

0

20

40

FEKODSP

(c)

Figure 4: (a) Cubic shell consists of three dielectric layers. (b) 𝜃𝜃-polarized bistatic RCS. (c) 𝜑𝜑-polarized bistatic RCS.

Table 1: Comparison of numerical performances of NFP, DSP, and No PC for the multilayer sphere.

CPU time for PC (s) Memory for PC (MB) Iteration number Iteration time (s) Total time (s)No PC — — 513 1,060 1,060NFP 445 3,693 15 47 492DSP 34 673 45 105 139

Table 2: Parameters of the multilayer spherical shell.

𝜀𝑟of 𝑅4

Element size in meshing 𝑆3, 𝑆4

Number of unknowns2 0.1𝜆

016,128

4 0.08𝜆0

19,2996 0.07𝜆

022,014

8 0.06𝜆0

26,16310 0.05𝜆

033,012

the residual error is 10−3. It is shown that the computationalvalues of DSP are in agreement with results from FEKO.

The second example is an ellipsoidal shell with radiusalong 𝑥-, 𝑦-, and 𝑧-axis being 2𝜆

0, 2𝜆0, and 𝜆

0, respectively,

as shown in Figure 5(a). It consists of three dielectric layersof 𝜀𝑟

= 2, 3, 4 from inside to outside. The thickness of theinnermost layer is 0.2𝜆

0and the others are both 0.05𝜆

0.

Bistatic RCS obtained using DSP with 45,558 unknowns iscompared with the result from FEKO in Figures 5(b) and 5(c)and good agreement is observed.

The last example is a cylindrical cavity as shown inFigure 6(a), which is a concave object. The radius of thecylindrical cavity at the base is 2.0𝜆

0and the height is 4.0𝜆

0.

It consists of three dielectric layers; each has thickness of


x

y

z

(a)

RCS

(dBs

m)

−10

0

10

20

30

40

0 30 90 120 150 18060

FEKODSP

𝜃 (degree)

(b)

RCS

(dBs

m)

FEKODSP

30 60 90 120 150 1800

𝜃 (degree)

−10

0

10

20

30

40

(c)

Figure 5: (a) Ellipsoidal shell with three dielectric layers. (b) 𝜃𝜃-polarized bistatic RCS. (c) 𝜑𝜑-polarized bistatic RCS.

0.1𝜆0. The middle layer has 𝜀

𝑟= 2 and the other two layers

have 𝜀𝑟

= 3. Bistatic RCS obtained using DSP with 80,250unknowns is compared with the result from FEKO in Figures6(b) and 6(c) and good agreement is observed.The computa-tion information of these three examples is listed in Table 3.The presented solution is more efficient than FEKO.

6. Conclusions

The surface integral solution of scattering from multilayerdielectric bodies is studied by using CTF. A simple precon-ditioner of DSP is presented, and its efficiency is validatedby numerical experiments. Compared with the conventionalnear field preconditioner, the proposed preconditioner ofDSP greatly reduce computation time and memory require-ment. An efficient MLFMA implementation is given formultilayer dielectric bodies. Numerical results show that the

Table 3: Comparison of total solution time and peak memory ofDSP and FEKO for the scattering problems.

Problem Method Total solution time (s) Peak memory (MB)

Figure 4 FEKO 9,582 42,812DSP 2,680 14,789

Figure 5 FEKO 2,742 12,481DSP 1,283 4,543

Figure 6 FEKO 24,448 24,090DSP 2,480 11,212

presented MLFMA implementation can save half iterationtime compared with the conventional MLFMA implementa-tion. Numerical results show that the presented CTF solutionis much more efficient than FEKO for multilayer dielectricbodies.


x

y

z

(a)

RCS

(dBs

m)

FEKODSP

0 60 90 120 150 18030

𝜃 (degree)

−20

0

20

40

(b)

RCS

(dBs

m)

FEKODSP

−20

0

20

40

30 60 90 120 150 1800

𝜃 (degree)

(c)

Figure 6: (a) Cylindrical cavity with three dielectric layers. (b) 𝜃𝜃-polarized bistatic RCS. (c) 𝜑𝜑-polarized bistatic RCS.

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was supported by the National Basic ResearchProgram (973) under Grants no. 61320602 and no. 61327301,the 111 Project of China under Grant B14010, and the NSFCunder Grant no. 61421001.

References

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